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numeric-prelude 0.2.2.1 → 0.4.4

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@@ -1,674 +1,26 @@-                    GNU GENERAL PUBLIC LICENSE-                       Version 3, 29 June 2007-- Copyright (C) 2007 Free Software Foundation, Inc. <http://fsf.org/>- Everyone is permitted to copy and distribute verbatim copies- of this license document, but changing it is not allowed.--                            Preamble--  The GNU General Public License is a free, copyleft license for-software and other kinds of works.--  The licenses for most software and other practical works are designed-to take away your freedom to share and change the works.  By contrast,-the GNU General Public License is intended to guarantee your freedom to-share and change all versions of a program--to make sure it remains free-software for all its users.  We, the Free Software Foundation, use the-GNU General Public License for most of our software; it applies also to-any other work released this way by its authors.  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Makefile view
@@ -1,84 +1,32 @@+HCI6 = ghci+HCI7 = ghci -XCPP -DNoImplicitPrelude=RebindableSyntax -OBJECT_DIR    := build/$(shell uname -s)-$(shell uname -m)-INTERFACE_DIR := build/Interface+.PHONY: ghci ghci6 ghci7 ghci-gauss ghci-compile -MODULES = $(wildcard src/*.hs) \-          $(wildcard src/NumericPrelude/*.hs) \-          $(wildcard src/Algebra/*.hs) \-          $(wildcard src/Algebra/NormedSpace/*.hs) \-          $(wildcard src/Number/*.hs) \-          $(wildcard src/Number/Physical/*.hs) \-          $(wildcard src/Number/DimensionTerm/*.hs) \-          $(wildcard src/Number/SI/*.hs) \-          $(wildcard src/Number/ResidueClass/*.hs) \-          $(wildcard src/Number/FixedPoint/*.hs) \-          $(wildcard src/Number/Positional/*.hs) \-          $(wildcard src/MathObj/*hs) \-          $(wildcard src/MathObj/Permutation/*.hs) \-          $(wildcard src/MathObj/Permutation/CycleList/*.hs) \-          $(wildcard src/MathObj/PowerSeries/*.hs)+ghci:	ghci7 -GHC_OPTIONS = -Wall -odir$(OBJECT_DIR) -hidir$(INTERFACE_DIR)+ghci6:+	$(HCI6) -Wall -i:src:test +RTS -M256m -c30 -RTS test/Demo.hs +ghci7:+	$(HCI7) -Wall -i:src:test +RTS -M256m -c30 -RTS test/Demo.hs -# names of literate modules after removing literary information-UNLIT_MODULES = $(patsubst %.lhs, %.hs, $(patsubst %.hs, , $(MODULES)))+ghci-gauss:+	$(HCI7) -Wall -i:src:test:gaussian +RTS -M256m -c30 -RTS test/Test/MathObj/Gaussian/Variance.hs -# names of all modules without literary information-HS_MODULES = $(patsubst %.lhs, %.hs, $(MODULES))+ghci-compile:+	$(HCI7) -Wall -i:src:test +RTS -M256m -c30 -RTS -fobject-code -O -hidir=dist/build -odir=dist/build test/Demo.hs -STDINTERFACES = base/base.haddock parsec/parsec.haddock -HADDOCK_INCL = $(patsubst %, -i /usr/local/share/ghc-6.2/html/libraries/%, \-                    $(STDINTERFACES))--HC = ghc--HCI = ghci+run-test:	update-test+	runhaskell Setup configure --user -fbuildExamples --enable-tests+	runhaskell Setup build+	runhaskell Setup haddock+	./dist/build/numeric-prelude-test/numeric-prelude-test +update-test:+	doctest-extract-0.1 -i src/ -i gaussian/ -i playground/ -o test/ --executable-main=Test/Run.hs $$(cat test-module.list)  -.INTERMEDIATE:	$(UNLIT_MODULES)--.PHONY:	all doc clean build test ghci publish--all:	build--clean:-	-rm `find $(OBJECT_DIR) -name "*.o"`-	-rm `find $(INTERFACE_DIR) -name "*.hi"`--test:	build-#	$(HC) -Wall -i:$(INTERFACE_DIR) -hide-package NumericPrelude -c test/Test.hs-	$(HC) $(GHC_OPTIONS) -i:src:test --make -hide-package numeric-prelude -o testsuite test/Test/Run.hs-	./testsuite--ghci:-	$(HCI) -Wall -i:src:test +RTS -M256m -c30 -RTS test/Test.hs--build:-	-mkdir $(OBJECT_DIR)-	$(HC) $(GHC_OPTIONS) -hide-package numeric-prelude --make -O $(MODULES)--doc:	$(HS_MODULES)-	haddock -o docs/html --dump-interface=docs/numericprelude.haddock $(HADDOCK_INCL) -h $(HS_MODULES)--%.hs:	%.lhs-	unlit $< $@--HASKELLORG_HTMLDIR = /home/darcs/numericprelude/docs/html--publish:-	scp -r dist/doc/html/* cvs.haskell.org:$(HASKELLORG_HTMLDIR)/-	#scp -r docs/html/* cvs.haskell.org:$(HASKELLORG_HTMLDIR)/-	ssh cvs.haskell.org chmod -R o+r $(HASKELLORG_HTMLDIR)-	#ssh cvs.haskell.org chmod o+x `find $(HASKELLORG_HTMLDIR) -type d`--# TARBALL = dist/numeric-prelude-0.2.1--# sdist:-#	cabal sdist-#	gunzip --stdout $(TARBALL).tar.gz >$(TARBALL)-ext.tar-#	tar rf --dereference $(TARBALL)-ext.tar src-ghc-6.12-#	gzip $(TARBALL)-ext.tar-##	gunzip --stdout $(TARBALL).tar.gz | tar r --dereference src-ghc-6.12 | gzip >$(TARBALL)-ext.tar.gz+%.html:	%.md+	pandoc $< --output=$@
+ README.md view
@@ -0,0 +1,139 @@+# Revisiting the Numeric Classes++## Introduction++The Prelude for Haskell 98 offers a well-considered set of numeric classes+which covers the standard numeric types+(`Integer`, `Int`, `Rational`, `Float`, `Double`, `Complex`) quite well.+But they offer limited extensibility and have a few other flaws.+In this proposal we will revisit these classes, addressing the following concerns:++1.  The current Prelude defines no semantics for the fundamental operations.+    For instance, presumably addition should be associative+    (or come as close as feasible),+    but this is not mentioned anywhere.++2.  There are some superfluous superclasses.+    For instance, `Eq` and `Show` are superclasses of `Num`.+    Consider the data type+    `   data IntegerFunction a = IF (a -> Integer) `.+    One can reasonably define all the methods of `Algebra.Ring.C` for+    `IntegerFunction a` (satisfying good semantics),+    but it is impossible to define non-bottom instances of `Eq` and `Show`.+    In general, superclass relationship should indicate+    some semantic connection between the two classes.++3.  In a few cases, there is a mix of semantic operations and+    representation-specific operations.+    `toInteger`, `toRational`,+    and the various operations in `RealFloating` (`decodeFloat`, ...)+    are the main examples.++4.  In some cases, the hierarchy is not finely-grained enough:+    Operations that are often defined independently are lumped together.+    For instance, in a financial application one might want a type "Dollar",+    or in a graphics application one might want a type "Vector".+    It is reasonable to add two Vectors or Dollars,+    but not, in general, reasonable to multiply them.+    But the programmer is currently forced to define a method for `(*)`+    when she defines a method for `(+)`.++In specifying the semantics of type classes,+I will state laws as follows:++~~~~+    (a + b) + c === a + (b + c)+~~~~++The intended meaning is extensional equality:+The rest of the program should behave in the same way+if one side is replaced with the other.+Unfortunately, the laws are frequently violated by standard instances;+the law above, for instance, fails for `Float`:++~~~~+    (1e20 + (-1e20)) + 1.0  = 1.0+     1e20 + ((-1e20) + 1.0) = 0.0+~~~~++For inexact number types like floating point types,+thus these laws should be interpreted as guidelines rather than absolute rules.+In particular, the compiler is not allowed to use them for optimization.+Unless stated otherwise, default definitions should also be taken as laws.++Thanks to Brian Boutel, Joe English, William Lee Irwin II, Marcin+Kowalczyk, Ketil Malde, Tom Schrijvers, Ken Shan, and Henning+Thielemann for helpful comments.+++## Usage++Write modules in the following style:++~~~~+    {-# LANGUAGE RebindableSyntax #-}+    module MyModule where++    ... various specific imports ...++    import NumericPrelude+~~~~++Importing `NumericPrelude` is almost the same as++~~~~+    import NumericPrelude.Numeric+    import NumericPrelude.Base   .+~~~~++Instead of the `NoImplicitPrelude` pragma+you could also write `import Prelude ()`+but this will yield problems with numeric literals.++There are two wrapper types that allow types+to be used with both Haskell98 and NumericPrelude type classes+that are initially implemented for only one of them.+++## Scope & Limitations/TODO++* It might be desireable to split `Ord` up into `Poset` and `Ord`+  (a total ordering).+  This is not addressed here.++* In some cases, this hierarchy may not yet be fine-grained enough.+  For instance, time spans ("5 minutes") can be added to times ("12:34"),+  but two times are not addable. ("12:34 + 8:23")+  As it stands,+  users have to use a different operator for adding time spans to times+  than for adding two time spans.+  Similar issues arise for vector space et al.+  This is a consciously-made tradeoff, but might be changed.+  This becomes most serious when dealing with quantities with units+  like `length/distance^2`, for which `(*)` as defined here is useless.+  (One way to see the issue: should+  `  f x y = iterate (x *) y  `+  have principal type+  `  (Ring.C a) => a -> a -> [a]  `+  or something like+  `  (Ring.C a, Module a b) => a -> b -> [b]  `+  ?)++* I stuck with the Haskell 98 names.+  In some cases I find them lacking.+  Neglecting backwards compatibility, we have renamed classes as follows:++    ~~~~+    Num           --> Additive, Ring, Absolute+    Integral      --> ToInteger, IntegralDomain, RealIntegral+    Fractional    --> Field+    Floating      --> Algebraic, Transcendental+    Real          --> ToRational+    RealFrac      --> RealRing, RealField+    RealFloat     --> RealTranscendental+    ~~~~+++Additional standard libraries might include `Enum`, `IEEEFloat`+(including the bulk of the functions in Haskell 98's `RealFloat` class),+`VectorSpace`, `Ratio`, and `Lattice`.
docs/NOTES view
@@ -1,3 +1,124 @@+* Positional: test suite++Test against 'compensated' package.++* Positional and zero++Represent zero with empty mantissa?+Or better have NonZero type with non-empty mantissa+and a full number type with optional zero?+Or something where we can have negative numbers and zero as option?+Problem is, that we allow negative digits+and thus even a Positive number type can represent zero and negative numbers.++We might at least define a NonEmptyMantissa type for interim computations,+like in 'divide'.++* Positional.Fixed++We could derive the base from digit type, e.g.+   Int32 -> 1000+   Int64 -> 1000000+   newtype Integer -> anything++* Algebra.Module++I think it should be a type family rather than a multi-parameter type class.+My main motivation for multi-paramter type class+was to allow complex numbers to be a vector space over both real and complex numbers.+This does not worked well and even more type inference often fails.+We should just have two different types of complex numbers:+One complex number type being a vector space over reals+and another complex type being a vector space over complex numbers.++* zipWithChecked++We could make the second operand lazy,+and this way we would get a version of 'zipWith'+that can be used for some tying-the-knot applications+(as in unique-logic and BurrowsWheeler).++* Algebra.Complex++conjugate method that works both on real and complex numbers.+This is need in ScalarProduct implementations,+as in Synthesizer.Test.Fourier.++* multiply vs. mul++Make identifiers consistent.+I think Gaussian.multiply can be dropped when (*) is removed from Ring.++* better support for Show instances in NumericPrelude.Text++Appearance of all objects with prefix constructor should be consistent.++* Show for infix operators++Verify Show instances of (%) and (+:).+E.g. Show (Complex Rational)+When a number has negative sign,+the result of 'show' is sometimes no valid Haskell expression.++* infix operator (*:) for Complex.scale++This would be nice to have in Gaussian.Bell and Gaussian.Polynomial.++* explicit export lists++* AdditiveSemiGroup, AdditiveMonoid, AdditiveGroup++Matrices with dynamic dimension have no 'zero' (because of unknown dimension)+   (that is, they have a commutative AdditiveSemiGroup and MultiplicativeSemiGroup)+NonNegative has no (-)++* MultiplicativeSemiGroup, MultiplicativeMonoid, MultiplicativeGroup++Gaussian function, Root, ComplexSquareRoot have no AdditiveGroup operations++* Decision class should be thrown out again++They do not scale well.+They should be divided into 'if' and 'select' parts.++* Eq superclass for Absolute?++If we have Absolute we can always define++  a == b   ===   isZero (a-b)++Maybe this reasoning is no longer valid,+when superclasses of Ring are finer grained.++* Ratios of polynomials++Q(pi) can be represented by Ratio (Polynomial Integer),+however the standard cancelling algorithm+using the GCD and Euclidean algorithm would not work,+because the GCD is a too strict notion here.+Thus currently Q(pi) must be represented by Ratio (Polynomial Rational),+which is redundant.+How can this be overcome?++Are there polynomials p and q over Z+such that p and q are relatively prime,+but for an integer multiple k·p and k·q they have a non-trivial common divisor?++* optimizations++http://www.haskell.org/pipermail/libraries/2010-September/014434.html++ #4101: constant folding for (**)+ #3676: realToFrac conversions+ #3744: comparisons against minBound and maxBound are not optimised away+ #3065: quot is sub-optimal+ #2269: Word type to Double or Float conversions+ #2271: floor, ceiling, round :: Double -> Int are awesomely slow+ #1434: slow conversion Double to Int++All are accessible as http://hackage.haskell.org/trac/ghc/ticket/4101, etc.+All need love.+ * sum (and mconcat)  How to provide a 'sum' function that works optimal for the strict and lazy types?@@ -314,6 +435,15 @@   +* Function type class++Useful for all function representations like Polynomials, Gaussian functions.+However polynomials can only be applied to Ring types,+and Gaussian functions can only be applied to Transcendental types.+This it would require the usual type tricks+for type constructor class with restricted type arguments.++ * Complex numbers  The module looks horrible because auxiliary type classes are introduced@@ -340,7 +470,7 @@    - Partial Fractions:       - introduce Indexable type class for allowing partial fractions of polynomials       - example decomposition (e.g. implemented in test suite)-          (n-2)*(n+2)/((n-4)*n*(n+4))+          (n-2)·(n+2)/((n-4)·n·(n+4))    - Hypercomplex numbers: Octonions    - matrices, vectors       - conversion of complex and quaternions to real matrices
+ gaussian/Gaussian.hs view
@@ -0,0 +1,6 @@+module Main where++import qualified MathObj.Gaussian.Example as Example++main :: IO ()+main = Example.polyApprox
+ gaussian/MathObj/Gaussian/Bell.hs view
@@ -0,0 +1,398 @@+{-# LANGUAGE RebindableSyntax #-}+{-+Complex translated and modulated Gaussian bell curve.++It could be extended to chirps+using a complex valued quadratic term with (real c >= 0).+This allows for a new test:+Express the Fourier transform in terms of a convolution with a chirp.+-}+module MathObj.Gaussian.Bell where++import qualified MathObj.Polynomial as Poly+import qualified Number.Complex as Complex++import qualified Algebra.Transcendental as Trans+import qualified Algebra.Field          as Field+import qualified Algebra.Absolute       as Absolute+import qualified Algebra.Ring           as Ring+import qualified Algebra.Additive       as Additive++import Number.Complex ((+:), )++import Test.QuickCheck (Arbitrary, arbitrary, )+import Control.Monad (liftM4, )++import NumericPrelude.Numeric+import NumericPrelude.Base hiding (reverse, )+++{- $setup+>>> import qualified MathObj.Gaussian.Bell as G+>>> import qualified Algebra.ZeroTestable as ZeroTestable+>>> import qualified Algebra.Laws as Laws+>>> import qualified Number.Complex as Complex+>>> import Number.Complex ((+:))+>>> import NumericPrelude.Base as P+>>> import NumericPrelude.Numeric as NP+>>> import Prelude ()+>>> import qualified Test.QuickCheck as QC+>>> import Data.Function.HT (Id, nest)+>>>+>>> asRational :: Id (G.T Rational)+>>> asRational = id+>>>+>>> withRational :: Id (G.T Rational -> a)+>>> withRational = id+>>>+>>> isConstant :: ZeroTestable.C a => G.T a -> Bool+>>> isConstant (G.Cons _amp _a b c) = isZero b && isZero c+-}+++data T a = Cons {amp :: a, c0, c1 :: Complex.T a, c2 :: a}+   deriving (Eq, Show)++instance (Absolute.C a, Arbitrary a) => Arbitrary (T a) where+   arbitrary =+      liftM4+         (\k a b c -> Cons (abs k) a b (1 + abs c))+         arbitrary arbitrary arbitrary arbitrary+++constant :: Ring.C a => T a+constant = Cons one zero zero zero++{- |+eigenfunction of 'fourier'+-}+unit :: Ring.C a => T a+unit = Cons one zero zero one++{-# INLINE evaluate #-}+evaluate :: (Trans.C a) =>+   T a -> a -> Complex.T a+evaluate f x =+   Complex.scale+     (sqrt (amp f))+     (Complex.exp $ Complex.scale (-pi) $+      c0 f + Complex.scale x (c1 f) + Complex.fromReal (c2 f * x^2))++evaluateSqRt :: (Trans.C a) =>+   T a -> a -> Complex.T a+evaluateSqRt f x0 =+   Complex.scale+     (sqrt (amp f))+     (let x = sqrt pi * x0+      in  Complex.exp $ negate $+          c0 f + Complex.scale x (c1 f) + Complex.fromReal (c2 f * x^2))++exponentPolynomial :: (Additive.C a) =>+   T a -> Poly.T (Complex.T a)+exponentPolynomial f =+   Poly.fromCoeffs [c0 f, c1 f, Complex.fromReal (c2 f)]+++{-+norm functions depend on interpretation+and would have to return both a rational and transcendental part+expressed as @exp a@.+-}++variance :: (Trans.C a) =>+   T a -> a+variance f =+   recip $ c2 f * 2*pi++{- |+prop> Laws.identity G.multiply G.constant . asRational+prop> Laws.commutative G.multiply . asRational+prop> Laws.associative G.multiply . asRational+-}+multiply :: (Ring.C a) =>+   T a -> T a -> T a+multiply f g =+   Cons+      (amp f * amp g)+      (c0 f + c0 g) (c1 f + c1 g) (c2 f + c2 g)++powerRing :: (Trans.C a) =>+   Integer -> T a -> T a+powerRing p f =+   let pa = fromInteger p+   in  Cons+          (amp f ^ p)+          (pa * c0 f) (pa * c1 f) (fromInteger p * c2 f)++{-+powerField does not makes sense,+since the reciprocal of a Gaussian diverges.+-}++powerAlgebraic :: (Trans.C a) =>+   Rational -> T a -> T a+powerAlgebraic p f =+   let pa = fromRational' p+   in  Cons+          (amp f ^/ p)+          (pa * c0 f) (pa * c1 f) (fromRational' p * c2 f)++powerTranscendental :: (Trans.C a) =>+   a -> T a -> T a+powerTranscendental p f =+   Cons+      (amp f ^? p)+      (Complex.scale p $ c0 f) (Complex.scale p $ c1 f) (p * c2 f)+++{- |+>>> let x=G.Cons 2 (1+:3) (4+:5) (7::Rational); y=G.Cons 7 (1+:4) (3+:2) (5::Rational) in G.convolve x y+Cons {amp = 7 % 6, c0 = 13 % 6 +: 55 % 8, c1 = 41 % 12 +: 13 % 4, c2 = 35 % 12}++prop> Laws.commutative G.convolve . asRational+prop> Laws.associative G.convolve . asRational++Would be nice to have something like:++> Laws.identity G.convolve G.dirac++but we cannot represent @G.dirac@.++prop> isConstant . G.convolve G.constant . asRational++Using a @G.norm1@ we could exactly compute the amplitude+of the resulting constant function.+But that would require transcendent operations.+-}+convolve :: (Field.C a) =>+   T a -> T a -> T a+convolve f g =+   let s = c2 f + c2 g+       {-+       fd = f1/(2*f2)+       gd = g1/(2*g2)+       c = f2*g2/(f2+g2)++       c*(fd+gd) = (f1*g2+f2*g1)/(2*(f2+g2)) = b/2++       c*(fd+gd)^2 - fd^2*f2 - gd^2*g2+         = f2*g2*(fd+gd)^2/(f2 + g2) - (fd^2*f2 + gd^2*g2)+         = (f2*g2*(fd+gd)^2 - (f2+g2)*(fd^2*f2+gd^2*g2)) / (f2 + g2)+         = (2*f2*g2*fd*gd - (fd^2*f2^2+gd^2*g2^2)) / (f2 + g2)+         = (2*f1*g1 - (f1^2+g1^2)) / (4*(f2 + g2))+         = -(f1 - g1)^2/(4*(f2 + g2))+       -}+   in  Cons+          (amp f * amp g / s)+          (c0 f + c0 g+              - Complex.scale (recip (4*s)) ((c1 f - c1 g)^2))+          (Complex.scale (c2 g / s) (c1 f) ++           Complex.scale (c2 f / s) (c1 g))+          (c2 f * c2 g / s)+            -- recip $ recip (c2 f) + recip (c2 g)+{-+   Cons+      (c0 f + c0 g) (c1 f + c1 g)+      (recip $ recip (c2 f) + recip (c2 g))+-}++{- |+prop> withRational $ \x y -> G.convolve x y == G.convolveByTranslation x y+-}+convolveByTranslation :: (Field.C a) =>+   T a -> T a -> T a+convolveByTranslation f0 g0 =+   let fd = Complex.scale (recip (2 * c2 f0)) $ c1 f0+       gd = Complex.scale (recip (2 * c2 g0)) $ c1 g0+       f1 = translateComplex fd f0+       g1 = translateComplex gd g0+       s = c2 f1 + c2 g1+   in  translateComplex (negate $ fd + gd) $+       Cons+          (amp f1 * amp g1 / s)+          (c0 f1 + c0 g1) zero+          (c2 f1 * c2 g1 / s)++{- |+prop> withRational $ \x y -> G.convolve x y == G.convolveByFourier x y+-}+convolveByFourier :: (Field.C a) =>+   T a -> T a -> T a+convolveByFourier f g =+   reverse $ fourier $ multiply (fourier f) (fourier g)++{- |+prop> withRational $ \x y -> G.fourier (G.convolve x y) == G.multiply (G.fourier x) (G.fourier y)+prop> withRational $ \x -> nest 2 G.fourier x == G.reverse x+prop> G.fourier G.unit == (asRational G.unit)+prop> withRational $ \x a -> G.fourier (G.translate a x) == G.modulate a (G.fourier x)+prop> withRational $ \x (QC.Positive a) -> G.fourier (G.dilate a x) == G.amplify a (G.shrink a (G.fourier x))+-}+fourier :: (Field.C a) =>+   T a -> T a+fourier f =+   let a = c0 f+       b = c1 f+       rc = recip $ c2 f+   in  Cons+          (amp f * rc)+          (Complex.scale (rc/4) (-b^2) + a)+          (Complex.scale rc $ Complex.quarterRight b)+          rc++{- |+prop> withRational $ \x -> G.fourier x == G.fourierByTranslation x+-}+fourierByTranslation :: (Field.C a) =>+   T a -> T a+fourierByTranslation f =+   translateComplex (Complex.scale (1/2) $ Complex.quarterLeft $ c1 f) $+   Cons (amp f / c2 f) (c0 f) zero (recip $ c2 f)++{-+a + b*x + c*x^2+ = c*(a/c + b/c*x + x^2)+ = c*((x-b/(2*c))^2 + (4*a*c+b^2)/(4*c^2))+ = c*(x-b/(2*c))^2 + (4*a*c+b^2)/(4*c)++fourier ->+   x^2/c - i*b/c*x + (4*a*c+b^2)/(4*c)++fourier (x -> exp(-pi*c*(x-t)^2))+ = fourier $ translate t $ shrink (sqrt c) $ x -> exp(-pi*x^2)+ = modulate t $ dilate (sqrt c) $ fourier $ x -> exp(-pi*x^2)+ = modulate t $ dilate (sqrt c) $ x -> exp(-pi*x^2)+ = modulate t $ x -> exp(-pi*x^2/c)+ = x -> exp(-pi*x^2/c) * exp(-2*pi*i*x*t)+ = x -> exp(-pi*(x^2/c - 2*i*x*t))+-}++{-+b*x + c*x^2+ = c*(b/c*x + x^2)+ = c*((x-br/(2*c))^2 + i*x*bi/c - br^2/(4*c^2))+ = c*(x-br/(2*c))^2 + i*x*bi - br^2/(4*c)++fourier ->+   (x+bi/2)^2/c - i*br/c*(x+bi/2) - br^2/(4*c)+ = (1/c) * ((x+bi/2)^2 - i*br*(x+bi/2) + (br/2)^2)+ = (1/c) * (x^2 - i*b*x + -(br/2)^2 + (bi/2)^2 - i*br*bi/2)+ = (1/c) * (x^2 - i*b*x - (br^2-bi^2+2*br*bi*i)^2 /4)+ = (1/c) * (x^2 - i*b*x - b^2 / 4)+ = (1/c) * (x^2 - i*b*x + (i*b/2)^2)+ = (1/c) * (x - i*b/2)^2++Example:+  (x-b)^2 = b^2 - 2*b*x + x^2+    ->  (- i*2*b*x + x^2)+++fourier (x -> exp(-pi*(c*(x-t)^2 + 2*i*m*x)))+ = fourier $ modulate m $ translate t $ shrink (sqrt c) $ x -> exp(-pi*x^2)+ = translate (-m) $ modulate t $ dilate (sqrt c) $ fourier $ x -> exp(-pi*x^2)+ = translate (-m) $ modulate t $ dilate (sqrt c) $ x -> exp(-pi*x^2)+ = translate (-m) $ modulate t $ x -> exp(-pi*x^2/c)+ = translate (-m) $ x -> exp(-pi*x^2/c) * exp(-2*pi*i*x*t)+ = x -> exp(-pi*(x+m)^2/c) * exp(-2*pi*i*(x+m)*t)+ = x -> exp(-pi*((x+m)^2/c - 2*i*(x+m)*t))+-}++{-+fourier (Cons a 0 0) =+  Cons a 0 infinity++fourier (Cons 0 0 c) =+  Cons 0 0 (recip c)++fourier (Cons 0 b 1) =+  Cons 0 (i*b) 1+-}++{- |+prop> withRational $ \x a b -> G.translate a (G.translate b x) == G.translate (a+b) x+-}+translate :: Ring.C a => a -> T a -> T a+translate d f =+   let a = c0 f+       b = c1 f+       c = c2 f+   in  Cons+          (amp f)+          (Complex.fromReal (c*d^2) - Complex.scale d b + a)+          (Complex.fromReal (-2*c*d) + b)+          c++{- |+prop> withRational $ \x a b -> G.translateComplex a (G.translateComplex b x) == G.translateComplex (a+b) x+prop> withRational $ \x a -> G.translateComplex (Complex.fromReal a) x == G.translate a x+-}+translateComplex :: Ring.C a => Complex.T a -> T a -> T a+translateComplex d f =+   let a = c0 f+       b = c1 f+       c = c2 f+   in  Cons+          (amp f)+          (Complex.scale c (d^2) - b*d + a)+          (Complex.scale (-2*c) d + b)+          c++{- |+prop> withRational $ \x a b -> G.modulate a (G.modulate b x) == G.modulate (a+b) x+prop> withRational $ \x a b -> G.modulate b (G.translate a x) == G.turn (a*b) (G.translate a (G.modulate b x))+-}+modulate :: Ring.C a => a -> T a -> T a+modulate d f =+   Cons+      (amp f)+      (c0 f)+      (c1 f + (zero +: 2*d))+      (c2 f)++turn :: Ring.C a => a -> T a -> T a+turn d f =+   Cons+      (amp f)+      (c0 f + (zero +: 2*d))+      (c1 f)+      (c2 f)++{- |+prop> withRational $ \x -> nest 2 G.reverse x == x+-}+reverse :: Additive.C a => T a -> T a+reverse f =+   f{c1 = negate $ c1 f}+++{- |+prop> withRational $ \x (QC.Positive a) (QC.Positive b) -> G.dilate a (G.dilate b x) == G.dilate (a*b) x+prop> withRational $ \x (QC.Positive a) -> G.shrink a x == G.dilate (recip a) x+-}+dilate :: Field.C a => a -> T a -> T a+dilate k f =+   Cons+      (amp f)+      (c0 f)+      (Complex.scale (recip k) $ c1 f)+      (c2 f / k^2)++{- |+prop> withRational $ \x (QC.Positive a) -> G.dilate a (G.shrink a x) == x+prop> withRational $ \x (QC.Positive a) -> G.shrink a (G.dilate a x) == x+-}+shrink :: Ring.C a => a -> T a -> T a+shrink k f =+   Cons+      (amp f)+      (c0 f)+      (Complex.scale k $ c1 f)+      (c2 f * k^2)++amplify :: (Ring.C a) => a -> T a -> T a+amplify k f =+   Cons+      (k^2 * amp f)+      (c0 f)+      (c1 f)+      (c2 f)
+ gaussian/MathObj/Gaussian/Example.hs view
@@ -0,0 +1,226 @@+{-# LANGUAGE RebindableSyntax #-}+{-+Reciprocal of variance of a Gaussian bell curve.+We describe the curve only in terms of its variance+thus we represent a bell curve at the coordinate origin+neglecting its amplitude.++We could also define the amplitude as @root 4 c@,+thus preserving L2 norm being one,+but then @dilate@ and @shrink@ also include an amplification.++We could do some projective geometry in the exponent+in order to also have zero variance,+which corresponds to the dirac impulse.+-}+module MathObj.Gaussian.Example where++import qualified MathObj.Gaussian.Polynomial as PolyBell+import qualified MathObj.Gaussian.Bell as Bell+import qualified MathObj.Gaussian.Variance as Var++import qualified MathObj.Polynomial as Poly++import qualified Algebra.Transcendental as Trans+import qualified Algebra.Algebraic      as Algebraic+import qualified Algebra.Field          as Field+import qualified Algebra.Ring           as Ring++import qualified Number.Complex as Complex+import qualified Number.Root as Root++import Algebra.Transcendental (pi, )+import Algebra.Algebraic (root, )+import Algebra.Ring ((*), (^), )++import Number.Complex ((+:), )++import qualified Numerics.Function as Func+import qualified Numerics.Fourier as Fourier+import qualified Numerics.Integration as Integ+import qualified Numerics.Differentiation as Diff++import qualified Graphics.Gnuplot.Simple as GP++import Control.Applicative (liftA2, )++import NumericPrelude.Numeric+import NumericPrelude.Base+import qualified Prelude as P+++curve0 :: Var.T Double+curve0 = curve0a++curve0a :: Var.T Double+curve0a = Var.Cons 1.4 3.3++curve0b :: Var.T Double+curve0b = Var.Cons 2.2 1.7++variance0 :: (Double, Double)+variance0 =+   (Var.variance curve0,+    (Integ.rectangular 1000 (-2,2) $ liftA2 (*) (^2) (Var.evaluate curve0)) /+    (Integ.rectangular 1000 (-2,2) $ Var.evaluate curve0))++norm10 :: (Double, Double, Double)+norm10 =+   (Integ.rectangular 1000 (-2,2) $ Var.evaluate curve0,+    Var.norm1 curve0,+    Root.toNumber (Var.norm1Root curve0))++norm20 :: (Double, Double, Double)+norm20 =+   (sqrt $ Integ.rectangular 1000 (-2,2) $ (^2) . Var.evaluate curve0,+    Var.norm2 curve0,+    Root.toNumber (Var.norm2Root curve0))++norm30 :: (Double, Double, Double)+norm30 =+   (root 3 $ Integ.rectangular 1000 (-2,2) $ (^3) . Var.evaluate curve0,+    Var.normP 3 curve0,+    Root.toNumber (Var.normPRoot 3 curve0))++fourier0 :: IO ()+fourier0 =+   GP.plotFuncs []+      (GP.linearScale 100 (-2,2))+      [Var.evaluate $ Var.fourier curve0,+       Fourier.analysisTransformOneReal 100 (-2,2) $ Var.evaluate curve0]++multiply0 :: IO ()+multiply0 =+   GP.plotFuncs []+      (GP.linearScale 100 (-1,1))+      [Var.evaluate $ Var.multiply curve0a curve0b,+       liftA2 (*) (Var.evaluate curve0a) (Var.evaluate curve0b)]++convolve0 :: IO ()+convolve0 =+   GP.plotFuncs []+      (GP.linearScale 100 (-2,2))+      [Var.evaluate $ Var.convolve curve0a curve0b,+       Integ.convolve 1000 (-3,3) (Var.evaluate curve0a) (Var.evaluate curve0b)]+++curve1 :: Bell.T Double+curve1 = curve1a++curve1a :: Bell.T Double+curve1a = Bell.Cons 1.4 (0.1+:0.3) ((-0.2)+:1.4) 2.3++curve1b :: Bell.T Double+curve1b = Bell.Cons 2.2 ((-0.3)+:2.1) (0.2+:(-0.4)) 1.7++variance1 :: (Double, Double)+variance1 =+   (Bell.variance curve1,+    (Integ.rectangular 1000 (-2,2) $+        liftA2 (*) (^2)+           (Complex.magnitudeSqr .+            Func.translateRight+               (Complex.real (Bell.c1 curve1) / (2 * Bell.c2 curve1))+               (Bell.evaluate curve1))) /+    (Integ.rectangular 1000 (-2,2) $ Complex.magnitude . Bell.evaluate curve1))++{- the norm depends on too much things+norm0vs1 :: (Double, Double)+norm0vs1 =+   ((Integ.rectangular 1000 (-5,5) $ Var.evaluate curve0)+         * exp (- Complex.real (Bell.c0 curve1)),+    Integ.rectangular 1000 (-5,5) $ Complex.magnitude . Bell.evaluate curve1)+-}++fourier1 :: IO ()+fourier1 =+   GP.plotFuncs []+      (GP.linearScale 100 (-5,5))+      [Complex.real . (Bell.evaluate $ Bell.fourier curve1),+       fourierAnalysisReal 100 (-2,2) $ Bell.evaluate curve1]+++curve2 :: PolyBell.T Double+curve2 =+   PolyBell.Cons+--      Bell.unit+--      (Bell.Cons 1.4 (0.1+:0.3) 0 1.2)+--      (Bell.Cons 1.4 (0.1+:0.3) ((-0.2)+:1.4) 1)+      curve1+--      (Poly.fromCoeffs [one])+--      (Poly.fromCoeffs [zero,one])+--      (Poly.fromCoeffs [zero,zero,one])+--      (Poly.fromCoeffs [0,Complex.imaginaryUnit])+      (Poly.fromCoeffs [1.4+:(-0.1),0.8+:(0.1),(-1.1)+:0.3])++differentiate2 :: IO ()+differentiate2 =+   GP.plotFuncs []+      (GP.linearScale 100 (-2,2))+      [Complex.real . (PolyBell.evaluateSqRt $ PolyBell.differentiate curve2),+       ((/ sqrt pi) . ) $ Diff.diff (1e-5) $ Complex.real . PolyBell.evaluateSqRt curve2]++fourier2 :: IO ()+fourier2 =+   GP.plotFuncs []+      (GP.linearScale 100 (-5,5))+      [Complex.real . (PolyBell.evaluateSqRt $ PolyBell.fourier curve2),+       fourierAnalysisReal 100 (-2,2) $ PolyBell.evaluateSqRt curve2]++++fourierAnalysisReal ::+   (P.Floating a) =>+   Integer -> (a, a) -> (a -> Complex.T a) -> a -> a+fourierAnalysisReal n rng f =+   liftA2 (P.-)+      (Fourier.analysisTransformOneReal n rng (Complex.real . f))+      (Fourier.analysisTransformOneImag n rng (Complex.imag . f))+++{- |+Try to approximate @\x -> exp (-x^2) * x@+by a difference of translated Gaussian bells.++exp(-x^2) * x+  ==  exp(-(a+b*x+c*x^2)) - exp(-(a-b*x+c*x^2))+  ==  exp(-(a+c*x^2)) * (exp(-b*x) - exp(b*x))+  ==  exp(-(a+c*x^2)) * 2*sinh (b*x)++It holds+  lim (\b x -> sinh (b*x) / b)  =  id+-}+diffApprox :: IO ()+diffApprox =+   let amp = (2*b)^- (-2)+       a = 0+       {-+       amp = 1+       a = log (2 * abs b)+       -}+       b = -0.1+       c = 1+       ac = Complex.fromReal a+       bc = Complex.fromReal b+   in  GP.plotFuncs []+          (GP.linearScale 100 (-2,2::Double))+          [Complex.real .+           (PolyBell.evaluateSqRt $+              PolyBell.Cons Bell.unit (Poly.fromCoeffs [zero,one])),+           Complex.real .+           liftA2 (-)+             (PolyBell.evaluateSqRt $+                PolyBell.Cons (Bell.Cons amp ac bc c) (Poly.fromCoeffs [one]))+             (PolyBell.evaluateSqRt $+                PolyBell.Cons (Bell.Cons amp ac (-bc) c) (Poly.fromCoeffs [one]))]+++polyApprox :: IO ()+polyApprox =+   GP.plotFuncs []+      (GP.linearScale 100 (-2,2::Double))+      [Complex.real .+         PolyBell.evaluateSqRt curve2,+       Complex.real . sum .+         mapM (\(amp,b) -> \x -> amp * Bell.evaluateSqRt b x)+         (PolyBell.approximateByBells 0.1 curve2)]
+ gaussian/MathObj/Gaussian/ExponentTuple.hs view
@@ -0,0 +1,114 @@+{-# LANGUAGE RebindableSyntax #-}+module MathObj.Gaussian.ExponentTuple where++import qualified Test.QuickCheck as QC++import Control.Applicative (liftA2, liftA3)++import Data.Function.HT (compose2)++import NumericPrelude.Base as P+import NumericPrelude.Numeric as NP+++{- $setup+>>> import MathObj.Gaussian.ExponentTuple (HoelderConjugates(HoelderConjugates))+>>> import MathObj.Gaussian.ExponentTuple (YoungConjugates(YoungConjugates))+>>> import NumericPrelude.Base as P+>>> import NumericPrelude.Numeric as NP+>>> import Prelude ()+-}+++{- |+For @(HoelderConjugates p q)@ it holds++prop> \(HoelderConjugates p q)  ->  p>=1 && q>=1 && 1/p + 1/q == 1+-}+data HoelderConjugates = HoelderConjugates Rational Rational+   deriving Show++instance QC.Arbitrary HoelderConjugates where+   arbitrary = genHoelderConjugates0++genHoelderConjugates0 :: QC.Gen HoelderConjugates+genHoelderConjugates0 =+   liftA2+      (\(QC.Positive p) (QC.Positive q) ->+         let s = p + q in HoelderConjugates (s % p) (s % q))+      QC.arbitrary QC.arbitrary++genHoelderConjugates1 :: QC.Gen HoelderConjugates+genHoelderConjugates1 =+   liftA2+      (\(QC.Positive p) (QC.Positive q) ->+         let s = 1%p + 1%q+         in HoelderConjugates (fromInteger p * s) (fromInteger q * s))+      QC.arbitrary QC.arbitrary+++{- |+For @(YoungConjugates p q r)@ it holds++prop> \(YoungConjugates p q r)  ->  p>=1 && q>=1 && r>=1 && 1/p + 1/q == 1/r + 1+-}+data YoungConjugates = YoungConjugates Rational Rational Rational+   deriving Show++instance QC.Arbitrary YoungConjugates where+   arbitrary = genYoungConjugates0++{-+Find positive natural numbers @a, b, c, d@ with++> a + b = c + d++and++> d >= a, d >= b, d >= c++then set++> p=d/a, q=d/b, r=d/c+++a+b<=c+b+c<=a+->  2b <= 0+-}+genYoungConjugates0 :: QC.Gen YoungConjugates+genYoungConjugates0 =+   liftA3+      (\(QC.Positive a0) (QC.Positive b0) (QC.Positive c0) ->+         let guardSwap cond (x,y) =+                if cond x y then (x,y) else (y,x)+             {-+             If a+b<=c, then from b>0 it follows a<c and thus c+b>a.+             Swapping a and c is enough and we have not to consider more cases.+             -}+             (a1,c1) = guardSwap (\a c -> a+b0>c) (a0,c0)+             b1 = b0+             d1 = a1+b1-c1+             ((a2,b2),(c2,d2)) =+                guardSwap (compose2 (<=) snd)+                   (guardSwap (<=) (a1,b1),+                    guardSwap (<=) (c1,d1))+         in  YoungConjugates (d2%a2) (d2%b2) (d2%c2))+      QC.arbitrary QC.arbitrary QC.arbitrary++{- |+This one is simpler, but may yield exponents smaller than 1.+-}+genYoungConjugates1 :: QC.Gen YoungConjugates+genYoungConjugates1 =+   liftA3+      (\(QC.Positive a0) (QC.Positive b0) (QC.Positive c0) ->+         let {-+             If a+b<=c, then from b>0 it follows a<c and thus c+b>a.+             Swapping a and c is enough and we have not to consider more cases.+             -}+             (a1,c1) = if a0+b0<=c0 then (c0,a0) else (a0,c0)+             b1 = b0+             d1 = a1+b1-c1+         in  YoungConjugates (d1%a1) (d1%b1) (d1%c1))+      QC.arbitrary QC.arbitrary QC.arbitrary
+ gaussian/MathObj/Gaussian/Polynomial.hs view
@@ -0,0 +1,584 @@+{-# LANGUAGE RebindableSyntax #-}+{-+Complex Gaussian bell multiplied with a polynomial.++In order to make this free of @pi@ factors,+we have to choose @recip (sqrt pi)@+as unit for translations and modulations,+for linear factors and in the differentiation.+-}+{-+ToDo:++* In order to avoid the weird @sqrt pi@ factor,+  use a polynomial expression in @pi@.++* sum of multiple bells using Data.Map from exponent polynomial to coefficient polynomial+  use of Algebra object.++* Discrete Fourier Transform and its eigenvectors++* Use projective geometry in order to support Dirac impulse.+  There are many open questions:+  1. What shall be the product of two Dirac impulses -+     whether they are at the same location or not.+  2. How to organize coefficients+     such that the constant function can be modulated+     and the Dirac impulse can be translated.+-}+module MathObj.Gaussian.Polynomial where++import qualified MathObj.Gaussian.Bell as Bell++import qualified MathObj.LaurentPolynomial as LPoly+import qualified MathObj.Polynomial.Core   as PolyCore+import qualified MathObj.Polynomial        as Poly+import qualified Number.Complex     as Complex++import qualified Algebra.ZeroTestable   as ZeroTestable+import qualified Algebra.Differential   as Differential+import qualified Algebra.Transcendental as Trans+import qualified Algebra.Field          as Field+import qualified Algebra.Absolute       as Absolute+import qualified Algebra.Ring           as Ring+import qualified Algebra.Additive       as Additive++import qualified Data.Record.HT as Rec+import qualified Data.List as List+import Data.Function.HT (nest, )+import Data.Eq.HT (equating, )+import Data.List.HT (mapAdjacent, )+import Data.Tuple.HT (forcePair, )++import Test.QuickCheck (Arbitrary, arbitrary, )+import Control.Monad (liftM2, )++import NumericPrelude.Numeric+import NumericPrelude.Base hiding (reverse, )+++{- $setup+>>> :set -XRebindableSyntax+>>>+>>> import qualified MathObj.Gaussian.Polynomial as G+>>> import qualified MathObj.Gaussian.Bell as Bell+>>> import qualified MathObj.Polynomial as Poly+>>> import qualified Algebra.Laws as Laws+>>> import qualified Number.Complex as Complex+>>> import Number.Complex ((+:))+>>> import NumericPrelude.Base as P+>>> import NumericPrelude.Numeric as NP+>>> import qualified Test.QuickCheck as QC+>>> import Data.Function.HT (Id, nest)+>>> import Data.Tuple.HT (mapSnd)+>>>+>>> asRational :: Id (G.T Rational)+>>> asRational = id+>>>+>>> withRational :: Id (G.T Rational -> a)+>>> withRational = id+>>>+>>> mulLinear2i :: Id (G.T Rational)+>>> mulLinear2i x =+>>>    x{G.polynomial = Poly.fromCoeffs [0, 0+:2] * G.polynomial x}+>>>+>>> rotateQuarter :: Int -> Id (G.T Rational)+>>> rotateQuarter n =+>>>    G.scaleComplex (negate Complex.imaginaryUnit ^ fromIntegral n)+-}+++data T a = Cons {bell :: Bell.T a, polynomial :: Poly.T (Complex.T a)}+   deriving (Show)++instance (Absolute.C a, ZeroTestable.C a, Eq a) => Eq (T a) where+   (==) = equal+++{-+Helper data type for 'equal',+that allows to call the (not quite trivial) polynomial equality check.+@RootProduct r a@ represents @sqrt r * a@.+The test using 'signum' works for real numbers,+and I do not know, whether it is correct for other mathematical objects.+However I cannot imagine other mathematical objects,+that make sense at all, here.+Maybe elements of a finite field.+-}+data RootProduct a = RootProduct a a++instance (Absolute.C a, ZeroTestable.C a, Eq a) => Eq (RootProduct a) where+   (RootProduct xr xa) == (RootProduct yr ya)  =+      let xp = xr*xa^2+          yp = yr*ya^2+      in  xp==yp &&+          (isZero xp || signum xa == signum ya)++instance (ZeroTestable.C a) => ZeroTestable.C (RootProduct a) where+   isZero (RootProduct r a) = isZero r || isZero a+++{-+The derived Eq is not correct.+We have to combine the amplitude of the bell with the polynomial,+respecting signs and the square root of the bell amplitude.+-}+equal :: (Absolute.C a, ZeroTestable.C a, Eq a) => T a -> T a -> Bool+equal x y =+   let bx = bell x+       by = bell y+       scaleSqr b =+          (\p ->+              (fmap (RootProduct (Bell.amp b) . Complex.real) p,+               fmap (RootProduct (Bell.amp b) . Complex.imag) p))+           . polynomial+   in  Rec.equal+          (equating Bell.c0 :+           equating Bell.c1 :+           equating Bell.c2 :+           [])+          bx by+       &&+       scaleSqr bx x == scaleSqr by y+++instance (Absolute.C a, ZeroTestable.C a, Arbitrary a) => Arbitrary (T a) where+   arbitrary =+--      liftM2 Cons arbitrary arbitrary+      liftM2 Cons+         arbitrary+         -- we have to restrict the number of polynomial coefficients,+         -- since with the quadratic time algorithms like fourier and convolve,+         -- in connection with Rational slow down tests too much.+         (fmap (Poly.fromCoeffs . take 5 . Poly.coeffs) arbitrary)++++{-# INLINE evaluateSqRt #-}+evaluateSqRt :: (Trans.C a) =>+   T a -> a -> Complex.T a+evaluateSqRt f x =+   Bell.evaluateSqRt (bell f) x *+   Poly.evaluate (polynomial f) (Complex.fromReal $ sqrt pi * x)+{- ToDo: evaluating a complex polynomial for a real argument can be optimized -}+++constant :: (Ring.C a) => T a+constant =+   Cons Bell.constant (Poly.const one)++scale :: (Ring.C a) => a -> T a -> T a+scale x f =+   f{polynomial = fmap (Complex.scale x) $ polynomial f}++scaleComplex :: (Ring.C a) => Complex.T a -> T a -> T a+scaleComplex x f =+   f{polynomial = fmap (x*) $ polynomial f}+++unit :: (Ring.C a) => T a+unit = eigenfunction0++{- |+This one does not hold for larger degrees, although it would be nice:++prop> QC.forAll (QC.choose (0,3)) $ \n -> G.eigenfunctionDifferential n == asRational (G.eigenfunctionIterative n)++Unfortunately, both implementations compute different eigenbases.+-}+eigenfunction :: (Field.C a) => Int -> T a+eigenfunction =+   eigenfunctionDifferential++-- | prop> G.eigenfunction0  ==  asRational (G.eigenfunctionDifferential 0)+eigenfunction0 :: (Ring.C a) => T a+eigenfunction0 =+   Cons Bell.unit (Poly.fromCoeffs [one])++-- | prop> G.eigenfunction1  ==  asRational (G.eigenfunctionDifferential 1)+eigenfunction1 :: (Ring.C a) => T a+eigenfunction1 =+   Cons Bell.unit (Poly.fromCoeffs [zero, one])++-- | prop> G.eigenfunction2  ==  asRational (G.eigenfunctionDifferential 2)+eigenfunction2 :: (Field.C a) => T a+eigenfunction2 =+   Cons Bell.unit (Poly.fromCoeffs [-(1/4), zero, one])++-- | prop> G.eigenfunction3  ==  asRational (G.eigenfunctionDifferential 3)+eigenfunction3 :: (Field.C a) => T a+eigenfunction3 =+   Cons Bell.unit (Poly.fromCoeffs [zero, -(3/4), zero, one])+++{- |+prop> QC.forAll (QC.choose (0,15)) $ \n -> let x = G.eigenfunctionDifferential n in G.fourier x  ==  rotateQuarter n x+-}+eigenfunctionDifferential :: (Field.C a) => Int -> T a+eigenfunctionDifferential n =+   (\f -> f{bell = Bell.unit}) $+   nest n (scale (-1/4) . differentiate) $+   Cons (Bell.Cons one zero zero 2) one++{- |+prop> QC.forAll (QC.choose (0,15)) $ \n -> let x = G.eigenfunctionIterative n in G.fourier x  ==  rotateQuarter n x+-}+eigenfunctionIterative ::+   (Field.C a, Absolute.C a, ZeroTestable.C a, Eq a) => Int -> T a+eigenfunctionIterative n =+   fst . head . dropWhile (uncurry (/=)) . mapAdjacent (,) $+   eigenfunctionIteration $+   Cons+      Bell.unit+      (Poly.fromCoeffs $ replicate n zero ++ [one])++eigenfunctionIteration :: (Field.C a) => T a -> [T a]+eigenfunctionIteration =+   iterate (\x ->+      let y = fourier x+          px = polynomial x+          py = polynomial y+          c = last (Poly.coeffs px) / last (Poly.coeffs py)+      in  y{polynomial = fmap (0.5*) (px + fmap (c*) py)})+++{- |+prop> withRational $ Laws.identity G.multiply G.constant+prop> withRational $ Laws.commutative G.multiply+prop> withRational $ Laws.associative G.multiply+-}+multiply :: (Ring.C a) =>+   T a -> T a -> T a+multiply f g =+   Cons+      (Bell.multiply (bell f) (bell g))+      (polynomial f * polynomial g)++{- |+prop> withRational $ Laws.commutative G.convolve+prop> withRational $ Laws.associative G.convolve+-}+convolve, {- convolveByDifferentiation, -} convolveByFourier :: (Field.C a) =>+   T a -> T a -> T a+convolve = convolveByFourier++{-+f <*> g =+   let (foff,fint) = integrate f+   in  fint <*> differentiate g + makeGaussPoly foff * g++In principle this would work,+but (makeGaussPoly foff * g) contains a lot of+convolutions of Gaussian with Gaussian-polynomial-product,+where the Gaussians have different parameters.++convolveByDifferentiation f g =+   case polynomial f of+      fpoly ->+         if null $ Poly.coeffs fpoly+           then ...+           else ...+-}++convolveByFourier f g =+   reverse $ fourier $ multiply (fourier f) (fourier g)++{-+We use a Horner like scheme+in order to translate multiplications with @id@+to differentations on the Fourier side.+Quadratic runtime.++fourier (Cons bell (Poly.const a + Poly.shift f))+  = fourier (Cons bell (Poly.const a)) + fourier (Cons bell (Poly.shift f))+  = fourier (Cons bell (Poly.const a)) + differentiate (fourier (Cons bell f))++We can certainly speed this up considerably+by decomposing the polynomial into four polynomials,+one for each of the four eigenvalues 1, i, -1, -i.+-}+{- |+prop> withRational $ \x y -> G.fourier (G.convolve x y) == G.multiply (G.fourier x) (G.fourier y)+prop> withRational $ \x -> nest 2 G.fourier x == G.reverse x+prop> withRational $ \x a -> G.fourier (G.translate a x) == G.modulate a (G.fourier x)+prop> withRational $ \x (QC.Positive a) -> G.fourier (G.dilate a x) == G.amplify a (G.shrink a (G.fourier x))+prop> withRational $ \x -> G.fourier (G.differentiate x) == mulLinear2i (G.fourier x)+-}+fourier :: (Field.C a) =>+   T a -> T a+fourier f =+   foldr+      (\c p ->+          let q = differentiate p+          in  q{polynomial =+                   Poly.const c ++                   fmap (Complex.scale (1/2) . Complex.quarterLeft) (polynomial q)})+      (Cons (Bell.fourier $ bell f) zero) $+   Poly.coeffs $ polynomial f++{- |+Differentiate and divide by @sqrt pi@ in order to stay in a ring.+This way, we do not need to fiddle with pi factors.++prop> withRational $ \x y -> G.convolve (G.differentiate x) y == G.convolve x (G.differentiate y)+-}+differentiate :: (Ring.C a) => T a -> T a+differentiate f =+   f{polynomial =+        Differential.differentiate (polynomial f)+        - Differential.differentiate (Bell.exponentPolynomial (bell f))+           * polynomial f}++{-+g = (bell f * poly f)'+  = bell f * ((poly f)' - (exppoly (bell f))' * poly f)+poly g = (poly f)' - (exppoly (bell f))' * poly f++Integration means we have g and ask for f.++poly f = ((poly f)' - poly g) / (exppoly (bell f))'++However must start with the highest term of 'poly f',+and thus we need to perform the division on reversed polynomials.+-}+{- |+>>> snd $ G.integrate $ G.differentiate $ G.Cons Bell.unit (Poly.fromCoeffs [7,7,7,7 :: Complex.T Rational])+Cons {bell = Cons {amp = 1 % 1, c0 = 0 % 1 +: 0 % 1, c1 = 0 % 1 +: 0 % 1, c2 = 1 % 1}, polynomial = Polynomial.fromCoeffs [7 % 1 +: 0 % 1,7 % 1 +: 0 % 1,7 % 1 +: 0 % 1,7 % 1 +: 0 % 1]}++prop> withRational $ \x -> G.integrate (G.differentiate x) == (zero, x)+prop> withRational $ \x@(G.Cons b p) -> let (xoff,xint) = G.integrate x in G.differentiate xint == G.Cons b (p + Poly.const xoff)+-}+integrate ::+   (Field.C a, ZeroTestable.C a) =>+   T a -> (Complex.T a, T a)+integrate f =+   let fs = Poly.coeffs $ polynomial f+       (ys,~[r]) =+          PolyCore.divModRev+             {-+             We need the shortening convention of 'zipWith'+             in order to limit the result list,+             we cannot use list instance for (-).+             -}+             (zipWith (-)+                (0 : 0 : diffRev ys)+                (List.reverse fs))+             (List.reverse $ Poly.coeffs $+              Differential.differentiate $+              Bell.exponentPolynomial $ bell f)+   in  forcePair $+       if null fs+         then (zero, f)+         else (r, f{polynomial = Poly.fromCoeffs $ List.reverse ys})++diffRev :: Ring.C a => [a] -> [a]+diffRev xs =+   zipWith (*) xs+      (drop 1 (iterate (subtract 1) (fromIntegral $ length xs)))++{-+integrateDefinite+   (maybe rename integrate to antiderivative and call this one integrate)++int(x^(2*n)*exp(-x^2),x=-infinity..infinity)+ = 2 * int(x^(2*n)*exp(-x^2),x=0..infinity)+     substitute t=x^2, dt = dx * 2 * sqrt t+ = int(t^(n-1/2)*exp(-t),x=0..infinity)+ = Gamma(n+1/2)+ = (2n-1)!!/2^n * sqrt pi++int(pi^n*x^(2*n)*exp(-pi*x^2),x=-infinity..infinity)+ = (2n-1)!!/2^n+++The remainder value of 'integrate'+is the coefficient of the error function+and this is the only part that does not vanish when approaching the limit.+++In order to stay in a field,+we have to return a rational number+and a transcendental part written es @exp a@.++It would be interesting to see how integral inequalities+translate to scalar inequalities containing exponential functions.+-}+++{- |+prop> withRational $ \x a b -> G.translate a (G.translate b x) == G.translate (a+b) x+-}+translate :: Ring.C a => a -> T a -> T a+translate d =+   translateComplex (Complex.fromReal d)++{- |+prop> withRational $ \x a b -> G.translateComplex a (G.translateComplex b x) == G.translateComplex (a+b) x+prop> withRational $ \x a -> G.translateComplex (Complex.fromReal a) x == G.translate a x+-}+translateComplex :: Ring.C a => Complex.T a -> T a -> T a+translateComplex d f =+   Cons+      (Bell.translateComplex d $ bell f)+      (Poly.translate d $ polynomial f)++{- |+prop> withRational $ \x a b -> G.modulate a (G.modulate b x) == G.modulate (a+b) x+prop> withRational $ \x a b -> G.modulate b (G.translate a x) == G.turn (a*b) (G.translate a (G.modulate b x))+-}+modulate :: Ring.C a => a -> T a -> T a+modulate d f =+   Cons+      (Bell.modulate d $ bell f)+      (polynomial f)++turn :: Ring.C a => a -> T a -> T a+turn d f =+   Cons+      (Bell.turn d $ bell f)+      (polynomial f)++{- |+prop> withRational $ \x -> nest 2 G.reverse x == x+-}+reverse :: Additive.C a => T a -> T a+reverse f =+   Cons+      (Bell.reverse $ bell f)+      (Poly.reverse $ polynomial f)++{- |+prop> withRational $ \x (QC.Positive a) (QC.Positive b) -> G.dilate a (G.dilate b x) == G.dilate (a*b) x+prop> withRational $ \x (QC.Positive a) -> G.shrink a x == G.dilate (recip a) x+-}+dilate :: Field.C a => a -> T a -> T a+dilate k f =+   Cons+      (Bell.dilate k $ bell f)+      (Poly.dilate (Complex.fromReal k) $ polynomial f)++{- |+prop> withRational $ \x (QC.Positive a) -> G.dilate a (G.shrink a x) == x+prop> withRational $ \x (QC.Positive a) -> G.shrink a (G.dilate a x) == x+-}+shrink :: Ring.C a => a -> T a -> T a+shrink k f =+   Cons+      (Bell.shrink k $ bell f)+      (Poly.shrink (Complex.fromReal k) $ polynomial f)++{-+We could also amplify the polynomial coefficients.+-}+amplify :: Ring.C a => a -> T a -> T a+amplify k f =+   Cons+      (Bell.amplify k $ bell f)+      (polynomial f)+++{- |+Approximate a @T a@ using a linear combination of translated @Bell.T a@.+The smaller the unit (e.g. 0.1, 0.01, 0.001)+the better the approximation but the worse the numeric properties.++We cannot put all information into @amp@ of @Bell@,+since @amp@ must be real, but is complex here by construction.+We really need at least signed amplitudes at this place,+since we want to represent differences of Gaussians.++prop> withRational $ \x (QC.NonZero unit) d -> G.approximateByBells unit (G.translateComplex d x) == map (mapSnd (Bell.translateComplex d)) (G.approximateByBells unit x)+prop> withRational $ \x (QC.NonZero unit) (QC.NonZero d) -> G.approximateByBells unit (G.dilate d x) == map (mapSnd (Bell.dilate d)) (G.approximateByBells (unit/d) x)+prop> withRational $ \x (QC.NonZero unit) (QC.NonZero d) -> G.approximateByBells unit (G.shrink d x) == map (mapSnd (Bell.shrink d)) (G.approximateByBells (unit*d) x)+-}+approximateByBells ::+   Field.C a =>+   a -> T a -> [(Complex.T a, Bell.T a)]+approximateByBells unit_ f =+   let b = bell f+       amps =+          -- approximateByBellsByTranslation+          approximateByBellsAtOnce+             unit_+             (Complex.scale (recip (2 * Bell.c2 b)) (Bell.c1 b))+             (recip (2*unit_*Bell.c2 b))+             (polynomial f)+   in  zip (LPoly.coeffs amps) $+       map+          (\d -> Bell.translate d b)+          (laurentAbscissas (unit_/2) amps)++{- |+prop> \(QC.NonZero unit) d s p0 -> let p = Poly.fromCoeffs $ take 10 p0 in G.approximateByBellsAtOnce unit d s p == G.approximateByBellsByTranslation unit d (s::Rational) p+-}+approximateByBellsAtOnce ::+   Field.C a =>+   a -> Complex.T a -> a -> Poly.T (Complex.T a) -> LPoly.T (Complex.T a)+approximateByBellsAtOnce unit_ d s p =+   foldr+      (\x amps0 ->+         {-+         Decompose (bell t * (t-d)) = bell t * t - bell t * d+         -}+         let y = fmap (Complex.scale s) amps0+         in  -- \t -> bell t * t+             --    ~   (translate unit_ bell - translate (-unit_) bell) / unit_+             LPoly.shift 1 y -+             LPoly.shift (-1) y ++             -- bell t * d+             zipWithAbscissas+                (\t z -> (Complex.fromReal t - d) * z)+                (unit_/2) amps0 ++             LPoly.const x)+      (LPoly.fromCoeffs [])+      (Poly.coeffs p)++approximateByBellsByTranslation ::+   Field.C a =>+   a -> Complex.T a -> a -> Poly.T (Complex.T a) -> LPoly.T (Complex.T a)+approximateByBellsByTranslation unit_ d s p =+   foldr+      (\x amps0 ->+         {-+         Decompose (bell t * (t-d)) = bell t * t - bell t * d+         -}+         let y = fmap (Complex.scale s) amps0+         in  -- \t -> bell t * t+             --    ~   (translate unit_ bell - translate (-unit_) bell) / unit_+             LPoly.shift 1 y -+             LPoly.shift (-1) y ++             -- bell t * d+             zipWithAbscissas Complex.scale (unit_/2) amps0 ++             LPoly.const x)+      (LPoly.fromCoeffs [])+      (Poly.coeffs $ Poly.translate d p)++zipWithAbscissas ::+   (Ring.C a) =>+   (a -> b -> c) -> a -> LPoly.T b -> LPoly.T c+zipWithAbscissas h unit_ y =+   LPoly.fromShiftCoeffs (LPoly.expon y) $+   zipWith h+      (laurentAbscissas unit_ y)+      (LPoly.coeffs y)++laurentAbscissas :: Ring.C a => a -> LPoly.T c -> [a]+laurentAbscissas unit_ =+   map (\d -> fromIntegral d * unit_) .+   iterate (1+) . LPoly.expon+++{- No Ring instance for Gaussians+instance (Ring.C a) => Differential.C (T a) where+   differentiate = differentiate+-}++{- laws+differentiate (f*g) =+   (differentiate f) * g + f * (differentiate g)++inequalities:++Heisenberg's uncertainty relation+   needs integrals and thus needs product of exponential numbers and roots+-}
+ gaussian/MathObj/Gaussian/Variance.hs view
@@ -0,0 +1,285 @@+{-# LANGUAGE RebindableSyntax #-}+{-+We represent a Gaussian bell curve in terms of the reciprocal of its variance+and its value at the origin.++We could do some projective geometry in the exponent+in order to also have zero variance,+which corresponds to the dirac impulse.++The Gaussians form a nice multiplicative commutative monoid.+Maybe we should have such a structure.+It would also be useful for the Root data type+and a new Exponential data type.+-}+module MathObj.Gaussian.Variance where++import qualified MathObj.Polynomial as Poly+import qualified Number.Root as Root++import qualified Algebra.Transcendental as Trans+import qualified Algebra.Algebraic      as Algebraic+import qualified Algebra.Field          as Field+import qualified Algebra.Absolute       as Absolute+import qualified Algebra.Ring           as Ring+import qualified Algebra.Additive       as Additive++import Test.QuickCheck (Arbitrary, arbitrary, )+import Control.Monad (liftM2, )++import NumericPrelude.Numeric+import NumericPrelude.Base+++{- $setup+>>> import qualified MathObj.Gaussian.Variance as G+>>> import MathObj.Gaussian.ExponentTuple (HoelderConjugates(HoelderConjugates))+>>> import MathObj.Gaussian.ExponentTuple (YoungConjugates(YoungConjugates))+>>> import qualified Algebra.Laws as Laws+>>> import qualified Number.Root as Root+>>> import NumericPrelude.Base as P+>>> import NumericPrelude.Numeric as NP+>>> import Prelude ()+>>> import qualified Test.QuickCheck as QC+>>> import Data.Function.HT (Id, nest)+>>>+>>> asRational :: Id (G.T Rational)+>>> asRational = id+>>>+>>> withRational :: Id (G.T Rational -> a)+>>> withRational = id+-}+++{- |+Since @amp@ is the square of the actual amplitude it must be non-negative.+-}+data T a = Cons {amp, c :: a}+   deriving (Eq, Show)++instance (Absolute.C a, Arbitrary a) => Arbitrary (T a) where+   arbitrary =+      liftM2 Cons+         (fmap abs arbitrary)+         (fmap ((1+) . abs) arbitrary)+++constant :: Ring.C a => T a+constant = Cons one zero++{- |+eigenfunction of 'fourier'+-}+unit :: Ring.C a => T a+unit = Cons one one++{-# INLINE evaluate #-}+evaluate :: (Trans.C a) =>+   T a -> a -> a+evaluate f x =+   sqrt (amp f) * exp (-pi * c f * x^2)++exponentPolynomial :: (Additive.C a) =>+   T a -> Poly.T a+exponentPolynomial f =+   Poly.fromCoeffs [zero, zero, c f]+++integrateRoot :: (Field.C a) => T a -> Root.T a+integrateRoot f =+   Root.sqrt $ Root.fromNumber $ amp f / c f++{- |+Cauchy-Schwarz inequality:++prop> withRational $ \x y -> G.scalarProductRoot x y <= G.norm2Root x `Root.mul` G.norm2Root y++Hoelder inequality:++prop> withRational $ \x y -> G.scalarProductRoot x y <= G.norm1Root x `Root.mul` G.normInfRoot y+prop> withRational $ \x y (HoelderConjugates p q) -> G.scalarProductRoot x y <= G.normPRoot p x `Root.mul` G.normPRoot q y+-}+scalarProductRoot :: (Field.C a) => T a -> T a -> Root.T a+scalarProductRoot f g =+   integrateRoot (multiply f g)+++{- |+prop> withRational $ \x -> G.norm1Root x == G.normPRoot 1 x+-}+norm1Root :: (Field.C a) => T a -> Root.T a+norm1Root = integrateRoot++{- |+prop> withRational $ \x -> G.norm2Root x == G.normPRoot 2 x+-}+norm2Root :: (Field.C a) => T a -> Root.T a+norm2Root f =+   Root.sqrt $+      Root.fromNumber (amp f)+      `Root.div`+      Root.sqrt (Root.fromNumber $ 2 * c f)++normInfRoot :: (Field.C a) => T a -> Root.T a+normInfRoot f =+   Root.sqrt $ Root.fromNumber $ amp f++{-+I would have liked to test for a monotony of norms.+Unfortunately, it does not hold.++Means contain a division by the size of the domain.+Norms do not have this division.+Means are monotonic with respect to the degree.+Norms are not.+We cannot turn the norms into means since the size of the domain+(the complete real axis) is infinitely large.++prop> :{ withRational $ \x p0 q0 ->+   let p = 1 + abs p0+       q = 1 + abs q0+   in  case compare p q of+          EQ -> G.normPRoot p x == G.normPRoot q x+          LT -> G.normPRoot p x <= G.normPRoot q x+          GT -> G.normPRoot p x >= G.normPRoot q x+:}++This should also fail,+but QuickCheck does not seem to try counterexamples.++prop> :{ withRational $ \x p0 ->+   let p = 1 + abs p0+   in  G.normPRoot p x <= G.normInfRoot x+:}+-}+normPRoot :: (Field.C a) => Rational -> T a -> Root.T a+normPRoot p f =+   Root.sqrt (Root.fromNumber (amp f))+   `Root.div`+   Root.rationalPower (recip (2*p)) (Root.fromNumber (fromRational' p * c f))+++-- ToDo: implement NormedSpace.Sum et.al.+norm1 :: (Algebraic.C a) => T a -> a+norm1 f =+   sqrt $ amp f / c f++norm2 :: (Algebraic.C a) => T a -> a+norm2 f =+   sqrt $ amp f / (sqrt $ 2 * c f)++normInf :: (Algebraic.C a) => T a -> a+normInf f =+   sqrt (amp f)++normP :: (Trans.C a) => a -> T a -> a+normP p f =+   sqrt (amp f) * (p * c f) ^? (- recip (2*p))+++variance :: (Trans.C a) =>+   T a -> a+variance f =+   recip $ c f * 2*pi++{- |+prop> withRational $ \x (QC.Positive a) -> G.varianceRational (G.dilate a x) == a^2 * G.varianceRational x+prop> withRational $ \x y -> G.varianceRational (G.convolve x y) == G.varianceRational x + G.varianceRational y+-}+varianceRational :: (Field.C a) => T a -> a+varianceRational f = recip $ c f++{- |+prop> Laws.identity G.multiply G.constant . asRational+prop> Laws.commutative G.multiply . asRational+prop> Laws.associative G.multiply . asRational+-}+multiply :: (Ring.C a) =>+   T a -> T a -> T a+multiply f g =+   Cons (amp f * amp g) (c f + c g)++powerRing :: (Trans.C a) =>+   Integer -> T a -> T a+powerRing p f =+   Cons (amp f ^ p) (fromInteger p * c f)++{-+powerField does not makes sense,+since the reciprocal of a Gaussian diverges.+-}++powerAlgebraic :: (Trans.C a) =>+   Rational -> T a -> T a+powerAlgebraic p f =+   Cons (amp f ^/ p) (fromRational' p * c f)++powerTranscendental :: (Trans.C a) =>+   a -> T a -> T a+powerTranscendental p f =+   Cons (amp f ^? p) (p * c f)++{- |+> convolve x y t =+>    integrate $ \s -> x s * y(t-s)++Convergence only for @c f + c g > 0@.++prop> Laws.commutative G.convolve . asRational+prop> Laws.associative G.convolve . asRational++Young inequality:++prop> withRational $ \x y -> G.normInfRoot (G.convolve x y) <= G.norm1Root x `Root.mul` G.normInfRoot y+prop> withRational $ \x y (HoelderConjugates p q) -> G.normInfRoot (G.convolve x y) <= G.normPRoot p x `Root.mul` G.normPRoot q y+prop> withRational $ \x y (YoungConjugates p q r) -> G.normPRoot r (G.convolve x y) <= G.normPRoot p x `Root.mul` G.normPRoot q y+-}+convolve :: (Field.C a) =>+   T a -> T a -> T a+convolve f g =+   let s = c f + c g+   in  Cons+          (amp f * amp g / s)+          (c f * c g / s)++{- |+> fourier x f =+>    integrate $ \t -> x t * cis (-2*pi*t*f)++Convergence only for @c f > 0@.++prop> withRational $ \x y -> G.fourier (G.convolve x y) == G.multiply (G.fourier x) (G.fourier y)+prop> withRational $ \x -> nest 4 G.fourier x == x+prop> withRational $ \x (QC.Positive a) -> G.fourier (G.dilate a x) == G.amplify a (G.shrink a (G.fourier x))+prop> withRational $ \x y -> G.scalarProductRoot x y == G.scalarProductRoot (G.fourier x) (G.fourier y)+-}+fourier :: (Field.C a) =>+   T a -> T a+fourier f =+   Cons (amp f / c f) (recip $ c f)+{-+fourier (t -> exp(-(a*t)^2))+-}++{- |+prop> withRational $ \x (QC.Positive a) (QC.Positive b) -> G.dilate a (G.dilate b x) == G.dilate (a*b) x+prop> withRational $ \x (QC.Positive a) -> G.shrink a x == G.dilate (recip a) x+-}+dilate :: (Field.C a) => a -> T a -> T a+dilate k f =+   Cons (amp f) $ c f / k^2++{- |+prop> withRational $ \x (QC.Positive a) -> G.dilate a (G.shrink a x) == x+prop> withRational $ \x (QC.Positive a) -> G.shrink a (G.dilate a x) == x+-}+shrink :: (Ring.C a) => a -> T a -> T a+shrink k f =+   Cons (amp f) $ c f * k^2++{- |+@amplify k@ scales by @abs k@!+-}+amplify :: (Ring.C a) => a -> T a -> T a+amplify k f =+   Cons (k^2 * amp f) $ c f
numeric-prelude.cabal view
@@ -1,412 +1,62 @@+Cabal-Version:  2.2 Name:           numeric-prelude-Version:        0.2.2.1-License:        GPL+Version:        0.4.4+License:        BSD-3-Clause License-File:   LICENSE Author:         Dylan Thurston <dpt@math.harvard.edu>, Henning Thielemann <numericprelude@henning-thielemann.de>, Mikael Johansson Maintainer:     Henning Thielemann <numericprelude@henning-thielemann.de> Homepage:       http://www.haskell.org/haskellwiki/Numeric_Prelude Category:       Math Stability:      Experimental-Cabal-Version:  >=1.6+Tested-With:    GHC==7.4.2, GHC==7.6.3, GHC==7.8.4, GHC==7.10.3+Tested-With:    GHC==8.4.4, GHC==8.6.5, GHC==9.0.1 Build-Type:     Simple-Tested-With:    GHC==6.4.1, GHC==6.8.2, GHC==6.10.4, GHC==6.12.3, GHC==7.0.2, GHC==7.2.1 Synopsis:       An experimental alternative hierarchy of numeric type classes Description:-  Revisiting the Numeric Classes-  .-  The Prelude for Haskell 98 offers a well-considered set of numeric classes-  which covers the standard numeric types-  ('Integer', 'Int', 'Rational', 'Float', 'Double', 'Complex') quite well.-  But they offer limited extensibility and have a few other flaws.-  In this proposal we will revisit these classes, addressing the following concerns:-  .-  [1] The current Prelude defines no semantics for the fundamental operations.-      For instance, presumably addition should be associative-      (or come as close as feasible),-      but this is not mentioned anywhere.-  .-  [2] There are some superfluous superclasses.-      For instance, 'Eq' and 'Show' are superclasses of 'Num'.-      Consider the data type-      @   data IntegerFunction a = IF (a -> Integer) @-      One can reasonably define all the methods of 'Algebra.Ring.C' for-      @IntegerFunction a@ (satisfying good semantics),-      but it is impossible to define non-bottom instances of 'Eq' and 'Show'.-      In general, superclass relationship should indicate-      some semantic connection between the two classes.-  .-  [3] In a few cases, there is a mix of semantic operations and-      representation-specific operations.-      'toInteger', 'toRational',-      and the various operations in 'RealFloating' ('decodeFloat', ...)-      are the main examples.-  .-  [4] In some cases, the hierarchy is not finely-grained enough:-      Operations that are often defined independently are lumped together.-      For instance, in a financial application one might want a type \"Dollar\",-      or in a graphics application one might want a type \"Vector\".-      It is reasonable to add two Vectors or Dollars,-      but not, in general, reasonable to multiply them.-      But the programmer is currently forced to define a method for '(*)'-      when she defines a method for '(+)'.-  .-  In specifying the semantics of type classes,-  I will state laws as follows:-  .-  >    (a + b) + c === a + (b + c)-  .-  The intended meaning is extensional equality:-  The rest of the program should behave in the same way-  if one side is replaced with the other.-  Unfortunately, the laws are frequently violated by standard instances;-  the law above, for instance, fails for 'Float':-  .-  >    (1e20 + (-1e20)) + 1.0  = 1.0-  >     1e20 + ((-1e20) + 1.0) = 0.0-  .-  For inexact number types like floating point types,-  thus these laws should be interpreted as guidelines rather than absolute rules.-  In particular, the compiler is not allowed to use them for optimization.-  Unless stated otherwise, default definitions should also be taken as laws.-  .-  Thanks to Brian Boutel, Joe English, William Lee Irwin II, Marcin-  Kowalczyk, Ketil Malde, Tom Schrijvers, Ken Shan, and Henning-  Thielemann for helpful comments.-  .-  .-  Usage:-  .-  Write modules in the following style:-  .-  > [-# NoImplicitPrelude #-]-  > module MyModule where-  >-  > ... various specific imports ...-  >-  > import NumericPrelude-  .-  Importing @NumericPrelude@ is almost the same as-  .-  > import NumericPrelude.Numeric-  > import NumericPrelude.Base   .-  .-  Instead of the @NoImplicitPrelude@ pragma-  you could also write @import Prelude ()@-  but this will yield problems with numeric literals.-  .-  .-  Scope & Limitations\/TODO:-  .-  * It might be desireable to split Ord up into Poset and Ord-    (a total ordering).-    This is not addressed here.-  .-  * In some cases, this hierarchy may not yet be fine-grained enough.-    For instance, time spans (\"5 minutes\") can be added to times (\"12:34\"),-    but two times are not addable. (\"12:34 + 8:23\")-    As it stands,-    users have to use a different operator for adding time spans to times-    than for adding two time spans.-    Similar issues arise for vector space et al.-    This is a consciously-made tradeoff, but might be changed.-    This becomes most serious when dealing with quantities with units-    like @length\/distance^2@, for which @(*)@ as defined here is useless.-    (One way to see the issue: should-    @  f x y = iterate (x *) y  @-    have principal type-    @  (Ring.C a) => a -> a -> [a]  @-    or something like-    @  (Ring.C a, Module a b) => a -> b -> [b]  @-    ?)-  .-  * I stuck with the Haskell 98 names.-    In some cases I find them lacking.-    Neglecting backwards compatibility, we have renamed classes as follows:-      Num           --> Additive, Ring, Absolute-      Integral      --> ToInteger, IntegralDomain, RealIntegral-      Fractional    --> Field-      Floating      --> Algebraic, Transcendental-      Real          --> ToRational-      RealFrac      --> RealRing, RealField-      RealFloat     --> RealTranscendental-  .-  .-  Additional standard libraries might include Enum, IEEEFloat (including-  the bulk of the functions in Haskell 98's RealFloat class),-  VectorSpace, Ratio, and Lattice.+  The package provides an experimental alternative hierarchy+  of numeric type classes.+  The type classes are more oriented at mathematical structures+  and their methods come with laws that the instances must fulfill.  Extra-Source-Files:   Makefile+  README.md   docs/NOTES   docs/README-  src/Algebra/Absolute.hs-  src/Algebra/Additive.hs-  src/Algebra/AffineSpace.hs-  src/Algebra/Algebraic.hs-  src/Algebra/Differential.hs-  src/Algebra/DimensionTerm.hs-  src/Algebra/DivisibleSpace.hs-  src/Algebra/EqualityDecision.hs-  src/Algebra/Field.hs   src/Algebra/GenerateRules.hs-  src/Algebra/Indexable.hs-  src/Algebra/IntegralDomain.hs-  src/Algebra/Lattice.hs-  src/Algebra/Laws.hs-  src/Algebra/Module.hs-  src/Algebra/ModuleBasis.hs-  src/Algebra/Monoid.hs-  src/Algebra/NonNegative.hs-  src/Algebra/NormedSpace/Euclidean.hs-  src/Algebra/NormedSpace/Maximum.hs-  src/Algebra/NormedSpace/Sum.hs-  src/Algebra/OccasionallyScalar.hs-  src/Algebra/OrderDecision.hs-  src/Algebra/PrincipalIdealDomain.hs-  src/Algebra/RealField.hs-  src/Algebra/RealIntegral.hs-  src/Algebra/RealRing.hs-  src/Algebra/RealTranscendental.hs-  src/Algebra/RightModule.hs-  src/Algebra/Ring.hs-  src/Algebra/ToInteger.hs-  src/Algebra/ToRational.hs-  src/Algebra/Transcendental.hs-  src/Algebra/Units.hs-  src/Algebra/Vector.hs-  src/Algebra/VectorSpace.hs-  src/Algebra/ZeroTestable.hs-  src/MathObj/Algebra.hs-  src/MathObj/DiscreteMap.hs-  src/MathObj/Gaussian/Bell.hs-  src/MathObj/Gaussian/Example.hs-  src/MathObj/Gaussian/Polynomial.hs-  src/MathObj/Gaussian/Variance.hs-  src/MathObj/LaurentPolynomial.hs-  src/MathObj/Matrix.hs-  src/MathObj/Monoid.hs-  src/MathObj/PartialFraction.hs-  src/MathObj/Permutation.hs-  src/MathObj/Permutation/CycleList.hs-  src/MathObj/Permutation/CycleList/Check.hs-  src/MathObj/Permutation/Table.hs-  src/MathObj/Polynomial.hs-  src/MathObj/Polynomial/Core.hs-  src/MathObj/PowerSeries.hs-  src/MathObj/PowerSeries/Core.hs-  src/MathObj/PowerSeries/DifferentialEquation.hs-  src/MathObj/PowerSeries/Example.hs-  src/MathObj/PowerSeries/Mean.hs-  src/MathObj/PowerSeries2.hs-  src/MathObj/PowerSeries2/Core.hs-  src/MathObj/PowerSum.hs-  src/MathObj/RefinementMask2.hs-  src/MathObj/RootSet.hs-  src/Number/Complex.hs-  src/Number/ComplexSquareRoot.hs-  src/Number/DimensionTerm.hs-  src/Number/DimensionTerm/SI.hs-  src/Number/FixedPoint.hs-  src/Number/FixedPoint/Check.hs-  src/Number/GaloisField2p32m5.hs-  src/Number/NonNegative.hs-  src/Number/NonNegativeChunky.hs-  src/Number/OccasionallyScalarExpression.hs-  src/Number/PartiallyTranscendental.hs-  src/Number/Peano.hs-  src/Number/Physical.hs-  src/Number/Physical/Read.hs-  src/Number/Physical/Show.hs-  src/Number/Physical/Unit.hs-  src/Number/Physical/UnitDatabase.hs-  src/Number/Positional.hs-  src/Number/Positional/Check.hs-  src/Number/Quaternion.hs-  src/Number/Ratio.hs-  src/Number/ResidueClass.hs-  src/Number/ResidueClass/Check.hs-  src/Number/ResidueClass/Func.hs-  src/Number/ResidueClass/Maybe.hs-  src/Number/ResidueClass/Reader.hs-  src/Number/Root.hs-  src/Number/SI.hs-  src/Number/SI/Unit.hs-  src/NumericPrelude.hs-  src/NumericPrelude/Base.hs-  src/NumericPrelude/Elementwise.hs-  src/NumericPrelude/List.hs-  src/NumericPrelude/List/Checked.hs-  src/NumericPrelude/List/Generic.hs-  src/NumericPrelude/Numeric.hs-  test/Gaussian.hs-  test/Test.hs-  test/Test/Algebra/IntegralDomain.hs-  test/Test/Algebra/RealRing.hs-  test/Test/MathObj/Gaussian/Bell.hs-  test/Test/MathObj/Gaussian/Polynomial.hs-  test/Test/MathObj/Gaussian/Variance.hs-  test/Test/MathObj/Matrix.hs-  test/Test/MathObj/PartialFraction.hs-  test/Test/MathObj/Polynomial.hs-  test/Test/MathObj/PowerSeries.hs-  test/Test/MathObj/RefinementMask2.hs-  test/Test/Number/ComplexSquareRoot.hs-  test/Test/Number/GaloisField2p32m5.hs-  test/Test/NumericPrelude/Utility.hs-  test/Test/Run.hs-  src-ghc-6.12/Algebra/Absolute.hs-  src-ghc-6.12/Algebra/Additive.hs-  src-ghc-6.12/Algebra/AffineSpace.hs-  src-ghc-6.12/Algebra/Algebraic.hs-  src-ghc-6.12/Algebra/Differential.hs-  src-ghc-6.12/Algebra/DimensionTerm.hs-  src-ghc-6.12/Algebra/DivisibleSpace.hs-  src-ghc-6.12/Algebra/EqualityDecision.hs-  src-ghc-6.12/Algebra/Field.hs-  src-ghc-6.12/Algebra/GenerateRules.hs-  src-ghc-6.12/Algebra/Indexable.hs-  src-ghc-6.12/Algebra/IntegralDomain.hs-  src-ghc-6.12/Algebra/Lattice.hs-  src-ghc-6.12/Algebra/Laws.hs-  src-ghc-6.12/Algebra/Module.hs-  src-ghc-6.12/Algebra/ModuleBasis.hs-  src-ghc-6.12/Algebra/Monoid.hs-  src-ghc-6.12/Algebra/NonNegative.hs-  src-ghc-6.12/Algebra/NormedSpace/Euclidean.hs-  src-ghc-6.12/Algebra/NormedSpace/Maximum.hs-  src-ghc-6.12/Algebra/NormedSpace/Sum.hs-  src-ghc-6.12/Algebra/OccasionallyScalar.hs-  src-ghc-6.12/Algebra/OrderDecision.hs-  src-ghc-6.12/Algebra/PrincipalIdealDomain.hs-  src-ghc-6.12/Algebra/RealField.hs-  src-ghc-6.12/Algebra/RealIntegral.hs-  src-ghc-6.12/Algebra/RealRing.hs-  src-ghc-6.12/Algebra/RealTranscendental.hs-  src-ghc-6.12/Algebra/RightModule.hs-  src-ghc-6.12/Algebra/Ring.hs-  src-ghc-6.12/Algebra/ToInteger.hs-  src-ghc-6.12/Algebra/ToRational.hs-  src-ghc-6.12/Algebra/Transcendental.hs-  src-ghc-6.12/Algebra/Units.hs-  src-ghc-6.12/Algebra/Vector.hs-  src-ghc-6.12/Algebra/VectorSpace.hs-  src-ghc-6.12/Algebra/ZeroTestable.hs-  src-ghc-6.12/MathObj/Algebra.hs-  src-ghc-6.12/MathObj/DiscreteMap.hs-  src-ghc-6.12/MathObj/Gaussian/Bell.hs-  src-ghc-6.12/MathObj/Gaussian/Example.hs-  src-ghc-6.12/MathObj/Gaussian/Polynomial.hs-  src-ghc-6.12/MathObj/Gaussian/Variance.hs-  src-ghc-6.12/MathObj/LaurentPolynomial.hs-  src-ghc-6.12/MathObj/Matrix.hs-  src-ghc-6.12/MathObj/Monoid.hs-  src-ghc-6.12/MathObj/PartialFraction.hs-  src-ghc-6.12/MathObj/Permutation.hs-  src-ghc-6.12/MathObj/Permutation/CycleList.hs-  src-ghc-6.12/MathObj/Permutation/CycleList/Check.hs-  src-ghc-6.12/MathObj/Permutation/Table.hs-  src-ghc-6.12/MathObj/Polynomial.hs-  src-ghc-6.12/MathObj/Polynomial/Core.hs-  src-ghc-6.12/MathObj/PowerSeries.hs-  src-ghc-6.12/MathObj/PowerSeries/Core.hs-  src-ghc-6.12/MathObj/PowerSeries/DifferentialEquation.hs-  src-ghc-6.12/MathObj/PowerSeries/Example.hs-  src-ghc-6.12/MathObj/PowerSeries/Mean.hs-  src-ghc-6.12/MathObj/PowerSeries2.hs-  src-ghc-6.12/MathObj/PowerSeries2/Core.hs-  src-ghc-6.12/MathObj/PowerSum.hs-  src-ghc-6.12/MathObj/RefinementMask2.hs-  src-ghc-6.12/MathObj/RootSet.hs-  src-ghc-6.12/Number/Complex.hs-  src-ghc-6.12/Number/ComplexSquareRoot.hs-  src-ghc-6.12/Number/DimensionTerm.hs-  src-ghc-6.12/Number/DimensionTerm/SI.hs-  src-ghc-6.12/Number/FixedPoint.hs-  src-ghc-6.12/Number/FixedPoint/Check.hs-  src-ghc-6.12/Number/GaloisField2p32m5.hs-  src-ghc-6.12/Number/NonNegative.hs-  src-ghc-6.12/Number/NonNegativeChunky.hs-  src-ghc-6.12/Number/OccasionallyScalarExpression.hs-  src-ghc-6.12/Number/PartiallyTranscendental.hs-  src-ghc-6.12/Number/Peano.hs-  src-ghc-6.12/Number/Physical.hs-  src-ghc-6.12/Number/Physical/Read.hs-  src-ghc-6.12/Number/Physical/Show.hs-  src-ghc-6.12/Number/Physical/Unit.hs-  src-ghc-6.12/Number/Physical/UnitDatabase.hs-  src-ghc-6.12/Number/Positional.hs-  src-ghc-6.12/Number/Positional/Check.hs-  src-ghc-6.12/Number/Quaternion.hs-  src-ghc-6.12/Number/Ratio.hs-  src-ghc-6.12/Number/ResidueClass.hs-  src-ghc-6.12/Number/ResidueClass/Check.hs-  src-ghc-6.12/Number/ResidueClass/Func.hs-  src-ghc-6.12/Number/ResidueClass/Maybe.hs-  src-ghc-6.12/Number/ResidueClass/Reader.hs-  src-ghc-6.12/Number/Root.hs-  src-ghc-6.12/Number/SI.hs-  src-ghc-6.12/Number/SI/Unit.hs-  src-ghc-6.12/NumericPrelude.hs-  src-ghc-6.12/NumericPrelude/Base.hs-  src-ghc-6.12/NumericPrelude/Elementwise.hs-  src-ghc-6.12/NumericPrelude/List.hs-  src-ghc-6.12/NumericPrelude/List/Checked.hs-  src-ghc-6.12/NumericPrelude/List/Generic.hs-  src-ghc-6.12/NumericPrelude/Numeric.hs-  test-ghc-6.12/Gaussian.hs-  test-ghc-6.12/Test.hs-  test-ghc-6.12/Test/Algebra/IntegralDomain.hs-  test-ghc-6.12/Test/Algebra/RealRing.hs-  test-ghc-6.12/Test/MathObj/Gaussian/Bell.hs-  test-ghc-6.12/Test/MathObj/Gaussian/Polynomial.hs-  test-ghc-6.12/Test/MathObj/Gaussian/Variance.hs-  test-ghc-6.12/Test/MathObj/Matrix.hs-  test-ghc-6.12/Test/MathObj/PartialFraction.hs-  test-ghc-6.12/Test/MathObj/Polynomial.hs-  test-ghc-6.12/Test/MathObj/PowerSeries.hs-  test-ghc-6.12/Test/MathObj/RefinementMask2.hs-  test-ghc-6.12/Test/Number/ComplexSquareRoot.hs-  test-ghc-6.12/Test/Number/GaloisField2p32m5.hs-  test-ghc-6.12/Test/NumericPrelude/Utility.hs-  test-ghc-6.12/Test/Run.hs -Flag splitBase-  description: Choose the new smaller, split-up base package.--Flag buildTests-  description: Build test executables+Flag buildExamples+  description: Build example executables   default:     False  Source-Repository this-  Tag:         0.2.2+  Tag:         0.4.4   Type:        darcs-  Location:    http://code.haskell.org/numeric-prelude/+  Location:    https://hub.darcs.net/thielema/numeric-prelude/  Source-Repository head   Type:        darcs-  Location:    http://code.haskell.org/numeric-prelude/+  Location:    https://hub.darcs.net/thielema/numeric-prelude/  Library   Build-Depends:     parsec >=1 && <4,-    QuickCheck >=1 && <3,+    QuickCheck >=2.10 && <3,     storable-record >=0.0.1 && <0.1,     non-negative >=0.0.5 && <0.2,-    utility-ht >=0.0.6 && <0.1-  If flag(splitBase)-    Build-Depends:-      base >= 2 && <6,-      array >=0.1 && <0.4,-      containers >=0.1 && <0.5,-      random >=1.0 && <1.1-  Else-    Build-Depends: base >= 1.0 && < 2+    semigroups >=0.1 && <1.0,+    utility-ht >=0.0.13 && <0.1,+    deepseq >=1.1 && <1.5 +  Build-Depends:+    array >=0.1 && <0.6,+    containers >=0.1 && <0.7,+    random >=1.0 && <1.3,+    base >=4.5 && <5++  Default-Language: Haskell98   GHC-Options:    -Wall-  If impl(ghc>=7.0)-    Hs-source-dirs: src-  Else-    Hs-source-dirs: src-ghc-6.12+  Hs-source-dirs: src   Exposed-modules:     Algebra.Absolute     Algebra.Additive@@ -415,6 +65,7 @@     Algebra.DimensionTerm     Algebra.DivisibleSpace     Algebra.Field+    Algebra.FloatingPoint     Algebra.Indexable     Algebra.IntegralDomain     Algebra.NonNegative@@ -463,6 +114,8 @@     MathObj.PowerSum     MathObj.RefinementMask2     MathObj.RootSet+    MathObj.Wrapper.Haskell98+    MathObj.Wrapper.NumericPrelude     Number.Complex     Number.DimensionTerm     Number.DimensionTerm.SI@@ -500,61 +153,87 @@   Other-modules:     NumericPrelude.List     Algebra.AffineSpace-    MathObj.Gaussian.Variance-    MathObj.Gaussian.Bell-    MathObj.Gaussian.Polynomial-    Number.ComplexSquareRoot+    Algebra.RealRing98     -- I think I won't add them this way.     -- It is certainly better to split the class into comparison and selection.     Algebra.EqualityDecision     Algebra.OrderDecision -Executable test-  If impl(ghc>=7.0)-    Hs-source-dirs: src, test-  Else-    Hs-source-dirs: src-ghc-6.12, test-ghc-6.12-  Main-Is: Test.hs-  If !flag(buildTests)-    Buildable:         False+Executable numeric-prelude-demo+  Hs-Source-Dirs: test+  GHC-Options:    -Wall+  Default-Language: Haskell98+  Main-Is: Demo.hs -Executable testsuite-  If impl(ghc>=7.0)-    Hs-source-dirs: src, test+  If flag(buildExamples)+    Build-Depends:+      numeric-prelude,+      base   Else-    Hs-source-dirs: src-ghc-6.12, test-ghc-6.12+    Buildable: False++Test-Suite numeric-prelude-test+  Type: exitcode-stdio-1.0   GHC-Options:    -Wall+  Default-Language: Haskell98+  Hs-Source-Dirs: test   Other-modules:     Test.NumericPrelude.Utility     Test.Number.GaloisField2p32m5     Test.Number.ComplexSquareRoot     Test.Algebra.IntegralDomain+    Test.Algebra.PrincipalIdealDomain     Test.Algebra.RealRing+    Test.Algebra.Additive     Test.MathObj.RefinementMask2     Test.MathObj.PartialFraction     Test.MathObj.Matrix     Test.MathObj.Polynomial+    Test.MathObj.Polynomial.Core     Test.MathObj.PowerSeries+    Test.MathObj.PowerSeries.Core+    Test.MathObj.PowerSeries.Example+    Test.MathObj.Gaussian.ExponentTuple     Test.MathObj.Gaussian.Variance     Test.MathObj.Gaussian.Bell     Test.MathObj.Gaussian.Polynomial+  Hs-Source-Dirs: playground+  Other-modules:+    Number.ComplexSquareRoot+  Hs-Source-Dirs: gaussian+  Other-Modules:+    MathObj.Gaussian.Bell+    MathObj.Gaussian.Polynomial+    MathObj.Gaussian.Variance+    MathObj.Gaussian.ExponentTuple   Main-Is: Test/Run.hs-  If flag(buildTests)-    Build-Depends: HUnit >=1 && <2-  Else-    Buildable: False -Executable test-gaussian-  If impl(ghc>=7.0)-    Hs-source-dirs: src, test-  Else-    Hs-source-dirs: src-ghc-6.12, test-ghc-6.12+  Build-Depends:+    doctest-exitcode-stdio >=0.0 && <0.1,+    doctest-lib >=0.1 && <0.1.1,+    numeric-prelude,+    QuickCheck,+    utility-ht,+    random,+    base++Executable numeric-prelude-gaussian+  Hs-Source-Dirs: gaussian   Main-Is: Gaussian.hs+  Default-Language: Haskell98   Other-Modules:     MathObj.Gaussian.Example-  If flag(buildTests)+    MathObj.Gaussian.Variance+    MathObj.Gaussian.Bell+    MathObj.Gaussian.Polynomial++  If flag(buildExamples)     Build-Depends:-      gnuplot >=0.3 && <0.5,-      HTam >=0.0.2 && <0.1+      gnuplot >=0.5 && <0.6,+      HTam >=0.0.2 && <0.2,+      numeric-prelude,+      QuickCheck,+      utility-ht,+      base   Else     Buildable: False
+ playground/Number/ComplexSquareRoot.hs view
@@ -0,0 +1,137 @@+module Number.ComplexSquareRoot where++import qualified Algebra.RealField as RealField+import qualified Algebra.RealRing as RealRing+import qualified Algebra.Ring as Ring+import qualified Algebra.Additive as Additive+import qualified Algebra.ZeroTestable as ZeroTestable++import qualified Number.Complex as Complex++import Test.QuickCheck (Arbitrary, arbitrary, )++import Control.Monad (liftM2, )++import qualified NumericPrelude.Numeric as NP+import NumericPrelude.Numeric hiding (recip, )+import NumericPrelude.Base+import Prelude ()+++{- $setup+>>> import qualified Number.ComplexSquareRoot as SR+>>> import qualified Number.Complex as Complex+>>> import qualified Algebra.Laws as Laws+>>> import Test.QuickCheck ((==>))+>>> import NumericPrelude.Numeric+>>> import NumericPrelude.Base+>>> import Prelude ()+>>>+>>> sr :: SR.T Rational -> SR.T Rational+>>> sr = id+-}++{- |+Represent the square root of a complex number+without actually having to compute a square root.+If the Bool is False,+then the square root is represented with positive real part+or zero real part and positive imaginary part.+If the Bool is True the square root is negated.++prop> Laws.identity SR.mul SR.one . sr+prop> Laws.commutative SR.mul . sr+prop> Laws.associative SR.mul . sr+prop> Laws.homomorphism SR.fromNumber (\x y -> x * (y :: Complex.T Rational)) SR.mul+prop> Laws.rightIdentity SR.div SR.one . sr+prop> \x -> not (isZero x) ==> SR.recip (SR.recip x) == sr x+prop> \x -> not (isZero x) ==> Laws.inverse SR.mul SR.recip SR.one (sr x)+-}+data T a = Cons Bool (Complex.T a)+   deriving (Show)++{- |+You must use @fmap@ only for number type conversion.+-}+instance Functor T where+   fmap f (Cons n x) = Cons n (fmap f x)++instance (ZeroTestable.C a) => ZeroTestable.C (T a) where+   isZero (Cons _b s) = isZero s++instance (ZeroTestable.C a, Eq a) => Eq (T a) where+   (Cons xb xs) == (Cons yb ys) =+      isZero xs && isZero ys  ||+      xb==yb && xs==ys++instance (Arbitrary a) => Arbitrary (T a) where+   arbitrary = liftM2 Cons arbitrary arbitrary+++fromNumber :: (RealRing.C a) => Complex.T a -> T a+fromNumber x =+   Cons+      (case compare zero (Complex.real x) of+         LT -> False+         GT -> True+         EQ -> Complex.imag x < zero)+      (x^2)++-- htam:Wavelet.DyadicResultant.parityFlip+toNumber :: (RealRing.C a, Complex.Power a) => T a -> Complex.T a+toNumber (Cons n x) =+   case sqrt x of y -> if n then NP.negate y else y+++one :: (Ring.C a) => T a+one = Cons False NP.one++inUpperHalfplane :: (Additive.C a, Ord a) => Complex.T a -> Bool+inUpperHalfplane x =+   case compare (Complex.imag x) zero of+      GT -> True+      LT -> False+      EQ -> Complex.real x < zero++mul, mulAlt, mulAlt2 :: (RealRing.C a) => T a -> T a -> T a+mul (Cons xb xs) (Cons yb ys) =+   let zs = xs*ys+   in  Cons+          ((xb /= yb) /=+             case (inUpperHalfplane xs,+                   inUpperHalfplane ys,+                   inUpperHalfplane zs) of+                (True,True,False) -> True+                (False,False,True) -> True+                _ -> False)+          zs++mulAlt (Cons xb xs) (Cons yb ys) =+   let zs = xs*ys+   in  Cons+          ((xb /= yb) /=+             let xi = Complex.imag xs+                 yi = Complex.imag ys+                 zi = Complex.imag zs+             in  (xi>=zero) /= (yi>=zero) &&+                 (xi>=zero) /= (zi>=zero))+          zs++mulAlt2 (Cons xb xs) (Cons yb ys) =+   let zs = xs*ys+   in  Cons+          ((xb /= yb) /=+             let xi = Complex.imag xs+                 yi = Complex.imag ys+                 zi = Complex.imag zs+             in  xi*yi<zero && xi*zi<zero)+          zs++div :: (RealField.C a) => T a -> T a -> T a+div x y = mul x (recip y)++recip :: (RealField.C a) => T a -> T a+recip (Cons b s) =+   Cons+      (b /= (Complex.imag s == zero && Complex.real s < zero))+      (NP.recip s)
− src-ghc-6.12/Algebra/Absolute.hs 
@@ -1,151 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Algebra.Absolute (-   C(abs, signum),-   absOrd, signumOrd,-   ) where--import qualified Algebra.Ring         as Ring-import qualified Algebra.Additive     as Additive-import qualified Algebra.ZeroTestable as ZeroTestable--import Algebra.Ring (one, ) -- fromInteger-import Algebra.Additive (zero, negate,)--import Data.Int  (Int,  Int8,  Int16,  Int32,  Int64,  )-import Data.Word (Word, Word8, Word16, Word32, Word64, )--import NumericPrelude.Base-import qualified Prelude as P-import Prelude (Integer, Float, Double, )---{- |-This is the type class of a ring with a notion of an absolute value,-satisfying the laws-->                        a * b === b * a->   a /= 0  =>  abs (signum a) === 1->             abs a * signum a === a--Minimal definition: 'abs', 'signum'.--If the type is in the 'Ord' class-we expect 'abs' = 'absOrd' and 'signum' = 'signumOrd'-and we expect the following laws to hold:-->      a + (max b c) === max (a+b) (a+c)->   negate (max b c) === min (negate b) (negate c)->      a * (max b c) === max (a*b) (a*c) where a >= 0->           absOrd a === max a (-a)--We do not require 'Ord' as superclass-since we also want to have "Number.Complex" as instance.-'abs' for complex numbers alone may have an inappropriate type,-because it does not reflect that the absolute value is a real number.-You might prefer 'Number.Complex.magnitude'.-This type class is intended for unifying algorithms-that work for both real and complex numbers.-Note the similarity to "Algebra.Units":-'abs' plays the role of @stdAssociate@-and 'signum' plays the role of @stdUnit@.--Actually, since 'abs' can be defined using 'max' and 'negate'-we could relax the superclasses to @Additive@ and 'Ord'-if his class would only contain 'signum'.--}-class (Ring.C a, ZeroTestable.C a) => C a where-    abs    :: a -> a-    signum :: a -> a---absOrd :: (Additive.C a, Ord a) => a -> a-absOrd x = max x (negate x)--signumOrd :: (Ring.C a, Ord a) => a -> a-signumOrd x =-   case compare x zero of-      GT ->        one-      EQ ->        zero-      LT -> negate one---instance C Integer where-   {-# INLINE abs #-}-   {-# INLINE signum #-}-   abs = P.abs-   signum = P.signum--instance C Float   where-   {-# INLINE abs #-}-   {-# INLINE signum #-}-   abs = P.abs-   signum = P.signum--instance C Double  where-   {-# INLINE abs #-}-   {-# INLINE signum #-}-   abs = P.abs-   signum = P.signum---instance C Int     where-   {-# INLINE abs #-}-   {-# INLINE signum #-}-   abs = P.abs-   signum = P.signum--instance C Int8    where-   {-# INLINE abs #-}-   {-# INLINE signum #-}-   abs = P.abs-   signum = P.signum--instance C Int16   where-   {-# INLINE abs #-}-   {-# INLINE signum #-}-   abs = P.abs-   signum = P.signum--instance C Int32   where-   {-# INLINE abs #-}-   {-# INLINE signum #-}-   abs = P.abs-   signum = P.signum--instance C Int64   where-   {-# INLINE abs #-}-   {-# INLINE signum #-}-   abs = P.abs-   signum = P.signum---instance C Word    where-   {-# INLINE abs #-}-   {-# INLINE signum #-}-   abs = P.abs-   signum = P.signum--instance C Word8   where-   {-# INLINE abs #-}-   {-# INLINE signum #-}-   abs = P.abs-   signum = P.signum--instance C Word16  where-   {-# INLINE abs #-}-   {-# INLINE signum #-}-   abs = P.abs-   signum = P.signum--instance C Word32  where-   {-# INLINE abs #-}-   {-# INLINE signum #-}-   abs = P.abs-   signum = P.signum--instance C Word64  where-   {-# INLINE abs #-}-   {-# INLINE signum #-}-   abs = P.abs-   signum = P.signum-
− src-ghc-6.12/Algebra/Additive.hs 
@@ -1,364 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Algebra.Additive (-    -- * Class-    C,-    zero,-    (+), (-),-    negate, subtract,--    -- * Complex functions-    sum, sum1,--    -- * Instance definition helpers-    elementAdd, elementSub, elementNeg,-    (<*>.+), (<*>.-), (<*>.-$),--    -- * Instances for atomic types-    propAssociative,-    propCommutative,-    propIdentity,-    propInverse,-  ) where--import qualified Algebra.Laws as Laws--import Data.Int  (Int,  Int8,  Int16,  Int32,  Int64,  )-import Data.Word (Word, Word8, Word16, Word32, Word64, )--import qualified NumericPrelude.Elementwise as Elem-import Control.Applicative (Applicative(pure, (<*>)), )-import Data.Tuple.HT (fst3, snd3, thd3, )--import qualified Data.Ratio as Ratio98-import qualified Prelude as P-import Prelude (Integer, Float, Double, fromInteger, )-import NumericPrelude.Base---infixl 6  +, ---{- |-Additive a encapsulates the notion of a commutative group, specified-by the following laws:--@-          a + b === b + a-    (a + b) + c === a + (b + c)-       zero + a === a-   a + negate a === 0-@--Typical examples include integers, dollars, and vectors.--Minimal definition: '+', 'zero', and ('negate' or '(-)')--}--class C a where-    -- | zero element of the vector space-    zero     :: a-    -- | add and subtract elements-    (+), (-) :: a -> a -> a-    -- | inverse with respect to '+'-    negate   :: a -> a--    {-# INLINE negate #-}-    negate a = zero - a-    {-# INLINE (-) #-}-    a - b    = a + negate b--{- |-'subtract' is @(-)@ with swapped operand order.-This is the operand order which will be needed in most cases-of partial application.--}-subtract :: C a => a -> a -> a-subtract = flip (-)-----{- |-Sum up all elements of a list.-An empty list yields zero.--This function is inappropriate for number types like Peano.-Maybe we should make 'sum' a method of Additive.-This would also make 'lengthLeft' and 'lengthRight' superfluous.--}-sum :: (C a) => [a] -> a-sum = foldl (+) zero--{- |-Sum up all elements of a non-empty list.-This avoids including a zero which is useful for types-where no universal zero is available.--}-sum1 :: (C a) => [a] -> a-sum1 = foldl1 (+)----{- |-Instead of baking the add operation into the element function,-we could use higher rank types-and pass a generic @uncurry (+)@ to the run function.-We do not do so in order to stay Haskell 98-at least for parts of NumericPrelude.--}-{-# INLINE elementAdd #-}-elementAdd ::-   (C x) =>-   (v -> x) -> Elem.T (v,v) x-elementAdd f =-   Elem.element (\(x,y) -> f x + f y)--{-# INLINE elementSub #-}-elementSub ::-   (C x) =>-   (v -> x) -> Elem.T (v,v) x-elementSub f =-   Elem.element (\(x,y) -> f x - f y)--{-# INLINE elementNeg #-}-elementNeg ::-   (C x) =>-   (v -> x) -> Elem.T v x-elementNeg f =-   Elem.element (negate . f)----- like <*>-infixl 4 <*>.+, <*>.-, <*>.-$--{- |-> addPair :: (Additive.C a, Additive.C b) => (a,b) -> (a,b) -> (a,b)-> addPair = Elem.run2 $ Elem.with (,) <*>.+  fst <*>.+  snd--}-{-# INLINE (<*>.+) #-}-(<*>.+) ::-   (C x) =>-   Elem.T (v,v) (x -> a) -> (v -> x) -> Elem.T (v,v) a-(<*>.+) f acc =-   f <*> elementAdd acc--{-# INLINE (<*>.-) #-}-(<*>.-) ::-   (C x) =>-   Elem.T (v,v) (x -> a) -> (v -> x) -> Elem.T (v,v) a-(<*>.-) f acc =-   f <*> elementSub acc--{-# INLINE (<*>.-$) #-}-(<*>.-$) ::-   (C x) =>-   Elem.T v (x -> a) -> (v -> x) -> Elem.T v a-(<*>.-$) f acc =-   f <*> elementNeg acc----- * Instances for atomic types--instance C Integer where-   {-# INLINE zero #-}-   {-# INLINE negate #-}-   {-# INLINE (+) #-}-   {-# INLINE (-) #-}-   zero   = P.fromInteger 0-   negate = P.negate-   (+)    = (P.+)-   (-)    = (P.-)--instance C Float   where-   {-# INLINE zero #-}-   {-# INLINE negate #-}-   {-# INLINE (+) #-}-   {-# INLINE (-) #-}-   zero   = P.fromInteger 0-   negate = P.negate-   (+)    = (P.+)-   (-)    = (P.-)--instance C Double  where-   {-# INLINE zero #-}-   {-# INLINE negate #-}-   {-# INLINE (+) #-}-   {-# INLINE (-) #-}-   zero   = P.fromInteger 0-   negate = P.negate-   (+)    = (P.+)-   (-)    = (P.-)---instance C Int     where-   {-# INLINE zero #-}-   {-# INLINE negate #-}-   {-# INLINE (+) #-}-   {-# INLINE (-) #-}-   zero   = P.fromInteger 0-   negate = P.negate-   (+)    = (P.+)-   (-)    = (P.-)--instance C Int8    where-   {-# INLINE zero #-}-   {-# INLINE negate #-}-   {-# INLINE (+) #-}-   {-# INLINE (-) #-}-   zero   = P.fromInteger 0-   negate = P.negate-   (+)    = (P.+)-   (-)    = (P.-)--instance C Int16   where-   {-# INLINE zero #-}-   {-# INLINE negate #-}-   {-# INLINE (+) #-}-   {-# INLINE (-) #-}-   zero   = P.fromInteger 0-   negate = P.negate-   (+)    = (P.+)-   (-)    = (P.-)--instance C Int32   where-   {-# INLINE zero #-}-   {-# INLINE negate #-}-   {-# INLINE (+) #-}-   {-# INLINE (-) #-}-   zero   = P.fromInteger 0-   negate = P.negate-   (+)    = (P.+)-   (-)    = (P.-)--instance C Int64   where-   {-# INLINE zero #-}-   {-# INLINE negate #-}-   {-# INLINE (+) #-}-   {-# INLINE (-) #-}-   zero   = P.fromInteger 0-   negate = P.negate-   (+)    = (P.+)-   (-)    = (P.-)---instance C Word    where-   {-# INLINE zero #-}-   {-# INLINE negate #-}-   {-# INLINE (+) #-}-   {-# INLINE (-) #-}-   zero   = P.fromInteger 0-   negate = P.negate-   (+)    = (P.+)-   (-)    = (P.-)--instance C Word8   where-   {-# INLINE zero #-}-   {-# INLINE negate #-}-   {-# INLINE (+) #-}-   {-# INLINE (-) #-}-   zero   = P.fromInteger 0-   negate = P.negate-   (+)    = (P.+)-   (-)    = (P.-)--instance C Word16  where-   {-# INLINE zero #-}-   {-# INLINE negate #-}-   {-# INLINE (+) #-}-   {-# INLINE (-) #-}-   zero   = P.fromInteger 0-   negate = P.negate-   (+)    = (P.+)-   (-)    = (P.-)--instance C Word32  where-   {-# INLINE zero #-}-   {-# INLINE negate #-}-   {-# INLINE (+) #-}-   {-# INLINE (-) #-}-   zero   = P.fromInteger 0-   negate = P.negate-   (+)    = (P.+)-   (-)    = (P.-)--instance C Word64  where-   {-# INLINE zero #-}-   {-# INLINE negate #-}-   {-# INLINE (+) #-}-   {-# INLINE (-) #-}-   zero   = P.fromInteger 0-   negate = P.negate-   (+)    = (P.+)-   (-)    = (P.-)------- * Instances for composed types--instance (C v0, C v1) => C (v0, v1) where-   {-# INLINE zero #-}-   {-# INLINE negate #-}-   {-# INLINE (+) #-}-   {-# INLINE (-) #-}-   zero   = (,) zero zero-   (+)    = Elem.run2 $ pure (,) <*>.+  fst <*>.+  snd-   (-)    = Elem.run2 $ pure (,) <*>.-  fst <*>.-  snd-   negate = Elem.run  $ pure (,) <*>.-$ fst <*>.-$ snd--instance (C v0, C v1, C v2) => C (v0, v1, v2) where-   {-# INLINE zero #-}-   {-# INLINE negate #-}-   {-# INLINE (+) #-}-   {-# INLINE (-) #-}-   zero   = (,,) zero zero zero-   (+)    = Elem.run2 $ pure (,,) <*>.+  fst3 <*>.+  snd3 <*>.+  thd3-   (-)    = Elem.run2 $ pure (,,) <*>.-  fst3 <*>.-  snd3 <*>.-  thd3-   negate = Elem.run  $ pure (,,) <*>.-$ fst3 <*>.-$ snd3 <*>.-$ thd3---instance (C v) => C [v] where-   zero   = []-   negate = map negate-   (+) (x:xs) (y:ys) = (+) x y : (+) xs ys-   (+) xs     []     = xs-   (+) []     ys     = ys-   (-) (x:xs) (y:ys) = (-) x y : (-) xs ys-   (-) xs     []     = xs-   (-) []     ys     = negate ys---instance (C v) => C (b -> v) where-   {-# INLINE zero #-}-   {-# INLINE negate #-}-   {-# INLINE (+) #-}-   {-# INLINE (-) #-}-   zero       _ = zero-   (+)    f g x = (+) (f x) (g x)-   (-)    f g x = (-) (f x) (g x)-   negate f   x = negate (f x)---- * Properties--propAssociative :: (Eq a, C a) => a -> a -> a -> Bool-propCommutative :: (Eq a, C a) => a -> a -> Bool-propIdentity    :: (Eq a, C a) => a -> Bool-propInverse     :: (Eq a, C a) => a -> Bool--propCommutative  =  Laws.commutative (+)-propAssociative  =  Laws.associative (+)-propIdentity     =  Laws.identity (+) zero-propInverse      =  Laws.inverse (+) negate zero------ legacy--instance (P.Integral a) => C (Ratio98.Ratio a) where-   {-# INLINE zero #-}-   {-# INLINE negate #-}-   {-# INLINE (+) #-}-   {-# INLINE (-) #-}-   zero                =  0-   (+)                 =  (P.+)-   (-)                 =  (P.-)-   negate              =  P.negate
− src-ghc-6.12/Algebra/AffineSpace.hs 
@@ -1,247 +0,0 @@-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-{- |-This module is not yet exported-since its interface is not mature.-There are two approaches for representing affine spaces:--[1] Two sets: A set of points and a set of vectors.-    Examples: Absolute potential and voltage,-    absolute temperature and temperature difference.-    Operations are-      add :: vector -> point -> point-      sub :: point -> point -> vector--[2] One set for points, no vectors.-    Examples: Interpolation-    Operation:-      combine :: [(coefficient, vector)] -> vector-    Where it must be asserted,-    that the coefficients sum up to 1.--The second one is the one we follow here.-It is more similar to Module and VectorSpace.--}-module Algebra.AffineSpace where--import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.Additive as Additive-import qualified Algebra.Module as Module-import qualified Number.Ratio as Ratio--import qualified Number.Complex as Complex--import Control.Applicative (Applicative(pure, (<*>)), )--import NumericPrelude.Numeric hiding (zero, )-import NumericPrelude.Base-import Prelude ()--{- |-The type class is for representing affine spaces via affine combinations.-However, we didn't find a way to both ensure the property-that the combination coefficients sum up to 1,-and keep it efficient.--We propose this class instead of a combination of Additive and Module-for interpolation for those types,-where scaling and addition alone makes no sense.-Such types are e.g. internal filter parameters in signal processing:-For these types interpolation makes definitely sense,-but addition and scaling make not.--That is, both classes are isomorphic-(you can define one in terms of the other),-but instances of this class are more easily defined,-and using an AffineSpace constraint instead of Module in a type signature-is important for documentation purposes.-AffineSpace should be superclass of Module.-(But then you may ask, why not adding another superclass for Convex spaces.-This class would provide a linear combination operation,-where the coefficients sum up to one-and all of them are non-negative.)--We may add a safety layer that ensures-that the coefficients sum up to 1,-using start points on the simplex-and functions to move on the simplex.-Start points have components that sum up to 1, e.g.--> (1, 0, 0, 0)-> (0, 1, 0, 0)-> (0, 0, 1, 0)-> (0, 0, 0, 1)-> (1/4, 1/4, 1/4, 1/4)--Then you may move along the simplex in the directions--> (1,  -1, 0,  0)-> (0,   1, 0, -1)-> (-1, -1, 3, -1)--which are characterized by components that sum up to 0.--For example linear combination is defined by--> lerp k (a,b) = (1-k)*>a + k*>b--that is the coefficients are (1-k) and k.-The pair (1-k, k) can be constructed-by starting at pair (1,0)-and moving k units in direction (-1,1).--> (1-k, k) = (1,0) + k*(-1,1)--It is however a challenge to manage the coefficient tuples-in a type safe and efficient way.-For small numbers of interpolation nodes-(constant, linear, cubic interpolation)-a type level list would appropriate,-but what to do for large tuples-like for Whittaker interpolation?---As side note:-In principle it would be sufficient-to provide an affine combination of two points,-since all affine combinations of more points-can be decomposed into such simple combinations.--> lerp a x y = (1-a)*>x + a*>y--E.g. @a*>x + b*>y + c*>z@ with @a+b+c=1@-can be written as @lerp c (lerp (b/(1-c)) x y) z@.-More generally you can use--> lerpnorm a b x y->    = lerp (b/(a+b)) x y->    = (a/(a+b))*>x + (b/(a+b))*>y--for writing--> a*>x + b*>y + c*>z ==->    lerpnorm (a+b) c (lerpnorm a b x y) z--or--> a*>x + b*>y + c*>z + d*>w ==->    lerpnorm (a+b+c) d (lerpnorm (a+b) c (lerpnorm a b x y) z) w--with @a+b+c+d=1@.--The downside is, that lerpnorm requires division, that is, a field,-whereas the computation of the coefficients-sometimes only requires ring operations.--}-class Zero v => C a v where-   multiplyAccumulate :: (a,v) -> v -> v--class Zero v where-   zero :: v---instance Zero Float where-   {-# INLINE zero #-}-   zero = Additive.zero--instance Zero Double where-   {-# INLINE zero #-}-   zero = Additive.zero--instance (Zero a) => Zero (Complex.T a) where-   {-# INLINE zero #-}-   zero = zero Complex.+: zero--instance (PID.C a) => Zero (Ratio.T a) where-   {-# INLINE zero #-}-   zero = Additive.zero---instance C Float Float where-   {-# INLINE multiplyAccumulate #-}-   multiplyAccumulate (a,x) y = a*x+y--instance C Double Double where-   {-# INLINE multiplyAccumulate #-}-   multiplyAccumulate (a,x) y = a*x+y--instance (C a v) => C a (Complex.T v) where-   {-# INLINE multiplyAccumulate #-}-   multiplyAccumulate =-      makeMac2 (Complex.+:) Complex.real Complex.imag--instance (PID.C a) => C (Ratio.T a) (Ratio.T a) where-   {-# INLINE multiplyAccumulate #-}-   multiplyAccumulate (a,x) y = a*x+y---infixl 6 *.+--{- |-Infix variant of 'multiplyAccumulate'.--}-{-# INLINE (*.+) #-}-(*.+) :: C a v => v -> (a,v) -> v-(*.+) = flip multiplyAccumulate----- * convenience functions for defining multiplyAccumulate--{-# INLINE multiplyAccumulateModule #-}-multiplyAccumulateModule ::-   Module.C a v =>-   (a,v) -> v -> v-multiplyAccumulateModule (a,x) y =-   a *> x + y---{- |-A special reader monad.--}-newtype MAC a v x = MAC {runMac :: (a,v) -> v -> x}--{-# INLINE element #-}-element ::-   (C a x) =>-   (v -> x) -> MAC a v x-element f =-   MAC (\(a,x) y -> multiplyAccumulate (a, f x) (f y))--instance Functor (MAC a v) where-   {-# INLINE fmap #-}-   fmap f (MAC x) =-      MAC $ \av v -> f $ x av v--instance Applicative (MAC a v) where-   {-# INLINE pure #-}-   {-# INLINE (<*>) #-}-   pure x = MAC $ \ _av _v -> x-   MAC f <*> MAC x =-      MAC $ \av v -> f av v $ x av v--{-# INLINE makeMac #-}-makeMac ::-   (C a x) =>-   (x -> v) ->-   (v -> x) ->-   (a,v) -> v -> v-makeMac cons x =-   runMac $ pure cons <*> element x--{-# INLINE makeMac2 #-}-makeMac2 ::-   (C a x, C a y) =>-   (x -> y -> v) ->-   (v -> x) -> (v -> y) ->-   (a,v) -> v -> v-makeMac2 cons x y =-   runMac $ pure cons <*> element x <*> element y--{-# INLINE makeMac3 #-}-makeMac3 ::-   (C a x, C a y, C a z) =>-   (x -> y -> z -> v) ->-   (v -> x) -> (v -> y) -> (v -> z) ->-   (a,v) -> v -> v-makeMac3 cons x y z =-   runMac $ pure cons <*> element x <*> element y <*> element z
− src-ghc-6.12/Algebra/Algebraic.hs 
@@ -1,65 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Algebra.Algebraic where--import qualified Algebra.Field as Field--- import qualified Algebra.Units as Units-import qualified Algebra.Laws as Laws-import qualified Algebra.ToRational as ToRational-import qualified Algebra.ToInteger  as ToInteger--import Number.Ratio (Rational, (%), numerator, denominator)-import Algebra.Field ((^-), recip, fromRational')-import Algebra.Ring ((*), (^), fromInteger)-import Algebra.Additive((+))--import NumericPrelude.Base-import qualified Prelude as P---infixr 8  ^/--{- | Minimal implementation: 'root' or '(^\/)'. -}--class (Field.C a) => C a where-    sqrt :: a -> a-    sqrt = root 2-    -- sqrt x  =  x ** (1/2)--    root :: P.Integer -> a -> a-    root n x = x ^/ (1 % n)--    (^/) :: a -> Rational -> a-    x ^/ y = root (denominator y) (x ^- numerator y)--genericRoot :: (C a, ToInteger.C b) => b -> a -> a-genericRoot n = root (ToInteger.toInteger n)--power :: (C a, ToRational.C b) => b -> a -> a-power r = (^/ ToRational.toRational r)--instance C P.Float where-    sqrt     = P.sqrt-    root n x = x P.** recip (P.fromInteger n)-    x ^/ y   = x P.** fromRational' y--instance C P.Double where-    sqrt     = P.sqrt-    root n x = x P.** recip (P.fromInteger n)-    x ^/ y   = x P.** fromRational' y---{- * Properties -}---- propSqrtSqr :: (Eq a, C a, Units.C a) => a -> Bool--- propSqrtSqr x = sqrt (x^2) == Units.stdAssociate x--propSqrSqrt :: (Eq a, C a) => a -> Bool-propSqrSqrt x = sqrt x ^ 2 == x--propPowerCascade      :: (Eq a, C a) => a -> Rational -> Rational -> Bool-propPowerProduct      :: (Eq a, C a) => a -> Rational -> Rational -> Bool-propPowerDistributive :: (Eq a, C a) => Rational -> a -> a -> Bool--propPowerCascade      x i j  =  Laws.rightCascade (*) (^/) x i j-propPowerProduct      x i j  =  Laws.homomorphism (x^/) (+) (*) i j-propPowerDistributive i x y  =  Laws.leftDistributive (^/) (*) i x y
− src-ghc-6.12/Algebra/Differential.hs 
@@ -1,19 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Algebra.Differential where--import qualified Algebra.Ring as Ring---- import NumericPrelude.Numeric--- import qualified Prelude--{- |-'differentiate' is a general differentation operation-It must fulfill the Leibnitz condition-->   differentiate (x * y) == differentiate x * y + x * differentiate y--Unfortunately, this scheme cannot be easily extended to more than two variables,-e.g. "MathObj.PowerSeries2".--}-class Ring.C a => C a where-   differentiate :: a -> a
− src-ghc-6.12/Algebra/DimensionTerm.hs 
@@ -1,225 +0,0 @@-{- |-Copyright   :  (c) Henning Thielemann 2008-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable---We already have the dynamically checked physical units-provided by "Number.Physical"-and the statically checked ones of the @dimensional@ package of Buckwalter,-which require multi-parameter type classes with functional dependencies.--Here we provide a poor man's approach:-The units are presented by type terms.-There is no canonical form and thus the type checker-can not automatically check for equal units.-However, if two unit terms represent the same unit,-then you can tell the type checker to rewrite one into the other.--You can add more dimensions by introducing more types of class 'C'.--This approach is not entirely safe-because you can write your own flawed rewrite rules.-It is however more safe than with no units at all.--}--module Algebra.DimensionTerm where--import Prelude hiding (recip)---{- Haddock does not like 'where' clauses before empty declarations -}-class Show a => C a -- where---noValue :: C a => a-noValue =-   let x = error ("there is no value of type " ++ show x)-   in  x--{- * Type constructors -}--data Scalar  = Scalar-data Mul a b = Mul-data Recip a = Recip-type Sqr   a = Mul a a--appPrec :: Int-appPrec = 10--instance Show Scalar where-   show _ = "scalar"--instance (Show a, Show b) => Show (Mul a b) where-   showsPrec p x =-      let disect :: Mul a b -> (a,b)-          disect _ = undefined-          (y,z) = disect x-      in  showParen (p >= appPrec)-            (showString "mul " . showsPrec appPrec y .-             showString " " . showsPrec appPrec z)--instance (Show a) => Show (Recip a) where-   showsPrec p x =-      let disect :: Recip a -> a-          disect _ = undefined-      in  showParen (p >= appPrec)-            (showString "recip " . showsPrec appPrec (disect x))---instance C Scalar -- where--instance (C a, C b) => C (Mul a b) -- where--instance (C a) => C (Recip a) -- where---scalar :: Scalar-scalar = noValue--mul :: (C a, C b) => a -> b -> Mul a b-mul _ _ = noValue--recip :: (C a) => a -> Recip a-recip _ = noValue---infixl 7 %*%-infixl 7 %/%--(%*%) :: (C a, C b) => a -> b -> Mul a b-(%*%) = mul--(%/%) :: (C a, C b) => a -> b -> Mul a (Recip b)-(%/%) x y = mul x (recip y)---{- * Rewrites -}--applyLeftMul :: (C u0, C u1, C v) => (u0 -> u1) -> Mul u0 v -> Mul u1 v-applyLeftMul _ _ = noValue-applyRightMul :: (C u0, C u1, C v) => (u0 -> u1) -> Mul v u0 -> Mul v u1-applyRightMul _ _ = noValue-applyRecip :: (C u0, C u1) => (u0 -> u1) -> Recip u0 -> Recip u1-applyRecip _ _ = noValue--commute :: (C u0, C u1) => Mul u0 u1 -> Mul u1 u0-commute _ = noValue-associateLeft :: (C u0, C u1, C u2) => Mul u0 (Mul u1 u2) -> Mul (Mul u0 u1) u2-associateLeft _ = noValue-associateRight :: (C u0, C u1, C u2) => Mul (Mul u0 u1) u2 -> Mul u0 (Mul u1 u2)-associateRight _ = noValue-recipMul :: (C u0, C u1) => Recip (Mul u0 u1) -> Mul (Recip u0) (Recip u1)-recipMul _ = noValue-mulRecip :: (C u0, C u1) => Mul (Recip u0) (Recip u1) -> Recip (Mul u0 u1)-mulRecip _ = noValue--identityLeft :: C u => Mul Scalar u -> u-identityLeft _ = noValue-identityRight :: C u => Mul u Scalar -> u-identityRight _ = noValue-cancelLeft :: C u => Mul (Recip u) u -> Scalar-cancelLeft _ = noValue-cancelRight :: C u => Mul u (Recip u) -> Scalar-cancelRight _ = noValue-invertRecip :: C u => Recip (Recip u) -> u-invertRecip _ = noValue-doubleRecip :: C u => u -> Recip (Recip u)-doubleRecip _ = noValue-recipScalar :: Recip Scalar -> Scalar-recipScalar _ = noValue---{- * Example dimensions -}--{- ** Scalar -}--{- |-This class allows defining instances that are exclusively for 'Scalar' dimension.-You won't want to define instances by yourself.--}-class C dim => IsScalar dim where-   toScalar :: dim -> Scalar-   fromScalar :: Scalar -> dim--instance IsScalar Scalar where-   toScalar = id-   fromScalar = id---{- ** Basis dimensions -}--data Length      = Length-data Time        = Time-data Mass        = Mass-data Charge      = Charge-data Angle       = Angle-data Temperature = Temperature-data Information = Information--length :: Length-length = noValue--time :: Time-time = noValue--mass :: Mass-mass = noValue--charge :: Charge-charge = noValue--angle :: Angle-angle = noValue--temperature :: Temperature-temperature = noValue--information :: Information-information = noValue---instance Show Length      where show _ = "length"-instance Show Time        where show _ = "time"-instance Show Mass        where show _ = "mass"-instance Show Charge      where show _ = "charge"-instance Show Angle       where show _ = "angle"-instance Show Temperature where show _ = "temperature"-instance Show Information where show _ = "information"--instance C Length      -- where-instance C Time        -- where-instance C Mass        -- where-instance C Charge      -- where-instance C Angle       -- where-instance C Temperature -- where-instance C Information -- where--{- ** Derived dimensions -}--type Frequency = Recip Time--frequency :: Frequency-frequency = noValue---data Voltage = Voltage--type VoltageAnalytical =-        Mul (Mul (Sqr Length) Mass) (Recip (Mul (Sqr Time) Charge))--voltage :: Voltage-voltage = noValue--instance Show Voltage where show _ = "voltage"--instance C Voltage -- where--unpackVoltage :: Voltage -> VoltageAnalytical-unpackVoltage _ = noValue--packVoltage :: VoltageAnalytical -> Voltage-packVoltage _ = noValue
− src-ghc-6.12/Algebra/DivisibleSpace.hs 
@@ -1,21 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-module Algebra.DivisibleSpace where--import qualified Algebra.VectorSpace as VectorSpace---- Is this right?-infix 7 </>--{-|-DivisibleSpace is used for free one-dimensional vector spaces.  It-satisfies-->  (a </> b) *> b = a--Examples include dollars and kilometers.--}-class (VectorSpace.C a b) => C a b where-    (</>) :: b -> b -> a-
− src-ghc-6.12/Algebra/EqualityDecision.hs 
@@ -1,110 +0,0 @@-{- |-Combination of @(==)@ and @if then else@-that can be instantiated for more types than @Eq@-or can be instantiated in a way-that allows more defined results (\"more total\" functions):--* Reader like types for representing a context-  like 'Number.ResidueClass.Reader'--* Expressions in an EDSL--* More generally every type based on an applicative functor--* Tuples and Vector types--* Positional and Peano numbers,-  a common prefix of two numbers can be emitted-  before the comparison is done.-  (This could also be done with an overloaded 'if',-   what we also do not have.)--}-module Algebra.EqualityDecision where--import qualified NumericPrelude.Elementwise as Elem-import Control.Applicative (Applicative(pure, (<*>)), )-import Data.Tuple.HT (fst3, snd3, thd3, )-import Data.List (zipWith4, )---{- |-For atomic types this could be a superclass of 'Eq'.-However for composed types like tuples, lists, functions-we do elementwise comparison-which is incompatible with the complete comparison performed by '(==)'.--}-class C a where-   {- |-   It holds--   > (a ==? b) eq noteq  ==  if a==b then eq else noteq--   for atomic types where the right hand side can be defined.-   -}-   (==?) :: a -> a -> a -> a -> a----{-# INLINE deflt #-}-deflt :: Eq a => a -> a -> a -> a -> a-deflt a b eq noteq =-   if a==b then eq else noteq----instance C Int where-   {-# INLINE (==?) #-}-   (==?) = deflt--instance C Integer where-   {-# INLINE (==?) #-}-   (==?) = deflt--instance C Float where-   {-# INLINE (==?) #-}-   (==?) = deflt--instance C Double where-   {-# INLINE (==?) #-}-   (==?) = deflt--instance C Bool where-   {-# INLINE (==?) #-}-   (==?) = deflt--instance C Ordering where-   {-# INLINE (==?) #-}-   (==?) = deflt----{-# INLINE element #-}-element ::-   (C x) =>-   (v -> x) -> Elem.T (v,v,v,v) x-element f =-   Elem.element (\(x,y,eq,noteq) -> (f x ==? f y) (f eq) (f noteq))--{-# INLINE (<*>.==?) #-}-(<*>.==?) ::-   (C x) =>-   Elem.T (v,v,v,v) (x -> a) -> (v -> x) -> Elem.T (v,v,v,v) a-(<*>.==?) f acc =-   f <*> element acc---instance (C a, C b) => C (a,b) where-   {-# INLINE (==?) #-}-   (==?) = Elem.run4 $ pure (,) <*>.==?  fst <*>.==?  snd--instance (C a, C b, C c) => C (a,b,c) where-   {-# INLINE (==?) #-}-   (==?) = Elem.run4 $ pure (,,) <*>.==?  fst3 <*>.==?  snd3 <*>.==?  thd3--instance C a => C [a] where-   {-# INLINE (==?) #-}-   (==?) = zipWith4 (==?)--instance (C a) => C (b -> a) where-   {-# INLINE (==?) #-}-   (==?) x y eq noteq c  =  (x c ==? y c) (eq c) (noteq c)
− src-ghc-6.12/Algebra/Field.hs 
@@ -1,161 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Algebra.Field (-    {- * Class -}-    C,--    (/),-    recip,-    fromRational',-    fromRational,-    (^-),--    {- * Properties -}-    propDivision,-    propReciprocal,-  ) where--import Number.Ratio (T((:%)), Rational, (%), numerator, denominator, )-import qualified Number.Ratio as Ratio-import qualified Data.Ratio as Ratio98-import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.Units as Unit--import qualified Algebra.Ring         as Ring--- import qualified Algebra.Additive     as Additive-import qualified Algebra.ZeroTestable as ZeroTestable--import Algebra.Ring ((*), (^), one, fromInteger)-import Algebra.Additive (zero, negate)-import Algebra.ZeroTestable (isZero)--import NumericPrelude.Base-import Prelude (Integer, Float, Double)-import qualified Prelude as P-import Test.QuickCheck ((==>), Property)---infixr 8 ^--infixl 7 /---{- |-Field again corresponds to a commutative ring.-Division is partially defined and satisfies-->    not (isZero b)  ==>  (a * b) / b === a->    not (isZero a)  ==>  a * recip a === one--when it is defined. -To safely call division,-the program must take type-specific action;-e.g., the following is appropriate in many cases:--> safeRecip :: (Integral a, Eq a, Field.C a) => a -> Maybe a-> safeRecip x =->     let (q,r) = one `divMod` x->     in  toMaybe (isZero r) q--Typical examples include rationals, the real numbers,-and rational functions (ratios of polynomial functions).-An instance should be typically declared-only if most elements are invertible.--Actually, we have also used this type class for non-fields-containing lots of units,-e.g. residue classes with respect to non-primes and power series.-So the restriction @not (isZero a)@ must be better @isUnit a@.--Minimal definition: 'recip' or ('/')--}--class (Ring.C a) => C a where-    (/)           :: a -> a -> a-    recip         :: a -> a-    fromRational' :: Rational -> a-    (^-)          :: a -> Integer -> a--    {-# INLINE recip #-}-    recip a = one / a-    {-# INLINE (/) #-}-    a / b = a * recip b-    {-# INLINE fromRational' #-}-    fromRational' r = fromInteger (numerator r) / fromInteger (denominator r)-    {-# INLINE (^-) #-}-    a ^- n = if n < zero-               then recip (a^(-n))-               else a^n- -- a ^ n | n < 0 = reduceRepeated (^) one (recip a) (negate (toInteger n))- --       | True  = reduceRepeated (^) one a (toInteger n)------ | Needed to work around shortcomings in GHC.--{-# INLINE fromRational #-}-fromRational :: (C a) => P.Rational -> a-fromRational x = fromRational' (Ratio98.numerator x :% Ratio98.denominator x)---{- * Instances for atomic types -}--{--fromRational must be implemented explicitly for Float and Double!-It may be that numerator or denominator cannot be represented as Float-due to size constraints, but the fraction can.--}--instance C Float where-    {-# INLINE (/) #-}-    {-# INLINE recip #-}-    (/)    = (P./)-    recip  = (P.recip)-    -- using Ratio98.:% would be more efficient but it is not exported.-    fromRational' x =-       P.fromRational (numerator x Ratio98.% denominator x)--instance C Double where-    {-# INLINE (/) #-}-    {-# INLINE recip #-}-    (/)    = (P./)-    recip  = (P.recip)-    fromRational' x =-       P.fromRational (numerator x Ratio98.% denominator x)--instance (PID.C a) => C (Ratio.T a) where-    {-# INLINE (/) #-}-    {-# INLINE recip #-}-    {-# INLINE fromRational' #-}---    (/)                  =  Ratio.liftOrd (%)-    x / y                =  x * recip y-{--This is efficient and almost correct in the sense,-that all admissible cases yield a correct result.-However it will hide division by zero and thus may hide bugs.-Unfortunately 'x' might not be a standard associate,-thus (y:%x) may deviate from the canonical representation.--    recip (x:%y)         =  (y:%x)--}-    recip (x:%y)         =-       if isZero y-         then error "Ratio./: division by zero"-         else (y * Unit.stdUnitInv x) :% Unit.stdAssociate x-    fromRational' (x:%y) =  fromInteger x % fromInteger y----- | the restriction on the divisor should be @isUnit a@ instead of @not (isZero a)@-propDivision   :: (Eq a, ZeroTestable.C a, C a) => a -> a -> Property-propReciprocal :: (Eq a, ZeroTestable.C a, C a) => a -> Property--propDivision   a b   =   not (isZero b)  ==>  (a * b) / b == a-propReciprocal a     =   not (isZero a)  ==>  a * recip a == one------ legacy--instance (P.Integral a) => C (Ratio98.Ratio a) where-    {-# INLINE (/) #-}-    {-# INLINE recip #-}-    (/)    = (P./)-    recip  = (P.recip)
− src-ghc-6.12/Algebra/GenerateRules.hs 
@@ -1,86 +0,0 @@-{- |-Poor man's Template Haskell:-Generate RULES for handling of primitive number types.--}-module Main where--import Data.Maybe (fromMaybe, )--import Prelude hiding (fromIntegral, )---pad :: Int -> String -> String-pad n str =-   zipWith fromMaybe-      (replicate n ' ')-      (map Just str ++ repeat Nothing)---machineIntegerTypes :: [String]-machineIntegerTypes =-   do typeSign <- "Int" : "Word" : []-      typeSize <- "" : "8" : "16" : "32" : "64" : []-      return $ typeSign ++ typeSize--functionSignature :: String -> String -> String -> String-functionSignature functionName sourceType targetType =-   functionName ++ " :: " ++ sourceType ++ " -> " ++ targetType--{--Simply replace NumericPrelude.roundFunc by Prelude98.roundFunc.-This is only sensible where Prelude functions are optimized.-Unfortunately there seems to be no optimization for target type Int8 et.al.--}-realField :: [String]-realField =-   do sourceType <- "Float" : "Double" : []-      targetType <- machineIntegerTypes-      method <- "round" : "truncate" : "floor" : "ceiling" : []-      let methodPad = pad 8 method-      let signature = functionSignature methodPad sourceType targetType-      return $ "     " ++-         pad 40 ("\"NP." ++ signature ++ "\"") ++-         methodPad ++ " = P." ++ signature ++ ";"--realFieldIndirect :: [String]-realFieldIndirect =-   do targetType <- tail machineIntegerTypes-      method <- "round" : "roundSimple" : "truncate" : "floor" : "ceiling" : []-      let methodPad = pad 11 method-      let signature = functionSignature methodPad "a" targetType-      return $ "     " ++-         pad 33 ("\"NP." ++ signature ++ "\"") ++-         methodPad ++ " = (" ++ functionSignature "P.fromIntegral" "Int" targetType ++ ") . "-             ++ method ++ ";"--splitFractionIndirect :: [String]-splitFractionIndirect =-   do targetType <- tail machineIntegerTypes-      method <- "splitFraction" : []-      let methodPad = pad 13 method-      let signature = functionSignature methodPad "a" ("("++targetType++",a)")-      return $ "     " ++-         pad 40 ("\"NP." ++ signature ++ "\"") ++-         methodPad ++ " = mapFst (" ++ functionSignature "P.fromIntegral" "Int" targetType ++ ") . "-             ++ method ++ ";"---fromIntegral :: [String]-fromIntegral =-   do sourceType <- "Integer" : machineIntegerTypes-      targetType <- "Int" : "Integer" : "Float" : "Double" : []-      let function = "fromIntegral"-      let signature = functionSignature function sourceType targetType-      return $ "     " ++-         pad 40 ("\"NP." ++ signature ++ "\"") ++-         function ++ " = P." ++ signature ++ ";"---main :: IO ()-main =-   putStrLn "module Algebra.RealRing" >>-   mapM_ putStrLn realFieldIndirect >>-   mapM_ putStrLn splitFractionIndirect >>--   putStrLn "module Algebra.ToInteger" >>-   mapM_ putStrLn fromIntegral
− src-ghc-6.12/Algebra/Indexable.hs 
@@ -1,76 +0,0 @@-{- |-Copyright    :   (c) Henning Thielemann 2007-Maintainer   :   numericprelude@henning-thielemann.de-Stability    :   provisional-Portability  :   portable--An alternative type class for Ord-which allows an ordering for dictionaries like "Data.Map" and "Data.Set"-independently from the ordering with respect to a magnitude.--}--module Algebra.Indexable (-    C(compare),-    ordCompare,-    liftCompare,-    ToOrd,-    toOrd,-    fromOrd,-    ) where--import Prelude hiding (compare)--import qualified Prelude as P---{- |-Definition of an alternative ordering of objects-independent from a notion of magnitude.-For an application see "MathObj.PartialFraction".--}-class Eq a => C a where-   compare :: a -> a -> Ordering--{- |-If the type has already an 'Ord' instance-it is certainly the most easiest to define 'Algebra.Indexable.compare'-to be equal to @Ord@'s 'compare'.--}-ordCompare :: Ord a => a -> a -> Ordering-ordCompare = P.compare--{- |-Lift 'compare' implementation from a wrapped object.--}-liftCompare :: C b => (a -> b) -> a -> a -> Ordering-liftCompare f x y = compare (f x) (f y)---instance (C a, C b) => C (a,b) where-   compare (x0,x1) (y0,y1) =-      let res = compare x0 y0-      in  case res of-             EQ -> compare x1 y1-             _  -> res--instance C a => C [a] where-   compare [] [] = EQ-   compare [] _  = LT-   compare _  [] = GT-   compare (x:xs) (y:ys) = compare (x,xs) (y,ys)--instance C Integer where-   compare = ordCompare---{- |-Wrap an indexable object such that it can be used in "Data.Map" and "Data.Set".--}-newtype ToOrd a = ToOrd {fromOrd :: a} deriving (Eq, Show)--toOrd :: a -> ToOrd a-toOrd = ToOrd---instance C a => Ord (ToOrd a) where-   compare (ToOrd x) (ToOrd y) = compare x y
− src-ghc-6.12/Algebra/IntegralDomain.hs 
@@ -1,339 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Algebra.IntegralDomain (-    {- * Class -}-    C,-    div, mod, divMod,--    {- * Derived functions -}-    divModZero,-    divides,-    sameResidueClass,-    divChecked, safeDiv,-    even,-    odd,--    divUp,-    roundDown,-    roundUp,--    {- * Algorithms -}-    decomposeVarPositional,-    decomposeVarPositionalInf,--    {- * Properties -}-    propInverse,-    propMultipleDiv,-    propMultipleMod,-    propProjectAddition,-    propProjectMultiplication,-    propUniqueRepresentative,-    propZeroRepresentative,-    propSameResidueClass,-  ) where--import qualified Algebra.Ring         as Ring--- import qualified Algebra.Additive     as Additive-import qualified Algebra.ZeroTestable as ZeroTestable--import Algebra.Ring     ((*), fromInteger, )-import Algebra.Additive (zero, (+), (-), negate, )-import Algebra.ZeroTestable (isZero, )--import Data.Bool.HT (implies, )-import Data.List (mapAccumL, )--import Test.QuickCheck ((==>), Property)--import Data.Int  (Int,  Int8,  Int16,  Int32,  Int64,  )-import Data.Word (Word, Word8, Word16, Word32, Word64, )--import NumericPrelude.Base-import Prelude (Integer, )-import qualified Prelude as P----infixl 7 `div`, `mod`---{--Shall we require-                   @ a `mod` 0 === a @   (divModZero)-or-                   @ a `mod` 0 === undefined @-?--}---{- |-@IntegralDomain@ corresponds to a commutative ring,-where @a `mod` b@ picks a canonical element-of the equivalence class of @a@ in the ideal generated by @b@.-'div' and 'mod' satisfy the laws-->                         a * b === b * a-> (a `div` b) * b + (a `mod` b) === a->               (a+k*b) `mod` b === a `mod` b->                     0 `mod` b === 0--Typical examples of @IntegralDomain@ include integers and-polynomials over a field.-Note that for a field, there is a canonical instance-defined by the above rules; e.g.,--> instance IntegralDomain.C Rational where->     divMod a b =->        if isZero b->          then (undefined,a)->          else (a\/b,0)--It shall be noted, that 'div', 'mod', 'divMod' have a parameter order-which is unfortunate for partial application.-But it is adapted to mathematical conventions,-where the operators are used in infix notation.--Minimal definition: 'divMod' or ('div' and 'mod')--}-class (Ring.C a) => C a where-    div, mod :: a -> a -> a-    divMod :: a -> a -> (a,a)--    {-# INLINE div #-}-    {-# INLINE mod #-}-    {-# INLINE divMod #-}-    div a b = fst (divMod a b)-    mod a b = snd (divMod a b)-    divMod a b = (div a b, mod a b)---{-# INLINE divides #-}-divides :: (C a, ZeroTestable.C a) => a -> a -> Bool-divides y x  =  isZero (mod x y)--{-# INLINE sameResidueClass #-}-sameResidueClass :: (C a, ZeroTestable.C a) => a -> a -> a -> Bool-sameResidueClass m x y = divides m (x-y)----{- |-@decomposeVarPositional [b0,b1,b2,...] x@-decomposes @x@ into a positional representation with mixed bases-@x0 + b0*(x1 + b1*(x2 + b2*x3))@-E.g. @decomposeVarPositional (repeat 10) 123 == [3,2,1]@--}-decomposeVarPositional :: (C a, ZeroTestable.C a) => [a] -> a -> [a]-decomposeVarPositional bs x =-   map fst $-   takeWhile (not . isZero . snd) $-   decomposeVarPositionalInfAux bs x--decomposeVarPositionalInf :: (C a) => [a] -> a -> [a]-decomposeVarPositionalInf bs =-   map fst . decomposeVarPositionalInfAux bs--decomposeVarPositionalInfAux :: (C a) => [a] -> a -> [(a,a)]-decomposeVarPositionalInfAux bs x =-   let (endN,digits) =-          mapAccumL-             (\n b -> let (q,r) = divMod n b in (q,(r,n)))-             x bs-   in  digits ++ [(endN,endN)]----{- |-Returns the result of the division, if divisible.-Otherwise undefined.--}-{-# INLINE divChecked #-}-divChecked, safeDiv :: (ZeroTestable.C a, C a) => a -> a -> a-divChecked a b =-   let (q,r) = divMod a b-   in  if isZero r-         then q-         else error "safeDiv: indivisible term"--{-# DEPRECATED safeDiv "use divChecked instead" #-}-safeDiv = divChecked--{- |-Allows division by zero.-If the divisor is zero, then the dividend is returned as remainder.--}-{-# INLINE divModZero #-}-divModZero :: (C a, ZeroTestable.C a) => a -> a -> (a,a)-divModZero x y =-   if isZero y-     then (zero,x)-     else divMod x y----{-# INLINE even #-}-{-# INLINE odd #-}-even, odd :: (C a, ZeroTestable.C a) => a -> Bool-even n    =  divides 2 n-odd       =  not . even---{- |-@roundDown n m@ rounds @n@ down to the next multiple of @m@.-That is, @roundDown n m@ is the greatest multiple of @m@-that is at most @n@.-The parameter order is consistent with @div@ and friends,-but maybe not useful for partial application.--}-roundDown :: C a => a -> a -> a-roundDown n m = n - mod n m--{- |-@roundUp n m@ rounds @n@ up to the next multiple of @m@.-That is, @roundUp n m@ is the greatest multiple of @m@-that is at most @n@.--}-roundUp :: C a => a -> a -> a-roundUp n m = n + mod (-n) m--{- |-@divUp n m@ is similar to @div@-but it rounds up the quotient,-such that @divUp n m * m = roundUp n m@.--}-divUp :: C a => a -> a -> a-divUp n m = - div (-n) m--{--What sign of the remainder is most appropriate?--divModUp :: C a => a -> a -> (a,a)-divModUp n m = mapFst negate $ divMod (-n) m--}---{- * Instances for atomic types -}--instance C Integer where-   {-# INLINE div #-}-   {-# INLINE mod #-}-   {-# INLINE divMod #-}-   div = P.div-   mod = P.mod-   divMod = P.divMod--instance C Int     where-   {-# INLINE div #-}-   {-# INLINE mod #-}-   {-# INLINE divMod #-}-   div = P.div-   mod = P.mod-   divMod = P.divMod--instance C Int8    where-   {-# INLINE div #-}-   {-# INLINE mod #-}-   {-# INLINE divMod #-}-   div = P.div-   mod = P.mod-   divMod = P.divMod--instance C Int16   where-   {-# INLINE div #-}-   {-# INLINE mod #-}-   {-# INLINE divMod #-}-   div = P.div-   mod = P.mod-   divMod = P.divMod--instance C Int32   where-   {-# INLINE div #-}-   {-# INLINE mod #-}-   {-# INLINE divMod #-}-   div = P.div-   mod = P.mod-   divMod = P.divMod--instance C Int64   where-   {-# INLINE div #-}-   {-# INLINE mod #-}-   {-# INLINE divMod #-}-   div = P.div-   mod = P.mod-   divMod = P.divMod---instance C Word    where-   {-# INLINE div #-}-   {-# INLINE mod #-}-   {-# INLINE divMod #-}-   div = P.div-   mod = P.mod-   divMod = P.divMod--instance C Word8   where-   {-# INLINE div #-}-   {-# INLINE mod #-}-   {-# INLINE divMod #-}-   div = P.div-   mod = P.mod-   divMod = P.divMod--instance C Word16  where-   {-# INLINE div #-}-   {-# INLINE mod #-}-   {-# INLINE divMod #-}-   div = P.div-   mod = P.mod-   divMod = P.divMod--instance C Word32  where-   {-# INLINE div #-}-   {-# INLINE mod #-}-   {-# INLINE divMod #-}-   div = P.div-   mod = P.mod-   divMod = P.divMod--instance C Word64  where-   {-# INLINE div #-}-   {-# INLINE mod #-}-   {-# INLINE divMod #-}-   div = P.div-   mod = P.mod-   divMod = P.divMod------- Ring.propCommutative and ...-propInverse               :: (Eq a, C a, ZeroTestable.C a) => a -> a -> Property-propMultipleDiv           :: (Eq a, C a, ZeroTestable.C a) => a -> a -> Property-propMultipleMod           :: (Eq a, C a, ZeroTestable.C a) => a -> a -> Property-propProjectAddition       :: (Eq a, C a, ZeroTestable.C a) => a -> a -> a -> Property-propProjectMultiplication :: (Eq a, C a, ZeroTestable.C a) => a -> a -> a -> Property-propSameResidueClass      :: (Eq a, C a, ZeroTestable.C a) => a -> a -> a -> Property-propUniqueRepresentative  :: (Eq a, C a, ZeroTestable.C a) => a -> a -> a -> Property-propZeroRepresentative    :: (Eq a, C a, ZeroTestable.C a) => a -> Property---propInverse     m a =-   not (isZero m) ==> (a `div` m) * m + (a `mod` m)  ==  a-propMultipleDiv m a =-   not (isZero m) ==>                 (a*m) `div` m  ==  a-propMultipleMod m a =-   not (isZero m) ==>                 (a*m) `mod` m  ==  0-propProjectAddition m a b =-   not (isZero m) ==>-      (a+b) `mod` m  ==  ((a`mod`m)+(b`mod`m)) `mod` m-propProjectMultiplication m a b =-   not (isZero m) ==>-      (a*b) `mod` m  ==  ((a`mod`m)*(b`mod`m)) `mod` m-propUniqueRepresentative m k a =-   not (isZero m) ==>-      (a+k*m) `mod` m  ==  a `mod` m-propZeroRepresentative m =-   not (isZero m) ==>-      zero `mod` m  ==  zero-propSameResidueClass m a b =-   not (isZero m) ==>-      a `mod` m == b `mod` m   `implies`   sameResidueClass m a b
− src-ghc-6.12/Algebra/Lattice.hs 
@@ -1,69 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Algebra.Lattice (-      C(up, dn)-    , max, min, abs-    , propUpCommutative, propDnCommutative-    , propUpAssociative, propDnAssociative-    , propUpDnDistributive, propDnUpDistributive-) where--import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.Additive as Additive-import qualified Number.Ratio     as Ratio--import qualified Algebra.Laws as Laws--import NumericPrelude.Numeric hiding (abs)-import NumericPrelude.Base hiding (max, min)-import qualified Prelude as P--infixl 5 `up`, `dn`--class C a where-    up, dn :: a -> a -> a---{- * Properties -}--propUpCommutative, propDnCommutative ::- (Eq a, C a) => a -> a -> Bool-propUpCommutative  =  Laws.commutative up-propDnCommutative  =  Laws.commutative dn--propUpAssociative, propDnAssociative ::- (Eq a, C a) => a -> a -> a -> Bool-propUpAssociative  =  Laws.associative up-propDnAssociative  =  Laws.associative dn--propUpDnDistributive, propDnUpDistributive ::- (Eq a, C a) => a -> a -> a -> Bool-propUpDnDistributive  =  Laws.leftDistributive up dn-propDnUpDistributive  =  Laws.leftDistributive dn up------- With  @up == gcd@  and  @dn == lcm@  we have also a lattice.-instance C Integer where-    up = P.max-    dn = P.min--instance (Ord a, PID.C a) => C (Ratio.T a) where-    up = P.max-    dn = P.min--instance C Bool where-    up = (P.||)-    dn = (P.&&)--instance (C a, C b) => C (a,b) where-    (x1,y1)`up`(x2,y2) = (x1`up`x2, y1`up`y2)-    (x1,y1)`dn`(x2,y2) = (x1`dn`x2, y1`dn`y2)---max, min :: (C a) => a -> a -> a-max = up-min = dn--abs :: (C a, Additive.C a) => a -> a-abs x = x `up` negate x
− src-ghc-6.12/Algebra/Laws.hs 
@@ -1,57 +0,0 @@-{- |-Define common properties that can be used e.g. for automated tests.-Cf. to "Test.QuickCheck.Utils".--}-module Algebra.Laws where---commutative :: Eq a => (b -> b -> a) -> b -> b -> Bool-commutative op x y  =  x `op` y == y `op` x--associative :: Eq a => (a -> a -> a) -> a -> a -> a -> Bool-associative op x y z  =  (x `op` y) `op` z == x `op` (y `op` z)--leftIdentity :: Eq a => (b -> a -> a) -> b -> a -> Bool-leftIdentity op y x  =  y `op` x == x--rightIdentity :: Eq a => (a -> b -> a) -> b -> a -> Bool-rightIdentity op y x  =  x `op` y == x--identity :: Eq a => (a -> a -> a) -> a -> a -> Bool-identity op x y  =  leftIdentity op x y &&  rightIdentity op x y--leftZero :: Eq a => (a -> a -> a) -> a -> a -> Bool-leftZero  =  flip . rightIdentity--rightZero :: Eq a => (a -> a -> a) -> a -> a -> Bool-rightZero  =  flip . leftIdentity--zero :: Eq a => (a -> a -> a) -> a -> a -> Bool-zero op x y  =  leftZero op x y  &&  rightZero op x y--leftInverse :: Eq a => (b -> b -> a) -> (b -> b) -> a -> b -> Bool-leftInverse op inv y x  =  inv x `op` x == y--rightInverse :: Eq a => (b -> b -> a) -> (b -> b) -> a -> b -> Bool-rightInverse op inv y x  =  x `op` inv x == y--inverse :: Eq a => (b -> b -> a) -> (b -> b) -> a -> b -> Bool-inverse op inv y x  =  leftInverse op inv y x && rightInverse op inv y x--leftDistributive :: Eq a => (a -> b -> a) -> (a -> a -> a) -> b -> a -> a -> Bool-leftDistributive ( # ) op x y z  =  (y `op` z) # x == (y # x) `op` (z # x)--rightDistributive :: Eq a => (b -> a -> a) -> (a -> a -> a) -> b -> a -> a -> Bool-rightDistributive ( # ) op x y z  =  x # (y `op` z) == (x # y) `op` (x # z)--homomorphism :: Eq a =>-   (b -> a) -> (b -> b -> b) -> (a -> a -> a) -> b -> b -> Bool-homomorphism f op0 op1 x y  =  f (x `op0` y) == f x `op1` f y--rightCascade :: Eq a =>-   (b -> b -> b) -> (a -> b -> a) -> a -> b -> b -> Bool-rightCascade ( # ) op x i j  =  (x `op` i) `op` j == x `op` (i#j)--leftCascade :: Eq a =>-   (b -> b -> b) -> (b -> a -> a) -> a -> b -> b -> Bool-leftCascade ( # ) op x i j  =  j `op` (i `op` x) == (j#i) `op` x
− src-ghc-6.12/Algebra/Module.hs 
@@ -1,153 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-{- |-Copyright   :  (c) Dylan Thurston, Henning Thielemann 2004-2005--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  requires multi-parameter type classes--Abstraction of modules--}--module Algebra.Module where--import qualified Number.Ratio as Ratio--import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.Ring      as Ring-import qualified Algebra.Additive  as Additive-import qualified Algebra.ToInteger as ToInteger--import qualified Algebra.Laws as Laws--import Algebra.Ring     ((*), fromInteger, )-import Algebra.Additive ((+), zero, sum, )--import qualified NumericPrelude.Elementwise as Elem-import Control.Applicative (Applicative(pure, (<*>)), )--import Data.Function.HT (powerAssociative, )-import Data.List (map, zipWith, )-import Data.Tuple.HT (fst3, snd3, thd3, )-import Data.Tuple (fst, snd, )--import Prelude((.), Eq, Bool, Int, Integer, Float, Double, ($), )--- import qualified Prelude as P----- Is this right?-infixr 7 *>--{--Functional dependency can't be used-since @Complex.T a@ is a module-with respect to both @a@ and @Complex.T a@.--class Algebra.Module.C a v | v -> a where--}--{-|-A Module over a ring satisfies:-->   a *> (b + c) === a *> b + a *> c->   (a * b) *> c === a *> (b *> c)->   (a + b) *> c === a *> c + b *> c--}-class (Ring.C a, Additive.C v) => C a v where-    -- | scale a vector by a scalar-    (*>) :: a -> v -> v---{-# INLINE (<*>.*>) #-}-(<*>.*>) ::-   (C a x) =>-   Elem.T (a,v) (x -> c) -> (v -> x) -> Elem.T (a,v) c-(<*>.*>) f acc =-   f <*> Elem.element (\(a,v) -> a *> acc v)----{-* Instances for atomic types -}--instance C Float Float where-   {-# INLINE (*>) #-}-   (*>) = (*)--instance C Double Double where-   {-# INLINE (*>) #-}-   (*>) = (*)--instance C Int Int where-   {-# INLINE (*>) #-}-   (*>) = (*)--instance C Integer Integer where-   {-# INLINE (*>) #-}-   (*>) = (*)--instance (PID.C a) => C (Ratio.T a) (Ratio.T a) where-   {-# INLINE (*>) #-}-   (*>) = (*)--instance (PID.C a) => C Integer (Ratio.T a) where-   {-# INLINE (*>) #-}-   x *> y = fromInteger x * y----{-* Instances for composed types -}--instance (C a b0, C a b1) => C a (b0, b1) where-   {-# INLINE (*>) #-}-   (*>) = Elem.run2 $ pure (,) <*>.*> fst <*>.*> snd-   -- s *> (x0,x1)   = (s *> x0, s *> x1)--instance (C a b0, C a b1, C a b2) => C a (b0, b1, b2) where-   {-# INLINE (*>) #-}-   (*>) = Elem.run2 $ pure (,,) <*>.*> fst3 <*>.*> snd3 <*>.*> thd3-   -- s *> (x0,x1,x2) = (s *> x0, s *> x1, s *> x2)--instance (C a v) => C a [v] where-   {-# INLINE (*>) #-}-   (*>) = map . (*>)--instance (C a v) => C a (c -> v) where-   {-# INLINE (*>) #-}-   (*>) s f = (*>) s . f---{-* Related functions -}--{-|-Compute the linear combination of a list of vectors.--ToDo:-Should it use 'NumericPrelude.List.Checked.zipWith' ?--}-linearComb :: C a v => [a] -> [v] -> v-linearComb c = sum . zipWith (*>) c--{-|-This function can be used to define any-'Additive.C' as a module over 'Integer'.--Better move to "Algebra.Additive"?--}-{-# INLINE integerMultiply #-}-integerMultiply :: (ToInteger.C a, Additive.C v) => a -> v -> v-integerMultiply a v =-   powerAssociative (+) zero v (ToInteger.toInteger a)---{- * Properties -}--propCascade :: (Eq v, C a v) => v -> a -> a -> Bool-propCascade  =  Laws.leftCascade (*) (*>)--propRightDistributive :: (Eq v, C a v) => a -> v -> v -> Bool-propRightDistributive  =  Laws.rightDistributive (*>) (+)--propLeftDistributive :: (Eq v, C a v) => v -> a -> a -> Bool-propLeftDistributive x  =  Laws.homomorphism (*>x) (+) (+)
− src-ghc-6.12/Algebra/ModuleBasis.hs 
@@ -1,95 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-{- |-Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  requires multi-parameter type classes--Abstraction of bases of finite dimensional modules--}--module Algebra.ModuleBasis where--import qualified Number.Ratio as Ratio--import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.Module   as Module--- import qualified Algebra.Additive as Additive-import Algebra.Ring     (one, fromInteger)-import Algebra.Additive ((+), zero)--import Data.List (map, length, (++))--import Prelude(Eq, (==), Bool, Int, Integer, Float, Double, asTypeOf, )--- import qualified Prelude as P--{- |-It must hold:-->   Module.linearComb (flatten v `asTypeOf` [a]) (basis a) == v->   dimension a v == length (flatten v `asTypeOf` [a])--}-class (Module.C a v) => C a v where-    {- | basis of the module with respect to the scalar type,-         the result must be independent of argument, 'Prelude.undefined' should suffice. -}-    basis :: a -> [v]-    -- | scale a vector by a scalar-    flatten :: v -> [a]-    {- | the size of the basis, should also work for undefined argument,-         the result must be independent of argument, 'Prelude.undefined' should suffice. -}-    dimension :: a -> v -> Int--{-* Instances for atomic types -}--instance C Float Float where-   basis _ = [one]-   flatten = (:[])-   dimension _ _ = 1--instance C Double Double where-   basis _ = [one]-   flatten = (:[])-   dimension _ _ = 1--instance C Int Int where-   basis _ = [one]-   flatten = (:[])-   dimension _ _ = 1--instance C Integer Integer where-   basis _ = [one]-   flatten = (:[])-   dimension _ _ = 1--instance (PID.C a) => C (Ratio.T a) (Ratio.T a) where-   basis _ = [one]-   flatten = (:[])-   dimension _ _ = 1----{-* Instances for composed types -}--instance (C a v0, C a v1) => C a (v0, v1) where-   basis s = map (\v -> (v,zero)) (basis s) ++-             map (\v -> (zero,v)) (basis s)-   flatten (x0,x1) = flatten x0 ++ flatten x1-   dimension s ~(x0,x1) = dimension s x0 + dimension s x1--instance (C a v0, C a v1, C a v2) => C a (v0, v1, v2) where-   basis s = map (\v -> (v,zero,zero)) (basis s) ++-             map (\v -> (zero,v,zero)) (basis s) ++-             map (\v -> (zero,zero,v)) (basis s)-   flatten (x0,x1,x2) = flatten x0 ++ flatten x1 ++ flatten x2-   dimension s ~(x0,x1,x2) = dimension s x0 + dimension s x1 + dimension s x2----{- * Properties -}--propFlatten :: (Eq v, C a v) => a -> v -> Bool-propFlatten a v  =  Module.linearComb (flatten v `asTypeOf` [a]) (basis a) == v--propDimension :: (C a v) => a -> v -> Bool-propDimension a v  =  dimension a v == length (flatten v `asTypeOf` [a])
− src-ghc-6.12/Algebra/Monoid.hs 
@@ -1,72 +0,0 @@-{- |-Copyright    :   (c) Henning Thielemann 2009-2010, Mikael Johansson 2006-Maintainer   :   numericprelude@henning-thielemann.de-Stability    :   provisional-Portability  :--Abstract concept of a Monoid.-Will be used in order to generate type classes for generic algebras.-An algebra is a vector space that also is a monoid.-Should we use the Monoid class from base library-despite its unfortunate method name @mappend@?--}--module Algebra.Monoid where--import qualified Algebra.Additive as Additive-import qualified Algebra.Ring as Ring--import Data.Monoid as Mn--{- |-We expect a monoid to adher to associativity and-the identity behaving decently.-Nothing more, really.--}-class C a where-  idt   :: a-  (<*>) :: a -> a -> a-  cumulate :: [a] -> a-  cumulate = foldr (<*>) idt---instance C All where-  idt = mempty-  (<*>) = mappend-  cumulate = mconcat--instance C Any where-  idt = mempty-  (<*>) = mappend-  cumulate = mconcat--instance C a => C (Dual a) where-  idt = Mn.Dual idt-  (Mn.Dual x) <*> (Mn.Dual y) = Mn.Dual (y <*> x)-  cumulate = Mn.Dual . cumulate . reverse . map Mn.getDual--instance C (Endo a) where-  idt = mempty-  (<*>) = mappend-  cumulate = mconcat--instance C (First a) where-  idt = mempty-  (<*>) = mappend-  cumulate = mconcat--instance C (Last a) where-  idt = mempty-  (<*>) = mappend-  cumulate = mconcat---instance Ring.C a => C (Product a) where-  idt = Mn.Product Ring.one-  (Mn.Product x) <*> (Mn.Product y) = Mn.Product (x Ring.* y)-  cumulate = Mn.Product . Ring.product . map Mn.getProduct--instance Additive.C a => C (Sum a) where-  idt = Mn.Sum Additive.zero-  (Mn.Sum x) <*> (Mn.Sum y) = Mn.Sum (x Additive.+ y)-  cumulate = Mn.Sum . Additive.sum . map Mn.getSum
− src-ghc-6.12/Algebra/NonNegative.hs 
@@ -1,130 +0,0 @@-{- |-Copyright   :  (c) Henning Thielemann 2007-2010--Maintainer  :  haskell@henning-thielemann.de-Stability   :  stable-Portability :  Haskell 98--A type class for non-negative numbers.-Prominent instances are 'Number.NonNegative.T' and 'Number.Peano.T' numbers.-This class cannot do any checks,-but it let you show to the user what arguments your function expects.-Thus you must define class instances with care.-In fact many standard functions ('take', '(!!)', ...)-should have this type class constraint.--}-module Algebra.NonNegative (-   C(..),-   splitDefault,--   (-|),---   (-?),-   zero,-   add,-   sum,-   ) where--import qualified Algebra.Additive as Additive--- import qualified Algebra.RealRing as RealRing--import qualified Algebra.Monoid as Monoid---- import Algebra.Absolute (abs, )-import Algebra.Additive ((-), )--import Prelude hiding (sum, (-), abs, )---infixl 6 -|  -- , -?---{- |-Instances of this class must ensure non-negative values.-We cannot enforce this by types, but the type class constraint @NonNegative.C@-avoids accidental usage of types which allow for negative numbers.--The Monoid superclass contributes a zero and an addition.--}-class (Ord a, Monoid.C a) => C a where-   {- |-   @split x y == (m,(b,d))@ means that-   @b == (x<=y)@,-   @m == min x y@,-   @d == max x y - min x y@, that is @d == abs(x-y)@.--   We have chosen this function as base function,-   since it provides comparison and subtraction in one go,-   which is important for replacing common structures like--   > if x<=y-   >   then f(x-y)-   >   else g(y-x)--   that lead to a memory leak for peano numbers.-   We have choosen the simple check @x<=y@-   instead of a full-blown @compare@,-   since we want @Zero <= undefined@ for peano numbers.-   Because of undefined values 'split' is in general-   not commutative in the sense--   > let (m0,(b0,d0)) = split x y-   >     (m1,(b1,d1)) = split y x-   > in  m0==m1 && d0==d1--   The result values are in the order-   in which they are generated for Peano numbers.-   We have chosen the nested pair instead of a triple-   in order to prevent a memory leak-   that occurs if you only use @b@ and @d@ and ignore @m@.-   This is demonstrated by test cases-   Chunky.splitSpaceLeak3 and Chunky.splitSpaceLeak4.-   -}-   split :: a -> a -> (a, (Bool, a))---{- |-Default implementation for wrapped types of 'Ord' and 'Num' class.--}-{-# INLINE splitDefault #-}-splitDefault ::-   (Ord b, Additive.C b) =>-   (a -> b) -> (b -> a) -> a -> a -> (a, (Bool, a))-splitDefault unpack pack px py =-   let x = unpack px-       y = unpack py-   in  if x<=y-         then (pack x, (True,  pack (y-x)))-         else (pack y, (False, pack (x-y)))---zero :: C a => a-zero = Monoid.idt---- like (+)-infixl 6 `add`--add :: C a => a -> a -> a-add = (Monoid.<*>)--sum :: C a => [a] -> a-sum = Monoid.cumulate---{- |-@x -| y == max 0 (x-y)@--The default implementation is not efficient,-because it compares the values and then subtracts, again, if safe.-@max 0 (x-y)@ is more elegant and efficient-but not possible in the general case,-since @x-y@ may already yield a negative number.--}-(-|) :: C a => a -> a -> a-x -| y  =-   let (b,d) = snd $ split y x-   in  if b then d else zero--{--(-?) :: (RealRing.C a) => a -> a -> (Bool, a)-(-?) x y  =  snd $ split y x--}
− src-ghc-6.12/Algebra/NormedSpace/Euclidean.hs 
@@ -1,126 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}--{- |-Copyright   :  (c) Henning Thielemann 2005-2010-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  requires multi-parameter type classes--Abstraction of normed vector spaces--}--module Algebra.NormedSpace.Euclidean where--import NumericPrelude.Base-import NumericPrelude.Numeric (sqr, abs, zero, (+), sum, Float, Double, Int, Integer, )--import qualified Number.Ratio as Ratio--import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.Algebraic as Algebraic-import qualified Algebra.Absolute      as Absolute-import qualified Algebra.Module    as Module--import qualified Data.Foldable as Fold---{-|-Helper class for 'C' that does not need an algebraic type @a@.--Minimal definition:-'normSqr'--}-class (Absolute.C a, Module.C a v) => Sqr a v where-  {-| Square of the Euclidean norm of a vector.-      This is sometimes easier to implement. -}-  normSqr :: v -> a---  normSqr = sqr . norm--{- |-Default definition for 'normSqr' that is based on 'Fold.Foldable' class.--}-{-# INLINE normSqrFoldable #-}-normSqrFoldable ::-   (Sqr a v, Fold.Foldable f) => f v -> a-normSqrFoldable =-   Fold.foldl (\a v -> a + normSqr v) zero--{- |-Default definition for 'normSqr' that is based on 'Fold.Foldable' class-and the argument vector has at least one component.--}-{-# INLINE normSqrFoldable1 #-}-normSqrFoldable1 ::-   (Sqr a v, Fold.Foldable f, Functor f) => f v -> a-normSqrFoldable1 =-   Fold.foldl1 (+) . fmap normSqr---{-|-A vector space equipped with an Euclidean or a Hilbert norm.--Minimal definition:-'norm'--}-class (Sqr a v) => C a v where-  {-| Euclidean norm of a vector. -}-  norm :: v -> a---defltNorm :: (Algebraic.C a, Sqr a v) => v -> a-defltNorm = Algebraic.sqrt . normSqr---{-* Instances for atomic types -}--instance Sqr Float Float where-  normSqr = sqr--instance C Float Float where-  norm    = abs--instance Sqr Double Double where-  normSqr = sqr--instance C Double Double where-  norm    = abs--instance Sqr Int Int where-  normSqr = sqr--instance C Int Int where-  norm    = abs--instance Sqr Integer Integer where-  normSqr = sqr--instance C Integer Integer where-  norm    = abs---{-* Instances for composed types -}--instance (Absolute.C a, PID.C a) => Sqr (Ratio.T a) (Ratio.T a) where-  normSqr = sqr--instance (Sqr a v0, Sqr a v1) => Sqr a (v0, v1) where-  normSqr (x0,x1) = normSqr x0 + normSqr x1--instance (Algebraic.C a, Sqr a v0, Sqr a v1) => C a (v0, v1) where-  norm    = defltNorm--instance (Sqr a v0, Sqr a v1, Sqr a v2) => Sqr a (v0, v1, v2) where-  normSqr (x0,x1,x2) = normSqr x0 + normSqr x1 + normSqr x2--instance (Algebraic.C a, Sqr a v0, Sqr a v1, Sqr a v2) => C a (v0, v1, v2) where-  norm    = defltNorm--instance (Sqr a v) => Sqr a [v] where-  normSqr = sum . map normSqr--instance (Algebraic.C a, Sqr a v) => C a [v] where-  norm    = defltNorm
− src-ghc-6.12/Algebra/NormedSpace/Maximum.hs 
@@ -1,85 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}--{- |-Copyright   :  (c) Henning Thielemann 2005-2010-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  requires multi-parameter type classes--Abstraction of normed vector spaces--}--module Algebra.NormedSpace.Maximum where--import NumericPrelude.Base-import NumericPrelude.Numeric--import qualified Number.Ratio as Ratio--import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.ToInteger as ToInteger-import qualified Algebra.RealRing as RealRing-import qualified Algebra.Module   as Module--import qualified Data.Foldable as Fold---class (RealRing.C a, Module.C a v) => C a v where-  norm :: v -> a--{- |-Default definition for 'norm' that is based on 'Fold.Foldable' class.--}-{-# INLINE normFoldable #-}-normFoldable ::-   (C a v, Fold.Foldable f) => f v -> a-normFoldable =-   Fold.foldl (\a v -> max a (norm v)) zero--{- |-Default definition for 'norm' that is based on 'Fold.Foldable' class-and the argument vector has at least one component.--}-{-# INLINE normFoldable1 #-}-normFoldable1 ::-   (C a v, Fold.Foldable f, Functor f) => f v -> a-normFoldable1 =-   Fold.foldl1 max . fmap norm--{--instance (Ring.C a, Algebra.Module a a) => C a a where-  norm = abs--}-instance C Float Float where-  norm = abs--instance C Double Double where-  norm = abs--instance C Int Int where-  norm = abs--instance C Integer Integer where-  norm = abs---instance (RealRing.C a, ToInteger.C a, PID.C a) => C (Ratio.T a) (Ratio.T a) where-  norm = abs--instance (Ord a, C a v0, C a v1) => C a (v0, v1) where-  norm (x0,x1) = max (norm x0) (norm x1)--instance (Ord a, C a v0, C a v1, C a v2) => C a (v0, v1, v2) where-  norm (x0,x1,x2) = (norm x0) `max` (norm x1) `max` (norm x2)--instance (Ord a, C a v) => C a [v] where-  norm = foldl max zero . map norm-{--Since the norm is always non-negative,-we can use zero as identity element.-  norm = maximum . map norm--}
− src-ghc-6.12/Algebra/NormedSpace/Sum.hs 
@@ -1,90 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}--{- |-Copyright   :  (c) Henning Thielemann 2005-2010-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  requires multi-parameter type classes--Abstraction of normed vector spaces--}--module Algebra.NormedSpace.Sum where--import NumericPrelude.Base-import NumericPrelude.Numeric--import qualified Number.Ratio as Ratio--import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.Absolute     as Absolute-import qualified Algebra.Additive as Additive-import qualified Algebra.Module   as Module--import qualified Data.Foldable as Fold---{-|-  The super class is only needed to state the laws-  @-     v == zero        ==   norm v == zero-     norm (scale x v) ==   abs x * norm v-     norm (u+v)       <=   norm u + norm v-  @--}-class (Absolute.C a, Module.C a v) => C a v where-  norm :: v -> a--{- |-Default definition for 'norm' that is based on 'Fold.Foldable' class.--}-{-# INLINE normFoldable #-}-normFoldable ::-   (C a v, Fold.Foldable f) => f v -> a-normFoldable =-   Fold.foldl (\a v -> a + norm v) zero--{- |-Default definition for 'norm' that is based on 'Fold.Foldable' class-and the argument vector has at least one component.--}-{-# INLINE normFoldable1 #-}-normFoldable1 ::-   (C a v, Fold.Foldable f, Functor f) => f v -> a-normFoldable1 =-   Fold.foldl1 (+) . fmap norm---{--instance (Ring.C a, Algebra.Module a a) => C a a where-  norm = abs--}--instance C Float Float where-  norm = abs--instance C Double Double where-  norm = abs--instance C Int Int where-  norm = abs--instance C Integer Integer where-  norm = abs---instance (Absolute.C a, PID.C a) => C (Ratio.T a) (Ratio.T a) where-  norm = abs--instance (Additive.C a, C a v0, C a v1) => C a (v0, v1) where-  norm (x0,x1) = norm x0 + norm x1--instance (Additive.C a, C a v0, C a v1, C a v2) => C a (v0, v1, v2) where-  norm (x0,x1,x2) = norm x0 + norm x1 + norm x2--instance (Additive.C a, C a v) => C a [v] where-  norm = sum . map norm
− src-ghc-6.12/Algebra/OccasionallyScalar.hs 
@@ -1,83 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}--{- |--There are several types of numbers-where a subset of numbers can be considered as set of scalars.-- * A '(Complex.T Double)' value can be converted to 'Double' if the imaginary part is zero.-- * A value with physical units can be converted to a scalar if there is no unit. --Of course this can be cascaded,-e.g. a complex number with physical units can be converted to a scalar-if there is both no imaginary part and no unit.--This is somewhat similar to the multi-type classes NormedMax.C and friends.--I hesitate to define an instance for lists-to avoid the mess known of MatLab.-But if you have an application where you think-you need this instance definitely-I'll think about that, again.---}--module Algebra.OccasionallyScalar where---- import qualified Algebra.RealRing    as RealRing-import qualified Algebra.ZeroTestable as ZeroTestable-import qualified Algebra.Additive     as Additive-import qualified Number.Complex       as Complex--import Data.Maybe (fromMaybe, )--import Number.Complex((+:))--import NumericPrelude.Base-import NumericPrelude.Numeric----- this is somehow similar to Normalized classes-class C a v where-   toScalar      :: v -> a-   toMaybeScalar :: v -> Maybe a-   fromScalar    :: a -> v--toScalarDefault :: (C a v) => v -> a-toScalarDefault v =-   fromMaybe (error ("The value is not scalar."))-             (toMaybeScalar v)--toScalarShow :: (C a v, Show v) => v -> a-toScalarShow v =-   fromMaybe (error (show v ++ " is not a scalar value."))-             (toMaybeScalar v)---instance C Float Float where-   toScalar      = id-   toMaybeScalar = Just-   fromScalar    = id--instance C Double Double where-   toScalar      = id-   toMaybeScalar = Just-   fromScalar    = id---- this instance should be defined in Number.Complex-instance (Show v, ZeroTestable.C v, Additive.C v, C a v) => C a (Complex.T v) where-   toScalar        = toScalarShow-   toMaybeScalar x = if isZero (Complex.imag x)-                       then toMaybeScalar (Complex.real x)-                       else Nothing-   fromScalar x    = fromScalar x +: zero--{- converting values automatically to integers is a bad idea-instance (Integral b, RealRing.C a)-      => C b a where-   toScalar        = toScalarDefault-   toMaybeScalar x = mapMaybe round (toMaybeScalar x)--}
− src-ghc-6.12/Algebra/OrderDecision.hs 
@@ -1,244 +0,0 @@-{- |-Combination of @compare@ and @if then else@-that can be instantiated for more types than @Ord@-or can be instantiated in a way-that allows more defined results (\"more total\" functions):--* Reader like types for representing a context-  like 'Number.ResidueClass.Reader'--* Expressions in an EDSL--* More generally every type based on an applicative functor--* Tuples and Vector types--* Positional and Peano numbers,-  a common prefix of two numbers can be emitted-  before the comparison is done.-  (This could also be done with an overloaded 'if',-   what we also do not have.)--}-module Algebra.OrderDecision where--import qualified Algebra.EqualityDecision as Equality-import Algebra.EqualityDecision ((==?), )--import qualified NumericPrelude.Elementwise as Elem-import Control.Applicative (Applicative(pure, (<*>)), )-import Data.Tuple.HT (fst3, snd3, thd3, )-import Data.List (zipWith4, zipWith5, )--import Prelude hiding (compare, min, max, minimum, maximum, )-import qualified Prelude as P----{- |-For atomic types this could be a superclass of 'Ord'.-However for composed types like tuples, lists, functions-we do elementwise comparison-which is incompatible with the complete comparison performed by 'P.compare'.--}-class Equality.C a => C a where-   {- |-   It holds--   > (compare a b) lt eq gt  ==-   >    case Prelude.compare a b of-   >       LT -> lt-   >       EQ -> eq-   >       GT -> gt--   for atomic types where the right hand side can be defined.--   Minimal complete definition:-   'compare' or '(<=?)'.-   -}-   compare :: a -> a -> a -> a -> a -> a-   compare x y lt eq gt =-      (x ==? y) eq ((x <=? y) lt gt)--   {-# INLINE (<=?) #-}-   (<=?) :: a -> a -> a -> a -> a-   (<=?) x y le gt =-      compare x y le le gt--   {-# INLINE (>=?) #-}-   (>=?) :: a -> a -> a -> a -> a-   (>=?) = flip (<=?)--   (<?) :: a -> a -> a -> a -> a-   (<?) x y = flip (x >=? y)--   {-# INLINE (>?) #-}-   (>?) :: a -> a -> a -> a -> a-   (>?) = flip (<?)--{--   (<?) :: a -> a -> a -> a -> a-   (<?) x y lt ge =-      compare x y lt ge ge--   (>?) :: a -> a -> a -> a -> a-   (>?) x y gt le =-      compare x y le le gt--   (<=?) :: a -> a -> a -> a -> a-   (<=?) x y le gt =-      compare x y le le gt--   (>=?) :: a -> a -> a -> a -> a-   (>=?) x y ge lt =-      compare x y lt ge ge--}---max :: C a => a -> a -> a-max x y = (x>?y) x y--min :: C a => a -> a -> a-min x y = (x<?y) x y--maximum :: C a => a -> [a] -> a-maximum x xs = foldl max x xs--minimum :: C a => a -> [a] -> a-minimum x xs = foldl min x xs----{-# INLINE compareOrd #-}-compareOrd :: Ord a => a -> a -> a -> a -> a -> a-compareOrd a b lt eq gt =-   case P.compare a b of-      LT -> lt-      EQ -> eq-      GT -> gt--instance C Int where-   {-# INLINE compare #-}-   compare = compareOrd--instance C Integer where-   {-# INLINE compare #-}-   compare = compareOrd--instance C Float where-   {-# INLINE compare #-}-   compare = compareOrd--instance C Double where-   {-# INLINE compare #-}-   compare = compareOrd--instance C Bool where-   {-# INLINE compare #-}-   compare = compareOrd--instance C Ordering where-   {-# INLINE compare #-}-   compare = compareOrd----{-# INLINE elementCompare #-}-elementCompare ::-   (C x) =>-   (v -> x) -> Elem.T (v,v,v,v,v) x-elementCompare f =-   Elem.element (\(x,y,lt,eq,gt) ->-      compare (f x) (f y) (f lt) (f eq) (f gt))--{-# INLINE (<*>.<=>?) #-}-(<*>.<=>?) ::-   (C x) =>-   Elem.T (v,v,v,v,v) (x -> a) -> (v -> x) -> Elem.T (v,v,v,v,v) a-(<*>.<=>?) f acc =-   f <*> elementCompare acc---{-# INLINE element #-}-element ::-   (C x) =>-   (x -> x -> x -> x -> x) ->-   (v -> x) -> Elem.T (v,v,v,v) x-element cmp f =-   Elem.element (\(x,y,true,false) -> cmp (f x) (f y) (f true) (f false))--{-# INLINE (<*>.<=?) #-}-(<*>.<=?) ::-   (C x) =>-   Elem.T (v,v,v,v) (x -> a) -> (v -> x) -> Elem.T (v,v,v,v) a-(<*>.<=?) f acc =-   f <*> element (<=?) acc--{-# INLINE (<*>.>=?) #-}-(<*>.>=?) ::-   (C x) =>-   Elem.T (v,v,v,v) (x -> a) -> (v -> x) -> Elem.T (v,v,v,v) a-(<*>.>=?) f acc =-   f <*> element (>=?) acc--{-# INLINE (<*>.<?) #-}-(<*>.<?) ::-   (C x) =>-   Elem.T (v,v,v,v) (x -> a) -> (v -> x) -> Elem.T (v,v,v,v) a-(<*>.<?) f acc =-   f <*> element (<?) acc--{-# INLINE (<*>.>?) #-}-(<*>.>?) ::-   (C x) =>-   Elem.T (v,v,v,v) (x -> a) -> (v -> x) -> Elem.T (v,v,v,v) a-(<*>.>?) f acc =-   f <*> element (>?) acc---instance (C a, C b) => C (a,b) where-   {-# INLINE compare #-}-   compare = Elem.run5 $ pure (,) <*>.<=>? fst <*>.<=>? snd-   {-# INLINE (<=?) #-}-   (<=?)   = Elem.run4 $ pure (,) <*>.<=?  fst <*>.<=?  snd-   {-# INLINE (>=?) #-}-   (>=?)   = Elem.run4 $ pure (,) <*>.>=?  fst <*>.>=?  snd-   {-# INLINE (<?) #-}-   (<?)    = Elem.run4 $ pure (,) <*>.<?   fst <*>.<?   snd-   {-# INLINE (>?) #-}-   (>?)    = Elem.run4 $ pure (,) <*>.>?   fst <*>.>?   snd--instance (C a, C b, C c) => C (a,b,c) where-   {-# INLINE compare #-}-   compare = Elem.run5 $ pure (,,) <*>.<=>? fst3 <*>.<=>? snd3 <*>.<=>? thd3-   {-# INLINE (<=?) #-}-   (<=?)   = Elem.run4 $ pure (,,) <*>.<=?  fst3 <*>.<=?  snd3 <*>.<=?  thd3-   {-# INLINE (>=?) #-}-   (>=?)   = Elem.run4 $ pure (,,) <*>.>=?  fst3 <*>.>=?  snd3 <*>.>=?  thd3-   {-# INLINE (<?) #-}-   (<?)    = Elem.run4 $ pure (,,) <*>.<?   fst3 <*>.<?   snd3 <*>.<?   thd3-   {-# INLINE (>?) #-}-   (>?)    = Elem.run4 $ pure (,,) <*>.>?   fst3 <*>.>?   snd3 <*>.>?   thd3--instance C a => C [a] where-   {-# INLINE compare #-}-   compare = zipWith5 compare-   {-# INLINE (<=?) #-}-   (<=?) = zipWith4 (<=?)-   {-# INLINE (>=?) #-}-   (>=?) = zipWith4 (>=?)-   {-# INLINE (<?) #-}-   (<?)  = zipWith4 (<?)-   {-# INLINE (>?) #-}-   (>?)  = zipWith4 (>?)--instance (C a) => C (b -> a) where-   {-# INLINE compare #-}-   compare x y lt eq gt c  =  compare (x c) (y c) (lt c) (eq c) (gt c)-   {-# INLINE (<=?) #-}-   (<=?) x y true false c  =  (x c <=? y c) (true c) (false c)-   {-# INLINE (>=?) #-}-   (>=?) x y true false c  =  (x c >=? y c) (true c) (false c)-   {-# INLINE (<?) #-}-   (<?)  x y true false c  =  (x c <?  y c) (true c) (false c)-   {-# INLINE (>?) #-}-   (>?)  x y true false c  =  (x c >?  y c) (true c) (false c)
− src-ghc-6.12/Algebra/PrincipalIdealDomain.hs 
@@ -1,384 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Algebra.PrincipalIdealDomain (-    {- * Class -}-    C,-    extendedGCD,-    gcd,-    lcm,-    coprime,--    {- * Standard implementations for instances -}-    euclid,-    extendedEuclid,--    {- * Algorithms -}-    extendedGCDMulti,-    diophantine,-    diophantineMin,-    diophantineMulti,-    chineseRemainder,-    chineseRemainderMulti,--    {- * Properties -}-    propMaximalDivisor,-    propGCDDiophantine,-    propExtendedGCDMulti,-    propDiophantine,-    propDiophantineMin,-    propDiophantineMulti,-    propDiophantineMultiMin,-    propChineseRemainder,-    propDivisibleGCD,-    propDivisibleLCM,-    propGCDIdentity,-    propGCDCommutative,-    propGCDAssociative,-    propGCDHomogeneous,-    propGCD_LCM,-  ) where--import qualified Algebra.Units          as Units-import qualified Algebra.IntegralDomain as Integral--- import qualified Algebra.Ring           as Ring--- import qualified Algebra.Additive       as Additive-import qualified Algebra.ZeroTestable   as ZeroTestable--import qualified Algebra.Laws as Laws--import Algebra.Units          (stdAssociate, stdUnitInv)-import Algebra.IntegralDomain (mod, divChecked, divMod, divides, divModZero)-import Algebra.Ring           (one, (*), scalarProduct)-import Algebra.Additive       (zero, (+), (-))-import Algebra.ZeroTestable   (isZero)--import Data.Maybe.HT (toMaybe, )--import Control.Monad (foldM, liftM)-import Data.List (mapAccumL, mapAccumR, unfoldr)--import Data.Int  (Int,  Int8,  Int16,  Int32,  Int64,  )--import NumericPrelude.Base-import Prelude (Integer, )-import Test.QuickCheck ((==>), Property)----{- |-A principal ideal domain is a ring in which every ideal-(the set of multiples of some generating set of elements)-is principal:-That is,-every element can be written as the multiple of some generating element.-@gcd a b@ gives a generator for the ideal generated by @a@ and @b@.-The algorithm above works whenever @mod x y@ is smaller-(in a suitable sense) than both @x@ and @y@;-otherwise the algorithm may run forever.--Laws:-->   divides x (lcm x y)->   x `gcd` (y `gcd` z) == (x `gcd` y) `gcd` z->   gcd x y * z == gcd (x*z) (y*z)->   gcd x y * lcm x y == x * y--(etc: canonical)--Minimal definition:- * nothing, if the standard Euclidean algorithm work- * if 'extendedGCD' is implemented customly, 'gcd' and 'lcm' make use of it--}-class (Units.C a, ZeroTestable.C a) => C a where-    {- |-    Compute the greatest common divisor and-    solve a respective Diophantine equation.--    >   (g,(a,b)) = extendedGCD x y ==>-    >        g==a*x+b*y   &&  g == gcd x y--    TODO: This method is not appropriate for the PID class,-          because there are rings like the one of the multivariate polynomials,-          where for all x and y greatest common divisors of x and y exist,-          but they cannot be represented as a linear combination of x and y.-    TODO: The definition of extendedGCD does not return the canonical associate.-    -}-    extendedGCD :: a -> a -> (a,(a,a))-    extendedGCD = extendedEuclid divMod--    {- |-    The Greatest Common Divisor is defined by:--    >   gcd x y == gcd y x-    >   divides z x && divides z y ==> divides z (gcd x y)   (specification)-    >   divides (gcd x y) x-    -}-    gcd         :: a -> a -> a-    gcd x y     = fst $ extendedGCD x y--    {- |-    Least common multiple-    -}-    lcm         :: a -> a -> a-    lcm x y     =-       if isZero x-         then x -- avoid computing undefined (gcd 0 0)-         else divChecked x (gcd x y) * y  -- avoid big temporary results-    -- lcm x y     = divChecked (x * y) (gcd x y)---{--These do only work if zero and one are really identity elements.--gcdMulti :: (C a) => [a] -> a-gcdMulti = foldl gcd zero--lcmMulti :: (C a) => [a] -> a-lcmMulti = foldl lcm one--}--coprime :: (C a) => a -> a -> Bool-coprime x y =-   Units.isUnit (gcd x y)----{--We could implement a helper function,-which exposes the temporary results.-This could be re-used for extendedEuclid.--}-euclid :: (Units.C a, ZeroTestable.C a) =>-   (a -> a -> a) -> a -> a -> a-euclid genMod =-   let aux x y =-          if isZero y-            then stdAssociate x-            else aux y (x `genMod` y)-   in  aux---- could be implemented in a tail-recursive manner-{--Unfortunately, with the normalization to the stdAssociate,-@gcd 0@ is no longer the identity function,-since @gcd 0 (-2) = 2@.--}-extendedEuclid :: (Units.C a, ZeroTestable.C a) =>-   (a -> a -> (a,a)) -> a -> a -> (a,(a,a))-extendedEuclid genDivMod =-   let aux x y =-          if isZero y-            then (stdAssociate x, (stdUnitInv x, zero))-            else-              let (d,m) = x `genDivMod` y   -- x == d*y + m-                  (g,(a,b)) = aux y m       -- g == a*y + b*m-              in  (g,(b,a-b*d))             -- g == a*y + b*(x-d*y)-   in aux---{- |-Compute the greatest common divisor for multiple numbers-by repeated application of the two-operand-gcd.--}-extendedGCDMulti :: C a => [a] -> (a,[a])-extendedGCDMulti xs =-   let (g,cs) = mapAccumL extendedGCD zero xs-   in  (g, snd $ mapAccumR (\acc (c0,c1) -> (acc*c0,acc*c1)) one cs)--{- |-A variant with small coefficients.--}---{- |-@Just (a,b) = diophantine z x y@-means-@a*x+b*y = z@.-It is required that @gcd(y,z) `divides` x@.--}-diophantine :: C a => a -> a -> a -> Maybe (a,a)-diophantine z x y =-   fmap snd $ diophantineAux z x y--{- |-Like 'diophantine', but @a@ is minimal-with respect to the measure function of the Euclidean algorithm.--}-diophantineMin :: C a => a -> a -> a -> Maybe (a,a)-diophantineMin z x y =-   fmap (uncurry (minimizeFirstOperand (x,y))) $-   diophantineAux z x y--minimizeFirstOperand :: C a => (a,a) -> a -> (a,a) -> (a,a)-minimizeFirstOperand (x,y) g (a,b) =-   if isZero g-     then (zero,zero)-     else-       let xl = divChecked x g-           yl = divChecked y g-           (d,aRed) = divModZero a yl-       in  (aRed, b + d*xl)--diophantineAux :: C a => a -> a -> a -> Maybe (a, (a,a))-diophantineAux z x y =-   let (g,(a,b)) = extendedGCD x y-       (q,r) = divModZero z g-   in  toMaybe (isZero r) (g, (q*a, q*b))---{- |--}-diophantineMulti :: C a => a -> [a] -> Maybe [a]-diophantineMulti z xs =-   let (g,as) = extendedGCDMulti xs-       (q,r)  = divModZero z g-   in  toMaybe (isZero r) (map (q*) as)--{- |-Not efficient because it requires duplicate computations of GCDs.-However GCDs of neighbouring list elements were not computed before.-It is also quite arbitrary,-because only neighbouring elements are used for balancing.-There are certainly more sophisticated solutions.--}-diophantineMultiMin :: C a => a -> [a] -> Maybe [a]-diophantineMultiMin z xs =-   do as <- diophantineMulti z xs-      return $ unfoldr-         (\as' -> case as' of-           ((x0,a0):(x1,a1):aRest) ->-              let (b0,b1) = minimizeFirstOperand (x0,x1) (gcd x0 x1) (a0,a1)-              in  Just (b0, (x1,b1):aRest)-           (_,a):[] -> Just (a,[])-           [] -> Nothing) $-         zip xs as--{--diophantineMultiMin :: C a => a -> [a] -> Maybe [a]-diophantineMultiMin z xs =-   do as <- diophantineMulti z xs-      return $-         case as of-           (c:cs'@(_:_)) ->-              let (cs,cLast) = splitLast cs'-                  (d,as') = mapAccumL (\a b -> swap $ minimizeFirstOperand (gcd a b) (a,b)) c cs-                  (d',cLast') = minimizeFirstOperand (gcd d cLast) d cLast-              in  as' ++ [d',cLast']-           _ -> as--}--{- |-Not efficient enough, because GCD\/LCM is computed twice.--}-chineseRemainder :: C a => (a,a) -> (a,a) -> Maybe (a,a)-chineseRemainder (m0,a0) (m1,a1) =-   liftM (\(k,_) -> let m = lcm m0 m1 in (m, mod (a0-k*m0) m)) $-   diophantineMin (a0-a1) m0 m1-{--a0-k*m0 == a1+l*m1-a0-a1 == k*m0+l*m1--}--{- |-For @Just (b,n) = chineseRemainder [(a0,m0), (a1,m1), ..., (an,mn)]@-and all @x@ with @x = b mod n@ the congruences-@x=a0 mod m0, x=a1 mod m1, ..., x=an mod mn@-are fulfilled.--}-chineseRemainderMulti :: C a => [(a,a)] -> Maybe (a,a)-chineseRemainderMulti congs =-   case congs of-      [] -> Nothing-      (c:cs) -> foldM chineseRemainder c cs----{- * Instances for atomic types -}---{--There is the binary GCD algorithm,-that is specialised for integers in binary representation.-It does not need a division.-However, since we have an optimized division,-the standard implementation is probably faster.--TODO: Can Integer make use of the GMP GCD routine?--}--instance C Integer where-    -- possibly more efficient than the default method-    gcd = euclid mod--instance C Int where-    gcd = euclid mod--instance C Int8 where-    gcd = euclid mod--instance C Int16 where-    gcd = euclid mod--instance C Int32 where-    gcd = euclid mod--instance C Int64 where-    gcd = euclid mod---propGCDIdentity     :: (Eq a, C a) => a -> Bool-propGCDAssociative :: (Eq a, C a) => a -> a -> a -> Bool-propGCDCommutative :: (Eq a, C a) => a -> a -> Bool-propGCDDiophantine :: (Eq a, C a) => a -> a -> Bool-propExtendedGCDMulti :: (Eq a, C a) => [a] -> Bool-propDiophantineGen :: (Eq a, C a) =>-   (a -> a -> a -> Maybe (a,a)) -> a -> a -> a -> a -> Bool-propDiophantine    :: (Eq a, C a) => a -> a -> a -> a -> Bool-propDiophantineMin :: (Eq a, C a) => a -> a -> a -> a -> Bool-propDiophantineMultiGen :: (Eq a, C a) =>-   (a -> [a] -> Maybe [a]) -> [(a,a)] -> Bool-propDiophantineMulti    :: (Eq a, C a) => [(a,a)] -> Bool-propDiophantineMultiMin :: (Eq a, C a) => [(a,a)] -> Bool-propDivisibleGCD   :: C a => a -> a -> Bool-propDivisibleLCM   :: C a => a -> a -> Bool-propGCD_LCM        :: (Eq a, C a) => a -> a -> Bool-propGCDHomogeneous :: (Eq a, C a) => a -> a -> a -> Bool-propMaximalDivisor :: C a => a -> a -> a -> Property-propChineseRemainder :: (Eq a, C a) => a -> a -> [a] -> Property--propMaximalDivisor x y z =-   divides z x && divides z y ==> divides z (gcd x y)-propGCDDiophantine x y =-   let (g,(a,b)) = extendedGCD x y-   in  g == gcd x y  &&  g == a*x+b*y-propExtendedGCDMulti xs =-   let (g,as) = extendedGCDMulti xs-   in  g == scalarProduct as xs  &&-       (isZero g || all (divides g) xs)-propDiophantineGen dio a b x y =-   let z = a*x+b*y-   in  maybe False (\(a',b') -> z == a'*x+b'*y) (dio z x y)-propDiophantine    = propDiophantineGen diophantine-propDiophantineMin = propDiophantineGen diophantineMin-propDiophantineMultiGen dio axs =-   let (as,xs) = unzip axs-       z = scalarProduct as xs-   in  maybe False (\as' -> z == scalarProduct as' xs) (dio z xs)-propDiophantineMulti    = propDiophantineMultiGen diophantineMulti-propDiophantineMultiMin = propDiophantineMultiGen diophantineMultiMin-propDivisibleGCD x y  =  divides (gcd x y) x-propDivisibleLCM x y  =  divides x (lcm x y)--propGCDIdentity     =  Laws.identity gcd zero . stdAssociate-propGCDCommutative  =  Laws.commutative gcd-propGCDAssociative  =  Laws.associative gcd-propGCDHomogeneous  =  Laws.leftDistributive (*) gcd . stdAssociate-propGCD_LCM x y     =  gcd x y * lcm x y == x * y-propChineseRemainder k x ms =-   not (null ms) && all (not . isZero) ms ==>-   -- cf. Useful.functionToGraph-   let congs = zip ms (map (mod x) ms)-   in  maybe False-          (\(mGlob,y) ->-             let yk = y+mGlob*k-             in  all (\(m,a) -> Integral.sameResidueClass m a yk) congs)-          (chineseRemainderMulti congs)
− src-ghc-6.12/Algebra/RealField.hs 
@@ -1,26 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Algebra.RealField (-   C,-   ) where--import qualified Algebra.Field as Field-import qualified Algebra.RealRing as RealRing-import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.ToInteger as ToInteger--import qualified Number.Ratio as Ratio---- import NumericPrelude.Base--- import qualified Prelude as P-import Prelude (Float, Double, )--{- |-This is a convenient class for common types-that both form a field and have a notion of ordering by magnitude.--}-class (RealRing.C a, Field.C a) => C a where--instance C Float where-instance C Double where--instance (ToInteger.C a, PID.C a) => C (Ratio.T a) where
− src-ghc-6.12/Algebra/RealIntegral.hs 
@@ -1,151 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Generally before using 'quot' and 'rem', think twice.-In most cases 'divMod' and friends are the right choice,-because they fulfill more of the wanted properties.-On some systems 'quot' and 'rem' are more efficient-and if you only use positive numbers, you may be happy with them.-But we cannot warrant the efficiency advantage.--See also:-Daan Leijen: Division and Modulus for Computer Scientists-<http://www.cs.uu.nl/%7Edaan/download/papers/divmodnote-letter.pdf>,-<http://www.haskell.org/pipermail/haskell-cafe/2007-August/030394.html>--}-module Algebra.RealIntegral (-   C(quot, rem, quotRem),-   ) where--import qualified Algebra.IntegralDomain as Integral-import qualified Algebra.Absolute       as Absolute--- import qualified Algebra.Ring           as Ring--- import qualified Algebra.Additive       as Additive--import Algebra.Absolute (signum, )-import Algebra.IntegralDomain (divMod, )-import Algebra.Ring (one, ) -- fromInteger-import Algebra.Additive (zero, (+), (-), )--import Data.Int  (Int,  Int8,  Int16,  Int32,  Int64,  )-import Data.Word (Word, Word8, Word16, Word32, Word64, )--import NumericPrelude.Base-import qualified Prelude as P-import Prelude (Integer, )---infixl 7 `quot`, `rem`--{- |-Remember that 'divMod' does not specify exactly what @a `quot` b@ should be,-mainly because there is no sensible way to define it in general.-For an instance of @Algebra.RealIntegral.C a@,-it is expected that @a `quot` b@ will round towards 0 and-@a `Prelude.div` b@ will round towards minus infinity.--Minimal definition: nothing required--}--class (Absolute.C a, Ord a, Integral.C a) => C a where-    quot, rem        :: a -> a -> a-    quotRem          :: a -> a -> (a,a)--    {-# INLINE quot #-}-    {-# INLINE rem #-}-    {-# INLINE quotRem #-}-    quot a b = fst (quotRem a b)-    rem a b  = snd (quotRem a b)-    quotRem a b = let (d,m) = divMod a b in-                   if (signum d < zero) then-                         (d+one,m-b) else (d,m)---instance C Integer where-   {-# INLINE quot #-}-   {-# INLINE rem #-}-   {-# INLINE quotRem #-}-   quot = P.quot-   rem = P.rem-   quotRem = P.quotRem--instance C Int     where-   {-# INLINE quot #-}-   {-# INLINE rem #-}-   {-# INLINE quotRem #-}-   quot = P.quot-   rem = P.rem-   quotRem = P.quotRem--instance C Int8    where-   {-# INLINE quot #-}-   {-# INLINE rem #-}-   {-# INLINE quotRem #-}-   quot = P.quot-   rem = P.rem-   quotRem = P.quotRem--instance C Int16   where-   {-# INLINE quot #-}-   {-# INLINE rem #-}-   {-# INLINE quotRem #-}-   quot = P.quot-   rem = P.rem-   quotRem = P.quotRem--instance C Int32   where-   {-# INLINE quot #-}-   {-# INLINE rem #-}-   {-# INLINE quotRem #-}-   quot = P.quot-   rem = P.rem-   quotRem = P.quotRem--instance C Int64   where-   {-# INLINE quot #-}-   {-# INLINE rem #-}-   {-# INLINE quotRem #-}-   quot = P.quot-   rem = P.rem-   quotRem = P.quotRem---instance C Word    where-   {-# INLINE quot #-}-   {-# INLINE rem #-}-   {-# INLINE quotRem #-}-   quot = P.quot-   rem = P.rem-   quotRem = P.quotRem--instance C Word8   where-   {-# INLINE quot #-}-   {-# INLINE rem #-}-   {-# INLINE quotRem #-}-   quot = P.quot-   rem = P.rem-   quotRem = P.quotRem--instance C Word16  where-   {-# INLINE quot #-}-   {-# INLINE rem #-}-   {-# INLINE quotRem #-}-   quot = P.quot-   rem = P.rem-   quotRem = P.quotRem--instance C Word32  where-   {-# INLINE quot #-}-   {-# INLINE rem #-}-   {-# INLINE quotRem #-}-   quot = P.quot-   rem = P.rem-   quotRem = P.quotRem--instance C Word64  where-   {-# INLINE quot #-}-   {-# INLINE rem #-}-   {-# INLINE quotRem #-}-   quot = P.quot-   rem = P.rem-   quotRem = P.quotRem-
− src-ghc-6.12/Algebra/RealRing.hs 
@@ -1,584 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Algebra.RealRing where--import qualified Algebra.Field              as Field-import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.Absolute           as Absolute-import qualified Algebra.Ring           as Ring-import qualified Algebra.ToRational     as ToRational-import qualified Algebra.ToInteger      as ToInteger--import qualified Algebra.OrderDecision as OrdDec-import Algebra.OrderDecision ((<?), (>=?), )--import Algebra.Field          (fromRational, )-import Algebra.RealIntegral   (quotRem, )-import Algebra.IntegralDomain (divMod, even, )-import Algebra.Ring           ((*), fromInteger, one, )-import Algebra.Additive       ((+), (-), negate, zero, )-import Algebra.ZeroTestable   (isZero, )-import Algebra.ToInteger      (fromIntegral, )--import qualified Number.Ratio as Ratio-import Number.Ratio (T((:%)), Rational)--import Data.Int  (Int,  Int8,  Int16,  Int32,  Int64,  )-import Data.Word (Word, Word8, Word16, Word32, Word64, )--import qualified GHC.Float as GHC-import Data.List as List-import Data.Tuple.HT (mapFst, mapPair, )-import Prelude (Integer, Float, Double, )-import qualified Prelude as P-import NumericPrelude.Base---{- |-Minimal complete definition:-     'splitFraction' or 'floor'--There are probably more laws, but some laws are--> splitFraction x === (fromInteger (floor x), fraction x)-> fromInteger (floor x) + fraction x === x-> floor x       <= x       x <  floor x + 1-> ceiling x - 1 <  x       x <= ceiling x-> 0 <= fraction x          fraction x < 1-->               - ceiling x === floor (-x)->                truncate x === signum x * floor (abs x)->    ceiling (toRational x) === ceiling x :: Integer->   truncate (toRational x) === truncate x :: Integer->      floor (toRational x) === floor x :: Integer--The new function 'fraction' doesn't return the integer part of the number.-This also removes a type ambiguity if the integer part is not needed.--Many people will associate rounding with fractional numbers,-and thus they are surprised about the superclass being @Ring@ not @Field@.-The reason is that all of these methods can be defined-exclusively with functions from @Ord@ and @Ring@.-The implementations of 'genericFloor' and other functions demonstrate that.-They implement power-of-two-algorithms-like the one for finding the number of digits of an 'Integer'-in FixedPoint-fractions module.-They are even reasonably efficient.--I am still uncertain whether it was a good idea-to add instances for @Integer@ and friends,-since calling @floor@ or @fraction@ on an integer may well indicate a bug.-The rounding functions are just the identity function-and 'fraction' is constant zero.-However, I decided to associate our class with @Ring@ rather than @Field@,-after I found myself using repeated subtraction and testing-rather than just calling @fraction@,-just in order to get the constraint @(Ring a, Ord a)@-that was more general than @(RealField a)@.--For the results of the rounding functions-we have chosen the constraint @Ring@ instead of @ToInteger@,-since this is more flexible to use,-but it still signals to the user that only integral numbers can be returned.-This is so, because the plain @Ring@ class only provides-@zero@, @one@ and operations that allow to reach all natural numbers but not more.---As an aside, let me note the similarities-between @splitFraction x@ and @divMod x 1@ (if that were defined).-In particular, it might make sense to unify the rounding modes somehow.--The new methods 'fraction' and 'splitFraction'-differ from 'Prelude.properFraction' semantics.-They always round to 'floor'.-This means that the fraction is always non-negative and-is always smaller than 1.-This is more useful in practice and-can be generalised to more than real numbers.-Since every 'Number.Ratio.T' denominator type-supports 'Algebra.IntegralDomain.divMod',-every 'Number.Ratio.T' can provide 'fraction' and 'splitFraction',-e.g. fractions of polynomials.-However the @Ring@ constraint for the ''integral'' part of 'splitFraction'-is too weak in order to generate polynomials.-After all, I am uncertain whether this would be useful or not.--Can there be a separate class for-'fraction', 'splitFraction', 'floor' and 'ceiling'-since they do not need reals and their ordering?--We might also add a round method,-that rounds 0.5 always up or always down.-This is much more efficient in inner loops-and is acceptable or even preferable for many applications.--}--class (Absolute.C a, Ord a) => C a where-    splitFraction    :: (Ring.C b) => a -> (b,a)-    fraction         ::               a -> a-    ceiling, floor   :: (Ring.C b) => a -> b-    truncate         :: (Ring.C b) => a -> b-    round            :: (ToInteger.C b) => a -> b---    splitFraction x   =  (floor x, fraction x)--    fraction x   =  x - fromInteger (floor x)--    floor x      =  fromInteger (fst (splitFraction x))--    ceiling x    =  - floor (-x)----    truncate x   =  signum x * floor (abs x)-    truncate x =-       if x>=0-         then floor x-         else ceiling x--    {--    The ToInteger constraint can be lifted to Ring-    if use Integer temporarily.-    I expect this would not be efficient in many cases.-    -}-    round x =-       let (n,r) = splitFraction x-       in  case compare (2*r) one of-              LT -> n-              EQ -> if even n then n else n+1-              GT -> n+1---{- |-This function rounds to the closest integer.-For @fraction x == 0.5@ it rounds away from zero.-This function is not the result of an ingenious mathematical insight,-but is simply a kind of rounding that is the fastest-on IEEE floating point architectures.--}-roundSimple :: (C a, Ring.C b) => a -> b-roundSimple x =-   let (n,r) = splitFraction x-   in  case compare (2*r) one of-          LT -> n-          EQ -> if x<0 then n else n+1-          GT -> n+1---instance (ToInteger.C a, PID.C a) => C (Ratio.T a) where-    splitFraction (x:%y) = (fromIntegral q, r:%y)-                               where (q,r) = divMod x y--instance C Int where-    {-# INLINE splitFraction #-}-    {-# INLINE fraction #-}-    {-# INLINE floor #-}-    {-# INLINE ceiling #-}-    {-# INLINE round #-}-    {-# INLINE truncate #-}-    splitFraction x = (fromIntegral x, zero)-    fraction      _ = zero-    floor         x = fromIntegral x-    ceiling       x = fromIntegral x-    round         x = fromIntegral x-    truncate      x = fromIntegral x--instance C Integer where-    {-# INLINE splitFraction #-}-    {-# INLINE fraction #-}-    {-# INLINE floor #-}-    {-# INLINE ceiling #-}-    {-# INLINE round #-}-    {-# INLINE truncate #-}-    splitFraction x = (fromInteger x, zero)-    fraction      _ = zero-    floor         x = fromInteger x-    ceiling       x = fromInteger x-    round         x = fromInteger x-    truncate      x = fromInteger x--instance C Float where-    {-# INLINE splitFraction #-}-    {-# INLINE fraction #-}-    {-# INLINE floor #-}-    {-# INLINE ceiling #-}-    {-# INLINE round #-}-    {-# INLINE truncate #-}-    splitFraction = fastSplitFraction GHC.float2Int GHC.int2Float-    fraction      = fastFraction (GHC.int2Float . GHC.float2Int)-    floor         = fromInteger . P.floor-    ceiling       = fromInteger . P.ceiling-    round         = fromInteger . P.round-    truncate      = fromInteger . P.truncate--instance C Double where-    {-# INLINE splitFraction #-}-    {-# INLINE fraction #-}-    {-# INLINE floor #-}-    {-# INLINE ceiling #-}-    {-# INLINE round #-}-    {-# INLINE truncate #-}-    splitFraction = fastSplitFraction GHC.double2Int GHC.int2Double-    fraction      = fastFraction (GHC.int2Double . GHC.double2Int)-    floor         = fromInteger . P.floor-    ceiling       = fromInteger . P.ceiling-    round         = fromInteger . P.round-    truncate      = fromInteger . P.truncate---{-# INLINE fastSplitFraction #-}-fastSplitFraction :: (P.RealFrac a, Absolute.C a, Ring.C b) =>-   (a -> Int) -> (Int -> a) -> a -> (b,a)-fastSplitFraction trunc toFloat x =-   fixSplitFraction $-   if toFloat minBound <= x && x <= toFloat maxBound-     then case trunc x of n -> (fromIntegral n, x - toFloat n)-     else case P.properFraction x of (n,f) -> (fromInteger n, f)--{-# INLINE fixSplitFraction #-}-fixSplitFraction :: (Ring.C a, Ring.C b, Ord a) => (b,a) -> (b,a)-fixSplitFraction (n,f) =-   --  if x>=0 || f==0-   if f>=0-     then (n,   f)-     else (n-1, f+1)--{-# INLINE fastFraction #-}-fastFraction :: (P.RealFrac a, Absolute.C a) => (a -> a) -> a -> a-fastFraction trunc x =-   fixFraction $-   if fromIntegral (minBound :: Int) <= x && x <= fromIntegral (maxBound :: Int)-     then x - trunc x-     else preludeFraction x--{-# INLINE preludeFraction #-}-preludeFraction :: (P.RealFrac a, Ring.C a) => a -> a-preludeFraction x =-   let second :: (Integer, a) -> a-       second = snd-   in  second (P.properFraction x)--{-# INLINE fixFraction #-}-fixFraction :: (Ring.C a, Ord a) => a -> a-fixFraction y =-   if y>=0 then y else y+1--{--mapM_ (\n -> let x = fromInteger n / 10 in print (x, floorInt GHC.double2Int GHC.int2Double x)) [-20,-19..20]--}--{-# INLINE splitFractionInt #-}-splitFractionInt :: (Ring.C a, Ord a) => (a -> Int) -> (Int -> a) -> a -> (Int, a)-splitFractionInt trunc toFloat x =-   let n = trunc x-   in  fixSplitFraction (n, x - toFloat n)--{-# INLINE floorInt #-}-floorInt :: (Ring.C a, Ord a) => (a -> Int) -> (Int -> a) -> a -> Int-floorInt trunc toFloat x =-   let n = trunc x-   in  if x >= toFloat n-         then n-         else pred n--{-# INLINE ceilingInt #-}-ceilingInt :: (Ring.C a, Ord a) => (a -> Int) -> (Int -> a) -> a -> Int-ceilingInt trunc toFloat x =-   let n = trunc x-   in  if x <= toFloat n-         then n-         else succ n--{-# INLINE roundInt #-}-roundInt :: (Field.C a, Ord a) => (a -> Int) -> (Int -> a) -> a -> Int-roundInt trunc toFloat x =-   let half = 0.5 -- P.fromRational-       halfUp = x+half-       n = floorInt trunc toFloat halfUp-   in  if toFloat n == halfUp  &&  P.odd n-         then pred n-         else n--{-# INLINE roundSimpleInt #-}-roundSimpleInt ::-   (Field.C a, Absolute.C a, Ord a) =>-   (a -> Int) -> (Int -> a) -> a -> Int-roundSimpleInt trunc _toFloat x =-   trunc (x + Absolute.signum x * 0.5)----{- RULES maybe used, when Prelude implementations become more efficient-     "NP.round    :: Float -> Int"    round    = P.round    :: Float -> Int;-     "NP.truncate :: Float -> Int"    truncate = P.truncate :: Float -> Int;-     "NP.floor    :: Float -> Int"    floor    = P.floor    :: Float -> Int;-     "NP.ceiling  :: Float -> Int"    ceiling  = P.ceiling  :: Float -> Int;-     "NP.round    :: Double -> Int"   round    = P.round    :: Double -> Int;-     "NP.truncate :: Double -> Int"   truncate = P.truncate :: Double -> Int;-     "NP.floor    :: Double -> Int"   floor    = P.floor    :: Double -> Int;-     "NP.ceiling  :: Double -> Int"   ceiling  = P.ceiling  :: Double -> Int;-  -}---- these rules will also be needed for Int16 et.al.-{-# RULES-     "NP.round       :: Float -> Int"    round    = roundInt       GHC.float2Int  GHC.int2Float;-     "NP.roundSimple :: Float -> Int"    round    = roundSimpleInt GHC.float2Int  GHC.int2Float;-     "NP.truncate    :: Float -> Int"    truncate =                GHC.float2Int               ;-     "NP.floor       :: Float -> Int"    floor    = floorInt       GHC.float2Int  GHC.int2Float;-     "NP.ceiling     :: Float -> Int"    ceiling  = ceilingInt     GHC.float2Int  GHC.int2Float;-     "NP.round       :: Double -> Int"   round    = roundInt       GHC.double2Int GHC.int2Double;-     "NP.roundSimple :: Double -> Int"   round    = roundSimpleInt GHC.double2Int GHC.int2Double;-     "NP.truncate    :: Double -> Int"   truncate =                GHC.double2Int               ;-     "NP.floor       :: Double -> Int"   floor    = floorInt       GHC.double2Int GHC.int2Double;-     "NP.ceiling     :: Double -> Int"   ceiling  = ceilingInt     GHC.double2Int GHC.int2Double;--     "NP.splitFraction :: Float ->  (Int, Float)"  splitFraction = splitFractionInt GHC.float2Int GHC.int2Float;-     "NP.splitFraction :: Double -> (Int, Double)" splitFraction = splitFractionInt GHC.double2Int GHC.int2Double;-  #-}---- generated by GenerateRules.hs-{-# RULES-     "NP.round       :: a -> Int8"    round       = (P.fromIntegral :: Int -> Int8) . round;-     "NP.roundSimple :: a -> Int8"    roundSimple = (P.fromIntegral :: Int -> Int8) . roundSimple;-     "NP.truncate    :: a -> Int8"    truncate    = (P.fromIntegral :: Int -> Int8) . truncate;-     "NP.floor       :: a -> Int8"    floor       = (P.fromIntegral :: Int -> Int8) . floor;-     "NP.ceiling     :: a -> Int8"    ceiling     = (P.fromIntegral :: Int -> Int8) . ceiling;-     "NP.round       :: a -> Int16"   round       = (P.fromIntegral :: Int -> Int16) . round;-     "NP.roundSimple :: a -> Int16"   roundSimple = (P.fromIntegral :: Int -> Int16) . roundSimple;-     "NP.truncate    :: a -> Int16"   truncate    = (P.fromIntegral :: Int -> Int16) . truncate;-     "NP.floor       :: a -> Int16"   floor       = (P.fromIntegral :: Int -> Int16) . floor;-     "NP.ceiling     :: a -> Int16"   ceiling     = (P.fromIntegral :: Int -> Int16) . ceiling;-     "NP.round       :: a -> Int32"   round       = (P.fromIntegral :: Int -> Int32) . round;-     "NP.roundSimple :: a -> Int32"   roundSimple = (P.fromIntegral :: Int -> Int32) . roundSimple;-     "NP.truncate    :: a -> Int32"   truncate    = (P.fromIntegral :: Int -> Int32) . truncate;-     "NP.floor       :: a -> Int32"   floor       = (P.fromIntegral :: Int -> Int32) . floor;-     "NP.ceiling     :: a -> Int32"   ceiling     = (P.fromIntegral :: Int -> Int32) . ceiling;-     "NP.round       :: a -> Int64"   round       = (P.fromIntegral :: Int -> Int64) . round;-     "NP.roundSimple :: a -> Int64"   roundSimple = (P.fromIntegral :: Int -> Int64) . roundSimple;-     "NP.truncate    :: a -> Int64"   truncate    = (P.fromIntegral :: Int -> Int64) . truncate;-     "NP.floor       :: a -> Int64"   floor       = (P.fromIntegral :: Int -> Int64) . floor;-     "NP.ceiling     :: a -> Int64"   ceiling     = (P.fromIntegral :: Int -> Int64) . ceiling;-     "NP.round       :: a -> Word"    round       = (P.fromIntegral :: Int -> Word) . round;-     "NP.roundSimple :: a -> Word"    roundSimple = (P.fromIntegral :: Int -> Word) . roundSimple;-     "NP.truncate    :: a -> Word"    truncate    = (P.fromIntegral :: Int -> Word) . truncate;-     "NP.floor       :: a -> Word"    floor       = (P.fromIntegral :: Int -> Word) . floor;-     "NP.ceiling     :: a -> Word"    ceiling     = (P.fromIntegral :: Int -> Word) . ceiling;-     "NP.round       :: a -> Word8"   round       = (P.fromIntegral :: Int -> Word8) . round;-     "NP.roundSimple :: a -> Word8"   roundSimple = (P.fromIntegral :: Int -> Word8) . roundSimple;-     "NP.truncate    :: a -> Word8"   truncate    = (P.fromIntegral :: Int -> Word8) . truncate;-     "NP.floor       :: a -> Word8"   floor       = (P.fromIntegral :: Int -> Word8) . floor;-     "NP.ceiling     :: a -> Word8"   ceiling     = (P.fromIntegral :: Int -> Word8) . ceiling;-     "NP.round       :: a -> Word16"  round       = (P.fromIntegral :: Int -> Word16) . round;-     "NP.roundSimple :: a -> Word16"  roundSimple = (P.fromIntegral :: Int -> Word16) . roundSimple;-     "NP.truncate    :: a -> Word16"  truncate    = (P.fromIntegral :: Int -> Word16) . truncate;-     "NP.floor       :: a -> Word16"  floor       = (P.fromIntegral :: Int -> Word16) . floor;-     "NP.ceiling     :: a -> Word16"  ceiling     = (P.fromIntegral :: Int -> Word16) . ceiling;-     "NP.round       :: a -> Word32"  round       = (P.fromIntegral :: Int -> Word32) . round;-     "NP.roundSimple :: a -> Word32"  roundSimple = (P.fromIntegral :: Int -> Word32) . roundSimple;-     "NP.truncate    :: a -> Word32"  truncate    = (P.fromIntegral :: Int -> Word32) . truncate;-     "NP.floor       :: a -> Word32"  floor       = (P.fromIntegral :: Int -> Word32) . floor;-     "NP.ceiling     :: a -> Word32"  ceiling     = (P.fromIntegral :: Int -> Word32) . ceiling;-     "NP.round       :: a -> Word64"  round       = (P.fromIntegral :: Int -> Word64) . round;-     "NP.roundSimple :: a -> Word64"  roundSimple = (P.fromIntegral :: Int -> Word64) . roundSimple;-     "NP.truncate    :: a -> Word64"  truncate    = (P.fromIntegral :: Int -> Word64) . truncate;-     "NP.floor       :: a -> Word64"  floor       = (P.fromIntegral :: Int -> Word64) . floor;-     "NP.ceiling     :: a -> Word64"  ceiling     = (P.fromIntegral :: Int -> Word64) . ceiling;--     "NP.splitFraction :: a -> (Int8,a)"     splitFraction = mapFst (P.fromIntegral :: Int -> Int8) . splitFraction;-     "NP.splitFraction :: a -> (Int16,a)"    splitFraction = mapFst (P.fromIntegral :: Int -> Int16) . splitFraction;-     "NP.splitFraction :: a -> (Int32,a)"    splitFraction = mapFst (P.fromIntegral :: Int -> Int32) . splitFraction;-     "NP.splitFraction :: a -> (Int64,a)"    splitFraction = mapFst (P.fromIntegral :: Int -> Int64) . splitFraction;-     "NP.splitFraction :: a -> (Word,a)"     splitFraction = mapFst (P.fromIntegral :: Int -> Word) . splitFraction;-     "NP.splitFraction :: a -> (Word8,a)"    splitFraction = mapFst (P.fromIntegral :: Int -> Word8) . splitFraction;-     "NP.splitFraction :: a -> (Word16,a)"   splitFraction = mapFst (P.fromIntegral :: Int -> Word16) . splitFraction;-     "NP.splitFraction :: a -> (Word32,a)"   splitFraction = mapFst (P.fromIntegral :: Int -> Word32) . splitFraction;-     "NP.splitFraction :: a -> (Word64,a)"   splitFraction = mapFst (P.fromIntegral :: Int -> Word64) . splitFraction;-  #-}---{- | TODO: Should be moved to a continued fraction module. -}--approxRational :: (ToRational.C a, C a) => a -> a -> Rational-approxRational rat eps    =  simplest (rat-eps) (rat+eps)-        where simplest x y | y < x      =  simplest y x-                           | x == y     =  xr-                           | x > 0      =  simplest' n d n' d'-                           | y < 0      =  - simplest' (-n') d' (-n) d-                           | otherwise  =  0 :% 1-                                        where xr@(n:%d) = ToRational.toRational x-                                              (n':%d')  = ToRational.toRational y--              simplest' n d n' d'       -- assumes 0 < n%d < n'%d'-                        | isZero r   =  q :% 1-                        | q /= q'    =  (q+1) :% 1-                        | otherwise  =  (q*n''+d'') :% n''-                                     where (q,r)      =  quotRem n d-                                           (q',r')    =  quotRem n' d'-                                           (n'':%d'') =  simplest' d' r' d r----- * generic implementation of round functions--powersOfTwo :: (Ring.C a) => [a]-powersOfTwo = iterate (2*) one--pairsOfPowersOfTwo :: (Ring.C a, Ring.C b) => [(a,b)]-pairsOfPowersOfTwo =-   zip powersOfTwo powersOfTwo--{- |-The generic rounding functions need a number of operations-proportional to the number of binary digits of the integer portion.-If operations like multiplication with two and comparison-need time proportional to the number of binary digits,-then the overall rounding requires quadratic time.--}-genericFloor :: (Ord a, Ring.C a, Ring.C b) => a -> b-genericFloor a =-   if a>=zero-     then genericPosFloor a-     else negate $ genericPosCeiling $ negate a--genericCeiling :: (Ord a, Ring.C a, Ring.C b) => a -> b-genericCeiling a =-   if a>=zero-     then genericPosCeiling a-     else negate $ genericPosFloor $ negate a--genericTruncate :: (Ord a, Ring.C a, Ring.C b) => a -> b-genericTruncate a =-   if a>=zero-     then genericPosFloor a-     else negate $ genericPosFloor $ negate a--genericRound :: (Ord a, Ring.C a, Ring.C b) => a -> b-genericRound a =-   if a>=zero-     then genericPosRound a-     else negate $ genericPosRound $ negate a--genericFraction :: (Ord a, Ring.C a) => a -> a-genericFraction a =-   if a>=zero-     then genericPosFraction a-     else fixFraction $ negate $ genericPosFraction $ negate a--genericSplitFraction :: (Ord a, Ring.C a, Ring.C b) => a -> (b,a)-genericSplitFraction a =-   if a>=zero-     then genericPosSplitFraction a-     else fixSplitFraction $ mapPair (negate, negate) $-          genericPosSplitFraction $ negate a---genericPosFloor :: (Ord a, Ring.C a, Ring.C b) => a -> b-genericPosFloor a =-   snd $-   foldr-      (\(pa,pb) acc@(accA,accB) ->-         let newA = accA+pa-         in  if newA>a then acc else (newA,accB+pb))-      (zero,zero) $-   takeWhile ((a>=) . fst) $-   pairsOfPowersOfTwo--genericPosCeiling :: (Ord a, Ring.C a, Ring.C b) => a -> b-genericPosCeiling a =-   snd $-   (\(ps,u:_) ->-      foldr-         (\(pa,pb) acc@(accA,accB) ->-            let newA = accA-pa-            in  if newA>=a then (newA,accB-pb) else acc)-         u ps) $-   span ((a>) . fst) $-   (zero,zero) : pairsOfPowersOfTwo--{--genericPosFloorDigits :: (Ord a, Ring.C a, Ring.C b) => a -> ((a,b), [Bool])-genericPosFloorDigits a =-   List.mapAccumR-      (\acc@(accA,accB) (pa,pb) ->-         let newA = accA+pa-             b = newA<=a-         in  (if b then (newA,accB+pb) else acc, b))-      (zero,zero) $-   takeWhile ((a>=) . fst) $-   pairsOfPowersOfTwo--}--genericHalfPosFloorDigits :: (Ord a, Ring.C a, Ring.C b) => a -> ((a,b), [Bool])-genericHalfPosFloorDigits a =-   List.mapAccumR-      (\acc@(accA,accB) (pa,pb) ->-         let newA = accA+pa-             b = newA<=a-         in  (if b then (newA,accB+pb) else acc, b))-      (zero,zero) $-   takeWhile ((a>=) . fst) $-   zip powersOfTwo (zero:powersOfTwo)--genericPosRound :: (Ord a, Ring.C a, Ring.C b) => a -> b-genericPosRound a =-   let a2 = 2*a-       ((ai,bi), ds) = genericHalfPosFloorDigits a2-   in  if ai==a2-         then-           case ds of-             True : True : _ -> bi+one-             _ -> bi-         else-           case ds of-             True : _ -> bi+one-             _ -> bi--genericPosFraction :: (Ord a, Ring.C a) => a -> a-genericPosFraction a =-   foldr-      (\p acc ->-         if p>acc then acc else acc-p)-      a $-   takeWhile (a>=) $-   powersOfTwo--genericPosSplitFraction :: (Ord a, Ring.C a, Ring.C b) => a -> (b,a)-genericPosSplitFraction a =-   foldr-      (\(pb,pa) acc@(accB,accA) ->-         if pa>accA then acc else (accB+pb,accA-pa))-      (zero,a) $-   takeWhile ((a>=) . snd) $-   pairsOfPowersOfTwo---{- |-Needs linear time with respect to the number of digits.--This and other functions using OrderDecision-like @floor@ where argument and result are the same-may be moved to a new module.--}-decisionPosFraction :: (OrdDec.C a, Ring.C a) => a -> a-decisionPosFraction a0 =-   (\ps ->-      foldr-         (\p cont a ->-            (a<?one) a $ cont $-            (a>=?p) (a-p) a)-         (error "decisionPosFraction: end of list should never be reached")-         ps a0) $-   concatMap (reverse . flip take powersOfTwo) powersOfTwo--{--Works but needs quadratic time with respect to the number of digits.-I feel that there must be something more efficient.--}-decisionPosFractionSqrTime :: (OrdDec.C a, Ring.C a) => a -> a-decisionPosFractionSqrTime a0 =-   (\ps ->-      foldr-         (\p cont a ->-            (a<?one) a $ cont $-            (a>=?p) (a-p) a)-         (error "decisionPosFraction: end of list should never be reached")-         ps a0) $-   concatMap reverse $-   inits powersOfTwo
− src-ghc-6.12/Algebra/RealTranscendental.hs 
@@ -1,37 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Algebra.RealTranscendental where--import qualified Algebra.Transcendental      as Trans-import qualified Algebra.RealField           as RealField--import Algebra.Transcendental (atan, pi)-import Algebra.Field          ((/))-import Algebra.Ring           (fromInteger)-import Algebra.Additive       ((+), negate)--import Data.Bool.HT (select, )--import qualified Prelude as P-import NumericPrelude.Base----{-|-This class collects all functions for _scalar_ floating point numbers.-E.g. computing 'atan2' for complex floating numbers makes certainly no sense.--}-class (RealField.C a, Trans.C a) => C a where-    atan2 :: a -> a -> a--    atan2 y x = select 0   -- must be after the other double zero tests-      [(x>0,          atan (y/x)),-       (x==0 && y>0,  pi/2),-       (x<0  && y>0,  pi + atan (y/x)),-       (x<=0 && y<0, -atan2 (-y) x),-       (y==0 && x<0,  pi)] -- must be after the previous test on zero y--instance C P.Float where-    atan2 = P.atan2--instance C P.Double where-    atan2 = P.atan2
− src-ghc-6.12/Algebra/RightModule.hs 
@@ -1,17 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-module Algebra.RightModule where--import qualified Algebra.Ring     as Ring-import qualified Algebra.Additive as Additive---- import NumericPrelude.Numeric--- import qualified Prelude----- Is this right?-infixl 7 <*--class (Ring.C a, Additive.C b) => C a b where-    (<*) :: b -> a -> b
− src-ghc-6.12/Algebra/Ring.hs 
@@ -1,257 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Algebra.Ring (-    {- * Class -}-    C,--    (*),-    one,-    fromInteger,-    (^), sqr,--    {- * Complex functions -}-    product, product1, scalarProduct,--    {- * Properties -}-    propAssociative,-    propLeftDistributive,-    propRightDistributive,-    propLeftIdentity,-    propRightIdentity,-    propPowerCascade,-    propPowerProduct,-    propPowerDistributive,-    propCommutative,-  ) where--import qualified Algebra.Additive as Additive-import qualified Algebra.Laws as Laws--import Algebra.Additive(zero, (+), negate, sum)--import Data.Function.HT (powerAssociative, )-import NumericPrelude.List (zipWithChecked, )--import Test.QuickCheck ((==>), Property)--import Data.Int  (Int,  Int8,  Int16,  Int32,  Int64,  )-import Data.Word (Word, Word8, Word16, Word32, Word64, )--import NumericPrelude.Base-import Prelude (Integer, Float, Double, )-import qualified Data.Ratio as Ratio98-import qualified Prelude as P--- import Test.QuickCheck---infixl 7 *-infixr 8 ^---{- |-Ring encapsulates the mathematical structure-of a (not necessarily commutative) ring, with the laws--@-  a * (b * c) === (a * b) * c-      one * a === a-      a * one === a-  a * (b + c) === a * b + a * c-@--Typical examples include integers, polynomials, matrices, and quaternions.--Minimal definition: '*', ('one' or 'fromInteger')--}--class (Additive.C a) => C a where-    (*)         :: a -> a -> a-    one         :: a-    fromInteger :: Integer -> a-    {- |-    The exponent has fixed type 'Integer' in order-    to avoid an arbitrarily limitted range of exponents,-    but to reduce the need for the compiler to guess the type (default type).-    In practice the exponent is most oftenly fixed, and is most oftenly @2@.-    Fixed exponents can be optimized away and-    thus the expensive computation of 'Integer's doesn't matter.-    The previous solution used a 'Algebra.ToInteger.C' constrained type-    and the exponent was converted to Integer before computation.-    So the current solution is not less efficient.--    A variant of '^' with more flexibility is provided by 'Algebra.Core.ringPower'.-    -}-    (^)         :: a -> Integer -> a--    {-# INLINE fromInteger #-}-    fromInteger n = if n < 0-                      then powerAssociative (+) zero (negate one) (negate n)-                      else powerAssociative (+) zero one n-    {-# INLINE (^) #-}-    a ^ n = if n >= zero-              then powerAssociative (*) one a n-              else error "(^): Illegal negative exponent"-    {-# INLINE one #-}-    one = fromInteger 1---sqr :: C a => a -> a-sqr x = x*x--product :: (C a) => [a] -> a-product = foldl (*) one--product1 :: (C a) => [a] -> a-product1 = foldl1 (*)---scalarProduct :: C a => [a] -> [a] -> a-scalarProduct as bs = sum (zipWithChecked (*) as bs)---{- * Instances for atomic types -}--instance C Integer where-   {-# INLINE one #-}-   {-# INLINE fromInteger #-}-   {-# INLINE (*) #-}-   one         = P.fromInteger 1-   fromInteger = P.fromInteger-   (*)         = (P.*)--instance C Float   where-   {-# INLINE one #-}-   {-# INLINE fromInteger #-}-   {-# INLINE (*) #-}-   one         = P.fromInteger 1-   fromInteger = P.fromInteger-   (*)         = (P.*)--instance C Double  where-   {-# INLINE one #-}-   {-# INLINE fromInteger #-}-   {-# INLINE (*) #-}-   one         = P.fromInteger 1-   fromInteger = P.fromInteger-   (*)         = (P.*)---instance C Int     where-   {-# INLINE one #-}-   {-# INLINE fromInteger #-}-   {-# INLINE (*) #-}-   one         = P.fromInteger 1-   fromInteger = P.fromInteger-   (*)         = (P.*)--instance C Int8    where-   {-# INLINE one #-}-   {-# INLINE fromInteger #-}-   {-# INLINE (*) #-}-   one         = P.fromInteger 1-   fromInteger = P.fromInteger-   (*)         = (P.*)--instance C Int16   where-   {-# INLINE one #-}-   {-# INLINE fromInteger #-}-   {-# INLINE (*) #-}-   one         = P.fromInteger 1-   fromInteger = P.fromInteger-   (*)         = (P.*)--instance C Int32   where-   {-# INLINE one #-}-   {-# INLINE fromInteger #-}-   {-# INLINE (*) #-}-   one         = P.fromInteger 1-   fromInteger = P.fromInteger-   (*)         = (P.*)--instance C Int64   where-   {-# INLINE one #-}-   {-# INLINE fromInteger #-}-   {-# INLINE (*) #-}-   one         = P.fromInteger 1-   fromInteger = P.fromInteger-   (*)         = (P.*)---instance C Word    where-   {-# INLINE one #-}-   {-# INLINE fromInteger #-}-   {-# INLINE (*) #-}-   one         = P.fromInteger 1-   fromInteger = P.fromInteger-   (*)         = (P.*)--instance C Word8   where-   {-# INLINE one #-}-   {-# INLINE fromInteger #-}-   {-# INLINE (*) #-}-   one         = P.fromInteger 1-   fromInteger = P.fromInteger-   (*)         = (P.*)--instance C Word16  where-   {-# INLINE one #-}-   {-# INLINE fromInteger #-}-   {-# INLINE (*) #-}-   one         = P.fromInteger 1-   fromInteger = P.fromInteger-   (*)         = (P.*)--instance C Word32  where-   {-# INLINE one #-}-   {-# INLINE fromInteger #-}-   {-# INLINE (*) #-}-   one         = P.fromInteger 1-   fromInteger = P.fromInteger-   (*)         = (P.*)--instance C Word64  where-   {-# INLINE one #-}-   {-# INLINE fromInteger #-}-   {-# INLINE (*) #-}-   one         = P.fromInteger 1-   fromInteger = P.fromInteger-   (*)         = (P.*)------propAssociative       :: (Eq a, C a) => a -> a -> a -> Bool-propLeftDistributive  :: (Eq a, C a) => a -> a -> a -> Bool-propRightDistributive :: (Eq a, C a) => a -> a -> a -> Bool-propLeftIdentity      :: (Eq a, C a) => a -> Bool-propRightIdentity     :: (Eq a, C a) => a -> Bool--propAssociative       =  Laws.associative (*)-propLeftDistributive  =  Laws.leftDistributive  (*) (+)-propRightDistributive =  Laws.rightDistributive (*) (+)-propLeftIdentity      =  Laws.leftIdentity  (*) one-propRightIdentity     =  Laws.rightIdentity (*) one--propPowerCascade      :: (Eq a, C a) => a -> Integer -> Integer -> Property-propPowerProduct      :: (Eq a, C a) => a -> Integer -> Integer -> Property-propPowerDistributive :: (Eq a, C a) => Integer -> a -> a -> Property--propPowerCascade      x i j  =  i>=0 && j>=0 ==> Laws.rightCascade (*) (^) x i j-propPowerProduct      x i j  =  i>=0 && j>=0 ==> Laws.homomorphism (x^) (+) (*) i j-propPowerDistributive i x y  =  i>=0 ==> Laws.leftDistributive (^) (*) i x y--{- | Commutativity need not be satisfied by all instances of 'Algebra.Ring.C'. -}-propCommutative :: (Eq a, C a) => a -> a -> Bool--propCommutative  =  Laws.commutative (*)----- legacy--instance (P.Integral a) => C (Ratio98.Ratio a) where-   {-# INLINE one #-}-   {-# INLINE fromInteger #-}-   {-# INLINE (*) #-}-   one                 =  1-   fromInteger         =  P.fromInteger-   (*)                 =  (P.*)
− src-ghc-6.12/Algebra/ToInteger.hs 
@@ -1,141 +0,0 @@-{-# OPTIONS_GHC -fno-warn-orphans #-}-{--The orphan instance could be fixed-by making this module mutually recursive with ToRational.hs,-but that's not worth the complication.--}--module Algebra.ToInteger where--import qualified Number.Ratio as Ratio--import qualified Algebra.ToRational     as ToRational-import qualified Algebra.Field          as Field-import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.RealIntegral   as RealIntegral-import qualified Algebra.Ring           as Ring--import Number.Ratio (T((:%)), )--import Algebra.Field ((^-), )-import Algebra.Ring ((^), fromInteger, )--import Data.Int  (Int,  Int8,  Int16,  Int32,  Int64,  )-import Data.Word (Word, Word8, Word16, Word32, Word64, )--import qualified Prelude as P-import NumericPrelude.Base-import Prelude (Integer, Float, Double, )---{- |-The two classes 'Algebra.ToInteger.C' and 'Algebra.ToRational.C'-exist to allow convenient conversions,-primarily between the built-in types.-They should satisfy-->   fromInteger .  toInteger === id->    toRational .  toInteger === toRational--Conversions must be lossless,-that is, they do not round in any way.-For rounding see "Algebra.RealRing".-With the instances for 'Prelude.Float' and 'Prelude.Double'-we acknowledge that these types actually represent rationals-rather than (approximated) real numbers.-However, this contradicts to the 'Algebra.Transcendental.C' instance.--}-class (ToRational.C a, RealIntegral.C a) => C a where-   toInteger :: a -> Integer---fromIntegral :: (C a, Ring.C b) => a -> b-fromIntegral = fromInteger . toInteger----- generated by GenerateRules.hs-{-# RULES-     "NP.fromIntegral :: Integer -> Int"     fromIntegral = P.fromIntegral :: Integer -> Int;-     "NP.fromIntegral :: Integer -> Integer" fromIntegral = P.fromIntegral :: Integer -> Integer;-     "NP.fromIntegral :: Integer -> Float"   fromIntegral = P.fromIntegral :: Integer -> Float;-     "NP.fromIntegral :: Integer -> Double"  fromIntegral = P.fromIntegral :: Integer -> Double;-     "NP.fromIntegral :: Int -> Int"         fromIntegral = P.fromIntegral :: Int -> Int;-     "NP.fromIntegral :: Int -> Integer"     fromIntegral = P.fromIntegral :: Int -> Integer;-     "NP.fromIntegral :: Int -> Float"       fromIntegral = P.fromIntegral :: Int -> Float;-     "NP.fromIntegral :: Int -> Double"      fromIntegral = P.fromIntegral :: Int -> Double;-     "NP.fromIntegral :: Int8 -> Int"        fromIntegral = P.fromIntegral :: Int8 -> Int;-     "NP.fromIntegral :: Int8 -> Integer"    fromIntegral = P.fromIntegral :: Int8 -> Integer;-     "NP.fromIntegral :: Int8 -> Float"      fromIntegral = P.fromIntegral :: Int8 -> Float;-     "NP.fromIntegral :: Int8 -> Double"     fromIntegral = P.fromIntegral :: Int8 -> Double;-     "NP.fromIntegral :: Int16 -> Int"       fromIntegral = P.fromIntegral :: Int16 -> Int;-     "NP.fromIntegral :: Int16 -> Integer"   fromIntegral = P.fromIntegral :: Int16 -> Integer;-     "NP.fromIntegral :: Int16 -> Float"     fromIntegral = P.fromIntegral :: Int16 -> Float;-     "NP.fromIntegral :: Int16 -> Double"    fromIntegral = P.fromIntegral :: Int16 -> Double;-     "NP.fromIntegral :: Int32 -> Int"       fromIntegral = P.fromIntegral :: Int32 -> Int;-     "NP.fromIntegral :: Int32 -> Integer"   fromIntegral = P.fromIntegral :: Int32 -> Integer;-     "NP.fromIntegral :: Int32 -> Float"     fromIntegral = P.fromIntegral :: Int32 -> Float;-     "NP.fromIntegral :: Int32 -> Double"    fromIntegral = P.fromIntegral :: Int32 -> Double;-     "NP.fromIntegral :: Int64 -> Int"       fromIntegral = P.fromIntegral :: Int64 -> Int;-     "NP.fromIntegral :: Int64 -> Integer"   fromIntegral = P.fromIntegral :: Int64 -> Integer;-     "NP.fromIntegral :: Int64 -> Float"     fromIntegral = P.fromIntegral :: Int64 -> Float;-     "NP.fromIntegral :: Int64 -> Double"    fromIntegral = P.fromIntegral :: Int64 -> Double;-     "NP.fromIntegral :: Word -> Int"        fromIntegral = P.fromIntegral :: Word -> Int;-     "NP.fromIntegral :: Word -> Integer"    fromIntegral = P.fromIntegral :: Word -> Integer;-     "NP.fromIntegral :: Word -> Float"      fromIntegral = P.fromIntegral :: Word -> Float;-     "NP.fromIntegral :: Word -> Double"     fromIntegral = P.fromIntegral :: Word -> Double;-     "NP.fromIntegral :: Word8 -> Int"       fromIntegral = P.fromIntegral :: Word8 -> Int;-     "NP.fromIntegral :: Word8 -> Integer"   fromIntegral = P.fromIntegral :: Word8 -> Integer;-     "NP.fromIntegral :: Word8 -> Float"     fromIntegral = P.fromIntegral :: Word8 -> Float;-     "NP.fromIntegral :: Word8 -> Double"    fromIntegral = P.fromIntegral :: Word8 -> Double;-     "NP.fromIntegral :: Word16 -> Int"      fromIntegral = P.fromIntegral :: Word16 -> Int;-     "NP.fromIntegral :: Word16 -> Integer"  fromIntegral = P.fromIntegral :: Word16 -> Integer;-     "NP.fromIntegral :: Word16 -> Float"    fromIntegral = P.fromIntegral :: Word16 -> Float;-     "NP.fromIntegral :: Word16 -> Double"   fromIntegral = P.fromIntegral :: Word16 -> Double;-     "NP.fromIntegral :: Word32 -> Int"      fromIntegral = P.fromIntegral :: Word32 -> Int;-     "NP.fromIntegral :: Word32 -> Integer"  fromIntegral = P.fromIntegral :: Word32 -> Integer;-     "NP.fromIntegral :: Word32 -> Float"    fromIntegral = P.fromIntegral :: Word32 -> Float;-     "NP.fromIntegral :: Word32 -> Double"   fromIntegral = P.fromIntegral :: Word32 -> Double;-     "NP.fromIntegral :: Word64 -> Int"      fromIntegral = P.fromIntegral :: Word64 -> Int;-     "NP.fromIntegral :: Word64 -> Integer"  fromIntegral = P.fromIntegral :: Word64 -> Integer;-     "NP.fromIntegral :: Word64 -> Float"    fromIntegral = P.fromIntegral :: Word64 -> Float;-     "NP.fromIntegral :: Word64 -> Double"   fromIntegral = P.fromIntegral :: Word64 -> Double;-  #-}---instance C Integer where {-#INLINE toInteger #-}; toInteger = id--instance C Int     where {-#INLINE toInteger #-}; toInteger = P.toInteger-instance C Int8    where {-#INLINE toInteger #-}; toInteger = P.toInteger-instance C Int16   where {-#INLINE toInteger #-}; toInteger = P.toInteger-instance C Int32   where {-#INLINE toInteger #-}; toInteger = P.toInteger-instance C Int64   where {-#INLINE toInteger #-}; toInteger = P.toInteger--instance C Word    where {-#INLINE toInteger #-}; toInteger = P.toInteger-instance C Word8   where {-#INLINE toInteger #-}; toInteger = P.toInteger-instance C Word16  where {-#INLINE toInteger #-}; toInteger = P.toInteger-instance C Word32  where {-#INLINE toInteger #-}; toInteger = P.toInteger-instance C Word64  where {-#INLINE toInteger #-}; toInteger = P.toInteger---instance (C a, PID.C a) => ToRational.C (Ratio.T a) where-   toRational (x:%y)   =  toInteger x :% toInteger y---{-|-A prefix function of '(Algebra.Ring.^)'-with a parameter order that fits the needs of partial application-and function composition.-It has generalised exponent.--See: Argument order of @expNat@ on-<http://www.haskell.org/pipermail/haskell-cafe/2006-September/018022.html>--}-ringPower :: (Ring.C a, C b) => b -> a -> a-ringPower exponent basis = basis ^ toInteger exponent--{- |-A prefix function of '(Algebra.Field.^-)'.-It has a generalised exponent.--}-fieldPower :: (Field.C a, C b) => b -> a -> a-fieldPower exponent basis = basis ^- toInteger exponent
− src-ghc-6.12/Algebra/ToRational.hs 
@@ -1,100 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Algebra.ToRational where--import qualified Algebra.Field    as Field-import qualified Algebra.Absolute as Absolute-import Algebra.Field (fromRational, )-import Algebra.Ring (fromInteger, )--import Number.Ratio (Rational, )--import Data.Int  (Int,  Int8,  Int16,  Int32,  Int64,  )-import Data.Word (Word, Word8, Word16, Word32, Word64, )--import qualified Prelude as P-import NumericPrelude.Base-import Prelude (Integer, Float, Double, )--{- |-This class allows lossless conversion-from any representation of a rational to the fixed 'Rational' type.-\"Lossless\" means - don't do any rounding.-For rounding see "Algebra.RealRing".-With the instances for 'Float' and 'Double'-we acknowledge that these types actually represent rationals-rather than (approximated) real numbers.-However, this contradicts to the 'Algebra.Transcendental' class.--Laws that must be satisfied by instances:-->  fromRational' . toRational === id--}-class (Absolute.C a) => C a where-   -- | Lossless conversion from any representation of a rational to 'Rational'-   toRational :: a -> Rational--instance C Integer where-   {-# INLINE toRational #-}-   toRational = fromInteger--instance C Float where-   {-# INLINE toRational #-}-   toRational = fromRational . P.toRational--instance C Double where-   {-# INLINE toRational #-}-   toRational = fromRational . P.toRational--instance C Int    where {-# INLINE toRational #-}; toRational = toRational . P.toInteger-instance C Int8   where {-# INLINE toRational #-}; toRational = toRational . P.toInteger-instance C Int16  where {-# INLINE toRational #-}; toRational = toRational . P.toInteger-instance C Int32  where {-# INLINE toRational #-}; toRational = toRational . P.toInteger-instance C Int64  where {-# INLINE toRational #-}; toRational = toRational . P.toInteger--instance C Word   where {-# INLINE toRational #-}; toRational = toRational . P.toInteger-instance C Word8  where {-# INLINE toRational #-}; toRational = toRational . P.toInteger-instance C Word16 where {-# INLINE toRational #-}; toRational = toRational . P.toInteger-instance C Word32 where {-# INLINE toRational #-}; toRational = toRational . P.toInteger-instance C Word64 where {-# INLINE toRational #-}; toRational = toRational . P.toInteger---{- |-It should hold--> realToField = fromRational' . toRational--but it should be much more efficient for particular pairs of types,-such as converting 'Float' to 'Double'.-This achieved by optimizer rules.--}-realToField :: (C a, Field.C b) => a -> b-realToField = Field.fromRational' . toRational--{-# RULES-     "NP.realToField :: Integer  -> Float "  realToField = P.realToFrac :: Integer  -> Float ;-     "NP.realToField :: Int      -> Float "  realToField = P.realToFrac :: Int      -> Float ;-     "NP.realToField :: Int8     -> Float "  realToField = P.realToFrac :: Int8     -> Float ;-     "NP.realToField :: Int16    -> Float "  realToField = P.realToFrac :: Int16    -> Float ;-     "NP.realToField :: Int32    -> Float "  realToField = P.realToFrac :: Int32    -> Float ;-     "NP.realToField :: Int64    -> Float "  realToField = P.realToFrac :: Int64    -> Float ;-     "NP.realToField :: Word     -> Float "  realToField = P.realToFrac :: Word     -> Float ;-     "NP.realToField :: Word8    -> Float "  realToField = P.realToFrac :: Word8    -> Float ;-     "NP.realToField :: Word16   -> Float "  realToField = P.realToFrac :: Word16   -> Float ;-     "NP.realToField :: Word32   -> Float "  realToField = P.realToFrac :: Word32   -> Float ;-     "NP.realToField :: Word64   -> Float "  realToField = P.realToFrac :: Word64   -> Float ;-     "NP.realToField :: Float    -> Float "  realToField = P.realToFrac :: Float    -> Float ;-     "NP.realToField :: Double   -> Float "  realToField = P.realToFrac :: Double   -> Float ;-     "NP.realToField :: Integer  -> Double"  realToField = P.realToFrac :: Integer  -> Double;-     "NP.realToField :: Int      -> Double"  realToField = P.realToFrac :: Int      -> Double;-     "NP.realToField :: Int8     -> Double"  realToField = P.realToFrac :: Int8     -> Double;-     "NP.realToField :: Int16    -> Double"  realToField = P.realToFrac :: Int16    -> Double;-     "NP.realToField :: Int32    -> Double"  realToField = P.realToFrac :: Int32    -> Double;-     "NP.realToField :: Int64    -> Double"  realToField = P.realToFrac :: Int64    -> Double;-     "NP.realToField :: Word     -> Double"  realToField = P.realToFrac :: Word     -> Double;-     "NP.realToField :: Word8    -> Double"  realToField = P.realToFrac :: Word8    -> Double;-     "NP.realToField :: Word16   -> Double"  realToField = P.realToFrac :: Word16   -> Double;-     "NP.realToField :: Word32   -> Double"  realToField = P.realToFrac :: Word32   -> Double;-     "NP.realToField :: Word64   -> Double"  realToField = P.realToFrac :: Word64   -> Double;-     "NP.realToField :: Float    -> Double"  realToField = P.realToFrac :: Float    -> Double;-     "NP.realToField :: Double   -> Double"  realToField = P.realToFrac :: Double   -> Double;-  #-}
− src-ghc-6.12/Algebra/Transcendental.hs 
@@ -1,200 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Algebra.Transcendental where--import qualified Algebra.Algebraic as Algebraic--- import qualified Algebra.Ring      as Ring--- import qualified Algebra.Additive  as Additive--import qualified Algebra.Laws as Laws--import Algebra.Algebraic (sqrt)-import Algebra.Field     ((/), recip)-import Algebra.Ring      ((*), (^), fromInteger)-import Algebra.Additive  ((+), (-), negate)--import qualified Prelude as P-import NumericPrelude.Base---infixr 8  **, ^?--{-|-Transcendental is the type of numbers supporting the elementary-transcendental functions.  Examples include real numbers, complex-numbers, and computable reals represented as a lazy list of rational-approximations.--Note the default declaration for a superclass.  See the comments-below, under "Instance declaractions for superclasses".--The semantics of these operations are rather ill-defined because of-branch cuts, etc.--Minimal complete definition:-     pi, exp, log, sin, cos, asin, acos, atan--}-class (Algebraic.C a) => C a where-    pi                  :: a-    exp, log            :: a -> a-    logBase, (**)       :: a -> a -> a-    sin, cos, tan       :: a -> a-    asin, acos, atan    :: a -> a-    sinh, cosh, tanh    :: a -> a-    asinh, acosh, atanh :: a -> a--    {-# INLINE pi #-}-    {-# INLINE exp #-}-    {-# INLINE log #-}-    {-# INLINE logBase #-}-    {-# INLINE (**) #-}-    {-# INLINE sin #-}-    {-# INLINE tan #-}-    {-# INLINE cos #-}-    {-# INLINE asin #-}-    {-# INLINE atan #-}-    {-# INLINE acos #-}-    {-# INLINE sinh #-}-    {-# INLINE tanh #-}-    {-# INLINE cosh #-}-    {-# INLINE asinh #-}-    {-# INLINE atanh #-}-    {-# INLINE acosh #-}--    x ** y           =  exp (log x * y)-    logBase x y      =  log y / log x--    tan  x           =  sin x / cos x--    asin x           =  atan (x / sqrt (1-x^2))-    acos x           =  pi/2 - asin x--    -- if these definitions have errors, then those in FMP.Types have them, too-    sinh x           =  (exp x - exp (-x)) / 2-    cosh x           =  (exp x + exp (-x)) / 2-    -- tanh x           =  (exp x - exp (-x)) / (exp x + exp (-x))-    tanh x           =  sinh x / cosh x--    asinh x          =  log (sqrt (x^2+1) + x)-    acosh x          =  log (sqrt (x^2-1) + x)-    atanh x          =  (log (1+x) - log (1-x)) / 2---instance C P.Float where-    {-# INLINE pi #-}-    {-# INLINE exp #-}-    {-# INLINE log #-}-    {-# INLINE logBase #-}-    {-# INLINE (**) #-}-    {-# INLINE sin #-}-    {-# INLINE tan #-}-    {-# INLINE cos #-}-    {-# INLINE asin #-}-    {-# INLINE atan #-}-    {-# INLINE acos #-}-    {-# INLINE sinh #-}-    {-# INLINE tanh #-}-    {-# INLINE cosh #-}-    {-# INLINE asinh #-}-    {-# INLINE atanh #-}-    {-# INLINE acosh #-}--    (**)  = (P.**)-    exp   = P.exp;   log   = P.log;   logBase = P.logBase-    pi    = P.pi;-    sin   = P.sin;   cos   = P.cos;   tan     = P.tan-    asin  = P.asin;  acos  = P.acos;  atan    = P.atan-    sinh  = P.sinh;  cosh  = P.cosh;  tanh    = P.tanh-    asinh = P.asinh; acosh = P.acosh; atanh   = P.atanh--instance C P.Double where-    {-# INLINE pi #-}-    {-# INLINE exp #-}-    {-# INLINE log #-}-    {-# INLINE logBase #-}-    {-# INLINE (**) #-}-    {-# INLINE sin #-}-    {-# INLINE tan #-}-    {-# INLINE cos #-}-    {-# INLINE asin #-}-    {-# INLINE atan #-}-    {-# INLINE acos #-}-    {-# INLINE sinh #-}-    {-# INLINE tanh #-}-    {-# INLINE cosh #-}-    {-# INLINE asinh #-}-    {-# INLINE atanh #-}-    {-# INLINE acosh #-}--    (**)  = (P.**)-    exp   = P.exp;   log   = P.log;   logBase = P.logBase-    pi    = P.pi;-    sin   = P.sin;   cos   = P.cos;   tan     = P.tan-    asin  = P.asin;  acos  = P.acos;  atan    = P.atan-    sinh  = P.sinh;  cosh  = P.cosh;  tanh    = P.tanh-    asinh = P.asinh; acosh = P.acosh; atanh   = P.atanh----{-# INLINE (^?) #-}-(^?) :: C a => a -> a -> a-(^?) = (**)---{-* Transcendental laws, will only hold approximately on floating point numbers -}--propExpLog      :: (Eq a, C a) => a -> Bool-propLogExp      :: (Eq a, C a) => a -> Bool-propExpNeg      :: (Eq a, C a) => a -> Bool-propLogRecip    :: (Eq a, C a) => a -> Bool-propExpProduct  :: (Eq a, C a) => a -> a -> Bool-propExpLogPower :: (Eq a, C a) => a -> a -> Bool-propLogSum      :: (Eq a, C a) => a -> a -> Bool--propExpLog      x   = exp (log x)     == x-propLogExp      x   = log (exp x)     == x-propExpNeg      x   = exp (negate x)  == recip (exp x)-propLogRecip    x   = log (recip x)   == negate (log x)-propExpProduct  x y = Laws.homomorphism exp (+) (*) x y-propExpLogPower x y = exp (log x * y) == x ** y-propLogSum      x y = Laws.homomorphism log (*) (+) x y---propPowerCascade      :: (Eq a, C a) => a -> a -> a -> Bool-propPowerProduct      :: (Eq a, C a) => a -> a -> a -> Bool-propPowerDistributive :: (Eq a, C a) => a -> a -> a -> Bool--propPowerCascade      x i j  =  Laws.rightCascade (*) (**) x i j-propPowerProduct      x i j  =  Laws.homomorphism (x**) (+) (*) i j-propPowerDistributive i x y  =  Laws.rightDistributive (**) (*) i x y--{- * Trigonometric laws, addition theorems -}--propTrigonometricPythagoras :: (Eq a, C a) => a -> Bool-propTrigonometricPythagoras x  =  cos x ^ 2 + sin x ^ 2 == 1--propSinPeriod   :: (Eq a, C a) => a -> Bool-propCosPeriod   :: (Eq a, C a) => a -> Bool-propTanPeriod   :: (Eq a, C a) => a -> Bool--propSinPeriod x = sin (x+2*pi) == sin x-propCosPeriod x = cos (x+2*pi) == cos x-propTanPeriod x = tan (x+2*pi) == tan x--propSinAngleSum  :: (Eq a, C a) => a -> a -> Bool-propCosAngleSum  :: (Eq a, C a) => a -> a -> Bool--propSinAngleSum x y  =  sin (x+y) == sin x * cos y + cos x * sin y-propCosAngleSum x y  =  cos (x+y) == cos x * cos y - sin x * sin y--propSinDoubleAngle :: (Eq a, C a) => a -> Bool-propCosDoubleAngle :: (Eq a, C a) => a -> Bool--propSinDoubleAngle x  =  sin (2*x) == 2 * sin x * cos x-propCosDoubleAngle x  =  cos (2*x) == 2 * cos x ^ 2 - 1--propSinSquare :: (Eq a, C a) => a -> Bool-propCosSquare :: (Eq a, C a) => a -> Bool--propSinSquare x  =  sin x ^ 2 == (1 - cos (2*x)) / 2-propCosSquare x  =  cos x ^ 2 == (1 + cos (2*x)) / 2-
− src-ghc-6.12/Algebra/Units.hs 
@@ -1,153 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Algebra.Units (-    {- * Class -}-    C,-    isUnit,-    stdAssociate,-    stdUnit,-    stdUnitInv,--    {- * Standard implementations for instances -}-    intQuery,-    intAssociate,-    intStandard,-    intStandardInverse,--    {- * Properties -}-    propComposition,-    propInverseUnit,-    propUniqueAssociate,-    propAssociateProduct,-  ) where--import qualified Algebra.IntegralDomain as Integral-import qualified Algebra.Ring           as Ring--- import qualified Algebra.Additive       as Additive-import qualified Algebra.ZeroTestable   as ZeroTestable--import qualified Algebra.Laws           as Laws--import Algebra.IntegralDomain (div)-import Algebra.Ring           (one, (*))-import Algebra.Additive       (negate)-import Algebra.ZeroTestable   (isZero)--import Data.Int  (Int,  Int8,  Int16,  Int32,  Int64,  )--import NumericPrelude.Base-import Prelude (Integer, )-import qualified Prelude as P-import Test.QuickCheck ((==>), Property)---{- |-This class lets us deal with the units in a ring.-'isUnit' tells whether an element is a unit.-The other operations let us canonically-write an element as a unit times another element.-Two elements a, b of a ring R are _associates_ if a=b*u for a unit u.-For an element a, we want to write it as a=b*u where b is an associate of a.-The map (a->b) is called-"StandardAssociate" by Gap,-"unitCanonical" by Axiom,-and "canAssoc" by DoCon.-The map (a->u) is called-"canInv" by DoCon and-"unitNormal(x).unit" by Axiom.--The laws are-->   stdAssociate x * stdUnit x === x->     stdUnit x * stdUnitInv x === 1->  isUnit u ==> stdAssociate x === stdAssociate (x*u)--Currently some algorithms assume-->  stdAssociate(x*y) === stdAssociate x * stdAssociate y--Minimal definition:-   'isUnit' and ('stdUnit' or 'stdUnitInv') and optionally 'stdAssociate'--}--class (Integral.C a) => C a where-  isUnit :: a -> Bool-  stdAssociate, stdUnit, stdUnitInv :: a -> a--  stdAssociate x = x * stdUnitInv x-  stdUnit      x = div one (stdUnitInv x)  -- should be divChecked-  stdUnitInv   x = div one (stdUnit x)-----{- * Instances for atomic types -}--intQuery :: (P.Integral a, Ring.C a) => a -> Bool-intQuery = flip elem [one, negate one]-{- constraint must be replaced by NumericPrelude.Absolute -}-intAssociate, intStandard, intStandardInverse ::-   (P.Integral a, Ring.C a, ZeroTestable.C a) => a -> a-intAssociate = P.abs-intStandard x = if isZero x then one else P.signum x-intStandardInverse = intStandard--instance C Int where-  isUnit       = intQuery-  stdAssociate = intAssociate-  stdUnit      = intStandard-  stdUnitInv   = intStandardInverse--instance C Integer where-  isUnit       = intQuery-  stdAssociate = intAssociate-  stdUnit      = intStandard-  stdUnitInv   = intStandardInverse--instance C Int8 where-  isUnit       = intQuery-  stdAssociate = intAssociate-  stdUnit      = intStandard-  stdUnitInv   = intStandardInverse--instance C Int16 where-  isUnit       = intQuery-  stdAssociate = intAssociate-  stdUnit      = intStandard-  stdUnitInv   = intStandardInverse--instance C Int32 where-  isUnit       = intQuery-  stdAssociate = intAssociate-  stdUnit      = intStandard-  stdUnitInv   = intStandardInverse--instance C Int64 where-  isUnit       = intQuery-  stdAssociate = intAssociate-  stdUnit      = intStandard-  stdUnitInv   = intStandardInverse---{--fieldQuery = not . isZero-fieldAssociate = 1-fieldStandard        x = if isZero x then 1 else x-fieldStandardInverse x = if isZero x then 1 else recip x--}----propComposition      :: (Eq a, C a) => a -> Bool-propInverseUnit      :: (Eq a, C a) => a -> Bool-propUniqueAssociate  :: (Eq a, C a) => a -> a -> Property-propAssociateProduct :: (Eq a, C a) => a -> a -> Bool--propComposition x  =  stdAssociate x * stdUnit x == x-propInverseUnit x  =    stdUnit x * stdUnitInv x == one-propUniqueAssociate u x =-                     isUnit u ==> stdAssociate x == stdAssociate (x*u)--{- | Currently some algorithms assume this property. -}-propAssociateProduct =-    Laws.homomorphism stdAssociate (*) (*)-
− src-ghc-6.12/Algebra/Vector.hs 
@@ -1,101 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Copyright   :  (c) Henning Thielemann 2004-2005--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable--Abstraction of vectors--}--module Algebra.Vector where--import qualified Algebra.Ring     as Ring-import qualified Algebra.Additive as Additive--import Algebra.Ring     ((*))-import Algebra.Additive ((+))--import Data.List (zipWith, foldl)--- import Data.Functor (Functor, fmap)--import Prelude((.), (==), Bool, Functor, fmap)-import qualified Prelude as P----- Is this right?-infixr 7 *>--{-|-A Module over a ring satisfies:-->   a *> (b + c) === a *> b + a *> c->   (a * b) *> c === a *> (b *> c)->   (a + b) *> c === a *> c + b *> c--}-class C v where-    -- duplicate some methods from Additive-    -- | zero element of the vector space-    zero  :: (Additive.C a) => v a-    -- | add and subtract elements-    (<+>) :: (Additive.C a) => v a -> v a -> v a-    -- | scale a vector by a scalar-    (*>)  :: (Ring.C a) => a -> v a -> v a--infixl 6 <+>---{- |-We need a Haskell 98 type class-which provides equality test for Vector type constructors.--}-class Eq v where-   eq :: P.Eq a => v a -> v a -> Bool---infix 4 `eq`---{-* Instances for standard type constructors -}--functorScale :: (Functor v, Ring.C a) => a -> v a -> v a-functorScale = fmap . (*)--instance C [] where-   zero  = Additive.zero-   (<+>) = (Additive.+)-   (*>)  = functorScale--instance C ((->) b) where-   zero     = Additive.zero-   (<+>)    = (Additive.+)-   (*>) s f = (s*) . f--instance Eq [] where-   eq = (==)----{-* Related functions -}--{-|-Compute the linear combination of a list of vectors.--}-linearComb :: (Ring.C a, C v) => [a] -> [v a] -> v a-linearComb c = foldl (<+>) zero . zipWith (*>) c---{- * Properties -}--propCascade :: (C v, Eq v, Ring.C a, P.Eq a) =>-   a -> a -> v a -> Bool-propCascade a b c           = (a * b) *> c  `eq`  a *> (b *> c)--propRightDistributive :: (C v, Eq v, Ring.C a, P.Eq a) =>-   a -> v a -> v a -> Bool-propRightDistributive a b c =   a *> (b <+> c)  `eq`  a*>b <+> a*>c--propLeftDistributive :: (C v, Eq v, Ring.C a, P.Eq a) =>-   a -> a -> v a -> Bool-propLeftDistributive a b c  =   (a+b) *> c  `eq`  a*>c <+> b*>c
− src-ghc-6.12/Algebra/VectorSpace.hs 
@@ -1,34 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-module Algebra.VectorSpace where--import qualified Algebra.Module as Module-import qualified Algebra.Field  as Field-import qualified Algebra.PrincipalIdealDomain as PID-import qualified Number.Ratio   as Ratio---- import NumericPrelude.Numeric-import qualified Prelude as P---class (Field.C a, Module.C a b) => C a b---{-* Instances for atomic types -}--instance C P.Float P.Float--instance C P.Double P.Double--{-* Instances for composed types -}--instance (PID.C a) => C (Ratio.T a) (Ratio.T a)--instance (C a b0, C a b1) => C a (b0, b1)--instance (C a b0, C a b1, C a b2) => C a (b0, b1, b2)--instance (C a b) => C a [b]--instance (C a b) => C a (c -> b)
− src-ghc-6.12/Algebra/ZeroTestable.hs 
@@ -1,65 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Algebra.ZeroTestable where--import qualified Algebra.Additive as Additive--import Data.Int  (Int,  Int8,  Int16,  Int32,  Int64,  )-import Data.Word (Word, Word8, Word16, Word32, Word64, )---- import qualified Prelude as P-import Prelude (Integer, Float, Double, )-import NumericPrelude.Base--{- |-Maybe the naming should be according to Algebra.Unit:-Algebra.Zero as module name, and @query@ as method name.--}-class C a where-   isZero :: a -> Bool--{- |-Checks if a number is the zero element.-This test is not possible for all 'Additive.C' types,-since e.g. a function type does not belong to Eq.-isZero is possible for some types where (==zero) fails-because there is no unique zero.-Examples are-vector (the length of the zero vector is unknown),-physical values (the unit of a zero quantity is unknown),-residue class (the modulus is unknown).--}-defltIsZero :: (Eq a, Additive.C a) => a -> Bool-defltIsZero = (Additive.zero==)---{-* Instances for atomic types -}--instance C Integer where isZero = defltIsZero-instance C Float   where isZero = defltIsZero-instance C Double  where isZero = defltIsZero--instance C Int     where isZero = defltIsZero-instance C Int8    where isZero = defltIsZero-instance C Int16   where isZero = defltIsZero-instance C Int32   where isZero = defltIsZero-instance C Int64   where isZero = defltIsZero--instance C Word    where isZero = defltIsZero-instance C Word8   where isZero = defltIsZero-instance C Word16  where isZero = defltIsZero-instance C Word32  where isZero = defltIsZero-instance C Word64  where isZero = defltIsZero----{-* Instances for composed types -}--instance (C v0, C v1) => C (v0, v1) where-    isZero (x0,x1) = isZero x0 && isZero x1--instance (C v0, C v1, C v2) => C (v0, v1, v2) where-    isZero (x0,x1,x2) = isZero x0 && isZero x1 && isZero x2---instance (C v) => C [v] where-    isZero = all isZero
− src-ghc-6.12/MathObj/Algebra.hs 
@@ -1,74 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Copyright    :   (c) Mikael Johansson 2006-Maintainer   :   mik@math.uni-jena.de-Stability    :   provisional-Portability  :   requires multi-parameter type classes--The generic case of a k-algebra generated by a monoid.--}--module MathObj.Algebra where--import qualified Algebra.Vector   as Vector-import qualified Algebra.Ring     as Ring-import qualified Algebra.Additive as Additive-import qualified Algebra.Monoid   as Monoid--import Algebra.Ring((*))-import Algebra.Additive((+),negate,zero)-import Algebra.Monoid((<*>))--import Control.Monad(liftM2,Functor,fmap)-import Data.Map(Map)-import qualified Data.Map as Map-import Data.List(intersperse)--import NumericPrelude.Base(Ord,Eq,{-Read,-}Show,(++),($),-                   concat,map,show)---newtype {- (Ord a, Monoid.C a, Ring.C b) => -}-     T a b = Cons (Map a b)-         deriving (Eq {- ,Read -} )--instance Functor (T a) where-   fmap f (Cons x) = Cons (fmap f x)---- is an Indexable instance better than an Ord instance here?--instance (Ord a, Additive.C b) => Additive.C (T a b) where-   (+) = zipWith (+)-   {- This implementation is attracting but wrong.-     It fails if terms are present in b that are missing in a.-     Default implementation is better here.-   (-) = zipWith (-)-   -}-   negate = fmap negate-   zero = Cons Map.empty--zipWith :: (Ord a) => (b -> b -> b) -> (T a b -> T a b -> T a b)-zipWith op (Cons ma) (Cons mb) = Cons (Map.unionWith op ma mb)--instance Ord a => Vector.C (T a) where-   zero  = zero-   (<+>) = (+)-   (*>)  = Vector.functorScale--instance (Ord a, Monoid.C a, Ring.C b) => Ring.C (T a b) where-   one = Cons $ Map.singleton Monoid.idt Ring.one-   (Cons ma) * (Cons mb) =-      Cons $ Map.fromListWith (+) $-         liftM2 mulMonomial (Map.toList ma) (Map.toList mb)--mulMonomial :: (Monoid.C a, Ring.C b) => (a,b) -> (a,b) -> (a,b)-mulMonomial (c1,m1) (c2,m2) = (c1<*>c2,m1*m2)--instance (Show a, Show b) => Show (T a b) where-   show (Cons ma) = concat $-           intersperse "+" $-           map (\(m,c) -> show c ++ "." ++ show m)-               (Map.toList ma)--monomial :: a -> b -> (T a b)-monomial index coefficient = Cons (Map.singleton index coefficient)
− src-ghc-6.12/MathObj/DiscreteMap.hs 
@@ -1,93 +0,0 @@-{-# OPTIONS_GHC -fno-warn-orphans #-}-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}--{- FIXME:-Rationale for -fno-warn-orphans:- * The orphan instances can't be put into Numeric.NonNegative.Wrapper-   since that's in another package.- * We had to spread the instance declarations-   over the modules defining the typeclasses instantiated.-   Do we want that?- * We could define the DiscreteMap as newtype.--}--{- |-DiscreteMap was originally intended as a type class-that unifies Map and Array.-One should be able to simply choose between- - Map for sparse arrays- - Array for full arrays.--However, the Edison package provides the class AssocX-which already exists for that purpose.--Currently I use this module for some numeric instances of Data.Map.--}-module MathObj.DiscreteMap where--import qualified Algebra.NormedSpace.Sum       as NormedSum-import qualified Algebra.NormedSpace.Euclidean as NormedEuc-import qualified Algebra.NormedSpace.Maximum   as NormedMax-import qualified Algebra.VectorSpace           as VectorSpace-import qualified Algebra.Module                as Module-import qualified Algebra.Vector                as Vector-import qualified Algebra.Algebraic             as Algebraic-import qualified Algebra.Additive              as Additive--import Algebra.Module   ((*>))-import Algebra.Additive (zero,(+),negate)-import qualified Data.Map as Map-import Data.Map (Map)---- import qualified Prelude as P-import NumericPrelude.Base---- FIXME: Should this be implemented by isZero?--- | Remove all zero values from the map.-strip :: (Ord i, Eq v, Additive.C v) => Map i v -> Map i v-strip = Map.filter (zero /=)---strip = Map.filter (((0 /=) .) . (flip const))--instance (Ord i, Eq v, Additive.C v) => Additive.C (Map i v) where-   zero = Map.empty-   (+)  = (strip.). Map.unionWith (+)-   --(+) y x = strip (Map.unionWith (+) y x)-   (-) x y = (+) x (negate y)-   {- won't work because Map.unionWith won't negate a value from y if no x value corresponds to it-   (-) x y = strip (Map.unionWith sub x y)-   -}-   negate  = fmap negate--instance Ord i => Vector.C (Map i) where-   zero  = Map.empty-   (<+>) = Map.unionWith (+)-   -- requires Eq instance for expo-   -- expo *> x = if expo == zero then zero else Vector.functorScale expo x-   (*>)  = Vector.functorScale--instance (Ord i, Eq a, Eq v, Module.C a v)-             => Module.C a (Map i v) where---   (*>) 0    = \_ -> zero---   (*>) expo = fmap ((*>) expo)-   (*>) expo x = if expo == zero then zero else fmap (expo *>) x--instance (Ord i, Eq a, Eq v, VectorSpace.C a v)-             => VectorSpace.C a (Map i v)--instance (Ord i, Eq a, Eq v, NormedSum.C a v)-             => NormedSum.C a (Map i v) where-   norm = foldl (+) zero . map NormedSum.norm . Map.elems--instance (Ord i, Eq a, Eq v, NormedEuc.Sqr a v)-             => NormedEuc.Sqr a (Map i v) where-   normSqr = foldl (+) zero . map NormedEuc.normSqr . Map.elems--instance (Ord i, Eq a, Eq v, Algebraic.C a, NormedEuc.Sqr a v)-             => NormedEuc.C a (Map i v) where-   norm = NormedEuc.defltNorm--instance (Ord i, Eq a, Eq v, NormedMax.C a v)-             => NormedMax.C a (Map i v) where-   norm = foldl max zero . map NormedMax.norm . Map.elems
− src-ghc-6.12/MathObj/Gaussian/Bell.hs 
@@ -1,314 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{--Complex translated Gaussian bell curve-with amplitude abstracted away.--}-module MathObj.Gaussian.Bell where--import qualified MathObj.Polynomial as Poly-import qualified Number.Complex as Complex--import qualified Algebra.Transcendental as Trans-import qualified Algebra.Field          as Field-import qualified Algebra.Absolute       as Absolute-import qualified Algebra.Ring           as Ring-import qualified Algebra.Additive       as Additive--import Number.Complex ((+:), )--import Test.QuickCheck (Arbitrary, arbitrary, )-import Control.Monad (liftM4, )---- import Prelude (($))-import NumericPrelude.Numeric-import NumericPrelude.Base hiding (reverse, )---data T a = Cons {amp :: a, c0, c1 :: Complex.T a, c2 :: a}-   deriving (Eq, Show)--instance (Absolute.C a, Arbitrary a) => Arbitrary (T a) where-   arbitrary =-      liftM4-         (\k a b c -> Cons (abs k) a b (1 + abs c))-         arbitrary arbitrary arbitrary arbitrary---constant :: Ring.C a => T a-constant = Cons one zero zero zero--{- |-eigenfunction of 'fourier'--}-unit :: Ring.C a => T a-unit = Cons one zero zero one--{-# INLINE evaluate #-}-evaluate :: (Trans.C a) =>-   T a -> a -> Complex.T a-evaluate f x =-   Complex.scale-     (sqrt (amp f))-     (Complex.exp $ Complex.scale (-pi) $-      c0 f + Complex.scale x (c1 f) + Complex.fromReal (c2 f * x^2))--evaluateSqRt :: (Trans.C a) =>-   T a -> a -> Complex.T a-evaluateSqRt f x0 =-   Complex.scale-     (sqrt (amp f))-     (let x = sqrt pi * x0-      in  Complex.exp $ negate $-          c0 f + Complex.scale x (c1 f) + Complex.fromReal (c2 f * x^2))--exponentPolynomial :: (Additive.C a) =>-   T a -> Poly.T (Complex.T a)-exponentPolynomial f =-   Poly.fromCoeffs [c0 f, c1 f, Complex.fromReal (c2 f)]---variance :: (Trans.C a) =>-   T a -> a-variance f =-   recip $ c2 f * 2*pi--multiply :: (Ring.C a) =>-   T a -> T a -> T a-multiply f g =-   Cons-      (amp f * amp g)-      (c0 f + c0 g) (c1 f + c1 g) (c2 f + c2 g)--powerRing :: (Trans.C a) =>-   Integer -> T a -> T a-powerRing p f =-   let pa = fromInteger p-   in  Cons-          (amp f ^ p)-          (pa * c0 f) (pa * c1 f) (fromInteger p * c2 f)--{--powerField does not makes sense,-since the reciprocal of a Gaussian diverges.--}--powerAlgebraic :: (Trans.C a) =>-   Rational -> T a -> T a-powerAlgebraic p f =-   let pa = fromRational' p-   in  Cons-          (amp f ^/ p)-          (pa * c0 f) (pa * c1 f) (fromRational' p * c2 f)--powerTranscendental :: (Trans.C a) =>-   a -> T a -> T a-powerTranscendental p f =-   Cons-      (amp f ^? p)-      (Complex.scale p $ c0 f) (Complex.scale p $ c1 f) (p * c2 f)---{--let x=Cons 2 (1+:3) (4+:5) (7::Rational); y=Cons 7 (1+:4) (3+:2) (5::Rational)--}-convolve :: (Field.C a) =>-   T a -> T a -> T a-convolve f g =-   let s = c2 f + c2 g-       {--       fd = f1/(2*f2)-       gd = g1/(2*g2)-       c = f2*g2/(f2+g2)--       c*(fd+gd) = (f1*g2+f2*g1)/(2*(f2+g2)) = b/2--       c*(fd+gd)^2 - fd^2*f2 - gd^2*g2-         = f2*g2*(fd+gd)^2/(f2 + g2) - (fd^2*f2 + gd^2*g2)-         = (f2*g2*(fd+gd)^2 - (f2+g2)*(fd^2*f2+gd^2*g2)) / (f2 + g2)-         = (2*f2*g2*fd*gd - (fd^2*f2^2+gd^2*g2^2)) / (f2 + g2)-         = (2*f1*g1 - (f1^2+g1^2)) / (4*(f2 + g2))-         = -(f1 - g1)^2/(4*(f2 + g2))-       -}-   in  Cons-          (amp f * amp g / s)-          (c0 f + c0 g-              - Complex.scale (recip (4*s)) ((c1 f - c1 g)^2))-          (Complex.scale (c2 g / s) (c1 f) +-           Complex.scale (c2 f / s) (c1 g))-          (c2 f * c2 g / s)-            -- recip $ recip (c2 f) + recip (c2 g)-{--   Cons-      (c0 f + c0 g) (c1 f + c1 g)-      (recip $ recip (c2 f) + recip (c2 g))--}--convolveByTranslation :: (Field.C a) =>-   T a -> T a -> T a-convolveByTranslation f0 g0 =-   let fd = Complex.scale (recip (2 * c2 f0)) $ c1 f0-       gd = Complex.scale (recip (2 * c2 g0)) $ c1 g0-       f1 = translateComplex fd f0-       g1 = translateComplex gd g0-       s = c2 f1 + c2 g1-   in  translateComplex (negate $ fd + gd) $-       Cons-          (amp f1 * amp g1 / s)-          (c0 f1 + c0 g1) zero-          (c2 f1 * c2 g1 / s)--convolveByFourier :: (Field.C a) =>-   T a -> T a -> T a-convolveByFourier f g =-   reverse $ fourier $ multiply (fourier f) (fourier g)--fourier :: (Field.C a) =>-   T a -> T a-fourier f =-   let a = c0 f-       b = c1 f-       rc = recip $ c2 f-   in  Cons-          (amp f * rc)-          (Complex.scale (rc/4) (-b^2) + a)-          (Complex.scale rc $ Complex.quarterRight b)-          rc--fourierByTranslation :: (Field.C a) =>-   T a -> T a-fourierByTranslation f =-   translateComplex (Complex.scale (1/2) $ Complex.quarterLeft $ c1 f) $-   Cons (amp f / c2 f) (c0 f) zero (recip $ c2 f)--{--a + b*x + c*x^2- = c*(a/c + b/c*x + x^2)- = c*((x-b/(2*c))^2 + (4*a*c+b^2)/(4*c^2))- = c*(x-b/(2*c))^2 + (4*a*c+b^2)/(4*c)--fourier ->-   x^2/c - i*b/c*x + (4*a*c+b^2)/(4*c)--fourier (x -> exp(-pi*c*(x-t)^2))- = fourier $ translate t $ shrink (sqrt c) $ x -> exp(-pi*x^2)- = modulate t $ dilate (sqrt c) $ fourier $ x -> exp(-pi*x^2)- = modulate t $ dilate (sqrt c) $ x -> exp(-pi*x^2)- = modulate t $ x -> exp(-pi*x^2/c)- = x -> exp(-pi*x^2/c) * exp(-2*pi*i*x*t)- = x -> exp(-pi*(x^2/c - 2*i*x*t))--}--{--b*x + c*x^2- = c*(b/c*x + x^2)- = c*((x-br/(2*c))^2 + i*x*bi/c - br^2/(4*c^2))- = c*(x-br/(2*c))^2 + i*x*bi - br^2/(4*c)--fourier ->-   (x+bi/2)^2/c - i*br/c*(x+bi/2) - br^2/(4*c)- = (1/c) * ((x+bi/2)^2 - i*br*(x+bi/2) + (br/2)^2)- = (1/c) * (x^2 - i*b*x + -(br/2)^2 + (bi/2)^2 - i*br*bi/2)- = (1/c) * (x^2 - i*b*x - (br^2-bi^2+2*br*bi*i)^2 /4)- = (1/c) * (x^2 - i*b*x - b^2 / 4)- = (1/c) * (x^2 - i*b*x + (i*b/2)^2)- = (1/c) * (x - i*b/2)^2--Example:-  (x-b)^2 = b^2 - 2*b*x + x^2-    ->  (- i*2*b*x + x^2)---fourier (x -> exp(-pi*(c*(x-t)^2 + 2*i*m*x)))- = fourier $ modulate m $ translate t $ shrink (sqrt c) $ x -> exp(-pi*x^2)- = translate (-m) $ modulate t $ dilate (sqrt c) $ fourier $ x -> exp(-pi*x^2)- = translate (-m) $ modulate t $ dilate (sqrt c) $ x -> exp(-pi*x^2)- = translate (-m) $ modulate t $ x -> exp(-pi*x^2/c)- = translate (-m) $ x -> exp(-pi*x^2/c) * exp(-2*pi*i*x*t)- = x -> exp(-pi*(x+m)^2/c) * exp(-2*pi*i*(x+m)*t)- = x -> exp(-pi*((x+m)^2/c - 2*i*(x+m)*t))--}--{--fourier (Cons a 0 0) =-  Cons a 0 infinity--fourier (Cons 0 0 c) =-  Cons 0 0 (recip c)--fourier (Cons 0 b 1) =-  Cons 0 (i*b) 1--}--translate :: Ring.C a => a -> T a -> T a-translate d f =-   let a = c0 f-       b = c1 f-       c = c2 f-   in  Cons-          (amp f)-          (Complex.fromReal (c*d^2) - Complex.scale d b + a)-          (Complex.fromReal (-2*c*d) + b)-          c--translateComplex :: Ring.C a => Complex.T a -> T a -> T a-translateComplex d f =-   let a = c0 f-       b = c1 f-       c = c2 f-   in  Cons-          (amp f)-          (Complex.scale c (d^2) - b*d + a)-          (Complex.scale (-2*c) d + b)-          c--modulate :: Ring.C a => a -> T a -> T a-modulate d f =-   Cons-      (amp f)-      (c0 f)-      (c1 f + (zero +: 2*d))-      (c2 f)--turn :: Ring.C a => a -> T a -> T a-turn d f =-   Cons-      (amp f)-      (c0 f + (zero +: 2*d))-      (c1 f)-      (c2 f)--reverse :: Additive.C a => T a -> T a-reverse f =-   f{c1 = negate $ c1 f}---dilate :: Field.C a => a -> T a -> T a-dilate k f =-   Cons-      (amp f)-      (c0 f)-      (Complex.scale (recip k) $ c1 f)-      (c2 f / k^2)--shrink :: Ring.C a => a -> T a -> T a-shrink k f =-   Cons-      (amp f)-      (c0 f)-      (Complex.scale k $ c1 f)-      (c2 f * k^2)--amplify :: (Ring.C a) => a -> T a -> T a-amplify k f =-   Cons-      (k^2 * amp f)-      (c0 f)-      (c1 f)-      (c2 f)---{- laws-fourier (convolve f g) = fourier f * fourier g--fourier (fourier f) = reverse f--}
− src-ghc-6.12/MathObj/Gaussian/Example.hs 
@@ -1,227 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{--Reciprocal of variance of a Gaussian bell curve.-We describe the curve only in terms of its variance-thus we represent a bell curve at the coordinate origin-neglecting its amplitude.--We could also define the amplitude as @root 4 c@,-thus preserving L2 norm being one,-but then @dilate@ and @shrink@ also include an amplification.--We could do some projective geometry in the exponent-in order to also have zero variance,-which corresponds to the dirac impulse.--}-module MathObj.Gaussian.Example where--import qualified MathObj.Gaussian.Polynomial as PolyBell-import qualified MathObj.Gaussian.Bell as Bell-import qualified MathObj.Gaussian.Variance as Var--import qualified MathObj.Polynomial as Poly--import qualified Algebra.Transcendental as Trans-import qualified Algebra.Algebraic      as Algebraic-import qualified Algebra.Field          as Field--- import qualified Algebra.Absolute           as Absolute-import qualified Algebra.Ring           as Ring--- import qualified Algebra.Additive       as Additive--import qualified Number.Complex as Complex--import Algebra.Transcendental (pi, )-import Algebra.Algebraic (root, )-import Algebra.Ring ((*), (^), )--import Number.Complex ((+:), )--import qualified Numerics.Function as Func-import qualified Numerics.Fourier as Fourier-import qualified Numerics.Integration as Integ-import qualified Numerics.Differentiation as Diff--import qualified Graphics.Gnuplot.Simple as GP--import Control.Applicative (liftA2, )---- import System.Exit (ExitCode, )---- import Prelude (($))-import NumericPrelude.Numeric-import NumericPrelude.Base-import qualified Prelude as P---curve0 :: Var.T Double-curve0 = curve0a--curve0a :: Var.T Double-curve0a = Var.Cons 1.4 3.3--curve0b :: Var.T Double-curve0b = Var.Cons 2.2 1.7--variance0 :: (Double, Double)-variance0 =-   (Var.variance curve0,-    (Integ.rectangular 1000 (-2,2) $ liftA2 (*) (^2) (Var.evaluate curve0)) /-    (Integ.rectangular 1000 (-2,2) $ Var.evaluate curve0))--norm10 :: (Double, Double)-norm10 =-   (Integ.rectangular 1000 (-2,2) $ Var.evaluate curve0,-    Var.norm1 curve0)--norm20 :: (Double, Double)-norm20 =-   (sqrt $ Integ.rectangular 1000 (-2,2) $ (^2) . Var.evaluate curve0,-    Var.norm2 curve0)--norm30 :: (Double, Double)-norm30 =-   (root 3 $ Integ.rectangular 1000 (-2,2) $ (^3) . Var.evaluate curve0,-    Var.normP 3 curve0)--fourier0 :: IO ()-fourier0 =-   GP.plotFuncs []-      (GP.linearScale 100 (-2,2))-      [Var.evaluate $ Var.fourier curve0,-       Fourier.analysisTransformOneReal 100 (-2,2) $ Var.evaluate curve0]--multiply0 :: IO ()-multiply0 =-   GP.plotFuncs []-      (GP.linearScale 100 (-1,1))-      [Var.evaluate $ Var.multiply curve0a curve0b,-       liftA2 (*) (Var.evaluate curve0a) (Var.evaluate curve0b)]--convolve0 :: IO ()-convolve0 =-   GP.plotFuncs []-      (GP.linearScale 100 (-2,2))-      [Var.evaluate $ Var.convolve curve0a curve0b,-       Integ.convolve 1000 (-3,3) (Var.evaluate curve0a) (Var.evaluate curve0b)]---curve1 :: Bell.T Double-curve1 = curve1a--curve1a :: Bell.T Double-curve1a = Bell.Cons 1.4 (0.1+:0.3) ((-0.2)+:1.4) 2.3--curve1b :: Bell.T Double-curve1b = Bell.Cons 2.2 ((-0.3)+:2.1) (0.2+:(-0.4)) 1.7--variance1 :: (Double, Double)-variance1 =-   (Bell.variance curve1,-    (Integ.rectangular 1000 (-2,2) $-        liftA2 (*) (^2)-           (Complex.magnitudeSqr .-            Func.translateRight-               (Complex.real (Bell.c1 curve1) / (2 * Bell.c2 curve1))-               (Bell.evaluate curve1))) /-    (Integ.rectangular 1000 (-2,2) $ Complex.magnitude . Bell.evaluate curve1))--{- the norm depends on too much things-norm0vs1 :: (Double, Double)-norm0vs1 =-   ((Integ.rectangular 1000 (-5,5) $ Var.evaluate curve0)-         * exp (- Complex.real (Bell.c0 curve1)),-    Integ.rectangular 1000 (-5,5) $ Complex.magnitude . Bell.evaluate curve1)--}--fourier1 :: IO ()-fourier1 =-   GP.plotFuncs []-      (GP.linearScale 100 (-5,5))-      [Complex.real . (Bell.evaluate $ Bell.fourier curve1),-       fourierAnalysisReal 100 (-2,2) $ Bell.evaluate curve1]---curve2 :: PolyBell.T Double-curve2 =-   PolyBell.Cons---      Bell.unit---      (Bell.Cons 1.4 (0.1+:0.3) 0 1.2)---      (Bell.Cons 1.4 (0.1+:0.3) ((-0.2)+:1.4) 1)-      curve1---      (Poly.fromCoeffs [one])---      (Poly.fromCoeffs [zero,one])---      (Poly.fromCoeffs [zero,zero,one])---      (Poly.fromCoeffs [0,Complex.imaginaryUnit])-      (Poly.fromCoeffs [1.4+:(-0.1),0.8+:(0.1),(-1.1)+:0.3])--differentiate2 :: IO ()-differentiate2 =-   GP.plotFuncs []-      (GP.linearScale 100 (-2,2))-      [Complex.real . (PolyBell.evaluateSqRt $ PolyBell.differentiate curve2),-       ((/ sqrt pi) . ) $ Diff.diff (1e-5) $ Complex.real . PolyBell.evaluateSqRt curve2]--fourier2 :: IO ()-fourier2 =-   GP.plotFuncs []-      (GP.linearScale 100 (-5,5))-      [Complex.real . (PolyBell.evaluateSqRt $ PolyBell.fourier curve2),-       fourierAnalysisReal 100 (-2,2) $ PolyBell.evaluateSqRt curve2]----fourierAnalysisReal ::-   (P.Floating a) =>-   Integer -> (a, a) -> (a -> Complex.T a) -> a -> a-fourierAnalysisReal n rng f =-   liftA2 (P.-)-      (Fourier.analysisTransformOneReal n rng (Complex.real . f))-      (Fourier.analysisTransformOneImag n rng (Complex.imag . f))---{- |-Try to approximate @\x -> exp (-x^2) * x@-by a difference of translated Gaussian bells.--exp(-x^2) * x-  ==  exp(-(a+b*x+c*x^2)) - exp(-(a-b*x+c*x^2))-  ==  exp(-(a+c*x^2)) * (exp(-b*x) - exp(b*x))-  ==  exp(-(a+c*x^2)) * 2*sinh (b*x)--It holds-  lim (\b x -> sinh (b*x) / b)  =  id--}-diffApprox :: IO ()-diffApprox =-   let amp = (2*b)^- (-2)-       a = 0-       {--       amp = 1-       a = log (2 * abs b)-       -}-       b = -0.1-       c = 1-       ac = Complex.fromReal a-       bc = Complex.fromReal b-   in  GP.plotFuncs []-          (GP.linearScale 100 (-2,2::Double))-          [Complex.real .-           (PolyBell.evaluateSqRt $-              PolyBell.Cons Bell.unit (Poly.fromCoeffs [zero,one])),-           Complex.real .-           liftA2 (-)-             (PolyBell.evaluateSqRt $-                PolyBell.Cons (Bell.Cons amp ac bc c) (Poly.fromCoeffs [one]))-             (PolyBell.evaluateSqRt $-                PolyBell.Cons (Bell.Cons amp ac (-bc) c) (Poly.fromCoeffs [one]))]---polyApprox :: IO ()-polyApprox =-   GP.plotFuncs []-      (GP.linearScale 100 (-2,2::Double))-      [Complex.real .-         PolyBell.evaluateSqRt curve2,-       Complex.real . sum .-         mapM (\(amp,b) -> \x -> amp * Bell.evaluateSqRt b x)-         (PolyBell.approximateByBells 0.1 curve2)]
− src-ghc-6.12/MathObj/Gaussian/Polynomial.hs 
@@ -1,435 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{--Complex Gaussian bell multiplied with a polynomial.--In order to make this free of @pi@ factors,-we have to choose @recip (sqrt pi)@-as unit for translations and modulations,-for linear factors and in the differentiation.--}-{--ToDo:--* In order to avoid the weird @sqrt pi@ factor,-  use a polynomial expression in @pi@.--* sum of multiple bells using Data.Map from exponent polynomial to coefficient polynomial-  use of Algebra object.--* Projective geometry in order to support Dirac impulse.--}-module MathObj.Gaussian.Polynomial where--import qualified MathObj.Gaussian.Bell as Bell--import qualified MathObj.LaurentPolynomial as LPoly-import qualified MathObj.Polynomial.Core   as PolyCore-import qualified MathObj.Polynomial        as Poly-import qualified Number.Complex     as Complex--import qualified Algebra.ZeroTestable   as ZeroTestable-import qualified Algebra.Differential   as Differential-import qualified Algebra.Transcendental as Trans-import qualified Algebra.Field          as Field-import qualified Algebra.Absolute           as Absolute-import qualified Algebra.Ring           as Ring-import qualified Algebra.Additive       as Additive--import qualified Data.Record.HT as Rec-import qualified Data.List as List-import Data.Function.HT (nest, )-import Data.Eq.HT (equating, )-import Data.List.HT (mapAdjacent, )-import Data.Tuple.HT (forcePair, )--import Test.QuickCheck (Arbitrary, arbitrary, )-import Control.Monad (liftM2, )--import NumericPrelude.Numeric-import NumericPrelude.Base hiding (reverse, )--- import Prelude ()---data T a = Cons {bell :: Bell.T a, polynomial :: Poly.T (Complex.T a)}-   deriving (Show)--instance (Absolute.C a, Eq a) => Eq (T a) where-   (==) = equal---{--Helper data type for 'equal',-that allows to call the (not quite trivial) polynomial equality check.-@RootProduct r a@ represents @sqrt r * a@.-The test using 'signum' works for real numbers,-and I do not know, whether it is correct for other mathematical objects.-However I cannot imagine other mathematical objects,-that make sense at all, here.-Maybe elements of a finite field.--}-data RootProduct a = RootProduct a a--instance (Absolute.C a, Eq a) => Eq (RootProduct a) where-   (RootProduct xr xa) == (RootProduct yr ya)  =-      let xp = xr*xa^2-          yp = yr*ya^2-      in  xp==yp &&-          (isZero xp || signum xa == signum ya)--instance (ZeroTestable.C a) => ZeroTestable.C (RootProduct a) where-   isZero (RootProduct r a) = isZero r || isZero a---{--The derived Eq is not correct.-We have to combine the amplitude of the bell with the polynomial,-respecting signs and the square root of the bell amplitude.--}-equal :: (Absolute.C a, Eq a) => T a -> T a -> Bool-equal x y =-   let bx = bell x-       by = bell y-       scaleSqr b =-          (\p ->-              (fmap (RootProduct (Bell.amp b) . Complex.real) p,-               fmap (RootProduct (Bell.amp b) . Complex.imag) p))-           . polynomial-   in  Rec.equal-          (equating Bell.c0 :-           equating Bell.c1 :-           equating Bell.c2 :-           [])-          bx by-       &&-       scaleSqr bx x == scaleSqr by y---instance (Absolute.C a, Arbitrary a) => Arbitrary (T a) where-   arbitrary =---      liftM2 Cons arbitrary arbitrary-      liftM2 Cons-         arbitrary-         -- we have to restrict the number of polynomial coefficients,-         -- since with the quadratic time algorithms like fourier and convolve,-         -- in connection with Rational slow down tests too much.-         (fmap (Poly.fromCoeffs . take 5 . Poly.coeffs) arbitrary)----{-# INLINE evaluateSqRt #-}-evaluateSqRt :: (Trans.C a) =>-   T a -> a -> Complex.T a-evaluateSqRt f x =-   Bell.evaluateSqRt (bell f) x *-   Poly.evaluate (polynomial f) (Complex.fromReal $ sqrt pi * x)-{- ToDo: evaluating a complex polynomial for a real argument can be optimized -}---constant :: (Ring.C a) => T a-constant =-   Cons Bell.constant (Poly.const one)--scale :: (Ring.C a) => a -> T a -> T a-scale x f =-   f{polynomial = fmap (Complex.scale x) $ polynomial f}--scaleComplex :: (Ring.C a) => Complex.T a -> T a -> T a-scaleComplex x f =-   f{polynomial = fmap (x*) $ polynomial f}---eigenfunction :: (Field.C a) => Int -> T a-eigenfunction =-   eigenfunctionDifferential--eigenfunction0 :: (Ring.C a) => T a-eigenfunction0 =-   Cons Bell.unit (Poly.fromCoeffs [one])--eigenfunction1 :: (Ring.C a) => T a-eigenfunction1 =-   Cons Bell.unit (Poly.fromCoeffs [zero, one])--eigenfunction2 :: (Field.C a) => T a-eigenfunction2 =-   Cons Bell.unit (Poly.fromCoeffs [-(1/4), zero, one])--eigenfunction3 :: (Field.C a) => T a-eigenfunction3 =-   Cons Bell.unit (Poly.fromCoeffs [zero, -(3/4), zero, one])---eigenfunctionDifferential :: (Field.C a) => Int -> T a-eigenfunctionDifferential n =-   (\f -> f{bell = Bell.unit}) $-   nest n (scale (-1/4) . differentiate) $-   Cons (Bell.Cons one zero zero 2) one--eigenfunctionIterative :: (Field.C a, Absolute.C a, Eq a) => Int -> T a-eigenfunctionIterative n =-   fst . head . dropWhile (uncurry (/=)) . mapAdjacent (,) $-   eigenfunctionIteration $-   Cons-      Bell.unit-      (Poly.fromCoeffs $ replicate n zero ++ [one])--eigenfunctionIteration :: (Field.C a) => T a -> [T a]-eigenfunctionIteration =-   iterate (\x ->-      let y = fourier x-          px = polynomial x-          py = polynomial y-          c = last (Poly.coeffs px) / last (Poly.coeffs py)-      in  y{polynomial = fmap (0.5*) (px + fmap (c*) py)})---multiply :: (Ring.C a) =>-   T a -> T a -> T a-multiply f g =-   Cons-      (Bell.multiply (bell f) (bell g))-      (polynomial f * polynomial g)--convolve, {- convolveByDifferentiation, -} convolveByFourier :: (Field.C a) =>-   T a -> T a -> T a-convolve = convolveByFourier--{--f <*> g =-   let (foff,fint) = integrate f-   in  fint <*> differentiate g + makeGaussPoly foff * g--In principle this would work,-but (makeGaussPoly foff * g) contains a lot of-convolutions of Gaussian with Gaussian-polynomial-product,-where the Gaussians have different parameters.--convolveByDifferentiation f g =-   case polynomial f of-      fpoly ->-         if null $ Poly.coeffs fpoly-           then ...-           else ...--}--convolveByFourier f g =-   reverse $ fourier $ multiply (fourier f) (fourier g)--{--We use a Horner like scheme-in order to translate multiplications with @id@-to differentations on the Fourier side.-Quadratic runtime.--fourier (Cons bell (Poly.const a + Poly.shift f))-  = fourier (Cons bell (Poly.const a)) + fourier (Cons bell (Poly.shift f))-  = fourier (Cons bell (Poly.const a)) + differentiate (fourier (Cons bell f))--}-fourier :: (Field.C a) =>-   T a -> T a-fourier f =-   foldr-      (\c p ->-          let q = differentiate p-          in  q{polynomial =-                   Poly.const c +-                   fmap (Complex.scale (1/2) . Complex.quarterLeft) (polynomial q)})-      (Cons (Bell.fourier $ bell f) zero) $-   Poly.coeffs $ polynomial f--{- |-Differentiate and divide by @sqrt pi@ in order to stay in a ring.-This way, we do not need to fiddle with pi factors.--}-differentiate :: (Ring.C a) => T a -> T a-differentiate f =-   f{polynomial =-        Differential.differentiate (polynomial f)-        - Differential.differentiate (Bell.exponentPolynomial (bell f))-           * polynomial f}--{--snd $ integrate $ differentiate (Cons Bell.unit (Poly.fromCoeffs [7,7,7,7]) :: T Double)--g = (bell f * poly f)'-  = bell f * ((poly f)' - (exppoly (bell f))' * poly f)-poly g = (poly f)' - (exppoly (bell f))' * poly f--Integration means we have g and ask for f.--poly f = ((poly f)' - poly g) / (exppoly (bell f))'--However must start with the highest term of 'poly f',-and thus we need to perform the division on reversed polynomials.--}-integrate ::-   (Field.C a, ZeroTestable.C a) =>-   T a -> (Complex.T a, T a)-integrate f =-   let fs = Poly.coeffs $ polynomial f-       (ys,~[r]) =-          PolyCore.divModRev-             {--             We need the shortening convention of 'zipWith'-             in order to limit the result list,-             we cannot use list instance for (-).-             -}-             (zipWith (-)-                (0 : 0 : diffRev ys)-                (List.reverse fs))-             (List.reverse $ Poly.coeffs $-              Differential.differentiate $-              Bell.exponentPolynomial $ bell f)-   in  forcePair $-       if null fs-         then (zero, f)-         else (r, f{polynomial = Poly.fromCoeffs $ List.reverse ys})--diffRev :: Ring.C a => [a] -> [a]-diffRev xs =-   zipWith (*) xs-      (drop 1 (iterate (subtract 1) (fromIntegral $ length xs)))--translate :: Ring.C a => a -> T a -> T a-translate d =-   translateComplex (Complex.fromReal d)--translateComplex :: Ring.C a => Complex.T a -> T a -> T a-translateComplex d f =-   Cons-      (Bell.translateComplex d $ bell f)-      (Poly.translate d $ polynomial f)--modulate :: Ring.C a => a -> T a -> T a-modulate d f =-   Cons-      (Bell.modulate d $ bell f)-      (polynomial f)--turn :: Ring.C a => a -> T a -> T a-turn d f =-   Cons-      (Bell.turn d $ bell f)-      (polynomial f)--reverse :: Additive.C a => T a -> T a-reverse f =-   Cons-      (Bell.reverse $ bell f)-      (Poly.reverse $ polynomial f)--dilate :: Field.C a => a -> T a -> T a-dilate k f =-   Cons-      (Bell.dilate k $ bell f)-      (Poly.dilate (Complex.fromReal k) $ polynomial f)--shrink :: Ring.C a => a -> T a -> T a-shrink k f =-   Cons-      (Bell.shrink k $ bell f)-      (Poly.shrink (Complex.fromReal k) $ polynomial f)--{--We could also amplify the polynomial coefficients.--}-amplify :: Ring.C a => a -> T a -> T a-amplify k f =-   Cons-      (Bell.amplify k $ bell f)-      (polynomial f)---{- |-Approximate a @T a@ using a linear combination of translated @Bell.T a@.-The smaller the unit (e.g. 0.1, 0.01, 0.001)-the better the approximation but the worse the numeric properties.--We cannot put all information into @amp@ of @Bell@,-since @amp@ must be real, but is complex here by construction.-We really need at least signed amplitudes at this place,-since we want to represent differences of Gaussians.--}-approximateByBells ::-   Field.C a =>-   a -> T a -> [(Complex.T a, Bell.T a)]-approximateByBells unit f =-   let b = bell f-       amps =-          -- approximateByBellsByTranslation-          approximateByBellsAtOnce-             unit-             (Complex.scale (recip (2 * Bell.c2 b)) (Bell.c1 b))-             (recip (2*unit*Bell.c2 b))-             (polynomial f)-   in  zip (LPoly.coeffs amps) $-       map-          (\d -> Bell.translate d b)-          (laurentAbscissas (unit/2) amps)--approximateByBellsAtOnce ::-   Field.C a =>-   a -> Complex.T a -> a -> Poly.T (Complex.T a) -> LPoly.T (Complex.T a)-approximateByBellsAtOnce unit d s p =-   foldr-      (\x amps0 ->-         {--         Decompose (bell t * (t-d)) = bell t * t - bell t * d-         -}-         let y = fmap (Complex.scale s) amps0-         in  -- \t -> bell t * t-             --    ~   (translate unit bell - translate (-unit) bell) / unit-             LPoly.shift 1 y --             LPoly.shift (-1) y +-             -- bell t * d-             zipWithAbscissas-                (\t z -> (Complex.fromReal t - d) * z)-                (unit/2) amps0 +-             LPoly.const x)-      (LPoly.fromCoeffs [])-      (Poly.coeffs p)--approximateByBellsByTranslation ::-   Field.C a =>-   a -> Complex.T a -> a -> Poly.T (Complex.T a) -> LPoly.T (Complex.T a)-approximateByBellsByTranslation unit d s p =-   foldr-      (\x amps0 ->-         {--         Decompose (bell t * (t-d)) = bell t * t - bell t * d-         -}-         let y = fmap (Complex.scale s) amps0-         in  -- \t -> bell t * t-             --    ~   (translate unit bell - translate (-unit) bell) / unit-             LPoly.shift 1 y --             LPoly.shift (-1) y +-             -- bell t * d-             zipWithAbscissas Complex.scale (unit/2) amps0 +-             LPoly.const x)-      (LPoly.fromCoeffs [])-      (Poly.coeffs $ Poly.translate d p)--zipWithAbscissas ::-   (Ring.C a) =>-   (a -> b -> c) -> a -> LPoly.T b -> LPoly.T c-zipWithAbscissas h unit y =-   LPoly.fromShiftCoeffs (LPoly.expon y) $-   zipWith h-      (laurentAbscissas unit y)-      (LPoly.coeffs y)--laurentAbscissas :: Ring.C a => a -> LPoly.T c -> [a]-laurentAbscissas unit =-   map (\d -> fromIntegral d * unit) .-   iterate (1+) . LPoly.expon---{- No Ring instance for Gaussians-instance (Ring.C a) => Differential.C (T a) where-   differentiate = differentiate--}--{- laws-differentiate (f*g) =-   (differentiate f) * g + f * (differentiate g)--}
− src-ghc-6.12/MathObj/Gaussian/Variance.hs 
@@ -1,194 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{--We represent a Gaussian bell curve in terms of the reciprocal of its variance-and its value at the origin.--We could do some projective geometry in the exponent-in order to also have zero variance,-which corresponds to the dirac impulse.--}-module MathObj.Gaussian.Variance where--import qualified MathObj.Polynomial as Poly-import qualified Number.Root as Root--import qualified Algebra.Transcendental as Trans-import qualified Algebra.Algebraic      as Algebraic-import qualified Algebra.Field          as Field-import qualified Algebra.Absolute       as Absolute-import qualified Algebra.Ring           as Ring-import qualified Algebra.Additive       as Additive--{--import Algebra.Transcendental (pi, )-import Algebra.Ring ((*), (^), )-import Algebra.Additive ((+))--}-import Test.QuickCheck (Arbitrary, arbitrary, )-import Control.Monad (liftM2, )---- import Prelude (($))-import NumericPrelude.Numeric-import NumericPrelude.Base---{- |-Since @amp@ is the square of the actual amplitude it must be non-negative.--}-data T a = Cons {amp, c :: a}-   deriving (Eq, Show)--instance (Absolute.C a, Arbitrary a) => Arbitrary (T a) where-   arbitrary =-      liftM2 Cons-         (fmap abs arbitrary)-         (fmap ((1+) . abs) arbitrary)---constant :: Ring.C a => T a-constant = Cons one zero--{-# INLINE evaluate #-}-evaluate :: (Trans.C a) =>-   T a -> a -> a-evaluate f x =-   sqrt (amp f) * exp (-pi * c f * x^2)--exponentPolynomial :: (Additive.C a) =>-   T a -> Poly.T a-exponentPolynomial f =-   Poly.fromCoeffs [zero, zero, c f]---integrateRoot :: (Field.C a) => T a -> Root.T a-integrateRoot f =-   Root.sqrt $ Root.fromNumber $ amp f / c f--scalarProductRoot :: (Field.C a) => T a -> T a -> Root.T a-scalarProductRoot f g =-   integrateRoot (multiply f g)---norm1Root :: (Field.C a) => T a -> Root.T a-norm1Root = integrateRoot--norm2Root :: (Field.C a) => T a -> Root.T a-norm2Root f =-   Root.sqrt $-      Root.fromNumber (amp f)-      `Root.div`-      Root.sqrt (Root.fromNumber $ 2 * c f)--normInfRoot :: (Field.C a) => T a -> Root.T a-normInfRoot f =-   Root.sqrt $ Root.fromNumber $ amp f--normPRoot :: (Field.C a) => Rational -> T a -> Root.T a-normPRoot p f =-   Root.sqrt (Root.fromNumber (amp f))-   `Root.div`-   Root.rationalPower (recip (2*p)) (Root.fromNumber (fromRational' p * c f))---norm1 :: (Algebraic.C a) => T a -> a-norm1 f =-   sqrt $ amp f / c f--norm2 :: (Algebraic.C a) => T a -> a-norm2 f =-   sqrt $ amp f / (sqrt $ 2 * c f)--normInf :: (Algebraic.C a) => T a -> a-normInf f =-   sqrt (amp f)--normP :: (Trans.C a) => a -> T a -> a-normP p f =-   sqrt (amp f) * (p * c f) ^? (- recip (2*p))---variance :: (Trans.C a) =>-   T a -> a-variance f =-   recip $ c f * 2*pi--multiply :: (Ring.C a) =>-   T a -> T a -> T a-multiply f g =-   Cons (amp f * amp g) (c f + c g)--powerRing :: (Trans.C a) =>-   Integer -> T a -> T a-powerRing p f =-   Cons (amp f ^ p) (fromInteger p * c f)--{--powerField does not makes sense,-since the reciprocal of a Gaussian diverges.--}--powerAlgebraic :: (Trans.C a) =>-   Rational -> T a -> T a-powerAlgebraic p f =-   Cons (amp f ^/ p) (fromRational' p * c f)--powerTranscendental :: (Trans.C a) =>-   a -> T a -> T a-powerTranscendental p f =-   Cons (amp f ^? p) (p * c f)--{- |-> convolve x y t =->    integrate $ \s -> x s * y(t-s)--Convergence only for @c f + c g > 0@.--}-convolve :: (Field.C a) =>-   T a -> T a -> T a-convolve f g =-   let s = c f + c g-   in  Cons-          (amp f * amp g / s)-          (c f * c g / s)--{- |-> fourier x f =->    integrate $ \t -> x t * cis (-2*pi*t*f)--Convergence only for @c f > 0@.--}-fourier :: (Field.C a) =>-   T a -> T a-fourier f =-   Cons (amp f / c f) (recip $ c f)-{--fourier (t -> exp(-(a*t)^2))--}--dilate :: (Field.C a) => a -> T a -> T a-dilate k f =-   Cons (amp f) $ c f / k^2--shrink :: (Ring.C a) => a -> T a -> T a-shrink k f =-   Cons (amp f) $ c f * k^2--{- |-@amplify k@ scales by @abs k@!--}-amplify :: (Ring.C a) => a -> T a -> T a-amplify k f =-   Cons (k^2 * amp f) $ c f---{- laws-fourier (convolve f g) = multiply (fourier f) (fourier g)--dilate k (dilate m f) = dilate (k*m) f--dilate k (shrink k f) = f--variance (dilate k f) = k^2 * variance f--variance (convolve f g) = variance f + variance g--}
− src-ghc-6.12/MathObj/LaurentPolynomial.hs 
@@ -1,288 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-{- |-Copyright   :  (c) Henning Thielemann 2004-2006--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  requires multi-parameter type classes--Polynomials with negative and positive exponents.--}-module MathObj.LaurentPolynomial where--import qualified MathObj.Polynomial  as Poly-import qualified MathObj.PowerSeries as PS-import qualified MathObj.PowerSeries.Core as PSCore--import qualified Algebra.VectorSpace  as VectorSpace-import qualified Algebra.Module       as Module-import qualified Algebra.Vector       as Vector-import qualified Algebra.Field        as Field-import qualified Algebra.Ring         as Ring-import qualified Algebra.Additive     as Additive-import qualified Algebra.ZeroTestable as ZeroTestable--import qualified Number.Complex as Complex--import Algebra.Module((*>))--import qualified NumericPrelude.Base as P-import qualified NumericPrelude.Numeric as NP--import NumericPrelude.Base    hiding (const, reverse, )-import NumericPrelude.Numeric hiding (div, negate, )--import qualified Data.List as List-import Data.List.HT (mapAdjacent)---{- | Polynomial including negative exponents -}--data T a = Cons {expon :: Int, coeffs :: [a]}---{- * Basic Operations -}--const :: a -> T a-const x = fromCoeffs [x]--(!) :: Additive.C a => T a -> Int -> a-(!) (Cons xt x) n =-   if n<xt-     then zero-     else head (drop (n-xt) x ++ [zero])--fromCoeffs :: [a] -> T a-fromCoeffs = fromShiftCoeffs 0--fromShiftCoeffs :: Int -> [a] -> T a-fromShiftCoeffs = Cons--fromPolynomial :: Poly.T a -> T a-fromPolynomial = fromCoeffs . Poly.coeffs--fromPowerSeries :: PS.T a -> T a-fromPowerSeries = fromCoeffs . PS.coeffs--bounds :: T a -> (Int, Int)-bounds (Cons xt x) = (xt, xt + length x - 1)--shift :: Int -> T a -> T a-shift t (Cons xt x) = Cons (xt+t) x--{-# DEPRECATED translate "In order to avoid confusion with Polynomial.translate, use 'shift' instead" #-}-translate :: Int -> T a -> T a-translate = shift---instance Functor T where-  fmap f (Cons xt xs) = Cons xt (map f xs)---{- * Show -}--appPrec :: Int-appPrec  = 10--instance (Show a) => Show (T a) where-  showsPrec p (Cons xt xs) =-    showParen (p >= appPrec)-       (showString "LaurentPolynomial.Cons " . shows xt .-        showString " " . shows xs)--{- * Additive -}--add :: Additive.C a => T a -> T a -> T a-add (Cons _ [])  y          = y-add  x          (Cons _ []) = x-add (Cons xt x) (Cons yt y) =-   if xt < yt-     then Cons xt (addShifted (yt-xt) x y)-     else Cons yt (addShifted (xt-yt) y x)--{--Compute the value of a series of Laurent polynomials.--Requires that the start exponents constitute a (weakly) rising sequence,-where each exponent occurs only finitely often.--Cf. Synthesizer.Cut.arrange--}-series :: (Additive.C a) => [T a] -> T a-series [] = fromCoeffs []-series ps =-   let es = map expon  ps-       xs = map coeffs ps-       ds = mapAdjacent subtract es-   in  Cons (head es) (addShiftedMany ds xs)--{- |-Add lists of numbers respecting a relative shift between the starts of the lists.-The shifts must be non-negative.-The list of relative shifts is one element shorter-than the list of summands.-Infinitely many summands are permitted,-provided that runs of zero shifts are all finite.---We could add the lists either with 'foldl' or with 'foldr',-'foldl' would be straightforward, but more time consuming (quadratic time)-whereas foldr is not so obvious but needs only linear time.--(stars denote the coefficients,- frames denote what is contained in the interim results)-'foldl' sums this way:--> | | | *******************************-> | | +---------------------------------> | |          ************************-> | +-----------------------------------> |                        ************-> +--------------------------------------I.e. 'foldl' would use much time find the time differences-by successive subtraction 1.--'foldr' mixes this way:-->     +--------------------------------->     | *******************************->     |      +-------------------------->     |      | ************************->     |      |           +-------------->     |      |           | ************---}-addShiftedMany :: (Additive.C a) => [Int] -> [[a]] -> [a]-addShiftedMany ds xss =-   foldr (uncurry addShifted) [] (zip (ds++[0]) xss)----addShifted :: Additive.C a => Int -> [a] -> [a] -> [a]-addShifted del px py =-   let recurse 0 x      = PSCore.add x py-       recurse d []     = replicate d zero ++ py-       recurse d (x:xs) = x : recurse (d-1) xs-   in  if del >= 0-         then recurse del px-         else error "LaurentPolynomial.addShifted: negative shift"---negate :: Additive.C a => T a -> T a-negate (Cons xt x) = Cons xt (map NP.negate x)--sub :: Additive.C a => T a -> T a -> T a-sub x y = add x (negate y)--instance Additive.C a => Additive.C (T a) where-   zero   = fromCoeffs []-   (+)    = add-   (-)    = sub-   negate = negate---{- * Module -}--instance Vector.C T where-   zero  = zero-   (<+>) = (+)-   (*>)  = Vector.functorScale--instance (Module.C a b) => Module.C a (T b) where-    x *> Cons yt ys = Cons yt (x *> ys)--instance (Field.C a, Module.C a b) => VectorSpace.C a (T b)---{- * Ring -}--mul :: Ring.C a => T a -> T a -> T a-mul (Cons xt x) (Cons yt y) = Cons (xt+yt) (PSCore.mul x y)--instance (Ring.C a) => Ring.C (T a) where-    one           = const one-    fromInteger n = const (fromInteger n)-    (*)           = mul---{- * Field.C -}--div :: (Field.C a, ZeroTestable.C a) => T a -> T a -> T a-div (Cons xt xs) (Cons yt ys) =-   let (xzero,x) = span isZero xs-       (yzero,y) = span isZero ys-   in  Cons (xt - yt + length xzero - length yzero)-            (PSCore.divide x y)--instance (Field.C a, ZeroTestable.C a) => Field.C (T a) where-   (/) = div--divExample :: T NP.Rational-divExample = div (Cons 1 [0,0,1,2,1]) (Cons 1 [0,0,0,1,1])-----{- * Comparisons -}--{- |-Two polynomials may be stored differently.-This function checks whether two values of type @LaurentPolynomial@-actually represent the same polynomial.--}-equivalent :: (Eq a, ZeroTestable.C a) => T a -> T a -> Bool-equivalent xs ys =-   let (Cons xt x, Cons yt y) =-          if expon xs <= expon ys-            then (xs,ys)-            else (ys,xs)-       (xPref, xSuf) = splitAt (yt-xt) x-       aux (a:as) (b:bs) = a == b && aux as bs-       aux []     bs     = all isZero bs-       aux as     []     = all isZero as-   in  all isZero xPref  &&  aux xSuf y--instance (Eq a, ZeroTestable.C a) => Eq (T a) where-   (==) = equivalent---identical :: (Eq a) => T a -> T a -> Bool-identical (Cons xt xs) (Cons yt ys) =-   xt==yt && xs == ys---{- |-Check whether a Laurent polynomial has only the absolute term,-that is, it represents the constant polynomial.--}-isAbsolute :: (ZeroTestable.C a) => T a -> Bool-isAbsolute (Cons xt x) =-   and (zipWith (\i xi -> isZero xi || i==0) [xt..] x)----{- * Transformations of arguments -}--{- | p(z) -> p(-z) -}-alternate :: Additive.C a => T a -> T a-alternate (Cons xt x) =-   Cons xt (zipWith id (drop (mod xt 2) (cycle [id,NP.negate])) x)--{- | p(z) -> p(1\/z) -}-reverse :: T a -> T a-reverse (Cons xt x) =-   Cons (1 - xt - length x) (List.reverse x)--{- |-p(exp(i·x)) -> conjugate(p(exp(i·x)))--If you interpret @(p*)@ as a linear operator on the space of Laurent polynomials,-then @(adjoint p *)@ is the adjoint operator.--}-adjoint :: Additive.C a => T (Complex.T a) -> T (Complex.T a)-adjoint x =-   let (Cons yt y) = reverse x-   in  (Cons yt (map Complex.conjugate y))
− src-ghc-6.12/MathObj/Matrix.hs 
@@ -1,278 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-{- |-Copyright    :   (c) Henning Thielemann 2009, Mikael Johansson 2006-Maintainer   :   numericprelude@henning-thielemann.de-Stability    :   provisional-Portability  :   requires multi-parameter type classes--Routines and abstractions for Matrices and-basic linear algebra over fields or rings.--We stick to simple Int indices.-Although advanced indices would be nice-e.g. for matrices with sub-matrices,-this is not easily implemented since arrays-do only support a lower and an upper bound-but no additional parameters.--ToDo:- - Matrix inverse, determinant--}--module MathObj.Matrix (-   T, Dimension,-   format,-   transpose,-   rows,-   columns,-   index,-   fromRows,-   fromColumns,-   fromList,-   dimension,-   numRows,-   numColumns,-   zipWith,-   zero,-   one,-   diagonal,-   scale,-   random,-   randomR,-   ) where--import qualified Algebra.Module   as Module-import qualified Algebra.Vector   as Vector-import qualified Algebra.Ring     as Ring-import qualified Algebra.Additive as Additive--import Algebra.Module((*>), )-import Algebra.Ring((*), fromInteger, scalarProduct, )-import Algebra.Additive((+), (-), subtract, )--import qualified System.Random as Rnd-import Data.Array (Array, array, listArray, accumArray, elems, bounds, (!), ixmap, range, )-import qualified Data.List as List--import Control.Monad (liftM2, )-import Control.Exception (assert, )--import Data.Function.HT (powerAssociative, )-import Data.Tuple.HT (swap, mapFst, )-import Data.List.HT (outerProduct, )-import Text.Show.HT (concatS, )--import NumericPrelude.Numeric (Int, )-import NumericPrelude.Base hiding (zipWith, )---{- |-A matrix is a twodimensional array, indexed by integers.--}-data T a =-   Cons (Array (Dimension, Dimension) a)-      deriving (Eq,Ord,Read)--type Dimension = Int--{- |-Transposition of matrices is just transposition in the sense of Data.List.--}-transpose :: T a -> T a-transpose (Cons m) =-   let (lower,upper) = bounds m-   in  Cons (ixmap (swap lower, swap upper) swap m)--rows :: T a -> [[a]]-rows mM@(Cons m) =-   let ((lr,lc), (ur,uc)) = bounds m-   in  outerProduct (index mM) (range (lr,ur)) (range (lc,uc))--columns :: T a -> [[a]]-columns mM@(Cons m) =-   let ((lr,lc), (ur,uc)) = bounds m-   in  outerProduct (flip (index mM)) (range (lc,uc)) (range (lr,ur))--index :: T a -> Dimension -> Dimension -> a-index (Cons m) i j = m ! (i,j)--fromRows :: Dimension -> Dimension -> [[a]] -> T a-fromRows m n =-   Cons .-   array (indexBounds m n) .-   concat .-   List.zipWith (\r -> map (\(c,x) -> ((r,c),x))) allIndices .-   map (zip allIndices)--fromColumns :: Dimension -> Dimension -> [[a]] -> T a-fromColumns m n =-   Cons .-   array (indexBounds m n) .-   concat .-   List.zipWith (\r -> map (\(c,x) -> ((c,r),x))) allIndices .-   map (zip allIndices)--fromList :: Dimension -> Dimension -> [a] -> T a-fromList m n xs = Cons (listArray (indexBounds m n) xs)--appPrec :: Int-appPrec = 10--instance (Show a) => Show (T a) where-   showsPrec p m =-      showParen (p >= appPrec)-         (showString "Matrix.fromRows " . showsPrec appPrec (rows m))--format :: (Show a) => T a -> String-format m = formatS m ""--formatS :: (Show a) => T a -> ShowS-formatS =-   concatS .-   map (\r -> showString "(" . concatS r . showString ")\n") .-   map (List.intersperse (' ':) . map (showsPrec 11)) .-   rows--dimension :: T a -> (Dimension,Dimension)-dimension (Cons m) = uncurry subtract (bounds m) + (1,1)--numRows :: T a -> Dimension-numRows = fst . dimension--numColumns :: T a -> Dimension-numColumns = snd . dimension---- These implementations may benefit from a better exception than--- just assertions to validate dimensionalities-instance (Additive.C a) => Additive.C (T a) where-   (+) = zipWith (+)-   (-) = zipWith (-)-   zero = zero 1 1--zipWith :: (a -> b -> c) -> T a -> T b -> T c-zipWith op mM@(Cons m) nM@(Cons n) =-   let d = dimension mM-       em = elems m-       en = elems n-   in  assert (d == dimension nM) $-         uncurry fromList d (List.zipWith op em en)--zero :: (Additive.C a) => Dimension -> Dimension -> T a-zero m n =-   fromList m n $-   List.repeat Additive.zero---    List.replicate (fromInteger (m*n)) zero--one :: (Ring.C a) => Dimension -> T a-one n =-   Cons $-   accumArray (flip const) Additive.zero-      (indexBounds n n)-      (map (\i -> ((i,i), Ring.one)) (indexRange n))--diagonal :: (Additive.C a) => [a] -> T a-diagonal xs =-   let n = List.length xs-   in  Cons $-       accumArray (flip const) Additive.zero-          (indexBounds n n)-          (zip (map (\i -> (i,i)) allIndices) xs)--scale :: (Ring.C a) => a -> T a -> T a-scale s = Vector.functorScale s--instance (Ring.C a) => Ring.C (T a) where-   mM * nM =-      assert (numColumns mM == numRows nM) $-      fromList (numRows mM) (numColumns nM) $-      liftM2 scalarProduct (rows mM) (columns nM)-   fromInteger n = fromList 1 1 [fromInteger n]-   mM ^ n =-      assert (numColumns mM == numRows mM) $-      assert (n >= Additive.zero) $-      powerAssociative (*) (one (numColumns mM)) mM n--instance Functor T where-   fmap f (Cons m) = Cons (fmap f m)--instance Vector.C T where-   zero  = Additive.zero-   (<+>) = (+)-   (*>)  = scale--instance Module.C a b => Module.C a (T b) where-   x *> m = fmap (x*>) m---random :: (Rnd.RandomGen g, Rnd.Random a) =>-   Dimension -> Dimension -> g -> (T a, g)-random =-   randomAux Rnd.random--randomR :: (Rnd.RandomGen g, Rnd.Random a) =>-   Dimension -> Dimension -> (a,a) -> g -> (T a, g)-randomR m n rng =-   randomAux (Rnd.randomR rng) m n--{--could be made nicer with the State monad,-but I like to keep dependencies minimal--}-randomAux :: (Rnd.RandomGen g, Rnd.Random a) =>-   (g -> (a,g)) -> Dimension -> Dimension -> g -> (T a, g)-randomAux rnd m n g0 =-   mapFst (fromList m n) $ swap $-   List.mapAccumL (\g _i -> swap $ rnd g) g0 (indexRange (m*n))--{--What more do we need from our matrix type? We have addition,-subtraction and multiplication, and thus composition of generic-free-module-maps. We're going to want to solve linear equations with-or without fields underneath, so we're going to want an implementation-of the Gaussian algorithm as well as most probably Smith normal-form. Determinants are cool, and these are to be calculated either-with the Gaussian algorithm or some other goodish method.--}--{--{- |- We'll want generic linear equation solving, returning one solution,-any solution really, or nothing. Basically, this is asking for the-preimage of a given vector over the given map, so--a_11 x_1 + .. + a_1n x_n = y_1- ...-a_m1 x_1 + .. + a_mn a_n = y_m--has really x_1,...,x_n as a preimage of the vector y_1,..,y_m under-the map (a_ij), since obviously y_1,..,y_m = (a_ij) x_1,..,x_n--So, generic linear equation solving boils down to the function--}-preimage :: (Ring.C a) => T a -> T a -> Maybe (T a)-preimage a y = assert-        (numRows a == numRows y &&     -- they match-         numColumns y == 1)               -- and y is a column vector-                Nothing--}--{--Cf. /usr/lib/hugs/demos/Matrix.hs--}----- these functions control whether we use 0 or 1 based indices--indexRange :: Dimension -> [Dimension]-indexRange n = [0..(n-1)]--indexBounds ::-   Dimension -> Dimension ->-   ((Dimension,Dimension), (Dimension,Dimension))-indexBounds m n =-   ((0,0), (m-1,n-1))--allIndices :: [Dimension]-allIndices = [0..]
− src-ghc-6.12/MathObj/Monoid.hs 
@@ -1,56 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module MathObj.Monoid where--import qualified Algebra.PrincipalIdealDomain as PID--import Algebra.PrincipalIdealDomain (gcd, lcm, )-import Algebra.Additive (zero, )-import Algebra.Monoid (C, idt, (<*>), )--import NumericPrelude.Base--{- |-It is only a monoid for non-negative numbers.--> idt <*> GCD (-2) = GCD 2--Thus, use this Monoid only for non-negative numbers!--}-newtype GCD a = GCD {runGCD :: a}-   deriving (Show, Eq)--instance PID.C a => C (GCD a) where-   idt = GCD zero-   (GCD x) <*> (GCD y) = GCD (gcd x y)---newtype LCM a = LCM {runLCM :: a}-   deriving (Show, Eq)--instance PID.C a => C (LCM a) where-   idt = LCM zero-   (LCM x) <*> (LCM y) = LCM (lcm x y)---{- |-@Nothing@ is the largest element.--}-newtype Min a = Min {runMin :: Maybe a}-   deriving (Show, Eq)--instance Ord a => C (Min a) where-   idt = Min Nothing-   (Min x) <*> (Min y) = Min $-      maybe y (\x' -> maybe x (Just . min x') y) x---{- |-@Nothing@ is the smallest element.--}-newtype Max a = Max {runMax :: Maybe a}-   deriving (Show, Eq)--instance Ord a => C (Max a) where-   idt = Max Nothing-   (Max x) <*> (Max y) = Max $-      maybe y (\x' -> maybe x (Just . max x') y) x
− src-ghc-6.12/MathObj/PartialFraction.hs 
@@ -1,399 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Copyright    :   (c) Henning Thielemann 2007-Maintainer   :   numericprelude@henning-thielemann.de-Stability    :   provisional-Portability  :   portable--Implementation of partial fractions.-Useful e.g. for fractions of integers and fractions of polynomials.--For the considered ring the prime factorization must be unique.--}--module MathObj.PartialFraction where--import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.IntegralDomain       as Integral-import qualified Number.Ratio                 as Ratio-import qualified Algebra.Ring                 as Ring-import qualified Algebra.Additive             as Additive-import qualified Algebra.ZeroTestable         as ZeroTestable-import qualified Algebra.Indexable            as Indexable--import Number.Ratio((%))-import Algebra.IntegralDomain(divMod, divModZero, decomposeVarPositionalInf)-import Algebra.Units(stdAssociate, stdUnitInv)-import Algebra.Field((/))-import Algebra.Ring((*), one, product)-import Algebra.Additive((+), zero, negate)-import Algebra.ZeroTestable (isZero)--import qualified Data.List as List--import Data.Map(Map)-import qualified Data.Map as Map-import Data.Maybe(fromMaybe, )-import qualified Data.List.Match as Match-import Data.List.HT (dropWhileRev, )-import Data.List (group, sortBy, mapAccumR, )--import NumericPrelude.Base hiding (zipWith)--import NumericPrelude.Numeric(Int, fromInteger)----{- |-@Cons z (indexMapFromList [(x0,[y00,y01]), (x1,[y10]), (x2,[y20,y21,y22])])@-represents the partial fraction-@z + y00/x0 + y01/x0^2 + y10/x1 + y20/x2 + y21/x2^2 + y22/x2^3@-The denominators @x0, x1, x2, ...@ must be irreducible,-but we can't check this in general.-It is also not enough to have relatively prime denominators,-because when adding two partial fraction representations-there might concur denominators that have non-trivial common divisors.--}-data T a =-   Cons a (Map (Indexable.ToOrd a) [a])-      deriving (Eq)--{- |-Unchecked construction.--}-fromFractionSum :: (Indexable.C a) => a -> [(a,[a])] -> T a-fromFractionSum z m =-   Cons z (indexMapFromList m)--toFractionSum :: (Indexable.C a) => T a -> (a, [(a,[a])])-toFractionSum (Cons z m) =-   (z, indexMapToList m)--appPrec :: Int-appPrec  = 10--instance (Show a) => Show (T a) where-  showsPrec p (Cons z m) =-    showParen (p >= appPrec)-       (showString "PartialFraction.fromFractionSum " .-        showsPrec (succ appPrec) z . showString " " .-        shows (indexMapToList m))---toFraction :: PID.C a => T a -> Ratio.T a-toFraction (Cons z m) =-   let fracs = map (uncurry multiToFraction) (indexMapToList m)-   in  foldl (+) (Ratio.fromValue z) fracs--{- |-'PrincipalIdealDomain.C' is not really necessary here and-only due to invokation of 'toFraction'.--}-toFactoredFraction :: (PID.C a) => T a -> ([a], a)-toFactoredFraction x@(Cons _ m) =-   let r = toFraction x-       denoms = concat $ Map.elems $ indexMapMapWithKey (flip Match.replicate) m-       numer = foldl (flip Ratio.scale) r denoms-       {- From the theory it must be Ratio.denominator denom==1.-          We could check this dynamically, if there would be an Eq instance.-          We could omit this completely,-          if we would reimplement Ratio addition. -}-   in  (denoms, Ratio.numerator numer)--{- |-'PrincipalIdealDomain.C' is not really necessary here and-only due to invokation of 'Ratio.%'.--}-multiToFraction :: PID.C a => a -> [a] -> Ratio.T a-multiToFraction denom =-   foldr (\numer acc ->-            (Ratio.fromValue numer + acc) / Ratio.fromValue denom) zero--hornerRev :: Ring.C a => a -> [a] -> a-hornerRev x = foldl (\val c -> val*x+c) zero---{- |-@fromFactoredFraction x y@-computes the partial fraction representation of @y % product x@,-where the elements of @x@ must be irreducible.-The function transforms the factors into their standard form-with respect to unit factors.--There are more direct methods for special cases-like polynomials over rational numbers-where the denominators are linear factors.--}-fromFactoredFraction :: (PID.C a, Indexable.C a) => [a] -> a -> T a-fromFactoredFraction denoms0 numer0 =-   let denoms = group $ sortBy Indexable.compare $ map stdAssociate denoms0-       numer  = foldl (*) numer0 $ map stdUnitInv denoms0-       denomPowers = map product denoms-          {- since the sub-lists contain the same value,-             the products are powers,-             which could be computed more efficiently -}-       partProdLeft         = scanl (*) one denomPowers-       (prod:partProdRight) = scanr (*) one denomPowers-       (intPart,numerRed) = divMod numer prod-       facs = List.zipWith (*) partProdLeft partProdRight-       numers =-          fromMaybe-             (error $ "PartialFraction.fromFactoredFraction: " ++-                      "denominators must be relatively prime")-             (PID.diophantineMulti numerRed facs)-       pairs = List.zipWith multiFromFraction denoms numers-       -- Is reduceHeads also necessary for polynomial partial fractions?-   in  removeZeros $ reduceHeads $ Cons intPart (indexMapFromList pairs)--fromFactoredFractionAlt :: (PID.C a, Indexable.C a) => [a] -> a -> T a-fromFactoredFractionAlt denoms numer =-   foldl (\p d -> scaleFrac (one%d) p) (fromValue numer) denoms--{- |-The list of denominators must contain equal elements.-Sorry for this hack.--}-multiFromFraction :: PID.C a => [a] -> a -> (a,[a])-multiFromFraction (d:ds) n =-   (d, reverse $ decomposeVarPositionalInf ds n)-multiFromFraction [] _ =-   error "PartialFraction.multiFromFraction: there must be one denominator"--fromValue :: a -> T a-fromValue x = Cons x Map.empty---{- |-A normalization step which separates the integer part-from the leading fraction of each sub-list.--}-reduceHeads :: Integral.C a => T a -> T a-reduceHeads (Cons z m0) =-   let m1 = indexMapMapWithKey (\x (y:ys) -> let (q,r) = divMod y x in (q,r:ys)) m0-   in  Cons-          (foldl (+) z (map fst $ Map.elems m1))-          (fmap snd m1)--{- |-Cf. Number.Positional--}-carryRipple :: Integral.C a => a -> [a] -> (a,[a])-carryRipple b =-   mapAccumR (\carry y -> divMod (y+carry) b) zero---{- |-A normalization step which reduces all elements in sub-lists-modulo their denominators.-Zeros might be the result, that must be remove with 'removeZeros'.--}-normalizeModulo :: Integral.C a => T a -> T a-normalizeModulo (Cons z0 m0) =-   let m1 = indexMapMapWithKey carryRipple m0-       -- would be nice to have a Map.unzip function-       ints = Map.elems $ fmap fst m1-   in  Cons (foldl (+) z0 ints) (fmap snd m1)----{- |-Remove trailing zeros in sub-lists-because if lists are converted to fractions by 'multiToFraction'-we must be sure that the denominator of the (cancelled) fraction-is indeed the stored power of the irreducible denominator.-Otherwise 'mulFrac' leads to wrong results.--}-removeZeros :: (Indexable.C a, ZeroTestable.C a) => T a -> T a-removeZeros (Cons z m) =-   Cons z $-   Map.filter (not . null) $-   Map.map (dropWhileRev isZero) m---{--instance Functor (T a) where-   fmap f (Cons x) = Cons (fmap f x)--}--zipWith :: (Indexable.C a) => (a -> a -> a) -> ([a] -> [a] -> [a]) ->-   (T a -> T a -> T a)-zipWith opS opV (Cons za ma) (Cons zb mb) =-   Cons (opS za zb) (Map.unionWith opV ma mb)--instance (Indexable.C a, Integral.C a, ZeroTestable.C a) => Additive.C (T a) where-   a + b = removeZeros $ normalizeModulo $ zipWith (+) (+) a b-   {- This implementation is attracting but wrong.-     It fails if terms are present in b that are missing in a.-     Default implementation is better here.-     a - b = removeZeros $ normalizeModulo $ zipWith (-) (-) a b-   -}-   negate (Cons z m) = Cons (negate z) (fmap negate m)-   zero = fromValue zero--{- |-Transforms a product of two partial fractions-into a sum of two fractions.-The denominators must be at least relatively prime.-Since 'T' requires irreducible denominators,-these are also relatively prime.--Example: @mulFrac (1%6) (1%4)@ fails because of the common divisor @2@.--}-mulFrac :: (PID.C a) => Ratio.T a -> Ratio.T a -> (a, a)-mulFrac x y =-   let dx = Ratio.denominator x-       dy = Ratio.denominator y-   in  fromMaybe-          (error "PartialFraction.mulFrac: denominators must be relatively prime")-          (PID.diophantine (Ratio.numerator x * Ratio.numerator y) dy dx)--{--nx/dx * ny/dy = a/dx + b/dy-nx*ny = a*dy + b*dx--}--mulFrac' :: (PID.C a) => Ratio.T a -> Ratio.T a -> (Ratio.T a, Ratio.T a)-mulFrac' x y =-   let (na,nb) = mulFrac x y-   in  (na % Ratio.denominator x, nb % Ratio.denominator y)--{--Also works if the operands share a non-trivial divisor.--mulFracOverlap :: (PID.C a) =>-   Ratio.T a -> Ratio.T a -> ((Ratio.T a, Ratio.T a), Ratio.T a)-mulFracOverlap x y =-   let dx = Ratio.denominator x-       dy = Ratio.denominator y-       (g,(a0,b0)) = extendedGCD dy dx-       (q,r) = divModZero (Ratio.numerator x * Ratio.numerator y) g-   in  if (isZero r)-         then ((q*a, q*b), zero)-         else-           let fx = divChecked dx g-               fy = divChecked dy g-               (g,(k,c)) = extendedGCD (g^2) (fx*fy)--given dx=fx*g and dy=fy*g with fx and fy are relatively prime:-nx/(g*fx) * ny/(g*fy) = a/fx + b/fy + c/g^2-nx*ny = a*fy*g^2 + b*fx*g^2 + c*fx*fy-      = a*dy*g   + b*dx*g   + c*fx*fy-a0*dy + b0*dx = g-a=a0*k-b=b0*k--This approach does still fail on 1%2 * 1%4.--}--{- |-Works always but simply puts the product into the last fraction.--}-mulFracStupid :: (PID.C a) =>-   Ratio.T a -> Ratio.T a -> ((Ratio.T a, Ratio.T a), Ratio.T a)-mulFracStupid x y =-   let dx = Ratio.denominator x-       dy = Ratio.denominator y-       [a,b,c] =-          fromMaybe-             (error "PartialFraction.mulFracOverlap: (gcd 1 x) must always be a unit")-             (PID.diophantineMulti-                 (Ratio.numerator x * Ratio.numerator y) [dy, dx, one])-   in  ((a % dx, b % dy), c%(dx*dy))--{- |-Also works if the operands share a non-trivial divisor.-However the results are quite arbitrary.--}-mulFracOverlap :: (PID.C a) =>-   Ratio.T a -> Ratio.T a -> ((Ratio.T a, Ratio.T a), Ratio.T a)-mulFracOverlap x y =-   let dx = Ratio.denominator x-       dy = Ratio.denominator y-       nx = Ratio.numerator x-       ny = Ratio.numerator y-       (g,(a,b)) = PID.extendedGCD dy dx-       (q,r) = divModZero (nx*ny) g-   in  (((q*a)%dx, (q*b)%dy), r%(dx*dy))---{- |-Expects an irreducible denominator as associate in standard form.--}-scaleFrac :: (PID.C a, Indexable.C a) => Ratio.T a -> T a -> T a-scaleFrac s (Cons z0 m) =-   let ns = Ratio.numerator s-       ds = Ratio.denominator s-       dsOrd = Indexable.toOrd ds-       -- (z,zr) = Ratio.split (Ratio.scale z0 s)-       (z,zr) = divMod (z0*ns) ds-       scaleFracs =-          (\(scs,fracs) ->-             Map.insert dsOrd [foldl (+) zr scs] $-                indexMapFromList $-                   map (uncurry multiFromFraction) fracs) .-          unzip .-          map (\(dis,r) ->-                 let (sc,rc) = mulFrac s r-                 in  (sc, (dis, rc))) .-          Map.elems .-          indexMapMapWithKey-             (\d l -> (Match.replicate l d, multiToFraction d l))-   in  removeZeros $ reduceHeads $ Cons z-          (mapApplySplit dsOrd (+)-             (uncurry (:) . carryRipple ds . map (ns*))-             scaleFracs m)--scaleInt :: (PID.C a, Indexable.C a) => a -> T a -> T a-scaleInt x (Cons z m) =-   removeZeros $ normalizeModulo $-      Cons (x*z) (Map.map (map (x*)) m)---mul :: (PID.C a, Indexable.C a) => T a -> T a -> T a-mul (Cons z m) a =-   foldl-      (+) (scaleInt z a)-      (map (\(d,l) ->-              -- cf. to multiToFraction-              foldr (\numer acc ->-                 scaleFrac (one%d) (scaleInt numer a + acc)) zero l)-           (indexMapToList m))--mulFast :: (PID.C a, Indexable.C a) => T a -> T a -> T a-mulFast pa pb =-   let ra = toFactoredFraction pa-       rb = toFactoredFraction pb-   in  fromFactoredFraction (fst ra ++ fst rb) (snd ra * snd rb)---instance (PID.C a, Indexable.C a) => Ring.C (T a) where-   one = fromValue one-   (*) = mulFast---{- * Helper functions for work with Maps with Indexable keys -}--indexMapMapWithKey :: (a -> b -> c)-                      -> Map (Indexable.ToOrd a) b-                      -> Map (Indexable.ToOrd a) c-indexMapMapWithKey f = Map.mapWithKey (f . Indexable.fromOrd)--indexMapToList :: Map (Indexable.ToOrd a) b -> [(a, b)]-indexMapToList = map (\(k,e) -> (Indexable.fromOrd k, e)) . Map.toList--indexMapFromList :: Indexable.C a => [(a, b)] -> Map (Indexable.ToOrd a) b-indexMapFromList = Map.fromList . map (\(k,e) -> (Indexable.toOrd k, e))--{- |-Apply a function on a specific element if it exists,-and another function to the rest of the map.--}-mapApplySplit :: Ord a =>-   a -> (c -> c -> c) -> -   (b -> c) -> (Map a b -> Map a c) -> Map a b -> Map a c-mapApplySplit key addOp f g m =-   maybe-      (g m)-      (\x -> Map.insertWith addOp key (f x) $ g (Map.delete key m))-      (Map.lookup key m)-
− src-ghc-6.12/MathObj/Permutation.hs 
@@ -1,32 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Copyright    :   (c) Henning Thielemann 2006-Maintainer   :   numericprelude@henning-thielemann.de-Stability    :   provisional-Portability  :--Routines and abstractions for permutations of Integers.--***-Seems to be a candidate for Algebra directory.-Algebra.PermutationGroup ?--}--module MathObj.Permutation where--import Data.Array(Ix)---- import NumericPrelude.Numeric (Integer)--- import NumericPrelude.Base---{- |-There are quite a few way we could represent elements of permutation-groups: the images in a row, a list of the cycles, et.c. All of these-differ highly in how complex various operations end up being.--}--class C p where-   domain  :: (Ix i) => p i -> (i, i)-   apply   :: (Ix i) => p i -> i -> i-   inverse :: (Ix i) => p i -> p i
− src-ghc-6.12/MathObj/Permutation/CycleList.hs 
@@ -1,103 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Copyright    :   (c) Mikael Johansson 2006-Maintainer   :   mik@math.uni-jena.de-Stability    :   provisional-Portability  :   requires multi-parameter type classes--Permutation of Integers represented by cycles.--}--module MathObj.Permutation.CycleList where--import Data.Set(Set)-import qualified Data.Set as Set--import Data.List (unfoldr)-import Data.Array(Ix)-import qualified Data.Array as Array--import qualified Data.List.Match as Match-import Data.Maybe.HT (toMaybe)-import NumericPrelude.Numeric (fromInteger)-import NumericPrelude.Base---type Cycle i = [i]-type T i = [Cycle i]----fromFunction :: (Ix i) =>-   (i, i) -> (i -> i) -> T i-fromFunction rng f =-   let extractCycle available =-          do el <- choose available-             let orb = orbit f el-             return (orb, Set.difference available (Set.fromList orb))-       cycles = unfoldr extractCycle (Set.fromList (Array.range rng))-   in  keepEssentials cycles------ right action of a cycle-cycleRightAction :: (Eq i) => i -> Cycle i -> i-x `cycleRightAction` c = cycleAction c x---- left action of a cycle-cycleLeftAction :: (Eq i) => Cycle i -> i -> i-c `cycleLeftAction` x = cycleAction (reverse c) x--cycleAction :: (Eq i) => [i] -> i -> i-cycleAction cyc x =-   case dropWhile (x/=) (cyc ++ [head cyc]) of-      _:y:_ -> y-      _ -> x---cycleOrbit :: (Ord i) => Cycle i -> i -> [i]-cycleOrbit cyc = orbit (flip cycleRightAction cyc)--{- |-Right (left?) group action on the Integers.-Close to, but not the same as the module action in Algebra.Module.--}-(*>) :: (Eq i) => T i -> i -> i-p *> x = foldr (flip cycleRightAction) x p--cyclesOrbit ::(Ord i) => T i -> i -> [i]-cyclesOrbit p = orbit (p *>)--orbit :: (Ord i) => (i -> i) -> i -> [i]-orbit op x0 = takeUntilRepetition (iterate op x0)---- | candidates for Utility ?-takeUntilRepetition :: Ord a => [a] -> [a]-takeUntilRepetition xs =-   let accs = scanl (flip Set.insert) Set.empty xs-       lenlist = takeWhile not (zipWith Set.member xs accs)-   in  Match.take lenlist xs--takeUntilRepetitionSlow :: Eq a => [a] -> [a]-takeUntilRepetitionSlow xs =-   let accs = scanl (flip (:)) [] xs-       lenlist = takeWhile not (zipWith elem xs accs)-   in  Match.take lenlist xs---{--Alternative to Data.Set.minView in GHC-6.6.--}-choose :: Set a -> Maybe a-choose set =-   toMaybe (not (Set.null set)) (Set.findMin set)--keepEssentials :: T i -> T i-keepEssentials = filter isEssential---- is more lazy than (length cyc > 1)-isEssential :: Cycle i -> Bool-isEssential = not . null . drop 1--inverse :: T i -> T i-inverse = map reverse
− src-ghc-6.12/MathObj/Permutation/CycleList/Check.hs 
@@ -1,125 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Copyright    :   (c) Henning Thielemann 2006-Maintainer   :   numericprelude@henning-thielemann.de-Stability    :   provisional-Portability  :   requires multi-parameter type classes--}--module MathObj.Permutation.CycleList.Check where--import qualified MathObj.Permutation.CycleList as PermCycle-import qualified MathObj.Permutation.Table     as PermTable-import qualified MathObj.Permutation           as Perm--{--import qualified Algebra.Ring as Ring-import qualified Algebra.Additive as Additive-import Algebra.Ring((*),one,fromInteger)-import Algebra.Additive((+))--}-import Algebra.Monoid((<*>))-import qualified Algebra.Monoid as Monoid--import Data.Array((!), Ix)-import qualified Data.Array as Array---- import NumericPrelude.Numeric (Integer)-import NumericPrelude.Base hiding (cycle)--{- |-We shall make a little bit of a hack here, enabling us to use additive-or multiplicative syntax for groups as we wish by simply instantiating-Num with both operations corresponding to the group operation of the-permutation group we're studying--}--{- |-There are quite a few way we could represent elements of permutation-groups: the images in a row, a list of the cycles, et.c. All of these-differ highly in how complex various operations end up being.--}--newtype Cycle i = Cycle { cycle :: [i] } deriving (Read,Eq)-data T i = Cons { range :: (i, i), cycles :: [Cycle i] }--{- |-Does not check whether the input values are in range.--}-fromCycles :: (i, i) -> [[i]] -> T i-fromCycles rng = Cons rng . map Cycle--toCycles :: T i -> [[i]]-toCycles = map cycle . cycles--toTable :: (Ix i) => T i -> PermTable.T i-toTable x = PermTable.fromCycles (range x) (toCycles x)--fromTable :: (Ix i) => PermTable.T i -> T i-fromTable x =-   let rng = Array.bounds x-   in  fromCycles rng (PermCycle.fromFunction rng (x!))---errIncompat :: a-errIncompat = error "Permutation.CycleList: Incompatible domains"--liftCmpTable2 :: (Ix i) =>-   (PermTable.T i -> PermTable.T i -> a) -> T i -> T i -> a-liftCmpTable2 f x y =-   if range x == range y-     then f (toTable x) (toTable y)-     else errIncompat--liftTable2 :: (Ix i) =>-   (PermTable.T i -> PermTable.T i -> PermTable.T i) -> T i -> T i -> T i-liftTable2 f x y = fromTable (liftCmpTable2 f x y)---closure :: (Ix i) => [T i] -> [T i]-closure = map fromTable . PermTable.closure . map toTable---instance Perm.C T where-   domain    = range-   apply   p = ((toCycles p) PermCycle.*>)-   inverse p = fromCycles (range p) (PermCycle.inverse (toCycles p))--instance Show i => Show (Cycle i) where-   show c = "(" ++-           (unwords $-            map show $-            cycle c) ++ ")"--instance Show i => Show (T i) where-   show p =-      case cycles p of-         []  -> "Id"-         cyc -> concatMap show cyc---{- |-These instances may need more work-They involve converting a permutation to a table.--}-instance Ix i => Eq (T i) where-   (==)  =  liftCmpTable2 (==)--instance Ix i => Ord (T i) where-   compare  =  liftCmpTable2 compare--{- Better: Group class and instances-instance Additive.C (T i) where-   p + q = p * q-   negate = inverse-   zero = one--instance Ring.C (T i) where-   (Cons op cp) * (Cons oq cq) = reduceCycles $-           Cons (max op oq) (cp ++ cq)-   one = Cons 1 []--}--instance Ix i => Monoid.C (T i) where-   (<*>) = liftTable2 PermTable.compose-   idt   = error "There is no generic unit element"
− src-ghc-6.12/MathObj/Permutation/Table.hs 
@@ -1,113 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Copyright    :   (c) Henning Thielemann 2006-Maintainer   :   numericprelude@henning-thielemann.de-Stability    :   provisional-Portability  :--Permutation represented by an array of the images.--}--module MathObj.Permutation.Table where--import qualified MathObj.Permutation as Perm--import Data.Set(Set)-import qualified Data.Set as Set--import Data.Array(Array,(!),(//),Ix)-import qualified Data.Array as Array--import Data.List ((\\), nub, unfoldr, )--import Data.Tuple.HT (swap, )-import Data.Maybe.HT (toMaybe, )---- import NumericPrelude.Numeric (Integer)-import NumericPrelude.Base hiding (cycle)---type T i = Array i i---fromFunction :: (Ix i) =>-   (i, i) -> (i -> i) -> T i-fromFunction rng f =-   Array.listArray rng (map f (Array.range rng))--toFunction :: (Ix i) => T i -> (i -> i)-toFunction = (!)--{--Create a permutation in table form-from any other permutation representation.--}-fromPermutation :: (Ix i, Perm.C p) => p i -> T i-fromPermutation x =-   let rng = Perm.domain x-   in  Array.listArray rng (map (Perm.apply x) (Array.range rng))--fromCycles :: (Ix i) => (i, i) -> [[i]] -> T i-fromCycles rng = foldl (flip cycle) (identity rng)---identity :: (Ix i) => (i, i) -> T i-identity rng = Array.listArray rng (Array.range rng)--cycle :: (Ix i) => [i] -> T i -> T i-cycle cyc p =-   p // zipWith (\i j -> (j,p!i)) cyc (tail (cyc++cyc))--inverse :: (Ix i) => T i -> T i-inverse p =-   let rng = Array.bounds p-   in  Array.array rng (map swap (Array.assocs p))--compose :: (Ix i) => T i -> T i -> T i-compose p q =-   let pRng = Array.bounds p-       qRng = Array.bounds q-   in  if pRng==qRng-         then fmap (p!) q-         else error "compose: ranges differ"---                     ++ show pRng ++ " /= " ++ show qRng)---{- |-Extremely naïve algorithm-to generate a list of all elements in a group.-Should be replaced by a Schreier-Sims system-if this code is ever used for anything bigger than .. say ..-groups of order 512 or so.--}-{--Alternative to Data.Set.minView in GHC-6.6.--}-choose :: Set a -> Maybe (a, Set a)-choose set =-   toMaybe (not (Set.null set)) (Set.deleteFindMin set)--closure :: (Ix i) => [T i] -> [T i]-closure [] = []-closure generators@(gen:_) =-   let genSet = Set.fromList generators-       idSet  = Set.singleton (identity (Array.bounds gen))-       generate (registered, candidates) =-          do (cand, remCands) <- choose candidates-             let newCands =-                    flip Set.difference registered $-                    Set.map (compose cand) genSet-             return (cand, (Set.union registered newCands,-                            Set.union remCands newCands))-   in  unfoldr generate (idSet, idSet)--closureSlow :: (Ix i) => [T i] -> [T i]-closureSlow [] = []-closureSlow generators@(gen:_) =-   let addElts grp [] = grp-       addElts grp cands@(cand:remCands) =-          let group'   = grp ++ [cand]-              newCands = map (compose cand) generators-              cands'   = nub (remCands ++ newCands) \\ (grp ++ cands)-          in  addElts group' cands'-   in  addElts [] [identity (Array.bounds gen)]
− src-ghc-6.12/MathObj/Polynomial.hs 
@@ -1,309 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}--{- |-Polynomials and rational functions in a single indeterminate.-Polynomials are represented by a list of coefficients.-All non-zero coefficients are listed, but there may be extra '0's at the end.--Usage:-Say you have the ring of 'Integer' numbers-and you want to add a transcendental element @x@,-that is an element, which does not allow for simplifications.-More precisely, for all positive integer exponents @n@-the power @x^n@ cannot be rewritten as a sum of powers with smaller exponents.-The element @x@ must be represented by the polynomial @[0,1]@.--In principle, you can have more than one transcendental element-by using polynomials whose coefficients are polynomials as well.-However, most algorithms on multi-variate polynomials-prefer a different (sparse) representation,-where the ordering of elements is not so fixed.--If you want division, you need "Number.Ratio"s-of polynomials with coefficients from a "Algebra.Field".--You can also compute with an algebraic element,-that is an element which satisfies an algebraic equation like-@x^3-x-1==0@.-Actually, powers of @x@ with exponents above @3@ can be simplified,-since it holds @x^3==x+1@.-You can perform these computations with "Number.ResidueClass" of polynomials,-where the divisor is the polynomial equation that determines @x@.-If the polynomial is irreducible-(in our case @x^3-x-1@ cannot be written as a non-trivial product)-then the residue classes also allow unrestricted division-(except by zero, of course).-That is, using residue classes of polynomials-you can work with roots of polynomial equations-without representing them by radicals-(powers with fractional exponents).-It is well-known, that roots of polynomials of degree above 4-may not be representable by radicals.--}--module MathObj.Polynomial-   (T, fromCoeffs, coeffs, degree,-    showsExpressionPrec, const,-    evaluate, evaluateCoeffVector, evaluateArgVector,-    collinear,-    integrate,-    compose, fromRoots, reverse,-    translate, dilate, shrink, )-where--import qualified MathObj.Polynomial.Core as Core--import qualified Algebra.Differential         as Differential-import qualified Algebra.VectorSpace          as VectorSpace-import qualified Algebra.Module               as Module-import qualified Algebra.Vector               as Vector-import qualified Algebra.Field                as Field-import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.Units                as Units-import qualified Algebra.IntegralDomain       as Integral-import qualified Algebra.Ring                 as Ring-import qualified Algebra.Additive             as Additive-import qualified Algebra.ZeroTestable         as ZeroTestable-import qualified Algebra.Indexable            as Indexable--import Algebra.Module((*>))-import Algebra.ZeroTestable(isZero)--import Control.Monad (liftM, )-import qualified Data.List as List--import Test.QuickCheck (Arbitrary(arbitrary))--import NumericPrelude.Base    hiding (const, reverse, )-import NumericPrelude.Numeric--import qualified Prelude as P98---newtype T a = Cons {coeffs :: [a]}---{-# INLINE fromCoeffs #-}-fromCoeffs :: [a] -> T a-fromCoeffs = lift0--{-# INLINE lift0 #-}-lift0 :: [a] -> T a-lift0 = Cons--{-# INLINE lift1 #-}-lift1 :: ([a] -> [a]) -> (T a -> T a)-lift1 f (Cons x0) = Cons (f x0)--{-# INLINE lift2 #-}-lift2 :: ([a] -> [a] -> [a]) -> (T a -> T a -> T a)-lift2 f (Cons x0) (Cons x1) = Cons (f x0 x1)--degree :: (ZeroTestable.C a) => T a -> Maybe Int-degree x =-   case Core.normalize (coeffs x) of-      [] -> Nothing-      (_:xs) -> Just $ length xs--{--Functor instance is e.g. useful for showing polynomials in residue rings.-@fmap (ResidueClass.concrete 7) (polynomial [1,4,4::ResidueClass.T Integer] * polynomial [1,5,6])@--}--instance Functor T where-   fmap f (Cons xs) = Cons (map f xs)--{-# INLINE plusPrec #-}-{-# INLINE appPrec #-}-plusPrec, appPrec :: Int-plusPrec = 6-appPrec  = 10--instance (Show a) => Show (T a) where-   showsPrec p (Cons xs) =-      showParen (p >= appPrec) (showString "Polynomial.fromCoeffs " . shows xs)--{-# INLINE showsExpressionPrec #-}-showsExpressionPrec :: (Show a, ZeroTestable.C a, Additive.C a) =>-   Int -> String -> T a -> String -> String-showsExpressionPrec p var poly =-    if isZero poly-      then showString "0"-      else-        let terms = filter (not . isZero . fst)-                       (zip (coeffs poly) monomials)-            monomials = id :-                        showString "*" . showString var :-                        map (\k -> showString "*" . showString var-                                 . showString "^" . shows k)-                            [(2::Int)..]-            showsTerm x showsMon = showsPrec (plusPrec+1) x . showsMon-        in showParen (p > plusPrec)-           (foldl (.) id $ List.intersperse (showString " + ") $-            map (uncurry showsTerm) terms)---{-# INLINE evaluate #-}-evaluate :: Ring.C a => T a -> a -> a-evaluate (Cons y) x = Core.horner x y--{- |-Here the coefficients are vectors,-for example the coefficients are real and the coefficents are real vectors.--}-{-# INLINE evaluateCoeffVector #-}-evaluateCoeffVector :: Module.C a v => T v -> a -> v-evaluateCoeffVector (Cons y) x = Core.hornerCoeffVector x y--{- |-Here the argument is a vector,-for example the coefficients are complex numbers or square matrices-and the coefficents are reals.--}-{-# INLINE evaluateArgVector #-}-evaluateArgVector :: (Module.C a v, Ring.C v) => T a -> v -> v-evaluateArgVector (Cons y) x = Core.hornerArgVector x y--{- |-'compose' is the functional composition of polynomials.--It fulfills-  @ eval x . eval y == eval (compose x y) @--}---- compose :: Module.C a b => T b -> T a -> T a--- compose (Cons x) y = Core.horner y (map const x)-{-# INLINE compose #-}-compose :: (Ring.C a) => T a -> T a -> T a-compose (Cons x) y = Core.horner y (map const x)--{-# INLINE const #-}-const :: a -> T a-const x = lift0 [x]---collinear :: (Eq a, Ring.C a) => T a -> T a -> Bool-collinear (Cons x) (Cons y) = Core.collinear x y---instance (Eq a, ZeroTestable.C a) => Eq (T a) where-   (Cons x) == (Cons y) = Core.equal x y--instance (Indexable.C a, ZeroTestable.C a) => Indexable.C (T a) where-   compare = Indexable.liftCompare coeffs--instance (ZeroTestable.C a) => ZeroTestable.C (T a) where-   isZero (Cons x) = isZero x---instance (Additive.C a) => Additive.C (T a) where-   (+)    = lift2 Core.add-   (-)    = lift2 Core.sub-   zero   = lift0 []-   negate = lift1 Core.negate---instance Vector.C T where-   zero  = zero-   (<+>) = (+)-   (*>)  = Vector.functorScale--instance (Module.C a b) => Module.C a (T b) where-   (*>) x = lift1 (x *>)--instance (Field.C a, Module.C a b) => VectorSpace.C a (T b)---instance (Ring.C a) => Ring.C (T a) where-   one         = const one-   fromInteger = const . fromInteger-   (*)         = lift2 Core.mul---{- |-The 'Integral.C' instance is intensionally built-from the 'Field.C' structure of the polynomial coefficients.-If we would use @Integral.C a@ superclass,-then the Euclidean algorithm could not determine-the greatest common divisor of e.g. @[1,1]@ and @[2]@.--}-instance (ZeroTestable.C a, Field.C a) => Integral.C (T a) where-   divMod (Cons x) (Cons y) =-      let (d,m) = Core.divMod x y-      in  (Cons d, Cons m)--instance (ZeroTestable.C a, Field.C a) => Units.C (T a) where-   isUnit (Cons []) = False-   isUnit (Cons (x0:xs)) = not (isZero x0) && all isZero xs-   stdUnit    (Cons x) = const        (Core.stdUnit x)-   stdUnitInv (Cons x) = const (recip (Core.stdUnit x))--{--Polynomials are a Euclidean domain, so no instance is necessary-(although it might be faster).--}--instance (ZeroTestable.C a, Field.C a) => PID.C (T a)---instance (Ring.C a) => Differential.C (T a) where-   differentiate = lift1 Core.differentiate---{-# INLINE integrate #-}-integrate :: (Field.C a) => a -> T a -> T a-integrate = lift1 . Core.integrate----{-# INLINE fromRoots #-}-fromRoots :: (Ring.C a) => [a] -> T a-fromRoots = Cons . foldl (flip Core.mulLinearFactor) [one]--{-# INLINE reverse #-}-reverse :: Additive.C a => T a -> T a-reverse = lift1 Core.alternate--translate :: Ring.C a => a -> T a -> T a-translate d =-   lift1 $ foldr (\c p -> [c] + Core.mulLinearFactor d p) []--shrink :: Ring.C a => a -> T a -> T a-shrink k =-   lift1 $ zipWith (*) (iterate (k*) one)--dilate :: Field.C a => a -> T a -> T a-dilate = shrink . Field.recip---instance (Arbitrary a, ZeroTestable.C a) => Arbitrary (T a) where-   arbitrary = liftM (fromCoeffs . Core.normalize) arbitrary---{- * legacy instances -}--{- |-It is disputable whether polynomials shall be represented by number literals or not.-An advantage is, that one can write-let x = polynomial [0,1]-in  (x^2+x+1)*(x-1)-However the output looks much different.--}-{-# INLINE legacyInstance #-}-legacyInstance :: a-legacyInstance =-   error "legacy Ring.C instance for simple input of numeric literals"--instance (Ring.C a, Eq a, Show a, ZeroTestable.C a) => P98.Num (T a) where-   fromInteger = const . fromInteger-   negate = Additive.negate -- for unary minus-   (+)    = legacyInstance-   (*)    = legacyInstance-   abs    = legacyInstance-   signum = legacyInstance--instance (Field.C a, Eq a, Show a, ZeroTestable.C a) => P98.Fractional (T a) where-   fromRational = const . fromRational-   (/) = legacyInstance
− src-ghc-6.12/MathObj/Polynomial/Core.hs 
@@ -1,224 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-This module implements polynomial functions on plain lists.-We use such functions in order to implement methods of other datatypes.--The module organization differs from that of @ResidueClass@:-Here the @Polynomial@ module exports the type-that fits to the NumericPrelude type classes,-whereas in @ResidueClass@ the sub-modules export various flavors of them.--}-module MathObj.Polynomial.Core (-   horner, hornerCoeffVector, hornerArgVector,-   normalize,-   shift, unShift,-   equal,-   add, sub, negate,-   scale, collinear,-   tensorProduct, tensorProductAlt,-   mul, mulShear, mulShearTranspose,-   divMod, divModRev,-   stdUnit,-   progression, differentiate, integrate, integrateInt,-   mulLinearFactor,-   alternate,-   ) where--import qualified Algebra.Module               as Module-import qualified Algebra.Field                as Field-import qualified Algebra.IntegralDomain       as Integral-import qualified Algebra.Ring                 as Ring-import qualified Algebra.Additive             as Additive-import qualified Algebra.ZeroTestable         as ZeroTestable--import qualified Data.List as List-import NumericPrelude.List (zipWithOverlap, )-import Data.Tuple.HT (mapPair, mapFst, forcePair, )-import Data.List.HT-          (dropWhileRev, switchL, shear, shearTranspose, outerProduct, )--import qualified NumericPrelude.Base as P-import qualified NumericPrelude.Numeric as NP--import NumericPrelude.Base    hiding (const, reverse, )-import NumericPrelude.Numeric hiding (divMod, negate, stdUnit, )---{- |-Horner's scheme for evaluating a polynomial in a ring.--}-{-# INLINE horner #-}-horner :: Ring.C a => a -> [a] -> a-horner x = foldr (\c val -> c+x*val) zero--{- |-Horner's scheme for evaluating a polynomial in a module.--}-{-# INLINE hornerCoeffVector #-}-hornerCoeffVector :: Module.C a v => a -> [v] -> v-hornerCoeffVector x = foldr (\c val -> c+x*>val) zero--{-# INLINE hornerArgVector #-}-hornerArgVector :: (Module.C a v, Ring.C v) => v -> [a] -> v-hornerArgVector x = foldr (\c val -> c*>one+val*x) zero---{- |-It's also helpful to put a polynomial in canonical form.-'normalize' strips leading coefficients that are zero.--}-{-# INLINE normalize #-}-normalize :: (ZeroTestable.C a) => [a] -> [a]-normalize = dropWhileRev isZero--{- |-Multiply by the variable, used internally.--}-{-# INLINE shift #-}-shift :: (Additive.C a) => [a] -> [a]-shift [] = []-shift l  = zero : l--{-# INLINE unShift #-}-unShift :: [a] -> [a]-unShift []     = []-unShift (_:xs) = xs--{-# INLINE equal #-}-equal :: (Eq a, ZeroTestable.C a) => [a] -> [a] -> Bool-equal x y = and (zipWithOverlap isZero isZero (==) x y)---add, sub :: (Additive.C a) => [a] -> [a] -> [a]-add = (+)-sub = (-)--{-# INLINE negate #-}-negate :: (Additive.C a) => [a] -> [a]-negate = map NP.negate---{-# INLINE scale #-}-scale :: Ring.C a => a -> [a] -> [a]-scale s = map (s*)---collinear :: (Eq a, Ring.C a) => [a] -> [a] -> Bool-collinear (x:xs) (y:ys) =-   if x==zero && y==zero-     then collinear xs ys-     else scale x ys == scale y xs--- here at least one of xs and ys is empty-collinear xs ys =-   all (==zero) xs && all (==zero) ys---{-# INLINE tensorProduct #-}-tensorProduct :: Ring.C a => [a] -> [a] -> [[a]]-tensorProduct = outerProduct (*)--tensorProductAlt :: Ring.C a => [a] -> [a] -> [[a]]-tensorProductAlt xs ys = map (flip scale ys) xs---{- |-'mul' is fast if the second argument is a short polynomial,-'MathObj.PowerSeries.**' relies on that fact.--}--{-# INLINE mul #-}-mul :: Ring.C a => [a] -> [a] -> [a]-{- prevent from generation of many zeros-   if the first operand is the empty list -}-mul [] = P.const []-mul xs = foldr (\y zs -> let (v:vs) = scale y xs in v : add vs zs) []--- this one fails on infinite lists---    mul xs = foldr (\y zs -> add (scale y xs) (shift zs)) []--{-# INLINE mulShear #-}-mulShear :: Ring.C a => [a] -> [a] -> [a]-mulShear xs ys = map sum (shear (tensorProduct xs ys))--{-# INLINE mulShearTranspose #-}-mulShearTranspose :: Ring.C a => [a] -> [a] -> [a]-mulShearTranspose xs ys = map sum (shearTranspose (tensorProduct xs ys))---divMod :: (ZeroTestable.C a, Field.C a) => [a] -> [a] -> ([a], [a])-divMod x y =-   mapPair (List.reverse, List.reverse) $-   divModRev (List.reverse x) (List.reverse y)--{--snd $ Poly.divMod (repeat (1::Double)) [1,1]--}-divModRev :: (ZeroTestable.C a, Field.C a) => [a] -> [a] -> ([a], [a])-divModRev x y =-   let (y0:ys) = dropWhile isZero y-       -- the second parameter represents lazily (length x - length y)-       aux xs' =-         forcePair .-         switchL-           ([], xs')-           (P.const $-              let (x0:xs) = xs'-                  q0      = x0/y0-              in  mapFst (q0:) . aux (sub xs (scale q0 ys)))-   in  if isZero y-         then error "MathObj.Polynomial: division by zero"-         else aux x (drop (length y - 1) x)--{-# INLINE stdUnit #-}-stdUnit :: (ZeroTestable.C a, Ring.C a) => [a] -> a-stdUnit x = case normalize x of-    [] -> one-    l  -> last l---{-# INLINE progression #-}-progression :: Ring.C a => [a]-progression = iterate (one+) one--{-# INLINE differentiate #-}-differentiate :: (Ring.C a) => [a] -> [a]-differentiate = zipWith (*) progression . drop 1--{-# INLINE integrate #-}-integrate :: (Field.C a) => a -> [a] -> [a]-integrate c x = c : zipWith (/) x progression--{- |-Integrates if it is possible to represent the integrated polynomial-in the given ring.-Otherwise undefined coefficients occur.--}-{-# INLINE integrateInt #-}-integrateInt :: (ZeroTestable.C a, Integral.C a) => a -> [a] -> [a]-integrateInt c x =-   c : zipWith Integral.divChecked x progression---{-# INLINE mulLinearFactor #-}-mulLinearFactor :: Ring.C a => a -> [a] -> [a]-mulLinearFactor x yt@(y:ys) = Additive.negate (x*y) : yt - scale x ys-mulLinearFactor _ [] = []--{-# INLINE alternate #-}-alternate :: Additive.C a => [a] -> [a]-alternate = zipWith ($) (cycle [id, Additive.negate])---{--see htam: Wavelet/DyadicResultant--resultant :: Ring.C a => [a] -> [a] -> [a]-resultant xs ys =--discriminant :: Ring.C a => [a] -> a-discriminant xs =-   let degree = genericLength xs-   in  parityFlip (divChecked (degree*(degree-1)) 2)-                  (resultant xs (differentiate xs))-          `divChecked` last xs--}-
− src-ghc-6.12/MathObj/PowerSeries.hs 
@@ -1,193 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}--{- |-Power series, either finite or unbounded.-(zipWith does exactly the right thing to make it work almost transparently.)--}-module MathObj.PowerSeries where--import qualified MathObj.PowerSeries.Core as Core-import qualified MathObj.Polynomial.Core as Poly--import qualified Algebra.Differential   as Differential-import qualified Algebra.IntegralDomain as Integral-import qualified Algebra.VectorSpace    as VectorSpace-import qualified Algebra.Module         as Module-import qualified Algebra.Vector         as Vector-import qualified Algebra.Transcendental as Transcendental-import qualified Algebra.Algebraic      as Algebraic-import qualified Algebra.Field          as Field-import qualified Algebra.Ring           as Ring-import qualified Algebra.Additive       as Additive-import qualified Algebra.ZeroTestable   as ZeroTestable--import Algebra.Module((*>))--import NumericPrelude.Base    hiding (const)-import NumericPrelude.Numeric---newtype T a = Cons {coeffs :: [a]} deriving (Ord)--{-# INLINE fromCoeffs #-}-fromCoeffs :: [a] -> T a-fromCoeffs = lift0--{-# INLINE lift0 #-}-lift0 :: [a] -> T a-lift0 = Cons--{-# INLINE lift1 #-}-lift1 :: ([a] -> [a]) -> (T a -> T a)-lift1 f (Cons x0) = Cons (f x0)--{-# INLINE lift2 #-}-lift2 :: ([a] -> [a] -> [a]) -> (T a -> T a -> T a)-lift2 f (Cons x0) (Cons x1) = Cons (f x0 x1)--{-# INLINE const #-}-const :: a -> T a-const x = lift0 [x]--{--Functor instance is e.g. useful for showing power series in residue rings.-@fmap (ResidueClass.concrete 7) (powerSeries [1,4,4::ResidueClass.T Integer] * powerSeries [1,5,6])@--}--instance Functor T where-   fmap f (Cons xs) = Cons (map f xs)--{-# INLINE appPrec #-}-appPrec :: Int-appPrec  = 10--instance (Show a) => Show (T a) where-   showsPrec p (Cons xs) =-     showParen (p >= appPrec) (showString "PowerSeries.fromCoeffs " . shows xs)---{-# INLINE truncate #-}-truncate :: Int -> T a -> T a-truncate n = lift1 (take n)--{- |-Evaluate (truncated) power series.--}-{-# INLINE evaluate #-}-evaluate :: Ring.C a => T a -> a -> a-evaluate (Cons y) = Core.evaluate y--{- |-Evaluate (truncated) power series.--}-{-# INLINE evaluateCoeffVector #-}-evaluateCoeffVector :: Module.C a v => T v -> a -> v-evaluateCoeffVector (Cons y) = Core.evaluateCoeffVector y---{-# INLINE evaluateArgVector #-}-evaluateArgVector :: (Module.C a v, Ring.C v) => T a -> v -> v-evaluateArgVector (Cons y) = Core.evaluateArgVector y--{- |-Evaluate approximations that is evaluate all truncations of the series.--}-{-# INLINE approximate #-}-approximate :: Ring.C a => T a -> a -> [a]-approximate (Cons y) = Core.approximate y---{- |-Evaluate approximations that is evaluate all truncations of the series.--}-{-# INLINE approximateCoeffVector #-}-approximateCoeffVector :: Module.C a v => T v -> a -> [v]-approximateCoeffVector (Cons y) = Core.approximateCoeffVector y---{- |-Evaluate approximations that is evaluate all truncations of the series.--}-{-# INLINE approximateArgVector #-}-approximateArgVector :: (Module.C a v, Ring.C v) => T a -> v -> [v]-approximateArgVector (Cons y) = Core.approximateArgVector y---{--Note that the derived instances only make sense for finite series.--}--instance (Eq a, ZeroTestable.C a) => Eq (T a) where-   (Cons x) == (Cons y) = Poly.equal x y--instance (Additive.C a) => Additive.C (T a) where-   negate = lift1 Poly.negate-   (+)    = lift2 Poly.add-   (-)    = lift2 Poly.sub-   zero   = lift0 []--instance (Ring.C a) => Ring.C (T a) where-   one           = const one-   fromInteger n = const (fromInteger n)-   (*)           = lift2 Core.mul--instance Vector.C T where-   zero  = zero-   (<+>) = (+)-   (*>)  = Vector.functorScale--instance (Module.C a b) => Module.C a (T b) where-   (*>) x = lift1 (x *>)--instance (Field.C a, Module.C a b) => VectorSpace.C a (T b)---instance (Field.C a) => Field.C (T a) where-   (/) = lift2 Core.divide---instance (ZeroTestable.C a, Field.C a) => Integral.C (T a) where-   divMod (Cons x) (Cons y) =-      let (d,m) = Core.divMod x y-      in  (Cons d, Cons m)---instance (Ring.C a) => Differential.C (T a) where-   differentiate = lift1 Core.differentiate---instance (Algebraic.C a) => Algebraic.C (T a) where-   sqrt   = lift1 (Core.sqrt Algebraic.sqrt)-   x ^/ y = lift1 (Core.pow (Algebraic.^/ y)-                       (fromRational' y)) x---instance (Transcendental.C a) =>-             Transcendental.C (T a) where-   pi = const Transcendental.pi-   exp = lift1 (Core.exp Transcendental.exp)-   sin = lift1 (Core.sin Core.sinCosScalar)-   cos = lift1 (Core.cos Core.sinCosScalar)-   tan = lift1 (Core.tan Core.sinCosScalar)-   x ** y = Transcendental.exp (Transcendental.log x * y)-                {- This order of multiplication is especially fast-                   when y is a singleton. -}-   log  = lift1 (Core.log  Transcendental.log)-   asin = lift1 (Core.asin Algebraic.sqrt Transcendental.asin)-   acos = lift1 (Core.acos Algebraic.sqrt Transcendental.acos)-   atan = lift1 (Core.atan Transcendental.atan)--{- |-It fulfills-  @ evaluate x . evaluate y == evaluate (compose x y) @--}--compose :: (Ring.C a, ZeroTestable.C a) => T a -> T a -> T a-compose (Cons [])    (Cons []) = Cons []-compose (Cons (x:_)) (Cons []) = Cons [x]-compose (Cons x) (Cons (y:ys)) =-   if isZero y-     then Cons (Core.compose x ys)-     else error "PowerSeries.compose: inner series must not have an absolute term."
− src-ghc-6.12/MathObj/PowerSeries/Core.hs 
@@ -1,279 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module MathObj.PowerSeries.Core where--import qualified MathObj.Polynomial.Core as Poly--import qualified Algebra.Module         as Module-import qualified Algebra.Transcendental as Transcendental-import qualified Algebra.Field          as Field-import qualified Algebra.Ring           as Ring-import qualified Algebra.Additive       as Additive-import qualified Algebra.ZeroTestable   as ZeroTestable--import qualified Data.List.Match as Match-import qualified NumericPrelude.Numeric as NP-import qualified NumericPrelude.Base as P--import NumericPrelude.Base    hiding (const)-import NumericPrelude.Numeric hiding (negate, stdUnit, divMod,-                              sqrt, exp, log,-                              sin, cos, tan, asin, acos, atan)---{-# INLINE evaluate #-}-evaluate :: Ring.C a => [a] -> a -> a-evaluate = flip Poly.horner--{-# INLINE evaluateCoeffVector #-}-evaluateCoeffVector :: Module.C a v => [v] -> a -> v-evaluateCoeffVector = flip Poly.hornerCoeffVector--{-# INLINE evaluateArgVector #-}-evaluateArgVector :: (Module.C a v, Ring.C v) => [a] -> v -> v-evaluateArgVector = flip Poly.hornerArgVector---{-# INLINE approximate #-}-approximate :: Ring.C a => [a] -> a -> [a]-approximate y x =-   scanl (+) zero (zipWith (*) (iterate (x*) 1) y)--{-# INLINE approximateCoeffVector #-}-approximateCoeffVector :: Module.C a v => [v] -> a -> [v]-approximateCoeffVector y x =-   scanl (+) zero (zipWith (*>) (iterate (x*) 1) y)--{-# INLINE approximateArgVector #-}-approximateArgVector :: (Module.C a v, Ring.C v) => [a] -> v -> [v]-approximateArgVector y x =-   scanl (+) zero (zipWith (*>) y (iterate (x*) 1))---{- * Simple series manipulation -}--{- |-For the series of a real function @f@-compute the series for @\x -> f (-x)@--}--alternate :: Additive.C a => [a] -> [a]-alternate = zipWith id (cycle [id, NP.negate])--{- |-For the series of a real function @f@-compute the series for @\x -> (f x + f (-x)) \/ 2@--}--holes2 :: Additive.C a => [a] -> [a]-holes2 = zipWith id (cycle [id, P.const zero])--{- |-For the series of a real function @f@-compute the real series for @\x -> (f (i*x) + f (-i*x)) \/ 2@--}-holes2alternate :: Additive.C a => [a] -> [a]-holes2alternate =-   zipWith id (cycle [id, P.const zero, NP.negate, P.const zero])---{- * Series arithmetic -}--add, sub :: (Additive.C a) => [a] -> [a] -> [a]-add = Poly.add-sub = Poly.sub--negate :: (Additive.C a) => [a] -> [a]-negate = Poly.negate--scale :: Ring.C a => a -> [a] -> [a]-scale = Poly.scale--mul :: Ring.C a => [a] -> [a] -> [a]-mul = Poly.mul---stripLeadZero :: (ZeroTestable.C a) => [a] -> [a] -> ([a],[a])-stripLeadZero (x:xs) (y:ys) =-  if isZero x && isZero y-    then stripLeadZero xs ys-    else (x:xs,y:ys)-stripLeadZero xs ys = (xs,ys)---divMod :: (ZeroTestable.C a, Field.C a) => [a] -> [a] -> ([a],[a])-divMod xs ys =-   let (yZero,yRem) = span isZero ys-       (xMod, xRem) = Match.splitAt yZero xs-   in  (divide xRem yRem, xMod)--{- |-Divide two series where the absolute term of the divisor is non-zero.-That is, power series with leading non-zero terms are the units-in the ring of power series.--Knuth: Seminumerical algorithms--}-divide :: (Field.C a) => [a] -> [a] -> [a]-divide (x:xs) (y:ys) =-   let zs = map (/y) (x : sub xs (mul zs ys))-   in  zs-divide [] _ = []-divide _ [] = error "PowerSeries.divide: division by empty series"--{- |-Divide two series also if the divisor has leading zeros.--}-divideStripZero :: (ZeroTestable.C a, Field.C a) => [a] -> [a] -> [a]-divideStripZero x' y' =-   let (x0,y0) = stripLeadZero x' y'-   in  if null y0 || isZero (head y0)-         then error "PowerSeries.divideStripZero: Division by zero."-         else divide x0 y0---progression :: Ring.C a => [a]-progression = Poly.progression--recipProgression :: (Field.C a) => [a]-recipProgression = map recip progression--differentiate :: (Ring.C a) => [a] -> [a]-differentiate = Poly.differentiate--integrate :: (Field.C a) => a -> [a] -> [a]-integrate = Poly.integrate---{- |-We need to compute the square root only of the first term.-That is, if the first term is rational,-then all terms of the series are rational.--}-sqrt :: Field.C a => (a -> a) -> [a] -> [a]-sqrt _ [] = []-sqrt f0 (x:xs) =-   let y  = f0 x-       ys = map (/(y+y)) (xs - (0 : mul ys ys))-   in  y:ys--{--pow alpha t = t^alpha-(pow alpha . x)' = alpha * (pow (alpha-1) . x) * x'-alpha * (pow alpha . x) = x * x' * (pow alpha . x)'-y = pow alpha . x-alpha * y = x * x' * y'--}--{- |-Input series must start with non-zero term.--}-pow :: (Field.C a) => (a -> a) -> a -> [a] -> [a]-pow f0 expon x =-   let y  = integrate (f0 (head x)) y'-       y' = scale expon (divide y (mul x (differentiate x)))-   in  y---{- |-The first term needs a transcendent computation but the others do not.-That's why we accept a function which computes the first term.--> (exp . x)' =   (exp . x) * x'-> (sin . x)' =   (cos . x) * x'-> (cos . x)' = - (sin . x) * x'--}-exp :: Field.C a => (a -> a) -> [a] -> [a]-exp f0 x =-   let x' = differentiate x-       y  = integrate (f0 (head x)) (mul y x')-   in  y--sinCos :: Field.C a => (a -> (a,a)) -> [a] -> ([a],[a])-sinCos f0 x =-   let (y0Sin, y0Cos) = f0 (head x)-       x'   = differentiate x-       ySin = integrate y0Sin         (mul yCos x')-       yCos = integrate y0Cos (negate (mul ySin x'))-   in  (ySin, yCos)--sinCosScalar :: Transcendental.C a => a -> (a,a)-sinCosScalar x = (Transcendental.sin x, Transcendental.cos x)--sin, cos :: Field.C a => (a -> (a,a)) -> [a] -> [a]-sin f0 = fst . sinCos f0-cos f0 = snd . sinCos f0--tan :: (Field.C a) => (a -> (a,a)) -> [a] -> [a]-tan f0 = uncurry divide . sinCos f0--{--(log x)' == x'/x-(asin x)' == (acos x) == x'/sqrt(1-x^2)-(atan x)' == x'/(1+x^2)--}--{- |-Input series must start with non-zero term.--}-log :: (Field.C a) => (a -> a) -> [a] -> [a]-log f0 x = integrate (f0 (head x)) (derivedLog x)--{- |-Computes @(log x)'@, that is @x'\/x@--}-derivedLog :: (Field.C a) => [a] -> [a]-derivedLog x = divide (differentiate x) x--atan :: (Field.C a) => (a -> a) -> [a] -> [a]-atan f0 x =-   let x' = differentiate x-   in  integrate (f0 (head x)) (divide x' ([1] + mul x x))--asin, acos :: (Field.C a) =>-   (a -> a) -> (a -> a) -> [a] -> [a]-asin sqrt0 f0 x =-   let x' = differentiate x-   in  integrate (f0 (head x))-                 (divide x' (sqrt sqrt0 ([1] - mul x x)))-acos = asin--{- |-Since the inner series must start with a zero,-the first term is omitted in y.--}-compose :: (Ring.C a) => [a] -> [a] -> [a]-compose xs y = foldr (\x acc -> x : mul y acc) [] xs---{- |-Compose two power series where the outer series-can be developed for any expansion point.-To be more precise:-The outer series must be expanded with respect to the leading term-of the inner series.--}-composeTaylor :: Ring.C a => (a -> [a]) -> [a] -> [a]-composeTaylor x (y:ys) = compose (x y) ys-composeTaylor x []     = x 0----{--(x . y) = id-(x' . y) * y' = 1-y' = 1 / (x' . y)--}--{- |-This function returns the series of the function in the form:-(point of the expansion, power series)--This is exceptionally slow and needs cubic run-time.--}--inv :: (Field.C a) => [a] -> (a, [a])-inv x =-   let y' = divide [1] (compose (differentiate x) (tail y))-       y  = integrate 0 y'-            -- the first term is zero, which is required for composition-   in  (head x, y)
− src-ghc-6.12/MathObj/PowerSeries/DifferentialEquation.hs 
@@ -1,81 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Lazy evaluation allows for the solution- of differential equations in terms of power series.-Whenever you can express the highest derivative of the solution- as explicit expression of the lower derivatives- where each coefficient of the solution series- depends only on lower coefficients,- the recursive algorithm will work.--}--module MathObj.PowerSeries.DifferentialEquation where--import qualified MathObj.PowerSeries.Core    as PS-import qualified MathObj.PowerSeries.Example as PSE--import qualified Algebra.Field        as Field-import qualified Algebra.ZeroTestable as ZeroTestable--import NumericPrelude.Numeric-import NumericPrelude.Base---{- |-Example for a linear equation:-   Setup a differential equation for @y@ with-->    y   t = (exp (-t)) * (sin t)->    y'  t = -(exp (-t)) * (sin t) + (exp (-t)) * (cos t)->    y'' t = -2 * (exp (-t)) * (cos t)--Thus the differential equation-->    y'' = -2 * (y' + y)--holds.--The following function generates-a power series for @exp (-t) * sin t@-by solving the differential equation.--}--solveDiffEq0 :: (Field.C a) => [a]-solveDiffEq0 =-   let -- the initial conditions are passed to "PS.integrate"-       y   = PS.integrate 0 y'-       y'  = PS.integrate 1 y''-       y'' = PS.scale (-2) (PS.add y' y)-   in  y--verifyDiffEq0 :: (Field.C a) => [a]-verifyDiffEq0 =-   PS.mul (zipWith (*) (iterate negate 1) PSE.exp) PSE.sin--propDiffEq0 :: Bool-propDiffEq0 =  solveDiffEq0 == (verifyDiffEq0 :: [Rational])---{- |-We are not restricted to linear equations!- Let the solution be y with-  y   t =   (1-t)^-1-  y'  t =   (1-t)^-2-  y'' t = 2*(1-t)^-3- then it holds-  y'' = 2 * y' * y--}--solveDiffEq1 :: (ZeroTestable.C a, Field.C a) => [a]-solveDiffEq1 =-   let -- the initial conditions are passed to "PS.integrate"-       y   = PS.integrate 1 y'-       y'  = PS.integrate 1 y''-       y'' = PS.scale 2 (PS.mul y' y)-   in  y--verifyDiffEq1 :: (ZeroTestable.C a, Field.C a) => [a]-verifyDiffEq1 = PS.divide [1] [1, -1]--propDiffEq1 :: Bool-propDiffEq1 =  solveDiffEq1 == (verifyDiffEq1 :: [Rational])
− src-ghc-6.12/MathObj/PowerSeries/Example.hs 
@@ -1,156 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module MathObj.PowerSeries.Example where--import qualified MathObj.PowerSeries.Core as PS--import qualified Algebra.Field          as Field-import qualified Algebra.Ring           as Ring--- import qualified Algebra.Additive       as Additive-import qualified Algebra.ZeroTestable   as ZeroTestable-import qualified Algebra.Transcendental as Transcendental--import Algebra.Additive (zero, subtract, negate)--import Data.List (intersperse, )-import Data.List.HT (sieve, )--import NumericPrelude.Numeric (one, (*), (/),-                       fromInteger, {-fromRational,-} pi)-import NumericPrelude.Base -- (Bool, const, map, zipWith, id, (&&), (==))---{- * Default implementations. -}--recip :: (Ring.C a) => [a]-recip = recipExpl--exp, sin, cos,-  log, asin, atan, sqrt :: (Field.C a) => [a]-acos :: (Transcendental.C a) => [a]-tan :: (ZeroTestable.C a, Field.C a) => [a]-exp = expODE-sin = sinODE-cos = cosODE-tan = tanExplSieve-log = logODE-asin = asinODE-acos = acosODE-atan = atanODE--sinh, cosh, atanh :: (Field.C a) => [a]-sinh  = sinhODE-cosh  = coshODE-atanh = atanhODE--pow :: (Field.C a) => a -> [a]-pow = powExpl-sqrt = sqrtExpl---{- * Generate Taylor series explicitly. -}--recipExpl :: (Ring.C a) => [a]-recipExpl = cycle [1,-1]--expExpl, sinExpl, cosExpl :: (Field.C a) => [a]-expExpl = scanl (*) one PS.recipProgression-sinExpl = zero : PS.holes2alternate (tail expExpl)-cosExpl =        PS.holes2alternate       expExpl--tanExpl, tanExplSieve :: (ZeroTestable.C a, Field.C a) => [a]-tanExpl = PS.divide sinExpl cosExpl--- ignore zero values-tanExplSieve =-   concatMap-      (\x -> [zero,x])-      (PS.divide (sieve 2 (tail sin)) (sieve 2 cos))--logExpl, atanExpl, sqrtExpl :: (Field.C a) => [a]-logExpl  = zero : PS.alternate       PS.recipProgression-atanExpl = zero : PS.holes2alternate PS.recipProgression--sinhExpl, coshExpl, atanhExpl :: (Field.C a) => [a]-sinhExpl  = zero : PS.holes2 (tail expExpl)-coshExpl  =        PS.holes2       expExpl-atanhExpl = zero : PS.holes2 PS.recipProgression--{- * Power series of (1+x)^expon using the binomial series. -}--powExpl :: (Field.C a) => a -> [a]-powExpl expon =-   scanl (*) 1 (zipWith (/)-      (iterate (subtract 1) expon) PS.progression)-sqrtExpl = powExpl (1/2)--{- |-Power series of error function (almost).-More precisely @ erf = 2 \/ sqrt pi * integrate (\x -> exp (-x^2)) @,-with @erf 0 = 0@.--}--erf :: (Field.C a) => [a]-erf = PS.integrate 0 $ intersperse 0 $ PS.alternate exp--{--integrate (\x -> exp (-x^2/2)) :--erf = PS.integrate 0 $ intersperse 0 $-    snd $ mapAccumL (\twoPow c -> (twoPow/(-2), twoPow*c)) 1 exp--}---{- * Generate Taylor series from differential equations. -}--{--exp' x == exp x-sin' x == cos x-cos' x == - sin x--tan' x == 1 + tan x ^ 2-       == cos x ^ (-2)--}--expODE, sinODE, cosODE, tanODE, tanODESieve :: (Field.C a) => [a]-expODE = PS.integrate 1 expODE-sinODE = PS.integrate 0 cosODE-cosODE = PS.integrate 1 (PS.negate sinODE)-tanODE = PS.integrate 0 (PS.add [1] (PS.mul tanODE tanODE))-tanODESieve =-   -- sieve is too strict here because it wants to detect end of lists-   let tan2 = map head (iterate (drop 2) (tail tanODESieve))-   in  PS.integrate 0 (intersperse zero (1 : PS.mul tan2 tan2))--{--log' (1+x) == 1/(1+x)-asin' x == acos' x == 1/sqrt(1-x^2)-atan' x == 1/(1+x^2)--}--logODE, recipCircle, asinODE, atanODE, sqrtODE :: (Field.C a) => [a]-logODE  = PS.integrate zero recip-recipCircle = intersperse zero (PS.alternate (powODE (-1/2)))-asinODE = PS.integrate 0 recipCircle-atanODE = PS.integrate zero (cycle [1,0,-1,0])-sqrtODE = powODE (1/2)--acosODE :: (Transcendental.C a) => [a]-acosODE = PS.integrate (pi/2) recipCircle--sinhODE, coshODE, atanhODE :: (Field.C a) => [a]-sinhODE = PS.integrate 0 coshODE-coshODE = PS.integrate 1 sinhODE-atanhODE = PS.integrate zero (cycle [1,0])---{--Power series for y with-   y x = (1+x) ** alpha-by solving the differential equation-   alpha * y x = (1+x) * y' x--}--powODE :: (Field.C a) => a -> [a]-powODE expon =-   let y  = PS.integrate 1 y'-       y' = PS.scale expon (scanl1 subtract y)-   in  y
− src-ghc-6.12/MathObj/PowerSeries/Mean.hs 
@@ -1,234 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-This module computes power series for-representing some means as generalized $f$-means.--}-module MathObj.PowerSeries.Mean where--import qualified MathObj.PowerSeries2        as PS2-import qualified MathObj.PowerSeries2.Core   as PS2Core-import qualified MathObj.PowerSeries         as PS-import qualified MathObj.PowerSeries.Core    as PSCore-import qualified MathObj.PowerSeries.Example as PSE--import qualified Algebra.Field as Field-import qualified Algebra.Ring  as Ring--import Data.List.HT (shearTranspose)--import NumericPrelude.Numeric-import NumericPrelude.Base--{--$M_f$ is a generalized $f$-mean (quasi-arithmetic) if-\[M_f x = f^{ -1}\right(\frac{1}{n}\cdot\sum_{k=1}^{n} f(x_k)\left)\]--For instance there is the logarithmic mean-defined by-\[\frac{x-y}{\ln x - \ln y}\]-whose definition is inherently bound to two variables.-If we find a representation as a generalized $f$-mean-we can generalize this mean to more than two variables.--Btw. we can easily see that the logarithmic mean is not a quasi-arithmetic mean,-because \[ \anonymfunc{(a,b,c,d)}{L(L(a,b),L(c,d))} \]-is not commutative, but quasi-arithmetic means are always commutative.--First we note that an arbitrary constant offset and-an arbitrary scaling of $f$ does not alter the mean.-Therefore we choose $f(1)=0, f'(1)=1$-and we expand $f$ into a Taylor series with respect to 1.--For the logarithmic mean we will choose $y=0$.-This way we might get additional virtual solutions,-but we can identify them afterwards by a test.-\begin{eqnarray*}-f^{ -1}\left(\frac{f(1+x)+f(1+y)}{2}\right)- &=& \frac{x-y}{\ln(1+x) - \ln(1+y)} \\-f^{ -1}\left(\frac{f(1+x)}{2}\right)- &=& \frac{x}{\ln(1+x)} \\-f(1+x)- &=& 2 \cdot f\left(\frac{x}{\ln(1+x)}\right)-\end{eqnarray*}-This cannot be solved immediately-because in the power series expansions on both sides-unknown coefficients occur at the same monomials.-We can resolve that by subtracting the series of $2\cdot f(1+x/2)$-off both sides.-\begin{eqnarray*}-f(1+x) - 2\cdot f(1+x/2)- &=& 2 \cdot (f\left(\frac{x}{\ln(1+x)}\right) - f(1+x/2))-\end{eqnarray*}-We note that $1+x/2$ is the truncated series of $\frac{x}{\ln(1+x)}$.-This is also necessary in order to obtain an equation.--Now we have to derive an implementation of the right-hand side.-This is a difference of two series compositions, namely-$f(x+a*x^2+b*x^3+\dots) - f(x)$ .-The implementation takes care that the vanishing terms are not computed-and thus allows solution of series fixed point equations.-It is just done by throwing away the leading terms of all powers-of the series $x+a*x^2+b*x^3+\dots$.-In $x$ the constant monomial is omitted,-in the result both the constant and the linear term are omitted.--}--diffComp :: (Ring.C a) => [a] -> [a] -> [a]-diffComp ys x =-   map sum (shearTranspose (tail (zipWith PSCore.scale ys-                    (map tail (iterate (PSCore.mul x) [1])))))--{--Now we solve-\[-\frac{1}{2}\cdot f(1+2\cdot x) - f(1+x)- &=& f\left(\frac{2\cdot x}{\ln(1+2\cdot x)}\right) - f(1+x)-\]--}--logarithmic :: (Field.C a) => [a]-logarithmic =-   let -- series for \frac{2\cdot x}{\ln(1+2\cdot x)}-       fracLn = PSCore.divide [2]-                      (tail (zipWith (*) (iterate (2*) 1) PSE.log))-       fDiffFracLn = diffComp f (tail fracLn)-       f = 0 : 1 : zipWith (/) fDiffFracLn-                      (map (subtract 1) (iterate (2*) 2))-   in  f--elemSym3_2 :: (Field.C a) => [a]-elemSym3_2 =-   let -- series for \frac{2\cdot x}{\ln(1+2\cdot x)}-       root = zipWith (*) (iterate (2*) 1) PSE.sqrt-       fDiffRoot = diffComp f (tail root)-       f = 0 : 1 : zipWith (/) fDiffRoot-                      (map (subtract 1) (iterate (3*) 3))-   in  f---{--Means constructed by mean value theorem.--\[ M(x,y) = f'^{ -1}((f(x)-f(y))/(x-y)) \]--\[ f(x) = x^2  \implies M - arithmetic mean \]-\[ f(x) = 1/x  \implies M - geometric mean \]--Try to find a power series for $f$ for $M(x,y) = \sqrt{(x^2+y^2)/2}$-(quadratic mean).-Expansion point: 1.-$M(1+t,1) = \sqrt{1+t+t^2/2}$--}-quadratic :: (Field.C a, Eq a) => [a]-quadratic = PSCore.sqrt (\1 -> 1) [1,1,1/2]--quadraticMVF :: (Field.C a) => [a]-quadraticMVF =-   -- [1,1,1,1,1/2,3/23,2/143]-   -- [1,1,1,1,1/2,1/2]-   [1,1,1,1,1/2,-1/14]---- map (\x -> PSCore.coeffs (meanValueDiff2 quadratic2 [1,1,1,1,1/2,x] !! 4) !! 2) (GNUPlot.linearScale 10 (-0.071429,-1/14::Double))--- take 20 $ Numerics.ZeroFinder.RegulaFalsi.zero (-1,0) (\x -> PSCore.coeffs (meanValueDiff2 quadratic2 [1::Double,1,1,1,1/2,x] !! 4) !! 2)--{--Result: It seems,-that we cannot find an appropriate coefficient for the 5th power.-This indicates that it is not possible to represent-the quadratic mean as mean value mean.--}--quadraticDiff :: (Field.C a, Eq a) => [a]-quadraticDiff =-   let divDiffPS = tail quadraticMVF -- (f(1+t)-f(1))/((1+t)-1)-       (1, invPS) = PSCore.inv (PSCore.differentiate quadraticMVF)-       meanValuePS = PSCore.composeTaylor (\1 -> invPS) divDiffPS-       {- instead of computing an inverse series-          we could also apply (compose) the derived series-          to the series of the quadratic mean. -}-   in  quadratic - meanValuePS--{--Represent quadratic mean with a two-variate power series.--$M(1+x,1+y) = \sqrt{1+x+y+(x^2+y^2)/2}$--}-quadratic2 :: (Field.C a, Eq a) => PS2Core.T a-quadratic2 =-   PS2Core.sqrt (\1 -> 1) [[1],[1,1],[1/2,0,1/2]]--quadraticDiff2 :: (Field.C a, Eq a) => PS2Core.T a-quadraticDiff2 =-   meanValueDiff2 quadratic2 quadraticMVF----{--We can alter the square coefficient,-but consequently we have to scale the sub-sequent coefficients.-If the square coefficient is zero then the equation is fulfilled,-but this is a non-solution because it is degenerate.--}-harmonicMVF :: (Field.C a) => [a]-harmonicMVF =-   -- [1,1,1,-2,7/2,-62/11]-   -- [1,1,2,-4,7,-124/11]-   [1,1,3,-6,21/2,-186/11]--{--$M(1+x,1+y) = 2/(recip (1+x) + recip (1+y))$--}-harmonic2 :: (Field.C a, Eq a) => PS2Core.T a-harmonic2 =-   let rec = PS.fromCoeffs PSE.recip-   in  PS2Core.divide [[2]] $-       PS2.coeffs $-          PS2.fromPowerSeries0 rec +-          PS2.fromPowerSeries1 rec--harmonicDiff2 :: (Field.C a, Eq a) => PS2Core.T a-harmonicDiff2 =-   meanValueDiff2 harmonic2 harmonicMVF----arithmeticMVF :: (Field.C a) => [a]-arithmeticMVF = [1,2,1]--{--$M(1+x,1+y) = 1+x/2+y/2$--}-arithmetic2 :: (Field.C a, Eq a) => PS2Core.T a-arithmetic2 = [[1],[1/2,1/2]]--arithmeticDiff2 :: (Field.C a, Eq a) => PS2Core.T a-arithmeticDiff2 =-   meanValueDiff2 arithmetic2 arithmeticMVF---geometricMVF :: (Field.C a) => [a]-geometricMVF = PSE.recip--{--$M(1+x,1+y) = \sqrt{(1+x)·(1+y)}$--}-geometric2 :: (Field.C a, Eq a) => PS2Core.T a-geometric2 =-   PS2Core.sqrt (\1 -> 1) [[1],[1,1],[0,1,0]]--geometricDiff2 :: (Field.C a, Eq a) => PS2Core.T a-geometricDiff2 =-   meanValueDiff2 geometric2 geometricMVF-----meanValueDiff2 :: (Field.C a, Eq a) =>-   PS2Core.T a -> [a] -> PS2Core.T a-meanValueDiff2 mean2 curve =-   let -- (f(1+x)-f(1+y)) / (x-y)-       divDiffPS =-          zipWith replicate [1..] $ tail curve-       meanValuePS =-          PS2Core.compose (PSCore.differentiate curve) (tail mean2)-   in  meanValuePS - divDiffPS
− src-ghc-6.12/MathObj/PowerSeries2.hs 
@@ -1,126 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}--{- |-Two-variate power series.--}--module MathObj.PowerSeries2 where--import qualified MathObj.PowerSeries2.Core as Core-import qualified MathObj.PowerSeries as PS-import qualified MathObj.Polynomial.Core as Poly--import qualified Algebra.Vector         as Vector-import qualified Algebra.Algebraic      as Algebraic-import qualified Algebra.Field          as Field-import qualified Algebra.Ring           as Ring-import qualified Algebra.Additive       as Additive-import qualified Algebra.ZeroTestable   as ZeroTestable--import qualified NumericPrelude.Numeric as NP-import qualified NumericPrelude.Base as P--import Data.List (isPrefixOf, )-import qualified Data.List.Match as Match--import NumericPrelude.Base    hiding (const)-import NumericPrelude.Numeric--{- |-In order to handle both variables equivalently-we maintain a list of coefficients for terms of the same total degree.-That is--> eval [[a], [b,c], [d,e,f]] (x,y) ==->    a + b*x+c*y + d*x^2+e*x*y+f*y^2--Although the sub-lists are always finite and thus are more like polynomials than power series,-division and square root computation are easier to implement for power series.--}-newtype T a = Cons {coeffs :: Core.T a} deriving (Ord)---isValid :: [[a]] -> Bool-isValid = flip isPrefixOf [1..] . map length--check :: [[a]] -> [[a]]-check xs =-   zipWith (\n x ->-      if Match.compareLength n x == EQ-        then x-        else error "PowerSeries2.check: invalid length of sub-list")-     (iterate (():) [()]) xs---fromCoeffs :: [[a]] -> T a-fromCoeffs  =  Cons . check--fromPowerSeries0 :: Ring.C a => PS.T a -> T a-fromPowerSeries0 x =-   fromCoeffs $-   zipWith (:) (PS.coeffs x) $-   iterate (0:) []--fromPowerSeries1 :: Ring.C a => PS.T a -> T a-fromPowerSeries1 x =-   fromCoeffs $-   zipWith (++) (iterate (0:) []) $-   map (:[]) (PS.coeffs x)---lift0 :: Core.T a -> T a-lift0 = Cons--lift1 :: (Core.T a -> Core.T a) -> (T a -> T a)-lift1 f (Cons x0) = Cons (f x0)--lift2 :: (Core.T a -> Core.T a -> Core.T a) -> (T a -> T a -> T a)-lift2 f (Cons x0) (Cons x1) = Cons (f x0 x1)---const :: a -> T a-const x = lift0 [[x]]---instance Functor T where-   fmap f (Cons xs) = Cons (map (map f) xs)--appPrec :: Int-appPrec  = 10--instance (Show a) => Show (T a) where-   showsPrec p (Cons xs) =-      showParen (p >= appPrec) (showString "PowerSeries2.fromCoeffs " . shows xs)---instance (Eq a, ZeroTestable.C a) => Eq (T a) where-   (Cons x) == (Cons y) = Poly.equal x y--instance (Additive.C a) => Additive.C (T a) where-   negate = lift1 Core.negate-   (+)    = lift2 Core.add-   (-)    = lift2 Core.sub-   zero   = lift0 []---instance (Ring.C a) => Ring.C (T a) where-   one           = const one-   fromInteger n = const (fromInteger n)-   (*)           = lift2 Core.mul--instance Vector.C T where-   zero  = zero-   (<+>) = (+)-   (*>)  = Vector.functorScale---instance (Field.C a) => Field.C (T a) where-   (/) = lift2 Core.divide---instance (Algebraic.C a) => Algebraic.C (T a) where-   sqrt   = lift1 (Core.sqrt Algebraic.sqrt)---   x ^/ y = lift1 (Core.pow (Algebraic.^/ y)---                       (fromRational' y)) x
− src-ghc-6.12/MathObj/PowerSeries2/Core.hs 
@@ -1,89 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module MathObj.PowerSeries2.Core where--import qualified MathObj.PowerSeries as PS-import qualified MathObj.PowerSeries.Core as PSCore--import qualified Algebra.Differential   as Differential-import qualified Algebra.Vector         as Vector-import qualified Algebra.Field          as Field-import qualified Algebra.Ring           as Ring-import qualified Algebra.Additive       as Additive--import NumericPrelude.Base--- import NumericPrelude.Numeric hiding (negate, sqrt, )---type T a = [[a]]---lift0fromPowerSeries :: [PS.T a] -> T a-lift0fromPowerSeries = map PS.coeffs--lift1fromPowerSeries ::-   ([PS.T a] -> [PS.T a]) -> (T a -> T a)-lift1fromPowerSeries f x0 =-   map PS.coeffs (f (map PS.fromCoeffs x0))--lift2fromPowerSeries ::-   ([PS.T a] -> [PS.T a] -> [PS.T a]) -> (T a -> T a -> T a)-lift2fromPowerSeries f x0 x1 =-   map PS.coeffs (f (map PS.fromCoeffs x0) (map PS.fromCoeffs x1))---{- * Series arithmetic -}--add, sub :: (Additive.C a) => T a -> T a -> T a-add = PSCore.add-sub = PSCore.sub--negate :: (Additive.C a) => T a -> T a-negate = PSCore.negate---scale :: Ring.C a => a -> T a -> T a-scale = map . (Vector.*>)--mul :: Ring.C a => T a -> T a -> T a-mul = lift2fromPowerSeries PSCore.mul---divide :: (Field.C a) =>-   T a -> T a -> T a-divide = lift2fromPowerSeries PSCore.divide---sqrt :: (Field.C a) =>-   (a -> a) -> T a -> T a-sqrt fSqRt =-   lift1fromPowerSeries $-   PSCore.sqrt (PS.const . (\[x] -> fSqRt x) . PS.coeffs)----swapVariables :: T a -> T a-swapVariables = map reverse---differentiate0 :: (Ring.C a) => T a -> T a-differentiate0 =-   swapVariables . differentiate1 . swapVariables--differentiate1 :: (Ring.C a) => T a -> T a-differentiate1 = lift1fromPowerSeries $ map Differential.differentiate--integrate0 :: (Field.C a) => [a] -> T a -> T a-integrate0 cs =-   swapVariables . integrate1 cs . swapVariables--integrate1 :: (Field.C a) => [a] -> T a -> T a-integrate1 = zipWith PSCore.integrate----{- |-Since the inner series must start with a zero,-the first term is omitted in y.--}-compose :: (Ring.C a) => [a] -> T a -> T a-compose = lift1fromPowerSeries . PSCore.compose . map PS.const
− src-ghc-6.12/MathObj/PowerSum.hs 
@@ -1,234 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-{- |-Copyright   :  (c) Henning Thielemann 2004-2005--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  requires multi-parameter type classes---For a multi-set of numbers,-we describe a sequence of the sums of powers of the numbers in the set.-These can be easily converted to polynomials and back.-Thus they provide an easy way for computations on the roots of a polynomial.--}-module MathObj.PowerSum where--import qualified MathObj.Polynomial as Poly-import qualified MathObj.Polynomial.Core as PolyCore-import qualified MathObj.PowerSeries.Core as PS--import qualified Algebra.VectorSpace  as VectorSpace-import qualified Algebra.Module       as Module-import qualified Algebra.Algebraic    as Algebraic-import qualified Algebra.Field        as Field-import qualified Algebra.IntegralDomain as Integral-import qualified Algebra.Ring         as Ring-import qualified Algebra.Additive     as Additive-import qualified Algebra.ZeroTestable as ZeroTestable--import Algebra.Module((*>))--import Control.Monad(liftM2)-import qualified Data.List as List-import Data.List.HT (shearTranspose, sieve)--import NumericPrelude.Base as P hiding (const)-import NumericPrelude.Numeric as NP---newtype T a = Cons {sums :: [a]}---{- * Conversions -}--lift0 :: [a] -> T a-lift0 = Cons--lift1 :: ([a] -> [a]) -> (T a -> T a)-lift1 f (Cons x0) = Cons (f x0)--lift2 :: ([a] -> [a] -> [a]) -> (T a -> T a -> T a)-lift2 f (Cons x0) (Cons x1) = Cons (f x0 x1)---const :: (Ring.C a) => a -> T a-const x = Cons [1,x]--{- Newton-Girard formulas,  cf. Modula-3: arithmetic/RootBasic.mg-   s'/s = p -}--{--  s[k] - the elementary symmetric polynomial of degree k-  p[k] - sum of the k-th power--  s[0](x0,x1,x2) = 1-  s[1](x0,x1,x2) = x0+x1+x2-  s[2](x0,x1,x2) = x0*x1+x1*x2+x2*x0-  s[3](x0,x1,x2) = x0*x1*x2-  s[4](x0,x1,x2) = 0--  p[0](x0,x1,x2) =  1   +  1   +  1-  p[1](x0,x1,x2) = x0   + x1   + x2-  p[2](x0,x1,x2) = x0^2 + x1^2 + x2^2-  p[3](x0,x1,x2) = x0^3 + x1^3 + x2^3-  p[4](x0,x1,x2) = x0^4 + x1^4 + x2^4--  s(t) := s[0] + s[1]*t + s[2]*t^2 + ...-  p(t) :=        p[1]*t + p[2]*t^2 + ...--  Then it holds-    t*s'(t) + p(-t)*s(t) = 0-  This can be proven by considering p as sum of geometric series-  and differentiating s in the root-wise factored form.--  Note that we index the coefficients the other way round-  and that the coefficients of the polynomial-  are not pure elementary symmetric polynomials of the roots-  but have alternating signs, too.--}-fromElemSym :: (Eq a, Ring.C a) => [a] -> [a]-fromElemSym s =-   fromIntegral (length s - 1) :-      PolyCore.alternate (divOneFlip s (PolyCore.differentiate s))--divOneFlip :: (Eq a, Ring.C a) => [a] -> [a] -> [a]-divOneFlip (1:xs) =-   let aux (y:ys) = y : aux (ys - PolyCore.scale y xs)-       aux [] = []-   in  aux-divOneFlip _ =-   error "divOneFlip: first element must be one"--fromElemSymDenormalized :: (Field.C a, ZeroTestable.C a) => [a] -> [a]-fromElemSymDenormalized s =-   fromIntegral (length s - 1) :-      PolyCore.alternate (PS.derivedLog s)---toElemSym :: (Field.C a, ZeroTestable.C a) => [a] -> [a]-toElemSym p =-   let s' = PolyCore.mul (PolyCore.alternate (tail p)) s-       s  = PolyCore.integrate 1 s'-   in  s--toElemSymInt :: (Integral.C a, ZeroTestable.C a) => [a] -> [a]-toElemSymInt p =-   let s' = PolyCore.mul (PolyCore.alternate (tail p)) s-       s  = PolyCore.integrateInt 1 s'-   in  s----fromPolynomial :: (Field.C a, ZeroTestable.C a) => Poly.T a -> [a]-fromPolynomial =-   let aux s =-          fromIntegral (length s - 1) :-             PolyCore.negate (PS.derivedLog s)-   in  aux . reverse . Poly.coeffs--elemSymFromPolynomial :: Additive.C a => Poly.T a -> [a]-elemSymFromPolynomial = PolyCore.alternate . reverse . Poly.coeffs--{- toPolynomial is not possible because this had to consume the whole sum sequence. -}----binomials :: Ring.C a => [[a]]-binomials = [1] : binomials + map (0:) binomials--{- * Show -}--appPrec :: Int-appPrec  = 10--instance (Show a) => Show (T a) where-  showsPrec p (Cons xs) =-    showParen (p >= appPrec)-       (showString "PowerSum.Cons " . shows xs)---{- * Additive -}--{- Use binomial expansion of (x+y)^n -}-add :: (Ring.C a) => [a] -> [a] -> [a]-add xs ys =-   let powers = shearTranspose (PolyCore.tensorProduct xs ys)-   in  zipWith Ring.scalarProduct binomials powers--instance (Ring.C a) => Additive.C (T a) where-   zero   = const zero-   (+)    = lift2 add-   negate = lift1 PolyCore.alternate---{- * Ring -}--mul :: (Ring.C a) => [a] -> [a] -> [a]-mul xs ys = zipWith (*) xs ys--pow :: Integer -> [a] -> [a]-pow n =-   if n<0-     then error "PowerSum.pow: negative exponent"-     else sieve (fromInteger n)-       -- map head . iterate (List.genericDrop (toInteger n))--instance (Ring.C a) => Ring.C (T a) where-   one           = const one-   fromInteger n = const (fromInteger n)-   (*)           = lift2 mul-   x^n           = lift1 (pow n) x---{- * Module -}--instance (Module.C a v, Ring.C v) => Module.C a (T v) where-   x *> y = lift1 (zipWith (*>) (iterate (x*) one)) y--instance (VectorSpace.C a v, Ring.C v) => VectorSpace.C a (T v)---{- * Field.C -}--instance (Field.C a, ZeroTestable.C a) => Field.C (T a) where-   recip = lift1 (fromElemSymDenormalized . reverse . toElemSym)---{- * Algebra -}--root :: (Ring.C a) => Integer -> [a] -> [a]-root n xs =-   let upsample m ys =-          concat (List.intersperse-             (List.genericReplicate (m - 1) zero)-             (map (:[]) ys))-   in  case compare n 0 of-          LT -> upsample (-n) (reverse xs)-          GT -> upsample n xs-          EQ -> [1]--instance (Field.C a, ZeroTestable.C a) => Algebraic.C (T a) where-   root n = lift1 (fromElemSymDenormalized . root n . toElemSym)---{- given the list of power sums @x1^j + ... + xn^j@-   and a power series for the function @f@,-   compute the series approximations of @f(x1) + ... + f(xn)@. -}-approxSeries :: Module.C a b => [b] -> [a] -> [b]-approxSeries y x =-   scanl (+) zero (zipWith (*>) x y)---{- input lists contain roots -}-propOp :: (Eq a, Field.C a, ZeroTestable.C a) =>-   ([a] -> [a] -> [a]) -> (a -> a -> a) -> [a] -> [a] -> [Bool]-propOp powerOp op xs ys =-   let zs = liftM2 op xs ys-       xp = fromPolynomial (Poly.fromRoots xs)-       yp = fromPolynomial (Poly.fromRoots ys)-       ze = elemSymFromPolynomial (Poly.fromRoots zs)-   in  zipWith (==) (toElemSym (powerOp xp yp)) ze-       -- PolyCore.equal (toElemSym (powerOp xp yp)) ze
− src-ghc-6.12/MathObj/RefinementMask2.hs 
@@ -1,171 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module MathObj.RefinementMask2 (-   T, coeffs, fromCoeffs,-   fromPolynomial,-   toPolynomial,-   toPolynomialFast,-   refinePolynomial,-   ) where--import qualified MathObj.Polynomial as Poly-import qualified Algebra.RealField as RealField-import qualified Algebra.Field  as Field-import qualified Algebra.Ring   as Ring-import qualified Algebra.Vector as Vector--import qualified Data.List as List-import qualified Data.List.HT as ListHT-import qualified Data.List.Match as Match-import Control.Monad (liftM2, )--import qualified Test.QuickCheck as QC--import qualified NumericPrelude.List.Generic as NPList-import NumericPrelude.Base-import NumericPrelude.Numeric---newtype T a = Cons {coeffs :: [a]}---{-# INLINE fromCoeffs #-}-fromCoeffs :: [a] -> T a-fromCoeffs = lift0--{-# INLINE lift0 #-}-lift0 :: [a] -> T a-lift0 = Cons--{--{-# INLINE lift1 #-}-lift1 :: ([a] -> [a]) -> (T a -> T a)-lift1 f (Cons x0) = Cons (f x0)--{-# INLINE lift2 #-}-lift2 :: ([a] -> [a] -> [a]) -> (T a -> T a -> T a)-lift2 f (Cons x0) (Cons x1) = Cons (f x0 x1)--}--{--Functor instance is e.g. useful for converting number types,-say 'Rational' to 'Double'.--}--instance Functor T where-   fmap f (Cons xs) = Cons (map f xs)--{-# INLINE appPrec #-}-appPrec :: Int-appPrec  = 10--instance (Show a) => Show (T a) where-   showsPrec p (Cons xs) =-      showParen (p >= appPrec)-         (showString "RefinementMask2.fromCoeffs " . shows xs)--instance (QC.Arbitrary a, Field.C a) => QC.Arbitrary (T a) where-   arbitrary =-      liftM2-         (\degree body ->-            let s = sum body-            in  Cons $ map ((2 ^- degree - s) / NPList.lengthLeft body +) body)-         (QC.choose (-5,0)) QC.arbitrary---{- |-Determine mask by Gauss elimination.--R - alternating binomial coefficients-L - differences of translated polynomials in columns--p2 = L * R^(-1) * m--R * L^(-1) * p2 = m--}-fromPolynomial ::-   (Field.C a) => Poly.T a -> T a-fromPolynomial poly =-   fromCoeffs $-   foldr (\p ps ->-      ListHT.mapAdjacent (-) (p:ps++[0]))-      [] $-   foldr (\(db,dp) cs ->-      ListHT.switchR-         (error "RefinementMask2.fromPolynomial: polynomial should be non-empty")-         (\dps dpe ->-            cs ++ [(db - Ring.scalarProduct dps cs) / dpe])-         dp) [] $-   zip-      (Poly.coeffs $ Poly.dilate 2 poly)-      (List.transpose $-       Match.take (Poly.coeffs poly) $-       map Poly.coeffs $-       iterate polynomialDifference poly)--polynomialDifference ::-   (Ring.C a) => Poly.T a -> Poly.T a-polynomialDifference poly =-   Poly.fromCoeffs $ init $ Poly.coeffs $-   Poly.translate 1 poly - poly--{- |-If the mask does not sum up to a power of @1/2@-then the function returns 'Nothing'.--}-toPolynomial ::-   (RealField.C a) => T a -> Maybe (Poly.T a)-toPolynomial (Cons []) = Just $ Poly.fromCoeffs []-toPolynomial mask =-   let s = sum $ coeffs mask-       ks = reverse $ takeWhile (<=1) $ iterate (2*) s-   in  case ks of-          1:ks0 ->-             Just $-             foldl-                (\p k ->-                   let ip = Poly.integrate zero p-                   in  ip + Poly.const (correctConstant (fmap (k/s*) mask) ip))-                (Poly.const 1) ks0-          _ -> Nothing-{--> fmap (6 Vector.*>) $ toPolynomial (Cons [0.1, 0.02, 0.005::Rational])-Just (Polynomial.fromCoeffs [-12732 % 109375, 272 % 625, -18 % 25, 1 % 1])--}--{--The constant term must be zero,-higher terms must already satisfy the refinement constraint.--}-correctConstant ::-   (Field.C a) => T a -> Poly.T a -> a-correctConstant mask poly =-   let refined = refinePolynomial mask poly-   in  head (Poly.coeffs refined) / (1 - sum (coeffs mask))--toPolynomialFast ::-   (RealField.C a) => T a -> Maybe (Poly.T a)-toPolynomialFast mask =-   let s = sum $ coeffs mask-       ks = reverse $ takeWhile (<=1) $ iterate (2*) s-   in  case ks of-          1:ks0 ->-             Just $-             foldl-                (\p k ->-                   let ip = Poly.integrate zero p-                       c = head (Poly.coeffs (refinePolynomial mask ip))-                   in  ip + Poly.const (c*k / ((1-k)*s)))-                (Poly.const 1) ks0-          _ -> Nothing--refinePolynomial ::-   (Ring.C a) => T a -> Poly.T a -> Poly.T a-refinePolynomial mask =-   Poly.shrink 2 .-   Vector.linearComb (coeffs mask) .-   iterate (Poly.translate 1)-{--> mapM_ print $ take 50 $ iterate (refinePolynomial (Cons [0.1, 0.02, 0.005])) (Poly.fromCoeffs [0,0,0,1::Double])-...-Polynomial.fromCoeffs [-0.11640685714285712,0.4351999999999999,-0.7199999999999999,1.0]--}
− src-ghc-6.12/MathObj/RootSet.hs 
@@ -1,171 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Copyright   :  (c) Henning Thielemann 2004-2005--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  requires multi-parameter type classes--Computations on the set of roots of a polynomial.-These are represented as the list of their elementar symmetric terms.-The difference between a polynomial and the list of elementar symmetric terms-is the reversed order and the alternated signs.--Cf. /MathObj.PowerSum/ .--}-module MathObj.RootSet where--import qualified MathObj.Polynomial      as Poly-import qualified MathObj.Polynomial.Core as PolyCore-import qualified MathObj.PowerSum        as PowerSum--import qualified Algebra.Algebraic    as Algebraic-import qualified Algebra.IntegralDomain as Integral-import qualified Algebra.Field        as Field-import qualified Algebra.Ring         as Ring-import qualified Algebra.Additive     as Additive-import qualified Algebra.ZeroTestable as ZeroTestable--import qualified Data.List.Match as Match-import Control.Monad (liftM2)--import NumericPrelude.Base as P hiding (const)-import NumericPrelude.Numeric as NP---newtype T a = Cons {coeffs :: [a]}---{- * Conversions -}--lift0 :: [a] -> T a-lift0 = Cons--lift1 :: ([a] -> [a]) -> (T a -> T a)-lift1 f (Cons x0) = Cons (f x0)--lift2 :: ([a] -> [a] -> [a]) -> (T a -> T a -> T a)-lift2 f (Cons x0) (Cons x1) = Cons (f x0 x1)---const :: (Ring.C a) => a -> T a-const x = Cons [1,x]---toPolynomial :: T a -> Poly.T a-toPolynomial (Cons xs) = Poly.fromCoeffs (reverse xs)--fromPolynomial :: Poly.T a -> T a-fromPolynomial xs = Cons (reverse (Poly.coeffs xs))----toPowerSums :: (Field.C a, ZeroTestable.C a) => [a] -> [a]-toPowerSums = PowerSum.fromElemSymDenormalized--fromPowerSums :: (Field.C a, ZeroTestable.C a) => [a] -> [a]-fromPowerSums = PowerSum.toElemSym---{- | cf. 'MathObj.Polynomial.mulLinearFactor' -}-addRoot :: Ring.C a => a -> [a] -> [a]-addRoot x yt@(y:ys) =-   y : (ys + PolyCore.scale x yt)-addRoot _ [] =-   error "addRoot: list of elementar symmetric terms must consist at least of a 1"--fromRoots :: Ring.C a => [a] -> [a]-fromRoots = foldl (flip addRoot) [1]----liftPowerSum1Gen :: ([a] -> [a]) -> ([a] -> [a]) ->-   ([a] -> [a]) -> ([a] -> [a])-liftPowerSum1Gen fromPS toPS op x =-   Match.take x (fromPS (op (toPS x)))--liftPowerSum2Gen :: ([a] -> [a]) -> ([a] -> [a]) ->-   ([a] -> [a] -> [a]) -> ([a] -> [a] -> [a])-liftPowerSum2Gen fromPS toPS op x y =-   Match.take (undefined : liftM2 (,) (tail x) (tail y))-             (fromPS (op (toPS x) (toPS y)))---liftPowerSum1 :: (Field.C a, ZeroTestable.C a) =>-   ([a] -> [a]) -> ([a] -> [a])-liftPowerSum1 = liftPowerSum1Gen fromPowerSums toPowerSums--liftPowerSum2 :: (Field.C a, ZeroTestable.C a) =>-   ([a] -> [a] -> [a]) -> ([a] -> [a] -> [a])-liftPowerSum2 = liftPowerSum2Gen fromPowerSums toPowerSums--liftPowerSumInt1 :: (Integral.C a, Eq a, ZeroTestable.C a) =>-   ([a] -> [a]) -> ([a] -> [a])-liftPowerSumInt1 = liftPowerSum1Gen PowerSum.toElemSymInt PowerSum.fromElemSym--liftPowerSumInt2 :: (Integral.C a, Eq a, ZeroTestable.C a) =>-   ([a] -> [a] -> [a]) -> ([a] -> [a] -> [a])-liftPowerSumInt2 = liftPowerSum2Gen PowerSum.toElemSymInt PowerSum.fromElemSym-----{- * Show -}--appPrec :: Int-appPrec  = 10--instance (Show a) => Show (T a) where-  showsPrec p (Cons xs) =-    showParen (p >= appPrec)-       (showString "RootSet.Cons " . shows xs)---{- * Additive -}--{- Use binomial expansion of (x+y)^n -}-add :: (Field.C a, ZeroTestable.C a) => [a] -> [a] -> [a]-add = liftPowerSum2 PowerSum.add--addInt :: (Integral.C a, Eq a, ZeroTestable.C a) => [a] -> [a] -> [a]-addInt = liftPowerSumInt2 PowerSum.add--instance (Field.C a, ZeroTestable.C a) => Additive.C (T a) where-   zero   = const zero-   (+)    = lift2 add-   negate = lift1 PolyCore.alternate---{- * Ring -}--mul :: (Field.C a, ZeroTestable.C a) => [a] -> [a] -> [a]-mul = liftPowerSum2 PowerSum.mul--mulInt :: (Integral.C a, Eq a, ZeroTestable.C a) => [a] -> [a] -> [a]-mulInt = liftPowerSumInt2 PowerSum.mul---pow :: (Field.C a, ZeroTestable.C a) => Integer -> [a] -> [a]-pow n = liftPowerSum1 (PowerSum.pow n)--powInt :: (Integral.C a, Eq a, ZeroTestable.C a) => Integer -> [a] -> [a]-powInt n = liftPowerSumInt1 (PowerSum.pow n)---instance (Field.C a, ZeroTestable.C a) => Ring.C (T a) where-   one           = const one-   fromInteger n = const (fromInteger n)-   (*)           = lift2 mul-   x^n           = lift1 (pow n) x---{- * Field.C -}--instance (Field.C a, ZeroTestable.C a) => Field.C (T a) where-   recip = lift1 reverse---{- * Algebra -}--instance (Field.C a, ZeroTestable.C a) => Algebraic.C (T a) where-   root n = lift1 (PowerSum.root n)
− src-ghc-6.12/Number/Complex.hs 
@@ -1,575 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-{- Rules should be processed -}-{- |-Module      :  Number.Complex-Copyright   :  (c) The University of Glasgow 2001-License     :  BSD-style (see the file libraries/base/LICENSE)--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable (?)--Complex numbers.--}--module Number.Complex-        (-        -- * Cartesian form-        T(real,imag),-        imaginaryUnit,-        fromReal,--        (+:),-        (-:),-        scale,-        exp,-        quarterLeft,-        quarterRight,--        -- * Polar form-        fromPolar,-        cis,-        signum,-        signumNorm,-        toPolar,-        magnitude,-        magnitudeSqr,-        phase,-        -- * Conjugate-        conjugate,--        -- * Properties-        propPolar,--        -- * Auxiliary classes-        Power(power),-        defltPow,-        )  where---- import qualified Number.Ratio as Ratio--import qualified Algebra.NormedSpace.Euclidean as NormedEuc-import qualified Algebra.NormedSpace.Sum       as NormedSum-import qualified Algebra.NormedSpace.Maximum   as NormedMax--import qualified Algebra.VectorSpace        as VectorSpace-import qualified Algebra.Module             as Module-import qualified Algebra.Vector             as Vector-import qualified Algebra.RealTranscendental as RealTrans-import qualified Algebra.Transcendental     as Trans-import qualified Algebra.Algebraic          as Algebraic-import qualified Algebra.Field              as Field-import qualified Algebra.Units              as Units-import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.IntegralDomain     as Integral-import qualified Algebra.RealRing           as RealRing-import qualified Algebra.Absolute           as Absolute-import qualified Algebra.Ring               as Ring-import qualified Algebra.Additive           as Additive-import qualified Algebra.ZeroTestable       as ZeroTestable-import qualified Algebra.Indexable          as Indexable--import Algebra.ZeroTestable(isZero)-import Algebra.Module((*>), (<*>.*>), )-import Algebra.Algebraic((^/), )--import qualified NumericPrelude.Elementwise as Elem-import Algebra.Additive ((<*>.+), (<*>.-), (<*>.-$), )--import Foreign.Storable (Storable (..), )-import qualified Foreign.Storable.Record as Store-import Control.Applicative (liftA2, )--import Test.QuickCheck (Arbitrary, arbitrary, )-import Control.Monad (liftM2, )--import qualified Prelude as P-import NumericPrelude.Base-import NumericPrelude.Numeric hiding (signum, exp, )-import Text.Show.HT (showsInfixPrec, )-import Text.Read.HT (readsInfixPrec, )----- import qualified Data.Typeable as Ty--infix  6  +:, `Cons`--{- * The Complex type -}---- | Complex numbers are an algebraic type.-data T a-  = Cons {real :: !a   -- ^ real part-         ,imag :: !a   -- ^ imaginary part-         }-  deriving (Eq)--{-# INLINE imaginaryUnit #-}-imaginaryUnit :: Ring.C a => T a-imaginaryUnit = zero +: one--{-# INLINE fromReal #-}-fromReal :: Additive.C a => a -> T a-fromReal x = Cons x zero---{-# INLINE plusPrec #-}-plusPrec :: Int-plusPrec = 6--instance (Show a) => Show (T a) where-   showsPrec prec (Cons x y) = showsInfixPrec "+:" plusPrec prec x y--instance (Read a) => Read (T a) where-   readsPrec prec = readsInfixPrec "+:" plusPrec prec (+:)--instance Functor T where-   {-# INLINE fmap #-}-   fmap f (Cons x y) = Cons (f x) (f y)--instance (Arbitrary a) => Arbitrary (T a) where-   {-# INLINE arbitrary #-}-   arbitrary = liftM2 Cons arbitrary arbitrary--instance (Storable a) => Storable (T a) where-   sizeOf    = Store.sizeOf store-   alignment = Store.alignment store-   peek      = Store.peek store-   poke      = Store.poke store--store ::-   (Storable a) =>-   Store.Dictionary (T a)-store =-   Store.run $-   liftA2 (+:)-      (Store.element real)-      (Store.element imag)----{- * Functions -}---- | Construct a complex number from real and imaginary part.-{-# INLINE (+:) #-}-(+:) :: a -> a -> T a-(+:) = Cons---- | Construct a complex number with negated imaginary part.-{-# INLINE (-:) #-}-(-:) :: Additive.C a => a -> a -> T a-(-:) x y = Cons x (-y)---- | The conjugate of a complex number.-{- SPECIALISE conjugate :: T Double -> T Double -}-{-# INLINE conjugate #-}-conjugate :: (Additive.C a) => T a -> T a-conjugate (Cons x y) =  Cons x (-y)---- | Scale a complex number by a real number.-{- SPECIALISE scale :: Double -> T Double -> T Double -}-{-# INLINE scale #-}-scale :: (Ring.C a) => a -> T a -> T a-scale r =  fmap (r*)---- | Exponential of a complex number with minimal type class constraints.-{-# INLINE exp #-}-exp :: (Trans.C a) => T a -> T a-exp (Cons x y) =  scale (Trans.exp x) (cis y)---- | Turn the point one quarter to the right.-{-# INLINE quarterRight #-}-{-# INLINE quarterLeft #-}-quarterRight, quarterLeft :: (Additive.C a) => T a -> T a-quarterRight (Cons x y) = Cons   y  (-x)-quarterLeft  (Cons x y) = Cons (-y)   x--{- | Scale a complex number to magnitude 1.--For a complex number @z@,-@'abs' z@ is a number with the magnitude of @z@,-but oriented in the positive real direction,-whereas @'signum' z@ has the phase of @z@, but unit magnitude.--}--{- SPECIALISE signum :: T Double -> T Double -}-signum :: (Algebraic.C a, ZeroTestable.C a) => T a -> T a-signum z =-   if isZero z-     then zero-     else scale (recip (magnitude z)) z--{- SPECIALISE signumNorm :: T Double -> T Double -}-{-# INLINE signumNorm #-}-signumNorm :: (Algebraic.C a, NormedEuc.C a a, ZeroTestable.C a) => T a -> T a-signumNorm z =-   if isZero z-     then zero-     else scale (recip (NormedEuc.norm z)) z---- | Form a complex number from polar components of magnitude and phase.-{- SPECIALISE fromPolar :: Double -> Double -> T Double -}-{-# INLINE fromPolar #-}-fromPolar :: (Trans.C a) => a -> a -> T a-fromPolar r theta =  scale r (cis theta)---- | @'cis' t@ is a complex value with magnitude @1@--- and phase @t@ (modulo @2*'pi'@).-{- SPECIALISE cis :: Double -> T Double -}-{-# INLINE cis #-}-cis :: (Trans.C a) => a -> T a-cis theta =  Cons (cos theta) (sin theta)--propPolar :: (RealTrans.C a) => T a -> Bool-propPolar z =  uncurry fromPolar (toPolar z) == z---{- |-The nonnegative magnitude of a complex number.-This implementation respects the limited range of floating point numbers.-The trivial implementation 'magnitude'-would overflow for floating point exponents greater than-the half of the maximum admissible exponent.-We automatically drop in this implementation for 'Float' and 'Double'-by optimizer rules.-You should do so for your custom floating point types.--}-{-# INLINE floatMagnitude #-}-floatMagnitude :: (P.RealFloat a, Algebraic.C a) => T a -> a-floatMagnitude (Cons x y) =-   let k  = max (P.exponent x) (P.exponent y)-       mk = - k-   in  P.scaleFloat k-           (sqrt (P.scaleFloat mk x ^ 2 +-                  P.scaleFloat mk y ^ 2))--{-# INLINE [1] magnitude #-}-magnitude :: (Algebraic.C a) => T a -> a-magnitude = sqrt . magnitudeSqr--{-# RULES-     "Complex.magnitude :: Double"-        magnitude = floatMagnitude :: T Double -> Double;--     "Complex.magnitude :: Float"-        magnitude = floatMagnitude :: T Float -> Float;-  #-}---- like NormedEuc.normSqr with lifted class constraints-{-# INLINE magnitudeSqr #-}-magnitudeSqr :: (Ring.C a) => T a -> a-magnitudeSqr (Cons x y) = x^2 + y^2---- | The phase of a complex number, in the range @(-'pi', 'pi']@.--- If the magnitude is zero, then so is the phase.-{-# INLINE phase #-}-phase :: (RealTrans.C a, ZeroTestable.C a) => T a -> a-phase z =-   if isZero z-     then zero   -- SLPJ July 97 from John Peterson-     else case z of (Cons x y) -> atan2 y x---{- |-The function 'toPolar' takes a complex number and-returns a (magnitude, phase) pair in canonical form:-the magnitude is nonnegative, and the phase in the range @(-'pi', 'pi']@;-if the magnitude is zero, then so is the phase.--}-toPolar :: (RealTrans.C a) => T a -> (a,a)-toPolar z = (magnitude z, phase z)----{- * Instances of T -}--{--complexTc = Ty.mkTyCon "Complex.T"-instance Ty.Typeable1 T where { typeOf1 _ = Ty.mkTyConApp complexTc [] }-instance Ty.Typeable a => Ty.Typeable (T a) where { typeOf = Ty.typeOfDefault }--}--instance  (Indexable.C a) => Indexable.C (T a) where-    {-# INLINE compare #-}-    compare (Cons x y) (Cons x' y')  =  Indexable.compare (x,y) (x',y')--instance  (ZeroTestable.C a) => ZeroTestable.C (T a)  where-    {-# INLINE isZero #-}-    isZero (Cons x y)  = isZero x && isZero y--instance  (Additive.C a) => Additive.C (T a)  where-    {- SPECIALISE instance Additive.C (T Float) -}-    {- SPECIALISE instance Additive.C (T Double) -}-    {-# INLINE zero #-}-    {-# INLINE negate #-}-    {-# INLINE (+) #-}-    {-# INLINE (-) #-}-    zero   = Cons zero zero-    (+)    = Elem.run2 $ Elem.with Cons <*>.+  real <*>.+  imag-    (-)    = Elem.run2 $ Elem.with Cons <*>.-  real <*>.-  imag-    negate = Elem.run  $ Elem.with Cons <*>.-$ real <*>.-$ imag--instance  (Ring.C a) => Ring.C (T a)  where-    {- SPECIALISE instance Ring.C (T Float) -}-    {- SPECIALISE instance Ring.C (T Double) -}-    {-# INLINE one #-}-    one                         =  Cons one zero-    {-# INLINE (*) #-}-    (Cons x y) * (Cons x' y')   =  Cons (x*x'-y*y') (x*y'+y*x')-    {-# INLINE fromInteger #-}-    fromInteger                 =  fromReal . fromInteger--instance  (Absolute.C a, Algebraic.C a) => Absolute.C (T a)  where-    {- SPECIALISE instance Absolute.C (T Float) -}-    {- SPECIALISE instance Absolute.C (T Double) -}-    {-# INLINE abs #-}-    {-# INLINE signum #-}-    abs x  = Cons (magnitude x) zero-    signum = signum--instance Vector.C T where-   {-# INLINE zero #-}-   zero  = zero-   {-# INLINE (<+>) #-}-   (<+>) = (+)-   {-# INLINE (*>) #-}-   (*>)  = scale---- | The '(*>)' method can't replace 'scale'---   because it requires the Algebra.Module constraint-instance (Module.C a b) => Module.C a (T b) where-   {-# INLINE (*>) #-}-   (*>) = Elem.run2 $ Elem.with Cons <*>.*> real <*>.*> imag-   -- s *> (Cons x y)  = Cons (s *> x) (s *> y)--instance (VectorSpace.C a b) => VectorSpace.C a (T b)--instance (Additive.C a, NormedSum.C a v) => NormedSum.C a (T v) where-   {-# INLINE norm #-}-   norm x = NormedSum.norm (real x) + NormedSum.norm (imag x)--instance (NormedEuc.Sqr a b) => NormedEuc.Sqr a (T b) where-   {-# INLINE normSqr #-}-   normSqr x = NormedEuc.normSqr (real x) + NormedEuc.normSqr (imag x)--instance (Algebraic.C a, NormedEuc.Sqr a b) => NormedEuc.C a (T b) where-   {-# INLINE norm #-}-   norm = NormedEuc.defltNorm--instance (Ord a, NormedMax.C a v) => NormedMax.C a (T v) where-   {-# INLINE norm #-}-   norm x = max (NormedMax.norm (real x)) (NormedMax.norm (imag x))---{--  In this implementation the complex plane is structured-  as an orthogonal grid induced by the divisor z'.-  The coordinate of a cell within this grid is returned as quotient-  and the position of the cell in the grid is returned as remainder.-  The magnitude of the remainder might be larger than that of the divisor-  thus the Euclidean algorithm can fail.--}--instance  (Integral.C a) => Integral.C (T a)  where-    divMod z z' =-       let denom = magnitudeSqr z'-           zBig  = z * conjugate z'-           q     = fmap (flip div denom) zBig-       in  (q, z-q*z')---{--  This variant of divMod tries to come close to the origin.-  Thus the remainder has smaller magnitude than the divisor.-  This variant of divModCent can be used for Euclidean's algorithm.--}-{-# INLINE divModCent #-}-divModCent :: (Ord a, Integral.C a) => T a -> T a -> (T a, T a)-divModCent z z' =-   let denom = magnitudeSqr z'-       zBig  = z * conjugate z'-       re    = divMod (real zBig) denom-       im    = divMod (imag zBig) denom-       q  = Cons (fst re) (fst im)-       r  = Cons (snd re) (snd im)-       q' = Cons-              (real q + if 2 * real r > denom then one else zero)-              (imag q + if 2 * imag r > denom then one else zero)-   in  (q', z-q'*z')--{-# INLINE modCent #-}-modCent :: (Ord a, Integral.C a) => T a -> T a -> T a-modCent z z' = snd (divModCent z z')--instance  (Ord a, Units.C a) => Units.C (T a)  where-    {-# INLINE isUnit #-}-    isUnit (Cons x y) =-       isUnit x && y==zero  ||-       isUnit y && x==zero-    {-# INLINE stdAssociate #-}-    stdAssociate z@(Cons x y) =-       let z' = if y<0  ||  y==0 && x<0 then negate z else z-       in  if real z'<=0 then quarterRight z' else z'-    {-# INLINE stdUnit #-}-    stdUnit z@(Cons x y) =-       if z==zero-         then 1-         else-           let (x',sgn') = if y<0  ||  y==0 && x<0-                             then (negate x, -1)-                             else (x, 1)-           in  if x'<=0 then quarterLeft sgn' else sgn'---instance  (Ord a, ZeroTestable.C a, Units.C a) => PID.C (T a) where-   {-# INLINE gcd #-}-   gcd         = euclid modCent-   {-# INLINE extendedGCD #-}-   extendedGCD = extendedEuclid divModCent---{-# INLINE [1] divide #-}-divide :: (Field.C a) => T a -> T a -> T a-divide (Cons x y) z'@(Cons x' y') =-   let d = magnitudeSqr z'-   in  Cons ((x*x'+y*y') / d) ((y*x'-x*y') / d)---- | Special implementation of @(\/)@ for floating point numbers---   which prevent intermediate overflows.-{-# INLINE floatDivide #-}-floatDivide :: (P.RealFloat a, Field.C a) => T a -> T a -> T a-floatDivide (Cons x y) (Cons x' y') =-   let k   = - max (P.exponent x') (P.exponent y')-       x'' = P.scaleFloat k x'-       y'' = P.scaleFloat k y'-       d   = x'*x'' + y'*y''-   in  Cons ((x*x''+y*y'') / d) ((y*x''-x*y'') / d)--{-# RULES-     "Complex.divide :: Double"-        divide = floatDivide :: T Double -> T Double -> T Double;--     "Complex.divide :: Float"-        divide = floatDivide :: T Float -> T Float -> T Float;-  #-}-----instance  (Field.C a) => Field.C (T a)  where-    {-# INLINE (/) #-}-    (/)                 =  divide-    {-# INLINE fromRational' #-}-    fromRational'       =  fromReal . fromRational'--{-|-   We like to build the Complex Algebraic instance-   on top of the Algebraic instance of the scalar type.-   This poses no problem to 'sqrt'.-   However, 'Number.Complex.root' requires computing the complex argument-   which is a transcendent operation.-   In order to keep the type class dependencies clean-   for more sophisticated algebraic number types,-   we introduce a type class which actually performs the radix operation.--}-class (Algebraic.C a) => (Power a) where-    power  ::  Rational -> T a -> T a---{-# INLINE defltPow #-}-defltPow :: (RealTrans.C a) =>-    Rational -> T a -> T a-defltPow r x =-    let (mag,arg) = toPolar x-    in  fromPolar (mag ^/ r)-                  (arg * fromRational' r)---instance  Power Float where-    {-# INLINE power #-}-    power  =  defltPow--instance  Power Double where-    {-# INLINE power #-}-    power  =  defltPow---instance  (RealRing.C a, Algebraic.C a, Power a) =>-          Algebraic.C (T a)  where-    -- | the real part of the result is always non-negative-    {-# INLINE sqrt #-}-    sqrt z@(Cons x y)  =  if z == zero-                            then zero-                            else-                              let u'    = sqrt ((magnitude z + abs x) / 2)-                                  v'    = abs y / (u'*2)-                                  (u,v) = if x < 0 then (v',u') else (u',v')-                              in  Cons u (if y < 0 then -v else v)-    {-# INLINE (^/) #-}-    (^/) = flip power---instance  (RealRing.C a, RealTrans.C a, Power a) =>-          Trans.C (T a)  where-    {- SPECIALISE instance Trans.C (T Float) -}-    {- SPECIALISE instance Trans.C (T Double) -}-    {-# INLINE pi #-}-    pi                 =  fromReal pi-    {-# INLINE exp #-}-    exp                =  exp-    {-# INLINE log #-}-    log z              =  let (m,p) = toPolar z in Cons (log m) p--    -- use defaults for tan, tanh--    {-# INLINE sin #-}-    sin (Cons x y)     =  Cons (sin x * cosh y) (  cos x * sinh y)-    {-# INLINE cos #-}-    cos (Cons x y)     =  Cons (cos x * cosh y) (- sin x * sinh y)--    {-# INLINE sinh #-}-    sinh (Cons x y)    =  Cons (cos y * sinh x) (sin y * cosh x)-    {-# INLINE cosh #-}-    cosh (Cons x y)    =  Cons (cos y * cosh x) (sin y * sinh x)--    {-# INLINE asin #-}-    asin z             =  quarterRight (log (quarterLeft z + sqrt (1 - z^2)))-    {-# INLINE acos #-}-    acos z             =  quarterRight (log (z + quarterLeft (sqrt (1 - z^2))))-    {-# INLINE atan #-}-    atan z@(Cons x y)  =  quarterRight (log (Cons (1-y) x / sqrt (1+z^2)))--{- use the default implementation-    asinh z        =  log (z + sqrt (1+z^2))-    acosh z        =  log (z + (z+1) * sqrt ((z-1)/(z+1)))-    atanh z        =  log ((1+z) / sqrt (1-z^2))--}---{- * legacy instances -}--{-# INLINE legacyInstance #-}-legacyInstance :: a-legacyInstance =-   error "legacy Ring.C instance for simple input of numeric literals"--instance (Ring.C a, Eq a, Show a) => P.Num (T a) where-   {-# INLINE fromInteger #-}-   fromInteger = fromReal . fromInteger-   {-# INLINE negate #-}-   negate = negate -- for unary minus-   {-# INLINE (+) #-}-   (+)    = legacyInstance-   {-# INLINE (*) #-}-   (*)    = legacyInstance-   {-# INLINE abs #-}-   abs    = legacyInstance-   {-# INLINE signum #-}-   signum = legacyInstance--instance (Field.C a, Eq a, Show a) => P.Fractional (T a) where-   {-# INLINE fromRational #-}-   fromRational = fromRational-   {-# INLINE (/) #-}-   (/) = legacyInstance
− src-ghc-6.12/Number/ComplexSquareRoot.hs 
@@ -1,119 +0,0 @@-module Number.ComplexSquareRoot where---- import qualified Algebra.Algebraic as Algebraic-import qualified Algebra.RealField as RealField-import qualified Algebra.RealRing as RealRing--- import qualified Algebra.Field as Field-import qualified Algebra.Ring as Ring-import qualified Algebra.Additive as Additive-import qualified Algebra.ZeroTestable as ZeroTestable--import qualified Number.Complex as Complex--import Algebra.ZeroTestable(isZero, )--import Test.QuickCheck (Arbitrary, arbitrary, )--import Control.Monad (liftM2, )--import qualified NumericPrelude.Numeric as NP-import NumericPrelude.Numeric hiding (recip, )-import NumericPrelude.Base-import Prelude ()--{- |-Represent the square root of a complex number-without actually having to compute a square root.-If the Bool is False,-then the square root is represented with positive real part-or zero real part and positive imaginary part.-If the Bool is True the square root is negated.--}-data T a = Cons Bool (Complex.T a)-   deriving (Show)--{- |-You must use @fmap@ only for number type conversion.--}-instance Functor T where-   fmap f (Cons n x) = Cons n (fmap f x)--instance (ZeroTestable.C a) => ZeroTestable.C (T a) where-   isZero (Cons _b s) = isZero s--instance (ZeroTestable.C a, Eq a) => Eq (T a) where-   (Cons xb xs) == (Cons yb ys) =-      isZero xs && isZero ys  ||-      xb==yb && xs==ys--instance (Arbitrary a) => Arbitrary (T a) where-   arbitrary = liftM2 Cons arbitrary arbitrary---fromNumber :: (RealRing.C a) => Complex.T a -> T a-fromNumber x =-   Cons-      (case compare zero (Complex.real x) of-         LT -> False-         GT -> True-         EQ -> Complex.imag x < zero)-      (x^2)---- htam:Wavelet.DyadicResultant.parityFlip-toNumber :: (RealRing.C a, Complex.Power a) => T a -> Complex.T a-toNumber (Cons n x) =-   case sqrt x of y -> if n then NP.negate y else y---one :: (Ring.C a) => T a-one = Cons False NP.one--inUpperHalfplane :: (Additive.C a, Ord a) => Complex.T a -> Bool-inUpperHalfplane x =-   case compare (Complex.imag x) zero of-      GT -> True-      LT -> False-      EQ -> Complex.real x < zero--mul, mulAlt, mulAlt2 :: (RealRing.C a) => T a -> T a -> T a-mul (Cons xb xs) (Cons yb ys) =-   let zs = xs*ys-   in  Cons-          ((xb /= yb) /=-             case (inUpperHalfplane xs,-                   inUpperHalfplane ys,-                   inUpperHalfplane zs) of-                (True,True,False) -> True-                (False,False,True) -> True-                _ -> False)-          zs--mulAlt (Cons xb xs) (Cons yb ys) =-   let zs = xs*ys-   in  Cons-          ((xb /= yb) /=-             let xi = Complex.imag xs-                 yi = Complex.imag ys-                 zi = Complex.imag zs-             in  (xi>=zero) /= (yi>=zero) &&-                 (xi>=zero) /= (zi>=zero))-          zs--mulAlt2 (Cons xb xs) (Cons yb ys) =-   let zs = xs*ys-   in  Cons-          ((xb /= yb) /=-             let xi = Complex.imag xs-                 yi = Complex.imag ys-                 zi = Complex.imag zs-             in  xi*yi<zero && xi*zi<zero)-          zs--div :: (RealField.C a) => T a -> T a -> T a-div x y = mul x (recip y)--recip :: (RealField.C a) => T a -> T a-recip (Cons b s) =-   Cons-      (b /= (Complex.imag s == zero && Complex.real s < zero))-      (NP.recip s)
− src-ghc-6.12/Number/DimensionTerm.hs 
@@ -1,216 +0,0 @@-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-{- |-Copyright   :  (c) Henning Thielemann 2008-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable---See "Algebra.DimensionTerm".--}--module Number.DimensionTerm where--import qualified Algebra.DimensionTerm as Dim--import qualified Algebra.OccasionallyScalar as OccScalar-import qualified Algebra.Module        as Module-import qualified Algebra.Algebraic     as Algebraic-import qualified Algebra.Field         as Field-import qualified Algebra.Absolute          as Absolute-import qualified Algebra.Ring          as Ring-import qualified Algebra.Additive      as Additive--import Algebra.Field    ((/), fromRational', )-import Algebra.Ring     ((*), one, fromInteger, )-import Algebra.Additive ((+), (-), zero, negate, )-import Algebra.Module   ((*>), )--import System.Random (Random, randomR, random)--import Data.Tuple.HT (mapFst, )-import NumericPrelude.Base-import Prelude ()---{- * Number type -}--newtype T u a = Cons a-   deriving (Eq, Ord)---instance (Dim.C u, Show a) => Show (T u a) where-   showsPrec p x =-      let disect :: T u a -> (u,a)-          disect (Cons y) = (undefined, y)-          (u,z) = disect x-      in  showParen (p >= Dim.appPrec)-            (showString "DimensionNumber.fromNumberWithDimension " . showsPrec Dim.appPrec u .-             showString " " . showsPrec Dim.appPrec z)---fromNumber :: a -> Scalar a-fromNumber = Cons--toNumber :: Scalar a -> a-toNumber (Cons x) = x--fromNumberWithDimension :: Dim.C u => u -> a -> T u a-fromNumberWithDimension _ = Cons--toNumberWithDimension :: Dim.C u => u -> T u a -> a-toNumberWithDimension _ (Cons x) = x---instance (Dim.C u, Additive.C a) => Additive.C (T u a) where-   zero                = Cons zero-   (Cons a) + (Cons b) = Cons (a+b)-   (Cons a) - (Cons b) = Cons (a-b)-   negate (Cons a)     = Cons (negate a)--instance (Dim.C u, Module.C a b) => Module.C a (T u b) where-   a *> (Cons b) = Cons (a *> b)--instance (Dim.IsScalar u, Ring.C a) => Ring.C (T u a) where-   one                 = Cons one-   (Cons a) * (Cons b) = Cons (a*b)-   fromInteger a       = Cons (fromInteger a)--instance (Dim.IsScalar u, Field.C a) => Field.C (T u a) where-   (Cons a) / (Cons b) = Cons (a/b)-   recip (Cons a)      = Cons (Field.recip a)-   fromRational' a     = Cons (fromRational' a)--instance (Dim.IsScalar u, OccScalar.C a b) => OccScalar.C a (T u b) where-   toScalar =-      OccScalar.toScalar . toNumber . rewriteDimension Dim.toScalar-   toMaybeScalar =-      OccScalar.toMaybeScalar . toNumber . rewriteDimension Dim.toScalar-   fromScalar =-      rewriteDimension Dim.fromScalar . fromNumber . OccScalar.fromScalar--instance (Dim.C u, Random a) => Random (T u a) where-  randomR (Cons l, Cons u) = mapFst Cons . randomR (l,u)-  random = mapFst Cons . random---infixl 7 &*&, *&-infixl 7 &/&--(&*&) :: (Dim.C u, Dim.C v, Ring.C a) =>-   T u a -> T v a -> T (Dim.Mul u v) a-(&*&) (Cons x) (Cons y) = Cons (x Ring.* y)--(&/&) :: (Dim.C u, Dim.C v, Field.C a) =>-   T u a -> T v a -> T (Dim.Mul u (Dim.Recip v)) a-(&/&) (Cons x) (Cons y) = Cons (x Field./ y)--mulToScalar :: (Dim.C u, Ring.C a) =>-   T u a -> T (Dim.Recip u) a -> a-mulToScalar x y = cancelToScalar (x &*& y)--divToScalar :: (Dim.C u, Field.C a) =>-   T u a -> T u a -> a-divToScalar x y = cancelToScalar (x &/& y)--cancelToScalar :: (Dim.C u) =>-   T (Dim.Mul u (Dim.Recip u)) a -> a-cancelToScalar =-   toNumber . rewriteDimension Dim.cancelRight---recip :: (Dim.C u, Field.C a) =>-   T u a -> T (Dim.Recip u) a-recip (Cons x) = Cons (Field.recip x)--unrecip :: (Dim.C u, Field.C a) =>-   T (Dim.Recip u) a -> T u a-unrecip (Cons x) = Cons (Field.recip x)--sqr :: (Dim.C u, Ring.C a) =>-   T u a -> T (Dim.Sqr u) a-sqr x = x &*& x--sqrt :: (Dim.C u, Algebraic.C a) =>-   T (Dim.Sqr u) a -> T u a-sqrt (Cons x) = Cons (Algebraic.sqrt x)---abs :: (Dim.C u, Absolute.C a) => T u a -> T u a-abs (Cons x) = Cons (Absolute.abs x)--absSignum :: (Dim.C u, Absolute.C a) => T u a -> (T u a, a)-absSignum x0@(Cons x) = (abs x0, Absolute.signum x)--scale, (*&) :: (Dim.C u, Ring.C a) =>-   a -> T u a -> T u a-scale x (Cons y) = Cons (x Ring.* y)--(*&) = scale---rewriteDimension :: (Dim.C u, Dim.C v) => (u -> v) -> T u a -> T v a-rewriteDimension _ (Cons x) = Cons x---{--type class for converting Dim types to Dim value is straight-forward-   class SIDimensionType u where-      dynamic :: DimensionNumber u a -> SIValue a--   instance SIDimensionType Scalar where-      dynamic (DimensionNumber.Cons x) = SIValue.scalar x--   instance SIDimensionType Length where-      dynamic (DimensionNumber.Cons x) = SIValue.meter * dynamic x--}---{- * Example constructors -}--type Scalar      a = T Dim.Scalar a-type Length      a = T Dim.Length a-type Time        a = T Dim.Time a-type Mass        a = T Dim.Mass a-type Charge      a = T Dim.Charge a-type Angle       a = T Dim.Angle a-type Temperature a = T Dim.Temperature a-type Information a = T Dim.Information a--type Frequency   a = T Dim.Frequency a-type Voltage     a = T Dim.Voltage a---scalar :: a -> Scalar a-scalar = fromNumber--length :: a -> Length a-length = Cons--time :: a -> Time a-time = Cons--mass :: a -> Mass a-mass = Cons--charge :: a -> Charge a-charge = Cons--frequency :: a -> Frequency a-frequency = Cons--angle :: a -> Angle a-angle = Cons--temperature :: a -> Temperature a-temperature = Cons--information :: a -> Information a-information = Cons---voltage :: a -> Voltage a-voltage = Cons
− src-ghc-6.12/Number/DimensionTerm/SI.hs 
@@ -1,125 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Copyright   :  (c) Henning Thielemann 2003-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable--Special physical units: SI unit system--}--module Number.DimensionTerm.SI (-    second, minute, hour, day, year,-    hertz,-    meter,-    -- liter,-    gramm, tonne,-    -- newton,-    -- pascal,-    -- bar,-    -- joule,-    -- watt,-    coulomb,-    -- ampere,-    volt,-    -- ohm,-    -- farad,-    kelvin,-    bit, byte,-    -- baud,--    inch, foot, yard, astronomicUnit, parsec,--    SI.yocto, SI.zepto, SI.atto,  SI.femto, SI.pico, SI.nano,-    SI.micro, SI.milli, SI.centi, SI.deci,  SI.one,  SI.deca,-    SI.hecto, SI.kilo,  SI.mega,  SI.giga,  SI.tera, SI.peta,-    SI.exa,   SI.zetta, SI.yotta,-    ) where---- import qualified Algebra.Transcendental      as Trans-import qualified Algebra.Field               as Field---- import qualified Algebra.DimensionTerm as Dim-import qualified Number.DimensionTerm  as DN-import qualified Number.SI.Unit as SI---- aimport NumericPrelude.Base hiding (length)-import NumericPrelude.Numeric hiding (one)---second  :: Field.C a => DN.Time        a-second  = DN.time        1e+0-minute  :: Field.C a => DN.Time        a-minute  = DN.time        SI.secondsPerMinute-hour    :: Field.C a => DN.Time        a-hour    = DN.time        SI.secondsPerHour-day     :: Field.C a => DN.Time        a-day     = DN.time        SI.secondsPerDay-year    :: Field.C a => DN.Time        a-year    = DN.time        SI.secondsPerYear-hertz   :: Field.C a => DN.Frequency a-hertz   = DN.frequency   1e+0-meter   :: Field.C a => DN.Length      a-meter   = DN.length      1e+0--- liter   :: Field.C a => DN.Volume      a--- liter   = DN.volume      1e-3-gramm   :: Field.C a => DN.Mass        a-gramm   = DN.mass        1e-3-tonne   :: Field.C a => DN.Mass        a-tonne   = DN.mass        1e+3--- newton  :: Field.C a => DN.Force       a--- newton  = DN.force       1e+0--- pascal  :: Field.C a => DN.Pressure    a--- pascal  = DN.pressure    1e+0--- bar     :: Field.C a => DN.Pressure    a--- bar     = DN.pressure    1e+5--- joule   :: Field.C a => DN.Energy      a--- joule   = DN.energy      1e+0--- watt    :: Field.C a => DN.Power       a--- watt    = DN.power       1e+0-coulomb :: Field.C a => DN.Charge      a-coulomb = DN.charge      1e+0--- ampere  :: Field.C a => DN.Current     a--- ampere  = DN.current     1e+0-volt    :: Field.C a => DN.Voltage     a-volt    = DN.voltage     1e+0--- ohm     :: Field.C a => DN.Resistance  a--- ohm     = DN.resistance  1e+0--- farad   :: Field.C a => DN.Capacitance a--- farad   = DN.capacitance 1e+0-kelvin  :: Field.C a => DN.Temperature a-kelvin  = DN.temperature 1e+0-bit     :: Field.C a => DN.Information a-bit     = DN.information 1e+0-byte    :: Field.C a => DN.Information a-byte    = DN.information SI.bytesize--- baud    :: Field.C a => DN.DataRate    a--- baud    = DN.dataRate    1e+0--inch, foot, yard, astronomicUnit, parsec-   :: Field.C a => DN.Length a--inch           = DN.length SI.meterPerInch-foot           = DN.length SI.meterPerFoot-yard           = DN.length SI.meterPerYard-astronomicUnit = DN.length SI.meterPerAstronomicUnit-parsec         = DN.length SI.meterPerParsec--{--accelerationOfEarthGravity :: Field.C a => DN.Acceleration    a-accelerationOfEarthGravity = DN.acceleration SI.accelerationOfEarthGravity--mach         :: Field.C a => DN.Speed a-speedOfLight :: Field.C a => DN.Speed a-electronVolt :: Field.C a => DN.Energy a-calorien     :: Field.C a => DN.Energy a-horsePower   :: Field.C a => DN.Power a--mach         = DN.speed        SI.mach-speedOfLight = DN.speed        SI.speedOfLight-electronVolt = DN.energy       SI.electronVolt-calorien     = DN.energy       SI.calorien-horsePower   = DN.power        SI.horsePower--}
− src-ghc-6.12/Number/FixedPoint.hs 
@@ -1,235 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Copyright   :  (c) Henning Thielemann 2006--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  requires multi-parameter type classes--Fixed point numbers.-They are implemented as ratios with fixed denominator.-Many routines fail for some arguments.-When they work,-they can be useful for obtaining approximations of some constants.-We have not paid attention to rounding errors-and thus some of the trailing digits may be wrong.--}-module Number.FixedPoint where--import qualified Algebra.RealRing    as RealRing--- import qualified Algebra.Additive       as Additive--- import qualified Algebra.ZeroTestable   as ZeroTestable-import qualified Algebra.Transcendental as Trans-import qualified MathObj.PowerSeries.Example as PSE--import NumericPrelude.List (mapLast, )-import Data.Function.HT (powerAssociative, )-import Data.List.HT (dropWhileRev, padLeft, )-import Data.Maybe.HT (toMaybe, )-import Data.List (transpose, unfoldr, )-import Data.Char (intToDigit, )--import NumericPrelude.Base-import NumericPrelude.Numeric hiding (recip, sqrt, exp, sin, cos, tan,-                              fromRational')--import qualified NumericPrelude.Numeric as NP---{- ** Conversion -}--{- ** other number types -}--fromFloat :: RealRing.C a => Integer -> a -> Integer-fromFloat den x =-   round (x * NP.fromInteger den)---- | denominator conversion-fromFixedPoint :: Integer -> Integer -> Integer -> Integer-fromFixedPoint denDst denSrc x = div (x*denDst) denSrc---{- ** text -}--{- |-very efficient because it can make use of the decimal output of 'show'--}-showPositionalDec :: Integer -> Integer -> String-showPositionalDec den = liftShowPosToInt $ \x ->-   let packetSize = 50  -- process digits in packets of this size-       basis = ringPower packetSize 10-       (int,frac) = toPositional basis den x-   in  show int ++ "." ++-          concat (mapLast (dropWhileRev ('0'==))-             (map (padLeft '0' packetSize . show) frac))--showPositionalHex :: Integer -> Integer -> String-showPositionalHex = showPositionalBasis 16--showPositionalBin :: Integer -> Integer -> String-showPositionalBin = showPositionalBasis 2--showPositionalBasis :: Integer -> Integer -> Integer -> String-showPositionalBasis basis den = liftShowPosToInt $ \x ->-   let (int,frac) = toPositional basis den x-   in  show int ++ "." ++ map (intToDigit . fromInteger) frac--liftShowPosToInt :: (Integer -> String) -> (Integer -> String)-liftShowPosToInt f n =-   if n>=0-     then       f   n-     else '-' : f (-n)--toPositional :: Integer -> Integer -> Integer -> (Integer, [Integer])-toPositional basis den x =-   let (int, frac) = divMod x den-   in  (int, unfoldr (\rm -> toMaybe (rm/=0) (divMod (basis*rm) den)) frac)---{- * Additive -}--add :: Integer -> Integer -> Integer -> Integer-add _ = (+)--sub :: Integer -> Integer -> Integer -> Integer-sub _ = (-)---{- * Ring -}--mul :: Integer -> Integer -> Integer -> Integer-mul den x y = div (x*y) den---{- * Field -}--divide :: Integer -> Integer -> Integer -> Integer-divide den x y = div (x*den) y--recip :: Integer -> Integer -> Integer-recip den x = div (den^2) x---{- * Algebra -}--{--Newton's method for computing roots.--}--magnitudes :: [Integer]-magnitudes =-   concat (transpose [iterate (^2) 4, iterate (^2) 8])--{--Maybe we can speed up the algorithm-by calling sqrt recursively on deflated arguments.--}-sqrt :: Integer -> Integer -> Integer-sqrt den x =-   let xden     = x*den-       initial  = fst (head (dropWhile ((<= xden) . snd)-                                (zip magnitudes (tail (tail magnitudes)))))-       approxs  = iterate (\y -> div (y + div xden y) 2) initial-       isRoot y = y^2 <= xden && xden < (y+1)^2-   in  head (dropWhile (not . isRoot) approxs)---- bug: needs too long:  root (12::Int) (fromIntegerBase 10 1000 2)-root :: Integer -> Integer -> Integer -> Integer-root n den x =-   let n1       = n-1-       xden     = x * den^n1-       initial  = fst (head (dropWhile ((\y -> y^n <= xden) . snd)-                                (zip magnitudes (tail magnitudes))))-       approxs  = iterate (\y -> div (n1*y + div xden (y^n1)) n) initial-       isRoot y = y^n <= xden && xden < (y+1)^n-   in  head (dropWhile (not . isRoot) approxs)----{- * Transcendental -}---- very simple evaluation by power series with lots of rounding errors-evalPowerSeries :: [Rational] -> Integer -> Integer -> Integer-evalPowerSeries series den x =-   let powers   = iterate (mul den x) den-       summands = zipWith (\c p -> round (c * fromInteger p)) series powers-   in  sum (map snd (takeWhile (\(c,s) -> s/=0 || c==0)-                               (zip series summands)))--cos, sin, tan :: Integer -> Integer -> Integer-cos = evalPowerSeries PSE.cos-sin = evalPowerSeries PSE.sin--- tan will suffer from inaccuracies for small cosine-tan den x = divide den (sin den x) (cos den x)---- it must abs x <= den-arctanSmall :: Integer -> Integer -> Integer-arctanSmall = evalPowerSeries PSE.atan---- will fail for large inputs-arctan :: Integer -> Integer -> Integer-arctan den x =-   let estimate = fromFloat den-                     (Trans.atan (NP.fromRational' (x % den)) :: Double)-       tanEst   = tan den estimate-       residue  = divide den (x-tanEst) (den + mul den x tanEst)-   in  estimate + arctanSmall den residue--piConst :: Integer -> Integer-piConst den =-   let den4 = 4*den-       stArcTan k x = let d = k*den4 in arctanSmall d (div d x)-   in  {- formula 4 * (8 * arctan (1/10) - arctan (1/239) - 4 * arctan (1/515))-             from "Bartsch: Mathematische Formeln" -}-       -- (stArcTan 8 10 - stArcTan 1 239 - stArcTan 4 515)-       -- formula by Stoermer-       (stArcTan 44 57 + stArcTan 7 239 - stArcTan 12 682 + stArcTan 24 12943)---expSmall :: Integer -> Integer -> Integer-expSmall = evalPowerSeries PSE.exp--eConst :: Integer -> Integer-eConst den = expSmall den den--recipEConst :: Integer -> Integer-recipEConst den = expSmall den (-den)--exp :: Integer -> Integer -> Integer-exp den x =-   let den2 = div den 2-       (int,frac) = divMod (x + den2) den-       expFrac = expSmall den (frac-den2)-   in  case compare int 0 of-          EQ -> expFrac-          GT -> powerAssociative (mul den) expFrac (eConst      den)   int-          LT -> powerAssociative (mul den) expFrac (recipEConst den) (-int)-          -- LT -> nest (-int) (divide den e) expFrac---approxLogBase :: Integer -> Integer -> (Int, Integer)-approxLogBase base x =-   until ((<=base) . snd) (\(xE,xM) -> (succ xE, div xM base)) (0,x)--lnSmall :: Integer -> Integer -> Integer-lnSmall den x =-   evalPowerSeries PSE.log den (x-den)---- uses Double's log for an estimate and dramatic speed up-ln :: Integer -> Integer -> Integer-ln den x =-   let fac = 10^50 {- A constant which is representable by Double-                      and which will quickly split our number it pieces-                      small enough for Double. -}-       (denE, denM) = approxLogBase fac den-       (xE,   xM)   = approxLogBase fac x-       approxDouble :: Double-       approxDouble =-          log (NP.fromInteger fac) * fromIntegral (xE-denE) +-          log (NP.fromInteger xM / NP.fromInteger denM)-       {- We convert first with respect to @fac@-          in order to keep in the range of Double values. -}-       approxFac = round (approxDouble * NP.fromInteger fac)-       approx    = fromFixedPoint den fac approxFac-       xSmall    = divide den x (exp den approx)-   in  add den approx (lnSmall den xSmall)
− src-ghc-6.12/Number/FixedPoint/Check.hs 
@@ -1,194 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Number.FixedPoint.Check where--import qualified Number.FixedPoint as FP--import qualified MathObj.PowerSeries.Example as PSE--import qualified Algebra.Transcendental as Trans-import qualified Algebra.Algebraic      as Algebraic-import qualified Algebra.RealRing      as RealRing-import qualified Algebra.Field          as Field-import qualified Algebra.Absolute           as Absolute-import qualified Algebra.Ring           as Ring-import qualified Algebra.Additive       as Additive-import qualified Algebra.ZeroTestable   as ZeroTestable--import NumericPrelude.Base-import NumericPrelude.Numeric   hiding (fromRational')--import qualified Prelude        as P98-import qualified NumericPrelude.Numeric as NP---{- * Types -}--data T = Cons {denominator :: Integer, numerator :: Integer}---{- * Conversion -}--cons :: Integer -> Integer -> T-cons = Cons--{- ** other number types -}--fromFloat :: RealRing.C a => Integer -> a -> T-fromFloat den x =-   cons den (FP.fromFloat den x)--fromInteger' :: Integer -> Integer -> T-fromInteger' den x =-   cons den (x * den)--fromRational' :: Integer -> Rational -> T-fromRational' den x =-   cons den (round (x * NP.fromInteger den))--fromFloatBasis :: RealRing.C a => Integer -> Int -> a -> T-fromFloatBasis basis numDigits =-   fromFloat (ringPower numDigits basis)--fromIntegerBasis :: Integer -> Int -> Integer -> T-fromIntegerBasis basis numDigits =-   fromInteger' (ringPower numDigits basis)--fromRationalBasis :: Integer -> Int -> Rational -> T-fromRationalBasis basis numDigits =-   fromRational' (ringPower numDigits basis)---- | denominator conversion-fromFixedPoint :: Integer -> T -> T-fromFixedPoint denDst (Cons denSrc x) =-   cons denDst (FP.fromFixedPoint denDst denSrc x)---{- * Lift core function -}--lift0 :: Integer -> (Integer -> Integer) -> T-lift0 den f = Cons den (f den)--lift1 :: (Integer -> Integer -> Integer) -> (T -> T)-lift1 f (Cons xd xn) = Cons xd (f xd xn)--lift2 :: (Integer -> Integer -> Integer -> Integer) -> (T -> T -> T)-lift2 f (Cons xd xn) (Cons yd yn) =-   commonDenominator xd yd $ Cons xd (f xd xn yn)--commonDenominator :: Integer -> Integer -> a -> a-commonDenominator xd yd z =-   if xd == yd-     then z-     else error "Number.FixedPoint: denominators differ"---{- * Show -}--appPrec :: Int-appPrec  = 10--instance Show T where-  showsPrec p (Cons den num) =-    showParen (p >= appPrec)-       (showString "FixedPoint.cons " . shows den-          . showString " " . shows num)---defltDenominator :: Integer-defltDenominator = 10^100--defltShow :: T -> String-defltShow (Cons den x) =-   FP.showPositionalDec den x----instance Additive.C T where-   zero   = cons defltDenominator zero-   (+)    = lift2 FP.add-   (-)    = lift2 FP.sub-   negate (Cons xd xn) = Cons xd (negate xn)---instance Ring.C T where-   one         = cons defltDenominator defltDenominator-   fromInteger = fromInteger' defltDenominator . NP.fromInteger-   (*)         = lift2 FP.mul-   -- the default instance of (^) cumulates rounding errors but is faster-   -- x^n           = lift1 (pow n) x---instance Field.C T where-   (/)   = lift2 FP.divide-   recip = lift1 FP.recip-   fromRational' = fromRational' defltDenominator . NP.fromRational'---instance Algebraic.C T where-   sqrt   = lift1 FP.sqrt-   root n = lift1 (FP.root n)----- these function are only implemented for the convergence radius of their Taylor expansions-instance Trans.C T where-   pi    = lift0 defltDenominator FP.piConst-   exp   = lift1 FP.exp-   log   = lift1 FP.ln-   {--   logBase-   (**)-   -}-   sin   = lift1 (FP.evalPowerSeries PSE.sin)-   cos   = lift1 (FP.evalPowerSeries PSE.cos)-   -- tan   = lift1 (FP.evalPowerSeries PSE.tan)-   asin  = lift1 (FP.evalPowerSeries PSE.asin)-   atan  = lift1 FP.arctan-   {--   acos  = lift1 (FP.evalPowerSeries PSE.acos)-   sinh  = lift1 (FP.evalPowerSeries PSE.sinh)-   tanh  = lift1 (FP.evalPowerSeries PSE.tanh)-   cosh  = lift1 (FP.evalPowerSeries PSE.cosh)-   asinh = lift1 (FP.evalPowerSeries PSE.asinh)-   atanh = lift1 (FP.evalPowerSeries PSE.atanh)-   acosh = lift1 (FP.evalPowerSeries PSE.acosh)-   -}---instance ZeroTestable.C T where-   isZero (Cons _ xn)  =  isZero xn--instance Eq T where-   (Cons xd xn) == (Cons yd yn) =-      commonDenominator xd yd (xn==yn)--instance Ord T where-   compare (Cons xd xn) (Cons yd yn) =-      commonDenominator xd yd (compare xn yn)--instance Absolute.C T where-   abs = lift1 (const abs)-   signum = Absolute.signumOrd--instance RealRing.C T where-   splitFraction (Cons xd xn) =-      let (int, frac) = divMod xd xn-      in  (fromInteger int, Cons xd frac)------ legacy instances for work with GHCi-legacyInstance :: a-legacyInstance =-   error "legacy Ring.C instance for simple input of numeric literals"--instance P98.Num T where-   fromInteger = fromInteger' defltDenominator-   negate = negate --for unary minus-   (+)    = legacyInstance-   (*)    = legacyInstance-   abs    = legacyInstance-   signum = legacyInstance--instance P98.Fractional T where-   fromRational = fromRational' defltDenominator . fromRational-   (/) = legacyInstance
− src-ghc-6.12/Number/GaloisField2p32m5.hs 
@@ -1,92 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{- |-This number type is intended for tests of functions over fields,-where the field elements need constant space.-This way we can provide a Storable instance.-For 'Rational' this would not be possible.--However, be aware that sums of non-zero elements may yield zero.-Thus division is not always safe, where it is for rational numbers.--}-module Number.GaloisField2p32m5 where--import qualified Number.ResidueClass as RC-import qualified Algebra.Module   as Module-import qualified Algebra.Field    as Field-import qualified Algebra.Ring     as Ring-import qualified Algebra.Additive as Additive--import Data.Int (Int64, )-import Data.Word (Word32, Word64, )--import qualified Foreign.Storable.Newtype as SN-import qualified Foreign.Storable as St--import Test.QuickCheck (Arbitrary(arbitrary), )--import NumericPrelude.Base-import NumericPrelude.Numeric---newtype T = Cons {decons :: Word32}-   deriving Eq--{-# INLINE appPrec #-}-appPrec :: Int-appPrec  = 10--instance Show T where-   showsPrec p (Cons x) =-      showsPrec p x-{--      showParen (p >= appPrec)-         (showString "GF2p32m5.Cons " . shows x)--}--instance Arbitrary T where-   arbitrary = fmap (Cons . fromInteger . flip mod base) arbitrary--instance St.Storable T where-   sizeOf = SN.sizeOf decons-   alignment = SN.alignment decons-   peek = SN.peek Cons-   poke = SN.poke decons---base :: Ring.C a => a-base = 2^32-5---{-# INLINE lift2 #-}-lift2 :: (Word64 -> Word64 -> Word64) -> (T -> T -> T)-lift2 f (Cons x) (Cons y) =-   Cons (fromIntegral (mod (f (fromIntegral x) (fromIntegral y)) base))--{-# INLINE lift2Integer #-}-lift2Integer :: (Int64 -> Int64 -> Int64) -> (T -> T -> T)-lift2Integer f (Cons x) (Cons y) =-   Cons (fromIntegral (mod (f (fromIntegral x) (fromIntegral y)) base))---instance Additive.C T where-   zero = Cons zero-   (+) = lift2 (+)---   (-) = lift2 (-)-   x-y = x + negate y-   negate n@(Cons x) =-      if x==0-        then n-        else Cons (base-x)--instance Ring.C T where-   one = Cons one-   (*) = lift2 (*)-   fromInteger =-      Cons . fromInteger . flip mod base--instance Field.C T where-   (/) = lift2Integer (RC.divide base)--instance Module.C T T where-   (*>) = (*)
− src-ghc-6.12/Number/NonNegative.hs 
@@ -1,214 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# OPTIONS_GHC -fno-warn-orphans #-}--{--Rationale for -fno-warn-orphans:- * The orphan instances can't be put into Numeric.NonNegative.Wrapper-   since that's in another package.- * We had to spread the instance declarations-   over the modules defining the typeclasses instantiated.-   Do we want that?--}--{- |-Copyright   :  (c) Henning Thielemann 2007--Maintainer  :  haskell@henning-thielemann.de-Stability   :  stable-Portability :  Haskell 98--A type for non-negative numbers.-It performs a run-time check at construction time (i.e. at run-time)-and is a member of the non-negative number type class-'Numeric.NonNegative.Class.C'.--}-module Number.NonNegative-   (T, fromNumber, fromNumberMsg, fromNumberClip, fromNumberUnsafe, toNumber,-    NonNegW.Int, NonNegW.Integer, NonNegW.Float, NonNegW.Double,-    Ratio, Rational) where--import Numeric.NonNegative.Wrapper-   (T, fromNumberUnsafe, toNumber, )-import qualified Numeric.NonNegative.Wrapper as NonNegW--import qualified Algebra.NonNegative        as NonNeg-import qualified Algebra.Transcendental     as Trans-import qualified Algebra.Algebraic          as Algebraic-import qualified Algebra.RealRing          as RealRing-import qualified Algebra.Field              as Field-import qualified Algebra.RealIntegral       as RealIntegral-import qualified Algebra.IntegralDomain     as Integral-import qualified Algebra.Absolute               as Absolute-import qualified Algebra.Ring               as Ring-import qualified Algebra.Additive           as Additive-import qualified Algebra.Monoid             as Monoid-import qualified Algebra.ZeroTestable       as ZeroTestable--import qualified Algebra.ToInteger          as ToInteger-import qualified Algebra.ToRational         as ToRational--- import Test.QuickCheck (Arbitrary(arbitrary))--import qualified Number.Ratio as R--import NumericPrelude.Base-import Data.Tuple.HT (mapSnd, mapPair, )-import NumericPrelude.Numeric hiding (Int, Integer, Float, Double, Rational, )---{- |-Convert a number to a non-negative number.-If a negative number is given, an error is raised.--}-fromNumber :: (Ord a, Additive.C a) =>-      a-   -> T a-fromNumber = fromNumberMsg "fromNumber"--fromNumberMsg :: (Ord a, Additive.C a) =>-      String  {- ^ name of the calling function to be used in the error message -}-   -> a-   -> T a-fromNumberMsg funcName x =-   if x>=zero-     then fromNumberUnsafe x-     else error (funcName++": negative number")--fromNumberWrap :: (Ord a, Additive.C a) =>-      String-   -> a-   -> T a-fromNumberWrap funcName =-   fromNumberMsg ("Number.NonNegative."++funcName)--{- |-Convert a number to a non-negative number.-A negative number will be replaced by zero.-Use this function with care since it may hide bugs.--}-fromNumberClip :: (Ord a, Additive.C a) =>-      a-   -> T a-fromNumberClip = fromNumberUnsafe . max zero----{- |-Results are not checked for positivity.--}-lift :: (a -> a) -> (T a -> T a)-lift f = fromNumberUnsafe . f . toNumber--liftWrap :: (Ord a, Additive.C a) => String -> (a -> a) -> (T a -> T a)-liftWrap msg f = fromNumberWrap msg . f . toNumber---{- |-Results are not checked for positivity.--}-lift2 :: (a -> a -> a) -> (T a -> T a -> T a)-lift2 f x y =-   fromNumberUnsafe $ f (toNumber x) (toNumber y)----instance ZeroTestable.C a => ZeroTestable.C (T a) where-   isZero = isZero . toNumber--instance (Additive.C a) => Monoid.C (T a) where-   idt = fromNumberUnsafe Additive.zero-   x <*> y = fromNumberUnsafe (toNumber x + toNumber y)---   mconcat = fromNumberUnsafe . sum . map toNumber--instance (Ord a, Additive.C a) => NonNeg.C (T a) where-   split = NonNeg.splitDefault toNumber fromNumberUnsafe--instance (Ord a, Additive.C a) => Additive.C (T a) where-   zero   = fromNumberUnsafe zero-   (+)    = lift2 (+)-   (-)    = liftWrap "-" . (-) . toNumber-   negate = liftWrap "negate" negate--instance (Ord a, Ring.C a) => Ring.C (T a) where-   (*)    = lift2 (*)-   fromInteger = fromNumberWrap "fromInteger" . fromInteger--instance (Ord a, ToRational.C a) => ToRational.C (T a) where-   toRational = ToRational.toRational . toNumber--instance ToInteger.C a => ToInteger.C (T a) where-   toInteger = toInteger . toNumber--{- already defined in the imported module-instance (Ord a, Additive.C a, Enum a) => Enum (T a) where-   toEnum   = fromNumberWrap "toEnum" . toEnum-   fromEnum = fromEnum . toNumber--instance (Ord a, Additive.C a, Bounded a) => Bounded (T a) where-   minBound = fromNumberClip minBound-   maxBound = fromNumberWrap "maxBound" maxBound--instance (Additive.C a, Arbitrary a) => Arbitrary (T a) where-   arbitrary = liftM (fromNumberUnsafe . abs) arbitrary--}--instance RealIntegral.C a => RealIntegral.C (T a) where-   quot = lift2 quot-   rem  = lift2 rem-   quotRem x y =-      mapPair-         (fromNumberUnsafe, fromNumberUnsafe)-         (quotRem (toNumber x) (toNumber y))--instance (Ord a, Integral.C a) => Integral.C (T a) where-   div  = lift2 div-   mod  = lift2 mod-   divMod x y =-      mapPair-         (fromNumberUnsafe, fromNumberUnsafe)-         (divMod (toNumber x) (toNumber y))--instance (Ord a, Field.C a) => Field.C (T a) where-   fromRational' = fromNumberWrap "fromRational" . fromRational'-   (/) = lift2 (/)---instance (ZeroTestable.C a, Ord a, Absolute.C a) => Absolute.C (T a) where-   abs    = lift abs-   signum = lift signum--instance (RealRing.C a) => RealRing.C (T a) where-   splitFraction = mapSnd fromNumberUnsafe . splitFraction . toNumber-   truncate = truncate . toNumber-   round    = round    . toNumber-   ceiling  = ceiling  . toNumber-   floor    = floor    . toNumber--instance (Ord a, Algebraic.C a) => Algebraic.C (T a) where-   sqrt = lift sqrt-   (^/) x r = lift (^/ r) x--instance (Ord a, Trans.C a) => Trans.C (T a) where-   pi = fromNumber pi-   exp  = lift exp-   log  = liftWrap "log" log-   (**) = lift2 (**)-   logBase = liftWrap "logBase" . logBase . toNumber-   sin = liftWrap "sin" sin-   tan = liftWrap "tan" tan-   cos = liftWrap "cos" cos-   asin = liftWrap "asin" asin-   atan = liftWrap "atan" atan-   acos = liftWrap "acos" acos-   sinh = liftWrap "sinh" sinh-   tanh = liftWrap "tanh" tanh-   cosh = liftWrap "cosh" cosh-   asinh = liftWrap "asinh" asinh-   atanh = liftWrap "atanh" atanh-   acosh = liftWrap "acosh" acosh---type Ratio a  = T (R.T a)-type Rational = T R.Rational---{- legacy instances already defined in non-negative package -}
− src-ghc-6.12/Number/NonNegativeChunky.hs 
@@ -1,311 +0,0 @@-{- |-Copyright   :  (c) Henning Thielemann 2007-2010--Maintainer  :  haskell@henning-thielemann.de-Stability   :  stable-Portability :  Haskell 98--A lazy number type, which is a generalization of lazy Peano numbers.-Comparisons can be made lazy and-thus computations are possible which are impossible with strict number types,-e.g. you can compute @let y = min (1+y) 2 in y@.-You can even work with infinite values.-However, depending on the granularity,-the memory consumption is higher than that for strict number types.-This number type is of interest for the merge operation of event lists,-which allows for co-recursive merges.--}-module Number.NonNegativeChunky-   (T, fromChunks, toChunks, fromNumber, toNumber, fromChunky98, toChunky98,-    minMaxDiff, normalize, isNull, isPositive,-    divModLazy, divModStrict, ) where--import qualified Numeric.NonNegative.Chunky as Chunky98-import qualified Numeric.NonNegative.Class as NonNeg98--import qualified Algebra.NonNegative  as NonNeg-import qualified Algebra.Field        as Field-import qualified Algebra.Absolute         as Absolute-import qualified Algebra.Ring         as Ring-import qualified Algebra.Additive     as Additive-import qualified Algebra.ToInteger    as ToInteger-import qualified Algebra.ToRational   as ToRational-import qualified Algebra.IntegralDomain as Integral-import qualified Algebra.RealIntegral as RealIntegral-import qualified Algebra.ZeroTestable as ZeroTestable-import Algebra.ZeroTestable (isZero, )--import qualified Algebra.Monoid as Monoid-import qualified Data.Monoid as Mn98--import Control.Monad (liftM, liftM2, )-import Data.Tuple.HT (mapFst, mapSnd, mapPair, )--import Test.QuickCheck (Arbitrary(arbitrary))--import NumericPrelude.Numeric-import NumericPrelude.Base-import qualified Prelude as P98 (Num(..), Fractional(..), )---{- |-A chunky non-negative number is a list of non-negative numbers.-It represents the sum of the list elements.-It is possible to represent a finite number with infinitely many chunks-by using an infinite number of zeros.--Note the following problems:--Addition is commutative only for finite representations.-E.g. @let y = min (1+y) 2 in y@ is defined,-@let y = min (y+1) 2 in y@ is not.--The type is equivalent to 'Numeric.NonNegative.Chunky'.--}-newtype T a = Cons {decons :: [a]}---fromChunks :: NonNeg.C a => [a] -> T a-fromChunks = Cons--toChunks :: NonNeg.C a => T a -> [a]-toChunks = decons--fromChunky98 :: (NonNeg.C a, NonNeg98.C a) => Chunky98.T a -> T a-fromChunky98 = fromChunks . Chunky98.toChunks--toChunky98 :: (NonNeg.C a, NonNeg98.C a) => T a -> Chunky98.T a-toChunky98 = Chunky98.fromChunks . toChunks--fromNumber :: NonNeg.C a => a -> T a-fromNumber = fromChunks . (:[])--toNumber :: NonNeg.C a => T a -> a-toNumber =  Monoid.cumulate . toChunks----lift2 :: NonNeg.C a => ([a] -> [a] -> [a]) -> (T a -> T a -> T a)-lift2 f x y =-   fromChunks $ f (toChunks x) (toChunks y)--{- |-Remove zero chunks.--}-normalize :: NonNeg.C a => T a -> T a-normalize = fromChunks . filter (> NonNeg.zero) . toChunks--isNullList :: NonNeg.C a => [a] -> Bool-isNullList = null . filter (> NonNeg.zero)--isNull :: NonNeg.C a => T a -> Bool-isNull = isNullList . toChunks-  -- null . toChunks . normalize--isPositive :: NonNeg.C a => T a -> Bool-isPositive = not . isNull----{--normalizeZT :: ZeroTestable.C a => T a -> T a-normalizeZT = fromChunks . filter (not . isZero) . toChunks--}--isNullListZT :: ZeroTestable.C a => [a] -> Bool-isNullListZT = null . filter (not . isZero)--isNullZT :: ZeroTestable.C a => T a -> Bool-isNullZT = isNullListZT . decons-  -- null . toChunks . normalize-{--isPositiveZT :: ZeroTestable.C a => T a -> Bool-isPositiveZT = not . isNull--}---check :: String -> Bool -> a -> a-check funcName b x =-   if b-     then x-     else error ("Numeric.NonNegative.Chunky."++funcName++": negative number")---glue :: (NonNeg.C a) => [a] -> [a] -> ([a], (Bool, [a]))-glue [] ys = ([], (True,  ys))-glue xs [] = ([], (False, xs))-glue (x:xs) (y:ys) =-   let (z,~(zs,brs)) =-          flip mapSnd (NonNeg.split x y) $-          \(b,d) ->-             if b-               then glue xs $-                    if NonNeg.zero == d-                      then ys else d:ys-               else glue (d:xs) ys-   in  (z:zs,brs)--minMaxDiff :: (NonNeg.C a) => T a -> T a -> (T a, (Bool, T a))-minMaxDiff (Cons xs) (Cons ys) =-   let (zs, (b, rs)) = glue xs ys-   in  (Cons zs, (b, Cons rs))--equalList :: (NonNeg.C a) => [a] -> [a] -> Bool-equalList x y =-   isNullList $ snd $ snd $ glue x y--compareList :: (NonNeg.C a) => [a] -> [a] -> Ordering-compareList x y =-   let (b,r) = snd $ glue x y-   in  if isNullList r-         then EQ-         else if b then LT else GT--minList :: (NonNeg.C a) => [a] -> [a] -> [a]-minList x y =-   fst $ glue x y--maxList :: (NonNeg.C a) => [a] -> [a] -> [a]-maxList x y =-   -- matching the inner pair lazily is important-   let (z,~(_,r)) = glue x y in z++r---instance (NonNeg.C a) => Eq (T a) where-   (Cons x) == (Cons y) = equalList x y--instance (NonNeg.C a) => Ord (T a) where-   compare (Cons x) (Cons y) = compareList x y-   min = lift2 minList-   max = lift2 maxList---instance (NonNeg.C a) => NonNeg.C (T a) where-   split (Cons xs) (Cons ys) =-      let (zs, ~(b, rs)) = glue xs ys-      in  (Cons zs, (b, Cons rs))--instance (ZeroTestable.C a) => ZeroTestable.C (T a) where-   isZero = isNullZT--instance (NonNeg.C a) => Additive.C (T a) where-   zero  = Monoid.idt-   (+)   = (Monoid.<*>)-   (Cons x) - (Cons y) =-      let (b,d) = snd $ glue x y-          d' = Cons d-      in check "-" (not b || isNull d') d'-   negate x = check "negate" (isNull x) x-{--   x0 - y0 =-      let d' = lift2 (\x y -> let (_,d,b) = glue x y in  d) x0 y0-      in  check "-" (not b || isNull d') d'--}--instance (Ring.C a, NonNeg.C a) => Ring.C (T a) where-   one   = fromNumber one-   (*)   = lift2 (liftM2 (*))-   fromInteger = fromNumber . fromInteger--instance (Ring.C a, ZeroTestable.C a, NonNeg.C a) => Absolute.C (T a) where-   abs    = id-   signum = fromNumber . (\b -> if b then one else zero) . isPositive--instance (ToInteger.C a, NonNeg.C a) => ToInteger.C (T a) where-   toInteger = sum . map toInteger . toChunks--instance (ToRational.C a, NonNeg.C a) => ToRational.C (T a) where-   toRational = sum . map toRational . toChunks---instance (RealIntegral.C a, NonNeg.C a) => RealIntegral.C (T a) where-   quot = div-   rem  = mod-   quotRem = divMod--{- |-'divMod' is implemented in terms of 'divModStrict'.-If it is needed we could also provide a function-that accesses the divisor first in a lazy way-and then uses a strict divisor for subsequent rounds of the subtraction loop.-This way we can handle the cases \"dividend smaller than divisor\"-and \"dividend greater than divisor\" in a lazy and efficient way.-However changing the way of operation within one number is also not nice.--}-instance (Ord a, Integral.C a, NonNeg.C a) => Integral.C (T a) where-   divMod x y =-      mapSnd fromNumber $-      divModStrict x (toNumber y)--{- |-divModLazy accesses the divisor in a lazy way.-However this is only relevant if the dividend is smaller than the divisor.-For large dividends the divisor will be accessed multiple times-but since it is already fully evaluated it could also be strict.--}-divModLazy ::-   (Ring.C a, NonNeg.C a) =>-   T a -> T a -> (T a, T a)-divModLazy x0 y0 =-   let y = toChunks y0-       recourse x =-          let (r,~(b,d)) = glue y x-          in  if not b-                then ([], r)-                else mapFst (one:) (recourse d)-   in  mapPair-          (fromChunks, fromChunks)-          (recourse (toChunks x0))--{- |-This function has a strict divisor-and maintains the chunk structure of the dividend at a smaller scale.--}-divModStrict ::-   (Integral.C a, NonNeg.C a) =>-   T a -> a -> (T a, a)-divModStrict x0 y =-   let recourse [] r = ([], r)-       recourse (x:xs) r0 =-          case divMod (x+r0) y of-             (q,r1) -> mapFst (q:) $ recourse xs r1-   in  mapFst fromChunks $ recourse (toChunks x0) zero----instance (Show a) => Show (T a) where-   showsPrec p x =-      showParen (p>10)-         (showString "Chunky.fromChunks " . showsPrec 10 (decons x))---instance (NonNeg.C a, Arbitrary a) => Arbitrary (T a) where-   arbitrary = liftM Cons arbitrary----{- * legacy instances -}--legacyInstance :: a-legacyInstance =-   error "legacy Ring.C instance for simple input of numeric literals"--instance (Ring.C a, Eq a, Show a, NonNeg.C a) => P98.Num (T a) where-   fromInteger = fromNumber . fromInteger-   negate = Additive.negate -- for unary minus-   (+)    = legacyInstance-   (*)    = legacyInstance-   abs    = legacyInstance-   signum = legacyInstance--instance (Field.C a, Eq a, Show a, NonNeg.C a) => P98.Fractional (T a) where-   fromRational = fromNumber . fromRational-   (/) = legacyInstance--instance (NonNeg.C a) => Mn98.Monoid (T a) where-   mempty  = Monoid.idt-   mappend = (Monoid.<*>)--instance (NonNeg.C a) => Monoid.C (T a) where-   idt   = Cons []-   (<*>) = lift2 (++)
− src-ghc-6.12/Number/OccasionallyScalarExpression.hs 
@@ -1,196 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-{- |-Copyright   :  (c) Henning Thielemann 2004-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  multi-type parameter classes (vector space)--Physical expressions track the operations made on physical values-so we are able to give detailed information on how to resolve-unit violations.--}--module Number.OccasionallyScalarExpression where--import qualified Algebra.Transcendental      as Trans-import qualified Algebra.Algebraic           as Algebraic-import qualified Algebra.Field               as Field-import qualified Algebra.Absolute                as Absolute-import qualified Algebra.Ring                as Ring-import qualified Algebra.Additive            as Additive-import qualified Algebra.ZeroTestable        as ZeroTestable--import Algebra.Algebraic (sqrt, (^/))-import qualified Algebra.OccasionallyScalar as OccScalar--import Data.Maybe(fromMaybe)-import Data.Array(listArray,(!))--import NumericPrelude.Base-import NumericPrelude.Numeric---{- | A value of type 'T' stores information on how to resolve unit violations.-     The main application of the module are certainly-     Number.Physical type instances-     but in principle it can also be applied to other occasionally scalar types. -}-data T a v = Cons (Term a v) v--data Term a v =-     Const-   | Add (T a v) (T a v)-   | Mul (T a v) (T a v)-   | Div (T a v) (T a v)--fromValue :: v -> T a v-fromValue = Cons Const---makeLine :: Int -> String -> String-makeLine indent str = replicate indent ' ' ++ str ++ "\n"--showUnitError :: (Show v) => Bool -> Int -> v -> T a v -> String-showUnitError divide indent x (Cons expr y) =-  let indent'   = indent+2-      showSub d = showUnitError d (indent'+2) x-      mulDivArr = listArray (False, True) ["multiply", "divide"]-  in  makeLine indent-         (mulDivArr ! divide ++-          " " ++ show y ++ " by " ++ show x) ++-      case expr of-        (Const) -> ""-        (Add y0 y1) ->-          makeLine indent' "e.g." ++-          showSub divide y0 ++-          makeLine indent' "and " ++-          showSub divide y1-        (Mul y0 y1) ->-          makeLine indent' "e.g." ++-          showSub divide y0 ++-          makeLine indent' "or  " ++-          showSub divide y1-        (Div y0 y1) ->-          makeLine indent' "e.g." ++-          showSub divide y0 ++-          makeLine indent' "or  " ++-          showSub (not divide) y1---lift :: (v -> v) -> (T a v -> T a v)-lift f (Cons xe x) = Cons xe (f x)--fromScalar :: (Show v, OccScalar.C a v) =>-   a -> T a v-fromScalar = OccScalar.fromScalar--scalarMap :: (Show v, OccScalar.C a v) =>-   (a -> a) -> (T a v -> T a v)-scalarMap f x = OccScalar.fromScalar (f (OccScalar.toScalar x))--scalarMap2 :: (Show v, OccScalar.C a v) =>-   (a -> a -> a) -> (T a v -> T a v -> T a v)-scalarMap2 f x y = OccScalar.fromScalar (f (OccScalar.toScalar x) (OccScalar.toScalar y))---instance (Show v) => Show (T a v) where-  show (Cons _ x) = show x--instance (Eq v) => Eq (T a v) where-  (Cons _ x) == (Cons _ y) = x==y--instance (Ord v) => Ord (T a v) where-  compare (Cons _ x) (Cons _ y) = compare x y--instance (Additive.C v) => Additive.C (T a v) where-  zero = Cons Const zero-  xe@(Cons _ x) + ye@(Cons _ y) = Cons (Add xe ye) (x+y)-  xe@(Cons _ x) - ye@(Cons _ y) = Cons (Add xe ye) (x-y)-  negate = lift negate--instance (Ring.C v) => Ring.C (T a v) where-  xe@(Cons _ x) * ye@(Cons _ y) = Cons (Mul xe ye) (x*y)--  fromInteger = fromValue . fromInteger--instance (Field.C v) => Field.C (T a v) where-  xe@(Cons _ x) / ye@(Cons _ y) = Cons (Div xe ye) (x/y)-  fromRational' = fromValue . fromRational'--instance (ZeroTestable.C v) => ZeroTestable.C (T a v) where-  isZero (Cons _ x) = isZero x--instance (Absolute.C v) => Absolute.C (T a v) where-  {- are these definitions sensible? -}-  abs    = lift abs-  signum = lift signum---{- This instance is not quite satisfying.-   The expression data structure should also keep track of powers-   in order to report according errors. -}-instance (Algebraic.C a, Field.C v, Show v, OccScalar.C a v) =>-    Algebraic.C (T a v) where-  sqrt    = scalarMap  sqrt-  x ^/ y  = scalarMap  (^/ y) x--instance (Trans.C a, Field.C v, Show v, OccScalar.C a v) =>-    Trans.C (T a v) where-  pi      = fromScalar pi-  log     = scalarMap  log-  exp     = scalarMap  exp-  logBase = scalarMap2 logBase-  (**)    = scalarMap2 (**)-  cos     = scalarMap  cos-  tan     = scalarMap  tan-  sin     = scalarMap  sin-  acos    = scalarMap  acos-  atan    = scalarMap  atan-  asin    = scalarMap  asin-  cosh    = scalarMap  cosh-  tanh    = scalarMap  tanh-  sinh    = scalarMap  sinh-  acosh   = scalarMap  acosh-  atanh   = scalarMap  atanh-  asinh   = scalarMap  asinh---instance (OccScalar.C a v, Show v)-      => OccScalar.C a (T a v) where-   toScalar xe@(Cons _ x) =-      fromMaybe-         (error (show xe ++ " is not a scalar value.\n" ++-                 showUnitError True 0 x xe))-         (OccScalar.toMaybeScalar x)-   toMaybeScalar (Cons _ x) = OccScalar.toMaybeScalar x-   fromScalar = fromValue . OccScalar.fromScalar---{--  I would like to use OccasionallyScalar.toScalar-  in fmap and (>>=) to allow more sophisticated error messages-  for types that support more descriptive error messages.-  But this requires constraints to the type arguments of-  Functor and Monad.--}---{- Operators for lifting scalar operations to-   operations on physical values -}-{--instance Functor (T i) where-  fmap f (Cons xu x) =-    if Unit.isScalar xu-    then OccScalar.fromScalar (f x)-    else error "Physics.Quantity.Value.fmap: function for scalars, only"--instance Monad (T i) where-  (>>=) (Cons xu x) f =-    if Unit.isScalar xu-    then f x-    else error "Physics.Quantity.Value.(>>=): function for scalars, only"-  return = OccScalar.fromScalar--}
− src-ghc-6.12/Number/PartiallyTranscendental.hs 
@@ -1,91 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Define Transcendental functions on arbitrary fields.-These functions are defined for only a few (in most cases only one) arguments,-that's why discourage making these types instances of 'Algebra.Transcendental.C'.-But instances of 'Algebra.Transcendental.C' can be useful when working with power series.-If you intent to work with power series with 'Rational' coefficients,-you might consider using @MathObj.PowerSeries.T (Number.PartiallyTranscendental.T Rational)@-instead of @MathObj.PowerSeries.T Rational@.--}-module Number.PartiallyTranscendental (T, fromValue, toValue) where--import qualified Algebra.Transcendental as Transcendental-import qualified Algebra.Algebraic      as Algebraic-import qualified Algebra.Field          as Field-import qualified Algebra.Ring           as Ring-import qualified Algebra.Additive       as Additive--- import qualified Algebra.ZeroTestable   as ZeroTestable--import NumericPrelude.Numeric-import NumericPrelude.Base--import qualified Prelude as P---newtype T a = Cons {toValue :: a}-   deriving (Eq, Ord, Show)--fromValue :: a -> T a-fromValue = lift0--lift0 :: a -> T a-lift0 = Cons--lift1 :: (a -> a) -> (T a -> T a)-lift1 f (Cons x0) = Cons (f x0)--lift2 :: (a -> a -> a) -> (T a -> T a -> T a)-lift2 f (Cons x0) (Cons x1) = Cons (f x0 x1)---instance (Additive.C a) => Additive.C (T a) where-    negate = lift1 negate-    (+)    = lift2 (+)-    (-)    = lift2 (-)-    zero   = lift0 zero--instance (Ring.C a) => Ring.C (T a) where-    one           = lift0 one-    fromInteger n = lift0 (fromInteger n)-    (*)           = lift2 (*)--instance (Field.C a) => Field.C (T a) where-    (/) = lift2 (/)--instance (Algebraic.C a) => Algebraic.C (T a) where-    sqrt x = lift1 sqrt x-    root n = lift1 (Algebraic.root n)-    (^/) x y = lift1 (^/y) x--instance (Algebraic.C a, Eq a) => Transcendental.C (T a) where-    pi = undefined-    exp = \0 -> 1-    sin = \0 -> 0-    cos = \0 -> 1-    tan = \0 -> 0-    x ** y = if x==1 || y==0-               then 1-               else error "partially transcendental power undefined"-    log  = \1 -> 0-    asin = \0 -> 0-    acos = \1 -> 0-    atan = \0 -> 0----legacyInstance :: a-legacyInstance = error "legacy Ring instance for simple input of numeric literals"---instance (P.Num a) => P.Num (T a) where-   fromInteger n = lift0 $ P.fromInteger n-   negate = P.negate -- for unary minus-   (+)    = legacyInstance-   (*)    = legacyInstance-   abs    = legacyInstance-   signum = legacyInstance--instance (P.Num a) => P.Fractional (T a) where-   fromRational = P.fromRational-   (/) = legacyInstance
− src-ghc-6.12/Number/Peano.hs 
@@ -1,432 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Copyright    :   (c) Henning Thielemann 2007-Maintainer   :   numericprelude@henning-thielemann.de-Stability    :   provisional-Portability  :   portable--Lazy Peano numbers represent natural numbers inclusive infinity.-Since they are lazily evaluated,-they are optimally for use as number type of 'Data.List.genericLength' et.al.--}-module Number.Peano where--import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.Units                as Units-import qualified Algebra.RealIntegral         as RealIntegral-import qualified Algebra.IntegralDomain       as Integral-import qualified Algebra.Absolute             as Absolute-import qualified Algebra.Ring                 as Ring-import qualified Algebra.Additive             as Additive-import qualified Algebra.ZeroTestable         as ZeroTestable-import qualified Algebra.Indexable            as Indexable-import qualified Algebra.Monoid               as Monoid--import qualified Algebra.ToInteger            as ToInteger-import qualified Algebra.ToRational           as ToRational-import qualified Algebra.NonNegative          as NonNeg--import qualified Algebra.EqualityDecision as EqDec-import qualified Algebra.OrderDecision    as OrdDec--import Data.Maybe (catMaybes, )-import Data.Array(Ix(..))--import qualified Prelude     as P98-import qualified NumericPrelude.Base as P-import qualified NumericPrelude.Numeric as NP-import Data.List.HT (mapAdjacent, shearTranspose, )-import Data.Tuple.HT (mapFst, )--import NumericPrelude.Base-import NumericPrelude.Numeric---data T = Zero-       | Succ T-   deriving (Show, Read, Eq)--infinity :: T-infinity = Succ infinity--err :: String -> String -> a-err func msg = error ("Number.Peano."++func++": "++msg)---instance ZeroTestable.C T where-   isZero Zero     = True-   isZero (Succ _) = False--add :: T -> T -> T-add Zero y = y-add (Succ x) y = Succ (add x y)--sub :: T -> T -> T-sub x y =-   let (sign,z) = subNeg y x-   in  if sign-         then err "sub" "negative difference"-         else z--subNeg :: T -> T -> (Bool, T)-subNeg Zero y = (False, y)-subNeg x Zero = (True,  x)-subNeg (Succ x) (Succ y) = subNeg x y---mul :: T -> T -> T-mul Zero _ = Zero-mul _ Zero = Zero-mul (Succ x) y = add y (mul x y)--fromPosEnum :: (ZeroTestable.C a, Enum a) => a -> T-fromPosEnum n =-   if isZero n-      then Zero-      else Succ (fromPosEnum (pred n))--toPosEnum :: (Additive.C a, Enum a) => T -> a-toPosEnum Zero = zero-toPosEnum (Succ x) = succ (toPosEnum x)--instance Additive.C T where-   zero = Zero-   (+) = add-   (-) = sub-   negate Zero     = Zero-   negate (Succ _) = err "negate" "cannot negate positive number"--instance Ring.C T where-   one = Succ Zero-   (*) = mul-   fromInteger n =-      if n<0-        then err "fromInteger" "Peano numbers are always non-negative"-        else fromPosEnum n--instance Enum T where-   pred Zero = err "pred" "Zero has no predecessor"-   pred (Succ x) = x-   succ = Succ-   toEnum n =-      if n<0-        then err "toEnum" "Peano numbers are always non-negative"-        else fromPosEnum n-   fromEnum = toPosEnum-   enumFrom x = iterate Succ x-   enumFromThen x y =-      let (sign,d) = subNeg x y-      in  if sign-            then iterate (sub d) x-            else iterate (add d) x-   {--   enumFromTo =-   enumFromThenTo =-   -}---{- |-If all values are completely defined,-then it holds--> if b then x else y == ifLazy b x y--However if @b@ is undefined,-then it is at least known that the result is larger than @min x y@.--}-ifLazy :: Bool -> T -> T -> T-ifLazy b (Succ x) (Succ y) = Succ (ifLazy b x y)-ifLazy b x y = if b then x else y--instance EqDec.C T where-   (==?) x y = ifLazy (x==y)--instance OrdDec.C T where-   (<=?) x y le gt = ifLazy (x<=y) le gt--{--The default instance is good for compare,-but fails for min and max.--}-instance Ord T where-   compare (Succ x) (Succ y) = compare x y-   compare Zero     (Succ _) = LT-   compare (Succ _) Zero     = GT-   compare Zero     Zero     = EQ--   min (Succ x) (Succ y) = Succ (min x y)-   min _        _        = Zero--   max (Succ x) (Succ y) = Succ (max x y)-   max Zero     y        = y-   max x        Zero     = x--   {--   This special implementation works also for undefined < Zero.-   Thanks to Peter Divianszky for the hint.-   -}-   _      < Zero   = False-   Zero   < _      = True-   Succ n < Succ m = n < m--   x > y  = y < x--   x <= y = not (y < x)--   x >= y = not (x < y)---{- | cf.-To how to find the shortest list in a list of lists efficiently,-this means, also in the presence of infinite lists.-<http://www.haskell.org/pipermail/haskell-cafe/2006-October/018753.html>--}-argMinFull :: (T,a) -> (T,a) -> (T,a)-argMinFull (x0,xv) (y0,yv) =-   let recourse (Succ x) (Succ y) =-          let (z,zv) = recourse x y-          in  (Succ z, zv)-       recourse Zero _ = (Zero,xv)-       recourse _    _ = (Zero,yv)-   in  recourse x0 y0--{- |-On equality the first operand is returned.--}-argMin :: (T,a) -> (T,a) -> a-argMin x y = snd $ argMinFull x y--argMinimum :: [(T,a)] -> a-argMinimum = snd . foldl1 argMinFull---argMaxFull :: (T,a) -> (T,a) -> (T,a)-argMaxFull (x0,xv) (y0,yv) =-   let recourse (Succ x) (Succ y) =-          let (z,zv) = recourse x y-          in  (Succ z, zv)-       recourse x Zero = (x,xv)-       recourse _ y    = (y,yv)-   in  recourse x0 y0--{- |-On equality the first operand is returned.--}-argMax :: (T,a) -> (T,a) -> a-argMax x y = snd $ argMaxFull x y--argMaximum :: [(T,a)] -> a-argMaximum = snd . foldl1 argMaxFull------ isAscending - naive implementations--{- |-@x0 <= x1 && x1 <= x2 ... @-for possibly infinite numbers in finite lists.--}-isAscendingFiniteList :: [T] -> Bool-isAscendingFiniteList [] = True-isAscendingFiniteList (x:xs) =-   let decrement (Succ y) = Just y-       decrement _ = Nothing-   in  case x of-         Zero -> isAscendingFiniteList xs-         Succ xd ->-           case mapM decrement xs of-             Nothing -> False-             Just xsd -> isAscendingFiniteList (xd : xsd)--isAscendingFiniteNumbers :: [T] -> Bool-isAscendingFiniteNumbers = and . mapAdjacent (<=)----- isAscending - sophisticated implementations - explicit--toListMaybe :: a -> T -> [Maybe a]-toListMaybe a =-   let recourse Zero     = [Just a]-       recourse (Succ x) = Nothing : recourse x-   in  recourse--{- |-In @glue x y == (z,(b,r))@-@z@ represents @min x y@,-@r@ represents @max x y - min x y@,-and @x<=y  ==  b@.--Cf. Numeric.NonNegative.Chunky--}-glue :: T -> T -> (T, (Bool, T))-glue Zero ys = (Zero, (True, ys))-glue xs Zero = (Zero, (False, xs))-glue (Succ xs) (Succ ys) =-   mapFst Succ $ glue xs ys--{--Implementation notes:-We check all pairs of adjacent numbers for correct order.-We obtain a set of booleans, which must all be True.-The order of checking these booleans is crucial.-Pairs of numbers that are infinitely big or infinitely far in the list-must be checked \"last\".-Thus we order the booleans according to their computation costs-(list position + magnitude of number)-using 'shearTranspose'.--}-isAscending :: [T] -> Bool-isAscending =-   and . catMaybes . concat .-   shearTranspose .-   mapAdjacent (\x y ->-      let (costs0,(le,_)) = glue x y-      in  toListMaybe le costs0)----- isAscending - use a cost measuring data type (could generalized to a monad, when considered as Writer monad, see htam and unique-logic packages---- following an idea of vixy http://moonpatio.com:8080/fastcgi/hpaste.fcgi/view?id=562--data Valuable a = Valuable {costs :: T, value :: a}-   deriving (Show, Eq, Ord)---increaseCosts :: T -> Valuable a -> Valuable a-increaseCosts inc ~(Valuable c x) = Valuable (inc+c) x--{- |-Compute '(&&)' with minimal costs.--}-infixr 3 &&~-(&&~) :: Valuable Bool -> Valuable Bool -> Valuable Bool-(&&~) (Valuable xc xb) (Valuable yc yb) =-   let (minc,~(le,difc)) = glue xc yc-       (bCheap,bExpensive) =-          if le-            then (xb,yb)-            else (yb,xb)-   in  increaseCosts minc $-       uncurry Valuable $-       if bCheap-         then (difc, bExpensive)-         else (Zero, False)--andW :: [Valuable Bool] -> Valuable Bool-andW =-   foldr-      (\b acc -> b &&~ increaseCosts one acc)-      (Valuable Zero True)--leW :: T -> T -> Valuable Bool-leW x y =-   let (minc,~(le,_difc)) = glue x y-   in  Valuable minc le--isAscendingW :: [T] -> Valuable Bool-isAscendingW =-   andW . mapAdjacent leW--{--test with--*Number.Peano> isAscendingW [0,infinity,infinity,5]-False--}----- instances--instance Absolute.C T where-   signum Zero     = zero-   signum (Succ _) = one-   abs             = id--instance ToInteger.C T where-   toInteger = toPosEnum--instance ToRational.C T where-   toRational = toRational . toInteger--instance RealIntegral.C T where-   quot = div-   rem  = mod-   quotRem = divMod--instance Integral.C T where-   div x y = fst (divMod x y)-   mod x y = snd (divMod x y)-   divMod x y =-      let (isNeg,d) = subNeg y x-      in  if isNeg-            then (zero,x)-            else let (q,r) = divMod d y in (succ q,r)--instance Monoid.C T where-   idt = zero-   (<*>) = add-   cumulate = foldr add Zero--instance NonNeg.C T where-   split = glue--instance Ix T where-   range = uncurry enumFromTo-   index (lower,_) i =-      let (sign,offset) = subNeg lower i-      in  if sign-            then err "index" "index out of range"-            else toPosEnum offset-   inRange (lower,upper) i =-      isAscending [lower, i, upper]-   rangeSize (lower,upper) =-      toPosEnum (sub lower (succ upper))--instance Indexable.C T where-   compare = Indexable.ordCompare--instance Units.C T where-   isUnit x  =  x == one-   stdAssociate  =  id-   stdUnit    _ = one-   stdUnitInv _ = one--instance PID.C T where-   gcd = PID.euclid mod-   extendedGCD = PID.extendedEuclid divMod--instance Bounded T where-   minBound = Zero-   maxBound = infinity----legacyInstance :: a-legacyInstance =-   error "legacy Ring.C instance for simple input of numeric literals"--instance P98.Num T where-   fromInteger = Ring.fromInteger-   negate = Additive.negate -- for unary minus-   (+) = add-   (-) = sub-   (*) = mul-   signum = legacyInstance-   abs = legacyInstance---- for use with genericLength et.al.-instance P98.Real T where-   toRational = P98.toRational . toInteger--instance P98.Integral T where-   rem  = div-   quot = mod-   quotRem = divMod-   div x y = fst (divMod x y)-   mod x y = snd (divMod x y)-   divMod x y =-      let (sign,d) = subNeg y x-      in  if sign-            then (0,x)-            else let (q,r) = divMod d y in (succ q,r)-   toInteger = toPosEnum
− src-ghc-6.12/Number/Physical.hs 
@@ -1,236 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-{- |-Copyright   :  (c) Henning Thielemann 2003-2006-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  generic instances--Numeric values combined with abstract Physical Units--}--module Number.Physical where--import qualified Number.Physical.Unit as Unit--import           Algebra.OccasionallyScalar  as OccScalar-import qualified Algebra.VectorSpace         as VectorSpace-import qualified Algebra.Module              as Module-import qualified Algebra.Vector              as Vector-import qualified Algebra.Transcendental      as Trans-import qualified Algebra.Algebraic           as Algebraic-import qualified Algebra.Field               as Field-import qualified Algebra.Absolute                as Absolute-import qualified Algebra.Ring                as Ring-import qualified Algebra.Additive            as Additive-import qualified Algebra.ZeroTestable        as ZeroTestable--import qualified Algebra.ToInteger      as ToInteger--import Algebra.Algebraic (sqrt, (^/))--import qualified Number.Ratio as Ratio--import Control.Monad(guard,liftM,liftM2)--import Data.Maybe.HT(toMaybe)-import Data.Maybe(fromMaybe)--import NumericPrelude.Numeric-import NumericPrelude.Base----- | A Physics.Quantity.Value.T combines a numeric value with a physical unit.-data T i a = Cons (Unit.T i) a---- | Construct a physical value from a numeric value and--- the full vector representation of a unit.-quantity :: (Ord i, Enum i, Ring.C a) => [Int] -> a -> T i a-quantity v = Cons (Unit.fromVector v)--fromScalarSingle :: a -> T i a-fromScalarSingle = Cons Unit.scalar---- | Test for the neutral Unit.T. Also a zero has a unit!-isScalar :: T i a -> Bool-isScalar (Cons u _) = Unit.isScalar u---{- Using (((join.).).liftM2) you can turn madd and msub-   into operations that map Maybes to Maybes -}---- | apply a function to the numeric value while preserving the unit-lift :: (a -> b) -> T i a -> T i b-lift f (Cons xu x) = Cons xu (f x)--lift2 :: (Eq i) => String -> (a -> b -> c) -> T i a -> T i b -> T i c-lift2 opName op x y =-   fromMaybe (errorUnitMismatch opName) (lift2Maybe op x y)--lift2Maybe :: (Eq i) => (a -> b -> c) -> T i a -> T i b -> Maybe (T i c)-lift2Maybe op (Cons xu x) (Cons yu y) =-   toMaybe (xu==yu) (Cons xu (op x y))--lift2Gen :: (Eq i) => String -> (a -> b -> c) -> T i a -> T i b -> c-lift2Gen opName op (Cons xu x) (Cons yu y) =-   if (xu==yu)-     then op x y-     else errorUnitMismatch opName--errorUnitMismatch :: String -> a-errorUnitMismatch opName =-   error ("Physics.Quantity.Value."++opName++": units mismatch")------ | Add two values if the units match, otherwise return Nothing-addMaybe :: (Eq i, Additive.C a) =>-  T i a -> T i a -> Maybe (T i a)-addMaybe = lift2Maybe (+)---- | Subtract two values if the units match, otherwise return Nothing-subMaybe :: (Eq i, Additive.C a) =>-  T i a -> T i a -> Maybe (T i a)-subMaybe = lift2Maybe (-)---scale :: (Ord i, Ring.C a) => a -> T i a -> T i a-scale x = lift (x*)--ratPow :: Trans.C a => Ratio.T Int -> T i a -> T i a-ratPow expo (Cons xu x) =-  Cons (Unit.ratScale expo xu) (x ** fromRatio expo)--ratPowMaybe :: (Trans.C a) =>-    Ratio.T Int -> T i a -> Maybe (T i a)-ratPowMaybe expo (Cons xu x) =-  fmap (flip Cons (x ** fromRatio expo)) (Unit.ratScaleMaybe expo xu)--fromRatio :: (Field.C b, ToInteger.C a) => Ratio.T a -> b-fromRatio expo = fromIntegral (numerator expo) /-                 fromIntegral (denominator expo)----instance (ZeroTestable.C v) => ZeroTestable.C (T a v) where-  isZero (Cons _ x) = isZero x--instance (Eq i, Eq a) => Eq (T i a) where-  (==) = lift2Gen "(==)" (==)--instance (Ord i, Enum i, Show a) => Show (T i a) where-  --show (Cons xu x) = show x ++ " !* " ++ show (Unit.toVector xu)-  show (Cons xu x) = "quantity " ++ show (Unit.toVector xu) ++ " " ++ show x--instance (Ord i, Additive.C a) => Additive.C (T i a) where-  zero   = fromScalarSingle zero-  -- Add two values if the units match, otherwise raise an error-  (+)    = lift2 "(+)" (+)-  -- Subtract two values if the units match, otherwise raise an error-  (-)    = lift2 "(-)" (-)-  negate = lift negate--instance (Ord i, Ring.C a) => Ring.C (T i a) where-  (Cons xu x) * (Cons yu y) = Cons (xu+yu) (x*y)-  fromInteger = fromScalarSingle . fromInteger--instance (Ord i, Ord a) => Ord (T i a) where-  max     = lift2    "max"     max-  min     = lift2    "min"     min-  compare = lift2Gen "compare" compare-  (<)     = lift2Gen "(<)"     (<)-  (>)     = lift2Gen "(>)"     (>)-  (<=)    = lift2Gen "(<=)"    (<=)-  (>=)    = lift2Gen "(>=)"    (>=)--{--  Are absolute value and signum sensible for unit values?-  What is the sign, what is the absolute value?-  We could see it this way:-  The absolute value has no unit and-  the signum contains the unit and the scalar's sign.-  However the units contain also information of magnitude.-  E.g. if the base unit would be gramm instead kilogramm-  then the scalars would grow to a factor thousand.--  So is it better to give-  the absolute value unit and the absolute value of the scalar and-  the signum has no unit and the signum of the scalar?-  But the unit may also carry a kind of 'negativity' inside,-  e.g. the electric charge.--  It seems that there is no clear answer.-  However in my synthesizer application-  I need absolute values for sample rates and amplitudes.-  There the second interpretation is needed.--}-instance (Ord i, Absolute.C a) => Absolute.C (T i a) where-  abs               = lift abs-  signum (Cons _ x) = fromScalarSingle (signum x)---instance (Ord i, Field.C a) => Field.C (T i a) where-  (Cons xu x) / (Cons yu y) = Cons (xu-yu) (x/y)-  fromRational' = fromScalarSingle . fromRational'--instance (Ord i, Algebraic.C a) => Algebraic.C (T i a) where-  sqrt (Cons xu x) = Cons (Unit.ratScale 0.5 xu) (sqrt x)-  Cons xu x ^/ y =-     Cons (Unit.ratScale (fromRational' y) xu) (x ^/ y)--instance (Ord i, Trans.C a) => Trans.C (T i a) where-  pi      = fromScalarSingle pi-  log     = liftM  log-  exp     = liftM  exp-  logBase = liftM2 logBase-  (**)    = liftM2 (**)-  cos     = liftM  cos-  tan     = liftM  tan-  sin     = liftM  sin-  acos    = liftM  acos-  atan    = liftM  atan-  asin    = liftM  asin-  cosh    = liftM  cosh-  tanh    = liftM  tanh-  sinh    = liftM  sinh-  acosh   = liftM  acosh-  atanh   = liftM  atanh-  asinh   = liftM  asinh--instance Ord i => Vector.C (T i) where-  zero  = zero-  (<+>) = (+)-  (*>)  = scale--instance (Ord i, Module.C a v) => Module.C a (T i v) where-  x *> (Cons yu y) = Cons yu (x Module.*> y)--instance (Ord i, VectorSpace.C a v) => VectorSpace.C a (T i v)---instance (OccScalar.C a v)-      => OccScalar.C a (T i v) where-   toScalar = toScalarDefault-   toMaybeScalar (Cons xu x)-            = guard (Unit.isScalar xu) >> toMaybeScalar x-   fromScalar = fromScalarSingle . fromScalar----{- Operators for lifting scalar operations to-   operations on physical values -}-instance Functor (T i) where-  fmap f (Cons xu x) =-    if Unit.isScalar xu-    then fromScalarSingle (f x)-    else error "Physics.Quantity.Value.fmap: function for scalars, only"--instance Monad (T i) where-  (>>=) (Cons xu x) f =-    if Unit.isScalar xu-    then f x-    else error "Physics.Quantity.Value.(>>=): function for scalars, only"-  return = fromScalarSingle
− src-ghc-6.12/Number/Physical/Read.hs 
@@ -1,99 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Copyright   :  (c) Henning Thielemann 2004-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  multi-parameter type classes (VectorSpace.hs)--Convert a human readable string to a physical value.--}--module Number.Physical.Read where--import qualified Number.Physical        as Value-import qualified Number.Physical.UnitDatabase as Db-import qualified Algebra.VectorSpace as VectorSpace--- import Algebra.Module((*>))-import qualified Algebra.Field       as Field-import qualified Data.Map as Map-import Data.Map (Map)-import Text.ParserCombinators.Parsec-import Control.Monad(liftM)--import NumericPrelude.Base-import NumericPrelude.Numeric--mulPrec :: Int-mulPrec = 7---- How to handle the 'prec' argument?-readsNat :: (Enum i, Ord i, Read v, VectorSpace.C a v) =>-   Db.T i a -> Int -> ReadS (Value.T i v)-readsNat db prec =-   readParen (prec>=mulPrec)-      (map (\(x, rest) ->-             let (Value.Cons cu c, rest') = readUnitPart (createDict db) rest-             in  (Value.Cons cu (c *> x), rest'))-       .-       readsPrec mulPrec)--readUnitPart :: (Ord i, Field.C a) =>-   Map String (Value.T i a)-      -> String -> (Value.T i a, String)-readUnitPart dict str =-   let parseUnit =-          do p    <- parseProduct-             rest <- many anyChar-             return (product (map (\(unit,n) ->-                        Map.findWithDefault-                           (error ("unknown unit '" ++ unit ++ "'")) unit dict-                           ^ n) p),-                     rest)-   in  case parse parseUnit "unit" str of-          Left  msg -> error (show msg)-          Right val -> val---{-| This function could also return the value,-    but a list of pairs (String, Integer) is easier for testing. -}-parseProduct :: Parser [(String, Integer)]-parseProduct =-   skipMany space >>-      ((do p <- ignoreSpace parsePower-           t <- parseProductTail-           return (p : t)) <|>-       parseProductTail)--parseProductTail :: Parser [(String, Integer)]-parseProductTail =-   let parseTail c f = -         do _ <- ignoreSpace (char c)-            p <- ignoreSpace parsePower-            t <- parseProductTail-            return (f p : t)-   in  parseTail '*' id <|>-       parseTail '/' (\(x,n) -> (x,-n)) <|>-       return []--parsePower :: Parser (String, Integer)-parsePower =-   do w <- ignoreSpace (many1 (letter <|> char '\181'))-      e <- liftM read (ignoreSpace (char '^') >> many1 digit) <|> return 1-      return (w,e)--{- Turns a parser into one that ignores subsequent whitespaces. -}-ignoreSpace :: Parser a -> Parser a-ignoreSpace p =-   do x <- p-      skipMany space-      return x---createDict :: Db.T i a -> Map String (Value.T i a)-createDict db =-   Map.fromList (concatMap-      (\Db.UnitSet {Db.unit = xu, Db.scales = s}-           -> map (\Db.Scale {Db.symbol = sym, Db.magnitude = x}-                       -> (sym, Value.Cons xu x)) s) db)
− src-ghc-6.12/Number/Physical/Show.hs 
@@ -1,105 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Copyright   :  (c) Henning Thielemann 2004-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  multi-parameter type classes (VectorSpace.hs, Normalization.hs)--Convert a physical value to a human readable string.--}--module Number.Physical.Show where--import qualified Number.Physical              as Value-import qualified Number.Physical.UnitDatabase as Db-import Number.Physical.UnitDatabase-          (UnitSet, Scale, reciprocal, magnitude, symbol, scales)--import qualified Algebra.NormedSpace.Maximum as NormedMax-import qualified Algebra.Field               as Field-import qualified Algebra.Ring                as Ring--import Data.List(find)-import Data.Maybe(mapMaybe)--import NumericPrelude.Numeric-import NumericPrelude.Base---mulPrec :: Int-mulPrec = 7--{-| Show the physical quantity in a human readable form-    with respect to a given unit data base. -}-showNat :: (Ord i, Show v, Field.C a, Ord a, NormedMax.C a v) =>-   Db.T i a -> Value.T i v -> String-showNat db x =-   let (y, unitStr) = showSplit db x-   in  if null unitStr-       then show y-       else showsPrec mulPrec y unitStr--{-| Returns the rescaled value as number-    and the unit as string.-    The value can be used re-scale connected values-    and display them under the label of the unit -}-showSplit :: (Ord i, Show v, Field.C a, Ord a, NormedMax.C a v) =>-   Db.T i a -> Value.T i v -> (v, String)-showSplit db (Value.Cons xu x) =-   showScaled x (Db.positiveToFront (Db.decompose xu db))---showScaled :: (Ord i, Show v, Ord a, Field.C a, NormedMax.C a v) =>-   v -> [UnitSet i a] -> (v, String)-showScaled x [] = (x, "")-showScaled x (us:uss) =-  let (scaledX, sc) = chooseScale x us-  in  (scaledX, showUnitPart False (reciprocal us) sc ++-                   concatMap (\us' ->-                      showUnitPart True (reciprocal us') (defScale us')) uss)--{-| Choose a scale where the number becomes handy-    and return the scaled number and the corresponding scale. -}-chooseScale :: (Ord i, Show v, Ord a, Field.C a, NormedMax.C a v) =>-   v -> UnitSet i a -> (v, Scale a)-chooseScale x us =-   let sc = findCloseScale (NormedMax.norm x) (-               {- you should not reverse earlier,-                  otherwise the index of the default unit is wrong -}-               if reciprocal us-               then scales us-               else reverse (scales us))-   in  ((1 / magnitude sc) *> x, sc)---showUnitPart :: Bool -> Bool -> Scale a -> String-showUnitPart multSign rec sc =-   if rec-   then "/" ++ symbol sc-   else -- the multiplication sign can be omitted before the first unit component-        (if multSign then "*" else " ") ++ symbol sc--defScale :: UnitSet i v -> Scale v-defScale Db.UnitSet{Db.defScaleIx=def, Db.scales=scs} = scs!!def--findCloseScale :: (Ord a, Field.C a) => a -> [Scale a] -> Scale a-findCloseScale _ [sc]     = sc-findCloseScale x (sc:scs) =-   if 0.9 * magnitude sc < x-   then sc-   else findCloseScale x scs-findCloseScale _ _        =-   error "There must be at least one scale for a unit."--{-| unused -}-totalDefScale :: Ring.C a => Db.T i a -> a-totalDefScale =-   foldr (\us -> (magnitude (defScale us) *)) 1--{-| unused -}-getUnit :: Ring.C a => String -> Db.T i a -> Value.T i a-getUnit sym = Db.extractOne .-   (mapMaybe (\Db.UnitSet{Db.unit=u, scales=scs} ->-      fmap (Value.Cons u . magnitude) (find ((sym==) . symbol) scs)))
− src-ghc-6.12/Number/Physical/Unit.hs 
@@ -1,84 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Copyright   :  (c) Henning Thielemann 2003-2006-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable--Abstract Physical Units--}--module Number.Physical.Unit where--import MathObj.DiscreteMap (strip)-import qualified Data.Map as Map-import Data.Map (Map)-import Data.Maybe(fromJust,fromMaybe)--import qualified Number.Ratio as Ratio--import Data.Maybe.HT(toMaybe)--import NumericPrelude.Base-import NumericPrelude.Numeric--{- | A Unit.T is a sparse vector with integer entries-   Each map n->m means that the unit of the n-th dimension-   is given m times.--   Example: Let the quantity of length (meter, m) be the zeroth dimension-   and let the quantity of time (second, s) be the first dimension,-   then the composed unit "m_s²" corresponds to the Map-   [(0,1),(1,-2)]--   In future I want to have more abstraction here,-   e.g. a type class from the Edison project-   that abstracts from the underlying implementation.-   Then one can easily switch between-   Arrays, Binary trees (like Map) and what know I.--}-type T i = Map i Int---- | The neutral Unit.T-scalar :: T i-scalar = Map.empty---- | Test for the neutral Unit.T-isScalar ::  T i -> Bool-isScalar = Map.null---- | Convert a List to sparse Map representation--- Example: [-1,0,-2] -> [(0,-1),(2,-2)]-fromVector :: (Enum i, Ord i) => [Int] -> T i-fromVector x = strip (Map.fromList (zip [toEnum 0 .. toEnum ((length x)-1)] x))---- | Convert Map to a List-toVector :: (Enum i, Ord i) => T i -> [Int]-toVector x = map (flip (Map.findWithDefault 0) x)-                     [(toEnum 0)..(maximum (Map.keys x))]---ratScale :: Ratio.T Int -> T i -> T i-ratScale expo =-   fmap (fromMaybe (error "Physics.Quantity.Unit.ratScale: fractional result")) .-   ratScaleMaybe2 expo--ratScaleMaybe :: Ratio.T Int -> T i -> Maybe (T i)-ratScaleMaybe expo u =-   let fmMaybe = ratScaleMaybe2 expo u-   in  toMaybe (not (Nothing `elem` Map.elems fmMaybe))-               (fmap fromJust fmMaybe)---- helper function for ratScale and ratScaleMaybe-ratScaleMaybe2 :: Ratio.T Int -> T i -> Map i (Maybe Int)-ratScaleMaybe2 expo =-   fmap (\c -> let y = Ratio.scale c expo-               in  toMaybe (denominator y == 1) (numerator y))---{- impossible because Unit.T is a type synonyme but not a data type-instance Show (Unit.T i) where-  show = show.toVector--}
− src-ghc-6.12/Number/Physical/UnitDatabase.hs 
@@ -1,186 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Copyright   :  (c) Henning Thielemann 2003-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable--Tools for creating a data base of physical units-and for extracting data from it--}--module Number.Physical.UnitDatabase where--import qualified Number.Physical.Unit as Unit-import qualified Algebra.Field as Field---- import Algebra.Module((*>))-import Algebra.NormedSpace.Sum(norm)--import Data.Maybe.HT (toMaybe)-import Data.List (findIndices, partition, unfoldr, find, minimumBy)--import NumericPrelude.Base-import NumericPrelude.Numeric--type T i a = [UnitSet i a]---- since field names are reused for accessor functions--- they are global identifiers and can't be reused-data InitUnitSet i a =-  InitUnitSet {-    initUnit        :: Unit.T i,-    initIndependent :: Bool,-    initScales      :: [InitScale a]-  }--data InitScale a =-  InitScale {-    initSymbol  :: String,-    initMag     :: a,-    initIsUnit  :: Bool,-    initDefault :: Bool-  }---- | An entry for a unit and there scalings.-data UnitSet i a =-  UnitSet {-    unit        :: Unit.T i,-    independent :: Bool,-    defScaleIx  :: Int,-    reciprocal  :: Bool,  {-^ If True the symbols must be preceded with a '/'.-                              Though it sounds like an attribute of Scale-                              it must be the same for all scales and we need it-                              to sort positive powered unitsets to the front-                              of the list of unit components. -}-    scales      :: [Scale a]-  }-  deriving Show---- | A common scaling for a unit.-data Scale a =-  Scale {-    symbol     :: String,-    magnitude  :: a-  }-  deriving Show----- extract the element from a list containing exact one element--- fails if there are zero or more than one element--- 'head' fails only if there are zero elements-extractOne :: [a] -> a-extractOne (x:[]) = x-extractOne _      = error "There must be exactly one default unit in the data base."--initScale   :: String -> a -> Bool -> Bool -> InitScale a-initScale   = InitScale-initUnitSet :: Unit.T i -> Bool -> [InitScale a] -> InitUnitSet i a-initUnitSet = InitUnitSet--createScale :: InitScale a -> Scale a-createScale (InitScale sym mg _ _) = (Scale sym mg)--createUnitSet :: InitUnitSet i a -> UnitSet i a-createUnitSet (InitUnitSet u ind scs) = (UnitSet u ind-    (extractOne (findIndices initDefault scs))-    False-    (map createScale scs)-  )--{- Filter out all scales intended for showing.-   If there is none return Nothing. -}-showableUnit :: InitUnitSet i a -> Maybe (InitUnitSet i a)-showableUnit (InitUnitSet u ind scs) =-   let sscs = filter initIsUnit scs-   in  toMaybe (not (null sscs)) (InitUnitSet u ind sscs)---{- | Raise all scales of a unit and the unit itself to the n-th power -}-powerOfUnitSet :: (Ord i, Field.C a) => Int -> UnitSet i a -> UnitSet i a-powerOfUnitSet n us@UnitSet { unit = u, reciprocal = rec, scales = scs } =-   us { unit = n *> u,-        reciprocal = rec == (n>0),  -- flip sign-        scales = map (powerOfScale n) scs }---powerOfScale :: Field.C a => Int -> Scale a -> Scale a-powerOfScale n Scale { symbol = sym, magnitude = mag } =-   if n>0-   then Scale { symbol = sym ++ showExp   n,  magnitude = ringPower  n mag }-   else Scale { symbol = sym ++ showExp (-n), magnitude = fieldPower n mag }--showExp :: Int -> String-showExp 1    = ""---showExp 2    = "²"---showExp 3    = "³"-showExp expo = "^" ++ show expo---{- | Reorder the unit components in a way-     that the units with positive exponents lead the list. -}-positiveToFront :: [UnitSet i a] -> [UnitSet i a]-positiveToFront = uncurry (++) . partition (not . reciprocal)---- | Decompose a complex unit into common ones-decompose :: (Ord i, Field.C a) => Unit.T i -> T i a -> [UnitSet i a]-decompose u db =-   case (findIndep u db) of-      Just us -> [us]-      Nothing ->-        unfoldr (\urem ->-          toMaybe (not (Unit.isScalar urem))-                  (let us = findClosest urem db-                   in  (us, subtract (unit us) urem))-        ) u--findIndep :: (Eq i) => Unit.T i -> T i a -> Maybe (UnitSet i a)-findIndep u = find (\UnitSet {unit=un} -> u==un) . filter independent--findClosest :: (Ord i, Field.C a) => Unit.T i -> T i a -> UnitSet i a-findClosest u =-   fst . minimumBy (\(_,dist0) (_,dist1) -> compare dist0 dist1) .-            evalDist u . filter (not.independent)--evalDist :: (Ord i, Field.C a)-   => Unit.T i-   -> T i a-   -> [(UnitSet i a, Int)] {-^ (UnitSet,distance)   the UnitSet may contain powered units -}-evalDist target = map (\us->-    let (expo,dist)=findBestExp target (unit us)-    in  (powerOfUnitSet expo us, dist)-  )--findBestExp :: (Ord i) => Unit.T i -> Unit.T i -> (Int, Int)-findBestExp target u =-  let bestl = findMinExp (distances target (listMultiples (subtract u) (-1)))-      bestr = findMinExp (distances target (listMultiples ((+)      u)   1 ))-  in  if distLE bestl bestr-      then bestl-      else bestr--{-|-  Find the exponent that lead to minimal distance-  Since the list is infinite 'maximum' will fail-  but the sequence is convex-  and thus we can abort when the distance stop falling--}-findMinExp :: [(Int, Int)] -> (Int, Int)-findMinExp (x0:x1:rest) =-  if distLE x0 x1-  then x0-  else findMinExp (x1:rest)-findMinExp _ = error "List of unit approximations with respect to the unit exponent must be infinite."--distLE :: (Int, Int) -> (Int, Int) -> Bool-distLE (_,dist0) (_,dist1) = dist0<=dist1---distLE (exp0,dist0) (exp1,dist1) = (dist0<dist1) || (dist0==dist1 && (abs exp0) <= (abs exp1))---- [(exponent,unit)] -> [(exponent,distance)]-distances :: (Ord i) => Unit.T i -> [(Int, Unit.T i)] -> [(Int, Int)]-distances targetu = map (\(expo,u)->(expo, norm (subtract u targetu)))--listMultiples :: (Unit.T i -> Unit.T i) -> Int -> [(Int, Unit.T i)]-listMultiples f dir = iterate (\(expo,u)->(expo+dir,f u)) (0,Unit.scalar)
− src-ghc-6.12/Number/Positional.hs 
@@ -1,1465 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Copyright   :  (c) Henning Thielemann 2006-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional---Exact Real Arithmetic - Computable reals.-Inspired by ''The most unreliable technique for computing pi.''-See also <http://www.haskell.org/haskellwiki/Exact_real_arithmetic> .--}-module Number.Positional where--import qualified MathObj.LaurentPolynomial as LPoly-import qualified MathObj.Polynomial.Core   as Poly--import qualified Algebra.IntegralDomain as Integral-import qualified Algebra.Ring           as Ring--- import qualified Algebra.Additive       as Additive-import qualified Algebra.ToInteger      as ToInteger--import qualified Prelude as P98-import qualified NumericPrelude.Numeric as NP--import NumericPrelude.Base-import NumericPrelude.Numeric hiding (sqrt, tan, one, zero, )--import qualified Data.List as List-import Data.Char (intToDigit)--import qualified Data.List.Match as Match-import Data.Function.HT (powerAssociative, nest, )-import Data.Tuple.HT (swap, )-import Data.Maybe.HT (toMaybe, )-import Data.Bool.HT (select, if', )-import NumericPrelude.List (mapLast, )-import Data.List.HT-          (sliceVertical, mapAdjacent,-           padLeft, padRight, )---{--FIXME:--defltBase = 10-defltExp = 4--(sqrt 0.5) -- wrong result, probably due to undetected overflows--}--{- * types -}--type T = (Exponent, Mantissa)-type FixedPoint = (Integer, Mantissa)-type Mantissa = [Digit]-type Digit    = Int-type Exponent = Int-type Basis    = Int---{- * basic helpers -}--moveToZero :: Basis -> Digit -> (Digit,Digit)-moveToZero b n =-   let b2 = div b 2-       (q,r) = divMod (n+b2) b-   in  (q,r-b2)--checkPosDigit :: Basis -> Digit -> Digit-checkPosDigit b d =-   if d>=0 && d<b-     then d-     else error ("digit " ++ show d ++ " out of range [0," ++ show b ++ ")")--checkDigit :: Basis -> Digit -> Digit-checkDigit b d =-   if abs d < b-     then d-     else error ("digit " ++ show d ++ " out of range ("-                   ++ show (-b) ++ "," ++ show b ++ ")")--{- |-Converts all digits to non-negative digits,-that is the usual positional representation.-However the conversion will fail-when the remaining digits are all zero.-(This cannot be improved!)--}-nonNegative :: Basis -> T -> T-nonNegative b x =-   let (xe,xm) = compress b x-   in  (xe, nonNegativeMant b xm)--{- |-Requires, that no digit is @(basis-1)@ or @(1-basis)@.-The leading digit might be negative and might be @-basis@ or @basis@.--}-nonNegativeMant :: Basis -> Mantissa -> Mantissa-nonNegativeMant b =-   let recurse (x0:x1:xs) =-          select (replaceZeroChain x0 (x1:xs))-             [(x1 >=  1,  x0    : recurse (x1:xs)),-              (x1 <= -1, (x0-1) : recurse ((x1+b):xs))]-       recurse xs = xs--       replaceZeroChain x xs =-          let (xZeros,xRem) = span (0==) xs-          in  case xRem of-                [] -> (x:xs)  -- keep trailing zeros, because they show precision in 'show' functions-                (y:ys) ->-                  if y>=0  -- equivalent to y>0-                    then x     : Match.replicate xZeros 0     ++ recurse xRem-                    else (x-1) : Match.replicate xZeros (b-1) ++ recurse ((y+b) : ys)--   in  recurse---{- |-May prepend a digit.--}-compress :: Basis -> T -> T-compress _ x@(_, []) = x-compress b (xe, xm) =-   let (hi:his,los) = unzip (map (moveToZero b) xm)-   in  prependDigit hi (xe, Poly.add his los)--{- |-Compress first digit.-May prepend a digit.--}-compressFirst :: Basis -> T -> T-compressFirst _ x@(_, []) = x-compressFirst b (xe, x:xs) =-   let (hi,lo) = moveToZero b x-   in  prependDigit hi (xe, lo:xs)--{- |-Does not prepend a digit.--}-compressMant :: Basis -> Mantissa -> Mantissa-compressMant _ [] = []-compressMant b (x:xs) =-   let (his,los) = unzip (map (moveToZero b) xs)-   in  Poly.add his (x:los)--{- |-Compress second digit.-Sometimes this is enough to keep the digits in the admissible range.-Does not prepend a digit.--}-compressSecondMant :: Basis -> Mantissa -> Mantissa-compressSecondMant b (x0:x1:xs) =-   let (hi,lo) = moveToZero b x1-   in  x0+hi : lo : xs-compressSecondMant _ xm = xm--prependDigit :: Basis -> T -> T-prependDigit 0 x = x-prependDigit x (xe, xs) = (xe+1, x:xs)--{- |-Eliminate leading zero digits.-This will fail for zero.--}-trim :: T -> T-trim (xe,xm) =-   let (xZero, xNonZero) = span (0 ==) xm-   in  (xe - length xZero, xNonZero)--{- |-Trim until a minimum exponent is reached.-Safe for zeros.--}-trimUntil :: Exponent -> T -> T-trimUntil e =-   until (\(xe,xm) -> xe<=e ||-              not (null xm || head xm == 0)) trimOnce--trimOnce :: T -> T-trimOnce (xe,[])   = (xe-1,[])-trimOnce (xe,0:xm) = (xe-1,xm)-trimOnce x = x--{- |-Accept a high leading digit for the sake of a reduced exponent.-This eliminates one leading digit.-Like 'pumpFirst' but with exponent management.--}-decreaseExp :: Basis -> T -> T-decreaseExp b (xe,xm) =-   (xe-1, pumpFirst b xm)--{- |-Merge leading and second digit.-This is somehow an inverse of 'compressMant'.--}-pumpFirst :: Basis -> Mantissa -> Mantissa-pumpFirst b xm =-   case xm of-     (x0:x1:xs) -> x0*b+x1:xs-     (x0:[])    -> x0*b:[]-     []         -> []--decreaseExpFP :: Basis -> (Exponent, FixedPoint) ->-                          (Exponent, FixedPoint)-decreaseExpFP b (xe,xm) =-   (xe-1, pumpFirstFP b xm)--pumpFirstFP :: Basis -> FixedPoint -> FixedPoint-pumpFirstFP b (x,xm) =-   let xb = x * fromIntegral b-   in  case xm of-         (x0:xs) -> (xb + fromIntegral x0, xs)-         []      -> (xb, [])--{- |-Make sure that a number with absolute value less than 1-has a (small) negative exponent.-Also works with zero because it chooses an heuristic exponent for stopping.--}-negativeExp :: Basis -> T -> T-negativeExp b x =-   let tx  = trimUntil (-10) x-   in  if fst tx >=0 then decreaseExp b tx else tx---{- * conversions -}--{- ** integer -}--fromBaseCardinal :: Basis -> Integer -> T-fromBaseCardinal b n =-   let mant = mantissaFromCard b n-   in  (length mant - 1, mant)--fromBaseInteger :: Basis -> Integer -> T-fromBaseInteger b n =-   if n<0-     then neg b (fromBaseCardinal b (negate n))-     else fromBaseCardinal b n--mantissaToNum :: Ring.C a => Basis -> Mantissa -> a-mantissaToNum bInt =-   let b = fromIntegral bInt-   in  foldl (\x d -> x*b + fromIntegral d) 0--mantissaFromCard :: (ToInteger.C a) => Basis -> a -> Mantissa-mantissaFromCard bInt n =-   let b = NP.fromIntegral bInt-   in  reverse (map NP.fromIntegral-          (Integral.decomposeVarPositional (repeat b) n))--mantissaFromInt :: (ToInteger.C a) => Basis -> a -> Mantissa-mantissaFromInt b n =-   if n<0-     then negate (mantissaFromCard b (negate n))-     else mantissaFromCard b n--mantissaFromFixedInt :: Basis -> Exponent -> Integer -> Mantissa-mantissaFromFixedInt bInt e n =-   let b = NP.fromIntegral bInt-   in  map NP.fromIntegral (uncurry (:) (List.mapAccumR-          (\x () -> divMod x b)-          n (replicate (pred e) ())))---{- ** rational -}--fromBaseRational :: Basis -> Rational -> T-fromBaseRational bInt x =-   let b = NP.fromIntegral bInt-       xDen = denominator x-       (xInt,xNum) = divMod (numerator x) xDen-       (xe,xm) = fromBaseInteger bInt xInt-       xFrac = List.unfoldr-                 (\num -> toMaybe (num/=0) (divMod (num*b) xDen)) xNum-   in  (xe, xm ++ map NP.fromInteger xFrac)--{- ** fixed point -}--{- |-Split into integer and fractional part.--}-toFixedPoint :: Basis -> T -> FixedPoint-toFixedPoint b (xe,xm) =-   if xe>=0-     then let (x0,x1) = splitAtPadZero (xe+1) xm-          in  (mantissaToNum b x0, x1)-     else (0, replicate (- succ xe) 0 ++ xm)--fromFixedPoint :: Basis -> FixedPoint -> T-fromFixedPoint b (xInt,xFrac) =-   let (xe,xm) = fromBaseInteger b xInt-   in  (xe, xm++xFrac)---{- ** floating point -}--toDouble :: Basis -> T -> Double-toDouble b (xe,xm) =-   let txm = take (mantLengthDouble b) xm-       bf  = fromIntegral b-       br  = recip bf-   in  fieldPower xe bf * foldr (\xi y -> fromIntegral xi + y*br) 0 txm--{- |-cf. 'Numeric.floatToDigits'--}-fromDouble :: Basis -> Double -> T-fromDouble b x =-   let (n,frac) = splitFraction x-       (mant,e) = P98.decodeFloat frac-       fracR    = alignMant b (-1)-                     (fromBaseRational b (mant % ringPower (-e) 2))-   in  fromFixedPoint b (n, fracR)--{- |-Only return as much digits as are contained in Double.-This will speedup further computations.--}-fromDoubleApprox :: Basis -> Double -> T-fromDoubleApprox b x =-   let (xe,xm) = fromDouble b x-   in  (xe, take (mantLengthDouble b) xm)--fromDoubleRough :: Basis -> Double -> T-fromDoubleRough b x =-   let (xe,xm) = fromDouble b x-   in  (xe, take 2 xm)--mantLengthDouble :: Basis -> Exponent-mantLengthDouble b =-   let fi = fromIntegral :: Int -> Double-       x  = undefined :: Double-   in  ceiling-          (logBase (fi b) (fromInteger (P98.floatRadix x)) *-             fi (P98.floatDigits x))--liftDoubleApprox :: Basis -> (Double -> Double) -> T -> T-liftDoubleApprox b f = fromDoubleApprox b . f . toDouble b--liftDoubleRough :: Basis -> (Double -> Double) -> T -> T-liftDoubleRough b f = fromDoubleRough b . f . toDouble b---{- ** text -}--{- |-Show a number with respect to basis @10^e@.--}-showDec :: Exponent -> T -> String-showDec = showBasis 10--showHex :: Exponent -> T -> String-showHex = showBasis 16--showBin :: Exponent -> T -> String-showBin = showBasis 2--showBasis :: Basis -> Exponent -> T -> String-showBasis b e xBig =-   let x = rootBasis b e xBig-       (sign,absX) =-          case cmp b x (fst x,[]) of-             LT -> ("-", neg b x)-             _  -> ("", x)-       (int, frac) = toFixedPoint b (nonNegative b absX)-       checkedFrac = map (checkPosDigit b) frac-       intStr =-          if b==10-            then show int-            else map intToDigit (mantissaFromInt b int)-   in  sign ++ intStr ++ '.' : map intToDigit checkedFrac---{- ** basis -}--{- |-Convert from a @b@ basis representation to a @b^e@ basis.--Works well with every exponent.--}-powerBasis :: Basis -> Exponent -> T -> T-powerBasis b e (xe,xm) =-   let (ye,r)  = divMod xe e-       (y0,y1) = splitAtPadZero (r+1) xm-       y1pad   = mapLast (padRight 0 e) (sliceVertical e y1)-   in  (ye, map (mantissaToNum b) (y0 : y1pad))--{- |-Convert from a @b^e@ basis representation to a @b@ basis.--Works well with every exponent.--}-rootBasis :: Basis -> Exponent -> T -> T-rootBasis b e (xe,xm) =-   let splitDigit d = padLeft 0 e (mantissaFromInt b d)-   in  nest (e-1) trimOnce-            ((xe+1)*e-1, concatMap splitDigit (map (checkDigit (ringPower e b)) xm))--{- |-Convert between arbitrary bases.-This conversion is expensive (quadratic time).--}-fromBasis :: Basis -> Basis -> T -> T-fromBasis bDst bSrc x =-   let (int,frac) = toFixedPoint bSrc x-   in  fromFixedPoint bDst (int, fromBasisMant bDst bSrc frac)--fromBasisMant :: Basis -> Basis -> Mantissa -> Mantissa-fromBasisMant _    _    [] = []-fromBasisMant bDst bSrc xm =-   let {- We use a counter that alerts us,-          when the digits are grown too much by Poly.scale.-          Then it is time to do some carry/compression.-          'inc' is essentially the fractional number digits-          needed to represent the destination base in the source base.-          It is multiplied by 'unit' in order to allow integer computation. -}-       inc = ceiling-                (logBase (fromIntegral bSrc) (fromIntegral bDst)-                     * unit * 1.1 :: Double)-          -- Without the correction factor, invalid digits are emitted - why?-       unit :: Ring.C a => a-       unit = 10000-       {--       This would create finite representations-       in some cases (input is finite, and the result is finite)-       but will cause infinite loop otherwise.-       dropWhileRev (0==) . compressMant bDst-       -}-       cmpr (mag,xs) = (mag - unit, compressMant bSrc xs)--       scl (_,[]) = Nothing-       scl (mag,xs) =-          let (newMag,y:ys) =-                 until ((<unit) . fst) cmpr-                       (mag + inc, Poly.scale bDst xs)-              (d,y0) = moveToZero bSrc y-          in  Just (d, (newMag, y0:ys))--   in  List.unfoldr scl (0::Int,xm)---{- * comparison -}--{- |-The basis must be at least ***.-Note: Equality cannot be asserted in finite time on infinite precise numbers.-If you want to assert, that a number is below a certain threshold,-you should not call this routine directly,-because it will fail on equality.-Better round the numbers before comparison.--}-cmp :: Basis -> T -> T -> Ordering-cmp b x y =-   let (_,dm) = sub b x y-       {- Only differences above 2 allow a safe decision,-          because 1(-9)(-9)(-9)(-9)... and (-1)9999...-          represent the same number, namely zero. -}-       recurse [] = EQ-       recurse (d:[]) = compare d 0-       recurse (d0:d1:ds) =-          select (recurse (d0*b+d1 : ds))-             [(d0 < -2, LT),-              (d0 >  2, GT)]-   in  recurse dm--{--Compare two numbers approximately.-This circumvents the infinite loop if both numbers are equal.-If @lessApprox bnd b x y@-then @x@ is definitely smaller than @y@.-The converse is not true.-You should use this one instead of 'cmp' for checking for bounds.--}-lessApprox :: Basis -> Exponent -> T -> T -> Bool-lessApprox b bnd x y =-   let tx = trunc bnd x-       ty = trunc bnd y-   in  LT == cmp b (liftLaurent2 LPoly.add (bnd,[2]) tx) ty--trunc :: Exponent -> T -> T-trunc bnd (xe, xm) =-   if bnd > xe-     then (bnd, [])-     else (xe, take (1+xe-bnd) xm)--equalApprox :: Basis -> Exponent -> T -> T -> Bool-equalApprox b bnd x y =-   fst (trimUntil bnd (sub b x y)) == bnd---{- |-If all values are completely defined,-then it holds--> if b then x else y == ifLazy b x y--However if @b@ is undefined,-then it is at least known that the result is between @x@ and @y@.--}-ifLazy :: Basis -> Bool -> T -> T -> T-ifLazy b c x@(xe, _) y@(ye, _) =-   let ze = max xe ye-       xm = alignMant b ze x-       ym = alignMant b ze y-       recurse :: Mantissa -> Mantissa -> Mantissa-       recurse xs0 ys0 =-          withTwoMantissas xs0 ys0 [] $ \(x0,xs1) (y0,ys1) ->-          if abs (y0-x0) > 2-            then if c then xs0 else ys0-            else-              {--              @x0==y0 || c@ means that in case of @x0==y0@-              we do not have to check @c@.-              -}-              withTwoMantissas xs1 ys1 ((if x0==y0 || c then x0 else y0) : []) $-                  \(x1,xs2) (y1,ys2) ->-                {--                We can choose @z0@ only when knowing also x1 and y1.-                Because of x0x1 = 09 and y0y1 = 19-                we may always choose the larger one of x0 and y0-                in order to get admissible digit z1.-                But this would be wrong for x0x1 = 0(-9) and y0y1 = 1(-9).-                -}-                let z0  = mean2 b (x0,x1) (y0,y1)-                    x1' = x1+(x0-z0)*b-                    y1' = y1+(y0-z0)*b-                in  if abs x1' < b  &&  abs y1' < b-                      then z0 : recurse (x1':xs2) (y1':ys2)-                      else if c then xs0 else ys0-   in  (ze, recurse xm ym)--{- |-> mean2 b (x0,x1) (y0,y1)--computes @ round ((x0.x1 + y0.y1)/2) @,-where @x0.x1@ and @y0.y1@ are positional rational numbers-with respect to basis @b@--}-{-# INLINE mean2 #-}-mean2 :: Basis -> (Digit,Digit) -> (Digit,Digit) -> Digit-mean2 b (x0,x1) (y0,y1) =-   ((x0+y0+1)*b + (x1+y1)) `div` (2*b)--{--In a first trial I used--> zipMantissas :: Mantissa -> Mantissa -> [(Digit, Digit)]--for implementation of ifLazy.-However, this required to extract digits from the pairs-after the decision for an argument.-With withTwoMantissas we can just return a pointer to the original list.--}-withTwoMantissas ::-   Mantissa -> Mantissa ->-   a ->-   ((Digit,Mantissa) -> (Digit,Mantissa) -> a) ->-   a-withTwoMantissas [] [] r _ = r-withTwoMantissas [] (y:ys) _ f = f (0,[]) (y,ys)-withTwoMantissas (x:xs) [] _ f = f (x,xs) (0,[])-withTwoMantissas (x:xs) (y:ys) _ f = f (x,xs) (y,ys)---align :: Basis -> Exponent -> T -> T-align b ye x = (ye, alignMant b ye x)--{- |-Get the mantissa in such a form-that it fits an expected exponent.--@x@ and @(e, alignMant b e x)@ represent the same number.--}-alignMant :: Basis -> Exponent -> T -> Mantissa-alignMant b e (xe,xm) =-   if e>=xe-     then replicate (e-xe) 0 ++ xm-     else let (xm0,xm1) = splitAtPadZero (xe-e+1) xm-          in  mantissaToNum b xm0 : xm1--absolute :: T -> T-absolute (xe,xm) = (xe, absMant xm)--absMant :: Mantissa -> Mantissa-absMant xa@(x:xs) =-   case compare x 0 of-      EQ -> x : absMant xs-      LT -> Poly.negate xa-      GT -> xa-absMant [] = []---{- * arithmetic -}--fromLaurent :: LPoly.T Int -> T-fromLaurent (LPoly.Cons nxe xm) = (NP.negate nxe, xm)--toLaurent :: T -> LPoly.T Int-toLaurent (xe, xm) = LPoly.Cons (NP.negate xe) xm--liftLaurent2 ::-   (LPoly.T Int -> LPoly.T Int -> LPoly.T Int) ->-      (T -> T -> T)-liftLaurent2 f x y =-   fromLaurent (f (toLaurent x) (toLaurent y))--liftLaurentMany ::-   ([LPoly.T Int] -> LPoly.T Int) ->-      ([T] -> T)-liftLaurentMany f =-   fromLaurent . f . map toLaurent--{- |-Add two numbers but do not eliminate leading zeros.--}-add :: Basis -> T -> T -> T-add b x y = compress b (liftLaurent2 LPoly.add x y)--sub :: Basis -> T -> T -> T-sub b x y = compress b (liftLaurent2 LPoly.sub x y)--neg :: Basis -> T -> T-neg _ (xe, xm) = (xe, Poly.negate xm)---{- |-Add at most @basis@ summands.-More summands will violate the allowed digit range.--}-addSome :: Basis -> [T] -> T-addSome b = compress b . liftLaurentMany sum--{- |-Add many numbers efficiently by computing sums of sub lists-with only little carry propagation.--}-addMany :: Basis -> [T] -> T-addMany _ [] = zero-addMany b ys =-   let recurse xs =-          case map (addSome b) (sliceVertical b xs) of-            [s]  -> s-            sums -> recurse sums-   in  recurse ys---type Series = [(Exponent, T)]--{- |-Add an infinite number of numbers.-You must provide a list of estimate of the current remainders.-The estimates must be given as exponents of the remainder.-If such an exponent is too small, the summation will be aborted.-If exponents are too big, computation will become inefficient.--}-series :: Basis -> Series -> T-series _ [] = error "empty series: don't know a good exponent"--- series _ [] = (0,[]) -- unfortunate choice of exponent-series b summands =-   {- Some pre-processing that asserts decreasing exponents.-      Increasing coefficients can appear legally-      due to non-unique number representation. -}-   let (es,xs)    = unzip summands-       safeSeries = zip (scanl1 min es) xs-   in  seriesPlain b safeSeries--seriesPlain :: Basis -> Series -> T-seriesPlain _ [] = error "empty series: don't know a good exponent"-seriesPlain b summands =-   let (es,m:ms) = unzip (map (uncurry (align b)) summands)-       eDifs     = mapAdjacent (-) es-       eDifLists = sliceVertical (pred b) eDifs-       mLists    = sliceVertical (pred b) ms-       accum sumM (eDifList,mList) =-          let subM = LPoly.addShiftedMany eDifList (sumM:mList)-              -- lazy unary sum-              len = concatMap (flip replicate ()) eDifList-              (high,low)  = splitAtMatchPadZero len subM-          {--          'compressMant' looks unsafe-          when the residue doesn't decrease for many summands.-          Then there is a leading digit of a chunk-          which is not compressed for long time.-          However, if the residue is estimated correctly-          there can be no overflow.-          -}-          in  (compressMant b low, high)-       (trailingDigits, chunks) =-          List.mapAccumL accum m (zip eDifLists mLists)-   in  compress b (head es, concat chunks ++ trailingDigits)--{--An alternative series implementation-could reduce carries by do the following cycle-(split, add sub-lists).-This would reduce carries to the minimum-but we must work hard in order to find out lazily-how many digits can be emitted.--}---{- |-Like 'splitAt',-but it pads with zeros if the list is too short.-This way it preserves- @ length (fst (splitAtPadZero n xs)) == n @--}-splitAtPadZero :: Int -> Mantissa -> (Mantissa, Mantissa)-splitAtPadZero n [] = (replicate n 0, [])-splitAtPadZero 0 xs = ([], xs)-splitAtPadZero n (x:xs) =-   let (ys, zs) = splitAtPadZero (n-1) xs-   in  (x:ys, zs)--- must get a case for negative index--splitAtMatchPadZero :: [()] -> Mantissa -> (Mantissa, Mantissa)-splitAtMatchPadZero n  [] = (Match.replicate n 0, [])-splitAtMatchPadZero [] xs = ([], xs)-splitAtMatchPadZero n (x:xs) =-   let (ys, zs) = splitAtMatchPadZero (tail n) xs-   in  (x:ys, zs)---{- |-help showing series summands--}-truncSeriesSummands :: Series -> Series-truncSeriesSummands = map (\(e,x) -> (e,trunc (-20) x))----scale :: Basis -> Digit -> T -> T-scale b y x = compress b (scaleSimple y x)--{--scaleSimple :: ToInteger.C a => a -> T -> T-scaleSimple y (xe, xm) = (xe, Poly.scale (fromIntegral y) xm)--}--scaleSimple :: Basis -> T -> T-scaleSimple y (xe, xm) = (xe, Poly.scale y xm)--scaleMant :: Basis -> Digit -> Mantissa -> Mantissa-scaleMant b y xm = compressMant b (Poly.scale y xm)----mulSeries :: Basis -> T -> T -> Series-mulSeries _ (xe,[]) (ye,_) = [(xe+ye, zero)]-mulSeries b (xe,xm) (ye,ym) =-   let zes = iterate pred (xe+ye+1)-       zs  = zipWith (\xd e -> scale b xd (e,ym)) xm (tail zes)-   in  zip zes zs--{- |-For obtaining n result digits it is mathematically sufficient-to know the first (n+1) digits of the operands.-However this implementation needs (n+2) digits,-because of calls to 'compress' in both 'scale' and 'series'.-We should fix that.--}-mul :: Basis -> T -> T -> T-mul b x y = trimOnce (seriesPlain b (mulSeries b x y))----{- |-Undefined if the divisor is zero - of course.-Because it is impossible to assert that a real is zero,-the routine will not throw an error in general.--ToDo: Rigorously derive the minimal required magnitude of the leading divisor digit.--}-divide :: Basis -> T -> T -> T-divide b (xe,xm) (ye',ym') =-   let (ye,ym) = until ((>=b) . abs . head . snd)-                       (decreaseExp b)-                       (ye',ym')-   in  nest 3 trimOnce (compress b (xe-ye, divMant b ym xm))--divMant :: Basis -> Mantissa -> Mantissa -> Mantissa-divMant _ [] _   = error "Number.Positional: division by zero"-divMant b ym xm0 =-   snd $-   List.mapAccumL-      (\xm fullCompress ->-       let z = div (head xm) (head ym)-           {- 'scaleMant' contains compression,-              which is not much of a problem,-              because it is always applied to @ym@.-              That is, there is no nested call. -}-           dif = pumpFirst b (Poly.sub xm (scaleMant b z ym))-           cDif = if fullCompress-                    then compressMant       b dif-                    else compressSecondMant b dif-       in  (cDif, z))-   xm0 (cycle (replicate (b-1) False ++ [True]))--divMantSlow :: Basis -> Mantissa -> Mantissa -> Mantissa-divMantSlow _ [] = error "Number.Positional: division by zero"-divMantSlow b ym =-   List.unfoldr-      (\xm ->-       let z = div (head xm) (head ym)-           d = compressMant b (pumpFirst b-                  (Poly.sub xm (Poly.scale z ym)))-       in  Just (z, d))--reciprocal :: Basis -> T -> T-reciprocal b = divide b one---{- |-Fast division for small integral divisors,-which occur for instance in summands of power series.--}-divIntMant :: Basis -> Int -> Mantissa -> Mantissa-divIntMant b y xInit =-   List.unfoldr (\(r,rxs) ->-             let rb = r*b-                 (rbx, xs', run) =-                    case rxs of-                       []     -> (rb,   [], r/=0)-                       (x:xs) -> (rb+x, xs, True)-                 (d,m) = divMod rbx y-             in  toMaybe run (d, (m, xs')))-           (0,xInit)---- this version is simple but ignores the possibility of a terminating result-divIntMantInf :: Basis -> Int -> Mantissa -> Mantissa-divIntMantInf b y =-   map fst . tail .-      scanl (\(_,r) x -> divMod (r*b+x) y) (undefined,0) .-         (++ repeat 0)--divInt :: Basis -> Digit -> T -> T-divInt b y (xe,xm) =-   -- (xe, divIntMant b y xm)-   let z  = (xe, divIntMant b y xm)-       {- Division by big integers may cause leading zeros.-          Eliminate as many as we can expect from the division.-          If the input number has leading zeros (it might be equal to zero),-          then the result may have, too. -}-       tz = until ((<=1) . fst) (\(yi,zi) -> (div yi b, trimOnce zi)) (y,z)-   in  snd tz---{- * algebraic functions -}---sqrt :: Basis -> T -> T-sqrt b = sqrtDriver b sqrtFP--sqrtDriver :: Basis -> (Basis -> FixedPoint -> Mantissa) -> T -> T-sqrtDriver _ _ (xe,[]) = (div xe 2, [])-sqrtDriver b sqrtFPworker x =-   let (exe,ex0:exm) = if odd (fst x) then decreaseExp b x else x-       (nxe,(nx0,nxm)) =-           until (\xi -> fst (snd xi) >= fromIntegral b ^ 2)-                 (nest 2 (decreaseExpFP b))-                 (exe, (fromIntegral ex0, exm))-   in  compress b (div nxe 2, sqrtFPworker b (nx0,nxm))---sqrtMant :: Basis -> Mantissa -> Mantissa-sqrtMant _ [] = []-sqrtMant b (x:xs) =-   sqrtFP b (fromIntegral x, xs)--{- |-Square root.--We need a leading digit of type Integer,-because we have to collect up to 4 digits.-This presentation can also be considered as 'FixedPoint'.--ToDo:-Rigorously derive the minimal required magnitude-of the leading digit of the root.--Mathematically the @n@th digit of the square root-depends roughly only on the first @n@ digits of the input.-This is because @sqrt (1+eps) `equalApprox` 1 + eps\/2@.-However this implementation requires @2*n@ input digits-for emitting @n@ digits.-This is due to the repeated use of 'compressMant'.-It would suffice to fully compress only every @basis@th iteration (digit)-and compress only the second leading digit in each iteration.---Can the involved operations be made lazy enough to solve-@y = (x+frac)^2@-by-@frac = (y-x^2-frac^2) \/ (2*x)@ ?--}-sqrtFP :: Basis -> FixedPoint -> Mantissa-sqrtFP b (x0,xs) =-   let y0   = round (NP.sqrt (fromInteger x0 :: Double))-       dyx0 = fromInteger (x0 - fromIntegral y0 ^ 2)--       accum dif (e,ty) =-          -- (e,ty) == xm - (trunc j y)^2-          let yj = div (head dif + y0) (2*y0)-              newDif = pumpFirst b $-                 LPoly.addShifted e-                    (Poly.sub dif (scaleMant b (2*yj) ty))-                    [yj*yj]-              {- We could always compress the full difference number,-                 but it is not necessary,-                 and we save dependencies on less significant digits. -}-              cNewDif =-                 if mod e b == 0-                   then compressMant       b newDif-                   else compressSecondMant b newDif-          in  (cNewDif, yj)--       truncs = lazyInits y-       y = y0 : snd (List.mapAccumL-                        accum-                        (pumpFirst b (dyx0 : xs))-                        (zip [1..] (drop 2 truncs)))-   in  y---sqrtNewton :: Basis -> T -> T-sqrtNewton b = sqrtDriver b sqrtFPNewton--{- |-Newton iteration doubles the number of correct digits in every step.-Thus we process the data in chunks of sizes of powers of two.-This way we get fastest computation possible with Newton-but also more dependencies on input than necessary.-The question arises whether this implementation still fits the needs-of computational reals.-The input is requested as larger and larger chunks,-and the input itself might be computed this way,-e.g. a repeated square root.-Requesting one digit too much,-requires the double amount of work for the input computation,-which in turn multiplies time consumption by a factor of four,-and so on.--Optimal fast implementation of one routine-does not preserve fast computation of composed computations.--The routine assumes, that the integer parts is at least @b^2.@--}-sqrtFPNewton :: Basis -> FixedPoint -> Mantissa-sqrtFPNewton bInt (x0,xs) =-   let b = fromIntegral bInt-       chunkLengths = iterate (2*) 1-       xChunks = map (mantissaToNum bInt) $ snd $-            List.mapAccumL (\x cl -> swap (splitAtPadZero cl x))-                           xs chunkLengths-       basisPowers = iterate (^2) b-       truncXs = scanl (\acc (bp,frac) -> acc*bp+frac) x0-                       (zip basisPowers xChunks)-       accum y (bp, x) =-          let ybp  = y * bp-              newY = div (ybp + div (x * div bp b) y) 2-          in  (newY, newY-ybp)-       y0 = round (NP.sqrt (fromInteger x0 :: Double))-       yChunks = snd $ List.mapAccumL accum-                         y0 (zip basisPowers (tail truncXs))-       yFrac = concat $ zipWith (mantissaFromFixedInt bInt) chunkLengths yChunks-   in  fromInteger y0 : yFrac---{- |-List.inits is defined by-@inits = foldr (\x ys -> [] : map (x:) ys) [[]]@--This is too strict for our application.--> Prelude> List.inits (0:1:2:undefined)-> [[],[0],[0,1]*** Exception: Prelude.undefined--The following routine is more lazy than 'List.inits'-and even lazier than 'Data.List.HT.inits' from @utility-ht@ package,-but it is restricted to infinite lists.-This degree of laziness is needed for @sqrtFP@.--> Prelude> lazyInits (0:1:2:undefined)-> [[],[0],[0,1],[0,1,2],[0,1,2,*** Exception: Prelude.undefined--}-lazyInits :: [a] -> [[a]]-lazyInits ~(x:xs)  =  [] : map (x:) (lazyInits xs)-{--The lazy match above is irrefutable,-so the pattern @[]@ would never be reached.--}----{- * transcendent functions -}--{- ** exponential functions -}--expSeries :: Basis -> T -> Series-expSeries b xOrig =-   let x   = negativeExp b xOrig-       xps = scanl (\p n -> divInt b n (mul b x p)) one [1..]-   in  map (\xp -> (fst xp, xp)) xps--{- |-Absolute value of argument should be below 1.--}-expSmall :: Basis -> T -> T-expSmall b x = series b (expSeries b x)---expSeriesLazy :: Basis -> T -> Series-expSeriesLazy b x@(xe,_) =-   let xps = scanl (\p n -> divInt b n (mul b x p)) one [1..]-       {- much effort for computing the residue exponents-          without touching the arguments mantissa -}-       es :: [Double]-       es = zipWith (-)-               (map fromIntegral (iterate ((1+xe)+) 0))-               (scanl (+) 0-                  (map (logBase (fromIntegral b)-                          . fromInteger) [1..]))-   in  zip (map ceiling es) xps--expSmallLazy :: Basis -> T -> T-expSmallLazy b x = series b (expSeriesLazy b x)---exp :: Basis -> T -> T-exp b x =-   let (xInt,xFrac) = toFixedPoint b (compress b x)-       yFrac = expSmall b (-1,xFrac)-       {--       (xFrac0,xFrac1) = splitAt 2 xFrac-       yFrac = mul b-                 -- slow convergence but simple argument-                 (expSmall b (-1, xFrac0))-                 -- fast convergence but big argument-                 (expSmall b (-3, xFrac1))-       -}-   in  intPower b xInt yFrac (recipEConst b) (eConst b)--intPower :: Basis -> Integer -> T -> T -> T -> T-intPower b expon neutral recipX x =-   if expon >= 0-     then cardPower b   expon  neutral x-     else cardPower b (-expon) neutral recipX--cardPower :: Basis -> Integer -> T -> T -> T-cardPower b expon neutral x =-   if expon >= 0-     then powerAssociative (mul b) neutral x expon-     else error "negative exponent - use intPower"---{- |-Residue estimates will only hold for exponents-with absolute value below one.--The computation is based on 'Int',-thus the denominator should not be too big.-(Say, at most 1000 for 1000000 digits.)--It is not optimal to split the power into pure root and pure power-(that means, with integer exponents).-The root series can nicely handle all exponents,-but for exponents above 1 the series summands rises at the beginning-and thus make the residue estimate complicated.-For powers with integer exponents the root series turns-into the binomial formula,-which is just a complicated way to compute a power-which can also be determined by simple multiplication.--}-powerSeries :: Basis -> Rational -> T -> Series-powerSeries b expon xOrig =-   let scaleRat ni yi =-          divInt b (fromInteger (denominator yi) * ni) .-          scaleSimple (fromInteger (numerator yi))-       x   = negativeExp b (sub b xOrig one)-       xps = scanl (\p fac -> uncurry scaleRat fac (mul b x p))-                   one (zip [1..] (iterate (subtract 1) expon))-   in  map (\xp -> (fst xp, xp)) xps--powerSmall :: Basis -> Rational -> T -> T-powerSmall b y x = series b (powerSeries b y x)--power :: Basis -> Rational -> T -> T-power b expon x =-   let num   = numerator   expon-       den   = denominator expon-       rootX = root b den x-   in  intPower b num one (reciprocal b rootX) rootX--root :: Basis -> Integer -> T -> T-root b expon x =-   let estimate = liftDoubleApprox b (** (1 / fromInteger expon)) x-       estPower = cardPower b expon one estimate-       residue  = divide b x estPower-   in  mul b estimate (powerSmall b (1 % fromIntegral expon) residue)----{- |-Absolute value of argument should be below 1.--}-cosSinhSmall :: Basis -> T -> (T, T)-cosSinhSmall b x =-   let (coshXps, sinhXps) = unzip (sliceVertPair (expSeries b x))-   in  (series b coshXps,-        series b sinhXps)--{- |-Absolute value of argument should be below 1.--}-cosSinSmall :: Basis -> T -> (T, T)-cosSinSmall b x =-   let (coshXps, sinhXps) = unzip (sliceVertPair (expSeries b x))-       alternate s =-          zipWith3 if' (cycle [True,False])-             s (map (\(e,y) -> (e, neg b y)) s)-   in  (series b (alternate coshXps),-        series b (alternate sinhXps))---{- |-Like 'cosSinSmall' but converges faster.-It calls @cosSinSmall@ with reduced arguments-using the trigonometric identities-@-cos (4*x) = 8 * cos x ^ 2 * (cos x ^ 2 - 1) + 1-sin (4*x) = 4 * sin x * cos x * (1 - 2 * sin x ^ 2)-@--Note that the faster convergence is hidden by the overhead.--The same could be achieved with a fourth power of a complex number.--}-cosSinFourth :: Basis -> T -> (T, T)-cosSinFourth b x =-   let (cosx, sinx) = cosSinSmall b (divInt b 4 x)-       sinx2   = mul b sinx sinx-       cosx2   = mul b cosx cosx-       sincosx = mul b sinx cosx-   in  (add b one (scale b 8 (mul b cosx2 (sub b cosx2 one))),-        scale b 4 (mul b sincosx (sub b one (scale b 2 sinx2))))---cosSin :: Basis -> T -> (T, T)-cosSin b x =-   let pi2 = divInt b 2 (piConst b)-       {- @compress@ ensures that the leading digit of the fractional part-          is close to zero -}-       (quadrant, frac) = toFixedPoint b (compress b (divide b x pi2))-       -- it's possibly faster if we subtract quadrant*pi/4-       wrapped = if quadrant==0 then x else mul b pi2 (-1, frac)-       (cosW,sinW) = cosSinSmall b wrapped-   in  case mod quadrant 4 of-          0 -> (      cosW,       sinW)-          1 -> (neg b sinW,       cosW)-          2 -> (neg b cosW, neg b sinW)-          3 -> (      sinW, neg b cosW)-          _ -> error "error in implementation of 'mod'"--tan :: Basis -> T -> T-tan b x = uncurry (flip (divide b)) (cosSin b x)--cot :: Basis -> T -> T-cot b x = uncurry (divide b) (cosSin b x)---{- ** logarithmic functions -}--lnSeries :: Basis -> T -> Series-lnSeries b xOrig =-   let x   = negativeExp b (sub b xOrig one)-       mx  = neg b x-       xps = zipWith (divInt b) [1..] (iterate (mul b mx) x)-   in  map (\xp -> (fst xp, xp)) xps--lnSmall :: Basis -> T -> T-lnSmall b x = series b (lnSeries b x)--{- |-@-x' = x - (exp x - y) \/ exp x-   = x + (y * exp (-x) - 1)-@--First, the dependencies on low-significant places are currently-much more than mathematically necessary.-Check-@-*Number.Positional> expSmall 1000 (-1,100 : replicate 16 0 ++ [undefined])-(0,[1,105,171,-82,76*** Exception: Prelude.undefined-@-Every multiplication cut off two trailing digits.-@-*Number.Positional> nest 8 (mul 1000 (-1,repeat 1)) (-1,100 : replicate 16 0 ++ [undefined])-(-9,[101,*** Exception: Prelude.undefined-@--Possibly the dependencies of expSmall-could be resolved by not computing @mul@ immediately-but computing @mul@ series which are merged and subsequently added.-But this would lead to an explosion of series.--Second, even if the dependencies of all atomic operations-are reduced to a minimum,-the mathematical dependencies of the whole iteration function-are less than the sums of the parts.-Lets demonstrate this with the square root iteration.-It is-@-(1.4140 + 2/1.4140) / 2 == 1.414213578500707-(1.4149 + 2/1.4149) / 2 == 1.4142137288854335-@-That is, the digits @213@ do not depend mathematically on @x@ of @1.414x@,-but their computation depends.-Maybe there is a glorious trick to reduce the computational dependencies-to the mathematical ones.--}-lnNewton :: Basis -> T -> T-lnNewton b y =-   let estimate = liftDoubleApprox b log y-       expRes   = mul b y (expSmall b (neg b estimate))-       -- try to reduce dependencies by feeding expSmall with a small argument-       residue =-          sub b (mul b expRes (expSmallLazy b (neg b resTrim))) one-       resTrim =-          -- (-3, replicate 4 0 ++ alignMant b (-7) residue)-          align b (- mantLengthDouble b) residue-       lazyAdd (xe,xm) (ye,ym) =-          (xe, LPoly.addShifted (xe-ye) xm ym)-       x = lazyAdd estimate resTrim-   in  x--lnNewton' :: Basis -> T -> T-lnNewton' b y =-   let estimate = liftDoubleApprox b log y-       residue  =-          sub b (mul b y (expSmall b (neg b x))) one-          -- sub b (mul b y (expSmall b (neg b estimate))) one-          -- sub b (mul b y (expSmall b (neg b-          --     (fst estimate, snd estimate ++ [undefined])))) one-       resTrim =-          -- align b (-6) residue-          align b (- mantLengthDouble b) residue-             -- align returns the new exponent immediately-          -- nest (mantLengthDouble b) trimOnce residue-          -- negativeExp b residue-       lazyAdd (xe,xm) (ye,ym) =-          (xe, LPoly.addShifted (xe-ye) xm ym)-       x = lazyAdd estimate resTrim-          -- add b estimate resTrim-                -- LPoly.add checks for empty lists and is thus too strict-   in  x---ln :: Basis -> T -> T-ln b x@(xe,_) =-   let e  = round (log (fromIntegral b) * fromIntegral xe :: Double)-       ei = fromIntegral e-       y  = trim $-          if e<0-            then powerAssociative (mul b) x (eConst b)    (-ei)-            else powerAssociative (mul b) x (recipEConst b) ei-       estimate = liftDoubleApprox b log y-       residue  = mul b (expSmall b (neg b estimate)) y-   in  addSome b [(0,[e]), estimate, lnSmall b residue]---{- |-This is an inverse of 'cosSin',-also known as @atan2@ with flipped arguments.-It's very slow because of the computation of sinus and cosinus.-However, because it uses the 'atan2' implementation as estimator,-the final application of arctan series should converge rapidly.--It could be certainly accelerated by not using cosSin-and its fiddling with pi.-Instead we could analyse quadrants before calling atan2,-then calling cosSinSmall immediately.--}-angle :: Basis -> (T,T) -> T-angle b (cosx, sinx) =-   let wd      = atan2 (toDouble b sinx) (toDouble b cosx)-       wApprox = fromDoubleApprox b wd-       (cosApprox, sinApprox) = cosSin b wApprox-       (cosD,sinD) =-          (add b (mul b cosx cosApprox)-                 (mul b sinx sinApprox),-           sub b (mul b sinx cosApprox)-                 (mul b cosx sinApprox))-       sinDSmall = negativeExp b sinD-   in  add b wApprox (arctanSmall b (divide b sinDSmall cosD))---{- |-Arcus tangens of arguments with absolute value less than @1 \/ sqrt 3@.--}-arctanSeries :: Basis -> T -> Series-arctanSeries b xOrig =-   let x   = negativeExp b xOrig-       mx2 = neg b (mul b x x)-       xps = zipWith (divInt b) [1,3..] (iterate (mul b mx2) x)-   in  map (\xp -> (fst xp, xp)) xps--arctanSmall :: Basis -> T -> T-arctanSmall b x = series b (arctanSeries b x)--{- |-Efficient computation of Arcus tangens of an argument of the form @1\/n@.--}-arctanStem :: Basis -> Int -> T-arctanStem b n =-   let x = (0, divIntMant b n [1])-       divN2 = divInt b n . divInt b (-n)-       {- this one can cause overflows in piConst too easily-       mn2 = - n*n-       divN2 = divInt b mn2-       -}-       xps = zipWith (divInt b) [1,3..] (iterate (trim . divN2) x)-   in  series b (map (\xp -> (fst xp, xp)) xps)---{- |-This implementation gets the first decimal place for free-by calling the arcus tangens implementation for 'Double's.--}-arctan :: Basis -> T -> T-arctan b x =-   let estimate = liftDoubleRough b atan x-       tanEst   = tan b estimate-       residue  = divide b (sub b x tanEst) (add b one (mul b x tanEst))-   in  addSome b [estimate, arctanSmall b residue]--{- |-A classic implementation without ''cheating''-with floating point implementations.--For @x < 1 \/ sqrt 3@-(@1 \/ sqrt 3 == tan (pi\/6)@)-use @arctan@ power series.-(@sqrt 3@ is approximately @19\/11@.)--For @x > sqrt 3@-use-@arctan x = pi\/2 - arctan (1\/x)@--For other @x@ use--@arctan x = pi\/4 - 0.5*arctan ((1-x^2)\/2*x)@-(which follows from-@arctan x + arctan y == arctan ((x+y) \/ (1-x*y))@-which in turn follows from complex multiplication-@(1:+x)*(1:+y) == ((1-x*y):+(x+y))@--If @x@ is close to @sqrt 3@ or @1 \/ sqrt 3@ the computation is quite inefficient.--}-arctanClassic :: Basis -> T -> T-arctanClassic b x =-   let absX = absolute x-       pi2  = divInt b 2 (piConst b)-   in  select-          (divInt b 2 (sub b pi2-              (arctanSmall b-                  (divInt b 2 (sub b (reciprocal b x) x)))))-          [(lessApprox b (-5) absX (fromBaseRational b (11%19)),-               arctanSmall b x),-           (lessApprox b (-5) (fromBaseRational b (19%11)) absX,-               sub b pi2 (arctanSmall b (reciprocal b x)))]----{- * constants -}--{- ** elementary -}--zero :: T-zero = (0,[])--one :: T-one = (0,[1])--minusOne :: T-minusOne = (0,[-1])---{- ** transcendental -}--eConst :: Basis -> T-eConst b = expSmall b one--recipEConst :: Basis -> T-recipEConst b = expSmall b minusOne--piConst :: Basis -> T-piConst b =-   let numCompress = takeWhile (0/=)-          (iterate (flip div b) (4*(44+7+12+24)))-       stArcTan k den = scaleSimple k (arctanStem b den)-       sum' = addSome b-                 [stArcTan   44     57,-                  stArcTan    7    239,-                  stArcTan (-12)   682,-                  stArcTan   24  12943]-   in  foldl (const . compress b)-             (scaleSimple 4 sum') numCompress----{- * auxilary functions -}--sliceVertPair :: [a] -> [(a,a)]-sliceVertPair (x0:x1:xs) = (x0,x1) : sliceVertPair xs-sliceVertPair [] = []-sliceVertPair _ = error "odd number of elements"----{--Pi as a zero of trigonometric functions. --  Is a corresponding computation that bad?-Newton converges quadratically,-  but the involved trigonometric series converge only slightly more than linearly.---- lift cos to higher frequencies, in order to shift the zero to smaller values, which let trigonometric series converge faster--take 10 $ Numerics.Newton.zero 0.7 (\x -> (cos (2*x), -2 * sin (2*x)))--(\x -> (2 * cos x ^ 2 - 1, -4 * cos x * sin x))-(\x -> (cos x ^ 2 - sin x ^ 2, -4 * cos x * sin x))-(\x -> (tan x ^ 2 - 1, 4 * tan x))----- compute arctan as inverse of tan by Newton--zero 0.7 (\x -> (tan x - 1, 1 + tan x ^ 2))-zero 0.7 (\x -> (tan x - 1, 1 / cos x ^ 2))-iterate (\x -> x + (cos x - sin x) * cos x) 0.7-iterate (\x -> x + (cos x - sin x) * sqrt 0.5) 0.7-iterate (\x -> x + cos x ^ 2 - sin x * cos x) 0.7-iterate (\x -> x + 0.5 - sin x * cos x) 0.7-iterate (\x -> x + cos x ^ 2 - 0.5) 0.7----- compute section of tan and cot--zero 0.7 (\x -> (tan x - 1 / tan x, (1 + tan x ^ 2) * (1 + 1 / tan x ^ 2))-zero 0.7 (\x -> ((tan x ^ 2 - 1) * tan x, (1 + tan x ^ 2) ^ 2)-iterate (\x -> x - (sin x ^ 2 - cos x ^ 2) * sin x * cos x) 0.7-iterate (\x -> x - (sin x ^ 2 - cos x ^ 2) * 0.5) 0.7-iterate (\x -> x + 1/2 - sin x ^ 2) 0.7--For using the last formula,-the n-th digit of (sin x) must depend only on the n-th digit of x.-The same holds for (^2).-This means that no interim carry compensation is possible.-This will certainly force usage of Integer for digits,-otherwise the multiplication will overflow sooner or later.--}
− src-ghc-6.12/Number/Positional/Check.hs 
@@ -1,260 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Copyright   :  (c) Henning Thielemann 2006-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional---Interface to "Number.Positional" which dynamically checks for equal bases.--}-module Number.Positional.Check where--import qualified Number.Positional as Pos--import qualified Number.Complex as Complex---- import qualified Algebra.Module             as Module-import qualified Algebra.RealTranscendental as RealTrans-import qualified Algebra.Transcendental     as Trans-import qualified Algebra.Algebraic          as Algebraic-import qualified Algebra.RealField          as RealField-import qualified Algebra.Field              as Field-import qualified Algebra.RealRing           as RealRing-import qualified Algebra.Absolute           as Absolute-import qualified Algebra.Ring               as Ring-import qualified Algebra.Additive           as Additive-import qualified Algebra.ZeroTestable       as ZeroTestable--import qualified Algebra.EqualityDecision as EqDec-import qualified Algebra.OrderDecision    as OrdDec--import qualified NumericPrelude.Base as P-import qualified Prelude     as P98--import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP---{- |-The value @Cons b e m@-represents the number @b^e * (m!!0 \/ 1 + m!!1 \/ b + m!!2 \/ b^2 + ...)@.-The interpretation of exponent is chosen such that-@floor (logBase b (Cons b e m)) == e@.-That is, it is good for multiplication and logarithms.-(Because of the necessity to normalize the multiplication result,-the alternative interpretation wouldn't be more complicated.)-However for base conversions, roots, conversion to fixed point and-working with the fractional part-the interpretation-@b^e * (m!!0 \/ b + m!!1 \/ b^2 + m!!2 \/ b^3 + ...)@-would fit better.-The digits in the mantissa range from @1-base@ to @base-1@.-The representation is not unique-and cannot be made unique in finite time.-This way we avoid infinite carry ripples.--}-data T = Cons {base :: Int, exponent :: Int, mantissa :: Pos.Mantissa}-   deriving (Show)---{- * basic helpers -}--{- |-Shift digits towards zero by partial application of carries.-E.g. 1.8 is converted to 2.(-2)-If the digits are in the range @(1-base, base-1)@-the resulting digits are in the range @((1-base)/2-2, (base-1)/2+2)@.-The result is still not unique,-but may be useful for further processing.--}-compress :: T -> T-compress = lift1 Pos.compress---{- | perfect carry resolution, works only on finite numbers -}-carry :: T -> T-carry (Cons b ex xs) =-   let ys = scanr (\x (c,_) -> divMod (x+c) b) (0,undefined) xs-       digits = map snd (init ys)-   in  prependDigit (fst (head ys)) (Cons b ex digits)---prependDigit :: Int -> T -> T-prependDigit 0 x = x-prependDigit x (Cons b ex xs) =-   Cons b (ex+1) (x:xs)----{- * conversions -}--lift0 :: (Int -> Pos.T) -> T-lift0 op =-   uncurry (Cons defltBase) (op defltBase)--lift1 :: (Int -> Pos.T -> Pos.T) -> T -> T-lift1 op (Cons xb xe xm) =-   uncurry (Cons xb) (op xb (xe, xm))--lift2 :: (Int -> Pos.T -> Pos.T -> Pos.T) -> T -> T -> T-lift2 op (Cons xb xe xm) (Cons yb ye ym) =-   let b = commonBasis xb yb-   in  uncurry (Cons b) (op b (xe, xm) (ye, ym))--{--lift4 :: (Int -> Pos.T -> Pos.T -> Pos.T -> Pos.T -> Pos.T) -> T -> T -> T -> T -> T-lift4 op (Cons xb xe xm) (Cons yb ye ym) (Cons zb ze zm) (Cons wb we wm) =-   let b = xb `commonBasis` yb `commonBasis` zb `commonBasis` wb-   in  uncurry (Cons b) (op b (xe, xm) (ye, ym) (ze, zm) (we, wm))--}--commonBasis :: Pos.Basis -> Pos.Basis -> Pos.Basis-commonBasis xb yb =-   if xb == yb-     then xb-     else error "Number.Positional: bases differ"--fromBaseInteger :: Int -> Integer -> T-fromBaseInteger b n =-   uncurry (Cons b) (Pos.fromBaseInteger b n)--fromBaseRational :: Int -> Rational -> T-fromBaseRational b r =-   uncurry (Cons b) (Pos.fromBaseRational b r)------defltBaseRoot :: Pos.Basis-defltBaseRoot = 10--defltBaseExp :: Pos.Exponent-defltBaseExp = 3--- exp 4   let  (sqrt 0.5) fail--defltBase :: Pos.Basis-defltBase = ringPower defltBaseExp defltBaseRoot----defltShow :: T -> String-defltShow (Cons xb xe xm) =-   if xb == defltBase-     then Pos.showBasis defltBaseRoot defltBaseExp (xe,xm)-     else error "defltShow: wrong base"---instance Additive.C T where-   zero   = fromBaseInteger defltBase 0-   (+)    = lift2 Pos.add-   (-)    = lift2 Pos.sub-   negate = lift1 Pos.neg--instance Ring.C T where-   one           = fromBaseInteger defltBase 1-   fromInteger n = fromBaseInteger defltBase n-   (*)           = lift2 Pos.mul--{--instance Module.C T T where-   (*>) = (*)--}--instance Field.C T where-   (/)   = lift2 Pos.divide-   recip = lift1 Pos.reciprocal--instance Algebraic.C T where-   sqrt   = lift1 Pos.sqrtNewton-   root n = lift1 (flip Pos.root n)-   x ^/ y = lift1 (flip Pos.power y) x--instance Trans.C T where-   pi     = lift0 Pos.piConst--   exp    = lift1 Pos.exp-   log    = lift1 Pos.ln--   sin    = lift1 (\b -> snd . Pos.cosSin b)-   cos    = lift1 (\b -> fst . Pos.cosSin b)-   tan    = lift1 Pos.tan--   atan   = lift1 Pos.arctan--   {--   sinh   = lift1 (\b -> snd . Pos.cosSinh b)-   cosh   = lift1 (\b -> snd . Pos.cosSinh b)-   -}--{--The way EqDec and OrdDec are instantiated-it is possible to have different bases-for the arguments for comparison-and the arguments between we decide.-However, I would not rely on this.--}-instance EqDec.C T where-   x==?y  =  lift2 (\b -> Pos.ifLazy b (x==y))--instance OrdDec.C T where-   x<=?y  =  lift2 (\b -> Pos.ifLazy b (x<=y))--instance ZeroTestable.C T where-   isZero (Cons xb xe xm) =-      Pos.cmp xb (xe,xm) Pos.zero == EQ--instance Eq T where-   (Cons xb xe xm) == (Cons yb ye ym) =-      Pos.cmp (commonBasis xb yb) (xe,xm) (ye,ym) == EQ--instance Ord T where-   compare (Cons xb xe xm) (Cons yb ye ym) =-      Pos.cmp (commonBasis xb yb) (xe,xm) (ye,ym)--instance Absolute.C T where-   abs = lift1 (const Pos.absolute)-   signum = Absolute.signumOrd--instance RealRing.C T where-   splitFraction (Cons xb xe xm) =-      let (int, frac) = Pos.toFixedPoint xb (xe,xm)-      in  (fromInteger int, Cons xb (-1) frac)--instance RealField.C T where--instance RealTrans.C T where-   atan2  = lift2 (curry . Pos.angle)----- for complex numbers--instance Complex.Power T where-   power     = Complex.defltPow------- legacy instances for work with GHCi-legacyInstance :: a-legacyInstance =-   error "legacy Ring.C instance for simple input of numeric literals"--instance P98.Num T where-   fromInteger = fromBaseInteger defltBase-   negate = negate --for unary minus-   (+)    = legacyInstance-   (*)    = legacyInstance-   abs    = legacyInstance-   signum = legacyInstance--instance P98.Fractional T where-   fromRational = fromBaseRational defltBase . fromRational-   (/) = legacyInstance---{--MathObj.PowerSeries.approx MathObj.PowerSeries.Example.exp (Number.Positional.fromBaseInteger 10 1) List.!! 30 :: Number.Positional.Check.T--}
− src-ghc-6.12/Number/Quaternion.hs 
@@ -1,296 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-{- |-Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable (?)--Quaternions--}--module Number.Quaternion-        (-        -- * Cartesian form-        T(real,imag),-        fromReal,-        (+::),--        -- * Conversions-        toRotationMatrix,-        fromRotationMatrix,-        fromRotationMatrixDenorm,-        toComplexMatrix,-        fromComplexMatrix,--        -- * Operations-        scalarProduct,-        crossProduct,-        conjugate,-        scale,-        norm,-        normSqr,-        normalize,-        similarity,-        slerp,-        )  where--import qualified Algebra.NormedSpace.Euclidean as NormedEuc-import qualified Algebra.VectorSpace  as VectorSpace-import qualified Algebra.Module       as Module-import qualified Algebra.Vector       as Vector-import qualified Algebra.Transcendental as Trans-import qualified Algebra.Algebraic    as Algebraic-import qualified Algebra.Field        as Field-import qualified Algebra.Ring         as Ring-import qualified Algebra.Additive     as Additive-import qualified Algebra.ZeroTestable as ZeroTestable--import Algebra.ZeroTestable(isZero)-import Algebra.Module((*>), (<*>.*>), )--import qualified Number.Complex as Complex--import Number.Complex ((+:))--import qualified NumericPrelude.Elementwise as Elem-import Algebra.Additive ((<*>.+), (<*>.-), (<*>.-$), )---- import qualified Data.Typeable as Ty-import Data.Array (Array, (!))-import qualified Data.Array as Array--import qualified Prelude as P-import NumericPrelude.Base-import NumericPrelude.Numeric hiding (signum)-import Text.Show.HT (showsInfixPrec, )-import Text.Read.HT (readsInfixPrec, )---{- TODO:-conversion to and from complex matrix--}---infix  6  +::, `Cons`--{- |-Quaternions could be defined based on Complex numbers.-However quaternions are often considered as real part and three imaginary parts.--}-data T a-  = Cons {real :: !a           -- ^ real part-         ,imag :: !(a, a, a)   -- ^ imaginary parts-         }-  deriving (Eq)--fromReal :: Additive.C a => a -> T a-fromReal x = Cons x zero---plusPrec :: Int-plusPrec = 6--instance (Show a) => Show (T a) where-   showsPrec prec (x `Cons` y) = showsInfixPrec "+::" plusPrec prec x y--instance (Read a) => Read (T a) where-   readsPrec prec = readsInfixPrec "+::" plusPrec prec (+::)----- | Construct a quaternion from real and imaginary part.-(+::) :: a -> (a,a,a) -> T a-(+::) = Cons---- | The conjugate of a quaternion.-{-# SPECIALISE conjugate :: T Double -> T Double #-}-conjugate	 :: (Additive.C a) => T a -> T a-conjugate (Cons r i) =  Cons r (negate i)---- | Scale a quaternion by a real number.-{-# SPECIALISE scale :: Double -> T Double -> T Double #-}-scale		 :: (Ring.C a) => a -> T a -> T a-scale r (Cons xr xi) =  Cons (r * xr) (scaleImag r xi)---- | like Module.*> but without additional class dependency-scaleImag	 :: (Ring.C a) => a -> (a,a,a) -> (a,a,a)-scaleImag r (xi,xj,xk) =  (r * xi, r * xj, r * xk)---- | the same as NormedEuc.normSqr but with a simpler type class constraint-normSqr		 :: (Ring.C a) => T a -> a-normSqr (Cons xr xi) = xr*xr + scalarProduct xi xi--norm		 :: (Algebraic.C a) => T a -> a-norm x = sqrt (normSqr x)---- | scale a quaternion into a unit quaternion-normalize	 :: (Algebraic.C a) => T a -> T a-normalize x = scale (recip (norm x)) x--scalarProduct	 :: (Ring.C a) => (a,a,a) -> (a,a,a) -> a-scalarProduct (xi,xj,xk) (yi,yj,yk) =-   xi*yi + xj*yj + xk*yk--crossProduct	 :: (Ring.C a) => (a,a,a) -> (a,a,a) -> (a,a,a)-crossProduct (xi,xj,xk) (yi,yj,yk) =-   (xj*yk - xk*yj, xk*yi - xi*yk, xi*yj - xj*yi)--{- | similarity mapping as needed for rotating 3D vectors--It holds-@similarity (cos(a\/2) +:: scaleImag (sin(a\/2)) v) (0 +:: x) == (0 +:: y)@-where @y@ results from rotating @x@ around the axis @v@ by the angle @a@.--}-similarity	 :: (Field.C a) => T a -> T a -> T a-similarity c x = c*x/c--{--rotate	 :: (Field.C a) =>-      (a,a,a)  {- ^ rotation axis, must be normalized -}-   -> T a-   -> T a-rotate c x = c*x/c--}--{- |-Let @c@ be a unit quaternion, then it holds-@similarity c (0+::x) == toRotationMatrix c * x@--}-toRotationMatrix :: (Ring.C a) => T a -> Array (Int,Int) a-toRotationMatrix (Cons r (i,j,k)) =-   let r2 = r^2-       i2 = i^2;   j2 = j^2;   k2 = k^2-       ri = 2*r*i; rj = 2*r*j; rk = 2*r*k-       jk = 2*j*k; ki = 2*k*i; ij = 2*i*j-   in  Array.listArray ((0,0),(2,2)) $ concat $-          [r2+i2-j2-k2, ij-rk,       ki+rj      ] :-          [ij+rk,       r2-i2+j2-k2, jk-ri      ] :-          [ki-rj,       jk+ri,       r2-i2-j2+k2] :-          []--fromRotationMatrix :: (Algebraic.C a) => Array (Int,Int) a -> T a-fromRotationMatrix =-   normalize . fromRotationMatrixDenorm---checkBounds :: (Int,Int) -> Array (Int,Int) a -> Array (Int,Int) a-checkBounds (c,r) arr =-   let bnds@((c0,r0), (c1,r1)) = Array.bounds arr-   in  if c1-c0==c && r1-r0==r-         then Array.listArray ((0,0), (c1-c0, r1-r0))-                              (Array.elems arr)-         else error ("Quaternion.checkBounds: invalid matrix size "-                         ++ show bnds)---{- |-The rotation matrix must be normalized.-(I.e. no rotation with scaling)-The computed quaternion is not normalized.--}-fromRotationMatrixDenorm :: (Ring.C a) => Array (Int,Int) a -> T a-fromRotationMatrixDenorm mat' =-   let mat = checkBounds (2,2) mat'-       trace = sum (map (\i -> mat ! (i,i)) [0..2])-       dif (i,j) = mat!(i,j) - mat!(j,i)-   in  Cons (trace+1) (dif (2,1), dif (0,2), dif (1,0))--{- |-Map a quaternion to complex valued 2x2 matrix,-such that quaternion addition and multiplication-is mapped to matrix addition and multiplication.-The determinant of the matrix equals the squared quaternion norm ('normSqr').-Since complex numbers can be turned into real (orthogonal) matrices,-a quaternion could also be converted into a real matrix.--}-toComplexMatrix :: (Additive.C a) =>-   T a -> Array (Int,Int) (Complex.T a)-toComplexMatrix (Cons r (i,j,k)) =-   Array.listArray ((0,0), (1,1))-      [r+:i, (-j)+:(-k), j+:(-k), r+:(-i)]---{- |-Revert 'toComplexMatrix'.--}-fromComplexMatrix :: (Field.C a) =>-   Array (Int,Int) (Complex.T a) -> T a-fromComplexMatrix mat =-   let xs = Array.elems (checkBounds (1,1) mat)-       [ar,br,cr,dr] = map Complex.real xs-       [ai,bi,ci,di] = map Complex.imag xs-   in  scale (1/2) (Cons (ar+dr) (ai-di, cr-br, -ci-bi))---{- |-Spherical Linear Interpolation--Can be generalized to any transcendent Hilbert space.-In fact, we should also include the real part in the interpolation.--}-slerp :: (Trans.C a) =>-      a   {- ^ For @0@ return vector @v@,-               for @1@ return vector @w@ -}-   -> (a,a,a)  {- ^ vector @v@, must be normalized -}-   -> (a,a,a)  {- ^ vector @w@, must be normalized -}-   -> (a,a,a)-slerp c v w =-   let scal  = scalarProduct v w /-                  sqrt (scalarProduct v v * scalarProduct w w)-       angle = Trans.acos scal-   in  scaleImag (recip (Algebraic.sqrt (1-scal^2)))-         (scaleImag (Trans.sin ((1-c)*angle)) v +-          scaleImag (Trans.sin (   c *angle)) w)----instance (NormedEuc.Sqr a b) => NormedEuc.Sqr a (T b) where-   normSqr (Cons r i) = NormedEuc.normSqr r + NormedEuc.normSqr i--instance (Algebraic.C a, NormedEuc.Sqr a b) => NormedEuc.C a (T b) where-   norm = NormedEuc.defltNorm----instance (ZeroTestable.C a) => ZeroTestable.C (T a)  where-   isZero (Cons r i)  = isZero r && isZero i--instance (Additive.C a) => Additive.C (T a)  where-   {-# SPECIALISE instance Additive.C (T Float) #-}-   {-# SPECIALISE instance Additive.C (T Double) #-}-   zero   = Cons zero zero-   (+)    = Elem.run2 $ Elem.with Cons <*>.+  real <*>.+  imag-   (-)    = Elem.run2 $ Elem.with Cons <*>.-  real <*>.-  imag-   negate = Elem.run  $ Elem.with Cons <*>.-$ real <*>.-$ imag--instance (Ring.C a) => Ring.C (T a)  where-   {-# SPECIALISE instance Ring.C (T Float) #-}-   {-# SPECIALISE instance Ring.C (T Double) #-}-   one				=  Cons one zero-   fromInteger			=  fromReal . fromInteger-   (Cons xr xi) * (Cons yr yi)	=-       Cons (xr*yr - scalarProduct xi yi)-            (scaleImag xr yi + scaleImag yr xi +-             crossProduct xi yi)--instance (Field.C a) => Field.C (T a)  where-   {-# SPECIALISE instance Field.C (T Float) #-}-   {-# SPECIALISE instance Field.C (T Double) #-}-   recip x = scale (recip (normSqr x)) (conjugate x)-   (Cons xr xi) / y@(Cons yr yi) =-       scale (recip (normSqr y))-          (Cons (xr*yr + scalarProduct xi yi)-                (scaleImag yr xi - scaleImag xr yi - crossProduct xi yi))--instance Vector.C T where-   zero  = zero-   (<+>) = (+)-   (*>)  = scale---- | The '(*>)' method can't replace 'scale'---   because it requires the Algebra.Module constraint-instance (Module.C a b) => Module.C a (T b) where-   (*>) = Elem.run2 $ Elem.with Cons <*>.*> real <*>.*> imag--instance (VectorSpace.C a b) => VectorSpace.C a (T b)-
− src-ghc-6.12/Number/Ratio.hs 
@@ -1,249 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Module      :  Number.Ratio-Copyright   :  (c) Henning Thielemann, Dylan Thurston 2006--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable (?)--Ratios of mathematical objects.--}--module Number.Ratio-	(-	  T((:%), numerator, denominator), (%),-          Rational,-          fromValue,--          scale,-          split,-          showsPrecAuto,--          toRational98,-        )  where--import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.Absolute             as Absolute-import qualified Algebra.Ring                 as Ring-import qualified Algebra.Additive             as Additive-import qualified Algebra.ZeroTestable         as ZeroTestable-import qualified Algebra.Indexable            as Indexable--import Algebra.PrincipalIdealDomain (gcd, )-import Algebra.Units (stdUnitInv, stdAssociate, )-import Algebra.IntegralDomain (div, divMod, )-import Algebra.Ring (one, (*), (^), fromInteger, )-import Algebra.Additive (zero, (+), (-), negate, )-import Algebra.ZeroTestable (isZero, )--import Control.Monad(liftM, liftM2, )--import Foreign.Storable (Storable (..), )-import qualified Foreign.Storable.Record as Store-import Control.Applicative (liftA2, )--import Test.QuickCheck (Arbitrary(arbitrary))-import System.Random (Random(..), RandomGen, )--import qualified Data.Ratio as Ratio98--import qualified Prelude as P-import NumericPrelude.Base---infixl 7 %--data  {- (PID.C a)  => -} T a = (:%) {-        numerator   :: !a,-        denominator :: !a-     } deriving (Eq)-type  Rational = T P.Integer---fromValue :: Ring.C a => a -> T a-fromValue x = x :% one--scale :: (PID.C a) => a -> T a -> T a-scale s (x:%y) =-   let {- x and y are cancelled,-          thus we can only have common divisors in s and y -}-       (n:%d) = s%y-   in  ((n*x):%d)--{- | similar to 'Algebra.RealRing.splitFraction' -}-split :: (PID.C a) => T a -> (a, T a)-split (x:%y) =-   let (q,r) = divMod x y-   in  (q, r:%y)--ratioPrec :: P.Int-ratioPrec = 7--(%) :: (PID.C a) => a -> a -> T a-x % y =-  if isZero y-    then error "NumericPrelude.% : zero denominator"-    else-      let d  = gcd x y-          y0 = div y d-          x0 = div x d-      in  (stdUnitInv y0 * x0) :% stdAssociate y0--instance (PID.C a) => Additive.C (T a) where-    zero                =  fromValue zero---    (x:%y) + (x':%y')   =  (x*y' + x'*y) % (y*y')-    {--    This version reduces the size of intermediate results.-    Is it also faster than the naive version?-    The final (%) includes another gcd computation,-    but it is still needed since e.g.-    5:%7 + (-5):%7 shall be simplified to 0:%1, not 0:%7 .-    -}-    (x:%y) + (x':%y')   =-       let d = gcd y y'-           y0  = div y  d-           y0' = div y' d-       in  (x*y0' + x'*y0) % (y0*y')-    negate (x:%y)       =  (-x) :% y--instance (PID.C a) => Ring.C (T a) where-    one                 =  fromValue one-    fromInteger x       =  fromValue $ fromInteger x-    (x:%y) * (x':%y')   =  (x * x') % (y * y')-    (x:%y) ^ n          =  (x ^ n) :% (y ^ n)--instance (Absolute.C a, PID.C a) => Absolute.C (T a) where-    abs (x:%y)          =  Absolute.abs x :% y-    signum (x:%_)       =  Absolute.signum x :% one---liftOrd :: Ring.C a => (a -> a -> b) -> (T a -> T a -> b)-liftOrd f (x:%y) (x':%y') = f (x * y') (x' * y)--instance (Ord a, PID.C a) => Ord (T a) where-    (<=)     =  liftOrd (<=)-    (<)      =  liftOrd (<)-    (>=)     =  liftOrd (>=)-    (>)      =  liftOrd (>)-    compare  =  liftOrd compare--instance (Ord a, PID.C a) => Indexable.C (T a) where-    compare  =  compare--instance (ZeroTestable.C a, PID.C a) => ZeroTestable.C (T a) where-    isZero  =  isZero . numerator--instance  (Read a, PID.C a)  => Read (T a)  where-    readsPrec p  =-       readParen (p >= ratioPrec)-                 (\r -> [(x%y,u) | (x,s)   <- readsPrec ratioPrec r,-                                   ("%",t) <- lex s,-                                   (y,u)   <- readsPrec ratioPrec t ])--instance  (Show a, PID.C a)  => Show (T a)  where-    showsPrec p (x:%y)  =  showParen (p >= ratioPrec)-                               (shows x . showString " % " . shows y)--{- |-This is an alternative show method-that is more user-friendly but also potentially more ambigious.--}--showsPrecAuto :: (Eq a, PID.C a, Show a) =>-   P.Int -> T a -> String -> String-showsPrecAuto p (x:%y) =-   if y == 1-     then showsPrec p x-     else showParen (p > ratioPrec)-             (showsPrec (ratioPrec+1) x . showString "/" .-              showsPrec (ratioPrec+1) y)---instance (Arbitrary a, PID.C a, ZeroTestable.C a) => Arbitrary (T a) where-{--   arbitrary = liftM2 (%) arbitrary (untilM (not . isZero) arbitrary)--This implementation leads to blocking:--*Main> Test.QuickCheck.test (\x -> x==(x::Rational))-Interrupted.--}-   arbitrary =-      liftM2 (%) arbitrary-         (liftM (\x -> if isZero x then one else x) arbitrary)---instance (Storable a, PID.C a) => Storable (T a) where-   sizeOf    = Store.sizeOf store-   alignment = Store.alignment store-   peek      = Store.peek store-   poke      = Store.poke store--store ::-   (Storable a, PID.C a) =>-   Store.Dictionary (T a)-store =-   Store.run $-   liftA2 (%)-      (Store.element numerator)-      (Store.element denominator)--{--This instance may not be appropriate for mathematical objects other than numbers.-If we encounter such a type of object-we should define an intermediate class-which provides the necessary functions.-I should remark that methods of Random like 'randomR'-cannot sensibly be defined for ratios of polynomials.--}-instance (Random a, PID.C a, ZeroTestable.C a) => Random (T a) where-   random g0 =-      let (numer, g1) = random g0-          (denom, g2) = random g1-      in  (numer % if isZero denom then one else denom, g2)-   randomR (lower,upper) g0 =-      let (k, g1) = randomR01 g0-      in  (lower + k*(upper-lower), g1)---randomR01 ::-   (Random a, PID.C a, RandomGen g) =>-   g -> (T a, g)-randomR01 g0 =-   let (denom0, g1) = random g0-       denom = if isZero denom0 then one else denom0-       (numer, g2) = randomR (zero,denom) g1-   in  (numer % denom, g2)----- * Legacy Instances----- | Necessary when mixing NumericPrelude.Numeric Rationals with Prelude98 Rationals--toRational98 :: (P.Integral a, PID.C a) => T a -> Ratio98.Ratio a-toRational98 x = numerator x Ratio98.% denominator x---legacyInstance :: String -> a-legacyInstance op =-   error ("Ratio." ++ op ++ ": legacy Ring instance for simple input of numeric literals")----- instance (P.Num a, PID.C a) => P.Num (T a) where-instance (P.Num a, PID.C a, Absolute.C a) => P.Num (T a) where-   fromInteger n = P.fromInteger n % 1-   negate = negate -- for unary minus-   (+)    = legacyInstance "(+)"-   (*)    = legacyInstance "(*)"-   abs    = Absolute.abs -- needed for Arbitrary instance of NonNegative.Ratio-   signum = legacyInstance "signum"---- instance (P.Num a, PID.C a) => P.Fractional (T a) where-instance (P.Num a, PID.C a, Absolute.C a) => P.Fractional (T a) where---   fromRational = Field.fromRational-   fromRational x =-      fromInteger (Ratio98.numerator x) :%-      fromInteger (Ratio98.denominator x)-   (/) = legacyInstance "(/)"
− src-ghc-6.12/Number/ResidueClass.hs 
@@ -1,47 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Number.ResidueClass where--import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.IntegralDomain as Integral--- import qualified Algebra.Additive       as Additive--- import qualified Algebra.ZeroTestable   as ZeroTestable--import NumericPrelude.Base-import NumericPrelude.Numeric hiding (recip)-import Data.Maybe.HT (toMaybe)-import Data.Maybe (fromMaybe)---add, sub :: (Integral.C a) => a -> a -> a -> a-add m x y = mod (x+y) m-sub m x y = mod (x-y) m--neg :: (Integral.C a) => a -> a -> a-neg m x = mod (-x) m--mul :: (Integral.C a) => a -> a -> a -> a-mul m x y = mod (x*y) m---{- |-The division may be ambiguous.-In this case an arbitrary quotient is returned.--@-0/:4 * 2/:4 == 0/:4-2/:4 * 2/:4 == 0/:4-@--}-divideMaybe :: (PID.C a) => a -> a -> a -> Maybe a-divideMaybe m x y =-   let (d,(_,z)) = extendedGCD m y-       (q,r)     = divMod x d-   in  toMaybe (isZero r) (mod (q*z) m)--divide :: (PID.C a) => a -> a -> a -> a-divide m x y =-   fromMaybe (error "ResidueClass.divide: indivisible")-             (divideMaybe m x y)--recip :: (PID.C a) => a -> a -> a-recip m = divide m one
− src-ghc-6.12/Number/ResidueClass/Check.hs 
@@ -1,118 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Number.ResidueClass.Check where--import qualified Number.ResidueClass as Res--import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.IntegralDomain as Integral-import qualified Algebra.Field          as Field-import qualified Algebra.Ring           as Ring-import qualified Algebra.Additive       as Additive-import qualified Algebra.ZeroTestable   as ZeroTestable--import Algebra.ZeroTestable(isZero)--import qualified Data.Function.HT as Func-import Data.Maybe.HT (toMaybe, )-import Text.Show.HT (showsInfixPrec, )-import Text.Read.HT (readsInfixPrec, )--import NumericPrelude.Base-import NumericPrelude.Numeric (Int, Integer, mod, (*), )---infix 7 /:, `Cons`--{- |-The best solution seems to let 'modulus' be part of the type.-This could happen with a phantom type for modulus-and a @run@ function like 'Control.Monad.ST.runST'.-Then operations with non-matching moduli could be detected at compile time-and 'zero' and 'one' could be generated with the correct modulus.-An alternative trial can be found in module ResidueClassMaybe.--}-data T a-  = Cons {modulus        :: !a-         ,representative :: !a-         }--factorPrec :: Int-factorPrec = read "7"--instance (Show a) => Show (T a) where-   showsPrec prec (Cons m r) = showsInfixPrec "/:" factorPrec prec r m--instance (Read a, Integral.C a) => Read (T a) where-   readsPrec prec = readsInfixPrec "/:" factorPrec prec (/:)----- | @r \/: m@ is the residue class containing @r@ with respect to the modulus @m@-(/:) :: (Integral.C a) => a -> a -> T a-(/:) r m = Cons m (mod r m)---- | Check if two residue classes share the same modulus-isCompatible :: (Eq a) => T a -> T a -> Bool-isCompatible x y  =  modulus x == modulus y--maybeCompatible :: (Eq a) => T a -> T a -> Maybe a-maybeCompatible x y =-   let mx = modulus x-       my = modulus y-   in  toMaybe (mx==my) mx---fromRepresentative :: (Integral.C a) => a -> a -> T a-fromRepresentative m x = Cons m (mod x m)--lift1 :: (Eq a) => (a -> a -> a) -> T a -> T a-lift1 f x =-   let m = modulus x-   in  Cons m (f m (representative x))--lift2 :: (Eq a) => (a -> a -> a -> a) -> T a -> T a -> T a-lift2 f x y =-   maybe-      (errIncompat)-      (\m -> Cons m (f (modulus x) (representative x) (representative y)))-      (maybeCompatible x y)--errIncompat :: a-errIncompat = error "Residue class: Incompatible operands"---zero :: (Additive.C a) => a -> T a-zero m = Cons m Additive.zero--one :: (Ring.C a) => a -> T a-one  m = Cons m Ring.one--fromInteger :: (Integral.C a) => a -> Integer -> T a-fromInteger m x = fromRepresentative m (Ring.fromInteger x)----instance  (Eq a) => Eq (T a)  where-    (==) x y  =-        maybe errIncompat-           (const (representative x == representative y))-           (maybeCompatible x y)--instance  (ZeroTestable.C a) => ZeroTestable.C (T a)  where-    isZero (Cons _ r)   =  isZero r--instance  (Eq a, Integral.C a) => Additive.C (T a)  where-    zero		=  error "no generic zero in a residue class, use ResidueClass.zero"-    (+)			=  lift2 Res.add-    (-)			=  lift2 Res.sub-    negate		=  lift1 Res.neg--instance  (Eq a, Integral.C a) => Ring.C (T a)  where-    one			=  error "no generic one in a residue class, use ResidueClass.one"-    (*)			=  lift2 Res.mul-    fromInteger		=  error "no generic integer in a residue class, use ResidueClass.fromInteger"-    x^n                 =  Func.powerAssociative (*) (one (modulus x)) x n--instance  (Eq a, PID.C a) => Field.C (T a)  where-    (/)			=  lift2 Res.divide-    recip               =  lift1 (flip Res.divide Ring.one)-    fromRational'	=  error "no conversion from rational to residue class"
− src-ghc-6.12/Number/ResidueClass/Func.hs 
@@ -1,102 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Number.ResidueClass.Func where--import qualified Number.ResidueClass as Res--import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.IntegralDomain   as Integral-import qualified Algebra.Field            as Field-import qualified Algebra.Ring             as Ring-import qualified Algebra.Additive         as Additive-import qualified Algebra.EqualityDecision as EqDec--import Algebra.EqualityDecision ((==?), )-import NumericPrelude.Base-import NumericPrelude.Numeric hiding (zero, one, )--import qualified Prelude        as P-import qualified NumericPrelude.Numeric as NP--{- |-Here a residue class is a representative-and the modulus is an argument.-You cannot show a value of type 'T',-you can only show it with respect to a concrete modulus.-Values cannot be compared,-because the comparison result depends on the modulus.--}-newtype T a = Cons (a -> a)--concrete :: a -> T a -> a-concrete m (Cons r) = r m--fromRepresentative :: (Integral.C a) => a -> T a-fromRepresentative = Cons . mod--lift0 :: (a -> a) -> T a-lift0 = Cons--lift1 :: (a -> a -> a) -> T a -> T a-lift1 f (Cons x) = Cons $ \m -> f m (x m)--lift2 :: (a -> a -> a -> a) -> T a -> T a -> T a-lift2 f (Cons x) (Cons y) = Cons $ \m -> f m (x m) (y m)---zero :: (Additive.C a) => T a-zero = Cons $ const Additive.zero--one :: (Ring.C a) => T a-one  = Cons $ const NP.one--fromInteger :: (Integral.C a) => Integer -> T a-fromInteger = fromRepresentative . NP.fromInteger--equal :: Eq a => a -> T a -> T a -> Bool-equal m (Cons x) (Cons y)  =  x m == y m---instance  (EqDec.C a) => EqDec.C (T a)  where-    (==?) (Cons x) (Cons y) (Cons eq) (Cons noteq) =-       Cons (\m -> (x m ==? y m) (eq m) (noteq m))--instance  (Integral.C a) => Additive.C (T a)  where-    zero		=  zero-    (+)			=  lift2 Res.add-    (-)			=  lift2 Res.sub-    negate		=  lift1 Res.neg--instance  (Integral.C a) => Ring.C (T a)  where-    one			=  one-    (*)			=  lift2 Res.mul-    fromInteger		=  Number.ResidueClass.Func.fromInteger--instance  (PID.C a) => Field.C (T a)  where-    (/)			=  lift2 Res.divide-    recip               =  (NP.one /)-    fromRational'	=  error "no conversion from rational to residue class"---{--NumericPrelude.fromInteger seems to be not available at GHCi's prompt sometimes.-But Prelude.fromInteger requires Prelude.Num instance.--}---- legacy instances for work with GHCi-legacyInstance :: a-legacyInstance =-   error "legacy Ring.C instance for simple input of numeric literals"--instance (P.Num a, Integral.C a) => P.Num (T a) where-   fromInteger = Number.ResidueClass.Func.fromInteger-   negate = negate --for unary minus-   (+)    = legacyInstance-   (*)    = legacyInstance-   abs    = legacyInstance-   signum = legacyInstance--instance Eq (T a) where-   (==) = error "ResidueClass.Func: (==) not implemented"--instance Show (T a) where-   show = error "ResidueClass.Func: 'show' not implemented"
− src-ghc-6.12/Number/ResidueClass/Maybe.hs 
@@ -1,80 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Number.ResidueClass.Maybe where--import qualified Number.ResidueClass as Res--import qualified Algebra.IntegralDomain as Integral-import qualified Algebra.Ring           as Ring-import qualified Algebra.Additive       as Additive-import qualified Algebra.ZeroTestable   as ZeroTestable--import NumericPrelude.Base-import NumericPrelude.Numeric--infix 7 /:, `Cons`---{- |-Here we try to provide implementations for 'zero' and 'one'-by making the modulus optional.-We have to provide non-modulus operations for the cases-where both operands have Nothing modulus.-This is problematic since operations like '(\/)'-depend essentially on the modulus.--A working version with disabled 'zero' and 'one' can be found ResidueClass.--}-data T a-  = Cons {modulus        :: !(Maybe a)  -- ^ the modulus can be Nothing to denote a generic constant like 'zero' and 'one' which could not be bound to a specific modulus so far-         ,representative :: !a-         }-  deriving (Show, Read)----- | @r \/: m@ is the residue class containing @r@ with respect to the modulus @m@-(/:) :: (Integral.C a) => a -> a -> T a-(/:) r m = Cons (Just m) (mod r m)---matchMaybe :: Maybe a -> Maybe a -> Maybe a-matchMaybe Nothing y = y-matchMaybe x       _ = x--isCompatibleMaybe :: (Eq a) => Maybe a -> Maybe a -> Bool-isCompatibleMaybe Nothing _ = True-isCompatibleMaybe _ Nothing = True-isCompatibleMaybe (Just x) (Just y) = x == y---- | Check if two residue classes share the same modulus-isCompatible :: (Eq a) => T a -> T a -> Bool-isCompatible x y  =  isCompatibleMaybe (modulus x) (modulus y)---lift2 :: (Eq a) => (a -> a -> a -> a) -> (a -> a -> a) -> (T a -> T a -> T a)-lift2 f g x y =-  if isCompatible x y-    then let m = matchMaybe (modulus x) (modulus y)-         in  Cons m-                  (maybe g f m (representative x) (representative y))-    else error "ResidueClass: Incompatible operands"---instance  (Eq a, ZeroTestable.C a, Integral.C a) => Eq (T a)  where-    (==) x y =-      if isCompatible x y-        then maybe (==)-                   (\m x' y' -> isZero (mod (x'-y') m))-                   (matchMaybe (modulus x) (modulus y))-                   (representative x) (representative y)-        else error "ResidueClass.(==): Incompatible operands"--instance  (Eq a, Integral.C a) => Additive.C (T a)  where-    zero		=  Cons Nothing zero-    (+)			=  lift2 Res.add (+)-    (-)			=  lift2 Res.sub (-)-    negate (Cons m r)	=  Cons m (negate r)--instance  (Eq a, Integral.C a) => Ring.C (T a)  where-    one			=  Cons Nothing one-    (*)			=  lift2 Res.mul (*)-    fromInteger		=  Cons Nothing . fromInteger
− src-ghc-6.12/Number/ResidueClass/Reader.hs 
@@ -1,96 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Number.ResidueClass.Reader where--import qualified Number.ResidueClass as Res--import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.IntegralDomain as Integral-import qualified Algebra.Ring           as Ring-import qualified Algebra.Additive       as Additive--import NumericPrelude.Base-import NumericPrelude.Numeric--import Control.Monad (liftM2, liftM4)--- import Control.Monad.Reader (MonadReader)--import qualified Prelude        as P-import qualified NumericPrelude.Numeric as NP---{- |-T is a Reader monad but does not need functional dependencies-like that from the Monad Transformer Library.--}-newtype T a b = Cons {toFunc :: a -> b}--concrete :: a -> T a b -> b-concrete m (Cons r) = r m--fromRepresentative :: (Integral.C a) => a -> T a a-fromRepresentative = Cons . mod---getZero :: (Additive.C a) => T a a-getZero = Cons $ const Additive.zero--getOne :: (Ring.C a) => T a a-getOne  = Cons $ const NP.one--fromInteger :: (Integral.C a) => Integer -> T a a-fromInteger = fromRepresentative . NP.fromInteger---instance Monad (T a) where-   (Cons x) >>= y  =  Cons (\r -> toFunc (y (x r)) r)-   return = Cons . const----getAdd :: Integral.C a => T a (a -> a -> a)-getAdd = Cons Res.add--getSub :: Integral.C a => T a (a -> a -> a)-getSub = Cons Res.sub--getNeg :: Integral.C a => T a (a -> a)-getNeg = Cons Res.neg--getAdditiveVars :: Integral.C a => T a (a, a -> a -> a, a -> a -> a, a -> a)-getAdditiveVars = liftM4 (,,,) getZero getAdd getSub getNeg----getMul :: Integral.C a => T a (a -> a -> a)-getMul = Cons Res.mul--getRingVars :: Integral.C a => T a (a, a -> a -> a)-getRingVars = liftM2 (,) getOne getMul----getDivide :: PID.C a => T a (a -> a -> a)-getDivide = Cons Res.divide--getRecip :: PID.C a => T a (a -> a)-getRecip = Cons Res.recip--getFieldVars :: PID.C a => T a (a -> a -> a, a -> a)-getFieldVars = liftM2 (,) getDivide getRecip--monadExample :: PID.C a => T a [a]-monadExample =-   do (zero',(+~),(-~),negate') <- getAdditiveVars-      (one',(*~)) <- getRingVars-      ((/~),recip') <- getFieldVars-      let three = one'+one'+one'  -- is easier if only NP.fromInteger is visible-      let seven = three+three+one'-      return [zero'*~three, one'/~three, recip' three,-              three *~ seven, one' +~ three +~ seven,-              zero' -~ three, negate' three +~ seven]--runExample :: [Integer]-runExample =-   let three = one+one+one-       eleven = three+three+three + one+one-   in  concrete eleven monadExample
− src-ghc-6.12/Number/Root.hs 
@@ -1,97 +0,0 @@-module Number.Root where--import qualified Algebra.Algebraic as Algebraic-import qualified Algebra.Field as Field-import qualified Algebra.Ring as Ring--import qualified MathObj.RootSet as RootSet-import qualified Number.Ratio as Ratio--import Algebra.IntegralDomain (divChecked, )--import qualified NumericPrelude.Numeric as NP-import NumericPrelude.Numeric hiding (recip, )-import NumericPrelude.Base-import Prelude ()--{- |-The root degree must be positive.-This way we can implement multiplication-using only multiplication from type @a@.--}-data T a = Cons Integer a-   deriving (Show)--{- |-When you use @fmap@ you must assert that-@forall n. fmap f (Cons d x) == fmap f (Cons (n*d) (x^n))@--}-instance Functor T where-   fmap f (Cons d x) = Cons d (f x)--fromNumber :: a -> T a-fromNumber = Cons 1--toNumber :: Algebraic.C a => T a -> a-toNumber (Cons n x) = Algebraic.root n x--toRootSet :: Ring.C a => T a -> RootSet.T a-toRootSet (Cons d x) =-   RootSet.lift0 ([negate x] ++ replicate (pred (fromInteger d)) zero ++ [one])---commonDegree :: Ring.C a => T a -> T a -> T (a,a)-commonDegree (Cons xd x) (Cons yd y) =-   let zd = lcm xd yd-   in  Cons zd (x ^ divChecked zd xd, y ^ divChecked zd yd)--instance (Eq a, Ring.C a) => Eq (T a) where-   x == y  =-      case commonDegree x y of-         Cons _ (xn,yn) -> xn==yn--instance (Ord a, Ring.C a) => Ord (T a) where-   compare x y  =-      case commonDegree x y of-         Cons _ (xn,yn) -> compare xn yn---mul :: Ring.C a => T a -> T a -> T a-mul x y = fmap (uncurry (*)) $ commonDegree x y--div :: Field.C a => T a -> T a -> T a-div x y = fmap (uncurry (/)) $ commonDegree x y--recip :: Field.C a => T a -> T a-recip = fmap NP.recip--{- |-exponent must be non-negative--}-cardinalPower :: Ring.C a => Integer -> T a -> T a-cardinalPower n (Cons d x) =-   let m = gcd n d-   in  Cons (divChecked d m) (x ^ divChecked n m)--{- |-exponent can be negative--}-integerPower :: Field.C a => Integer -> T a -> T a-integerPower n =-   if n<0-     then cardinalPower (-n) . recip-     else cardinalPower n--rationalPower :: Field.C a => Rational -> T a -> T a-rationalPower n =-   integerPower (Ratio.numerator n) .-   root (Ratio.denominator n)--{- |-exponent must be positive--}-root :: Ring.C a => Integer -> T a -> T a-root n (Cons d x) = Cons (d*n) x--sqrt :: Ring.C a => T a -> T a-sqrt = root 2
− src-ghc-6.12/Number/SI.hs 
@@ -1,271 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-{- |-Copyright   :  (c) Henning Thielemann 2003-2006-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable--Numerical values equipped with SI units.-This is considered as the user front-end.--}--module Number.SI where--import qualified Number.SI.Unit       as SIUnit-import           Number.SI.Unit (Dimension, bytesize)--import qualified Number.Physical      as Value-import qualified Number.Physical.Unit as Unit-import qualified Number.Physical.Show as PVShow-import qualified Number.Physical.Read as PVRead-import qualified Number.Physical.UnitDatabase as UnitDatabase--import           Algebra.OccasionallyScalar  as OccScalar-import qualified Algebra.NormedSpace.Maximum as NormedMax--import qualified Algebra.VectorSpace         as VectorSpace-import qualified Algebra.Module              as Module-import qualified Algebra.Vector              as Vector-import qualified Algebra.Transcendental      as Trans-import qualified Algebra.Algebraic           as Algebraic-import qualified Algebra.Field               as Field-import qualified Algebra.Absolute                as Absolute-import qualified Algebra.Ring                as Ring-import qualified Algebra.Additive            as Additive-import qualified Algebra.ZeroTestable        as ZeroTestable--import Algebra.Algebraic (sqrt, (^/), )--import Data.Tuple.HT (mapFst, )--import qualified Prelude as P--import NumericPrelude.Numeric-import NumericPrelude.Base---newtype T a v = Cons (PValue v)-{- LANGUAGE GeneralizedNewtypeDeriving allows even this-   deriving (Monad, Functor)--}--type PValue v = Value.T Dimension v--{--import Control.Monad--instance Functor (SIValue.T a) where-  fmap f (SIValue.Cons x) = SIValue.Cons (f x)--instance Monad (SIValue.T a) where-  (>>=) (SIValue.Cons x) f = f x-  return = SIValue.Cons--}--{- I hoped it would be possible to replace these functions-   by fmap and monadic liftM, liftM2, return --   but SIValue.Cons lifts from the base type 'v' to 'SIValue.T a v'-   rather than the type 'PValue v' to 'SIValue.T a v'.--   I.e.-     fmap :: (v -> v) -> SIValue.T a v -> SIValue.T a v--}-lift :: (PValue v0 -> PValue v1) ->-            (T a v0 -> T a v1)-lift f (Cons x) = (Cons (f x))--lift2 :: (PValue v0 -> PValue v1 -> PValue v2) ->-            (T a v0 -> T a v1 -> T a v2)-lift2 f (Cons x) (Cons y) = (Cons (f x y))--liftGen :: (PValue v -> x) -> (T a v -> x)-liftGen f (Cons x) = f x--lift2Gen :: (PValue v0 -> PValue v1 -> x) ->-               (T a v0 -> T a v1 -> x)-lift2Gen f (Cons x) (Cons y) = f x y---{- There is almost nothing new to implement for SIValues.-   We have to lift existing functions to SIValues mainly. -}--scale :: Ring.C v => v -> T a v -> T a v-scale = lift . Value.scale--fromScalarSingle :: v -> T a v-fromScalarSingle = Cons . Value.fromScalarSingle---instance (ZeroTestable.C v) => ZeroTestable.C (T a v) where-  isZero = liftGen isZero--instance Eq v => Eq (T a v) where-  (==)  =  lift2Gen (==)--showNat :: (Show v, Field.C a, Ord a, NormedMax.C a v) =>-   UnitDatabase.T Dimension a -> T a v -> String-showNat db =-   liftGen (PVShow.showNat db)--instance (Show v, Ord a, Trans.C a, NormedMax.C a v) =>-    Show (T a v) where-  showsPrec prec x =-    showParen (prec > PVShow.mulPrec)-       (showNat SIUnit.databaseShow x ++)--readsNat :: (Read v, VectorSpace.C a v) =>-   UnitDatabase.T Dimension a -> Int -> ReadS (T a v)-readsNat db prec =-   map (mapFst Cons) . PVRead.readsNat db prec--instance (Read v, Ord a, Trans.C a, VectorSpace.C a v) =>-    Read (T a v) where-  readsPrec = readsNat SIUnit.databaseRead--instance (Additive.C v) => Additive.C (T a v) where-  zero   = Cons zero-  (+)    = lift2 (+)-  (-)    = lift2 (-)-  negate = lift negate--instance (Ring.C v) => Ring.C (T a v) where-  (*) = lift2 (*)-  fromInteger = Cons . fromInteger--instance (Ord v) => Ord (T a v) where-  max     = lift2    max-  min     = lift2    min-  compare = lift2Gen compare-  (<)     = lift2Gen (<)-  (>)     = lift2Gen (>)-  (<=)    = lift2Gen (<=)-  (>=)    = lift2Gen (>=)--instance (Absolute.C v) => Absolute.C (T a v) where-  abs    = lift abs-  signum = lift signum--instance (Field.C v) => Field.C (T a v) where-  (/) = lift2 (/)-  fromRational' = Cons . fromRational'--instance (Algebraic.C v) => Algebraic.C (T a v) where-  sqrt    = lift  sqrt-  x ^/ y  = lift  (^/ y) x--instance (Trans.C v) => Trans.C (T a v) where-  pi      = Cons pi-  log     = lift  log-  exp     = lift  exp-  logBase = lift2 logBase-  (**)    = lift2 (**)-  cos     = lift  cos-  tan     = lift  tan-  sin     = lift  sin-  acos    = lift  acos-  atan    = lift  atan-  asin    = lift  asin-  cosh    = lift  cosh-  tanh    = lift  tanh-  sinh    = lift  sinh-  acosh   = lift  acosh-  atanh   = lift  atanh-  asinh   = lift  asinh---instance Vector.C (T a) where-  zero  = zero-  (<+>) = (+)-  (*>)  = scale--instance (Module.C a v) => Module.C a (T b v) where-  (*>) x = lift (x Module.*>)--instance (VectorSpace.C a v) => VectorSpace.C a (T b v)--instance (Trans.C a, Ord a, OccScalar.C a v,-          Show v, NormedMax.C a v)-      => OccScalar.C a (T a v) where-   toScalar      = toScalarShow-   toMaybeScalar = liftGen toMaybeScalar-   fromScalar    = Cons . fromScalar----quantity :: (Field.C a, Field.C v) => Unit.T Dimension -> v -> T a v-quantity xu = Cons . Value.Cons xu--hertz, second, minute, hour, day, year,- meter, liter, gramm, tonne,- newton, pascal, bar, joule, watt,- kelvin,- coulomb, ampere, volt, ohm, farad,- bit, byte, baud,- inch, foot, yard, astronomicUnit, parsec,- mach, speedOfLight, electronVolt,- calorien, horsePower, accelerationOfEarthGravity ::-    (Field.C a, Field.C v) => T a v--hertz   = quantity SIUnit.frequency   1e+0-second  = quantity SIUnit.time        1e+0-minute  = quantity SIUnit.time        SIUnit.secondsPerMinute-hour    = quantity SIUnit.time        SIUnit.secondsPerHour-day     = quantity SIUnit.time        SIUnit.secondsPerDay-year    = quantity SIUnit.time        SIUnit.secondsPerYear-meter   = quantity SIUnit.length      1e+0-liter   = quantity SIUnit.volume      1e-3-gramm   = quantity SIUnit.mass        1e-3-tonne   = quantity SIUnit.mass        1e+3-newton  = quantity SIUnit.force       1e+0-pascal  = quantity SIUnit.pressure    1e+0-bar     = quantity SIUnit.pressure    1e+5-joule   = quantity SIUnit.energy      1e+0-watt    = quantity SIUnit.power       1e+0-coulomb = quantity SIUnit.charge      1e+0-ampere  = quantity SIUnit.current     1e+0-volt    = quantity SIUnit.voltage     1e+0-ohm     = quantity SIUnit.resistance  1e+0-farad   = quantity SIUnit.capacitance 1e+0-kelvin  = quantity SIUnit.temperature 1e+0-bit     = quantity SIUnit.information 1e+0-byte    = quantity SIUnit.information bytesize-baud    = quantity SIUnit.dataRate    1e+0--inch           = quantity SIUnit.length SIUnit.meterPerInch-foot           = quantity SIUnit.length SIUnit.meterPerFoot-yard           = quantity SIUnit.length SIUnit.meterPerYard-astronomicUnit = quantity SIUnit.length SIUnit.meterPerAstronomicUnit-parsec         = quantity SIUnit.length SIUnit.meterPerParsec--accelerationOfEarthGravity-             = quantity SIUnit.acceleration SIUnit.accelerationOfEarthGravity-mach         = quantity SIUnit.speed        SIUnit.mach-speedOfLight = quantity SIUnit.speed        SIUnit.speedOfLight-electronVolt = quantity SIUnit.energy       SIUnit.electronVolt-calorien     = quantity SIUnit.energy       SIUnit.calorien-horsePower   = quantity SIUnit.power        SIUnit.horsePower------ legacy instances for work with GHCi-legacyInstance :: a-legacyInstance =-   error "legacy Ring.C instance for simple input of numeric literals"--instance (Ord a, Trans.C a, NormedMax.C a v, P.Num v, Ring.C v) =>-      P.Num (T a v) where-   fromInteger = fromInteger-   negate = negate -- for unary minus-   (+)    = legacyInstance-   (*)    = legacyInstance-   abs    = legacyInstance-   signum = legacyInstance--instance (Ord a, Trans.C a, NormedMax.C a v, P.Num v, Field.C v) =>-      P.Fractional (T a v) where-   fromRational = fromRational-   (/) = legacyInstance
− src-ghc-6.12/Number/SI/Unit.hs 
@@ -1,293 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Copyright   :  (c) Henning Thielemann 2003-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable--Special physical units: SI unit system--}--module Number.SI.Unit where--import qualified Algebra.Transcendental      as Trans-import qualified Algebra.Field               as Field--import qualified Number.Physical.Unit         as Unit-import qualified Number.Physical.UnitDatabase as UnitDatabase-import Number.Physical.UnitDatabase(initScale, initUnitSet)-import Data.Maybe(catMaybes)--import NumericPrelude.Base hiding (length)-import NumericPrelude.Numeric hiding (one)--data Dimension =-   Length | Time | Mass | Charge |-   Angle | Temperature | Information-      deriving (Eq, Ord, Enum, Show)----- | Some common quantity classes.-angle, angularSpeed, -- needs explicit signature because it does not occur in the database- length, distance, area, volume, time,- frequency, speed, acceleration, mass,- force, pressure, energy, power,- charge, current, voltage, resistance,- capacitance, temperature,- information, dataRate-  :: Unit.T Dimension--length       = Unit.fromVector [ 1, 0, 0, 0, 0, 0, 0]--- synonym for 'length' which is distinct from List.length-distance     = Unit.fromVector [ 1, 0, 0, 0, 0, 0, 0]-area         = Unit.fromVector [ 2, 0, 0, 0, 0, 0, 0]-volume       = Unit.fromVector [ 3, 0, 0, 0, 0, 0, 0]-time         = Unit.fromVector [ 0, 1, 0, 0, 0, 0, 0]-frequency    = Unit.fromVector [ 0,-1, 0, 0, 0, 0, 0]-speed        = Unit.fromVector [ 1,-1, 0, 0, 0, 0, 0]-acceleration = Unit.fromVector [ 1,-2, 0, 0, 0, 0, 0]-mass         = Unit.fromVector [ 0, 0, 1, 0, 0, 0, 0]-force        = Unit.fromVector [ 1,-2, 1, 0, 0, 0, 0]-pressure     = Unit.fromVector [-1,-2, 1, 0, 0, 0, 0]-energy       = Unit.fromVector [ 2,-2, 1, 0, 0, 0, 0]-power        = Unit.fromVector [ 2,-3, 1, 0, 0, 0, 0]-charge       = Unit.fromVector [ 0, 0, 0, 1, 0, 0, 0]-current      = Unit.fromVector [ 0,-1, 0, 1, 0, 0, 0]-voltage      = Unit.fromVector [ 2,-2, 1,-1, 0, 0, 0]-resistance   = Unit.fromVector [ 2,-1, 1,-2, 0, 0, 0]-capacitance  = Unit.fromVector [-2, 2,-1, 2, 0, 0, 0]-angle        = Unit.fromVector [ 0, 0, 0, 0, 1, 0, 0]-angularSpeed = Unit.fromVector [ 0,-1, 0, 0, 1, 0, 0]-temperature  = Unit.fromVector [ 0, 0, 0, 0, 0, 1, 0]-information  = Unit.fromVector [ 0, 0, 0, 0, 0, 0, 1]-dataRate     = Unit.fromVector [ 0,-1, 0, 0, 0, 0, 1]---percent, fourth, half, threeFourth   :: Field.C a => a--secondsPerMinute, secondsPerHour, secondsPerDay, secondsPerYear, - meterPerInch, meterPerFoot, meterPerYard,- meterPerAstronomicUnit, meterPerParsec, - accelerationOfEarthGravity,- k2, deg180, grad200, bytesize       :: Field.C a => a--radPerDeg, radPerGrad                :: Trans.C a => a--mach, speedOfLight, electronVolt,- calorien, horsePower                :: Field.C a => a--yocto, zepto, atto, femto, pico,- nano, micro, milli, centi, deci,- one, deca, hecto, kilo, mega,- giga, tera, peta, exa, zetta, yotta :: Field.C a => a---- | Common constants-percent      = 0.01-fourth       = 0.25-half         = 0.50-threeFourth  = 0.75---- | Conversion factors-secondsPerMinute = 60-secondsPerHour   = 60*secondsPerMinute-secondsPerDay    = 24*secondsPerHour  -- 86400.0-secondsPerYear   = 365.2422*secondsPerDay--meterPerInch           = 0.0254-meterPerFoot           = 0.3048-meterPerYard           = 0.9144-meterPerAstronomicUnit = 149.6e6-meterPerParsec         = 30.857e12--k2           = 1024-deg180       = 180-grad200      = 200-radPerDeg    = pi/deg180;-radPerGrad   = pi/grad200;-bytesize     = 8------ | Physical constants-accelerationOfEarthGravity = 9.80665-mach                       = 332.0-speedOfLight               = 299792458.0-electronVolt               = 1.602e-19-calorien                   = 4.19-horsePower                 = 736.0---- | Prefixes used for SI units-yocto = 1.0e-24-zepto = 1.0e-21-atto  = 1.0e-18-femto = 1.0e-15-pico  = 1.0e-12-nano  = 1.0e-9-micro = 1.0e-6-milli = 1.0e-3-centi = 1.0e-2-deci  = 1.0e-1-one   = 1.0e0-deca  = 1.0e1-hecto = 1.0e2-kilo  = 1.0e3-mega  = 1.0e6-giga  = 1.0e9-tera  = 1.0e12-peta  = 1.0e15-exa   = 1.0e18-zetta = 1.0e21-yotta = 1.0e24----{- | UnitDatabase.T of units and their common scalings -}-databaseRead, databaseShow :: Trans.C a => UnitDatabase.T Dimension a-databaseRead = map UnitDatabase.createUnitSet database-databaseShow =-   map UnitDatabase.createUnitSet $-      catMaybes $ map UnitDatabase.showableUnit database--database :: Trans.C a => [UnitDatabase.InitUnitSet Dimension a]-database = [-    (initUnitSet Unit.scalar False [-      (initScale "pi"    pi                        False False),-      (initScale "e"     (exp 1)                   False False),-      (initScale "i"     (sqrt (-1))               False False),-      (initScale "%"     percent                   False False),-      (initScale "\188"  fourth                    False False),-      (initScale "\189"  half                      False False),-      (initScale "\190"  threeFourth               False False)-    ]),-    (initUnitSet angle False [-      (initScale "''"    (radPerDeg/secondsPerHour)   True  False),-      (initScale "'"     (radPerDeg/secondsPerMinute) True  False),-      (initScale "grad"  radPerGrad                False False),-      (initScale "\176"  radPerDeg                 True  True ),-      (initScale "rad"   one                       False False)-    ]),-    (initUnitSet frequency True [-      (initScale "bpm"   (one/secondsPerMinute)    False False),-      (initScale "Hz"    one                       True  True ),-      (initScale "kHz"   kilo                      True  False),-      (initScale "MHz"   mega                      True  False),-      (initScale "GHz"   giga                      True  False)-    ]),-    (initUnitSet time False [-      (initScale "ns"    nano                      True  False),-      (initScale "\181s" micro                     True  False),-      (initScale "ms"    milli                     True  False),-      (initScale "s"     one                       True  True ),-      (initScale "min"   secondsPerMinute          True  False),-      (initScale "h"     secondsPerHour            True  False),-      (initScale "d"     secondsPerDay             True  False),-      (initScale "a"     secondsPerYear            True  False)-    ]),---    (initUnitSet distance False [-    (initUnitSet length False [-      (initScale "nm"    nano                      True  False),-      (initScale "\181m" micro                     True  False),-      (initScale "mm"    milli                     True  False),-      (initScale "cm"    centi                     True  False),-      (initScale "dm"    deci                      True  False),-      (initScale "m"     one                       True  True ),-      (initScale "km"    kilo                      True  False)-    ]),-    (initUnitSet area False [-      (initScale "ha"    (hecto*hecto)             False False)-    ]),-    (initUnitSet volume False [-      (initScale "ml"    (milli*milli)             False False),-      (initScale "cl"    (milli*centi)             False False),-      (initScale "l"     milli                     False False)-    ]),-    (initUnitSet speed False [-      (initScale "mach"  mach                      False False),-      (initScale "c"     speedOfLight              False False)-    ]),-    (initUnitSet acceleration False [-      (initScale "G"     accelerationOfEarthGravity False False)-    ]),-    (initUnitSet mass False [-      (initScale "\181g" nano                      True  False),-      (initScale "mg"    micro                     True  False),-      (initScale "g"     milli                     True  False),-      (initScale "kg"    one                       True  True ),-      (initScale "dt"    hecto                     True  False),-      (initScale "t"     kilo                      True  False),-      (initScale "kt"    mega                      True  False)-    ]),-    (initUnitSet force False [-      (initScale "N"     one                       True  True ),-      (initScale "kp"    accelerationOfEarthGravity False False),-      (initScale "kN"    kilo                      True  False)-    ]),-    (initUnitSet pressure False [-      (initScale "Pa"    one                       True  True ),-      (initScale "mbar"  hecto                     False False),-      (initScale "kPa"   kilo                      True  False),-      (initScale "bar"   (hecto*kilo)              False False)-    ]),-    (initUnitSet energy False [-      (initScale "eV"    electronVolt              False False),-      (initScale "J"     one                       True  True ),-      (initScale "cal"   calorien                  False False),-      (initScale "kJ"    kilo                      True  False),-      (initScale "kcal"  (kilo*calorien)           False False),-      (initScale "MJ"    mega                      True  False)-    ]),-    (initUnitSet power False [-      (initScale "mW"    milli                     True  False),-      (initScale "W"     one                       True  True ),-      (initScale "HP"    horsePower                False False),-      (initScale "kW"    kilo                      True  False),-      (initScale "MW"    mega                      True  False)-    ]),-    (initUnitSet charge False [-      (initScale "C"     one                       True  True )-    ]),-    (initUnitSet current False [-      (initScale "\181A" micro                     True  False),-      (initScale "mA"    milli                     True  False),-      (initScale "A"     one                       True  True )-    ]),-    (initUnitSet voltage False [-      (initScale "mV"    milli                     True  False),-      (initScale "V"     one                       True  True ),-      (initScale "kV"    kilo                      True  False),-      (initScale "MV"    mega                      True  False),-      (initScale "GV"    giga                      True  False)-    ]),-    (initUnitSet resistance False [-      (initScale "Ohm"   one                       True  True ),-      (initScale "kOhm"  kilo                      True  False),-      (initScale "MOhm"  mega                      True  False)-    ]),-    (initUnitSet capacitance False [-      (initScale "pF"    pico                      True  False),-      (initScale "nF"    nano                      True  False),-      (initScale "\181F" micro                     True  False),-      (initScale "mF"    milli                     True  False),-      (initScale "F"     one                       True  True )-    ]),-    (initUnitSet temperature False [-      (initScale "K"     one                       True  True )-    ]),-    (initUnitSet information False [-      (initScale "bit"   one                       True  True ),-      (initScale "B"     bytesize                  True  False),-      (initScale "kB"    (kilo*bytesize)           False False),-      (initScale "KB"    (k2*bytesize)             True  False),-      (initScale "MB"    (k2*k2*bytesize)          True  False),-      (initScale "GB"    (k2*k2*k2*bytesize)       True  False)-    ]),-    (initUnitSet dataRate True [-      (initScale "baud"  one                       True  True ),-      (initScale "kbaud" kilo                      False False),-      (initScale "Kbaud" k2                        True  False),-      (initScale "Mbaud" (k2*k2)                   True  False),-      (initScale "Gbaud" (k2*k2*k2)                True  False)-    ])-  ]
− src-ghc-6.12/NumericPrelude.hs 
@@ -1,9 +0,0 @@-module NumericPrelude-   (module NumericPrelude.Numeric,-    module NumericPrelude.Base,-    max, min, abs, ) where--import NumericPrelude.Numeric hiding (abs, )-import NumericPrelude.Base    hiding (max, min, )-import Prelude ()-import Algebra.Lattice (max, min, abs, )
− src-ghc-6.12/NumericPrelude/Base.hs 
@@ -1,12 +0,0 @@-{- |-The only point of this module is-to reexport items that we want from the standard Prelude.--}--module NumericPrelude.Base (module Prelude) where-import Prelude hiding (-       Int, Integer, Float, Double, Rational, Num(..), Real(..),-       Integral(..), Fractional(..), Floating(..), RealFrac(..),-       RealFloat(..), subtract, even, odd,-       gcd, lcm, (^), (^^), sum, product,-       fromIntegral, fromRational, )
− src-ghc-6.12/NumericPrelude/Elementwise.hs 
@@ -1,54 +0,0 @@-module NumericPrelude.Elementwise where--import Control.Applicative (Applicative(pure, (<*>)), )--{- |-A reader monad for the special purpose-of defining instances of certain operations on tuples and records.-It does not add any new functionality to the common Reader monad,-but it restricts the functions to the required ones-and exports them from one module.-That is you do not have to import-both Control.Monad.Trans.Reader and Control.Applicative.-The type also tells the user, for what the Reader monad is used.-We can more easily replace or extend the implementation when needed.--}-newtype T v a = Cons {run :: v -> a}--{-# INLINE with #-}-with :: a -> T v a-with e = Cons $ \ _v -> e--{-# INLINE element #-}-element :: (v -> a) -> T v a-element = Cons---{-# INLINE run2 #-}-run2 :: T (x,y) a -> x -> y -> a-run2 = curry . run--{-# INLINE run3 #-}-run3 :: T (x,y,z) a -> x -> y -> z -> a-run3 e x y z = run e (x,y,z)--{-# INLINE run4 #-}-run4 :: T (x,y,z,w) a -> x -> y -> z -> w -> a-run4 e x y z w = run e (x,y,z,w)--{-# INLINE run5 #-}-run5 :: T (x,y,z,u,w) a -> x -> y -> z -> u -> w -> a-run5 e x y z u w = run e (x,y,z,u,w)---instance Functor (T v) where-   {-# INLINE fmap #-}-   fmap f (Cons e) =-      Cons $ \v -> f $ e v--instance Applicative (T v) where-   {-# INLINE pure #-}-   {-# INLINE (<*>) #-}-   pure = with-   Cons f <*> Cons e =-      Cons $ \v -> f v $ e v
− src-ghc-6.12/NumericPrelude/List.hs 
@@ -1,71 +0,0 @@-module NumericPrelude.List where--import Data.List.HT (switchL, switchR, )---{- * Zip lists -}--{- | zip two lists using an arbitrary function, the shorter list is padded -}-{-# INLINE zipWithPad #-}-zipWithPad :: a               {-^ padding value -}-           -> (a -> a -> b)   {-^ function applied to corresponding elements of the lists -}-           -> [a]-           -> [a]-           -> [b]-zipWithPad z f =-   let aux l []          = map (\x -> f x z) l-       aux [] l          = map (\y -> f z y) l-       aux (x:xs) (y:ys) = f x y : aux xs ys-   in  aux--{-# INLINE zipWithOverlap #-}-zipWithOverlap :: (a -> c) -> (b -> c) -> (a -> b -> c) -> [a] -> [b] -> [c]-zipWithOverlap fa fb fab =-   let aux (x:xs) (y:ys) = fab x y : aux xs ys-       aux xs [] = map fa xs-       aux [] ys = map fb ys-   in  aux--{--This is exported Checked.zipWith.-We need to define it here in order to prevent an import cycle.--}-zipWithChecked-   :: (a -> b -> c)   {-^ function applied to corresponding elements of the lists -}-   -> [a]-   -> [b]-   -> [c]-zipWithChecked f =-   let aux (x:xs) (y:ys) = f x y : aux xs ys-       aux []     []     = []-       aux _      _      = error "Checked.zipWith: lists must have the same length"-   in  aux---{- |-Apply a function to the last element of a list.-If the list is empty, nothing changes.--}-{-# INLINE mapLast #-}-mapLast :: (a -> a) -> [a] -> [a]-mapLast f =-   switchL []-      (\x xs ->-         uncurry (:) $-         foldr (\x1 k x0 -> (x0, uncurry (:) (k x1)))-            (\x0 -> (f x0, [])) xs x)--mapLast' :: (a -> a) -> [a] -> [a]-mapLast' f =-   let recourse [] = [] -- behaviour as needed in powerBasis-          -- otherwise: error "mapLast: empty list"-       recourse (x:xs) =-          uncurry (:) $-          if null xs-            then (f x, [])-            else (x, recourse xs)-   in  recourse--mapLast'' :: (a -> a) -> [a] -> [a]-mapLast'' f =-   switchR [] (\xs x -> xs ++ [f x])
− src-ghc-6.12/NumericPrelude/List/Checked.hs 
@@ -1,94 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Some functions that are counterparts of functions from "Data.List"-using NumericPrelude.Numeric type classes.-They are distinct in that they check for valid arguments,-e.g. the length argument of 'take' must be at most the length of the input list.-However, since many Haskell programs rely on the absence of such checks,-we did not make these the default implementations-as in "NumericPrelude.List.Generic".--}-module NumericPrelude.List.Checked-   (take, drop, splitAt, (!!), zipWith,-   ) where--import qualified Algebra.ToInteger  as ToInteger--- import qualified Algebra.Ring       as Ring-import Algebra.Ring (one, )-import Algebra.Additive (zero, (-), )--import Data.Tuple.HT (mapFst, )--import qualified NumericPrelude.List as NPList--import NumericPrelude.Base hiding (take, drop, splitAt, length, replicate, (!!), zipWith, )---moduleError :: String -> String -> a-moduleError name msg =-   error $ "NumericPrelude.List.Left." ++ name ++ ": " ++ msg--{- |-Taken number of elements must be at most the length of the list,-otherwise the end of the list is undefined.--}-take :: (ToInteger.C n) => n -> [a] -> [a]-take n =-   if n<=zero-     then const []-     else \xt ->-       case xt of-          [] -> moduleError "take" "index out of range"-          (x:xs) -> x : take (n-one) xs--{- |-Dropped number of elements must be at most the length of the list,-otherwise the end of the list is undefined.--}-drop :: (ToInteger.C n) => n -> [a] -> [a]-drop n =-   if n<=zero-     then id-     else \xt ->-       case xt of-          [] -> moduleError "drop" "index out of range"-          (_:xs) -> drop (n-one) xs--{- |-Split position must be at most the length of the list,-otherwise the end of the first list and the second list are undefined.--}-splitAt :: (ToInteger.C n) => n -> [a] -> ([a], [a])-splitAt n xt =-   if n<=zero-     then ([], xt)-     else-       case xt of-          [] -> moduleError "splitAt" "index out of range"-          (x:xs) -> mapFst (x:) $ splitAt (n-one) xs--{- |-The index must be smaller than the length of the list,-otherwise the result is undefined.--}-(!!) :: (ToInteger.C n) => [a] -> n -> a-(!!) [] _ = moduleError "(!!)" "index out of range"-(!!) (x:xs) n =-   if n<=zero-     then x-     else (!!) xs (n-one)---{- |-Zip two lists which must be of the same length.-This is checked only lazily, that is unequal lengths are detected only-if the list is evaluated completely.-But it is more strict than @zipWithPad undefined f@-since the latter one may succeed on unequal length list if @f@ is lazy.--}-zipWith-   :: (a -> b -> c)   {-^ function applied to corresponding elements of the lists -}-   -> [a]-   -> [b]-   -> [c]-zipWith = NPList.zipWithChecked
− src-ghc-6.12/NumericPrelude/List/Generic.hs 
@@ -1,84 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{- |-Functions that are counterparts of the @generic@ functions in "Data.List"-using NumericPrelude.Numeric type classes.-For input arguments we use the restrictive @ToInteger@ constraint,-although in principle @RealRing@ would be enough.-However we think that @take 0.5 xs@ is rather a bug than a feature,-thus we forbid fractional types.-On the other hand fractional types as result can be quite handy,-e.g. in @average xs = sum xs / length xs@.--}-module NumericPrelude.List.Generic-   ((!!), lengthLeft, lengthRight, replicate,-    take, drop, splitAt,-    findIndex, elemIndex, findIndices, elemIndices,-   ) where--import NumericPrelude.List.Checked ((!!), )--import qualified Algebra.ToInteger  as ToInteger-import qualified Algebra.Ring       as Ring-import Algebra.Ring (one, )-import Algebra.Additive (zero, (+), (-), )--import qualified Data.Maybe         as Maybe-import Data.Tuple.HT (mapFst, )--import NumericPrelude.Base as List-   hiding (take, drop, splitAt, length, replicate, (!!), )---replicate :: (ToInteger.C n) => n -> a -> [a]-replicate n x = take n (List.repeat x)--take :: (ToInteger.C n) => n -> [a] -> [a]-take _ [] = []-take n (x:xs) =-   if n<=zero-     then []-     else x : take (n-one) xs--drop :: (ToInteger.C n) => n -> [a] -> [a]-drop _ [] = []-drop n xt@(_:xs) =-   if n<=zero-     then xt-     else drop (n-one) xs--splitAt :: (ToInteger.C n) => n -> [a] -> ([a], [a])-splitAt _ [] = ([], [])-splitAt n xt@(x:xs) =-   if n<=zero-     then ([], xt)-     else mapFst (x:) $ splitAt (n-one) xs---{- |-Left associative length computation-that is appropriate for types like @Integer@.--}-lengthLeft :: (Ring.C n) => [a] -> n-lengthLeft = List.foldl (\n _ -> n+one) zero--{- |-Right associative length computation-that is appropriate for types like @Peano@ number.--}-lengthRight :: (Ring.C n) => [a] -> n-lengthRight = List.foldr (\_ n -> one+n) zero--elemIndex :: (Ring.C n, Eq a) => a -> [a] -> Maybe n-elemIndex e = findIndex (e==)--elemIndices :: (Ring.C n, Eq a) => a -> [a] -> [n]-elemIndices e = findIndices (e==)--findIndex :: Ring.C n => (a -> Bool) -> [a] -> Maybe n-findIndex p = Maybe.listToMaybe . findIndices p--findIndices :: Ring.C n => (a -> Bool) -> [a] -> [n]-findIndices p =-   map fst .-   filter (p . snd) .-   zip (iterate (one+) zero)
− src-ghc-6.12/NumericPrelude/Numeric.hs 
@@ -1,44 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module NumericPrelude.Numeric (-    {- Additive -} (+), (-), negate, zero, subtract, sum, sum1,-    {- ZeroTestable -} isZero,-    {- Ring -} (*), one, fromInteger, (^), ringPower, sqr, product, product1,-    {- IntegralDomain -} div, mod, divMod, divides, even, odd,-    {- Field -} (/), recip, fromRational', (^-), fieldPower, fromRational,-    {- Algebraic -} (^/), sqrt,-    {- Transcendental -}-        pi, exp, log, logBase, (**), (^?), sin, cos, tan,-        asin, acos, atan, sinh, cosh, tanh, asinh, acosh, atanh,-    {- Absolute -} abs, signum,-    {- RealIntegral -} quot, rem, quotRem,-    {- RealFrac -} splitFraction, fraction, truncate, round, ceiling, floor, approxRational,-    {- RealTrans -} atan2,-    {- ToRational -} toRational,-    {- ToInteger -} toInteger, fromIntegral,-    {- Units -} isUnit, stdAssociate, stdUnit, stdUnitInv,-    {- PID -} extendedGCD, gcd, lcm, euclid, extendedEuclid,-    {- Ratio -} Rational, (%), numerator, denominator,-    Integer, Int, Float, Double,-    {- Module -} (*>)-) where--import Number.Ratio (Rational, (%), numerator, denominator)--import Algebra.Module((*>))-import Algebra.RealTranscendental(atan2)-import Algebra.Transcendental-import Algebra.Algebraic((^/), sqrt)-import Algebra.RealRing(splitFraction, fraction, truncate, round, ceiling, floor, approxRational, )-import Algebra.Field((/), (^-), recip, fromRational', fromRational, )-import Algebra.PrincipalIdealDomain (extendedGCD, gcd, lcm, euclid, extendedEuclid)-import Algebra.Units (isUnit, stdAssociate, stdUnit, stdUnitInv)-import Algebra.RealIntegral (quot, rem, quotRem, )-import Algebra.IntegralDomain (div, mod, divMod, divides, even, odd)-import Algebra.Absolute (abs, signum, )-import Algebra.Ring (one, fromInteger, (*), (^), sqr, product, product1)-import Algebra.Additive (zero, (+), (-), negate, subtract, sum, sum1)-import Algebra.ZeroTestable (isZero)-import Algebra.ToInteger (ringPower, fieldPower, toInteger, fromIntegral, )-import Algebra.ToRational (toRational, )--import Prelude (Int, Integer, Float, Double)
src/Algebra/Absolute.hs view
@@ -6,9 +6,8 @@  import qualified Algebra.Ring         as Ring import qualified Algebra.Additive     as Additive-import qualified Algebra.ZeroTestable as ZeroTestable -import Algebra.Ring (one, ) -- fromInteger+import Algebra.Ring (one, ) import Algebra.Additive (zero, negate,)  import Data.Int  (Int,  Int8,  Int16,  Int32,  Int64,  )@@ -38,8 +37,17 @@ >      a * (max b c) === max (a*b) (a*c) where a >= 0 >           absOrd a === max a (-a) +If the type is @ZeroTestable@, then it should hold++>  isZero a  ===  signum a == signum (negate a)+ We do not require 'Ord' as superclass since we also want to have "Number.Complex" as instance.+We also do not require @ZeroTestable@ as superclass,+because we like to have expressions of foreign languages+to be instances (cf. embedded domain specific language approach, EDSL),+as well as function types.+ 'abs' for complex numbers alone may have an inappropriate type, because it does not reflect that the absolute value is a real number. You might prefer 'Number.Complex.magnitude'.@@ -53,7 +61,7 @@ we could relax the superclasses to @Additive@ and 'Ord' if his class would only contain 'signum'. -}-class (Ring.C a, ZeroTestable.C a) => C a where+class (Ring.C a) => C a where     abs    :: a -> a     signum :: a -> a
src/Algebra/Additive.hs view
@@ -8,6 +8,8 @@      -- * Complex functions     sum, sum1,+    sumNestedAssociative,+    sumNestedCommutative,      -- * Instance definition helpers     elementAdd, elementSub, elementNeg,@@ -28,13 +30,21 @@ import qualified NumericPrelude.Elementwise as Elem import Control.Applicative (Applicative(pure, (<*>)), ) import Data.Tuple.HT (fst3, snd3, thd3, )+import qualified Data.List.Match as Match +import qualified Data.Complex as Complex98 import qualified Data.Ratio as Ratio98 import qualified Prelude as P import Prelude (Integer, Float, Double, fromInteger, ) import NumericPrelude.Base  +{- $setup+>>> import qualified Algebra.Additive as A+>>> import qualified Test.QuickCheck as QC+-}++ infixl 6  +, -  {- |@@ -54,6 +64,7 @@ -}  class C a where+    {-# MINIMAL zero, (+), ((-) | negate) #-}     -- | zero element of the vector space     zero     :: a     -- | add and subtract elements@@ -92,13 +103,64 @@ Sum up all elements of a non-empty list. This avoids including a zero which is useful for types where no universal zero is available.+ToDo: Should have NonEmpty type.++prop> \(QC.NonEmpty ns) -> A.sum ns == (A.sum1 ns :: Integer) -} sum1 :: (C a) => [a] -> a sum1 = foldl1 (+)  +{- |+Sum the operands in an order,+such that the dependencies are minimized.+Does this have a measurably effect on speed? +Requires associativity.++prop> \ns -> A.sum ns == (A.sumNestedAssociative ns :: Integer)+-}+sumNestedAssociative :: (C a) => [a] -> a+sumNestedAssociative [] = zero+sumNestedAssociative [x] = x+sumNestedAssociative xs = sumNestedAssociative (sum2 xs)+ {- |+Make sure that the last entries in the list+are equally often part of an addition.+Maybe this can reduce rounding errors.+The list that sum2 computes is a breadth-first-flattened binary tree.++Requires associativity and commutativity.++prop> \ns -> A.sum ns == (A.sumNestedCommutative ns :: Integer)+-}+sumNestedCommutative :: (C a) => [a] -> a+sumNestedCommutative [] = zero+sumNestedCommutative xs@(_:rs) =+   let ys = xs ++ Match.take rs (sum2 ys)+   in  last ys++_sumNestedCommutative :: (C a) => [a] -> a+_sumNestedCommutative [] = zero+_sumNestedCommutative xs@(_:rs) =+   let ys = xs ++ take (length rs) (sum2 ys)+   in  last ys++{-+[a,b,c, a+b,c+(a+b)]+[a,b,c,d, a+b,c+d,(a+b)+(c+d)]+[a,b,c,d,e, a+b,c+d,e+(a+b),(c+d)+e+(a+b)]+[a,b,c,d,e,f, a+b,c+d,e+f,(a+b)+(c+d),(e+f)+((a+b)+(c+d))]+-}++sum2 :: (C a) => [a] -> [a]+sum2 (x:y:rest) = (x+y) : sum2 rest+sum2 xs = xs++++{- | Instead of baking the add operation into the element function, we could use higher rank types and pass a generic @uncurry (+)@ to the run function.@@ -316,6 +378,13 @@    negate = Elem.run  $ pure (,,) <*>.-$ fst3 <*>.-$ snd3 <*>.-$ thd3  +{- |+The 'Additive' instantiations treat lists+as prefixes of infinite lists with zero filled tail.+This interpretation is not always appropriate.+The end of a list may just mean: End of available data.+In this case the shortening 'zip' semantics would be more appropriate.+-} instance (C v) => C [v] where    zero   = []    negate = map negate@@ -358,7 +427,17 @@    {-# INLINE negate #-}    {-# INLINE (+) #-}    {-# INLINE (-) #-}-   zero                =  0+   zero                =  P.fromInteger 0+   (+)                 =  (P.+)+   (-)                 =  (P.-)+   negate              =  P.negate++instance (P.RealFloat a) => C (Complex98.Complex a) where+   {-# INLINE zero #-}+   {-# INLINE negate #-}+   {-# INLINE (+) #-}+   {-# INLINE (-) #-}+   zero                =  P.fromInteger 0    (+)                 =  (P.+)    (-)                 =  (P.-)    negate              =  P.negate
src/Algebra/Algebraic.hs view
@@ -2,7 +2,6 @@ module Algebra.Algebraic where  import qualified Algebra.Field as Field--- import qualified Algebra.Units as Units import qualified Algebra.Laws as Laws import qualified Algebra.ToRational as ToRational import qualified Algebra.ToInteger  as ToInteger@@ -21,6 +20,7 @@ {- | Minimal implementation: 'root' or '(^\/)'. -}  class (Field.C a) => C a where+    {-# MINIMAL root | (^/) #-}     sqrt :: a -> a     sqrt = root 2     -- sqrt x  =  x ** (1/2)
src/Algebra/Differential.hs view
@@ -3,8 +3,6 @@  import qualified Algebra.Ring as Ring --- import NumericPrelude.Numeric--- import qualified Prelude  {- | 'differentiate' is a general differentation operation
src/Algebra/DimensionTerm.hs view
@@ -1,12 +1,4 @@ {- |-Copyright   :  (c) Henning Thielemann 2008-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable-- We already have the dynamically checked physical units provided by "Number.Physical" and the statically checked ones of the @dimensional@ package of Buckwalter,
src/Algebra/Field.hs view
@@ -16,12 +16,11 @@  import Number.Ratio (T((:%)), Rational, (%), numerator, denominator, ) import qualified Number.Ratio as Ratio+import qualified Data.Complex as Complex98 import qualified Data.Ratio as Ratio98 import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.Units as Unit  import qualified Algebra.Ring         as Ring--- import qualified Algebra.Additive     as Additive import qualified Algebra.ZeroTestable as ZeroTestable  import Algebra.Ring ((*), (^), one, fromInteger)@@ -69,6 +68,7 @@ -}  class (Ring.C a) => C a where+    {-# MINIMAL recip | (/) #-}     (/)           :: a -> a -> a     recip         :: a -> a     fromRational' :: Rational -> a@@ -136,10 +136,7 @@      recip (x:%y)         =  (y:%x) -}-    recip (x:%y)         =-       if isZero y-         then error "Ratio./: division by zero"-         else (y * Unit.stdUnitInv x) :% Unit.stdAssociate x+    recip = Ratio.recip     fromRational' (x:%y) =  fromInteger x % fromInteger y  @@ -155,6 +152,12 @@ -- legacy  instance (P.Integral a) => C (Ratio98.Ratio a) where+    {-# INLINE (/) #-}+    {-# INLINE recip #-}+    (/)    = (P./)+    recip  = (P.recip)++instance (P.RealFloat a) => C (Complex98.Complex a) where     {-# INLINE (/) #-}     {-# INLINE recip #-}     (/)    = (P./)
+ src/Algebra/FloatingPoint.hs view
@@ -0,0 +1,57 @@+{-# LANGUAGE RebindableSyntax #-}+module Algebra.FloatingPoint where++import qualified Algebra.RealRing as RealRing+import NumericPrelude.Base++import qualified Prelude as P+import Prelude (Int, Integer, Float, Double, )+++{- |+Counterpart of 'Prelude.RealFloat' but with NumericPrelude superclass.+-}+class RealRing.C a => C a where+   radix :: a -> Integer+   digits :: a -> Int+   range :: a -> (Int, Int)+   decode :: a -> (Integer, Int)+   encode :: Integer -> Int -> a+   exponent :: a -> Int+   significand :: a -> a+   scale :: Int -> a -> a+   isNaN :: a -> Bool+   isInfinite :: a -> Bool+   isDenormalized :: a -> Bool+   isNegativeZero :: a -> Bool+   isIEEE :: a -> Bool++instance C Float where+   radix = P.floatRadix+   digits = P.floatDigits+   range = P.floatRange+   decode = P.decodeFloat+   encode = P.encodeFloat+   exponent = P.exponent+   significand = P.significand+   scale = P.scaleFloat+   isNaN = P.isNaN+   isInfinite = P.isInfinite+   isDenormalized = P.isDenormalized+   isNegativeZero = P.isNegativeZero+   isIEEE = P.isIEEE++instance C Double where+   radix = P.floatRadix+   digits = P.floatDigits+   range = P.floatRange+   decode = P.decodeFloat+   encode = P.encodeFloat+   exponent = P.exponent+   significand = P.significand+   scale = P.scaleFloat+   isNaN = P.isNaN+   isInfinite = P.isInfinite+   isDenormalized = P.isDenormalized+   isNegativeZero = P.isNegativeZero+   isIEEE = P.isIEEE
src/Algebra/IntegralDomain.hs view
@@ -32,7 +32,6 @@   ) where  import qualified Algebra.Ring         as Ring--- import qualified Algebra.Additive     as Additive import qualified Algebra.ZeroTestable as ZeroTestable  import Algebra.Ring     ((*), fromInteger, )@@ -52,7 +51,15 @@ import qualified Prelude as P  +{- $setup+>>> import Algebra.IntegralDomain (roundDown, roundUp, divUp)+>>> import qualified Test.QuickCheck as QC+>>> import NumericPrelude.Base as P+>>> import NumericPrelude.Numeric as NP+>>> import Prelude ()+-} + infixl 7 `div`, `mod`  @@ -95,7 +102,11 @@ Minimal definition: 'divMod' or ('div' and 'mod') -} class (Ring.C a) => C a where+    {-# MINIMAL divMod | (div, mod) #-}     div, mod :: a -> a -> a+    {- |+    prop> \n (QC.NonZero m) -> let (q,r) = divMod n m in n == (q*m+r :: Integer)+    -}     divMod :: a -> a -> (a,a)      {-# INLINE div #-}@@ -183,6 +194,8 @@ that is at most @n@. The parameter order is consistent with @div@ and friends, but maybe not useful for partial application.++prop> \n (QC.NonZero m) -> div n m * m == (roundDown n m :: Integer) -} roundDown :: C a => a -> a -> a roundDown n m = n - mod n m@@ -191,6 +204,10 @@ @roundUp n m@ rounds @n@ up to the next multiple of @m@. That is, @roundUp n m@ is the greatest multiple of @m@ that is at most @n@.++prop> \n (QC.NonZero m) -> divUp n m * m == (roundUp n m :: Integer)+prop> \n (QC.Positive m) -> let x = roundDown n m in  n-m < x && x <= (n :: Integer)+prop> \n (QC.NonZero m) -> - roundDown n m == (roundUp (-n) m :: Integer) -} roundUp :: C a => a -> a -> a roundUp n m = n + mod (-n) m@@ -321,7 +338,7 @@ propMultipleDiv m a =    not (isZero m) ==>                 (a*m) `div` m  ==  a propMultipleMod m a =-   not (isZero m) ==>                 (a*m) `mod` m  ==  zero+   not (isZero m) ==>                 (a*m) `mod` m  ==  0 propProjectAddition m a b =    not (isZero m) ==>       (a+b) `mod` m  ==  ((a`mod`m)+(b`mod`m)) `mod` m
src/Algebra/Module.hs view
@@ -28,13 +28,16 @@ import qualified NumericPrelude.Elementwise as Elem import Control.Applicative (Applicative(pure, (<*>)), ) +import qualified Data.Complex as Complex98+import Data.Int (Int, Int8, Int16, Int32, Int64, )+ import Data.Function.HT (powerAssociative, ) import Data.List (map, zipWith, ) import Data.Tuple.HT (fst3, snd3, thd3, ) import Data.Tuple (fst, snd, ) -import Prelude((.), Eq, Bool, Int, Integer, Float, Double, ($), )--- import qualified Prelude as P+import qualified Prelude as P+import Prelude((.), Eq, Bool, Integer, Float, Double, ($), )   -- Is this right?@@ -83,6 +86,22 @@    {-# INLINE (*>) #-}    (*>) = (*) +instance C Int8 Int8 where+   {-# INLINE (*>) #-}+   (*>) = (*)++instance C Int16 Int16 where+   {-# INLINE (*>) #-}+   (*>) = (*)++instance C Int32 Int32 where+   {-# INLINE (*>) #-}+   (*>) = (*)++instance C Int64 Int64 where+   {-# INLINE (*>) #-}+   (*>) = (*)+ instance C Integer Integer where    {-# INLINE (*>) #-}    (*>) = (*)@@ -116,6 +135,11 @@ instance (C a v) => C a (c -> v) where    {-# INLINE (*>) #-}    (*>) s f = (*>) s . f+++instance (C a b, P.RealFloat b) => C a (Complex98.Complex b) where+   {-# INLINE (*>) #-}+   s *> (x Complex98.:+ y)  =  (s *> x) Complex98.:+ (s *> y)   {-* Related functions -}
src/Algebra/ModuleBasis.hs view
@@ -15,14 +15,12 @@  import qualified Algebra.PrincipalIdealDomain as PID import qualified Algebra.Module   as Module--- import qualified Algebra.Additive as Additive import Algebra.Ring     (one, fromInteger) import Algebra.Additive ((+), zero)  import Data.List (map, length, (++))  import Prelude(Eq, (==), Bool, Int, Integer, Float, Double, asTypeOf, )--- import qualified Prelude as P  {- | It must hold:
src/Algebra/Monoid.hs view
@@ -18,6 +18,11 @@  import Data.Monoid as Mn +import Data.Function ((.))+import Data.List (foldr, reverse, map)+import Prelude ()++ {- | We expect a monoid to adher to associativity and the identity behaving decently.
src/Algebra/NonNegative.hs view
@@ -25,11 +25,9 @@    ) where  import qualified Algebra.Additive as Additive--- import qualified Algebra.RealRing as RealRing  import qualified Algebra.Monoid as Monoid --- import Algebra.Absolute (abs, ) import Algebra.Additive ((-), )  import Prelude hiding (sum, (-), abs, )
src/Algebra/NormedSpace/Euclidean.hs view
@@ -3,13 +3,6 @@ {-# LANGUAGE FlexibleInstances #-}  {- |-Copyright   :  (c) Henning Thielemann 2005-2010-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  requires multi-parameter type classes- Abstraction of normed vector spaces -} @@ -17,6 +10,7 @@  import NumericPrelude.Base import NumericPrelude.Numeric (sqr, abs, zero, (+), sum, Float, Double, Int, Integer, )+import qualified Prelude as P  import qualified Number.Ratio as Ratio @@ -25,6 +19,7 @@ import qualified Algebra.Absolute      as Absolute import qualified Algebra.Module    as Module +import qualified Data.Complex as Complex98 import qualified Data.Foldable as Fold  @@ -123,4 +118,13 @@   normSqr = sum . map normSqr  instance (Algebraic.C a, Sqr a v) => C a [v] where+  norm    = defltNorm+++instance (Sqr a v, P.RealFloat v) => Sqr a (Complex98.Complex v) where+  normSqr (x0 Complex98.:+ x1) = normSqr x0 + normSqr x1++instance+  (Algebraic.C a, Sqr a v, P.RealFloat v) =>+    C a (Complex98.Complex v) where   norm    = defltNorm
src/Algebra/NormedSpace/Maximum.hs view
@@ -3,13 +3,6 @@ {-# LANGUAGE FlexibleInstances #-}  {- |-Copyright   :  (c) Henning Thielemann 2005-2010-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  requires multi-parameter type classes- Abstraction of normed vector spaces -} @@ -17,6 +10,7 @@  import NumericPrelude.Base import NumericPrelude.Numeric+import qualified Prelude as P  import qualified Number.Ratio as Ratio @@ -25,6 +19,7 @@ import qualified Algebra.RealRing as RealRing import qualified Algebra.Module   as Module +import qualified Data.Complex as Complex98 import qualified Data.Foldable as Fold  @@ -83,3 +78,7 @@ we can use zero as identity element.   norm = maximum . map norm -}+++instance (C a v, P.RealFloat v) => C a (Complex98.Complex v) where+  norm (x0 Complex98.:+ x1) = max (norm x0) (norm x1)
src/Algebra/NormedSpace/Sum.hs view
@@ -3,13 +3,6 @@ {-# LANGUAGE FlexibleInstances #-}  {- |-Copyright   :  (c) Henning Thielemann 2005-2010-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  requires multi-parameter type classes- Abstraction of normed vector spaces -} @@ -17,6 +10,7 @@  import NumericPrelude.Base import NumericPrelude.Numeric+import qualified Prelude as P  import qualified Number.Ratio as Ratio @@ -25,6 +19,7 @@ import qualified Algebra.Additive as Additive import qualified Algebra.Module   as Module +import qualified Data.Complex as Complex98 import qualified Data.Foldable as Fold  @@ -88,3 +83,7 @@  instance (Additive.C a, C a v) => C a [v] where   norm = sum . map norm+++instance (C a v, P.RealFloat v) => C a (Complex98.Complex v) where+  norm (x0 Complex98.:+ x1) = norm x0 + norm x1
src/Algebra/OccasionallyScalar.hs view
@@ -27,15 +27,8 @@  module Algebra.OccasionallyScalar where --- import qualified Algebra.RealRing    as RealRing-import qualified Algebra.ZeroTestable as ZeroTestable-import qualified Algebra.Additive     as Additive-import qualified Number.Complex       as Complex- import Data.Maybe (fromMaybe, ) -import Number.Complex((+:))- import NumericPrelude.Base import NumericPrelude.Numeric @@ -66,14 +59,6 @@    toScalar      = id    toMaybeScalar = Just    fromScalar    = id---- this instance should be defined in Number.Complex-instance (Show v, ZeroTestable.C v, Additive.C v, C a v) => C a (Complex.T v) where-   toScalar        = toScalarShow-   toMaybeScalar x = if isZero (Complex.imag x)-                       then toMaybeScalar (Complex.real x)-                       else Nothing-   fromScalar x    = fromScalar x +: zero  {- converting values automatically to integers is a bad idea instance (Integral b, RealRing.C a)
src/Algebra/PrincipalIdealDomain.hs view
@@ -39,8 +39,6 @@  import qualified Algebra.Units          as Units import qualified Algebra.IntegralDomain as Integral--- import qualified Algebra.Ring           as Ring--- import qualified Algebra.Additive       as Additive import qualified Algebra.ZeroTestable   as ZeroTestable  import qualified Algebra.Laws as Laws@@ -63,7 +61,19 @@ import Test.QuickCheck ((==>), Property)  +{- $setup+>>> import qualified Algebra.PrincipalIdealDomain as PID+>>> import Test.NumericPrelude.Utility ((/\))+>>> import qualified Test.QuickCheck as QC+>>>+>>> genResidueClass :: QC.Gen (Integer,Integer)+>>> genResidueClass = do+>>>    m <- fmap QC.getNonZero $ QC.arbitrary+>>>    a <- QC.choose (min 0 $ 1+m, max 0 $ m-1)+>>>    return (m,a)+-} + {- | A principal ideal domain is a ring in which every ideal (the set of multiples of some generating set of elements)@@ -235,9 +245,9 @@  {- | Not efficient because it requires duplicate computations of GCDs.-However GCDs of neighbouring list elements were not computed before.+However GCDs of adjacent list elements were not computed before. It is also quite arbitrary,-because only neighbouring elements are used for balancing.+because only adjacent elements are used for balancing. There are certainly more sophisticated solutions. -} diophantineMultiMin :: C a => a -> [a] -> Maybe [a]@@ -279,10 +289,21 @@ -}  {- |-For @Just (b,n) = chineseRemainder [(a0,m0), (a1,m1), ..., (an,mn)]@-and all @x@ with @x = b mod n@ the congruences-@x=a0 mod m0, x=a1 mod m1, ..., x=an mod mn@+For @Just (n,b) = chineseRemainderMulti [(m0,a0), (m1,a1), ..., (mk,ak)]@+and all @x@ with @x = b mod n@, the congruences+@x=a0 mod m0, x=a1 mod m1, ..., x=ak mod mk@ are fulfilled.+Also, @n@ is the least common multiplier of all @mi@.++>>> PID.chineseRemainderMulti [(100,21), (10000,2021::Integer)]+Just (10000,2021)+>>> PID.chineseRemainderMulti [(97,90),(99,10),(100,0::Integer)]+Just (960300,100000)+>>> PID.chineseRemainderMulti [(95,30),(97,27),(98,8),(99,1::Integer)]+Just (89403930,1000000)++prop> QC.listOf genResidueClass /\ \xs -> case PID.chineseRemainderMulti xs of Nothing -> True; Just (n,b) -> abs n == abs (foldl lcm 1 (map fst xs)) && map snd xs == map (mod b . fst) xs+prop> \(QC.NonEmpty ms) b -> let xs = map (\(QC.NonZero m) -> (m, mod b m)) ms in case PID.chineseRemainderMulti xs of Nothing -> False; Just (n,c) -> abs n == abs (foldl lcm 1 (map QC.getNonZero ms)) && mod b n == (c::Integer) -} chineseRemainderMulti :: C a => [(a,a)] -> Maybe (a,a) chineseRemainderMulti congs =
src/Algebra/RealField.hs view
@@ -10,8 +10,6 @@  import qualified Number.Ratio as Ratio --- import NumericPrelude.Base--- import qualified Prelude as P import Prelude (Float, Double, )  {- |
src/Algebra/RealIntegral.hs view
@@ -16,10 +16,9 @@    C(quot, rem, quotRem),    ) where +import qualified Algebra.ZeroTestable   as ZeroTestable import qualified Algebra.IntegralDomain as Integral import qualified Algebra.Absolute       as Absolute--- import qualified Algebra.Ring           as Ring--- import qualified Algebra.Additive       as Additive  import Algebra.Absolute (signum, ) import Algebra.IntegralDomain (divMod, )@@ -46,7 +45,7 @@ Minimal definition: nothing required -} -class (Absolute.C a, Ord a, Integral.C a) => C a where+class (Absolute.C a, ZeroTestable.C a, Ord a, Integral.C a) => C a where     quot, rem        :: a -> a -> a     quotRem          :: a -> a -> (a,a)
src/Algebra/RealRing.hs view
@@ -1,12 +1,14 @@ {-# LANGUAGE RebindableSyntax #-} module Algebra.RealRing where -import qualified Algebra.Field              as Field+import qualified Algebra.RealRing98 as RealRing98+ import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.Absolute           as Absolute+import qualified Algebra.Field          as Field import qualified Algebra.Ring           as Ring import qualified Algebra.ToRational     as ToRational import qualified Algebra.ToInteger      as ToInteger+import qualified Algebra.Absolute       as Absolute  import qualified Algebra.OrderDecision as OrdDec import Algebra.OrderDecision ((<?), (>=?), )@@ -33,6 +35,20 @@ import NumericPrelude.Base  +{- $setup+>>> import qualified Algebra.RealRing as RealRing+>>> import Data.Tuple.HT (mapFst)+>>> import NumericPrelude.Numeric as NP+>>> import NumericPrelude.Base+>>> import Prelude ()+>>>+>>> infix 4 =~=+>>>+>>> (=~=) :: (Eq b) => (a -> b) -> (a -> b) -> a -> Bool+>>> (f =~= g) x = f x == g x+-}++ {- | Minimal complete definition:      'splitFraction' or 'floor'@@ -113,8 +129,24 @@ -}  class (Absolute.C a, Ord a) => C a where+    {-# MINIMAL splitFraction | floor #-}+    {- |+    prop> \x -> (x::Rational) == (uncurry (+) $ mapFst fromInteger $ splitFraction x)+    prop> \x -> uncurry (==) $ mapFst (((x::Double)-) . fromInteger) $ splitFraction x+    prop> \x -> uncurry (==) $ mapFst (((x::Rational)-) . fromInteger) $ splitFraction x+    prop> \x -> splitFraction x == (floor (x::Double) :: Integer, fraction x)+    prop> \x -> splitFraction x == (floor (x::Rational) :: Integer, fraction x)+    -}     splitFraction    :: (Ring.C b) => a -> (b,a)-    fraction         ::               a -> a+    {- |+    prop> \x -> let y = fraction (x::Double) in 0<=y && y<1+    prop> \x -> let y = fraction (x::Rational) in 0<=y && y<1+    -}+    fraction :: a -> a+    {- |+    prop> \x -> ceiling (-x) == negate (floor (x::Double) :: Integer)+    prop> \x -> ceiling (-x) == negate (floor (x::Rational) :: Integer)+    -}     ceiling, floor   :: (Ring.C b) => a -> b     truncate         :: (Ring.C b) => a -> b     round            :: (ToInteger.C b) => a -> b@@ -154,6 +186,7 @@ but is simply a kind of rounding that is the fastest on IEEE floating point architectures. -}+{-# NOINLINE [2] roundSimple #-} roundSimple :: (C a, Ring.C b) => a -> b roundSimple x =    let (n,r) = splitFraction x@@ -167,6 +200,20 @@     splitFraction (x:%y) = (fromIntegral q, r:%y)                                where (q,r) = divMod x y +instance C Integer where+    {-# INLINE splitFraction #-}+    {-# INLINE fraction #-}+    {-# INLINE floor #-}+    {-# INLINE ceiling #-}+    {-# INLINE round #-}+    {-# INLINE truncate #-}+    splitFraction x = (fromInteger x, zero)+    fraction      _ = zero+    floor         x = fromInteger x+    ceiling       x = fromInteger x+    round         x = fromInteger x+    truncate      x = fromInteger x+ instance C Int where     {-# INLINE splitFraction #-}     {-# INLINE fraction #-}@@ -181,20 +228,118 @@     round         x = fromIntegral x     truncate      x = fromIntegral x -instance C Integer where+instance C Int8 where     {-# INLINE splitFraction #-}     {-# INLINE fraction #-}     {-# INLINE floor #-}     {-# INLINE ceiling #-}     {-# INLINE round #-}     {-# INLINE truncate #-}-    splitFraction x = (fromInteger x, zero)+    splitFraction x = (fromIntegral x, zero)     fraction      _ = zero-    floor         x = fromInteger x-    ceiling       x = fromInteger x-    round         x = fromInteger x-    truncate      x = fromInteger x+    floor         x = fromIntegral x+    ceiling       x = fromIntegral x+    round         x = fromIntegral x+    truncate      x = fromIntegral x +instance C Int16 where+    {-# INLINE splitFraction #-}+    {-# INLINE fraction #-}+    {-# INLINE floor #-}+    {-# INLINE ceiling #-}+    {-# INLINE round #-}+    {-# INLINE truncate #-}+    splitFraction x = (fromIntegral x, zero)+    fraction      _ = zero+    floor         x = fromIntegral x+    ceiling       x = fromIntegral x+    round         x = fromIntegral x+    truncate      x = fromIntegral x++instance C Int32 where+    {-# INLINE splitFraction #-}+    {-# INLINE fraction #-}+    {-# INLINE floor #-}+    {-# INLINE ceiling #-}+    {-# INLINE round #-}+    {-# INLINE truncate #-}+    splitFraction x = (fromIntegral x, zero)+    fraction      _ = zero+    floor         x = fromIntegral x+    ceiling       x = fromIntegral x+    round         x = fromIntegral x+    truncate      x = fromIntegral x++instance C Int64 where+    {-# INLINE splitFraction #-}+    {-# INLINE fraction #-}+    {-# INLINE floor #-}+    {-# INLINE ceiling #-}+    {-# INLINE round #-}+    {-# INLINE truncate #-}+    splitFraction x = (fromIntegral x, zero)+    fraction      _ = zero+    floor         x = fromIntegral x+    ceiling       x = fromIntegral x+    round         x = fromIntegral x+    truncate      x = fromIntegral x++instance C Word8 where+    {-# INLINE splitFraction #-}+    {-# INLINE fraction #-}+    {-# INLINE floor #-}+    {-# INLINE ceiling #-}+    {-# INLINE round #-}+    {-# INLINE truncate #-}+    splitFraction x = (fromIntegral x, zero)+    fraction      _ = zero+    floor         x = fromIntegral x+    ceiling       x = fromIntegral x+    round         x = fromIntegral x+    truncate      x = fromIntegral x++instance C Word16 where+    {-# INLINE splitFraction #-}+    {-# INLINE fraction #-}+    {-# INLINE floor #-}+    {-# INLINE ceiling #-}+    {-# INLINE round #-}+    {-# INLINE truncate #-}+    splitFraction x = (fromIntegral x, zero)+    fraction      _ = zero+    floor         x = fromIntegral x+    ceiling       x = fromIntegral x+    round         x = fromIntegral x+    truncate      x = fromIntegral x++instance C Word32 where+    {-# INLINE splitFraction #-}+    {-# INLINE fraction #-}+    {-# INLINE floor #-}+    {-# INLINE ceiling #-}+    {-# INLINE round #-}+    {-# INLINE truncate #-}+    splitFraction x = (fromIntegral x, zero)+    fraction      _ = zero+    floor         x = fromIntegral x+    ceiling       x = fromIntegral x+    round         x = fromIntegral x+    truncate      x = fromIntegral x++instance C Word64 where+    {-# INLINE splitFraction #-}+    {-# INLINE fraction #-}+    {-# INLINE floor #-}+    {-# INLINE ceiling #-}+    {-# INLINE round #-}+    {-# INLINE truncate #-}+    splitFraction x = (fromIntegral x, zero)+    fraction      _ = zero+    floor         x = fromIntegral x+    ceiling       x = fromIntegral x+    round         x = fromIntegral x+    truncate      x = fromIntegral x+ instance C Float where     {-# INLINE splitFraction #-}     {-# INLINE fraction #-}@@ -203,7 +348,7 @@     {-# INLINE round #-}     {-# INLINE truncate #-}     splitFraction = fastSplitFraction GHC.float2Int GHC.int2Float-    fraction      = fastFraction (GHC.int2Float . GHC.float2Int)+    fraction      = RealRing98.fastFraction (GHC.int2Float . GHC.float2Int)     floor         = fromInteger . P.floor     ceiling       = fromInteger . P.ceiling     round         = fromInteger . P.round@@ -217,7 +362,7 @@     {-# INLINE round #-}     {-# INLINE truncate #-}     splitFraction = fastSplitFraction GHC.double2Int GHC.int2Double-    fraction      = fastFraction (GHC.int2Double . GHC.double2Int)+    fraction      = RealRing98.fastFraction (GHC.int2Double . GHC.double2Int)     floor         = fromInteger . P.floor     ceiling       = fromInteger . P.ceiling     round         = fromInteger . P.round@@ -241,21 +386,6 @@      then (n,   f)      else (n-1, f+1) -{-# INLINE fastFraction #-}-fastFraction :: (P.RealFrac a, Absolute.C a) => (a -> a) -> a -> a-fastFraction trunc x =-   fixFraction $-   if fromIntegral (minBound :: Int) <= x && x <= fromIntegral (maxBound :: Int)-     then x - trunc x-     else preludeFraction x--{-# INLINE preludeFraction #-}-preludeFraction :: (P.RealFrac a, Ring.C a) => a -> a-preludeFraction x =-   let second :: (Integer, a) -> a-       second = snd-   in  second (P.properFraction x)- {-# INLINE fixFraction #-} fixFraction :: (Ring.C a, Ord a) => a -> a fixFraction y =@@ -430,6 +560,9 @@ If operations like multiplication with two and comparison need time proportional to the number of binary digits, then the overall rounding requires quadratic time.++prop> RealRing.genericFloor =~= (NP.floor :: Double -> Integer)+prop> RealRing.genericFloor =~= (NP.floor :: Rational -> Integer) -} genericFloor :: (Ord a, Ring.C a, Ring.C b) => a -> b genericFloor a =@@ -437,30 +570,50 @@      then genericPosFloor a      else negate $ genericPosCeiling $ negate a +{- |+prop> RealRing.genericCeiling =~= (NP.ceiling :: Double -> Integer)+prop> RealRing.genericCeiling =~= (NP.ceiling :: Rational -> Integer)+-} genericCeiling :: (Ord a, Ring.C a, Ring.C b) => a -> b genericCeiling a =    if a>=zero      then genericPosCeiling a      else negate $ genericPosFloor $ negate a +{- |+prop> RealRing.genericTruncate =~= (NP.truncate :: Double -> Integer)+prop> RealRing.genericTruncate =~= (NP.truncate :: Rational -> Integer)+-} genericTruncate :: (Ord a, Ring.C a, Ring.C b) => a -> b genericTruncate a =    if a>=zero      then genericPosFloor a      else negate $ genericPosFloor $ negate a +{- |+prop> RealRing.genericRound =~= (NP.round :: Double -> Integer)+prop> RealRing.genericRound =~= (NP.round :: Rational -> Integer)+-} genericRound :: (Ord a, Ring.C a, Ring.C b) => a -> b genericRound a =    if a>=zero      then genericPosRound a      else negate $ genericPosRound $ negate a +{- |+prop> RealRing.genericFraction =~= (NP.fraction :: Double -> Double)+prop> RealRing.genericFraction =~= (NP.fraction :: Rational -> Rational)+-} genericFraction :: (Ord a, Ring.C a) => a -> a genericFraction a =    if a>=zero      then genericPosFraction a      else fixFraction $ negate $ genericPosFraction $ negate a +{- |+prop> RealRing.genericSplitFraction =~= (NP.splitFraction :: Double -> (Integer,Double))+prop> RealRing.genericSplitFraction =~= (NP.splitFraction :: Rational -> (Integer,Rational))+-} genericSplitFraction :: (Ord a, Ring.C a, Ring.C b) => a -> (b,a) genericSplitFraction a =    if a>=zero
+ src/Algebra/RealRing98.hs view
@@ -0,0 +1,39 @@+module Algebra.RealRing98 where++{-# INLINE fastSplitFraction #-}+fastSplitFraction :: (RealFrac a, Integral b) =>+   (a -> Int) -> (Int -> a) -> a -> (b,a)+fastSplitFraction trunc toFloat x =+   fixSplitFraction $+   if toFloat minBound <= x && x <= toFloat maxBound+     then case trunc x of n -> (fromIntegral n, x - toFloat n)+     else case properFraction x of (n,f) -> (fromInteger n, f)++{-# INLINE fixSplitFraction #-}+fixSplitFraction :: (Num a, Num b, Ord a) => (b,a) -> (b,a)+fixSplitFraction (n,f) =+   --  if x>=0 || f==0+   if f>=0+     then (n,   f)+     else (n-1, f+1)+++{-# INLINE fastFraction #-}+fastFraction :: (RealFrac a) => (a -> a) -> a -> a+fastFraction trunc x =+   fixFraction $+   if fromIntegral (minBound :: Int) <= x && x <= fromIntegral (maxBound :: Int)+     then x - trunc x+     else signedFraction x++{-# INLINE signedFraction #-}+signedFraction :: (RealFrac a) => a -> a+signedFraction x =+   let second :: (Integer, a) -> a+       second = snd+   in  second (properFraction x)++{-# INLINE fixFraction #-}+fixFraction :: (Real a) => a -> a+fixFraction y =+   if y>=0 then y else y+1
src/Algebra/RightModule.hs view
@@ -6,8 +6,6 @@ import qualified Algebra.Ring     as Ring import qualified Algebra.Additive as Additive --- import NumericPrelude.Numeric--- import qualified Prelude   -- Is this right?
src/Algebra/Ring.hs view
@@ -38,9 +38,9 @@  import NumericPrelude.Base import Prelude (Integer, Float, Double, )+import qualified Data.Complex as Complex98 import qualified Data.Ratio as Ratio98 import qualified Prelude as P--- import Test.QuickCheck   infixl 7 *@@ -64,6 +64,7 @@ -}  class (Additive.C a) => C a where+    {-# MINIMAL (*), (one | fromInteger) #-}     (*)         :: a -> a -> a     one         :: a     fromInteger :: Integer -> a@@ -252,6 +253,14 @@    {-# INLINE one #-}    {-# INLINE fromInteger #-}    {-# INLINE (*) #-}-   one                 =  1+   one                 =  P.fromInteger 1+   fromInteger         =  P.fromInteger+   (*)                 =  (P.*)++instance (P.RealFloat a) => C (Complex98.Complex a) where+   {-# INLINE one #-}+   {-# INLINE fromInteger #-}+   {-# INLINE (*) #-}+   one                 =  P.fromInteger 1    fromInteger         =  P.fromInteger    (*)                 =  (P.*)
src/Algebra/ToInteger.hs view
@@ -40,15 +40,16 @@ Conversions must be lossless, that is, they do not round in any way. For rounding see "Algebra.RealRing".-With the instances for 'Prelude.Float' and 'Prelude.Double'-we acknowledge that these types actually represent rationals-rather than (approximated) real numbers.-However, this contradicts to the 'Algebra.Transcendental.C' instance.++I think that the RealIntegral superclass is too restrictive.+Non-negative numbers are not a ring,+but can be easily converted to Integers. -} class (ToRational.C a, RealIntegral.C a) => C a where    toInteger :: a -> Integer  +{-# NOINLINE [2] fromIntegral #-} fromIntegral :: (C a, Ring.C b) => a -> b fromIntegral = fromInteger . toInteger
src/Algebra/ToRational.hs view
@@ -1,6 +1,7 @@ {-# LANGUAGE RebindableSyntax #-} module Algebra.ToRational where +import qualified Algebra.ZeroTestable as ZeroTestable import qualified Algebra.Field    as Field import qualified Algebra.Absolute as Absolute import Algebra.Field (fromRational, )@@ -29,7 +30,7 @@  >  fromRational' . toRational === id -}-class (Absolute.C a) => C a where+class (Absolute.C a, ZeroTestable.C a, Ord a) => C a where    -- | Lossless conversion from any representation of a rational to 'Rational'    toRational :: a -> Rational @@ -67,6 +68,7 @@ such as converting 'Float' to 'Double'. This achieved by optimizer rules. -}+{-# NOINLINE [2] realToField #-} realToField :: (C a, Field.C b) => a -> b realToField = Field.fromRational' . toRational
src/Algebra/Transcendental.hs view
@@ -2,8 +2,6 @@ module Algebra.Transcendental where  import qualified Algebra.Algebraic as Algebraic--- import qualified Algebra.Ring      as Ring--- import qualified Algebra.Additive  as Additive  import qualified Algebra.Laws as Laws @@ -31,9 +29,10 @@ branch cuts, etc.  Minimal complete definition:-     pi, exp, log, sin, cos, asin, acos, atan+     pi, exp, (log or logBase), sin, cos, atan -} class (Algebraic.C a) => C a where+    {-# MINIMAL pi, exp, (log | logBase), sin, cos, atan #-}     pi                  :: a     exp, log            :: a -> a     logBase, (**)       :: a -> a -> a@@ -56,6 +55,7 @@      x ** y           =  exp (log x * y)     logBase x y      =  log y / log x+    log              =  logBase (exp 1)      tan  x           =  sin x / cos x
src/Algebra/Units.hs view
@@ -22,7 +22,6 @@  import qualified Algebra.IntegralDomain as Integral import qualified Algebra.Ring           as Ring--- import qualified Algebra.Additive       as Additive import qualified Algebra.ZeroTestable   as ZeroTestable  import qualified Algebra.Laws           as Laws@@ -70,6 +69,7 @@ -}  class (Integral.C a) => C a where+  {-# MINIMAL isUnit, (stdUnit | stdUnitInv) #-}   isUnit :: a -> Bool   stdAssociate, stdUnit, stdUnitInv :: a -> a
src/Algebra/Vector.hs view
@@ -18,7 +18,6 @@ import Algebra.Additive ((+))  import Data.List (zipWith, foldl)--- import Data.Functor (Functor, fmap)  import Prelude((.), (==), Bool, Functor, fmap) import qualified Prelude as P
src/Algebra/VectorSpace.hs view
@@ -8,7 +8,8 @@ import qualified Algebra.PrincipalIdealDomain as PID import qualified Number.Ratio   as Ratio --- import NumericPrelude.Numeric+import qualified Data.Complex as Complex98+ import qualified Prelude as P  @@ -32,3 +33,5 @@ instance (C a b) => C a [b]  instance (C a b) => C a (c -> b)++instance (C a b, P.RealFloat b) => C a (Complex98.Complex b)
src/Algebra/ZeroTestable.hs view
@@ -6,7 +6,6 @@ import Data.Int  (Int,  Int8,  Int16,  Int32,  Int64,  ) import Data.Word (Word, Word8, Word16, Word32, Word64, ) --- import qualified Prelude as P import Prelude (Integer, Float, Double, ) import NumericPrelude.Base
src/MathObj/DiscreteMap.hs view
@@ -41,7 +41,6 @@ import qualified Data.Map as Map import Data.Map (Map) --- import qualified Prelude as P import NumericPrelude.Base  -- FIXME: Should this be implemented by isZero?
− src/MathObj/Gaussian/Bell.hs 
@@ -1,314 +0,0 @@-{-# LANGUAGE RebindableSyntax #-}-{--Complex translated Gaussian bell curve-with amplitude abstracted away.--}-module MathObj.Gaussian.Bell where--import qualified MathObj.Polynomial as Poly-import qualified Number.Complex as Complex--import qualified Algebra.Transcendental as Trans-import qualified Algebra.Field          as Field-import qualified Algebra.Absolute       as Absolute-import qualified Algebra.Ring           as Ring-import qualified Algebra.Additive       as Additive--import Number.Complex ((+:), )--import Test.QuickCheck (Arbitrary, arbitrary, )-import Control.Monad (liftM4, )---- import Prelude (($))-import NumericPrelude.Numeric-import NumericPrelude.Base hiding (reverse, )---data T a = Cons {amp :: a, c0, c1 :: Complex.T a, c2 :: a}-   deriving (Eq, Show)--instance (Absolute.C a, Arbitrary a) => Arbitrary (T a) where-   arbitrary =-      liftM4-         (\k a b c -> Cons (abs k) a b (1 + abs c))-         arbitrary arbitrary arbitrary arbitrary---constant :: Ring.C a => T a-constant = Cons one zero zero zero--{- |-eigenfunction of 'fourier'--}-unit :: Ring.C a => T a-unit = Cons one zero zero one--{-# INLINE evaluate #-}-evaluate :: (Trans.C a) =>-   T a -> a -> Complex.T a-evaluate f x =-   Complex.scale-     (sqrt (amp f))-     (Complex.exp $ Complex.scale (-pi) $-      c0 f + Complex.scale x (c1 f) + Complex.fromReal (c2 f * x^2))--evaluateSqRt :: (Trans.C a) =>-   T a -> a -> Complex.T a-evaluateSqRt f x0 =-   Complex.scale-     (sqrt (amp f))-     (let x = sqrt pi * x0-      in  Complex.exp $ negate $-          c0 f + Complex.scale x (c1 f) + Complex.fromReal (c2 f * x^2))--exponentPolynomial :: (Additive.C a) =>-   T a -> Poly.T (Complex.T a)-exponentPolynomial f =-   Poly.fromCoeffs [c0 f, c1 f, Complex.fromReal (c2 f)]---variance :: (Trans.C a) =>-   T a -> a-variance f =-   recip $ c2 f * 2*pi--multiply :: (Ring.C a) =>-   T a -> T a -> T a-multiply f g =-   Cons-      (amp f * amp g)-      (c0 f + c0 g) (c1 f + c1 g) (c2 f + c2 g)--powerRing :: (Trans.C a) =>-   Integer -> T a -> T a-powerRing p f =-   let pa = fromInteger p-   in  Cons-          (amp f ^ p)-          (pa * c0 f) (pa * c1 f) (fromInteger p * c2 f)--{--powerField does not makes sense,-since the reciprocal of a Gaussian diverges.--}--powerAlgebraic :: (Trans.C a) =>-   Rational -> T a -> T a-powerAlgebraic p f =-   let pa = fromRational' p-   in  Cons-          (amp f ^/ p)-          (pa * c0 f) (pa * c1 f) (fromRational' p * c2 f)--powerTranscendental :: (Trans.C a) =>-   a -> T a -> T a-powerTranscendental p f =-   Cons-      (amp f ^? p)-      (Complex.scale p $ c0 f) (Complex.scale p $ c1 f) (p * c2 f)---{--let x=Cons 2 (1+:3) (4+:5) (7::Rational); y=Cons 7 (1+:4) (3+:2) (5::Rational)--}-convolve :: (Field.C a) =>-   T a -> T a -> T a-convolve f g =-   let s = c2 f + c2 g-       {--       fd = f1/(2*f2)-       gd = g1/(2*g2)-       c = f2*g2/(f2+g2)--       c*(fd+gd) = (f1*g2+f2*g1)/(2*(f2+g2)) = b/2--       c*(fd+gd)^2 - fd^2*f2 - gd^2*g2-         = f2*g2*(fd+gd)^2/(f2 + g2) - (fd^2*f2 + gd^2*g2)-         = (f2*g2*(fd+gd)^2 - (f2+g2)*(fd^2*f2+gd^2*g2)) / (f2 + g2)-         = (2*f2*g2*fd*gd - (fd^2*f2^2+gd^2*g2^2)) / (f2 + g2)-         = (2*f1*g1 - (f1^2+g1^2)) / (4*(f2 + g2))-         = -(f1 - g1)^2/(4*(f2 + g2))-       -}-   in  Cons-          (amp f * amp g / s)-          (c0 f + c0 g-              - Complex.scale (recip (4*s)) ((c1 f - c1 g)^2))-          (Complex.scale (c2 g / s) (c1 f) +-           Complex.scale (c2 f / s) (c1 g))-          (c2 f * c2 g / s)-            -- recip $ recip (c2 f) + recip (c2 g)-{--   Cons-      (c0 f + c0 g) (c1 f + c1 g)-      (recip $ recip (c2 f) + recip (c2 g))--}--convolveByTranslation :: (Field.C a) =>-   T a -> T a -> T a-convolveByTranslation f0 g0 =-   let fd = Complex.scale (recip (2 * c2 f0)) $ c1 f0-       gd = Complex.scale (recip (2 * c2 g0)) $ c1 g0-       f1 = translateComplex fd f0-       g1 = translateComplex gd g0-       s = c2 f1 + c2 g1-   in  translateComplex (negate $ fd + gd) $-       Cons-          (amp f1 * amp g1 / s)-          (c0 f1 + c0 g1) zero-          (c2 f1 * c2 g1 / s)--convolveByFourier :: (Field.C a) =>-   T a -> T a -> T a-convolveByFourier f g =-   reverse $ fourier $ multiply (fourier f) (fourier g)--fourier :: (Field.C a) =>-   T a -> T a-fourier f =-   let a = c0 f-       b = c1 f-       rc = recip $ c2 f-   in  Cons-          (amp f * rc)-          (Complex.scale (rc/4) (-b^2) + a)-          (Complex.scale rc $ Complex.quarterRight b)-          rc--fourierByTranslation :: (Field.C a) =>-   T a -> T a-fourierByTranslation f =-   translateComplex (Complex.scale (1/2) $ Complex.quarterLeft $ c1 f) $-   Cons (amp f / c2 f) (c0 f) zero (recip $ c2 f)--{--a + b*x + c*x^2- = c*(a/c + b/c*x + x^2)- = c*((x-b/(2*c))^2 + (4*a*c+b^2)/(4*c^2))- = c*(x-b/(2*c))^2 + (4*a*c+b^2)/(4*c)--fourier ->-   x^2/c - i*b/c*x + (4*a*c+b^2)/(4*c)--fourier (x -> exp(-pi*c*(x-t)^2))- = fourier $ translate t $ shrink (sqrt c) $ x -> exp(-pi*x^2)- = modulate t $ dilate (sqrt c) $ fourier $ x -> exp(-pi*x^2)- = modulate t $ dilate (sqrt c) $ x -> exp(-pi*x^2)- = modulate t $ x -> exp(-pi*x^2/c)- = x -> exp(-pi*x^2/c) * exp(-2*pi*i*x*t)- = x -> exp(-pi*(x^2/c - 2*i*x*t))--}--{--b*x + c*x^2- = c*(b/c*x + x^2)- = c*((x-br/(2*c))^2 + i*x*bi/c - br^2/(4*c^2))- = c*(x-br/(2*c))^2 + i*x*bi - br^2/(4*c)--fourier ->-   (x+bi/2)^2/c - i*br/c*(x+bi/2) - br^2/(4*c)- = (1/c) * ((x+bi/2)^2 - i*br*(x+bi/2) + (br/2)^2)- = (1/c) * (x^2 - i*b*x + -(br/2)^2 + (bi/2)^2 - i*br*bi/2)- = (1/c) * (x^2 - i*b*x - (br^2-bi^2+2*br*bi*i)^2 /4)- = (1/c) * (x^2 - i*b*x - b^2 / 4)- = (1/c) * (x^2 - i*b*x + (i*b/2)^2)- = (1/c) * (x - i*b/2)^2--Example:-  (x-b)^2 = b^2 - 2*b*x + x^2-    ->  (- i*2*b*x + x^2)---fourier (x -> exp(-pi*(c*(x-t)^2 + 2*i*m*x)))- = fourier $ modulate m $ translate t $ shrink (sqrt c) $ x -> exp(-pi*x^2)- = translate (-m) $ modulate t $ dilate (sqrt c) $ fourier $ x -> exp(-pi*x^2)- = translate (-m) $ modulate t $ dilate (sqrt c) $ x -> exp(-pi*x^2)- = translate (-m) $ modulate t $ x -> exp(-pi*x^2/c)- = translate (-m) $ x -> exp(-pi*x^2/c) * exp(-2*pi*i*x*t)- = x -> exp(-pi*(x+m)^2/c) * exp(-2*pi*i*(x+m)*t)- = x -> exp(-pi*((x+m)^2/c - 2*i*(x+m)*t))--}--{--fourier (Cons a 0 0) =-  Cons a 0 infinity--fourier (Cons 0 0 c) =-  Cons 0 0 (recip c)--fourier (Cons 0 b 1) =-  Cons 0 (i*b) 1--}--translate :: Ring.C a => a -> T a -> T a-translate d f =-   let a = c0 f-       b = c1 f-       c = c2 f-   in  Cons-          (amp f)-          (Complex.fromReal (c*d^2) - Complex.scale d b + a)-          (Complex.fromReal (-2*c*d) + b)-          c--translateComplex :: Ring.C a => Complex.T a -> T a -> T a-translateComplex d f =-   let a = c0 f-       b = c1 f-       c = c2 f-   in  Cons-          (amp f)-          (Complex.scale c (d^2) - b*d + a)-          (Complex.scale (-2*c) d + b)-          c--modulate :: Ring.C a => a -> T a -> T a-modulate d f =-   Cons-      (amp f)-      (c0 f)-      (c1 f + (zero +: 2*d))-      (c2 f)--turn :: Ring.C a => a -> T a -> T a-turn d f =-   Cons-      (amp f)-      (c0 f + (zero +: 2*d))-      (c1 f)-      (c2 f)--reverse :: Additive.C a => T a -> T a-reverse f =-   f{c1 = negate $ c1 f}---dilate :: Field.C a => a -> T a -> T a-dilate k f =-   Cons-      (amp f)-      (c0 f)-      (Complex.scale (recip k) $ c1 f)-      (c2 f / k^2)--shrink :: Ring.C a => a -> T a -> T a-shrink k f =-   Cons-      (amp f)-      (c0 f)-      (Complex.scale k $ c1 f)-      (c2 f * k^2)--amplify :: (Ring.C a) => a -> T a -> T a-amplify k f =-   Cons-      (k^2 * amp f)-      (c0 f)-      (c1 f)-      (c2 f)---{- laws-fourier (convolve f g) = fourier f * fourier g--fourier (fourier f) = reverse f--}
− src/MathObj/Gaussian/Example.hs 
@@ -1,227 +0,0 @@-{-# LANGUAGE RebindableSyntax #-}-{--Reciprocal of variance of a Gaussian bell curve.-We describe the curve only in terms of its variance-thus we represent a bell curve at the coordinate origin-neglecting its amplitude.--We could also define the amplitude as @root 4 c@,-thus preserving L2 norm being one,-but then @dilate@ and @shrink@ also include an amplification.--We could do some projective geometry in the exponent-in order to also have zero variance,-which corresponds to the dirac impulse.--}-module MathObj.Gaussian.Example where--import qualified MathObj.Gaussian.Polynomial as PolyBell-import qualified MathObj.Gaussian.Bell as Bell-import qualified MathObj.Gaussian.Variance as Var--import qualified MathObj.Polynomial as Poly--import qualified Algebra.Transcendental as Trans-import qualified Algebra.Algebraic      as Algebraic-import qualified Algebra.Field          as Field--- import qualified Algebra.Absolute           as Absolute-import qualified Algebra.Ring           as Ring--- import qualified Algebra.Additive       as Additive--import qualified Number.Complex as Complex--import Algebra.Transcendental (pi, )-import Algebra.Algebraic (root, )-import Algebra.Ring ((*), (^), )--import Number.Complex ((+:), )--import qualified Numerics.Function as Func-import qualified Numerics.Fourier as Fourier-import qualified Numerics.Integration as Integ-import qualified Numerics.Differentiation as Diff--import qualified Graphics.Gnuplot.Simple as GP--import Control.Applicative (liftA2, )---- import System.Exit (ExitCode, )---- import Prelude (($))-import NumericPrelude.Numeric-import NumericPrelude.Base-import qualified Prelude as P---curve0 :: Var.T Double-curve0 = curve0a--curve0a :: Var.T Double-curve0a = Var.Cons 1.4 3.3--curve0b :: Var.T Double-curve0b = Var.Cons 2.2 1.7--variance0 :: (Double, Double)-variance0 =-   (Var.variance curve0,-    (Integ.rectangular 1000 (-2,2) $ liftA2 (*) (^2) (Var.evaluate curve0)) /-    (Integ.rectangular 1000 (-2,2) $ Var.evaluate curve0))--norm10 :: (Double, Double)-norm10 =-   (Integ.rectangular 1000 (-2,2) $ Var.evaluate curve0,-    Var.norm1 curve0)--norm20 :: (Double, Double)-norm20 =-   (sqrt $ Integ.rectangular 1000 (-2,2) $ (^2) . Var.evaluate curve0,-    Var.norm2 curve0)--norm30 :: (Double, Double)-norm30 =-   (root 3 $ Integ.rectangular 1000 (-2,2) $ (^3) . Var.evaluate curve0,-    Var.normP 3 curve0)--fourier0 :: IO ()-fourier0 =-   GP.plotFuncs []-      (GP.linearScale 100 (-2,2))-      [Var.evaluate $ Var.fourier curve0,-       Fourier.analysisTransformOneReal 100 (-2,2) $ Var.evaluate curve0]--multiply0 :: IO ()-multiply0 =-   GP.plotFuncs []-      (GP.linearScale 100 (-1,1))-      [Var.evaluate $ Var.multiply curve0a curve0b,-       liftA2 (*) (Var.evaluate curve0a) (Var.evaluate curve0b)]--convolve0 :: IO ()-convolve0 =-   GP.plotFuncs []-      (GP.linearScale 100 (-2,2))-      [Var.evaluate $ Var.convolve curve0a curve0b,-       Integ.convolve 1000 (-3,3) (Var.evaluate curve0a) (Var.evaluate curve0b)]---curve1 :: Bell.T Double-curve1 = curve1a--curve1a :: Bell.T Double-curve1a = Bell.Cons 1.4 (0.1+:0.3) ((-0.2)+:1.4) 2.3--curve1b :: Bell.T Double-curve1b = Bell.Cons 2.2 ((-0.3)+:2.1) (0.2+:(-0.4)) 1.7--variance1 :: (Double, Double)-variance1 =-   (Bell.variance curve1,-    (Integ.rectangular 1000 (-2,2) $-        liftA2 (*) (^2)-           (Complex.magnitudeSqr .-            Func.translateRight-               (Complex.real (Bell.c1 curve1) / (2 * Bell.c2 curve1))-               (Bell.evaluate curve1))) /-    (Integ.rectangular 1000 (-2,2) $ Complex.magnitude . Bell.evaluate curve1))--{- the norm depends on too much things-norm0vs1 :: (Double, Double)-norm0vs1 =-   ((Integ.rectangular 1000 (-5,5) $ Var.evaluate curve0)-         * exp (- Complex.real (Bell.c0 curve1)),-    Integ.rectangular 1000 (-5,5) $ Complex.magnitude . Bell.evaluate curve1)--}--fourier1 :: IO ()-fourier1 =-   GP.plotFuncs []-      (GP.linearScale 100 (-5,5))-      [Complex.real . (Bell.evaluate $ Bell.fourier curve1),-       fourierAnalysisReal 100 (-2,2) $ Bell.evaluate curve1]---curve2 :: PolyBell.T Double-curve2 =-   PolyBell.Cons---      Bell.unit---      (Bell.Cons 1.4 (0.1+:0.3) 0 1.2)---      (Bell.Cons 1.4 (0.1+:0.3) ((-0.2)+:1.4) 1)-      curve1---      (Poly.fromCoeffs [one])---      (Poly.fromCoeffs [zero,one])---      (Poly.fromCoeffs [zero,zero,one])---      (Poly.fromCoeffs [0,Complex.imaginaryUnit])-      (Poly.fromCoeffs [1.4+:(-0.1),0.8+:(0.1),(-1.1)+:0.3])--differentiate2 :: IO ()-differentiate2 =-   GP.plotFuncs []-      (GP.linearScale 100 (-2,2))-      [Complex.real . (PolyBell.evaluateSqRt $ PolyBell.differentiate curve2),-       ((/ sqrt pi) . ) $ Diff.diff (1e-5) $ Complex.real . PolyBell.evaluateSqRt curve2]--fourier2 :: IO ()-fourier2 =-   GP.plotFuncs []-      (GP.linearScale 100 (-5,5))-      [Complex.real . (PolyBell.evaluateSqRt $ PolyBell.fourier curve2),-       fourierAnalysisReal 100 (-2,2) $ PolyBell.evaluateSqRt curve2]----fourierAnalysisReal ::-   (P.Floating a) =>-   Integer -> (a, a) -> (a -> Complex.T a) -> a -> a-fourierAnalysisReal n rng f =-   liftA2 (P.-)-      (Fourier.analysisTransformOneReal n rng (Complex.real . f))-      (Fourier.analysisTransformOneImag n rng (Complex.imag . f))---{- |-Try to approximate @\x -> exp (-x^2) * x@-by a difference of translated Gaussian bells.--exp(-x^2) * x-  ==  exp(-(a+b*x+c*x^2)) - exp(-(a-b*x+c*x^2))-  ==  exp(-(a+c*x^2)) * (exp(-b*x) - exp(b*x))-  ==  exp(-(a+c*x^2)) * 2*sinh (b*x)--It holds-  lim (\b x -> sinh (b*x) / b)  =  id--}-diffApprox :: IO ()-diffApprox =-   let amp = (2*b)^- (-2)-       a = 0-       {--       amp = 1-       a = log (2 * abs b)-       -}-       b = -0.1-       c = 1-       ac = Complex.fromReal a-       bc = Complex.fromReal b-   in  GP.plotFuncs []-          (GP.linearScale 100 (-2,2::Double))-          [Complex.real .-           (PolyBell.evaluateSqRt $-              PolyBell.Cons Bell.unit (Poly.fromCoeffs [zero,one])),-           Complex.real .-           liftA2 (-)-             (PolyBell.evaluateSqRt $-                PolyBell.Cons (Bell.Cons amp ac bc c) (Poly.fromCoeffs [one]))-             (PolyBell.evaluateSqRt $-                PolyBell.Cons (Bell.Cons amp ac (-bc) c) (Poly.fromCoeffs [one]))]---polyApprox :: IO ()-polyApprox =-   GP.plotFuncs []-      (GP.linearScale 100 (-2,2::Double))-      [Complex.real .-         PolyBell.evaluateSqRt curve2,-       Complex.real . sum .-         mapM (\(amp,b) -> \x -> amp * Bell.evaluateSqRt b x)-         (PolyBell.approximateByBells 0.1 curve2)]
− src/MathObj/Gaussian/Polynomial.hs 
@@ -1,435 +0,0 @@-{-# LANGUAGE RebindableSyntax #-}-{--Complex Gaussian bell multiplied with a polynomial.--In order to make this free of @pi@ factors,-we have to choose @recip (sqrt pi)@-as unit for translations and modulations,-for linear factors and in the differentiation.--}-{--ToDo:--* In order to avoid the weird @sqrt pi@ factor,-  use a polynomial expression in @pi@.--* sum of multiple bells using Data.Map from exponent polynomial to coefficient polynomial-  use of Algebra object.--* Projective geometry in order to support Dirac impulse.--}-module MathObj.Gaussian.Polynomial where--import qualified MathObj.Gaussian.Bell as Bell--import qualified MathObj.LaurentPolynomial as LPoly-import qualified MathObj.Polynomial.Core   as PolyCore-import qualified MathObj.Polynomial        as Poly-import qualified Number.Complex     as Complex--import qualified Algebra.ZeroTestable   as ZeroTestable-import qualified Algebra.Differential   as Differential-import qualified Algebra.Transcendental as Trans-import qualified Algebra.Field          as Field-import qualified Algebra.Absolute           as Absolute-import qualified Algebra.Ring           as Ring-import qualified Algebra.Additive       as Additive--import qualified Data.Record.HT as Rec-import qualified Data.List as List-import Data.Function.HT (nest, )-import Data.Eq.HT (equating, )-import Data.List.HT (mapAdjacent, )-import Data.Tuple.HT (forcePair, )--import Test.QuickCheck (Arbitrary, arbitrary, )-import Control.Monad (liftM2, )--import NumericPrelude.Numeric-import NumericPrelude.Base hiding (reverse, )--- import Prelude ()---data T a = Cons {bell :: Bell.T a, polynomial :: Poly.T (Complex.T a)}-   deriving (Show)--instance (Absolute.C a, Eq a) => Eq (T a) where-   (==) = equal---{--Helper data type for 'equal',-that allows to call the (not quite trivial) polynomial equality check.-@RootProduct r a@ represents @sqrt r * a@.-The test using 'signum' works for real numbers,-and I do not know, whether it is correct for other mathematical objects.-However I cannot imagine other mathematical objects,-that make sense at all, here.-Maybe elements of a finite field.--}-data RootProduct a = RootProduct a a--instance (Absolute.C a, Eq a) => Eq (RootProduct a) where-   (RootProduct xr xa) == (RootProduct yr ya)  =-      let xp = xr*xa^2-          yp = yr*ya^2-      in  xp==yp &&-          (isZero xp || signum xa == signum ya)--instance (ZeroTestable.C a) => ZeroTestable.C (RootProduct a) where-   isZero (RootProduct r a) = isZero r || isZero a---{--The derived Eq is not correct.-We have to combine the amplitude of the bell with the polynomial,-respecting signs and the square root of the bell amplitude.--}-equal :: (Absolute.C a, Eq a) => T a -> T a -> Bool-equal x y =-   let bx = bell x-       by = bell y-       scaleSqr b =-          (\p ->-              (fmap (RootProduct (Bell.amp b) . Complex.real) p,-               fmap (RootProduct (Bell.amp b) . Complex.imag) p))-           . polynomial-   in  Rec.equal-          (equating Bell.c0 :-           equating Bell.c1 :-           equating Bell.c2 :-           [])-          bx by-       &&-       scaleSqr bx x == scaleSqr by y---instance (Absolute.C a, Arbitrary a) => Arbitrary (T a) where-   arbitrary =---      liftM2 Cons arbitrary arbitrary-      liftM2 Cons-         arbitrary-         -- we have to restrict the number of polynomial coefficients,-         -- since with the quadratic time algorithms like fourier and convolve,-         -- in connection with Rational slow down tests too much.-         (fmap (Poly.fromCoeffs . take 5 . Poly.coeffs) arbitrary)----{-# INLINE evaluateSqRt #-}-evaluateSqRt :: (Trans.C a) =>-   T a -> a -> Complex.T a-evaluateSqRt f x =-   Bell.evaluateSqRt (bell f) x *-   Poly.evaluate (polynomial f) (Complex.fromReal $ sqrt pi * x)-{- ToDo: evaluating a complex polynomial for a real argument can be optimized -}---constant :: (Ring.C a) => T a-constant =-   Cons Bell.constant (Poly.const one)--scale :: (Ring.C a) => a -> T a -> T a-scale x f =-   f{polynomial = fmap (Complex.scale x) $ polynomial f}--scaleComplex :: (Ring.C a) => Complex.T a -> T a -> T a-scaleComplex x f =-   f{polynomial = fmap (x*) $ polynomial f}---eigenfunction :: (Field.C a) => Int -> T a-eigenfunction =-   eigenfunctionDifferential--eigenfunction0 :: (Ring.C a) => T a-eigenfunction0 =-   Cons Bell.unit (Poly.fromCoeffs [one])--eigenfunction1 :: (Ring.C a) => T a-eigenfunction1 =-   Cons Bell.unit (Poly.fromCoeffs [zero, one])--eigenfunction2 :: (Field.C a) => T a-eigenfunction2 =-   Cons Bell.unit (Poly.fromCoeffs [-(1/4), zero, one])--eigenfunction3 :: (Field.C a) => T a-eigenfunction3 =-   Cons Bell.unit (Poly.fromCoeffs [zero, -(3/4), zero, one])---eigenfunctionDifferential :: (Field.C a) => Int -> T a-eigenfunctionDifferential n =-   (\f -> f{bell = Bell.unit}) $-   nest n (scale (-1/4) . differentiate) $-   Cons (Bell.Cons one zero zero 2) one--eigenfunctionIterative :: (Field.C a, Absolute.C a, Eq a) => Int -> T a-eigenfunctionIterative n =-   fst . head . dropWhile (uncurry (/=)) . mapAdjacent (,) $-   eigenfunctionIteration $-   Cons-      Bell.unit-      (Poly.fromCoeffs $ replicate n zero ++ [one])--eigenfunctionIteration :: (Field.C a) => T a -> [T a]-eigenfunctionIteration =-   iterate (\x ->-      let y = fourier x-          px = polynomial x-          py = polynomial y-          c = last (Poly.coeffs px) / last (Poly.coeffs py)-      in  y{polynomial = fmap (0.5*) (px + fmap (c*) py)})---multiply :: (Ring.C a) =>-   T a -> T a -> T a-multiply f g =-   Cons-      (Bell.multiply (bell f) (bell g))-      (polynomial f * polynomial g)--convolve, {- convolveByDifferentiation, -} convolveByFourier :: (Field.C a) =>-   T a -> T a -> T a-convolve = convolveByFourier--{--f <*> g =-   let (foff,fint) = integrate f-   in  fint <*> differentiate g + makeGaussPoly foff * g--In principle this would work,-but (makeGaussPoly foff * g) contains a lot of-convolutions of Gaussian with Gaussian-polynomial-product,-where the Gaussians have different parameters.--convolveByDifferentiation f g =-   case polynomial f of-      fpoly ->-         if null $ Poly.coeffs fpoly-           then ...-           else ...--}--convolveByFourier f g =-   reverse $ fourier $ multiply (fourier f) (fourier g)--{--We use a Horner like scheme-in order to translate multiplications with @id@-to differentations on the Fourier side.-Quadratic runtime.--fourier (Cons bell (Poly.const a + Poly.shift f))-  = fourier (Cons bell (Poly.const a)) + fourier (Cons bell (Poly.shift f))-  = fourier (Cons bell (Poly.const a)) + differentiate (fourier (Cons bell f))--}-fourier :: (Field.C a) =>-   T a -> T a-fourier f =-   foldr-      (\c p ->-          let q = differentiate p-          in  q{polynomial =-                   Poly.const c +-                   fmap (Complex.scale (1/2) . Complex.quarterLeft) (polynomial q)})-      (Cons (Bell.fourier $ bell f) zero) $-   Poly.coeffs $ polynomial f--{- |-Differentiate and divide by @sqrt pi@ in order to stay in a ring.-This way, we do not need to fiddle with pi factors.--}-differentiate :: (Ring.C a) => T a -> T a-differentiate f =-   f{polynomial =-        Differential.differentiate (polynomial f)-        - Differential.differentiate (Bell.exponentPolynomial (bell f))-           * polynomial f}--{--snd $ integrate $ differentiate (Cons Bell.unit (Poly.fromCoeffs [7,7,7,7]) :: T Double)--g = (bell f * poly f)'-  = bell f * ((poly f)' - (exppoly (bell f))' * poly f)-poly g = (poly f)' - (exppoly (bell f))' * poly f--Integration means we have g and ask for f.--poly f = ((poly f)' - poly g) / (exppoly (bell f))'--However must start with the highest term of 'poly f',-and thus we need to perform the division on reversed polynomials.--}-integrate ::-   (Field.C a, ZeroTestable.C a) =>-   T a -> (Complex.T a, T a)-integrate f =-   let fs = Poly.coeffs $ polynomial f-       (ys,~[r]) =-          PolyCore.divModRev-             {--             We need the shortening convention of 'zipWith'-             in order to limit the result list,-             we cannot use list instance for (-).-             -}-             (zipWith (-)-                (0 : 0 : diffRev ys)-                (List.reverse fs))-             (List.reverse $ Poly.coeffs $-              Differential.differentiate $-              Bell.exponentPolynomial $ bell f)-   in  forcePair $-       if null fs-         then (zero, f)-         else (r, f{polynomial = Poly.fromCoeffs $ List.reverse ys})--diffRev :: Ring.C a => [a] -> [a]-diffRev xs =-   zipWith (*) xs-      (drop 1 (iterate (subtract 1) (fromIntegral $ length xs)))--translate :: Ring.C a => a -> T a -> T a-translate d =-   translateComplex (Complex.fromReal d)--translateComplex :: Ring.C a => Complex.T a -> T a -> T a-translateComplex d f =-   Cons-      (Bell.translateComplex d $ bell f)-      (Poly.translate d $ polynomial f)--modulate :: Ring.C a => a -> T a -> T a-modulate d f =-   Cons-      (Bell.modulate d $ bell f)-      (polynomial f)--turn :: Ring.C a => a -> T a -> T a-turn d f =-   Cons-      (Bell.turn d $ bell f)-      (polynomial f)--reverse :: Additive.C a => T a -> T a-reverse f =-   Cons-      (Bell.reverse $ bell f)-      (Poly.reverse $ polynomial f)--dilate :: Field.C a => a -> T a -> T a-dilate k f =-   Cons-      (Bell.dilate k $ bell f)-      (Poly.dilate (Complex.fromReal k) $ polynomial f)--shrink :: Ring.C a => a -> T a -> T a-shrink k f =-   Cons-      (Bell.shrink k $ bell f)-      (Poly.shrink (Complex.fromReal k) $ polynomial f)--{--We could also amplify the polynomial coefficients.--}-amplify :: Ring.C a => a -> T a -> T a-amplify k f =-   Cons-      (Bell.amplify k $ bell f)-      (polynomial f)---{- |-Approximate a @T a@ using a linear combination of translated @Bell.T a@.-The smaller the unit (e.g. 0.1, 0.01, 0.001)-the better the approximation but the worse the numeric properties.--We cannot put all information into @amp@ of @Bell@,-since @amp@ must be real, but is complex here by construction.-We really need at least signed amplitudes at this place,-since we want to represent differences of Gaussians.--}-approximateByBells ::-   Field.C a =>-   a -> T a -> [(Complex.T a, Bell.T a)]-approximateByBells unit f =-   let b = bell f-       amps =-          -- approximateByBellsByTranslation-          approximateByBellsAtOnce-             unit-             (Complex.scale (recip (2 * Bell.c2 b)) (Bell.c1 b))-             (recip (2*unit*Bell.c2 b))-             (polynomial f)-   in  zip (LPoly.coeffs amps) $-       map-          (\d -> Bell.translate d b)-          (laurentAbscissas (unit/2) amps)--approximateByBellsAtOnce ::-   Field.C a =>-   a -> Complex.T a -> a -> Poly.T (Complex.T a) -> LPoly.T (Complex.T a)-approximateByBellsAtOnce unit d s p =-   foldr-      (\x amps0 ->-         {--         Decompose (bell t * (t-d)) = bell t * t - bell t * d-         -}-         let y = fmap (Complex.scale s) amps0-         in  -- \t -> bell t * t-             --    ~   (translate unit bell - translate (-unit) bell) / unit-             LPoly.shift 1 y --             LPoly.shift (-1) y +-             -- bell t * d-             zipWithAbscissas-                (\t z -> (Complex.fromReal t - d) * z)-                (unit/2) amps0 +-             LPoly.const x)-      (LPoly.fromCoeffs [])-      (Poly.coeffs p)--approximateByBellsByTranslation ::-   Field.C a =>-   a -> Complex.T a -> a -> Poly.T (Complex.T a) -> LPoly.T (Complex.T a)-approximateByBellsByTranslation unit d s p =-   foldr-      (\x amps0 ->-         {--         Decompose (bell t * (t-d)) = bell t * t - bell t * d-         -}-         let y = fmap (Complex.scale s) amps0-         in  -- \t -> bell t * t-             --    ~   (translate unit bell - translate (-unit) bell) / unit-             LPoly.shift 1 y --             LPoly.shift (-1) y +-             -- bell t * d-             zipWithAbscissas Complex.scale (unit/2) amps0 +-             LPoly.const x)-      (LPoly.fromCoeffs [])-      (Poly.coeffs $ Poly.translate d p)--zipWithAbscissas ::-   (Ring.C a) =>-   (a -> b -> c) -> a -> LPoly.T b -> LPoly.T c-zipWithAbscissas h unit y =-   LPoly.fromShiftCoeffs (LPoly.expon y) $-   zipWith h-      (laurentAbscissas unit y)-      (LPoly.coeffs y)--laurentAbscissas :: Ring.C a => a -> LPoly.T c -> [a]-laurentAbscissas unit =-   map (\d -> fromIntegral d * unit) .-   iterate (1+) . LPoly.expon---{- No Ring instance for Gaussians-instance (Ring.C a) => Differential.C (T a) where-   differentiate = differentiate--}--{- laws-differentiate (f*g) =-   (differentiate f) * g + f * (differentiate g)--}
− src/MathObj/Gaussian/Variance.hs 
@@ -1,194 +0,0 @@-{-# LANGUAGE RebindableSyntax #-}-{--We represent a Gaussian bell curve in terms of the reciprocal of its variance-and its value at the origin.--We could do some projective geometry in the exponent-in order to also have zero variance,-which corresponds to the dirac impulse.--}-module MathObj.Gaussian.Variance where--import qualified MathObj.Polynomial as Poly-import qualified Number.Root as Root--import qualified Algebra.Transcendental as Trans-import qualified Algebra.Algebraic      as Algebraic-import qualified Algebra.Field          as Field-import qualified Algebra.Absolute       as Absolute-import qualified Algebra.Ring           as Ring-import qualified Algebra.Additive       as Additive--{--import Algebra.Transcendental (pi, )-import Algebra.Ring ((*), (^), )-import Algebra.Additive ((+))--}-import Test.QuickCheck (Arbitrary, arbitrary, )-import Control.Monad (liftM2, )---- import Prelude (($))-import NumericPrelude.Numeric-import NumericPrelude.Base---{- |-Since @amp@ is the square of the actual amplitude it must be non-negative.--}-data T a = Cons {amp, c :: a}-   deriving (Eq, Show)--instance (Absolute.C a, Arbitrary a) => Arbitrary (T a) where-   arbitrary =-      liftM2 Cons-         (fmap abs arbitrary)-         (fmap ((1+) . abs) arbitrary)---constant :: Ring.C a => T a-constant = Cons one zero--{-# INLINE evaluate #-}-evaluate :: (Trans.C a) =>-   T a -> a -> a-evaluate f x =-   sqrt (amp f) * exp (-pi * c f * x^2)--exponentPolynomial :: (Additive.C a) =>-   T a -> Poly.T a-exponentPolynomial f =-   Poly.fromCoeffs [zero, zero, c f]---integrateRoot :: (Field.C a) => T a -> Root.T a-integrateRoot f =-   Root.sqrt $ Root.fromNumber $ amp f / c f--scalarProductRoot :: (Field.C a) => T a -> T a -> Root.T a-scalarProductRoot f g =-   integrateRoot (multiply f g)---norm1Root :: (Field.C a) => T a -> Root.T a-norm1Root = integrateRoot--norm2Root :: (Field.C a) => T a -> Root.T a-norm2Root f =-   Root.sqrt $-      Root.fromNumber (amp f)-      `Root.div`-      Root.sqrt (Root.fromNumber $ 2 * c f)--normInfRoot :: (Field.C a) => T a -> Root.T a-normInfRoot f =-   Root.sqrt $ Root.fromNumber $ amp f--normPRoot :: (Field.C a) => Rational -> T a -> Root.T a-normPRoot p f =-   Root.sqrt (Root.fromNumber (amp f))-   `Root.div`-   Root.rationalPower (recip (2*p)) (Root.fromNumber (fromRational' p * c f))---norm1 :: (Algebraic.C a) => T a -> a-norm1 f =-   sqrt $ amp f / c f--norm2 :: (Algebraic.C a) => T a -> a-norm2 f =-   sqrt $ amp f / (sqrt $ 2 * c f)--normInf :: (Algebraic.C a) => T a -> a-normInf f =-   sqrt (amp f)--normP :: (Trans.C a) => a -> T a -> a-normP p f =-   sqrt (amp f) * (p * c f) ^? (- recip (2*p))---variance :: (Trans.C a) =>-   T a -> a-variance f =-   recip $ c f * 2*pi--multiply :: (Ring.C a) =>-   T a -> T a -> T a-multiply f g =-   Cons (amp f * amp g) (c f + c g)--powerRing :: (Trans.C a) =>-   Integer -> T a -> T a-powerRing p f =-   Cons (amp f ^ p) (fromInteger p * c f)--{--powerField does not makes sense,-since the reciprocal of a Gaussian diverges.--}--powerAlgebraic :: (Trans.C a) =>-   Rational -> T a -> T a-powerAlgebraic p f =-   Cons (amp f ^/ p) (fromRational' p * c f)--powerTranscendental :: (Trans.C a) =>-   a -> T a -> T a-powerTranscendental p f =-   Cons (amp f ^? p) (p * c f)--{- |-> convolve x y t =->    integrate $ \s -> x s * y(t-s)--Convergence only for @c f + c g > 0@.--}-convolve :: (Field.C a) =>-   T a -> T a -> T a-convolve f g =-   let s = c f + c g-   in  Cons-          (amp f * amp g / s)-          (c f * c g / s)--{- |-> fourier x f =->    integrate $ \t -> x t * cis (-2*pi*t*f)--Convergence only for @c f > 0@.--}-fourier :: (Field.C a) =>-   T a -> T a-fourier f =-   Cons (amp f / c f) (recip $ c f)-{--fourier (t -> exp(-(a*t)^2))--}--dilate :: (Field.C a) => a -> T a -> T a-dilate k f =-   Cons (amp f) $ c f / k^2--shrink :: (Ring.C a) => a -> T a -> T a-shrink k f =-   Cons (amp f) $ c f * k^2--{- |-@amplify k@ scales by @abs k@!--}-amplify :: (Ring.C a) => a -> T a -> T a-amplify k f =-   Cons (k^2 * amp f) $ c f---{- laws-fourier (convolve f g) = multiply (fourier f) (fourier g)--dilate k (dilate m f) = dilate (k*m) f--dilate k (shrink k f) = f--variance (dilate k f) = k^2 * variance f--variance (convolve f g) = variance f + variance g--}
src/MathObj/LaurentPolynomial.hs view
@@ -26,9 +26,6 @@  import qualified Number.Complex as Complex -import Algebra.Module((*>))--import qualified NumericPrelude.Base as P import qualified NumericPrelude.Numeric as NP  import NumericPrelude.Base    hiding (const, reverse, )
src/MathObj/Matrix.hs view
@@ -18,7 +18,7 @@ but no additional parameters.  ToDo:- - Matrix inverse, determinant+ - Matrix inverse, determinant (see htam:Matrix) -}  module MathObj.Matrix (@@ -68,17 +68,56 @@ import NumericPrelude.Base hiding (zipWith, )  +{- $setup+>>> import qualified MathObj.Matrix as Matrix+>>> import qualified Algebra.Ring as Ring+>>> import qualified Algebra.Laws as Laws+>>> import Test.NumericPrelude.Utility ((/\))+>>> import qualified Test.QuickCheck as QC+>>> import NumericPrelude.Numeric as NP+>>> import NumericPrelude.Base as P+>>> import Prelude ()+>>>+>>> import Control.Monad (replicateM, join)+>>> import Control.Applicative (liftA2)+>>> import Data.Function.HT (nest)+>>>+>>> genDimension :: QC.Gen Int+>>> genDimension = QC.choose (0,20)+>>>+>>> genMatrixFor :: (QC.Arbitrary a) => Int -> Int -> QC.Gen (Matrix.T a)+>>> genMatrixFor m n =+>>>    fmap (Matrix.fromList m n) $ replicateM (m*n) QC.arbitrary+>>>+>>> genMatrix :: (QC.Arbitrary a) => QC.Gen (Matrix.T a)+>>> genMatrix = join $ liftA2 genMatrixFor genDimension genDimension+>>>+>>> genIntMatrix :: QC.Gen (Matrix.T Integer)+>>> genIntMatrix = genMatrix+>>>+>>> genFactorMatrix :: (QC.Arbitrary a) => Matrix.T a -> QC.Gen (Matrix.T a)+>>> genFactorMatrix a = genMatrixFor (Matrix.numColumns a) =<< genDimension+>>>+>>> genSameMatrix :: (QC.Arbitrary a) => Matrix.T a -> QC.Gen (Matrix.T a)+>>> genSameMatrix = uncurry genMatrixFor . Matrix.dimension+-}++ {- | A matrix is a twodimensional array, indexed by integers. -}-data T a =+newtype T a =    Cons (Array (Dimension, Dimension) a)-      deriving (Eq,Ord,Read)+      deriving (Eq, Ord, Read)  type Dimension = Int  {- | Transposition of matrices is just transposition in the sense of Data.List.++prop> genIntMatrix /\ \a -> Matrix.rows a == Matrix.columns (Matrix.transpose a)+prop> genIntMatrix /\ \a -> Matrix.columns a == Matrix.rows (Matrix.transpose a)+prop> genIntMatrix /\ \a -> genSameMatrix a /\ \b -> Laws.homomorphism Matrix.transpose (+) (+) a b -} transpose :: T a -> T a transpose (Cons m) =@@ -98,6 +137,9 @@ index :: T a -> Dimension -> Dimension -> a index (Cons m) i j = m ! (i,j) +{- |+prop> genIntMatrix /\ \a -> a == uncurry Matrix.fromRows (Matrix.dimension a) (Matrix.rows a)+-} fromRows :: Dimension -> Dimension -> [[a]] -> T a fromRows m n =    Cons .@@ -106,6 +148,9 @@    List.zipWith (\r -> map (\(c,x) -> ((r,c),x))) allIndices .    map (zip allIndices) +{- |+prop> genIntMatrix /\ \a -> a == uncurry Matrix.fromColumns (Matrix.dimension a) (Matrix.columns a)+-} fromColumns :: Dimension -> Dimension -> [[a]] -> T a fromColumns m n =    Cons .@@ -146,6 +191,10 @@  -- These implementations may benefit from a better exception than -- just assertions to validate dimensionalities+{- |+prop> genIntMatrix /\ \a -> genSameMatrix a /\ \b -> Laws.commutative (+) a b+prop> genIntMatrix /\ \a -> genSameMatrix a /\ \b -> genSameMatrix b /\ \c -> Laws.associative (+) a b c+-} instance (Additive.C a) => Additive.C (T a) where    (+) = zipWith (+)    (-) = zipWith (-)@@ -159,6 +208,9 @@    in  assert (d == dimension nM) $          uncurry fromList d (List.zipWith op em en) +{- |+prop> genIntMatrix /\ \a -> Laws.identity (+) (uncurry Matrix.zero $ Matrix.dimension a) a+-} zero :: (Additive.C a) => Dimension -> Dimension -> T a zero m n =    fromList m n $@@ -172,6 +224,9 @@       (indexBounds n n)       (map (\i -> ((i,i), Ring.one)) (indexRange n)) +{- |+prop> genDimension /\ \n -> Matrix.one n == Matrix.diagonal (replicate n Ring.one :: [Integer])+-} diagonal :: (Additive.C a) => [a] -> T a diagonal xs =    let n = List.length xs@@ -183,6 +238,15 @@ scale :: (Ring.C a) => a -> T a -> T a scale s = Vector.functorScale s +{- |+prop> genIntMatrix /\ \a -> Laws.leftIdentity  (*) (Matrix.one (Matrix.numRows a)) a+prop> genIntMatrix /\ \a -> Laws.rightIdentity (*) (Matrix.one (Matrix.numColumns a)) a+prop> genIntMatrix /\ \a -> genFactorMatrix a /\ \b -> Laws.homomorphism Matrix.transpose (*) (flip (*)) a b+prop> genIntMatrix /\ \a -> genFactorMatrix a /\ \b -> genFactorMatrix b /\ \c -> Laws.associative (*) a b c+prop> genIntMatrix /\ \b -> genSameMatrix b /\ \c -> genFactorMatrix b /\ \a -> Laws.leftDistributive (*) (+) a b c+prop> genIntMatrix /\ \a -> genFactorMatrix a /\ \b -> genSameMatrix b /\ \c -> Laws.rightDistributive (*) (+) a b c+prop> QC.choose (0,10) /\ \k -> genDimension /\ \n -> genMatrixFor n n /\ \a -> a^k == nest (fromInteger k) ((a::Matrix.T Integer)*) (Matrix.one n)+-} instance (Ring.C a) => Ring.C (T a) where    mM * nM =       assert (numColumns mM == numRows nM) $
src/MathObj/PartialFraction.hs view
@@ -29,21 +29,89 @@ import Algebra.Additive((+), zero, negate) import Algebra.ZeroTestable (isZero) +import qualified Data.List.Reverse.StrictSpine as Rev+import qualified Data.List.Match as Match import qualified Data.List as List--import Data.Map(Map) import qualified Data.Map as Map-import Data.Maybe(fromMaybe, )-import qualified Data.List.Match as Match-import Data.List.HT (dropWhileRev, )-import Data.List (group, sortBy, mapAccumR, )+import Data.Map (Map)+import Data.List (group, sortBy, mapAccumR)+import Data.Maybe (fromMaybe)  import NumericPrelude.Base hiding (zipWith)  import NumericPrelude.Numeric(Int, fromInteger)  +{- $setup+>>> import qualified MathObj.PartialFraction as PartialFraction+>>> import qualified MathObj.Polynomial.Core as PolyCore+>>> import qualified MathObj.Polynomial as Poly+>>> import qualified Algebra.PrincipalIdealDomain as PID+>>> import qualified Algebra.Indexable as Indexable+>>> import qualified Algebra.Laws as Laws+>>> import qualified Number.Ratio as Ratio+>>> import Test.NumericPrelude.Utility ((/\))+>>> import qualified Test.QuickCheck as QC+>>> import NumericPrelude.Numeric as NP+>>> import NumericPrelude.Base as P+>>> import Prelude ()+>>>+>>> import Control.Applicative (liftA2)+>>>+>>> {- |+>>> Generator of irreducible elements for tests.+>>> Choosing from a list of examples is a simple yet effective design.+>>> If we would construct irreducible elements by a clever algorithm+>>> we might obtain multiple primes only rarely.+>>> -} --+>>> genSmallPrime :: QC.Gen Integer+>>> genSmallPrime =+>>>    let primes = [2,3,5,7,11,13]+>>>    in  QC.elements (primes ++ map negate primes)+>>>+>>> genPartialFractionInt :: QC.Gen (PartialFraction.T Integer)+>>> genPartialFractionInt =+>>>    liftA2 PartialFraction.fromFactoredFraction+>>>       (QC.listOf genSmallPrime) QC.arbitrary+>>>+>>>+>>> genIrreduciblePolynomial :: QC.Gen (Poly.T Rational)+>>> genIrreduciblePolynomial = do+>>>    QC.NonZero unit <- QC.arbitrary+>>>    fmap (Poly.fromCoeffs . map (unit*)) $+>>>       QC.elements [[2,3],[2,0,1],[3,0,1],[1,-3,0,1]]+>>>+>>> genPartialFractionPoly :: QC.Gen (PartialFraction.T (Poly.T Rational))+>>> genPartialFractionPoly =+>>>    liftA2 PartialFraction.fromFactoredFraction+>>>       (fmap (take 3) $ QC.listOf genIrreduciblePolynomial)+>>>       (fmap (Poly.fromCoeffs . PolyCore.normalize . take 5) QC.arbitrary)+>>>+>>>+>>> fractionConv :: (PID.C a, Indexable.C a) => [a] -> a -> Bool+>>> fractionConv xs y =+>>>    PartialFraction.toFraction (PartialFraction.fromFactoredFraction xs y) ==+>>>    y % product xs+>>>+>>> fractionConvAlt :: (PID.C a, Indexable.C a) => [a] -> a -> Bool+>>> fractionConvAlt xs y =+>>>    PartialFraction.fromFactoredFraction xs y ==+>>>    PartialFraction.fromFactoredFractionAlt xs y+>>>+>>> scaleInt :: (PID.C a, Indexable.C a) => a -> PartialFraction.T a -> Bool+>>> scaleInt k a =+>>>    PartialFraction.toFraction (PartialFraction.scaleInt k a) ==+>>>    Ratio.scale k (PartialFraction.toFraction a)+>>>+>>> add, sub, mul ::+>>>    (PID.C a, Indexable.C a) =>+>>>    PartialFraction.T a -> PartialFraction.T a -> Bool+>>> add = Laws.homomorphism PartialFraction.toFraction (+) (+)+>>> sub = Laws.homomorphism PartialFraction.toFraction (-) (-)+>>> mul = Laws.homomorphism PartialFraction.toFraction (*) (*)+-} + {- | @Cons z (indexMapFromList [(x0,[y00,y01]), (x1,[y10]), (x2,[y20,y21,y22])])@ represents the partial fraction@@ -123,6 +191,9 @@ There are more direct methods for special cases like polynomials over rational numbers where the denominators are linear factors.++prop> QC.listOf genSmallPrime /\ fractionConv+prop> fmap (take 3) (QC.listOf genIrreduciblePolynomial) /\ fractionConv -} fromFactoredFraction :: (PID.C a, Indexable.C a) => [a] -> a -> T a fromFactoredFraction denoms0 numer0 =@@ -145,6 +216,10 @@        -- Is reduceHeads also necessary for polynomial partial fractions?    in  removeZeros $ reduceHeads $ Cons intPart (indexMapFromList pairs) +{- |+prop> QC.listOf genSmallPrime /\ fractionConvAlt+prop> fmap (take 3) (QC.listOf genIrreduciblePolynomial) /\ fractionConvAlt+-} fromFactoredFractionAlt :: (PID.C a, Indexable.C a) => [a] -> a -> T a fromFactoredFractionAlt denoms numer =    foldl (\p d -> scaleFrac (one%d) p) (fromValue numer) denoms@@ -205,9 +280,7 @@ -} removeZeros :: (Indexable.C a, ZeroTestable.C a) => T a -> T a removeZeros (Cons z m) =-   Cons z $-   Map.filter (not . null) $-   Map.map (dropWhileRev isZero) m+   Cons z $ Map.filter (not . null) $ Map.map (Rev.dropWhile isZero) m   {-@@ -220,7 +293,16 @@ zipWith opS opV (Cons za ma) (Cons zb mb) =    Cons (opS za zb) (Map.unionWith opV ma mb) -instance (Indexable.C a, Integral.C a, ZeroTestable.C a) => Additive.C (T a) where+{- |+prop> genPartialFractionInt /\ \x -> genPartialFractionInt /\ \y -> add x y+prop> genPartialFractionInt /\ \x -> genPartialFractionInt /\ \y -> sub x y++prop> genPartialFractionPoly /\ \x -> genPartialFractionPoly /\ \y -> add x y+prop> genPartialFractionPoly /\ \x -> genPartialFractionPoly /\ \y -> sub x y+-}+instance+   (Indexable.C a, Integral.C a, ZeroTestable.C a) =>+      Additive.C (T a) where    a + b = removeZeros $ normalizeModulo $ zipWith (+) (+) a b    {- This implementation is attracting but wrong.      It fails if terms are present in b that are missing in a.@@ -343,6 +425,10 @@              (uncurry (:) . carryRipple ds . map (ns*))              scaleFracs m) +{- |+prop> genPartialFractionInt /\ \x k -> scaleInt k x+prop> genPartialFractionPoly /\ \x k -> scaleInt k x+-} scaleInt :: (PID.C a, Indexable.C a) => a -> T a -> T a scaleInt x (Cons z m) =    removeZeros $ normalizeModulo $@@ -359,6 +445,10 @@                  scaleFrac (one%d) (scaleInt numer a + acc)) zero l)            (indexMapToList m)) +{- |+prop> genPartialFractionInt /\ \x -> genPartialFractionInt /\ \y -> mul x y+prop> genPartialFractionPoly /\ \x -> genPartialFractionPoly /\ \y -> mul x y+-} mulFast :: (PID.C a, Indexable.C a) => T a -> T a -> T a mulFast pa pb =    let ra = toFactoredFraction pa
src/MathObj/Permutation.hs view
@@ -16,8 +16,6 @@  import Data.Array(Ix) --- import NumericPrelude.Numeric (Integer)--- import NumericPrelude.Base   {- |
src/MathObj/Permutation/CycleList/Check.hs view
@@ -12,19 +12,12 @@ import qualified MathObj.Permutation.Table     as PermTable import qualified MathObj.Permutation           as Perm -{--import qualified Algebra.Ring as Ring-import qualified Algebra.Additive as Additive-import Algebra.Ring((*),one,fromInteger)-import Algebra.Additive((+))--}-import Algebra.Monoid((<*>)) import qualified Algebra.Monoid as Monoid+import Algebra.Monoid((<*>)) -import Data.Array((!), Ix) import qualified Data.Array as Array+import Data.Array((!), Ix) --- import NumericPrelude.Numeric (Integer) import NumericPrelude.Base hiding (cycle)  {- |
src/MathObj/Permutation/Table.hs view
@@ -23,7 +23,6 @@ import Data.Tuple.HT (swap, ) import Data.Maybe.HT (toMaybe, ) --- import NumericPrelude.Numeric (Integer) import NumericPrelude.Base hiding (cycle)
src/MathObj/Polynomial.hs view
@@ -68,23 +68,51 @@ import qualified Algebra.ZeroTestable         as ZeroTestable import qualified Algebra.Indexable            as Indexable -import Algebra.Module((*>))-import Algebra.ZeroTestable(isZero)- import Control.Monad (liftM, ) import qualified Data.List as List  import Test.QuickCheck (Arbitrary(arbitrary)) +import qualified MathObj.Wrapper.Haskell98 as W98+ import NumericPrelude.Base    hiding (const, reverse, ) import NumericPrelude.Numeric  import qualified Prelude as P98  +{- $setup+>>> import qualified MathObj.Polynomial as Poly+>>> import qualified Algebra.IntegralDomain as Integral+>>> import qualified Algebra.Laws as Laws+>>> import NumericPrelude.Numeric+>>> import NumericPrelude.Base+>>> import Prelude ()+>>>+>>> intPoly :: Poly.T Integer -> Poly.T Integer+>>> intPoly = id+>>>+>>> ratioPoly :: Poly.T Rational -> Poly.T Rational+>>> ratioPoly = id+-}++{- |+prop> Laws.identity (+) zero . intPoly+prop> Laws.commutative (+) . intPoly+prop> Laws.associative (+) . intPoly+prop> Laws.identity (*) one . intPoly+prop> Laws.commutative (*) . intPoly+prop> Laws.associative (*) . intPoly+prop> Laws.leftDistributive (*) (+) . intPoly+prop> Integral.propInverse . ratioPoly+-} newtype T a = Cons {coeffs :: [a]} +{-+>>> import Test.QuickCheck ((==>))+-} + {-# INLINE fromCoeffs #-} fromCoeffs :: [a] -> T a fromCoeffs = lift0@@ -271,18 +299,17 @@    lift1 $ foldr (\c p -> [c] + Core.mulLinearFactor d p) []  shrink :: Ring.C a => a -> T a -> T a-shrink k =-   lift1 $ zipWith (*) (iterate (k*) one)+shrink = lift1 . Core.shrink  dilate :: Field.C a => a -> T a -> T a-dilate = shrink . Field.recip+dilate = lift1 . Core.dilate   instance (Arbitrary a, ZeroTestable.C a) => Arbitrary (T a) where    arbitrary = liftM (fromCoeffs . Core.normalize) arbitrary  -{- * legacy instances -}+-- * Haskell 98 legacy instances  {- | It is disputable whether polynomials shall be represented by number literals or not.@@ -291,19 +318,20 @@ in  (x^2+x+1)*(x-1) However the output looks much different. -}-{-# INLINE legacyInstance #-}-legacyInstance :: a-legacyInstance =-   error "legacy Ring.C instance for simple input of numeric literals"+{-# INLINE notImplemented #-}+notImplemented :: String -> a+notImplemented name =+   error $ "MathObj.Polynomial: method " ++ name ++ " cannot be implemented" -instance (Ring.C a, Eq a, Show a, ZeroTestable.C a) => P98.Num (T a) where-   fromInteger = const . fromInteger-   negate = Additive.negate -- for unary minus-   (+)    = legacyInstance-   (*)    = legacyInstance-   abs    = legacyInstance-   signum = legacyInstance+-- legacy instances for use of numeric literals in GHCi+instance (P98.Num a) => P98.Num (T a) where+   fromInteger = const . P98.fromInteger+   negate = W98.unliftF1 Additive.negate+   (+)    = W98.unliftF2 (Additive.+)+   (*)    = W98.unliftF2 (Ring.*)+   abs    = notImplemented "abs"+   signum = notImplemented "signum" -instance (Field.C a, Eq a, Show a, ZeroTestable.C a) => P98.Fractional (T a) where-   fromRational = const . fromRational-   (/) = legacyInstance+instance (P98.Fractional a) => P98.Fractional (T a) where+   fromRational = const . P98.fromRational+   (/) = notImplemented "(/)"
src/MathObj/Polynomial/Core.hs view
@@ -21,7 +21,7 @@    stdUnit,    progression, differentiate, integrate, integrateInt,    mulLinearFactor,-   alternate,+   alternate, dilate, shrink,    ) where  import qualified Algebra.Module               as Module@@ -31,11 +31,11 @@ import qualified Algebra.Additive             as Additive import qualified Algebra.ZeroTestable         as ZeroTestable +import qualified Data.List.Reverse.StrictSpine as Rev import qualified Data.List as List import NumericPrelude.List (zipWithOverlap, ) import Data.Tuple.HT (mapPair, mapFst, forcePair, )-import Data.List.HT-          (dropWhileRev, switchL, shear, shearTranspose, outerProduct, )+import Data.List.HT (switchL, shear, shearTranspose, outerProduct)  import qualified NumericPrelude.Base as P import qualified NumericPrelude.Numeric as NP@@ -44,6 +44,25 @@ import NumericPrelude.Numeric hiding (divMod, negate, stdUnit, )  +{- $setup+>>> import qualified MathObj.Polynomial.Core as PolyCore+>>> import qualified MathObj.Polynomial as Poly+>>> import qualified Data.List as List+>>> import qualified Test.QuickCheck as QC+>>> import Test.QuickCheck ((==>))+>>> import Data.Tuple.HT (mapPair, mapSnd)+>>> import NumericPrelude.Numeric+>>> import NumericPrelude.Base+>>> import Prelude ()+>>>+>>> intPoly :: [Integer] -> [Integer]+>>> intPoly = id+>>>+>>> ratioPoly :: [Rational] -> [Rational]+>>> ratioPoly = id+-}++ {- | Horner's scheme for evaluating a polynomial in a ring. -}@@ -69,7 +88,7 @@ -} {-# INLINE normalize #-} normalize :: (ZeroTestable.C a) => [a] -> [a]-normalize = dropWhileRev isZero+normalize = Rev.dropWhile isZero  {- | Multiply by the variable, used internally.@@ -113,6 +132,9 @@    all (==zero) xs && all (==zero) ys  +{- |+prop> \(QC.NonEmpty xs) (QC.NonEmpty ys) -> PolyCore.tensorProduct xs ys == List.transpose (PolyCore.tensorProduct ys (intPoly xs))+-} {-# INLINE tensorProduct #-} tensorProduct :: Ring.C a => [a] -> [a] -> [[a]] tensorProduct = outerProduct (*)@@ -135,6 +157,9 @@ -- this one fails on infinite lists --    mul xs = foldr (\y zs -> add (scale y xs) (shift zs)) [] +{- |+prop> \xs ys  ->  PolyCore.equal (intPoly $ PolyCore.mul xs ys) (PolyCore.mulShear xs ys)+-} {-# INLINE mulShear #-} mulShear :: Ring.C a => [a] -> [a] -> [a] mulShear xs ys = map sum (shear (tensorProduct xs ys))@@ -144,6 +169,11 @@ mulShearTranspose xs ys = map sum (shearTranspose (tensorProduct xs ys))  +{- |+prop> \x y -> case (PolyCore.normalize x, PolyCore.normalize y) of (nx, ny) -> not (null (ratioPoly ny)) ==> mapSnd PolyCore.normalize (PolyCore.divMod nx ny) == mapPair (PolyCore.normalize, PolyCore.normalize) (PolyCore.divMod x y)+prop> \x y -> not (isZero (ratioPoly y)) ==> let z = fst $ PolyCore.divMod (Poly.coeffs x) y in  PolyCore.normalize z == z+prop> \x y -> case PolyCore.normalize $ ratioPoly y of ny -> not (null ny) ==> List.length (snd $ PolyCore.divMod x y) < List.length ny+-} divMod :: (ZeroTestable.C a, Field.C a) => [a] -> [a] -> ([a], [a]) divMod x y =    mapPair (List.reverse, List.reverse) $@@ -152,21 +182,26 @@ {- snd $ Poly.divMod (repeat (1::Double)) [1,1] -}+{- |+The modulus will always have one element less than the divisor.+This means that the modulus will be denormalized in some cases,+e.g. @mod [2,1,1] [1,1,1] == [1,0]@ instead of @[1]@.+-} divModRev :: (ZeroTestable.C a, Field.C a) => [a] -> [a] -> ([a], [a]) divModRev x y =-   let (y0:ys) = dropWhile isZero y-       -- the second parameter represents lazily (length x - length y)-       aux xs' =-         forcePair .-         switchL-           ([], xs')-           (P.const $-              let (x0:xs) = xs'-                  q0      = x0/y0-              in  mapFst (q0:) . aux (sub xs (scale q0 ys)))-   in  if isZero y-         then error "MathObj.Polynomial: division by zero"-         else aux x (drop (length y - 1) x)+   case dropWhile isZero y of+      [] -> error "MathObj.Polynomial: division by zero"+      y0:ys ->+         let -- the second parameter represents lazily (length x - length (normalize y))+             aux xs' =+               forcePair .+               switchL+                 ([], xs')+                 (P.const $+                    let (x0:xs) = xs'+                        q0      = x0/y0+                    in  mapFst (q0:) . aux (sub xs (scale q0 ys)))+         in  aux x (drop (length ys) x)  {-# INLINE stdUnit #-} stdUnit :: (ZeroTestable.C a, Ring.C a) => [a] -> a@@ -206,6 +241,14 @@ {-# INLINE alternate #-} alternate :: Additive.C a => [a] -> [a] alternate = zipWith ($) (cycle [id, Additive.negate])++{-# INLINE shrink #-}+shrink :: Ring.C a => a -> [a] -> [a]+shrink k = zipWith (*) (iterate (k*) one)++{-# INLINE dilate #-}+dilate :: Field.C a => a -> [a] -> [a]+dilate = shrink . Field.recip   {-
src/MathObj/PowerSeries.hs view
@@ -23,12 +23,21 @@ import qualified Algebra.Additive       as Additive import qualified Algebra.ZeroTestable   as ZeroTestable -import Algebra.Module((*>))- import NumericPrelude.Base    hiding (const) import NumericPrelude.Numeric  +{- $setup+>>> import qualified MathObj.PowerSeries.Core as PS+>>> import qualified MathObj.PowerSeries as PST+>>> import qualified Test.QuickCheck as QC+>>> import Test.NumericPrelude.Utility (equalTrunc, (/\))+>>> import NumericPrelude.Numeric as NP+>>> import NumericPrelude.Base as P+>>> import Prelude ()+-}++ newtype T a = Cons {coeffs :: [a]} deriving (Ord)  {-# INLINE fromCoeffs #-}@@ -128,6 +137,9 @@    (-)    = lift2 Poly.sub    zero   = lift0 [] +{- |+prop> QC.choose (1,10) /\ \expon (QC.Positive x) xs -> let xt = x:xs in  equalTrunc 15 (PS.pow (const x) (1 % expon) (PST.coeffs (PST.fromCoeffs xt ^ expon)) ++ repeat zero) (xt ++ repeat zero)+-} instance (Ring.C a) => Ring.C (T a) where    one           = const one    fromInteger n = const (fromInteger n)@@ -191,3 +203,9 @@    if isZero y      then Cons (Core.compose x ys)      else error "PowerSeries.compose: inner series must not have an absolute term."++shrink :: Ring.C a => a -> T a -> T a+shrink = lift1 . Poly.shrink++dilate :: Field.C a => a -> T a -> T a+dilate = lift1 . Poly.dilate
src/MathObj/PowerSeries/Core.hs view
@@ -20,6 +20,32 @@                               sin, cos, tan, asin, acos, atan)  +{- $setup+>>> import qualified MathObj.PowerSeries.Core as PS+>>> import qualified MathObj.PowerSeries.Example as PSE+>>> import Test.NumericPrelude.Utility (equalTrunc, (/\))+>>> import qualified Test.QuickCheck as QC+>>> import NumericPrelude.Numeric as NP+>>> import NumericPrelude.Base as P+>>> import Prelude ()+>>> import Control.Applicative (liftA3)+>>>+>>> checkHoles ::+>>>    Int -> ([Rational] -> [Rational]) ->+>>>    Rational -> [Rational] -> QC.Property+>>> checkHoles trunc f x xs =+>>>    QC.choose (1,10) /\ \expon ->+>>>    equalTrunc trunc+>>>       (f (PS.insertHoles expon (x:xs)) ++ repeat zero)+>>>       (PS.insertHoles expon (f (x:xs)) ++ repeat zero)+>>>+>>> genInvertible :: QC.Gen [Rational]+>>> genInvertible =+>>>    liftA3 (\x0 x1 xs -> x0:x1:xs)+>>>       QC.arbitrary (fmap QC.getNonZero QC.arbitrary) QC.arbitrary+-}++ {-# INLINE evaluate #-} evaluate :: Ring.C a => [a] -> a -> a evaluate = flip Poly.horner@@ -76,6 +102,18 @@    zipWith id (cycle [id, P.const zero, NP.negate, P.const zero])  +{- |+For power series of @f x@, compute the power series of @f(x^n)@.++prop> QC.choose (1,10) /\ \m -> QC.choose (1,10) /\ \n xs -> equalTrunc 100 (PS.insertHoles m $ PS.insertHoles n xs) (PS.insertHoles (m*n) xs)+-}+insertHoles :: Additive.C a => Int -> [a] -> [a]+insertHoles n =+   if n<=0+     then error $ "insertHoles requires positive exponent, but got " ++ show n+     else concatMap (\x -> x : replicate (n-1) zero)++ {- * Series arithmetic -}  add, sub :: (Additive.C a) => [a] -> [a] -> [a]@@ -148,6 +186,10 @@ We need to compute the square root only of the first term. That is, if the first term is rational, then all terms of the series are rational.++prop> equalTrunc 50 PSE.sqrtExpl (PS.sqrt (\1 -> 1) [1,1])+prop> equalTrunc 500 (1:1:repeat 0) (PS.sqrt (\1 -> 1) (PS.mul [1,1] [1,1]))+prop> checkHoles 50 (PS.sqrt (\1 -> 1)) 1 -} sqrt :: Field.C a => (a -> a) -> [a] -> [a] sqrt _ [] = []@@ -159,18 +201,28 @@ {- pow alpha t = t^alpha (pow alpha . x)' = alpha * (pow (alpha-1) . x) * x'-alpha * (pow alpha . x) = x * x' * (pow alpha . x)'+(pow alpha . x)' * x = alpha * (pow alpha . x) * x'+ y = pow alpha . x-alpha * y = x * x' * y'+y' * x = alpha * y * x'++This yields an implementation that is a fused+exp (alpha * log x) -}  {- |-Input series must start with non-zero term.+Input series must start with a non-zero term,+even better with a positive one.++prop> equalTrunc 100 (PSE.powExpl (-1/3)) (PS.pow (\1 -> 1) (-1/3) [1,1])+prop> equalTrunc 50 (PSE.powExpl (-1/3)) (PS.exp (\0 -> 1) (PS.scale (-1/3) PSE.log))+prop> checkHoles 30 (PS.pow (\1 -> 1) (1/3)) 1+prop> checkHoles 30 (PS.pow (\1 -> 1) (2/5)) 1 -} pow :: (Field.C a) => (a -> a) -> a -> [a] -> [a] pow f0 expon x =    let y  = integrate (f0 (head x)) y'-       y' = scale expon (divide y (mul x (differentiate x)))+       y' = scale expon (mul y (derivedLog x))    in  y  @@ -181,6 +233,10 @@ > (exp . x)' =   (exp . x) * x' > (sin . x)' =   (cos . x) * x' > (cos . x)' = - (sin . x) * x'++prop> equalTrunc 500 PSE.expExpl (PS.exp (\0 -> 1) [0,1])+prop> equalTrunc 100 (1:1:repeat 0) (PS.exp (\0 -> 1) PSE.log)+prop> checkHoles 30 (PS.exp (\0 -> 1)) 0 -} exp :: Field.C a => (a -> a) -> [a] -> [a] exp f0 x =@@ -199,10 +255,25 @@ sinCosScalar :: Transcendental.C a => a -> (a,a) sinCosScalar x = (Transcendental.sin x, Transcendental.cos x) -sin, cos :: Field.C a => (a -> (a,a)) -> [a] -> [a]+{- |+prop> equalTrunc 500 PSE.sinExpl (PS.sin (\0 -> (0,1)) [0,1])+prop> equalTrunc 50 (0:1:repeat 0) (PS.sin (\0 -> (0,1)) PSE.asin)+prop> checkHoles 20 (PS.sin (\0 -> (0,1))) 0+-}+sin :: Field.C a => (a -> (a,a)) -> [a] -> [a] sin f0 = fst . sinCos f0+{- |+prop> equalTrunc 500 PSE.cosExpl (PS.cos (\0 -> (0,1)) [0,1])+prop> checkHoles 20 (PS.cos (\0 -> (0,1))) 0+-}+cos :: Field.C a => (a -> (a,a)) -> [a] -> [a] cos f0 = snd . sinCos f0 +{- |+prop> equalTrunc 50 PSE.tanExpl (PS.tan (\0 -> (0,1)) [0,1])+prop> equalTrunc 50 (0:1:repeat 0) (PS.tan (\0 -> (0,1)) PSE.atan)+prop> checkHoles 20 (PS.tan (\0 -> (0,1))) 0+-} tan :: (Field.C a) => (a -> (a,a)) -> [a] -> [a] tan f0 = uncurry divide . sinCos f0 @@ -214,6 +285,10 @@  {- | Input series must start with non-zero term.++prop> equalTrunc 500 PSE.logExpl (PS.log (\1 -> 0) [1,1])+prop> equalTrunc 100 (0:1:repeat 0) (PS.log (\1 -> 0) PSE.exp)+prop> checkHoles 30 (PS.log (\1 -> 0)) 1 -} log :: (Field.C a) => (a -> a) -> [a] -> [a] log f0 x = integrate (f0 (head x)) (derivedLog x)@@ -224,17 +299,33 @@ derivedLog :: (Field.C a) => [a] -> [a] derivedLog x = divide (differentiate x) x +{- |+prop> equalTrunc 500 PSE.atan (PS.atan (\0 -> 0) [0,1])+prop> equalTrunc 50 (0:1:repeat 0) (PS.atan (\0 -> 0) PSE.tan)+prop> checkHoles 20 (PS.atan (\0 -> 0)) 0+-} atan :: (Field.C a) => (a -> a) -> [a] -> [a] atan f0 x =    let x' = differentiate x    in  integrate (f0 (head x)) (divide x' ([1] + mul x x)) -asin, acos :: (Field.C a) =>-   (a -> a) -> (a -> a) -> [a] -> [a]+{- |+prop> equalTrunc 100 (0:1:repeat 0) (PS.asin (\1 -> 1) (\0 -> 0) PSE.sin)+prop> equalTrunc 50 PSE.asin (PS.asin (\1 -> 1) (\0 -> 0) [0,1])+prop> checkHoles 30 (PS.asin (\1 -> 1) (\0 -> 0)) 0+-}+asin :: (Field.C a) => (a -> a) -> (a -> a) -> [a] -> [a] asin sqrt0 f0 x =    let x' = differentiate x    in  integrate (f0 (head x))                  (divide x' (sqrt sqrt0 ([1] - mul x x)))++{- |+Would be a nice test, but we cannot compute exactly with 'pi':++> equalTrunc 50 PSE.acos (PS.acos (\1 -> 1) (\0 -> pi/2) [0,1])+-}+acos :: (Field.C a) => (a -> a) -> (a -> a) -> [a] -> [a] acos = asin  {- |@@ -257,22 +348,58 @@ composeTaylor x []     = x 0  +{-+X(t) = t*x(t)+R(t) = t*r(t) +r(t) = 1 / (x(r(t)*t))+R(t)/t+   = 1 / (x(R(t)))+   = 1 / (X(R(t)) / R(t))+   = 1 / (t / R(t))+-}++{- |+This function returns the series of the inverse function in the form:+(point of the expansion, power series).++That is, say we have the equation:++> y = a + f(x)++where function f is given by a power series with f(0) = 0.+We want to solve for x:++> x = f^-1(y-a)++If you pass the power series of @a+f(x)@ to 'inv',+you get @(a, f^-1)@ as answer, where @f^-1@ is a power series.++The linear term of @f@ (the coefficient of @x@) must be non-zero.++This needs cubic run-time and thus is exceptionally slow.+Computing inverse series for special power series might be faster.++prop> genInvertible /\ \xs -> let (y,ys) = PS.inv xs; (z,zs) = PS.invDiff xs in y==z && equalTrunc 15 ys zs+-}+-- how about NonEmpty.T here?+inv :: (Eq a, Field.C a) => [a] -> (a, [a])+inv [] = error "inv: power series must be non-zero"+inv (x:xs) =+   (x, let r = divide [1] (compose xs r) in 0 : r)++ {- (x . y) = id (x' . y) * y' = 1 y' = 1 / (x' . y) -} -{- |-This function returns the series of the function in the form:-(point of the expansion, power series)--This is exceptionally slow and needs cubic run-time.+{-+Like 'inv' but with a slightly cumbersome implementation. -}--inv :: (Field.C a) => [a] -> (a, [a])-inv x =+invDiff :: (Field.C a) => [a] -> (a, [a])+invDiff x =    let y' = divide [1] (compose (differentiate x) (tail y))        y  = integrate 0 y'             -- the first term is zero, which is required for composition
src/MathObj/PowerSeries/Example.hs view
@@ -5,7 +5,6 @@  import qualified Algebra.Field          as Field import qualified Algebra.Ring           as Ring--- import qualified Algebra.Additive       as Additive import qualified Algebra.ZeroTestable   as ZeroTestable import qualified Algebra.Transcendental as Transcendental @@ -19,6 +18,16 @@ import NumericPrelude.Base -- (Bool, const, map, zipWith, id, (&&), (==))  +{- $setup+>>> import qualified MathObj.PowerSeries.Core as PS+>>> import qualified MathObj.PowerSeries.Example as PSE+>>> import Test.NumericPrelude.Utility (equalTrunc)+>>> import NumericPrelude.Numeric as NP+>>> import NumericPrelude.Base as P+>>> import Prelude ()+-}++ {- * Default implementations. -}  recip :: (Ring.C a) => [a]@@ -42,6 +51,8 @@ cosh  = coshODE atanh = atanhODE ++-- | prop> \m n -> equalTrunc 30 (PS.mul (PSE.pow m) (PSE.pow n)) (PSE.pow (m+n)) pow :: (Field.C a) => a -> [a] pow = powExpl sqrt = sqrtExpl@@ -52,34 +63,54 @@ recipExpl :: (Ring.C a) => [a] recipExpl = cycle [1,-1] -expExpl, sinExpl, cosExpl :: (Field.C a) => [a]+-- | prop> equalTrunc 500 PSE.expExpl PSE.expODE+expExpl :: (Field.C a) => [a] expExpl = scanl (*) one PS.recipProgression+-- | prop> equalTrunc 500 PSE.sinExpl PSE.sinODE+sinExpl :: (Field.C a) => [a] sinExpl = zero : PS.holes2alternate (tail expExpl)-cosExpl =        PS.holes2alternate       expExpl+-- | prop> equalTrunc 500 PSE.cosExpl PSE.cosODE+cosExpl :: (Field.C a) => [a]+cosExpl = PS.holes2alternate expExpl -tanExpl, tanExplSieve :: (ZeroTestable.C a, Field.C a) => [a]+-- | prop> equalTrunc 50 PSE.tanExpl PSE.tanODE+tanExpl :: (ZeroTestable.C a, Field.C a) => [a] tanExpl = PS.divide sinExpl cosExpl -- ignore zero values+-- | prop> equalTrunc 50 PSE.tanExpl PSE.tanExplSieve+tanExplSieve :: (ZeroTestable.C a, Field.C a) => [a] tanExplSieve =    concatMap       (\x -> [zero,x])       (PS.divide (sieve 2 (tail sin)) (sieve 2 cos)) -logExpl, atanExpl, sqrtExpl :: (Field.C a) => [a]+-- | prop> equalTrunc 500 PSE.logExpl PSE.logODE+logExpl :: (Field.C a) => [a] logExpl  = zero : PS.alternate       PS.recipProgression+-- | prop> equalTrunc 500 PSE.atanExpl PSE.atanODE+atanExpl :: (Field.C a) => [a] atanExpl = zero : PS.holes2alternate PS.recipProgression -sinhExpl, coshExpl, atanhExpl :: (Field.C a) => [a]+-- | prop> equalTrunc 500 PSE.sinhExpl PSE.sinhODE+sinhExpl :: (Field.C a) => [a] sinhExpl  = zero : PS.holes2 (tail expExpl)+-- | prop> equalTrunc 500 PSE.coshExpl PSE.coshODE+coshExpl :: (Field.C a) => [a] coshExpl  =        PS.holes2       expExpl+-- | prop> equalTrunc 500 PSE.atanhExpl PSE.atanhODE+atanhExpl :: (Field.C a) => [a] atanhExpl = zero : PS.holes2 PS.recipProgression  {- * Power series of (1+x)^expon using the binomial series. -} +-- | prop> \expon -> equalTrunc 50 (PSE.powODE expon) (PSE.powExpl expon) powExpl :: (Field.C a) => a -> [a] powExpl expon =    scanl (*) 1 (zipWith (/)       (iterate (subtract 1) expon) PS.progression)++-- | prop> equalTrunc 100 PSE.sqrtExpl PSE.sqrtODE+sqrtExpl :: (Field.C a) => [a] sqrtExpl = powExpl (1/2)  {- |@@ -110,11 +141,13 @@        == cos x ^ (-2) -} -expODE, sinODE, cosODE, tanODE, tanODESieve :: (Field.C a) => [a]+expODE, sinODE, cosODE, tanODE :: (Field.C a) => [a] expODE = PS.integrate 1 expODE sinODE = PS.integrate 0 cosODE cosODE = PS.integrate 1 (PS.negate sinODE) tanODE = PS.integrate 0 (PS.add [1] (PS.mul tanODE tanODE))+-- | prop> equalTrunc 50 PSE.tanODE PSE.tanODESieve+tanODESieve :: (Field.C a) => [a] tanODESieve =    -- sieve is too strict here because it wants to detect end of lists    let tan2 = map head (iterate (drop 2) (tail tanODESieve))@@ -126,9 +159,11 @@ atan' x == 1/(1+x^2) -} -logODE, recipCircle, asinODE, atanODE, sqrtODE :: (Field.C a) => [a]+logODE, recipCircle, atanODE, sqrtODE :: (Field.C a) => [a] logODE  = PS.integrate zero recip recipCircle = intersperse zero (PS.alternate (powODE (-1/2)))+-- | prop> equalTrunc 50 PSE.asinODE (snd $ PS.inv PSE.sinODE)+asinODE :: (Field.C a) => [a] asinODE = PS.integrate 0 recipCircle atanODE = PS.integrate zero (cycle [1,0,-1,0]) sqrtODE = powODE (1/2)
src/MathObj/PowerSeries2.hs view
@@ -19,9 +19,6 @@ import qualified Algebra.Additive       as Additive import qualified Algebra.ZeroTestable   as ZeroTestable -import qualified NumericPrelude.Numeric as NP-import qualified NumericPrelude.Base as P- import Data.List (isPrefixOf, ) import qualified Data.List.Match as Match @@ -84,6 +81,11 @@ const x = lift0 [[x]]  +{-# INLINE truncate #-}+truncate :: Int -> T a -> T a+truncate n = lift1 (take n)++ instance Functor T where    fmap f (Cons xs) = Cons (map (map f) xs) @@ -122,5 +124,4 @@  instance (Algebraic.C a) => Algebraic.C (T a) where    sqrt   = lift1 (Core.sqrt Algebraic.sqrt)---   x ^/ y = lift1 (Core.pow (Algebraic.^/ y)---                       (fromRational' y)) x+   x ^/ y = lift1 (Core.pow (Algebraic.^/ y) (fromRational' y)) x
src/MathObj/PowerSeries2/Core.hs view
@@ -11,7 +11,6 @@ import qualified Algebra.Additive       as Additive  import NumericPrelude.Base--- import NumericPrelude.Numeric hiding (negate, sqrt, )   type T a = [[a]]@@ -59,6 +58,11 @@    lift1fromPowerSeries $    PSCore.sqrt (PS.const . (\[x] -> fSqRt x) . PS.coeffs) +pow :: (Field.C a) =>+   (a -> a) -> a -> T a -> T a+pow fPow expon =+   lift1fromPowerSeries $+   PSCore.pow (PS.const . (\[x] -> fPow x) . PS.coeffs) (PS.const expon)   swapVariables :: T a -> T a
src/MathObj/PowerSum.hs view
@@ -29,8 +29,6 @@ import qualified Algebra.Additive     as Additive import qualified Algebra.ZeroTestable as ZeroTestable -import Algebra.Module((*>))- import Control.Monad(liftM2) import qualified Data.List as List import Data.List.HT (shearTranspose, sieve)
src/MathObj/RefinementMask2.hs view
@@ -5,9 +5,12 @@    toPolynomial,    toPolynomialFast,    refinePolynomial,+   convolvePolynomial,+   convolveTruncatedPowerPolynomials,    ) where  import qualified MathObj.Polynomial as Poly+import qualified MathObj.Polynomial.Core as PolyCore import qualified Algebra.RealField as RealField import qualified Algebra.Field  as Field import qualified Algebra.Ring   as Ring@@ -16,6 +19,7 @@ import qualified Data.List as List import qualified Data.List.HT as ListHT import qualified Data.List.Match as Match+import Data.Maybe (fromMaybe, ) import Control.Monad (liftM2, )  import qualified Test.QuickCheck as QC@@ -25,6 +29,43 @@ import NumericPrelude.Numeric  +{- $setup+>>> import qualified MathObj.RefinementMask2 as Mask+>>> import qualified MathObj.Polynomial      as Poly+>>> import qualified MathObj.Polynomial.Core as PolyCore+>>>+>>> import qualified Algebra.Differential as D+>>> import qualified Algebra.Ring as Ring+>>> import Test.NumericPrelude.Utility ((/\))+>>> import qualified Test.QuickCheck as QC+>>> import NumericPrelude.Numeric as NP+>>> import NumericPrelude.Base as P+>>> import Prelude ()+>>>+>>> import Data.Function.HT (nest)+>>> import Data.Maybe (fromMaybe)+>>>+>>>+>>> hasMultipleZero :: (Ring.C a, Eq a) => Int -> a -> Poly.T a -> Bool+>>> hasMultipleZero n x poly =+>>>    all (zero==) $ take n $+>>>    map (flip Poly.evaluate x) $+>>>    iterate D.differentiate poly+>>>+>>> genAdmissibleMask :: QC.Gen (Mask.T Rational, Poly.T Rational)+>>> genAdmissibleMask =+>>>    QC.suchThatMap QC.arbitrary $+>>>       \mask -> fmap ((,) mask) $ Mask.toPolynomial mask+>>>+>>> polyFromMask :: Mask.T a -> Poly.T a+>>> polyFromMask = Poly.fromCoeffs . Mask.coeffs+>>>+>>> genShortPolynomial :: Int -> QC.Gen (Poly.T Rational)+>>> genShortPolynomial n =+>>>    fmap (Poly.fromCoeffs . PolyCore.normalize . take n) $ QC.arbitrary+-}++ newtype T a = Cons {coeffs :: [a]}  @@ -81,6 +122,11 @@ p2 = L * R^(-1) * m  R * L^(-1) * p2 = m+++prop> genAdmissibleMask /\ \(mask,poly) -> hasMultipleZero (fromMaybe 0 $ Poly.degree poly) 1 (polyFromMask (Mask.fromPolynomial poly) - polyFromMask mask)++prop> genShortPolynomial 5 /\ \poly -> maybe False (Poly.collinear poly) $ Mask.toPolynomial $ Mask.fromPolynomial poly -} fromPolynomial ::    (Field.C a) => Poly.T a -> T a@@ -111,6 +157,9 @@ {- | If the mask does not sum up to a power of @1/2@ then the function returns 'Nothing'.++>>> fmap ((6::Rational) *>) $ Mask.toPolynomial (Mask.fromCoeffs [0.1, 0.02, 0.005::Rational])+Just (Polynomial.fromCoeffs [-12732 % 109375,272 % 625,-18 % 25,1 % 1]) -} toPolynomial ::    (RealField.C a) => T a -> Maybe (Poly.T a)@@ -127,10 +176,6 @@                    in  ip + Poly.const (correctConstant (fmap (k/s*) mask) ip))                 (Poly.const 1) ks0           _ -> Nothing-{--> fmap (6 Vector.*>) $ toPolynomial (Cons [0.1, 0.02, 0.005::Rational])-Just (Polynomial.fromCoeffs [-12732 % 109375, 272 % 625, -18 % 25, 1 % 1])--}  {- The constant term must be zero,@@ -158,14 +203,66 @@                 (Poly.const 1) ks0           _ -> Nothing +{- |+prop> genShortPolynomial 5 /\ \poly -> poly == Mask.refinePolynomial (Mask.fromPolynomial poly) poly++>>> fmap (round :: Double -> Integer) $ fmap (1000000*) $ nest 50 (Mask.refinePolynomial (Mask.fromCoeffs [0.1, 0.02, 0.005])) (Poly.fromCoeffs [0,0,0,1])+Polynomial.fromCoeffs [-116407,435200,-720000,1000000]+-} refinePolynomial ::    (Ring.C a) => T a -> Poly.T a -> Poly.T a refinePolynomial mask =    Poly.shrink 2 .    Vector.linearComb (coeffs mask) .    iterate (Poly.translate 1)++convolve ::+   (Ring.C a) => T a -> T a -> T a+convolve x y =+   fromCoeffs $+   PolyCore.mul (coeffs x) (coeffs y)++{- |+Convolve polynomials via refinement mask.++(mask x + ux*(-1,1)^degree x) * (mask y + uy*(-1,1)^degree y)+-}+convolvePolynomial ::+   (RealField.C a) =>+   Poly.T a -> Poly.T a -> Poly.T a+convolvePolynomial x y =+   fromMaybe+      (error "RefinementMask2.convolvePolynomial: leading term should always be correct") $+   toPolynomial $ fmap (/2) $+   convolve (fromPolynomial x) (fromPolynomial y)+ {--> mapM_ print $ take 50 $ iterate (refinePolynomial (Cons [0.1, 0.02, 0.005])) (Poly.fromCoeffs [0,0,0,1::Double])-...-Polynomial.fromCoeffs [-0.11640685714285712,0.4351999999999999,-0.7199999999999999,1.0]+This function interprets all monomials as truncated power functions,+that is power functions that are set to zero for negative arguments.+However the convolution implied by this interpretation+cannot be represented by means of mask convolution.+See for instance:++*MathObj.RefinementMask2> let x = Poly.fromCoeffs [1,1] :: Poly.T Rational+*MathObj.RefinementMask2> fromPolynomial $ convolvePolynomial2 x x+RefinementMask2.fromCoeffs [1 % 3,-1 % 8,-1 % 8,1 % 24]++The obtained mask cannot be factored,+thus it is not a complete square.+But maybe it becomes a square if we add u*(-1,1)^4.+However this mask has sum 1/8 and the added term has sum 0,+thus the sum of the modified mask is still 1/8 and thus not a square. -}+convolveTruncatedPowerPolynomials ::+   (RealField.C a) =>+   Poly.T a -> Poly.T a -> Poly.T a+convolveTruncatedPowerPolynomials x y =+   let facs = scanl (*) 1 $ iterate (1+) 1+       xl = Poly.coeffs x+       yl = Poly.coeffs y+   in  Poly.integrate 0 $+       Poly.fromCoeffs $+       zipWith (flip (/)) facs $+       PolyCore.mul+          (zipWith (*) facs xl)+          (zipWith (*) facs yl)
src/MathObj/RootSet.hs view
@@ -26,8 +26,11 @@ import qualified Algebra.Additive     as Additive import qualified Algebra.ZeroTestable as ZeroTestable +import qualified Algebra.RealRing     as RealRing+ import qualified Data.List.Match as Match-import Control.Monad (liftM2)+import qualified Data.List.Key as Key+import Control.Monad (liftM2, replicateM, )  import NumericPrelude.Base as P hiding (const) import NumericPrelude.Numeric as NP@@ -169,3 +172,32 @@  instance (Field.C a, ZeroTestable.C a) => Algebraic.C (T a) where    root n = lift1 (PowerSum.root n)++++{- |+Given an approximation of a root,+the degree of the polynomial and maximum value of coefficients,+find candidates of polynomials that have approximately this root+and show the actual value of the polynomial at the given root approximation.++This algorithm runs easily into a stack overflow, I do not know why.+We may also employ a more sophisticated integer relation algorithm,+like PSLQ and friends.+-}+{-# SPECIALISE approxPolynomial ::+       Int -> Integer -> Double -> (Double, Poly.T Double) #-}+{-# SPECIALISE approxPolynomial ::+       Int -> Integer -> Float -> (Float, Poly.T Float) #-}+approxPolynomial ::+   (RealRing.C a) =>+   Int -> Integer -> a -> (a, Poly.T a)+approxPolynomial d maxCoeff x =+   let powers = take (d+1) $ iterate (x*) one+   in  -- List.minimumBy (\a b -> compare (abs (fst a)) (abs (fst b))) $+       Key.minimum (abs . fst) $+       map+          ((\cs -> (sum $ zipWith (*) powers cs, Poly.fromCoeffs cs)) . reverse)+          (liftM2 (:)+             (map fromInteger [1 .. maxCoeff])+             (replicateM d $ map fromInteger [-maxCoeff .. maxCoeff]))
+ src/MathObj/Wrapper/Haskell98.hs view
@@ -0,0 +1,188 @@+{-# LANGUAGE GeneralizedNewtypeDeriving #-}+{- |+A wrapper that provides instances of Haskell 98 and NumericPrelude+numeric type classes+for types that have Haskell 98 instances.+-}+module MathObj.Wrapper.Haskell98 where++import qualified Algebra.Absolute as Absolute+import qualified Algebra.Additive as Additive+import qualified Algebra.Algebraic as Algebraic+import qualified Algebra.Field as Field+import qualified Algebra.FloatingPoint as Float+import qualified Algebra.IntegralDomain as Integral+import qualified Algebra.PrincipalIdealDomain as PID+import qualified Algebra.RealField as RealField+import qualified Algebra.RealIntegral as RealIntegral+import qualified Algebra.RealRing as RealRing+import qualified Algebra.RealTranscendental as RealTrans+import qualified Algebra.Ring as Ring+import qualified Algebra.ToInteger as ToInteger+import qualified Algebra.ToRational as ToRational+import qualified Algebra.Transcendental as Trans+import qualified Algebra.Units as Units+import qualified Algebra.ZeroTestable as ZeroTestable++import qualified Number.Ratio as Ratio++import qualified Algebra.RealRing98 as RealRing98++import Data.Ix (Ix, )++import Data.Tuple.HT (mapPair, )+++{- |+This makes a type usable in the NumericPrelude framework+that was initially implemented for Haskell98 typeclasses.+E.g. if @a@ is in class 'Num',+then @T a@ is both in class 'Num' and in 'Ring.C'.++You can even lift container types.+If @Polynomial a@ is in 'Num' for all types @a@ that are in 'Num',+then @T (Polynomial (MathObj.Wrapper.NumericPrelude.T a))@+is in 'Ring.C' for all types @a@ that are in 'Ring.C'.+-}+newtype T a = Cons {decons :: a}+   deriving+      (Show, Eq, Ord, Ix, Bounded, Enum,+       Num, Integral, Fractional, Floating,+       Real, RealFrac, RealFloat)+++{-# INLINE lift1 #-}+lift1 :: (a -> b) -> T a -> T b+lift1 f (Cons a) = Cons (f a)++{-# INLINE lift2 #-}+lift2 :: (a -> b -> c) -> T a -> T b -> T c+lift2 f (Cons a) (Cons b) = Cons (f a b)+++{-# INLINE unliftF1 #-}+unliftF1 :: Functor f => (f (T a) -> f (T b)) -> f a -> f b+unliftF1 f a = fmap decons $ f (fmap Cons a)++{-# INLINE unliftF2 #-}+unliftF2 :: Functor f => (f (T a) -> f (T b) -> f (T c)) -> f a -> f b -> f c+unliftF2 f a b = fmap decons $ f (fmap Cons a) (fmap Cons b)+++instance Functor T where+   {-# INLINE fmap #-}+   fmap f (Cons a) = Cons (f a)+++instance Num a => Additive.C (T a) where+   zero = 0+   (+) = lift2 (+)+   (-) = lift2 (-)+   negate = lift1 negate++instance (Num a) => Ring.C (T a) where+   fromInteger = Cons . fromInteger+   (*) = lift2 (*)+   (^) a n = lift1 (^n) a++instance (Fractional a) => Field.C (T a) where+   fromRational' r = Cons (fromRational (Ratio.toRational98 r))+   (/) = lift2 (/)+   recip = lift1 recip+   (^-) a n = lift1 (^^n) a++instance (Floating a) => Algebraic.C (T a) where+   sqrt = lift1 sqrt+   (^/) a r = lift1 (** fromRational (Ratio.toRational98 r)) a+   root n a = lift1 (** recip (fromInteger n)) a++instance (Floating a) => Trans.C (T a) where+   pi      = Cons pi+   log     = lift1 log+   exp     = lift1 exp+   logBase = lift2 logBase+   (**)    = lift2 (**)+   cos     = lift1 cos+   tan     = lift1 tan+   sin     = lift1 sin+   acos    = lift1 acos+   atan    = lift1 atan+   asin    = lift1 asin+   cosh    = lift1 cosh+   tanh    = lift1 tanh+   sinh    = lift1 sinh+   acosh   = lift1 acosh+   atanh   = lift1 atanh+   asinh   = lift1 asinh++instance (Integral a) => Integral.C (T a) where+   div = lift2 div+   mod = lift2 mod+   divMod (Cons a) (Cons b) =+      mapPair (Cons, Cons) (divMod a b)++instance (Integral a) => Units.C (T a) where+   isUnit = unimplemented "isUnit"+   stdAssociate = unimplemented "stdAssociate"+   stdUnit = unimplemented "stdUnit"+   stdUnitInv = unimplemented "stdUnitInv"++instance (Integral a) => PID.C (T a) where+   gcd = gcd+   lcm = lcm++instance (Eq a, Num a) => ZeroTestable.C (T a) where+   isZero (Cons a) = a==0++instance (Num a) => Absolute.C (T a) where+   abs = abs+   signum = signum++instance (RealFrac a) => RealRing.C (T a) where+   splitFraction (Cons a) =+      mapPair (Ring.fromInteger, Cons)+         (RealRing98.fixSplitFraction (properFraction a))+   fraction (Cons a) =+      Cons (RealRing98.fixFraction (RealRing98.signedFraction a))+   ceiling (Cons a) = Ring.fromInteger (ceiling a)+   floor (Cons a) = Ring.fromInteger (floor a)+   truncate (Cons a) = Ring.fromInteger (truncate a)+   round (Cons a) = Ring.fromInteger (round a)++instance (RealFrac a) => RealField.C (T a) where++instance (RealFloat a) => RealTrans.C (T a) where+   atan2 = atan2++instance (Integral a) => RealIntegral.C (T a) where+   quot = lift2 quot+   rem = lift2 rem+   quotRem (Cons a) (Cons b) =+      mapPair (Cons, Cons) (quotRem a b)++instance (Integral a) => ToInteger.C (T a) where+   toInteger (Cons a) = toInteger a++instance (Real a) => ToRational.C (T a) where+   toRational (Cons a) = Field.fromRational (toRational a)++instance (RealFloat a) => Float.C (T a) where+   radix = floatRadix . decons+   digits = floatDigits . decons+   range = floatRange . decons+   decode = decodeFloat . decons+   encode m = Cons . encodeFloat m+   exponent = exponent . decons+   significand = lift1 significand+   scale = lift1 . scaleFloat+   isNaN = isNaN . decons+   isInfinite = isInfinite . decons+   isDenormalized = isDenormalized . decons+   isNegativeZero = isNegativeZero . decons+   isIEEE = isIEEE . decons++++unimplemented :: String -> a+unimplemented name =+   error (name ++ "cannot be implemented in terms of Haskell98 type classes")
+ src/MathObj/Wrapper/NumericPrelude.hs view
@@ -0,0 +1,221 @@+{-# LANGUAGE GeneralizedNewtypeDeriving #-}+{-# LANGUAGE MultiParamTypeClasses #-}+{- |+A wrapper that provides instances of Haskell 98 and NumericPrelude+numeric type classes+for types that have NumericPrelude instances.+-}+module MathObj.Wrapper.NumericPrelude where++import qualified Algebra.Absolute as Absolute+import qualified Algebra.Additive as Additive+import qualified Algebra.Algebraic as Algebraic+import qualified Algebra.Field as Field+import qualified Algebra.FloatingPoint as Float+import qualified Algebra.IntegralDomain as Integral+import qualified Algebra.PrincipalIdealDomain as PID+import qualified Algebra.RealField as RealField+import qualified Algebra.RealIntegral as RealIntegral+import qualified Algebra.RealRing as RealRing+import qualified Algebra.RealTranscendental as RealTrans+import qualified Algebra.Ring as Ring+import qualified Algebra.ToInteger as ToInteger+import qualified Algebra.ToRational as ToRational+import qualified Algebra.Transcendental as Trans+import qualified Algebra.Units as Units+import qualified Algebra.ZeroTestable as ZeroTestable++import qualified Algebra.NormedSpace.Euclidean as NormEuc+import qualified Algebra.NormedSpace.Maximum as NormMax+import qualified Algebra.NormedSpace.Sum as NormSum+import qualified Algebra.OccasionallyScalar as OccScalar+import qualified Algebra.Differential as Differential+import qualified Algebra.DivisibleSpace as Divisible+import qualified Algebra.VectorSpace as VectorSpace+import qualified Algebra.Module as Module++import qualified Number.Ratio as Ratio++import Data.Ix (Ix, )++import Data.Tuple.HT (mapPair, )+++{- |+This makes a type usable with Haskell98 type classes+that was initially implemented for NumericPrelude typeclasses.+E.g. if @a@ is in class 'Ring.C',+then @T a@ is both in class 'Num' and in 'Ring.C'.++You can even lift container types.+If @Polynomial a@ is in 'Ring.C' for all types @a@ that are in 'Ring.C',+then @T (Polynomial (MathObj.Wrapper.Haskell98.T a))@+is in 'Num' for all types @a@ that are in 'Num'.+-}+newtype T a = Cons {decons :: a}+   deriving+      (Show, Eq, Ord, Ix, Bounded, Enum,+       Ring.C, Additive.C, Field.C, Algebraic.C, Trans.C,+       Integral.C, PID.C, Units.C,+       Absolute.C, ZeroTestable.C,+       RealField.C, RealIntegral.C, RealRing.C, RealTrans.C,+       ToInteger.C, ToRational.C, Float.C,+       Differential.C)++{-# INLINE lift1 #-}+lift1 :: (a -> b) -> T a -> T b+lift1 f (Cons a) = Cons (f a)++{-# INLINE lift2 #-}+lift2 :: (a -> b -> c) -> T a -> T b -> T c+lift2 f (Cons a) (Cons b) = Cons (f a b)+++instance Functor T where+   {-# INLINE fmap #-}+   fmap f (Cons a) = Cons (f a)+++{-+instance Enum a => Enum (T a) where+   succ (Cons n) = Cons (succ n)+   pred (Cons n) = Cons (pred n)+   toEnum n = Cons (toEnum n)+   fromEnum (Cons n) = fromEnum n+   enumFrom (Cons n) =+      map Cons (enumFrom n)+   enumFromThen (Cons n) (Cons m) =+      map Cons (enumFromThen n m)+   enumFromTo (Cons n) (Cons m) =+      map Cons (enumFromTo n m)+   enumFromThenTo (Cons n) (Cons m) (Cons p) =+      map Cons (enumFromThenTo n m p)+-}++instance (Ring.C a, Absolute.C a, Eq a, Show a) => Num (T a) where+   (+) = lift2 (Additive.+)+   (-) = lift2 (Additive.-)+   negate = lift1 Additive.negate++   fromInteger = Cons . Ring.fromInteger+   (*) = lift2 (Ring.*)++   abs = lift1 Absolute.abs+   signum = lift1 Absolute.signum++instance (RealIntegral.C a, Absolute.C a, ToInteger.C a, Ord a, Enum a, Show a) => Integral (T a) where+   quot = lift2 RealIntegral.quot+   rem = lift2 RealIntegral.rem+   quotRem (Cons a) (Cons b) =+      mapPair (Cons, Cons) (RealIntegral.quotRem a b)+   div = lift2 Integral.div+   mod = lift2 Integral.mod+   divMod (Cons a) (Cons b) =+      mapPair (Cons, Cons) (Integral.divMod a b)+   toInteger (Cons a) = ToInteger.toInteger a++instance (Field.C a, Absolute.C a, Eq a, Show a) => Fractional (T a) where+   (/) = lift2 (Field./)+   recip = lift1 Field.recip+   fromRational = Cons . Field.fromRational++instance (Trans.C a, Absolute.C a, Eq a, Show a) => Floating (T a) where+   sqrt    = lift1 Algebraic.sqrt+   pi      = Cons Trans.pi+   log     = lift1 Trans.log+   exp     = lift1 Trans.exp+   logBase = lift2 Trans.logBase+   (**)    = lift2 (Trans.**)+   cos     = lift1 Trans.cos+   tan     = lift1 Trans.tan+   sin     = lift1 Trans.sin+   acos    = lift1 Trans.acos+   atan    = lift1 Trans.atan+   asin    = lift1 Trans.asin+   cosh    = lift1 Trans.cosh+   tanh    = lift1 Trans.tanh+   sinh    = lift1 Trans.sinh+   acosh   = lift1 Trans.acosh+   atanh   = lift1 Trans.atanh+   asinh   = lift1 Trans.asinh++instance (ToRational.C a, Absolute.C a, Ord a, Show a) => Real (T a) where+   toRational (Cons a) =+      Ratio.toRational98 (ToRational.toRational a)++instance (Field.C a, RealRing.C a, ToRational.C a, Absolute.C a, Ord a, Show a) => RealFrac (T a) where+   properFraction (Cons a) =+      let b = RealRing.truncate a+      in  (fromInteger b, Cons (a Additive.- Ring.fromInteger b))+   ceiling (Cons a) = fromInteger (RealRing.ceiling a)+   floor (Cons a) = fromInteger (RealRing.floor a)+   truncate (Cons a) = fromInteger (RealRing.truncate a)+   round (Cons a) = fromInteger (RealRing.round a)++instance (RealTrans.C a, Float.C a, ToRational.C a, Absolute.C a, Ord a, Show a) => RealFloat (T a) where+   atan2 = RealTrans.atan2+   floatRadix = Float.radix . decons+   floatDigits = Float.digits . decons+   floatRange = Float.range . decons+   decodeFloat = Float.decode . decons+   encodeFloat m = Cons . Float.encode m+   exponent = Float.exponent . decons+   significand = lift1 Float.significand+   scaleFloat = lift1 . Float.scale+   isNaN = Float.isNaN . decons+   isInfinite = Float.isInfinite . decons+   isDenormalized = Float.isDenormalized . decons+   isNegativeZero = Float.isNegativeZero . decons+   isIEEE = Float.isIEEE . decons++{-+instance Additive.C (T a) where+instance Ring.C (T a) where+instance Field.C (T a) where+instance Algebraic.C (T a) where+instance Trans.C (T a) where++instance Units.C (T a) where+instance Integral.C (T a) where+instance PID.C (T a) where++instance ZeroTestable.C (T a) where+instance Absolute.C (T a) where+instance (Ord a) => RealField.C (T a) where+instance (Ord a) => RealIntegral.C (T a) where+instance (Ord a) => RealRing.C (T a) where+instance (Ord a) => RealTrans.C (T a) where++instance (Ord a) => ToInteger.C (T a) where+instance (Ord a) => ToRational.C (T a) where+-}++instance Module.C a v => Module.C (T a) (T v) where+   (*>) = lift2 (Module.*>)++instance VectorSpace.C a v => VectorSpace.C (T a) (T v) where++instance Divisible.C a v => Divisible.C (T a) (T v) where+   (</>) = lift2 (Divisible.</>)++instance OccScalar.C a v => OccScalar.C (T a) (T v) where+   toScalar = lift1 OccScalar.toScalar+   toMaybeScalar (Cons a) = fmap Cons (OccScalar.toMaybeScalar a)+   fromScalar = lift1 OccScalar.fromScalar++instance NormEuc.Sqr a v => NormEuc.Sqr (T a) (T v) where+   normSqr = lift1 NormEuc.normSqr++instance NormEuc.C a v => NormEuc.C (T a) (T v) where+   norm = lift1 NormEuc.norm++instance NormMax.C a v => NormMax.C (T a) (T v) where+   norm = lift1 NormMax.norm++instance NormSum.C a v => NormSum.C (T a) (T v) where+   norm = lift1 NormSum.norm+++unimplemented :: String -> a+unimplemented name =+   error (name ++ "cannot be implemented in terms of NumericPrelude type classes")
src/Number/Complex.hs view
@@ -48,12 +48,12 @@         defltPow,         )  where --- import qualified Number.Ratio as Ratio  import qualified Algebra.NormedSpace.Euclidean as NormedEuc import qualified Algebra.NormedSpace.Sum       as NormedSum import qualified Algebra.NormedSpace.Maximum   as NormedMax +import qualified Algebra.OccasionallyScalar as OccScalar import qualified Algebra.VectorSpace        as VectorSpace import qualified Algebra.Module             as Module import qualified Algebra.Vector             as Vector@@ -71,9 +71,7 @@ import qualified Algebra.ZeroTestable       as ZeroTestable import qualified Algebra.Indexable          as Indexable -import Algebra.ZeroTestable(isZero)-import Algebra.Module((*>), (<*>.*>), )-import Algebra.Algebraic((^/), )+import Algebra.Module((<*>.*>), )  import qualified NumericPrelude.Elementwise as Elem import Algebra.Additive ((<*>.+), (<*>.-), (<*>.-$), )@@ -83,8 +81,10 @@ import Control.Applicative (liftA2, )  import Test.QuickCheck (Arbitrary, arbitrary, )-import Control.Monad (liftM2, )+import Control.Monad (liftM2, guard, ) +import qualified MathObj.Wrapper.Haskell98 as W98+ import qualified Prelude as P import NumericPrelude.Base import NumericPrelude.Numeric hiding (signum, exp, )@@ -92,7 +92,6 @@ import Text.Read.HT (readsInfixPrec, )  --- import qualified Data.Typeable as Ty  infix  6  +:, `Cons` @@ -221,7 +220,7 @@ cis :: (Trans.C a) => a -> T a cis theta =  Cons (cos theta) (sin theta) -propPolar :: (RealTrans.C a) => T a -> Bool+propPolar :: (RealTrans.C a, ZeroTestable.C a) => T a -> Bool propPolar z =  uncurry fromPolar (toPolar z) == z  @@ -277,7 +276,7 @@ the magnitude is nonnegative, and the phase in the range @(-'pi', 'pi']@; if the magnitude is zero, then so is the phase. -}-toPolar :: (RealTrans.C a) => T a -> (a,a)+toPolar :: (RealTrans.C a, ZeroTestable.C a) => T a -> (a,a) toPolar z = (magnitude z, phase z)  @@ -320,7 +319,7 @@     {-# INLINE fromInteger #-}     fromInteger                 =  fromReal . fromInteger -instance  (Absolute.C a, Algebraic.C a) => Absolute.C (T a)  where+instance  (Absolute.C a, Algebraic.C a, ZeroTestable.C a) => Absolute.C (T a)  where     {- SPECIALISE instance Absolute.C (T Float) -}     {- SPECIALISE instance Absolute.C (T Double) -}     {-# INLINE abs #-}@@ -361,7 +360,14 @@    {-# INLINE norm #-}    norm x = max (NormedMax.norm (real x)) (NormedMax.norm (imag x)) +instance (Show v, ZeroTestable.C v, Additive.C v, OccScalar.C a v) => OccScalar.C a (T v) where+   toScalar        = OccScalar.toScalarShow+   toMaybeScalar x =+      guard (isZero (imag x)) >>+      OccScalar.toMaybeScalar (real x)+   fromScalar      = fromReal . OccScalar.fromScalar + {-   In this implementation the complex plane is structured   as an orthogonal grid induced by the divisor z'.@@ -478,7 +484,7 @@   {-# INLINE defltPow #-}-defltPow :: (RealTrans.C a) =>+defltPow :: (RealTrans.C a, ZeroTestable.C a) =>     Rational -> T a -> T a defltPow r x =     let (mag,arg) = toPolar x@@ -510,7 +516,7 @@     (^/) = flip power  -instance  (RealRing.C a, RealTrans.C a, Power a) =>+instance  (RealRing.C a, RealTrans.C a, ZeroTestable.C a, Power a) =>           Trans.C (T a)  where     {- SPECIALISE instance Trans.C (T Float) -}     {- SPECIALISE instance Trans.C (T Double) -}@@ -547,29 +553,25 @@ -}  -{- * legacy instances -}--{-# INLINE legacyInstance #-}-legacyInstance :: a-legacyInstance =-   error "legacy Ring.C instance for simple input of numeric literals"+-- * Haskell 98 legacy instances -instance (Ring.C a, Eq a, Show a) => P.Num (T a) where+-- legacy instances for use of numeric literals in GHCi+instance (P.Floating a, Eq a) => P.Num (T a) where    {-# INLINE fromInteger #-}-   fromInteger = fromReal . fromInteger+   fromInteger n = Cons (P.fromInteger n) (P.fromInteger 0)    {-# INLINE negate #-}-   negate = negate -- for unary minus+   negate = W98.unliftF1 Additive.negate    {-# INLINE (+) #-}-   (+)    = legacyInstance+   (+)    = W98.unliftF2 (Additive.+)    {-# INLINE (*) #-}-   (*)    = legacyInstance+   (*)    = W98.unliftF2 (Ring.*)    {-# INLINE abs #-}-   abs    = legacyInstance+   abs    = W98.unliftF1 Absolute.abs    {-# INLINE signum #-}-   signum = legacyInstance+   signum = W98.unliftF1 Absolute.signum -instance (Field.C a, Eq a, Show a) => P.Fractional (T a) where+instance (P.Floating a, Eq a) => P.Fractional (T a) where    {-# INLINE fromRational #-}-   fromRational = fromRational+   fromRational x = Cons (P.fromRational x) (P.fromInteger 0)    {-# INLINE (/) #-}-   (/) = legacyInstance+   (/) = W98.unliftF2 (Field./)
− src/Number/ComplexSquareRoot.hs 
@@ -1,119 +0,0 @@-module Number.ComplexSquareRoot where---- import qualified Algebra.Algebraic as Algebraic-import qualified Algebra.RealField as RealField-import qualified Algebra.RealRing as RealRing--- import qualified Algebra.Field as Field-import qualified Algebra.Ring as Ring-import qualified Algebra.Additive as Additive-import qualified Algebra.ZeroTestable as ZeroTestable--import qualified Number.Complex as Complex--import Algebra.ZeroTestable(isZero, )--import Test.QuickCheck (Arbitrary, arbitrary, )--import Control.Monad (liftM2, )--import qualified NumericPrelude.Numeric as NP-import NumericPrelude.Numeric hiding (recip, )-import NumericPrelude.Base-import Prelude ()--{- |-Represent the square root of a complex number-without actually having to compute a square root.-If the Bool is False,-then the square root is represented with positive real part-or zero real part and positive imaginary part.-If the Bool is True the square root is negated.--}-data T a = Cons Bool (Complex.T a)-   deriving (Show)--{- |-You must use @fmap@ only for number type conversion.--}-instance Functor T where-   fmap f (Cons n x) = Cons n (fmap f x)--instance (ZeroTestable.C a) => ZeroTestable.C (T a) where-   isZero (Cons _b s) = isZero s--instance (ZeroTestable.C a, Eq a) => Eq (T a) where-   (Cons xb xs) == (Cons yb ys) =-      isZero xs && isZero ys  ||-      xb==yb && xs==ys--instance (Arbitrary a) => Arbitrary (T a) where-   arbitrary = liftM2 Cons arbitrary arbitrary---fromNumber :: (RealRing.C a) => Complex.T a -> T a-fromNumber x =-   Cons-      (case compare zero (Complex.real x) of-         LT -> False-         GT -> True-         EQ -> Complex.imag x < zero)-      (x^2)---- htam:Wavelet.DyadicResultant.parityFlip-toNumber :: (RealRing.C a, Complex.Power a) => T a -> Complex.T a-toNumber (Cons n x) =-   case sqrt x of y -> if n then NP.negate y else y---one :: (Ring.C a) => T a-one = Cons False NP.one--inUpperHalfplane :: (Additive.C a, Ord a) => Complex.T a -> Bool-inUpperHalfplane x =-   case compare (Complex.imag x) zero of-      GT -> True-      LT -> False-      EQ -> Complex.real x < zero--mul, mulAlt, mulAlt2 :: (RealRing.C a) => T a -> T a -> T a-mul (Cons xb xs) (Cons yb ys) =-   let zs = xs*ys-   in  Cons-          ((xb /= yb) /=-             case (inUpperHalfplane xs,-                   inUpperHalfplane ys,-                   inUpperHalfplane zs) of-                (True,True,False) -> True-                (False,False,True) -> True-                _ -> False)-          zs--mulAlt (Cons xb xs) (Cons yb ys) =-   let zs = xs*ys-   in  Cons-          ((xb /= yb) /=-             let xi = Complex.imag xs-                 yi = Complex.imag ys-                 zi = Complex.imag zs-             in  (xi>=zero) /= (yi>=zero) &&-                 (xi>=zero) /= (zi>=zero))-          zs--mulAlt2 (Cons xb xs) (Cons yb ys) =-   let zs = xs*ys-   in  Cons-          ((xb /= yb) /=-             let xi = Complex.imag xs-                 yi = Complex.imag ys-                 zi = Complex.imag zs-             in  xi*yi<zero && xi*zi<zero)-          zs--div :: (RealField.C a) => T a -> T a -> T a-div x y = mul x (recip y)--recip :: (RealField.C a) => T a -> T a-recip (Cons b s) =-   Cons-      (b /= (Complex.imag s == zero && Complex.real s < zero))-      (NP.recip s)
src/Number/DimensionTerm.hs view
@@ -1,14 +1,6 @@ {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} {- |-Copyright   :  (c) Henning Thielemann 2008-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable-- See "Algebra.DimensionTerm". -} @@ -20,7 +12,7 @@ import qualified Algebra.Module        as Module import qualified Algebra.Algebraic     as Algebraic import qualified Algebra.Field         as Field-import qualified Algebra.Absolute          as Absolute+import qualified Algebra.Absolute      as Absolute import qualified Algebra.Ring          as Ring import qualified Algebra.Additive      as Additive @@ -31,6 +23,8 @@  import System.Random (Random, randomR, random) +import Control.DeepSeq (NFData(rnf), )+ import Data.Tuple.HT (mapFst, ) import NumericPrelude.Base import Prelude ()@@ -50,6 +44,9 @@       in  showParen (p >= Dim.appPrec)             (showString "DimensionNumber.fromNumberWithDimension " . showsPrec Dim.appPrec u .              showString " " . showsPrec Dim.appPrec z)++instance NFData a => NFData (T u a) where+   rnf (Cons x) = rnf x   fromNumber :: a -> Scalar a
src/Number/DimensionTerm/SI.hs view
@@ -1,12 +1,5 @@ {-# LANGUAGE RebindableSyntax #-} {- |-Copyright   :  (c) Henning Thielemann 2003-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable- Special physical units: SI unit system -} @@ -38,10 +31,8 @@     SI.exa,   SI.zetta, SI.yotta,     ) where --- import qualified Algebra.Transcendental      as Trans import qualified Algebra.Field               as Field --- import qualified Algebra.DimensionTerm as Dim import qualified Number.DimensionTerm  as DN import qualified Number.SI.Unit as SI
src/Number/FixedPoint.hs view
@@ -17,14 +17,13 @@ module Number.FixedPoint where  import qualified Algebra.RealRing    as RealRing--- import qualified Algebra.Additive       as Additive--- import qualified Algebra.ZeroTestable   as ZeroTestable import qualified Algebra.Transcendental as Trans import qualified MathObj.PowerSeries.Example as PSE +import qualified Data.List.Reverse.StrictElement as Rev import NumericPrelude.List (mapLast, ) import Data.Function.HT (powerAssociative, )-import Data.List.HT (dropWhileRev, padLeft, )+import Data.List.HT (padLeft) import Data.Maybe.HT (toMaybe, ) import Data.List (transpose, unfoldr, ) import Data.Char (intToDigit, )@@ -60,7 +59,7 @@        basis = ringPower packetSize 10        (int,frac) = toPositional basis den x    in  show int ++ "." ++-          concat (mapLast (dropWhileRev ('0'==))+          concat (mapLast (Rev.dropWhile ('0'==))              (map (padLeft '0' packetSize . show) frac))  showPositionalHex :: Integer -> Integer -> String@@ -123,6 +122,12 @@ {- Maybe we can speed up the algorithm by calling sqrt recursively on deflated arguments.++ToDo:+The algorithm just computes floor(sqrt(den*x)).+We might factor out the algorithm for (floor.sqrt)+and move it to a different module+together with Fermat factors and so on. -} sqrt :: Integer -> Integer -> Integer sqrt den x =
src/Number/FixedPoint/Check.hs view
@@ -176,19 +176,15 @@   --- legacy instances for work with GHCi-legacyInstance :: a-legacyInstance =-   error "legacy Ring.C instance for simple input of numeric literals"-+-- legacy instances for use of numeric literals in GHCi instance P98.Num T where    fromInteger = fromInteger' defltDenominator-   negate = negate --for unary minus-   (+)    = legacyInstance-   (*)    = legacyInstance-   abs    = legacyInstance-   signum = legacyInstance+   negate = negate -- for unary minus+   (+)    = (+)+   (*)    = (*)+   abs    = abs+   signum = signum  instance P98.Fractional T where    fromRational = fromRational' defltDenominator . fromRational-   (/) = legacyInstance+   (/) = (/)
src/Number/GaloisField2p32m5.hs view
@@ -7,11 +7,12 @@ For 'Rational' this would not be possible.  However, be aware that sums of non-zero elements may yield zero.-Thus division is not always safe, where it is for rational numbers.+Thus division is not always defined, where it is for rational numbers. -} module Number.GaloisField2p32m5 where  import qualified Number.ResidueClass as RC+import qualified Algebra.ZeroTestable as ZeroTestable import qualified Algebra.Module   as Module import qualified Algebra.Field    as Field import qualified Algebra.Ring     as Ring@@ -29,6 +30,30 @@ import NumericPrelude.Numeric  +{- $setup+>>> import qualified Number.GaloisField2p32m5 as GF+>>> import qualified Algebra.Laws as Laws+>>> import Test.QuickCheck ((==>))+>>> import NumericPrelude.Numeric+>>> import NumericPrelude.Base+>>> import Prelude ()+>>>+>>> gf :: GF.T -> GF.T+>>> gf = id+-}++{- |+prop> Laws.identity (+) zero . gf+prop> Laws.commutative (+) . gf+prop> Laws.associative (+) . gf+prop> Laws.inverse (+) negate zero . gf+prop> \x -> Laws.inverse (+) (x-) (gf x)+prop> Laws.identity (*) one . gf+prop> Laws.commutative (*) . gf+prop> Laws.associative (*) . gf+prop> \y -> gf y /= zero ==> Laws.inverse (*) recip one y+prop> \y x -> gf y /= zero ==> Laws.inverse (*) (x/) x y+-} newtype T = Cons {decons :: Word32}    deriving Eq @@ -90,3 +115,6 @@  instance Module.C T T where    (*>) = (*)++instance ZeroTestable.C T where+   isZero x  =  zero == x
src/Number/NonNegative.hs view
@@ -46,7 +46,6 @@  import qualified Algebra.ToInteger          as ToInteger import qualified Algebra.ToRational         as ToRational--- import Test.QuickCheck (Arbitrary(arbitrary))  import qualified Number.Ratio as R @@ -176,7 +175,7 @@    abs    = lift abs    signum = lift signum -instance (RealRing.C a) => RealRing.C (T a) where+instance (ZeroTestable.C a, RealRing.C a) => RealRing.C (T a) where    splitFraction = mapSnd fromNumberUnsafe . splitFraction . toNumber    truncate = truncate . toNumber    round    = round    . toNumber
src/Number/NonNegativeChunky.hs view
@@ -24,8 +24,7 @@ import qualified Numeric.NonNegative.Class as NonNeg98  import qualified Algebra.NonNegative  as NonNeg-import qualified Algebra.Field        as Field-import qualified Algebra.Absolute         as Absolute+import qualified Algebra.Absolute     as Absolute import qualified Algebra.Ring         as Ring import qualified Algebra.Additive     as Additive import qualified Algebra.ToInteger    as ToInteger@@ -33,10 +32,10 @@ import qualified Algebra.IntegralDomain as Integral import qualified Algebra.RealIntegral as RealIntegral import qualified Algebra.ZeroTestable as ZeroTestable-import Algebra.ZeroTestable (isZero, )  import qualified Algebra.Monoid as Monoid import qualified Data.Monoid as Mn98+import qualified Data.Semigroup as Sg98  import Control.Monad (liftM, liftM2, ) import Data.Tuple.HT (mapFst, mapSnd, mapPair, )@@ -45,9 +44,10 @@  import NumericPrelude.Numeric import NumericPrelude.Base-import qualified Prelude as P98 (Num(..), Fractional(..), ) +import qualified Prelude as P98 + {- | A chunky non-negative number is a list of non-negative numbers. It represents the sum of the list elements.@@ -284,27 +284,53 @@   -{- * legacy instances -}+-- * Haskell 98 legacy instances -legacyInstance :: a-legacyInstance =-   error "legacy Ring.C instance for simple input of numeric literals"+fromChunky98_ :: (NonNeg98.C a) => Chunky98.T a -> T a+fromChunky98_ = Cons . Chunky98.toChunks -instance (Ring.C a, Eq a, Show a, NonNeg.C a) => P98.Num (T a) where-   fromInteger = fromNumber . fromInteger-   negate = Additive.negate -- for unary minus-   (+)    = legacyInstance-   (*)    = legacyInstance-   abs    = legacyInstance-   signum = legacyInstance+toChunky98_ :: (NonNeg98.C a) => T a -> Chunky98.T a+toChunky98_ = Chunky98.fromChunks . decons -instance (Field.C a, Eq a, Show a, NonNeg.C a) => P98.Fractional (T a) where-   fromRational = fromNumber . fromRational-   (/) = legacyInstance+fromNumber_ :: a -> T a+fromNumber_ = Cons . (:[]) +{-# INLINE lift98_1 #-}+lift98_1 ::+   (NonNeg98.C a, NonNeg98.C b) =>+   (Chunky98.T a -> Chunky98.T b) -> T a -> T b+lift98_1 f a = fromChunky98_ (f (toChunky98_ a))++{-# INLINE lift98_2 #-}+lift98_2 ::+   (NonNeg98.C a, NonNeg98.C b, NonNeg98.C c) =>+   (Chunky98.T a -> Chunky98.T b -> Chunky98.T c) -> T a -> T b -> T c+lift98_2 f a b = fromChunky98_ (f (toChunky98_ a) (toChunky98_ b))+++{-# INLINE notImplemented #-}+notImplemented :: String -> a+notImplemented name =+   error $ "Number.NonNegativeChunky: method " ++ name ++ " cannot be implemented"++instance (NonNeg98.C a, P98.Num a) => P98.Num (T a) where+   fromInteger = fromNumber_ . P98.fromInteger+   negate = lift98_1 P98.negate+   (+)    = lift98_2 (P98.+)+   (*)    = lift98_2 (P98.*)+   abs    = lift98_1 P98.abs+   signum = lift98_1 P98.signum++instance (NonNeg98.C a, P98.Fractional a) => P98.Fractional (T a) where+   fromRational = fromNumber_ . P98.fromRational+   (/) = notImplemented "(/)"++instance (NonNeg.C a) => Sg98.Semigroup (T a) where+   (<>) = (Monoid.<*>)+ instance (NonNeg.C a) => Mn98.Monoid (T a) where    mempty  = Monoid.idt-   mappend = (Monoid.<*>)+   mappend = (Sg98.<>)  instance (NonNeg.C a) => Monoid.C (T a) where    idt   = Cons []
src/Number/OccasionallyScalarExpression.hs view
@@ -2,13 +2,6 @@ {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} {- |-Copyright   :  (c) Henning Thielemann 2004-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  multi-type parameter classes (vector space)- Physical expressions track the operations made on physical values so we are able to give detailed information on how to resolve unit violations.@@ -24,7 +17,6 @@ import qualified Algebra.Additive            as Additive import qualified Algebra.ZeroTestable        as ZeroTestable -import Algebra.Algebraic (sqrt, (^/)) import qualified Algebra.OccasionallyScalar as OccScalar  import Data.Maybe(fromMaybe)
src/Number/PartiallyTranscendental.hs view
@@ -2,9 +2,9 @@ {- | Define Transcendental functions on arbitrary fields. These functions are defined for only a few (in most cases only one) arguments,-that's why discourage making these types instances of 'Algebra.Transcendental.C'.+that's why we discourage making these types instances of 'Algebra.Transcendental.C'. But instances of 'Algebra.Transcendental.C' can be useful when working with power series.-If you intent to work with power series with 'Rational' coefficients,+If you intend to work with power series with 'Rational' coefficients, you might consider using @MathObj.PowerSeries.T (Number.PartiallyTranscendental.T Rational)@ instead of @MathObj.PowerSeries.T Rational@. -}@@ -15,7 +15,6 @@ import qualified Algebra.Field          as Field import qualified Algebra.Ring           as Ring import qualified Algebra.Additive       as Additive--- import qualified Algebra.ZeroTestable   as ZeroTestable  import NumericPrelude.Numeric import NumericPrelude.Base@@ -74,18 +73,15 @@   -legacyInstance :: a-legacyInstance = error "legacy Ring instance for simple input of numeric literals"-- instance (P.Num a) => P.Num (T a) where-   fromInteger n = lift0 $ P.fromInteger n-   negate = P.negate -- for unary minus-   (+)    = legacyInstance-   (*)    = legacyInstance-   abs    = legacyInstance-   signum = legacyInstance+   fromInteger = lift0 . P.fromInteger+   negate = lift1 P.negate+   (+)    = lift2 (P.+)+   (-)    = lift2 (P.-)+   (*)    = lift2 (P.*)+   abs    = lift1 P.abs+   signum = lift1 P.signum -instance (P.Num a) => P.Fractional (T a) where+instance (P.Fractional a) => P.Fractional (T a) where    fromRational = P.fromRational-   (/) = legacyInstance+   (/) = lift2 (P./)
src/Number/Peano.hs view
@@ -1,6 +1,6 @@ {-# LANGUAGE RebindableSyntax #-} {- |-Copyright    :   (c) Henning Thielemann 2007+Copyright    :   (c) Henning Thielemann 2007-2012 Maintainer   :   numericprelude@henning-thielemann.de Stability    :   provisional Portability  :   portable@@ -32,12 +32,10 @@ import Data.Maybe (catMaybes, ) import Data.Array(Ix(..)) -import qualified Prelude     as P98-import qualified NumericPrelude.Base as P-import qualified NumericPrelude.Numeric as NP import Data.List.HT (mapAdjacent, shearTranspose, ) import Data.Tuple.HT (mapFst, ) +import qualified Prelude as P98 import NumericPrelude.Base import NumericPrelude.Numeric @@ -401,9 +399,10 @@   -legacyInstance :: a-legacyInstance =-   error "legacy Ring.C instance for simple input of numeric literals"+{-# INLINE notImplemented #-}+notImplemented :: String -> a+notImplemented name =+   error $ "Number.Peano: method " ++ name ++ " cannot be implemented"  instance P98.Num T where    fromInteger = Ring.fromInteger@@ -411,8 +410,8 @@    (+) = add    (-) = sub    (*) = mul-   signum = legacyInstance-   abs = legacyInstance+   abs    = notImplemented "abs"+   signum = notImplemented "signum"  -- for use with genericLength et.al. instance P98.Real T where
src/Number/Physical.hs view
@@ -2,13 +2,6 @@ {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} {- |-Copyright   :  (c) Henning Thielemann 2003-2006-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  generic instances- Numeric values combined with abstract Physical Units -} @@ -23,18 +16,17 @@ import qualified Algebra.Transcendental      as Trans import qualified Algebra.Algebraic           as Algebraic import qualified Algebra.Field               as Field-import qualified Algebra.Absolute                as Absolute+import qualified Algebra.Absolute            as Absolute import qualified Algebra.Ring                as Ring import qualified Algebra.Additive            as Additive import qualified Algebra.ZeroTestable        as ZeroTestable  import qualified Algebra.ToInteger      as ToInteger -import Algebra.Algebraic (sqrt, (^/))- import qualified Number.Ratio as Ratio -import Control.Monad(guard,liftM,liftM2)+import Control.Monad (guard, liftM, liftM2, ap)+import Control.Applicative (Applicative(pure, (<*>)))  import Data.Maybe.HT(toMaybe) import Data.Maybe(fromMaybe)@@ -228,9 +220,13 @@     then fromScalarSingle (f x)     else error "Physics.Quantity.Value.fmap: function for scalars, only" +instance Applicative (T a) where+   (<*>) = ap+   pure = fromScalarSingle+ instance Monad (T i) where-  (>>=) (Cons xu x) f =+   (>>=) (Cons xu x) f =     if Unit.isScalar xu     then f x     else error "Physics.Quantity.Value.(>>=): function for scalars, only"-  return = fromScalarSingle+   return = pure
src/Number/Physical/Read.hs view
@@ -1,12 +1,5 @@ {-# LANGUAGE RebindableSyntax #-} {- |-Copyright   :  (c) Henning Thielemann 2004-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  multi-parameter type classes (VectorSpace.hs)- Convert a human readable string to a physical value. -} @@ -15,7 +8,6 @@ import qualified Number.Physical        as Value import qualified Number.Physical.UnitDatabase as Db import qualified Algebra.VectorSpace as VectorSpace--- import Algebra.Module((*>)) import qualified Algebra.Field       as Field import qualified Data.Map as Map import Data.Map (Map)
src/Number/Physical/Show.hs view
@@ -1,12 +1,5 @@ {-# LANGUAGE RebindableSyntax #-} {- |-Copyright   :  (c) Henning Thielemann 2004-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  multi-parameter type classes (VectorSpace.hs, Normalization.hs)- Convert a physical value to a human readable string. -}
src/Number/Physical/Unit.hs view
@@ -1,12 +1,5 @@ {-# LANGUAGE RebindableSyntax #-} {- |-Copyright   :  (c) Henning Thielemann 2003-2006-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable- Abstract Physical Units -} @@ -30,8 +23,8 @@     Example: Let the quantity of length (meter, m) be the zeroth dimension    and let the quantity of time (second, s) be the first dimension,-   then the composed unit "m_s²" corresponds to the Map-   [(0,1),(1,-2)]+   then the composed unit @m/s^2@ corresponds to the Map+   @[(0,1),(1,-2)]@.     In future I want to have more abstraction here,    e.g. a type class from the Edison project@@ -78,7 +71,7 @@                in  toMaybe (denominator y == 1) (numerator y))  -{- impossible because Unit.T is a type synonyme but not a data type+{- impossible because Unit.T is a type synonym but not a data type instance Show (Unit.T i) where   show = show.toVector -}
src/Number/Physical/UnitDatabase.hs view
@@ -1,12 +1,5 @@ {-# LANGUAGE RebindableSyntax #-} {- |-Copyright   :  (c) Henning Thielemann 2003-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable- Tools for creating a data base of physical units and for extracting data from it -}@@ -16,7 +9,6 @@ import qualified Number.Physical.Unit as Unit import qualified Algebra.Field as Field --- import Algebra.Module((*>)) import Algebra.NormedSpace.Sum(norm)  import Data.Maybe.HT (toMaybe)
src/Number/Positional.hs view
@@ -1,12 +1,5 @@ {-# LANGUAGE RebindableSyntax #-} {- |-Copyright   :  (c) Henning Thielemann 2006-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-- Exact Real Arithmetic - Computable reals. Inspired by ''The most unreliable technique for computing pi.'' See also <http://www.haskell.org/haskellwiki/Exact_real_arithmetic> .@@ -18,7 +11,6 @@  import qualified Algebra.IntegralDomain as Integral import qualified Algebra.Ring           as Ring--- import qualified Algebra.Additive       as Additive import qualified Algebra.ToInteger      as ToInteger  import qualified Prelude as P98@@ -428,7 +420,7 @@        This would create finite representations        in some cases (input is finite, and the result is finite)        but will cause infinite loop otherwise.-       dropWhileRev (0==) . compressMant bDst+       Rev.dropWhile (0==) . compressMant bDst        -}        cmpr (mag,xs) = (mag - unit, compressMant bSrc xs) @@ -596,20 +588,20 @@  {- * arithmetic -} -fromLaurent :: LPoly.T Int -> T+fromLaurent :: LPoly.T Digit -> T fromLaurent (LPoly.Cons nxe xm) = (NP.negate nxe, xm) -toLaurent :: T -> LPoly.T Int+toLaurent :: T -> LPoly.T Digit toLaurent (xe, xm) = LPoly.Cons (NP.negate xe) xm  liftLaurent2 ::-   (LPoly.T Int -> LPoly.T Int -> LPoly.T Int) ->+   (LPoly.T Digit -> LPoly.T Digit -> LPoly.T Digit) ->       (T -> T -> T) liftLaurent2 f x y =    fromLaurent (f (toLaurent x) (toLaurent y))  liftLaurentMany ::-   ([LPoly.T Int] -> LPoly.T Int) ->+   ([LPoly.T Digit] -> LPoly.T Digit) ->       ([T] -> T) liftLaurentMany f =    fromLaurent . f . map toLaurent@@ -780,7 +772,9 @@    let (ye,ym) = until ((>=b) . abs . head . snd)                        (decreaseExp b)                        (ye',ym')-   in  nest 3 trimOnce (compress b (xe-ye, divMant b ym xm))+   in  if null xm+         then (xe,xm)+         else nest 3 trimOnce (compress b (xe-ye, divMant b ym xm))  divMant :: Basis -> Mantissa -> Mantissa -> Mantissa divMant _ [] _   = error "Number.Positional: division by zero"@@ -818,7 +812,7 @@ Fast division for small integral divisors, which occur for instance in summands of power series. -}-divIntMant :: Basis -> Int -> Mantissa -> Mantissa+divIntMant :: Basis -> Digit -> Mantissa -> Mantissa divIntMant b y xInit =    List.unfoldr (\(r,rxs) ->              let rb = r*b@@ -831,7 +825,7 @@            (0,xInit)  -- this version is simple but ignores the possibility of a terminating result-divIntMantInf :: Basis -> Int -> Mantissa -> Mantissa+divIntMantInf :: Basis -> Digit -> Mantissa -> Mantissa divIntMantInf b y =    map fst . tail .       scanl (\(_,r) x -> divMod (r*b+x) y) (undefined,0) .@@ -1315,7 +1309,7 @@ {- | Efficient computation of Arcus tangens of an argument of the form @1\/n@. -}-arctanStem :: Basis -> Int -> T+arctanStem :: Basis -> Digit -> T arctanStem b n =    let x = (0, divIntMant b n [1])        divN2 = divInt b n . divInt b (-n)
src/Number/Positional/Check.hs view
@@ -1,12 +1,5 @@ {-# LANGUAGE RebindableSyntax #-} {- |-Copyright   :  (c) Henning Thielemann 2006-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-- Interface to "Number.Positional" which dynamically checks for equal bases. -} module Number.Positional.Check where@@ -15,7 +8,6 @@  import qualified Number.Complex as Complex --- import qualified Algebra.Module             as Module import qualified Algebra.RealTranscendental as RealTrans import qualified Algebra.Transcendental     as Trans import qualified Algebra.Algebraic          as Algebraic@@ -30,7 +22,6 @@ import qualified Algebra.EqualityDecision as EqDec import qualified Algebra.OrderDecision    as OrdDec -import qualified NumericPrelude.Base as P import qualified Prelude     as P98  import NumericPrelude.Base as P@@ -55,7 +46,7 @@ and cannot be made unique in finite time. This way we avoid infinite carry ripples. -}-data T = Cons {base :: Int, exponent :: Int, mantissa :: Pos.Mantissa}+data T = Cons {base :: Pos.Basis, exponent :: Int, mantissa :: Pos.Mantissa}    deriving (Show)  @@ -81,7 +72,7 @@    in  prependDigit (fst (head ys)) (Cons b ex digits)  -prependDigit :: Int -> T -> T+prependDigit :: Pos.Digit -> T -> T prependDigit 0 x = x prependDigit x (Cons b ex xs) =    Cons b (ex+1) (x:xs)@@ -90,15 +81,15 @@  {- * conversions -} -lift0 :: (Int -> Pos.T) -> T+lift0 :: (Pos.Basis -> Pos.T) -> T lift0 op =    uncurry (Cons defltBase) (op defltBase) -lift1 :: (Int -> Pos.T -> Pos.T) -> T -> T+lift1 :: (Pos.Basis -> Pos.T -> Pos.T) -> T -> T lift1 op (Cons xb xe xm) =    uncurry (Cons xb) (op xb (xe, xm)) -lift2 :: (Int -> Pos.T -> Pos.T -> Pos.T) -> T -> T -> T+lift2 :: (Pos.Basis -> Pos.T -> Pos.T -> Pos.T) -> T -> T -> T lift2 op (Cons xb xe xm) (Cons yb ye ym) =    let b = commonBasis xb yb    in  uncurry (Cons b) (op b (xe, xm) (ye, ym))@@ -116,11 +107,11 @@      then xb      else error "Number.Positional: bases differ" -fromBaseInteger :: Int -> Integer -> T+fromBaseInteger :: Pos.Basis -> Integer -> T fromBaseInteger b n =    uncurry (Cons b) (Pos.fromBaseInteger b n) -fromBaseRational :: Int -> Rational -> T+fromBaseRational :: Pos.Basis -> Rational -> T fromBaseRational b r =    uncurry (Cons b) (Pos.fromBaseRational b r) @@ -237,22 +228,18 @@   --- legacy instances for work with GHCi-legacyInstance :: a-legacyInstance =-   error "legacy Ring.C instance for simple input of numeric literals"-+-- legacy instances for use of numeric literals in GHCi instance P98.Num T where    fromInteger = fromBaseInteger defltBase-   negate = negate --for unary minus-   (+)    = legacyInstance-   (*)    = legacyInstance-   abs    = legacyInstance-   signum = legacyInstance+   negate = negate -- for unary minus+   (+)    = (+)+   (*)    = (*)+   abs    = abs+   signum = signum  instance P98.Fractional T where    fromRational = fromBaseRational defltBase . fromRational-   (/) = legacyInstance+   (/) = (/)   {-
src/Number/Quaternion.hs view
@@ -46,8 +46,7 @@ import qualified Algebra.Additive     as Additive import qualified Algebra.ZeroTestable as ZeroTestable -import Algebra.ZeroTestable(isZero)-import Algebra.Module((*>), (<*>.*>), )+import Algebra.Module((<*>.*>), )  import qualified Number.Complex as Complex @@ -56,11 +55,9 @@ import qualified NumericPrelude.Elementwise as Elem import Algebra.Additive ((<*>.+), (<*>.-), (<*>.-$), ) --- import qualified Data.Typeable as Ty import Data.Array (Array, (!)) import qualified Data.Array as Array -import qualified Prelude as P import NumericPrelude.Base import NumericPrelude.Numeric hiding (signum) import Text.Show.HT (showsInfixPrec, )@@ -104,34 +101,34 @@  -- | The conjugate of a quaternion. {-# SPECIALISE conjugate :: T Double -> T Double #-}-conjugate	 :: (Additive.C a) => T a -> T a+conjugate        :: (Additive.C a) => T a -> T a conjugate (Cons r i) =  Cons r (negate i)  -- | Scale a quaternion by a real number. {-# SPECIALISE scale :: Double -> T Double -> T Double #-}-scale		 :: (Ring.C a) => a -> T a -> T a+scale            :: (Ring.C a) => a -> T a -> T a scale r (Cons xr xi) =  Cons (r * xr) (scaleImag r xi)  -- | like Module.*> but without additional class dependency-scaleImag	 :: (Ring.C a) => a -> (a,a,a) -> (a,a,a)+scaleImag        :: (Ring.C a) => a -> (a,a,a) -> (a,a,a) scaleImag r (xi,xj,xk) =  (r * xi, r * xj, r * xk)  -- | the same as NormedEuc.normSqr but with a simpler type class constraint-normSqr		 :: (Ring.C a) => T a -> a+normSqr          :: (Ring.C a) => T a -> a normSqr (Cons xr xi) = xr*xr + scalarProduct xi xi -norm		 :: (Algebraic.C a) => T a -> a+norm             :: (Algebraic.C a) => T a -> a norm x = sqrt (normSqr x)  -- | scale a quaternion into a unit quaternion-normalize	 :: (Algebraic.C a) => T a -> T a+normalize        :: (Algebraic.C a) => T a -> T a normalize x = scale (recip (norm x)) x -scalarProduct	 :: (Ring.C a) => (a,a,a) -> (a,a,a) -> a+scalarProduct    :: (Ring.C a) => (a,a,a) -> (a,a,a) -> a scalarProduct (xi,xj,xk) (yi,yj,yk) =    xi*yi + xj*yj + xk*yk -crossProduct	 :: (Ring.C a) => (a,a,a) -> (a,a,a) -> (a,a,a)+crossProduct     :: (Ring.C a) => (a,a,a) -> (a,a,a) -> (a,a,a) crossProduct (xi,xj,xk) (yi,yj,yk) =    (xj*yk - xk*yj, xk*yi - xi*yk, xi*yj - xj*yi) @@ -141,11 +138,11 @@ @similarity (cos(a\/2) +:: scaleImag (sin(a\/2)) v) (0 +:: x) == (0 +:: y)@ where @y@ results from rotating @x@ around the axis @v@ by the angle @a@. -}-similarity	 :: (Field.C a) => T a -> T a -> T a+similarity       :: (Field.C a) => T a -> T a -> T a similarity c x = c*x/c  {--rotate	 :: (Field.C a) =>+rotate   :: (Field.C a) =>       (a,a,a)  {- ^ rotation axis, must be normalized -}    -> T a    -> T a@@ -266,9 +263,9 @@ instance (Ring.C a) => Ring.C (T a)  where    {-# SPECIALISE instance Ring.C (T Float) #-}    {-# SPECIALISE instance Ring.C (T Double) #-}-   one				=  Cons one zero-   fromInteger			=  fromReal . fromInteger-   (Cons xr xi) * (Cons yr yi)	=+   one                          =  Cons one zero+   fromInteger                  =  fromReal . fromInteger+   (Cons xr xi) * (Cons yr yi)  =        Cons (xr*yr - scalarProduct xi yi)             (scaleImag xr yi + scaleImag yr xi +              crossProduct xi yi)
src/Number/Ratio.hs view
@@ -1,7 +1,8 @@ {-# LANGUAGE RebindableSyntax #-} {- | Module      :  Number.Ratio-Copyright   :  (c) Henning Thielemann, Dylan Thurston 2006+Copyright   :  (c) Henning Thielemann 2011-2012+               (c) Dylan Thurston 2006  Maintainer  :  numericprelude@henning-thielemann.de Stability   :  provisional@@ -11,10 +12,11 @@ -}  module Number.Ratio-	(-	  T((:%), numerator, denominator), (%),+        (+          T((:%), numerator, denominator), (%),           Rational,           fromValue,+          recip,            scale,           split,@@ -24,6 +26,7 @@         )  where  import qualified Algebra.PrincipalIdealDomain as PID+import qualified Algebra.Units                as Unit import qualified Algebra.Absolute             as Absolute import qualified Algebra.Ring                 as Ring import qualified Algebra.Additive             as Additive@@ -117,7 +120,13 @@     abs (x:%y)          =  Absolute.abs x :% y     signum (x:%_)       =  Absolute.signum x :% one +recip :: (ZeroTestable.C a, Unit.C a) => T a -> T a+recip (x:%y) =+   if isZero y+     then error "Ratio.recip: division by zero"+     else (y * stdUnitInv x) :% stdAssociate x + liftOrd :: Ring.C a => (a -> a -> b) -> (T a -> T a -> b) liftOrd f (x:%y) (x':%y') = f (x * y') (x' * y) @@ -222,28 +231,54 @@  -- | Necessary when mixing NumericPrelude.Numeric Rationals with Prelude98 Rationals -toRational98 :: (P.Integral a, PID.C a) => T a -> Ratio98.Ratio a+toRational98 :: (P.Integral a) => T a -> Ratio98.Ratio a toRational98 x = numerator x Ratio98.% denominator x +fromRational98 :: (P.Integral a) => Ratio98.Ratio a -> T a+fromRational98 x = Ratio98.numerator x :% Ratio98.denominator x -legacyInstance :: String -> a-legacyInstance op =-   error ("Ratio." ++ op ++ ": legacy Ring instance for simple input of numeric literals") +{-# INLINE lift1 #-}+lift1 ::+   (P.Integral a, P.Integral b) =>+   (Ratio98.Ratio a -> Ratio98.Ratio b) -> T a -> T b+lift1 f a = fromRational98 (f (toRational98 a)) --- instance (P.Num a, PID.C a) => P.Num (T a) where-instance (P.Num a, PID.C a, Absolute.C a) => P.Num (T a) where+{-# INLINE lift2 #-}+lift2 ::+   (P.Integral a, P.Integral b, P.Integral c) =>+   (Ratio98.Ratio a -> Ratio98.Ratio b -> Ratio98.Ratio c) -> T a -> T b -> T c+lift2 f a b = fromRational98 (f (toRational98 a) (toRational98 b))+++instance (P.Integral a) => P.Num (T a) where+   fromInteger n = P.fromInteger n :% P.fromInteger 1+   negate = lift1 P.negate+   (+)    = lift2 (P.+)+   (*)    = lift2 (P.*)+   abs    = lift1 P.abs+   signum = lift1 P.signum++instance (P.Integral a) => P.Fractional (T a) where+   fromRational x =+      P.fromInteger (Ratio98.numerator x) :%+      P.fromInteger (Ratio98.denominator x)+   (/) = lift2 (P./)+   recip = lift1 P.recip++{- causes an import cycle+instance (P.Integral a) => P.Num (T a) where    fromInteger n = P.fromInteger n % 1-   negate = negate -- for unary minus-   (+)    = legacyInstance "(+)"-   (*)    = legacyInstance "(*)"-   abs    = Absolute.abs -- needed for Arbitrary instance of NonNegative.Ratio-   signum = legacyInstance "signum"+   negate = W98.unliftF1 P.negate+   (+)    = W98.unliftF2 (+)+   (*)    = W98.unliftF2 (*)+   abs    = W98.unliftF1 abs+   signum = W98.unliftF1 P.signum --- instance (P.Num a, PID.C a) => P.Fractional (T a) where-instance (P.Num a, PID.C a, Absolute.C a) => P.Fractional (T a) where+instance (P.Integral a) => P.Fractional (T a) where --   fromRational = Field.fromRational    fromRational x =       fromInteger (Ratio98.numerator x) :%       fromInteger (Ratio98.denominator x)-   (/) = legacyInstance "(/)"+   recip = recip+-}
src/Number/ResidueClass.hs view
@@ -3,8 +3,6 @@  import qualified Algebra.PrincipalIdealDomain as PID import qualified Algebra.IntegralDomain as Integral--- import qualified Algebra.Additive       as Additive--- import qualified Algebra.ZeroTestable   as ZeroTestable  import NumericPrelude.Base import NumericPrelude.Numeric hiding (recip)
src/Number/ResidueClass/Check.hs view
@@ -101,18 +101,18 @@     isZero (Cons _ r)   =  isZero r  instance  (Eq a, Integral.C a) => Additive.C (T a)  where-    zero		=  error "no generic zero in a residue class, use ResidueClass.zero"-    (+)			=  lift2 Res.add-    (-)			=  lift2 Res.sub-    negate		=  lift1 Res.neg+    zero                =  error "no generic zero in a residue class, use ResidueClass.zero"+    (+)                 =  lift2 Res.add+    (-)                 =  lift2 Res.sub+    negate              =  lift1 Res.neg  instance  (Eq a, Integral.C a) => Ring.C (T a)  where-    one			=  error "no generic one in a residue class, use ResidueClass.one"-    (*)			=  lift2 Res.mul-    fromInteger		=  error "no generic integer in a residue class, use ResidueClass.fromInteger"+    one                 =  error "no generic one in a residue class, use ResidueClass.one"+    (*)                 =  lift2 Res.mul+    fromInteger         =  error "no generic integer in a residue class, use ResidueClass.fromInteger"     x^n                 =  Func.powerAssociative (*) (one (modulus x)) x n  instance  (Eq a, PID.C a) => Field.C (T a)  where-    (/)			=  lift2 Res.divide+    (/)                 =  lift2 Res.divide     recip               =  lift1 (flip Res.divide Ring.one)-    fromRational'	=  error "no conversion from rational to residue class"+    fromRational'       =  error "no conversion from rational to residue class"
src/Number/ResidueClass/Func.hs view
@@ -11,6 +11,9 @@ import qualified Algebra.EqualityDecision as EqDec  import Algebra.EqualityDecision ((==?), )++import qualified MathObj.Wrapper.Haskell98 as W98+ import NumericPrelude.Base import NumericPrelude.Numeric hiding (zero, one, ) @@ -61,20 +64,20 @@        Cons (\m -> (x m ==? y m) (eq m) (noteq m))  instance  (Integral.C a) => Additive.C (T a)  where-    zero		=  zero-    (+)			=  lift2 Res.add-    (-)			=  lift2 Res.sub-    negate		=  lift1 Res.neg+    zero                =  zero+    (+)                 =  lift2 Res.add+    (-)                 =  lift2 Res.sub+    negate              =  lift1 Res.neg  instance  (Integral.C a) => Ring.C (T a)  where-    one			=  one-    (*)			=  lift2 Res.mul-    fromInteger		=  Number.ResidueClass.Func.fromInteger+    one                 =  one+    (*)                 =  lift2 Res.mul+    fromInteger         =  Number.ResidueClass.Func.fromInteger  instance  (PID.C a) => Field.C (T a)  where-    (/)			=  lift2 Res.divide+    (/)                 =  lift2 Res.divide     recip               =  (NP.one /)-    fromRational'	=  error "no conversion from rational to residue class"+    fromRational'       =  error "no conversion from rational to residue class"   {-@@ -82,21 +85,32 @@ But Prelude.fromInteger requires Prelude.Num instance. -} --- legacy instances for work with GHCi-legacyInstance :: a-legacyInstance =-   error "legacy Ring.C instance for simple input of numeric literals"+{-# INLINE notImplemented #-}+notImplemented :: String -> a+notImplemented name =+   error $ "ResidueClass.Func: method " ++ name ++ " cannot be implemented" -instance (P.Num a, Integral.C a) => P.Num (T a) where-   fromInteger = Number.ResidueClass.Func.fromInteger-   negate = negate --for unary minus-   (+)    = legacyInstance-   (*)    = legacyInstance-   abs    = legacyInstance-   signum = legacyInstance +lift98_1 :: (W98.T a -> W98.T a -> W98.T a) -> T a -> T a+lift98_1 f (Cons x) =+   Cons $ \m -> W98.decons $ f (W98.Cons m) (W98.Cons $ x m)++lift98_2 :: (W98.T a -> W98.T a -> W98.T a -> W98.T a) -> T a -> T a -> T a+lift98_2 f (Cons x) (Cons y) =+   Cons $ \m -> W98.decons $ f (W98.Cons m) (W98.Cons $ x m) (W98.Cons $ y m)+++-- legacy instances for use of numeric literals in GHCi+instance (P.Integral a) => P.Num (T a) where+   fromInteger = Cons . P.mod . P.fromInteger+   negate = lift98_1 Res.neg+   (+)    = lift98_2 Res.add+   (*)    = lift98_2 Res.mul+   abs    = notImplemented "abs"+   signum = notImplemented "signum"+ instance Eq (T a) where-   (==) = error "ResidueClass.Func: (==) not implemented"+   (==) = notImplemented "(==)"  instance Show (T a) where-   show = error "ResidueClass.Func: 'show' not implemented"+   show = notImplemented "show"
src/Number/ResidueClass/Maybe.hs view
@@ -69,12 +69,12 @@         else error "ResidueClass.(==): Incompatible operands"  instance  (Eq a, Integral.C a) => Additive.C (T a)  where-    zero		=  Cons Nothing zero-    (+)			=  lift2 Res.add (+)-    (-)			=  lift2 Res.sub (-)-    negate (Cons m r)	=  Cons m (negate r)+    zero                =  Cons Nothing zero+    (+)                 =  lift2 Res.add (+)+    (-)                 =  lift2 Res.sub (-)+    negate (Cons m r)   =  Cons m (negate r)  instance  (Eq a, Integral.C a) => Ring.C (T a)  where-    one			=  Cons Nothing one-    (*)			=  lift2 Res.mul (*)-    fromInteger		=  Cons Nothing . fromInteger+    one                 =  Cons Nothing one+    (*)                 =  lift2 Res.mul (*)+    fromInteger         =  Cons Nothing . fromInteger
src/Number/ResidueClass/Reader.hs view
@@ -11,10 +11,9 @@ import NumericPrelude.Base import NumericPrelude.Numeric -import Control.Monad (liftM2, liftM4)--- import Control.Monad.Reader (MonadReader)+import Control.Monad (liftM, liftM2, liftM4, ap)+import Control.Applicative (Applicative(pure, (<*>))) -import qualified Prelude        as P import qualified NumericPrelude.Numeric as NP  @@ -41,9 +40,16 @@ fromInteger = fromRepresentative . NP.fromInteger  +instance Functor (T a) where+   fmap = liftM++instance Applicative (T a) where+   (<*>) = ap+   pure = Cons . const+ instance Monad (T a) where    (Cons x) >>= y  =  Cons (\r -> toFunc (y (x r)) r)-   return = Cons . const+   return = pure
src/Number/Root.hs view
@@ -1,3 +1,8 @@+{-+ToDo:+having the root exponent as type-level number would be nice+there is a package for basic type-level number support+-} module Number.Root where  import qualified Algebra.Algebraic as Algebraic
src/Number/SI.hs view
@@ -1,14 +1,8 @@ {-# LANGUAGE RebindableSyntax #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-}+{-# LANGUAGE GeneralizedNewtypeDeriving #-} {- |-Copyright   :  (c) Henning Thielemann 2003-2006-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable- Numerical values equipped with SI units. This is considered as the user front-end. -}@@ -38,10 +32,10 @@ import qualified Algebra.Additive            as Additive import qualified Algebra.ZeroTestable        as ZeroTestable -import Algebra.Algebraic (sqrt, (^/), )- import Data.Tuple.HT (mapFst, ) +import qualified MathObj.Wrapper.Haskell98 as W98+ import qualified Prelude as P  import NumericPrelude.Numeric@@ -49,9 +43,7 @@   newtype T a v = Cons (PValue v)-{- LANGUAGE GeneralizedNewtypeDeriving allows even this-   deriving (Monad, Functor)--}+   deriving (Functor)  type PValue v = Value.T Dimension v @@ -251,21 +243,14 @@   --- legacy instances for work with GHCi-legacyInstance :: a-legacyInstance =-   error "legacy Ring.C instance for simple input of numeric literals"--instance (Ord a, Trans.C a, NormedMax.C a v, P.Num v, Ring.C v) =>-      P.Num (T a v) where-   fromInteger = fromInteger-   negate = negate -- for unary minus-   (+)    = legacyInstance-   (*)    = legacyInstance-   abs    = legacyInstance-   signum = legacyInstance+instance (P.Num v) => P.Num (T a v) where+   fromInteger = fromScalarSingle . P.fromInteger+   negate = W98.unliftF1 Additive.negate+   (+)    = W98.unliftF2 (Additive.+)+   (*)    = W98.unliftF2 (Ring.*)+   abs    = W98.unliftF1 Absolute.abs+   signum = W98.unliftF1 Absolute.signum -instance (Ord a, Trans.C a, NormedMax.C a v, P.Num v, Field.C v) =>-      P.Fractional (T a v) where-   fromRational = fromRational-   (/) = legacyInstance+instance (P.Fractional v) => P.Fractional (T a v) where+   fromRational = fromScalarSingle . P.fromRational+   (/) = W98.unliftF2 (Field./)
src/Number/SI/Unit.hs view
@@ -1,12 +1,5 @@ {-# LANGUAGE RebindableSyntax #-} {- |-Copyright   :  (c) Henning Thielemann 2003-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable- Special physical units: SI unit system -}
src/NumericPrelude/Base.hs view
@@ -3,11 +3,136 @@ to reexport items that we want from the standard Prelude. -} -module NumericPrelude.Base (module Prelude, ifThenElse, ) where-import Prelude hiding (-       Int, Integer, Float, Double, Rational, Num(..), Real(..),-       Integral(..), Fractional(..), Floating(..), RealFrac(..),-       RealFloat(..), subtract, even, odd,-       gcd, lcm, (^), (^^), sum, product,-       fromIntegral, fromRational, )+module NumericPrelude.Base (+   (P.!!),+   (P.$),+   (P.$!),+   (P.&&),+   (P.++),+   (P..),+   (P.=<<),+   P.Bool(..),+   P.Bounded(..),+   P.Char,+   P.Either(..),+   P.Enum(..),+   P.Eq(..),+   P.FilePath,+   P.Functor(..),+   P.IO,+   P.IOError,+   P.Maybe(..),+   P.Monad(..), P.fail,+   P.Ord(..),+   P.Ordering(..),+   P.Read(..),+   P.ReadS,+   P.Show(..),+   P.ShowS,+   P.String,+   P.all,+   P.and,+   P.any,+   P.appendFile,+   P.asTypeOf,+   P.break,+   P.concat,+   P.concatMap,+   P.const,+   P.curry,+   P.cycle,+   P.drop,+   P.dropWhile,+   P.either,+   P.elem,+   P.error,+   P.filter,+   P.flip,+   P.foldl,+   P.foldl1,+   P.foldr,+   P.foldr1,+   P.fst,+   P.getChar,+   P.getContents,+   P.getLine,+   P.head,+   P.id,+   P.init,+   P.interact,+   P.ioError,+   P.iterate,+   P.last,+   P.length,+   P.lex,+   P.lines,+   P.lookup,+   P.map,+   P.mapM,+   P.mapM_,+   P.maximum,+   P.maybe,+   P.minimum,+   P.not,+   P.notElem,+   P.null,+   P.or,+   P.otherwise,+   P.print,+   P.putChar,+   P.putStr,+   P.putStrLn,+   P.read,+   P.readFile,+   P.readIO,+   P.readLn,+   P.readParen,+   P.reads,+   P.realToFrac,+   P.repeat,+   P.replicate,+   P.reverse,+   P.scanl,+   P.scanl1,+   P.scanr,+   P.scanr1,+   P.seq,+   P.sequence,+   P.sequence_,+   P.showChar,+   P.showParen,+   P.showString,+   P.shows,+   P.snd,+   P.span,+   P.splitAt,+   P.tail,+   P.take,+   P.takeWhile,+   P.uncurry,+   P.undefined,+   P.unlines,+   P.until,+   P.unwords,+   P.unzip,+   P.unzip3,+   P.userError,+   P.words,+   P.writeFile,+   P.zip,+   P.zip3,+   P.zipWith,+   P.zipWith3,+   (P.||),++   catch,+   ifThenElse,+   ) where++import qualified System.IO.Error as IOError+import qualified Prelude as P+import Prelude (IO) import Data.Bool.HT (ifThenElse, )++catch :: IO a -> (P.IOError -> IO a) -> IO a+catch = IOError.catchIOError
src/NumericPrelude/List.hs view
@@ -27,7 +27,7 @@    in  aux  {--This is exported Checked.zipWith.+This is exported as Checked.zipWith. We need to define it here in order to prevent an import cycle. -} zipWithChecked
src/NumericPrelude/List/Checked.hs view
@@ -13,7 +13,6 @@    ) where  import qualified Algebra.ToInteger  as ToInteger--- import qualified Algebra.Ring       as Ring import Algebra.Ring (one, ) import Algebra.Additive (zero, (-), )
− test-ghc-6.12/Gaussian.hs 
@@ -1,6 +0,0 @@-module Main where--import qualified MathObj.Gaussian.Example as Example--main :: IO ()-main = Example.polyApprox
− test-ghc-6.12/Test.hs 
@@ -1,173 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Main where--import Number.Complex((+:), (-:), )-import qualified Number.Complex as Complex-import Number.Physical      as Value-import Number.SI            as SIValue -- units-import Number.SI.Unit       as SIUnit  -- unit prefixes-          (pico, nano, micro, milli, centi, deci,-           deca, hecto, kilo, mega, giga, tera, peta)-import Number.OccasionallyScalarExpression as Expr--import qualified Number.Positional.Check  as Absolute-import qualified Number.FixedPoint.Check  as FixedPoint-import qualified Number.ResidueClass.Func as ResidueClass-import qualified Number.Peano             as Peano--import qualified MathObj.Polynomial          as Polynomial-import qualified MathObj.LaurentPolynomial   as LaurentPolynomial-import qualified MathObj.PowerSeries         as PowerSeries-import qualified MathObj.PowerSeries.Example as PowerSeriesExample-import qualified MathObj.PartialFraction     as PartialFraction--import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.Field                as Field-import qualified Algebra.ZeroTestable         as ZeroTestable-import qualified Algebra.Indexable            as Indexable--import Data.List (genericTake, genericLength)--import NumericPrelude.Base-import NumericPrelude.Numeric---{- * Physical units -}---- some shorthands for common usage-type SIDouble  = SIValue.T Double Double-type SIComplex = SIValue.T Double (Complex.T Double)--{- this advice seems not to be targeted to ghc's interactive mode-default (SIDouble)--}-----test :: [SIDouble]-test =-   let lengthScales = map (\n->10^-n*meter) [-10..6]-       areaScales = map (\n->10^-n*meter^2) [-10..6]-   in  lengthScales ++ map recip lengthScales ++-       areaScales   ++ map recip areaScales ++-       map ((meter*gramm/second)^-) [-5..5] ++-       take 16 (iterate (10*) (10e-10*meter/gramm)) ++-       [1/meter^2, 1/meter, meter, meter^2,-        second, hertz,-        meter*second, second/meter, meter/second, 1/meter/second,-        volt/meter,newton/meter,-        gramm]--testComplex :: SIComplex-testComplex = (2 :: Double) *> (SIValue.fromScalarSingle (3+:4)*milli*second)--testMagnitude :: SIDouble-testMagnitude = SIValue.lift (Value.lift Complex.magnitude) testComplex--testExpr :: Expr.T Double SIDouble-testExpr = sin (5 / (3+1) * fromValue meter)--testPrefixes :: [SIDouble]-testPrefixes =-   [pico, nano, micro, milli, centi, deci,-    deca, hecto, kilo, mega, giga, tera, peta]---{- * Reals -}--testReal :: String-testReal = Absolute.defltShow (sqrt 2 + log 2 * pi)--testComplexReal :: Complex.T Absolute.T-testComplexReal = exp (0 +: pi) + exp (0 -: pi)--showReal :: Absolute.T -> String-showReal = Absolute.defltShow---{- * Fixed point numbers -}--testFixedPoint :: String-testFixedPoint = FixedPoint.defltShow (sqrt 2 + log 2 * pi)--showFixedPoint :: FixedPoint.T -> String-showFixedPoint = FixedPoint.defltShow---{- * Residue classes -}--testResidueClass :: Integer-testResidueClass = ResidueClass.concrete 7 (5*3/2)--polyResidueClass :: (ZeroTestable.C a, Field.C a) =>-   [a] -> ResidueClass.T (Polynomial.T a)-polyResidueClass = ResidueClass.fromRepresentative . polynomial--{- That's strange:-The residue class implementation should constantly compute mod-and thus should be much faster.-I assume that this is an effect of missing sharing.-The functions which represent a residue class are shared,-but not their values.--*Main> mod (3^3000000) 2 :: Integer-1-(2.47 secs, 24541080 bytes)-*Main> ResidueClass.concrete 2 (3^3000000) :: Integer-1-(7.33 secs, 515047072 bytes)--}---{- * Polynomials and power series -}--polynomial :: [a] -> Polynomial.T a-polynomial = Polynomial.fromCoeffs--powerSeries :: [a] -> PowerSeries.T a-powerSeries = PowerSeries.fromCoeffs--laurentPolynomial :: Int -> [a] -> LaurentPolynomial.T a-laurentPolynomial = LaurentPolynomial.fromShiftCoeffs--tanSeries :: PowerSeries.T Rational-tanSeries = powerSeries PowerSeriesExample.tan---{- * Partial fractions -}--partialFraction :: (PID.C a, Indexable.C a) =>-   [a] -> a -> PartialFraction.T a-partialFraction = PartialFraction.fromFactoredFraction--{- |-An example from wavelet theory: lifting coefficients of the CDF wavelet family.--}-cdfFraction :: PartialFraction.T (Polynomial.T Rational)-cdfFraction =-   partialFraction-      (map polynomial [[-4,1],[0,1],[4,1]])-      (product (map polynomial [[-2,1],[2,1]]))--{- |-The same example with different notation,-that relies on numerical literals being used for polynomials.--}-cdfFractionNum :: PartialFraction.T (Polynomial.T Rational)-cdfFractionNum =-   let x = polynomial [0,1]-   in  partialFraction [x-4,x,x+4] ((x-2)*(x+2))---{- * Peano numbers -}-testPeano :: Peano.T-testPeano = minimum [Peano.infinity, 2, Peano.infinity, 4]--testPeanoList :: [Char]-testPeanoList =-   genericTake (genericLength (repeat 'a') :: Peano.T) ['a'..'z']---main :: IO ()-main = print test
− test-ghc-6.12/Test/Algebra/IntegralDomain.hs 
@@ -1,41 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Test.Algebra.IntegralDomain where--import Algebra.IntegralDomain (roundDown, roundUp, divUp, )--import Test.NumericPrelude.Utility (testUnit, )-import Test.QuickCheck (Testable, quickCheck, (==>), )-import qualified Test.HUnit as HUnit--import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP---test ::-   (Testable t) =>-   (Integer -> t) -> IO ()-test = quickCheck---tests :: HUnit.Test-tests =-   HUnit.TestLabel "integral domain functions" $-   HUnit.TestList $-   map testUnit $-   testList--testList :: [(String, IO ())]-testList =-   ("divMod", test $ \n m ->-      m/=0 ==> let (q,r) = divMod n m in n == q*m+r) :-   ("divRound", test $ \n m ->-      m/=0 ==> div n m * m == roundDown n m) :-   ("divUpRound", test $ \n m ->-      m/=0 ==> divUp n m * m == roundUp n m) :-   ("floorLimit", test $ \n m0 ->-      let m = 1 + abs m0-          x = roundDown n m-      in  n-m < x && x <=n) :-   ("floorCeiling", test $ \n m ->-      m/=0 ==> - roundDown n m == roundUp (-n) m) :-   []
− test-ghc-6.12/Test/Algebra/RealRing.hs 
@@ -1,40 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Test.Algebra.RealRing where--import qualified Algebra.RealRing as RealRing--import Test.NumericPrelude.Utility (testUnit, )-import Test.QuickCheck (quickCheck, )-import qualified Test.HUnit as HUnit--import Data.Tuple.HT (mapFst, )--import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP---test :: (Eq a) => (Double -> a) -> (Double -> a) -> IO ()-test f g =-   quickCheck (\x -> f x == g x)---tests :: HUnit.Test-tests =-   HUnit.TestLabel "rounding functions" $-   HUnit.TestList $-   map testUnit $-      ("round",         test RealRing.genericRound    (NP.round :: Double -> Integer)) :-      ("truncate",      test RealRing.genericTruncate (NP.truncate :: Double -> Integer)) :-      ("ceiling",       test RealRing.genericCeiling  (NP.ceiling :: Double -> Integer)) :-      ("floor",         test RealRing.genericFloor    (NP.floor :: Double -> Integer)) :-      ("fraction",      test RealRing.genericFraction (NP.fraction :: Double -> Double)) :-      ("splitFraction", test RealRing.genericSplitFraction (NP.splitFraction :: Double -> (Integer, Double))) :--{--      ("splitFractionId", quickCheck (\x -> (x::Double) == (uncurry (+) $ mapFst fromInteger $ splitFraction x))) :--}-      ("splitFractionId", quickCheck (\x ->  uncurry (==) $ mapFst (((x::Double)-) . fromInteger) $ splitFraction x)) :-      ("splitFractionFloorFraction", quickCheck (\x -> (floor (x::Double) :: Integer, fraction x) == splitFraction x)) :-      ("fractionBound", quickCheck (\x -> let y = fraction (x::Double) in 0<=y && y<1)) :-      ("floorCeiling", quickCheck (\x -> negate (floor (x::Double) :: Integer) == ceiling (-x))) :-      []
− test-ghc-6.12/Test/MathObj/Gaussian/Bell.hs 
@@ -1,96 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-module Test.MathObj.Gaussian.Bell where--import qualified MathObj.Gaussian.Bell as G---- import qualified Algebra.Ring           as Ring--import qualified Algebra.Laws as Laws--import qualified Number.Complex as Complex--import Test.NumericPrelude.Utility (testUnit)-import Test.QuickCheck (Testable, quickCheck, (==>))-import qualified Test.HUnit as HUnit--import Data.Function.HT (nest, )--import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP---simple ::-   (Testable t) =>-   (G.T Rational -> t) -> IO ()-simple f =-   quickCheck (\x -> f (x :: G.T Rational))--tests :: HUnit.Test-tests =-   HUnit.TestLabel "polynomial" $-   HUnit.TestList $-   map testUnit $-{--      ("convolution, dirac",-          simple $ Laws.identity (+) zero) :--}-      ("convolution, commutative",-          simple $ Laws.commutative G.convolve) :-      ("convolution, associative",-          simple $ Laws.associative G.convolve) :-      ("multiplication, one",-          simple $ Laws.identity G.multiply G.constant) :-      ("multiplication, commutative",-          simple $ Laws.commutative G.multiply) :-      ("multiplication, associative",-          simple $ Laws.associative G.multiply) :-      ("convolution, multplication, fourier",-          simple $ \x y ->-             G.fourier (G.convolve x y)-              == G.multiply (G.fourier x) (G.fourier y)) :-      ("convolution via translation",-          simple $ \x y ->-             G.convolve x y-              == G.convolveByTranslation x y) :-      ("convolution via fourier",-          simple $ \x y ->-             G.convolve x y-              == G.convolveByFourier x y) :-      ("fourier reverse",-          simple $ \x -> nest 2 G.fourier x == G.reverse x) :-      ("reverse identity",-          simple $ \x -> nest 2 G.reverse x == x) :-      ("fourier unit",-          quickCheck $ G.fourier G.unit == (G.unit :: G.T Rational)) :-      ("translate additive",-          simple $ \x a b ->-             G.translate a (G.translate b x) == G.translate (a+b) x) :-      ("translateComplex additive",-          simple $ \x a b ->-             G.translateComplex a (G.translateComplex b x) == G.translateComplex (a+b) x) :-      ("translateComplex real",-          simple $ \x a ->-             G.translateComplex (Complex.fromReal a) x == G.translate a x) :-      ("modulate additive",-          simple $ \x a b ->-             G.modulate a (G.modulate b x) == G.modulate (a+b) x) :-      ("commute translate modulate",-          simple $ \x a b ->-             G.modulate b (G.translate a x)-              == G.turn (a*b) (G.translate a (G.modulate b x))) :-      ("fourier translate",-          simple $ \x a ->-             G.fourier (G.translate a x)-              == G.modulate a (G.fourier x)) :-      ("dilate multiplicative",-          simple $ \x a b -> a>0 && b>0 ==>-             G.dilate a (G.dilate b x) == G.dilate (a*b) x) :-      ("dilate by shrink",-          simple $ \x a -> a>0 ==>-             G.shrink a x == G.dilate (recip a) x) :-      ("fourier dilate",-          simple $ \x a -> a>0 ==>-             G.fourier (G.dilate a x) == G.amplify a (G.shrink a (G.fourier x))) :-      []
− test-ghc-6.12/Test/MathObj/Gaussian/Polynomial.hs 
@@ -1,158 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-module Test.MathObj.Gaussian.Polynomial where--import qualified MathObj.Gaussian.Polynomial as G-import qualified MathObj.Gaussian.Bell as B--import qualified MathObj.Polynomial as Poly---- import qualified Algebra.Ring           as Ring--import qualified Algebra.Laws as Laws--import qualified Number.Complex as Complex--import Test.NumericPrelude.Utility (testUnit)-import Test.QuickCheck (Testable, quickCheck, (==>))-import qualified Test.HUnit as HUnit--import qualified Number.NonNegative as NonNeg-import Data.Function.HT (nest, )-import Data.Tuple.HT (mapSnd, )---- import Debug.Trace (trace, )--import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP---simple ::-   (Testable t) =>-   (G.T Rational -> t) -> IO ()-simple f =-   quickCheck (\x -> f (x :: G.T Rational))--tests :: HUnit.Test-tests =-   HUnit.TestLabel "polynomial" $-   HUnit.TestList $-   map testUnit $-   testList--testList :: [(String, IO ())]-testList =-{--      ("convolution, dirac",-          simple $ Laws.identity (+) zero) :--}-      ("convolution, commutative",-          simple $ Laws.commutative G.convolve) :---          simple $ \x -> Laws.commutative G.convolve (trace (show x) x)) :-      ("convolution, associative",-          simple $ Laws.associative G.convolve) :-{--      ("convolution by differentiation vs. fourier",-          simple $ \x y ->-             G.convolveByDifferentiation x y-              == G.convolveByFourier x y) :--}-      ("multiplication, one",-          simple $ Laws.identity G.multiply G.constant) :-      ("multiplication, commutative",-          simple $ Laws.commutative G.multiply) :-      ("multiplication, associative",-          simple $ Laws.associative G.multiply) :-      ("convolution, multplication, fourier",-          simple $ \x y ->-             G.fourier (G.convolve x y)-              == G.multiply (G.fourier x) (G.fourier y)) :-      ("fourier reverse",-          simple $ \x -> nest 2 G.fourier x == G.reverse x) :-      ("reverse identity",-          simple $ \x -> nest 2 G.reverse x == x) :-      ("fourier eigenfunction differential",-          quickCheck $ \m ->-             m <= 15 ==>-                let n = NonNeg.toNumber m-                    x = G.eigenfunctionDifferential n :: G.T Rational-                    k = Complex.conjugate Complex.imaginaryUnit ^ fromIntegral n-                in  G.fourier x  ==  G.scaleComplex k x) :-      ("fourier eigenfunction iterative",-          quickCheck $ \m ->-             m <= 15 ==>-                let n = NonNeg.toNumber m-                    x = G.eigenfunctionIterative n :: G.T Rational-                    k = Complex.conjugate Complex.imaginaryUnit ^ fromIntegral n-                in  G.fourier x  ==  G.scaleComplex k x) :-{- this does not hold, both functions compute different eigenbases-      ("fourier eigenfunction diff vs. iterative",-          quickCheck $ \n ->-             n <= 15 ==>-                G.eigenfunctionDifferential n ==-                (G.eigenfunctionIterative n :: G.T Rational)) :--}-      ("translate additive",-          simple $ \x a b ->-             G.translate a (G.translate b x) == G.translate (a+b) x) :-      ("translateComplex additive",-          simple $ \x a b ->-             G.translateComplex a (G.translateComplex b x) == G.translateComplex (a+b) x) :-      ("translateComplex real",-          simple $ \x a ->-             G.translateComplex (Complex.fromReal a) x == G.translate a x) :-      ("modulate additive",-          simple $ \x a b ->-             G.modulate a (G.modulate b x) == G.modulate (a+b) x) :-      ("commute translate modulate",-          simple $ \x a b ->-             G.modulate b (G.translate a x)-              == G.turn (a*b) (G.translate a (G.modulate b x))) :-      ("fourier translate",-          simple $ \x a ->-             G.fourier (G.translate a x)-              == G.modulate a (G.fourier x)) :-      ("dilate multiplicative",-          simple $ \x a b -> a>0 && b>0 ==>-             G.dilate a (G.dilate b x) == G.dilate (a*b) x) :-      ("dilate by shrink",-          simple $ \x a -> a>0 ==>-             G.shrink a x == G.dilate (recip a) x) :-      ("fourier dilate",-          simple $ \x a -> a>0 ==>-             G.fourier (G.dilate a x) == G.amplify a (G.shrink a (G.fourier x))) :-      ("integrate differentiate",-          simple $ \x ->-             G.integrate (G.differentiate x) == (zero, x)) :-      ("differentiate integrate",-          simple $ \x@(G.Cons b p) ->-             let (xoff,xint) = G.integrate x-             in  G.differentiate xint == G.Cons b (p + Poly.const xoff)) :-      ("fourier differentiate",-          simple $ \x ->-             G.fourier (G.differentiate x) ==-              let y = G.fourier x-              in  y{G.polynomial =-                      Poly.fromCoeffs [0, 0 Complex.+: 2] * G.polynomial y}) :-      ("differentiate convolve",-          simple $ \x y ->-             G.convolve (G.differentiate x) y ==-             G.convolve x (G.differentiate y)) :-      ("approximate by bells, translate",-          simple $ \x unit d -> unit/=0 ==>-             G.approximateByBells unit (G.translateComplex d x) ==-             map (mapSnd (B.translateComplex d)) (G.approximateByBells unit x)) :-      ("approximate by bells, dilate",-          simple $ \x unit d -> unit/=0 && d/=0 ==>-             G.approximateByBells unit (G.dilate d x) ==-             map (mapSnd (B.dilate d)) (G.approximateByBells (unit/d) x)) :-      ("approximate by bells, shrink",-          simple $ \x unit d -> unit/=0 && d/=0 ==>-             G.approximateByBells unit (G.shrink d x) ==-             map (mapSnd (B.shrink d)) (G.approximateByBells (unit*d) x)) :-      ("approximate by bells, different implementations",-          quickCheck $ \unit d s p -> unit/=0 ==>-             G.approximateByBellsAtOnce unit d s (p::Poly.T (Complex.T Rational)) ==-             G.approximateByBellsByTranslation unit d s p) :-      []
− test-ghc-6.12/Test/MathObj/Gaussian/Variance.hs 
@@ -1,210 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-module Test.MathObj.Gaussian.Variance where--import qualified MathObj.Gaussian.Variance as G-import qualified Number.Root as Root---- import qualified Algebra.Ring           as Ring--import qualified Algebra.Laws as Laws--import Test.NumericPrelude.Utility (testUnit)-import Test.QuickCheck (Testable, quickCheck, (==>), Arbitrary, arbitrary, )-import qualified Test.HUnit as HUnit--import Control.Monad (liftM2, liftM3, )--import Data.Function.HT (nest, compose2, )--import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP---newtype PositiveInteger = PositiveInteger Integer-   deriving Show--instance Arbitrary PositiveInteger where-   arbitrary =-      fmap (\p -> PositiveInteger $ 1 + abs p) arbitrary---{- |-For @(HoelderConjugates p q)@ it holds--> 1/p + 1/q = 1--}-data HoelderConjugates = HoelderConjugates Rational Rational-   deriving Show--{--instance Arbitrary HoelderConjugates where-   arbitrary = liftM2-      (\(PositiveInteger p) (PositiveInteger q) ->-         let s  = 1%p + 1%q-         in  HoelderConjugates (fromInteger p * s) (fromInteger q * s))-      arbitrary arbitrary--}-instance Arbitrary HoelderConjugates where-   arbitrary = liftM2-      (\(PositiveInteger p) (PositiveInteger q) ->-         let s = p + q-         in  HoelderConjugates (s % p) (s % q))-      arbitrary arbitrary--{- |-For @(YoungConjugates p q r)@ it holds--> 1/p + 1/q = 1/r + 1--}-data YoungConjugates = YoungConjugates Rational Rational Rational-   deriving Show--{--Find positive natural numbers @a, b, c, d@ with--> a + b = c + d--and--> d >= a, d >= b, d >= c--then set--> p=d/a, q=d/b, r=d/c---a+b<=c-b+c<=a-->  2b <= 0--}-instance Arbitrary YoungConjugates where-   arbitrary = liftM3-      (\(PositiveInteger a0) (PositiveInteger b0) (PositiveInteger c0) ->-         let guardSwap cond (x,y) =-                if cond x y then (x,y) else (y,x)-             {--             If a+b<=c, then from b>0 it follows a<c and thus c+b>a.-             Swapping a and c is enough and we have not to consider more cases.-             -}-             (a1,c1) = guardSwap (\a c -> a+b0>c) (a0,c0)-             b1 = b0-             d1 = a1+b1-c1-             ((a2,b2),(c2,d2)) =-                guardSwap (compose2 (<=) snd)-                   (guardSwap (<=) (a1,b1),-                    guardSwap (<=) (c1,d1))-         in  YoungConjugates (d2%a2) (d2%b2) (d2%c2))-      arbitrary arbitrary arbitrary---simple ::-   (Testable t) =>-   (G.T Rational -> t) -> IO ()-simple f =-   quickCheck (\x -> f (x :: G.T Rational))--tests :: HUnit.Test-tests =-   HUnit.TestLabel "variance" $-   HUnit.TestList $-   map testUnit $-   testList--testList :: [(String, IO ())]-testList =-{--      ("convolution, dirac",-          simple $ Laws.identity (+) zero) :--}-      ("convolution, commutative",-          simple $ Laws.commutative G.convolve) :-      ("convolution, associative",-          simple $ Laws.associative G.convolve) :-      ("multiplication, one",-          simple $ Laws.identity G.multiply G.constant) :-      ("multiplication, commutative",-          simple $ Laws.commutative G.multiply) :-      ("multiplication, associative",-          simple $ Laws.associative G.multiply) :-      ("convolution via fourier",-          simple $ \x y ->-             G.fourier (G.convolve x y)-              == G.multiply (G.fourier x) (G.fourier y)) :-      ("fourier identity",-          simple $ \x -> nest 4 G.fourier x == x) :-      ("dilate multiplicative",-          simple $ \x a b -> a>0 && b>0 ==>-             G.dilate a (G.dilate b x) == G.dilate (a*b) x) :-      ("dilate by shrink",-          simple $ \x a -> a>0 ==>-             G.shrink a x == G.dilate (recip a) x) :-      ("fourier dilate",-          simple $ \x a -> a>0 ==>-             G.fourier (G.dilate a x) == G.amplify a (G.shrink a (G.fourier x))) :-      ("fourier, unitary",-          simple $ \x y ->-             G.scalarProductRoot x y-              == G.scalarProductRoot (G.fourier x) (G.fourier y)) :-      ("norm1 vs. normP 1",-          simple $ \x -> G.norm1Root x == G.normPRoot 1 x) :-      ("norm2 vs. normP 2",-          simple $ \x -> G.norm2Root x == G.normPRoot 2 x) :-{--I would have liked to test for a monotony of norms.-Unfortunately, it does not hold.--Means contain a division by the size of the domain.-Norms do not have this division.-Means are monotonic with respect to the degree.-Norms are not.-We cannot turn the norms into means since the size of the domain-(the complete real axis) is infinitely large.-      ("norm monotony",-          simple $ \x p0 q0 ->-             let p = 1 + abs p0-                 q = 1 + abs q0-             in  case compare p q of-                    EQ -> G.normPRoot p x == G.normPRoot q x-                    LT -> G.normPRoot p x <= G.normPRoot q x-                    GT -> G.normPRoot p x >= G.normPRoot q x) :--This should also fail,-but QuickCheck does not seem to try counterexamples.-      ("infinity norm upper bound",-          simple $ \x p0 ->-             let p = 1 + abs p0-             in  G.normPRoot p x <= G.normInfRoot x) :--}-      ("Cauchy-Schwarz inequality",-          simple $ \x y ->-             G.scalarProductRoot x y-                <= G.norm2Root x `Root.mul` G.norm2Root y) :-      ("Hoelder conjugates",-          quickCheck $ \(HoelderConjugates p q) ->-             p>=1 && q>=1 && 1/p + 1/q == 1) :-      ("Hoelder inequality with infinity norm",-          simple $ \x y ->-             G.scalarProductRoot x y-                <= G.norm1Root x `Root.mul` G.normInfRoot y) :-      ("Hoelder inequality",-          simple $ \x y (HoelderConjugates p q) ->-             G.scalarProductRoot x y-                <= G.normPRoot p x `Root.mul` G.normPRoot q y) :-      ("Young inequality with two infinity norms",-          simple $ \x y ->-             G.normInfRoot (G.convolve x y)-                <= G.norm1Root x `Root.mul` G.normInfRoot y) :-      ("Young inequality with infinity norm",-          simple $ \x y (HoelderConjugates p q) ->-             G.normInfRoot (G.convolve x y)-                <= G.normPRoot p x `Root.mul` G.normPRoot q y) :-      ("Young conjugates",-          quickCheck $ \(YoungConjugates p q r) ->-             p>=1 && q>=1 && r>=1 && 1/p + 1/q == 1/r + 1) :-      ("Young inequality",-          simple $ \x y (YoungConjugates p q r) ->-             G.normPRoot r (G.convolve x y)-                <= G.normPRoot p x `Root.mul` G.normPRoot q y) :-      []
− test-ghc-6.12/Test/MathObj/Matrix.hs 
@@ -1,103 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-module Test.MathObj.Matrix where--import qualified MathObj.Matrix as Matrix--import qualified Algebra.Ring           as Ring--import qualified Algebra.Laws as Laws--import qualified Number.NonNegative as NonNeg--import qualified System.Random as Random--import Data.Function.HT (nest, )--import Test.NumericPrelude.Utility (testUnit, )-import Test.QuickCheck (quickCheck, )-import qualified Test.HUnit as HUnit---import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP---type Seed = Int-type Dimension = NonNeg.Int--random :: Dimension -> Dimension -> Seed -> Matrix.T Integer-random mn nn seed =-   fst $-   Matrix.random (NonNeg.toNumber mn) (NonNeg.toNumber nn) $-   Random.mkStdGen seed---tests :: HUnit.Test-tests =-   HUnit.TestLabel "matrix" $-   HUnit.TestList $-   map testUnit $-      ("dimension",-          quickCheck (\m n a ->-             (NonNeg.toNumber m, NonNeg.toNumber n) == Matrix.dimension (random m n a))) :-      ("to and from rows",-          quickCheck (\m n a' ->-             let a = random m n a'-             in  a == Matrix.fromRows (NonNeg.toNumber m) (NonNeg.toNumber n) (Matrix.rows a))) :-      ("to and from columns",-          quickCheck (\m n a' ->-             let a = random m n a'-             in  a == Matrix.fromColumns (NonNeg.toNumber m) (NonNeg.toNumber n) (Matrix.columns a))) :-      ("transpose, rows, columns",-          quickCheck (\m n a' ->-             let a = random m n a'-             in  Matrix.rows a == Matrix.columns (Matrix.transpose a))) :-      ("transpose, columns, rows",-          quickCheck (\m n a' ->-             let a = random m n a'-             in  Matrix.columns a == Matrix.rows (Matrix.transpose a))) :-      ("addition, zero",-          quickCheck (\m n a ->-             Laws.identity (+) (Matrix.zero (NonNeg.toNumber m) (NonNeg.toNumber n)) (random m n a))) :-      ("addition, commutative",-          quickCheck (\m n a b ->-             Laws.commutative (+) (random m n a) (random m n b))) :-      ("addition, associative",-          quickCheck (\m n a b c ->-             Laws.associative (+) (random m n a) (random m n b) (random m n c))) :-      ("addition, transpose",-          quickCheck (\m n a b ->-             Laws.homomorphism Matrix.transpose (+) (+) (random m n a) (random m n b))) :-      ("one, diagonal",-          quickCheck (\n' ->-             let n = NonNeg.toNumber n'-             in Matrix.one n == (Matrix.diagonal $ replicate n Ring.one :: Matrix.T Integer))) :-      ("multiplication, one left",-          quickCheck (\m n a ->-             Laws.leftIdentity  (*) (Matrix.one (NonNeg.toNumber m)) (random m n a))) :-      ("multiplication, one right",-          quickCheck (\m n a ->-             Laws.rightIdentity (*) (Matrix.one (NonNeg.toNumber n)) (random m n a))) :-      ("multiplication, associative",-          quickCheck (\k l m n a b c ->-             Laws.associative (*) (random k l a) (random l m b) (random m n c))) :-      ("multiplication and addition, distributive left",-          quickCheck (\l m n a b c ->-             Laws.leftDistributive (*) (+) (random n l a) (random m n b) (random m n c))) :-      ("multiplication and addition, distributive right",-          quickCheck (\l m n a b c ->-             Laws.rightDistributive (*) (+) (random l m a) (random m n b) (random m n c))) :-      ("multiplication, transpose",-          quickCheck (\l m n a b ->-             Laws.homomorphism Matrix.transpose (*) (flip (*)) (random l m a) (random m n b))) :-      ("multiplication vs. power",-          quickCheck (\m a n0 ->-             let x = random m m a-                 n = mod n0 10-             in  x^n == nest (fromInteger n) (x*) (Matrix.one (NonNeg.toNumber m)))) :-{--      ("division",       quickCheck (\x -> Integral.propInverse (x :: Poly.T Rational))) :--}-      []
− test-ghc-6.12/Test/MathObj/PartialFraction.hs 
@@ -1,205 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-module Test.MathObj.PartialFraction where--import qualified MathObj.PartialFraction      as PartialFraction-import qualified MathObj.Polynomial           as Poly-import qualified Number.Ratio                 as Ratio--import qualified Algebra.PrincipalIdealDomain as PID--- import qualified Algebra.Ring                 as Ring-import qualified Algebra.Indexable            as Indexable-import qualified Algebra.Vector               as Vector--- import Algebra.Vector((*>))--import qualified Algebra.Laws as Laws-import qualified Test.QuickCheck as QC--import Control.Monad.HT as M-import Test.NumericPrelude.Utility (testUnit)-import Test.QuickCheck (quickCheck)-import qualified Test.HUnit as HUnit---import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP---{- * Properties for generic types -}--fractionConv :: (PID.C a, Indexable.C a) => [a] -> a -> Bool-fractionConv xs y =-   PartialFraction.toFraction (PartialFraction.fromFactoredFraction xs y) ==-   y % product xs--fractionConvAlt :: (PID.C a, Indexable.C a) => [a] -> a -> Bool-fractionConvAlt xs y =-   PartialFraction.fromFactoredFraction xs y ==-   PartialFraction.fromFactoredFractionAlt xs y--scaleInt :: (PID.C a, Indexable.C a) => a -> PartialFraction.T a -> Bool-scaleInt k a =-   PartialFraction.toFraction (PartialFraction.scaleInt k a) ==-   Ratio.scale k (PartialFraction.toFraction a)--add :: (PID.C a, Indexable.C a) => PartialFraction.T a -> PartialFraction.T a -> Bool-add = Laws.homomorphism PartialFraction.toFraction (+) (+)--sub :: (PID.C a, Indexable.C a) => PartialFraction.T a -> PartialFraction.T a -> Bool-sub = Laws.homomorphism PartialFraction.toFraction (-) (-)--mul :: (PID.C a, Indexable.C a) => PartialFraction.T a -> PartialFraction.T a -> Bool-mul = Laws.homomorphism PartialFraction.toFraction (*) (*)----{- * Properties for Integers -}--{- |-Arbitrary instance of that type generates irreducible elements for tests.-Choosing from a list of examples is a simple yet effective design.-If we would construct irreducible elements by a clever algorithm-we might obtain multiple primes only rarely.--}-newtype SmallPrime = SmallPrime {intFromSmallPrime :: Integer}--type IntFraction = ([SmallPrime],Integer)--instance QC.Arbitrary SmallPrime where-   arbitrary =-      let primes = [2,3,5,7,11,13]-      in  fmap SmallPrime $ QC.elements (primes ++ map negate primes)--instance Show SmallPrime where-   show = show . intFromSmallPrime---fractionConvInt :: [SmallPrime] -> Integer -> Bool-fractionConvInt =-   fractionConv . map intFromSmallPrime--fractionConvAltInt :: [SmallPrime] -> Integer -> Bool-fractionConvAltInt =-   fractionConvAlt . map intFromSmallPrime--fromSmallPrimes :: IntFraction -> PartialFraction.T Integer-fromSmallPrimes (xs,y) =-   PartialFraction.fromFactoredFraction (map intFromSmallPrime xs) y---scaleIntInt :: Integer -> IntFraction -> Bool-scaleIntInt k a =-   scaleInt k (fromSmallPrimes a)--addInt :: IntFraction -> IntFraction -> Bool-addInt q0 q1 =-   add-      (fromSmallPrimes q0)-      (fromSmallPrimes q1)--subInt :: IntFraction -> IntFraction -> Bool-subInt q0 q1 =-   sub-      (fromSmallPrimes q0)-      (fromSmallPrimes q1)--mulInt :: IntFraction -> IntFraction -> Bool-mulInt q0 q1 =-   mul-      (fromSmallPrimes q0)-      (fromSmallPrimes q1)---intTests :: HUnit.Test-intTests =-   HUnit.TestLabel "integer" $-   HUnit.TestList $-   map testUnit $-      ("conversion between partial and ordinary fraction", quickCheck fractionConvInt) :-      ("two conversion routines from factored fractions", quickCheck fractionConvAltInt) :-      ("integer scaling", quickCheck scaleIntInt) :-      ("addition", quickCheck addInt) :-      ("subtraction", quickCheck subInt) :-      ("multiplication", quickCheck mulInt) :-      []---{- * Properties for Polynomials -}--newtype IrredPoly = IrredPoly {polyFromIrredPoly :: Poly.T Rational}--type RatPolynomial = Poly.T Rational-type PolyFraction = ([IrredPoly],RatPolynomial)--instance QC.Arbitrary IrredPoly where-   arbitrary =-      do poly <- QC.elements (map Poly.fromCoeffs [[2,3],[2,0,1],[3,0,1],[1,-3,0,1]])-         unit <- M.until (not. isZero) QC.arbitrary-         return (IrredPoly (unit Vector.*> poly))--instance Show IrredPoly where-   show = show . polyFromIrredPoly---fractionConvPoly :: [IrredPoly] -> RatPolynomial -> Bool-fractionConvPoly =-   fractionConv . map polyFromIrredPoly--fractionConvAltPoly :: [IrredPoly] -> RatPolynomial -> Bool-fractionConvAltPoly =-   fractionConvAlt . map polyFromIrredPoly--fromIrredPolys :: PolyFraction -> PartialFraction.T RatPolynomial-fromIrredPolys (xs,y) =-   PartialFraction.fromFactoredFraction (map polyFromIrredPoly xs) y---scaleIntPoly :: RatPolynomial -> PolyFraction -> Bool-scaleIntPoly k a =-   scaleInt k (fromIrredPolys a)--addPoly :: PolyFraction -> PolyFraction -> Bool-addPoly q0 q1 =-   add-      (fromIrredPolys q0)-      (fromIrredPolys q1)--subPoly :: PolyFraction -> PolyFraction -> Bool-subPoly q0 q1 =-   sub-      (fromIrredPolys q0)-      (fromIrredPolys q1)--mulPoly :: PolyFraction -> PolyFraction -> Bool-mulPoly q0 q1 =-   mul-      (fromIrredPolys q0)-      (fromIrredPolys q1)----polyTests :: HUnit.Test-polyTests =-   HUnit.TestLabel "polynomial" $-   HUnit.TestList $-   map testUnit $-{- this test fails, because addition may result in leading zero coefficients,-      that is, polynomial addition does not contain a normalization-      if it would contain one, we would exclude computable reals -}--- wrong     ("conversion between partial and ordinary fraction", quickCheck fractionConvPoly) :--- wrong     ("two conversion routines from factored fractions", quickCheck fractionConvAltPoly) :--- too slow      ("integer scaling", quickCheck scaleIntPoly) :--- too slow      ("addition", quickCheck addPoly) :--- too slow      ("subtraction", quickCheck subPoly) :--- too slow      ("multiplication", quickCheck mulPoly) :-      []---tests :: HUnit.Test-tests =-   HUnit.TestLabel "partial fraction" $-   HUnit.TestList $-      intTests :---      polyTests :-      []
− test-ghc-6.12/Test/MathObj/Polynomial.hs 
@@ -1,56 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Test.MathObj.Polynomial where--import qualified MathObj.Polynomial      as Poly-import qualified MathObj.Polynomial.Core as PolyCore--import qualified Algebra.IntegralDomain as Integral-import qualified Algebra.Ring           as Ring--import qualified Algebra.ZeroTestable   as ZeroTestable-import qualified Algebra.Laws as Laws--import qualified Data.List as List--import Test.NumericPrelude.Utility (testUnit)-import Test.QuickCheck (Property, quickCheck, (==>), Testable, )-import qualified Test.HUnit as HUnit---import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP---tensorProductTranspose :: (Ring.C a, Eq a) => [a] -> [a] -> Property-tensorProductTranspose xs ys =-   not (null xs) && not (null ys) ==>-      PolyCore.tensorProduct xs ys == List.transpose (PolyCore.tensorProduct ys xs)---mul :: (Ring.C a, Eq a, ZeroTestable.C a) => [a] -> [a] -> Bool-mul xs ys  =  PolyCore.equal (PolyCore.mul xs ys) (PolyCore.mulShear xs ys)---test :: Testable a => (Poly.T Integer -> a) -> IO ()-test = quickCheck--testRat :: Testable a => (Poly.T Rational -> a) -> IO ()-testRat = quickCheck---tests :: HUnit.Test-tests =-   HUnit.TestLabel "polynomial" $-   HUnit.TestList $-   map testUnit $-      ("tensor product", quickCheck (tensorProductTranspose :: [Integer] -> [Integer] -> Property)) :-      ("mul speed",      quickCheck (mul                    :: [Integer] -> [Integer] -> Bool)) :-      ("addition, zero",         test (Laws.identity (+) zero)) :-      ("addition, commutative",  test (Laws.commutative (+))) :-      ("addition, associative",  test (Laws.associative (+))) :-      ("multiplication, one",          test (Laws.identity (*) one)) :-      ("multiplication, commutative",  test (Laws.commutative (*))) :-      ("multiplication, associative",  test (Laws.associative (*))) :-      ("multiplication and addition, distributive",   test (Laws.leftDistributive (*) (+))) :-      ("division",       testRat (Integral.propInverse)) :-      []
− test-ghc-6.12/Test/MathObj/PowerSeries.hs 
@@ -1,103 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-module Test.MathObj.PowerSeries where--import qualified MathObj.PowerSeries.Core    as PS-import qualified MathObj.PowerSeries.Example as PSE--import Test.NumericPrelude.Utility (equalInfLists {- , testUnit -} )--- import Test.QuickCheck (Property, quickCheck, (==>))-import qualified Test.HUnit as HUnit---import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP---identitiesExplODE, identitiesSeriesFunction, identitiesInverses ::-   [(String, Int, [Rational],[Rational])]--identitiesExplODE =-   ("exp",   500, PSE.expExpl,   PSE.expODE) :-   ("sin",   500, PSE.sinExpl,   PSE.sinODE) :-   ("cos",   500, PSE.cosExpl,   PSE.cosODE) :-   ("tan",    50, PSE.tanExpl,   PSE.tanODE) :-   ("tan",    50, PSE.tanExpl,   PSE.tanExplSieve) :-   ("tan",    50, PSE.tanODE,    PSE.tanODESieve) :-   ("log",   500, PSE.logExpl,   PSE.logODE) :-   ("asin",   50, PSE.asinODE,   snd (PS.inv PSE.sinODE)) :-   ("atan",  500, PSE.atanExpl,  PSE.atanODE) :-   ("sinh",  500, PSE.sinhExpl,  PSE.sinhODE) :-   ("cosh",  500, PSE.coshExpl,  PSE.coshODE) :-   ("atanh", 500, PSE.atanhExpl, PSE.atanhODE) :-   ("sqrt",  100, PSE.sqrtExpl,  PSE.sqrtODE) :-   []--identitiesSeriesFunction =-   ("exp",   500, PSE.expExpl,  PS.exp (\0 -> 1) [0,1]) :-   ("sin",   500, PSE.sinExpl,  PS.sin (\0 -> (0,1)) [0,1]) :-   ("cos",   500, PSE.cosExpl,  PS.cos (\0 -> (0,1)) [0,1]) :-   ("tan",    50, PSE.tanExpl,  PS.tan (\0 -> (0,1)) [0,1]) :-   ("sqrt",   50, PSE.sqrtExpl, PS.sqrt (\1 -> 1) [1,1]) :-   ("power", 500, PSE.powExpl (-1/3), PS.pow (\1 -> 1) (-1/3) [1,1]) :-   ("power",  50, PSE.powExpl (-1/3), PS.exp (\0 -> 1) (PS.scale (-1/3) PSE.log)) :-   ("log",   500, PSE.logExpl, PS.log (\1 -> 0) [1,1]) :-   ("asin",   50, PSE.asin, PS.asin (\1 -> 1) (\0 -> 0) [0,1]) :- --  ("acos",  50, PSE.acos, PS.acos (\1 -> 1) (\0 -> pi/2) [0,1]) :-   ("atan",  500, PSE.atan, PS.atan (\0 -> 0) [0,1]) :-   []--identitiesInverses =-   ("exp",   100, 1:1:repeat 0, PS.exp  (\0 -> 1) PSE.log) :-   ("log",   100, 0:1:repeat 0, PS.log  (\1 -> 0) PSE.exp) :-   ("tan",    50, 0:1:repeat 0, PS.tan  (\0 -> (0,1)) PSE.atan) :-   ("atan",   50, 0:1:repeat 0, PS.atan (\0 -> 0) PSE.tan) :-   ("sin",    50, 0:1:repeat 0, PS.sin  (\0 -> (0,1)) PSE.asin) :-   ("asin",  100, 0:1:repeat 0, PS.asin (\1 -> 1) (\0 -> 0) PSE.sin) :-   ("sqrt",  500, 1:1:repeat 0, PS.sqrt (\1 -> 1) (PS.mul [1,1] [1,1])) :-   []--testSeriesIdentity :: (String, Int, [Rational], [Rational]) -> HUnit.Test-testSeriesIdentity (label, len, x, y) =-   HUnit.test (HUnit.assertBool label (equalInfLists len [x,y]))--testSeriesIdentities ::-   String -> [(String, Int, [Rational], [Rational])] -> HUnit.Test-testSeriesIdentities label ids =-   HUnit.TestLabel label $-     HUnit.TestList $ map testSeriesIdentity ids--checkSeriesIdentities ::-   [(String, Int, [Rational], [Rational])] -> [(String,Bool)]-checkSeriesIdentities =-   map (\(label, len, x, y) -> (label, equalInfLists len [x,y]))-----powerMult :: Rational -> Rational -> Bool-powerMult exp0 exp1 =-   PS.mul (PSE.pow exp0) (PSE.pow exp1)  ==  PSE.pow (exp0+exp1)--powerExplODE :: Rational -> Bool-powerExplODE expon =-   PSE.powODE expon == PSE.powExpl expon---tests :: HUnit.Test-tests =-   HUnit.TestLabel "power series" $-   HUnit.TestList [-      testSeriesIdentities "explicit vs. ODE solution" identitiesExplODE,-      testSeriesIdentities "transcendent functions of series" identitiesSeriesFunction,-      testSeriesIdentities "inverses of some series" identitiesInverses-{--      HUnit.TestLabel "laws" $-      HUnit.TestList $-         map testUnit $-            ("products of powers",     quickCheck (powerMult)) :-            ("power explicit vs. ODE", quickCheck (powerExplODE)) :-            []--}-    ]
− test-ghc-6.12/Test/MathObj/RefinementMask2.hs 
@@ -1,78 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Test.MathObj.RefinementMask2 where--import qualified MathObj.RefinementMask2 as Mask-import qualified Algebra.Differential    as D--import qualified MathObj.Polynomial      as Poly-import qualified MathObj.Polynomial.Core as PolyCore--import qualified Algebra.RealField      as RealField-import qualified Algebra.Ring           as Ring--import qualified Algebra.ZeroTestable   as ZeroTestable--import Data.Maybe (fromMaybe, )--import Test.NumericPrelude.Utility (testUnit)-import Test.QuickCheck (Property, quickCheck, (==>), Testable, )-import qualified Test.HUnit as HUnit---import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP----hasMultipleZero :: (Ring.C a, Eq a) => Int -> a -> Poly.T a -> Bool-hasMultipleZero n x poly =-   all (zero==) $ take n $-   map (flip Poly.evaluate x) $-   iterate D.differentiate poly--inverse0 :: (RealField.C a) => Mask.T a -> Property-inverse0 mask0 =-   let (b,poly) =-          case Mask.toPolynomial mask0 of-             Just p -> (True, p)-             Nothing -> (False, error "RefinementMask2.inverse0: no admissible mask")-       mask1 = Mask.fromPolynomial poly-       maskD =-          Poly.fromCoeffs (Mask.coeffs mask1) --          Poly.fromCoeffs (Mask.coeffs mask0)-   in  b ==>-          hasMultipleZero (fromMaybe 0 $ Poly.degree poly)-             1 maskD--truncatePolynomial :: (ZeroTestable.C a) => Int -> Poly.T a -> Poly.T a-truncatePolynomial n =-   Poly.fromCoeffs . PolyCore.normalize . take n . Poly.coeffs--inverse1 :: (RealField.C a) => Poly.T a -> Bool-inverse1 poly0 =-   case Mask.toPolynomial (Mask.fromPolynomial poly0) of-      Just poly1 -> Poly.collinear poly0 poly1-      Nothing -> False--refining :: (RealField.C a) => Poly.T a -> Bool-refining poly =-   poly == Mask.refinePolynomial (Mask.fromPolynomial poly) poly----test :: Testable a => (Poly.T Integer -> a) -> IO ()-test = quickCheck--testRat :: Testable a => (Poly.T Rational -> a) -> IO ()-testRat = quickCheck---tests :: HUnit.Test-tests =-   HUnit.TestLabel "refinement mask" $-   HUnit.TestList $-   map testUnit $-      ("inverse0", quickCheck (inverse0 :: Mask.T Rational -> Property)) :-      ("inverse1", quickCheck (inverse1 . truncatePolynomial 5 :: Poly.T Rational -> Bool)) :-      ("refining", quickCheck (refining . truncatePolynomial 5 :: Poly.T Rational -> Bool)) :-      []
− test-ghc-6.12/Test/Number/ComplexSquareRoot.hs 
@@ -1,50 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-module Test.Number.ComplexSquareRoot where--import qualified Number.ComplexSquareRoot as S-import qualified Number.Complex as Complex---- import qualified Algebra.Ring           as Ring--import qualified Algebra.Laws as Laws--import Test.NumericPrelude.Utility (testUnit)-import Test.QuickCheck (Testable, quickCheck, (==>), )-import qualified Test.HUnit as HUnit--import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP---simple ::-   (Testable t) =>-   (S.T Rational -> t) -> IO ()-simple = quickCheck--tests :: HUnit.Test-tests =-   HUnit.TestLabel "complex square root" $-   HUnit.TestList $-   map testUnit $-   testList--testList :: [(String, IO ())]-testList =-   ("multiplication, one",-      simple $ Laws.identity S.mul S.one) :-   ("multiplication, commutative",-      simple $ Laws.commutative S.mul) :-   ("multiplication, associative",-      simple $ Laws.associative S.mul) :-   ("multiplication, homomorphism",-      quickCheck $ Laws.homomorphism S.fromNumber-         (\x y -> (x :: Complex.T Rational) * y) S.mul) :-   ("division, one",-      simple $ Laws.rightIdentity S.div S.one) :-   ("recip recip",-      simple $ \x -> not (isZero x) ==> S.recip (S.recip x) == x) :-   ("recip inverts multiplication",-      simple $ \x -> not (isZero x) ==> Laws.inverse S.mul S.recip S.one x) :-   []
− test-ghc-6.12/Test/Number/GaloisField2p32m5.hs 
@@ -1,37 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Test.Number.GaloisField2p32m5 where--import qualified Number.GaloisField2p32m5 as GF--import qualified Algebra.Laws as Laws--import Test.NumericPrelude.Utility (testUnit)-import Test.QuickCheck (Testable, quickCheck, (==>))-import qualified Test.HUnit as HUnit---import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP---test :: Testable a => (GF.T -> a) -> IO ()-test = quickCheck---tests :: HUnit.Test-tests =-   HUnit.TestLabel "galois field 2^32-5" $-   HUnit.TestList $-   map testUnit $-      ("addition, zero",         test (Laws.identity (+) zero)) :-      ("addition, commutative",  test (Laws.commutative (+))) :-      ("addition, associative",  test (Laws.associative (+))) :-      ("addition, negate",       test (Laws.inverse (+) negate zero)) :-      ("addition, subtract",     test (\x -> Laws.inverse (+) (x-) x)) :-      ("multiplication, one",          test (Laws.identity (*) one)) :-      ("multiplication, commutative",  test (Laws.commutative (*))) :-      ("multiplication, associative",  test (Laws.associative (*))) :-      ("multiplication, recip",        test (\y -> y /= 0 ==> Laws.inverse (*) recip one y)) :-      ("multiplication, division",     test (\y x -> y /= 0 ==> Laws.inverse (*) (x/) x y)) :-      ("multiplication and addition, distributive",   test (Laws.leftDistributive (*) (+))) :-      []
− test-ghc-6.12/Test/NumericPrelude/Utility.hs 
@@ -1,21 +0,0 @@--- cf. utility-ht Test.Utility-module Test.NumericPrelude.Utility where--import Data.List.HT (mapAdjacent, )-import qualified Data.List as List-import qualified Test.HUnit as HUnit---testUnit :: (String, IO ()) -> HUnit.Test-testUnit (label, check) =-   HUnit.TestLabel label (HUnit.TestCase check)---- compare the lists simultaneously-equalLists :: Eq a => [[a]] -> Bool-equalLists xs =-   let equalElems ys =-          and (mapAdjacent (==) ys)  &&  length xs == length ys-   in  all equalElems (List.transpose xs)--equalInfLists :: Eq a => Int -> [[a]] -> Bool-equalInfLists n xs = equalLists (map (take n) xs)
− test-ghc-6.12/Test/Run.hs 
@@ -1,34 +0,0 @@-module Main where--import qualified Test.MathObj.RefinementMask2 as RefinementMask2-import qualified Test.Algebra.RealRing as RealRing-import qualified Test.Algebra.IntegralDomain as Integral-import qualified Test.MathObj.Gaussian.Polynomial as GaussPoly-import qualified Test.MathObj.Gaussian.Variance as GaussVariance-import qualified Test.MathObj.Gaussian.Bell as GaussBell-import qualified Test.MathObj.PartialFraction as PartialFraction-import qualified Test.MathObj.Matrix  as Matrix-import qualified Test.MathObj.Polynomial  as Polynomial-import qualified Test.MathObj.PowerSeries as PowerSeries-import qualified Test.Number.ComplexSquareRoot as CSqRt-import qualified Test.Number.GaloisField2p32m5 as GF-import qualified Test.HUnit.Text as HUnitText-import qualified Test.HUnit as HUnit--main :: IO ()-main =-   print =<<-      HUnitText.runTestTT (HUnit.TestList $-         RefinementMask2.tests :-         RealRing.tests :-         Integral.tests :-         GaussVariance.tests :-         GaussBell.tests :-         GaussPoly.tests :-         PartialFraction.tests :-         Matrix.tests :-         Polynomial.tests :-         PowerSeries.tests :-         CSqRt.tests :-         GF.tests :-         [])
+ test/Demo.hs view
@@ -0,0 +1,178 @@+{-# LANGUAGE RebindableSyntax #-}+module Main where++import Number.Complex((+:), (-:), )+import qualified Number.Complex as Complex+import Number.Physical      as Value+import Number.SI            as SIValue -- units+import Number.SI.Unit       as SIUnit  -- unit prefixes+          (pico, nano, micro, milli, centi, deci,+           deca, hecto, kilo, mega, giga, tera, peta)+import Number.OccasionallyScalarExpression as Expr++import qualified Number.NonNegativeChunky as Chunky+import qualified Number.NonNegative       as NonNegW+import qualified Number.Positional.Check  as Real+import qualified Number.FixedPoint.Check  as FixedPoint+import qualified Number.ResidueClass.Func as ResidueClass+import qualified Number.Peano             as Peano++import qualified MathObj.Polynomial          as Polynomial+import qualified MathObj.LaurentPolynomial   as LaurentPolynomial+import qualified MathObj.PowerSeries         as PowerSeries+import qualified MathObj.PowerSeries.Example as PowerSeriesExample+import qualified MathObj.PartialFraction     as PartialFraction++import qualified Algebra.PrincipalIdealDomain as PID+import qualified Algebra.Field                as Field+import qualified Algebra.ZeroTestable         as ZeroTestable+import qualified Algebra.Indexable            as Indexable++import Data.List (genericTake, genericLength)++import NumericPrelude.Base+import NumericPrelude.Numeric+++{- * Physical units -}++-- some shorthands for common usage+type SIDouble  = SIValue.T Double Double+type SIComplex = SIValue.T Double (Complex.T Double)++{- this advice seems not to be targeted to ghc's interactive mode+default (SIDouble)+-}+++++test :: [SIDouble]+test =+   let lengthScales = map (\n->10^-n*meter) [-10..6]+       areaScales = map (\n->10^-n*meter^2) [-10..6]+   in  lengthScales ++ map recip lengthScales +++       areaScales   ++ map recip areaScales +++       map ((meter*gramm/second)^-) [-5..5] +++       take 16 (iterate (10*) (10e-10*meter/gramm)) +++       [1/meter^2, 1/meter, meter, meter^2,+        second, hertz,+        meter*second, second/meter, meter/second, 1/meter/second,+        volt/meter,newton/meter,+        gramm]++testComplex :: SIComplex+testComplex = (2 :: Double) *> (SIValue.fromScalarSingle (3+:4)*milli*second)++testMagnitude :: SIDouble+testMagnitude = SIValue.lift (Value.lift Complex.magnitude) testComplex++testExpr :: Expr.T Double SIDouble+testExpr = sin (5 / (3+1) * fromValue meter)++testPrefixes :: [SIDouble]+testPrefixes =+   [pico, nano, micro, milli, centi, deci,+    deca, hecto, kilo, mega, giga, tera, peta]+++{- * Reals -}++testReal :: String+testReal = Real.defltShow (sqrt 2 + log 2 * pi)++testComplexReal :: Complex.T Real.T+testComplexReal = exp (0 +: pi) + exp (0 -: pi)++showReal :: Real.T -> String+showReal = Real.defltShow+++{- * Fixed point numbers -}++testFixedPoint :: String+testFixedPoint = FixedPoint.defltShow (sqrt 2 + log 2 * pi)++showFixedPoint :: FixedPoint.T -> String+showFixedPoint = FixedPoint.defltShow+++{- * Residue classes -}++testResidueClass :: Integer+testResidueClass = ResidueClass.concrete 7 (5*3/2)++polyResidueClass :: (ZeroTestable.C a, Field.C a) =>+   [a] -> ResidueClass.T (Polynomial.T a)+polyResidueClass = ResidueClass.fromRepresentative . polynomial++{- That's strange:+The residue class implementation should constantly compute mod+and thus should be much faster.+I assume that this is an effect of missing sharing.+The functions which represent a residue class are shared,+but not their values.++*Main> mod (3^3000000) 2 :: Integer+1+(2.47 secs, 24541080 bytes)+*Main> ResidueClass.concrete 2 (3^3000000) :: Integer+1+(7.33 secs, 515047072 bytes)+-}+++{- * Polynomials and power series -}++polynomial :: [a] -> Polynomial.T a+polynomial = Polynomial.fromCoeffs++powerSeries :: [a] -> PowerSeries.T a+powerSeries = PowerSeries.fromCoeffs++laurentPolynomial :: Int -> [a] -> LaurentPolynomial.T a+laurentPolynomial = LaurentPolynomial.fromShiftCoeffs++tanSeries :: PowerSeries.T Rational+tanSeries = powerSeries PowerSeriesExample.tan+++{- * Partial fractions -}++partialFraction :: (PID.C a, Indexable.C a) =>+   [a] -> a -> PartialFraction.T a+partialFraction = PartialFraction.fromFactoredFraction++{- |+An example from wavelet theory: lifting coefficients of the CDF wavelet family.+-}+cdfFraction :: PartialFraction.T (Polynomial.T Rational)+cdfFraction =+   partialFraction+      (map polynomial [[-4,1],[0,1],[4,1]])+      (product (map polynomial [[-2,1],[2,1]]))++{- |+The same example with different notation,+that relies on numerical literals being used for polynomials.+-}+cdfFractionNum :: PartialFraction.T (Polynomial.T Rational)+cdfFractionNum =+   let x = polynomial [0,1]+   in  partialFraction [x-4,x,x+4] ((x-2)*(x+2))+++{- * Peano numbers -}+testPeano :: Peano.T+testPeano = minimum [Peano.infinity, 2, Peano.infinity, 4]++testPeanoList :: [Char]+testPeanoList =+   genericTake (genericLength (repeat 'a') :: Peano.T) ['a'..'z']++testChunky :: Chunky.T NonNegW.Int+testChunky = (2+3)*(1+5)+++main :: IO ()+main = print test
− test/Gaussian.hs 
@@ -1,6 +0,0 @@-module Main where--import qualified MathObj.Gaussian.Example as Example--main :: IO ()-main = Example.polyApprox
− test/Test.hs 
@@ -1,173 +0,0 @@-{-# LANGUAGE RebindableSyntax #-}-module Main where--import Number.Complex((+:), (-:), )-import qualified Number.Complex as Complex-import Number.Physical      as Value-import Number.SI            as SIValue -- units-import Number.SI.Unit       as SIUnit  -- unit prefixes-          (pico, nano, micro, milli, centi, deci,-           deca, hecto, kilo, mega, giga, tera, peta)-import Number.OccasionallyScalarExpression as Expr--import qualified Number.Positional.Check  as Absolute-import qualified Number.FixedPoint.Check  as FixedPoint-import qualified Number.ResidueClass.Func as ResidueClass-import qualified Number.Peano             as Peano--import qualified MathObj.Polynomial          as Polynomial-import qualified MathObj.LaurentPolynomial   as LaurentPolynomial-import qualified MathObj.PowerSeries         as PowerSeries-import qualified MathObj.PowerSeries.Example as PowerSeriesExample-import qualified MathObj.PartialFraction     as PartialFraction--import qualified Algebra.PrincipalIdealDomain as PID-import qualified Algebra.Field                as Field-import qualified Algebra.ZeroTestable         as ZeroTestable-import qualified Algebra.Indexable            as Indexable--import Data.List (genericTake, genericLength)--import NumericPrelude.Base-import NumericPrelude.Numeric---{- * Physical units -}---- some shorthands for common usage-type SIDouble  = SIValue.T Double Double-type SIComplex = SIValue.T Double (Complex.T Double)--{- this advice seems not to be targeted to ghc's interactive mode-default (SIDouble)--}-----test :: [SIDouble]-test =-   let lengthScales = map (\n->10^-n*meter) [-10..6]-       areaScales = map (\n->10^-n*meter^2) [-10..6]-   in  lengthScales ++ map recip lengthScales ++-       areaScales   ++ map recip areaScales ++-       map ((meter*gramm/second)^-) [-5..5] ++-       take 16 (iterate (10*) (10e-10*meter/gramm)) ++-       [1/meter^2, 1/meter, meter, meter^2,-        second, hertz,-        meter*second, second/meter, meter/second, 1/meter/second,-        volt/meter,newton/meter,-        gramm]--testComplex :: SIComplex-testComplex = (2 :: Double) *> (SIValue.fromScalarSingle (3+:4)*milli*second)--testMagnitude :: SIDouble-testMagnitude = SIValue.lift (Value.lift Complex.magnitude) testComplex--testExpr :: Expr.T Double SIDouble-testExpr = sin (5 / (3+1) * fromValue meter)--testPrefixes :: [SIDouble]-testPrefixes =-   [pico, nano, micro, milli, centi, deci,-    deca, hecto, kilo, mega, giga, tera, peta]---{- * Reals -}--testReal :: String-testReal = Absolute.defltShow (sqrt 2 + log 2 * pi)--testComplexReal :: Complex.T Absolute.T-testComplexReal = exp (0 +: pi) + exp (0 -: pi)--showReal :: Absolute.T -> String-showReal = Absolute.defltShow---{- * Fixed point numbers -}--testFixedPoint :: String-testFixedPoint = FixedPoint.defltShow (sqrt 2 + log 2 * pi)--showFixedPoint :: FixedPoint.T -> String-showFixedPoint = FixedPoint.defltShow---{- * Residue classes -}--testResidueClass :: Integer-testResidueClass = ResidueClass.concrete 7 (5*3/2)--polyResidueClass :: (ZeroTestable.C a, Field.C a) =>-   [a] -> ResidueClass.T (Polynomial.T a)-polyResidueClass = ResidueClass.fromRepresentative . polynomial--{- That's strange:-The residue class implementation should constantly compute mod-and thus should be much faster.-I assume that this is an effect of missing sharing.-The functions which represent a residue class are shared,-but not their values.--*Main> mod (3^3000000) 2 :: Integer-1-(2.47 secs, 24541080 bytes)-*Main> ResidueClass.concrete 2 (3^3000000) :: Integer-1-(7.33 secs, 515047072 bytes)--}---{- * Polynomials and power series -}--polynomial :: [a] -> Polynomial.T a-polynomial = Polynomial.fromCoeffs--powerSeries :: [a] -> PowerSeries.T a-powerSeries = PowerSeries.fromCoeffs--laurentPolynomial :: Int -> [a] -> LaurentPolynomial.T a-laurentPolynomial = LaurentPolynomial.fromShiftCoeffs--tanSeries :: PowerSeries.T Rational-tanSeries = powerSeries PowerSeriesExample.tan---{- * Partial fractions -}--partialFraction :: (PID.C a, Indexable.C a) =>-   [a] -> a -> PartialFraction.T a-partialFraction = PartialFraction.fromFactoredFraction--{- |-An example from wavelet theory: lifting coefficients of the CDF wavelet family.--}-cdfFraction :: PartialFraction.T (Polynomial.T Rational)-cdfFraction =-   partialFraction-      (map polynomial [[-4,1],[0,1],[4,1]])-      (product (map polynomial [[-2,1],[2,1]]))--{- |-The same example with different notation,-that relies on numerical literals being used for polynomials.--}-cdfFractionNum :: PartialFraction.T (Polynomial.T Rational)-cdfFractionNum =-   let x = polynomial [0,1]-   in  partialFraction [x-4,x,x+4] ((x-2)*(x+2))---{- * Peano numbers -}-testPeano :: Peano.T-testPeano = minimum [Peano.infinity, 2, Peano.infinity, 4]--testPeanoList :: [Char]-testPeanoList =-   genericTake (genericLength (repeat 'a') :: Peano.T) ['a'..'z']---main :: IO ()-main = print test
+ test/Test/Algebra/Additive.hs view
@@ -0,0 +1,28 @@+-- Do not edit! Automatically created with doctest-extract from src/Algebra/Additive.hs+{-# LINE 42 "src/Algebra/Additive.hs" #-}++module Test.Algebra.Additive where++import qualified Test.DocTest.Driver as DocTest++{-# LINE 43 "src/Algebra/Additive.hs" #-}+import     qualified Algebra.Additive as A+import     qualified Test.QuickCheck as QC++test :: DocTest.T ()+test = do+ DocTest.printPrefix "Algebra.Additive:108: "+{-# LINE 108 "src/Algebra/Additive.hs" #-}+ DocTest.property+{-# LINE 108 "src/Algebra/Additive.hs" #-}+     (\(QC.NonEmpty ns) -> A.sum ns == (A.sum1 ns :: Integer))+ DocTest.printPrefix "Algebra.Additive:121: "+{-# LINE 121 "src/Algebra/Additive.hs" #-}+ DocTest.property+{-# LINE 121 "src/Algebra/Additive.hs" #-}+     (\ns -> A.sum ns == (A.sumNestedAssociative ns :: Integer))+ DocTest.printPrefix "Algebra.Additive:136: "+{-# LINE 136 "src/Algebra/Additive.hs" #-}+ DocTest.property+{-# LINE 136 "src/Algebra/Additive.hs" #-}+     (\ns -> A.sum ns == (A.sumNestedCommutative ns :: Integer))
test/Test/Algebra/IntegralDomain.hs view
@@ -1,41 +1,41 @@-{-# LANGUAGE RebindableSyntax #-}-module Test.Algebra.IntegralDomain where--import Algebra.IntegralDomain (roundDown, roundUp, divUp, )--import Test.NumericPrelude.Utility (testUnit, )-import Test.QuickCheck (Testable, quickCheck, (==>), )-import qualified Test.HUnit as HUnit--import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP-+-- Do not edit! Automatically created with doctest-extract from src/Algebra/IntegralDomain.hs+{-# LINE 54 "src/Algebra/IntegralDomain.hs" #-} -test ::-   (Testable t) =>-   (Integer -> t) -> IO ()-test = quickCheck+module Test.Algebra.IntegralDomain where +import qualified Test.DocTest.Driver as DocTest -tests :: HUnit.Test-tests =-   HUnit.TestLabel "integral domain functions" $-   HUnit.TestList $-   map testUnit $-   testList+{-# LINE 55 "src/Algebra/IntegralDomain.hs" #-}+import     Algebra.IntegralDomain (roundDown, roundUp, divUp)+import     qualified Test.QuickCheck as QC+import     NumericPrelude.Base as P+import     NumericPrelude.Numeric as NP+import     Prelude () -testList :: [(String, IO ())]-testList =-   ("divMod", test $ \n m ->-      m/=0 ==> let (q,r) = divMod n m in n == q*m+r) :-   ("divRound", test $ \n m ->-      m/=0 ==> div n m * m == roundDown n m) :-   ("divUpRound", test $ \n m ->-      m/=0 ==> divUp n m * m == roundUp n m) :-   ("floorLimit", test $ \n m0 ->-      let m = 1 + abs m0-          x = roundDown n m-      in  n-m < x && x <=n) :-   ("floorCeiling", test $ \n m ->-      m/=0 ==> - roundDown n m == roundUp (-n) m) :-   []+test :: DocTest.T ()+test = do+ DocTest.printPrefix "Algebra.IntegralDomain:108: "+{-# LINE 108 "src/Algebra/IntegralDomain.hs" #-}+ DocTest.property+{-# LINE 108 "src/Algebra/IntegralDomain.hs" #-}+         (\n (QC.NonZero m) -> let (q,r) = divMod n m in n == (q*m+r :: Integer))+ DocTest.printPrefix "Algebra.IntegralDomain:198: "+{-# LINE 198 "src/Algebra/IntegralDomain.hs" #-}+ DocTest.property+{-# LINE 198 "src/Algebra/IntegralDomain.hs" #-}+     (\n (QC.NonZero m) -> div n m * m == (roundDown n m :: Integer))+ DocTest.printPrefix "Algebra.IntegralDomain:208: "+{-# LINE 208 "src/Algebra/IntegralDomain.hs" #-}+ DocTest.property+{-# LINE 208 "src/Algebra/IntegralDomain.hs" #-}+     (\n (QC.NonZero m) -> divUp n m * m == (roundUp n m :: Integer))+ DocTest.printPrefix "Algebra.IntegralDomain:209: "+{-# LINE 209 "src/Algebra/IntegralDomain.hs" #-}+ DocTest.property+{-# LINE 209 "src/Algebra/IntegralDomain.hs" #-}+     (\n (QC.Positive m) -> let x = roundDown n m in  n-m < x && x <= (n :: Integer))+ DocTest.printPrefix "Algebra.IntegralDomain:210: "+{-# LINE 210 "src/Algebra/IntegralDomain.hs" #-}+ DocTest.property+{-# LINE 210 "src/Algebra/IntegralDomain.hs" #-}+     (\n (QC.NonZero m) -> - roundDown n m == (roundUp (-n) m :: Integer))
+ test/Test/Algebra/PrincipalIdealDomain.hs view
@@ -0,0 +1,49 @@+-- Do not edit! Automatically created with doctest-extract from src/Algebra/PrincipalIdealDomain.hs+{-# LINE 64 "src/Algebra/PrincipalIdealDomain.hs" #-}++module Test.Algebra.PrincipalIdealDomain where++import Test.DocTest.Base+import qualified Test.DocTest.Driver as DocTest++{-# LINE 65 "src/Algebra/PrincipalIdealDomain.hs" #-}+import     qualified Algebra.PrincipalIdealDomain as PID+import     Test.NumericPrelude.Utility ((/\))+import     qualified Test.QuickCheck as QC++genResidueClass     :: QC.Gen (Integer,Integer)+genResidueClass     = do+       m <- fmap QC.getNonZero $ QC.arbitrary+       a <- QC.choose (min 0 $ 1+m, max 0 $ m-1)+       return (m,a)++test :: DocTest.T ()+test = do+ DocTest.printPrefix "Algebra.PrincipalIdealDomain:305: "+{-# LINE 305 "src/Algebra/PrincipalIdealDomain.hs" #-}+ DocTest.property+{-# LINE 305 "src/Algebra/PrincipalIdealDomain.hs" #-}+     (QC.listOf genResidueClass /\ \xs -> case PID.chineseRemainderMulti xs of Nothing -> True; Just (n,b) -> abs n == abs (foldl lcm 1 (map fst xs)) && map snd xs == map (mod b . fst) xs)+ DocTest.printPrefix "Algebra.PrincipalIdealDomain:306: "+{-# LINE 306 "src/Algebra/PrincipalIdealDomain.hs" #-}+ DocTest.property+{-# LINE 306 "src/Algebra/PrincipalIdealDomain.hs" #-}+     (\(QC.NonEmpty ms) b -> let xs = map (\(QC.NonZero m) -> (m, mod b m)) ms in case PID.chineseRemainderMulti xs of Nothing -> False; Just (n,c) -> abs n == abs (foldl lcm 1 (map QC.getNonZero ms)) && mod b n == (c::Integer))+ DocTest.printPrefix "Algebra.PrincipalIdealDomain:298: "+{-# LINE 298 "src/Algebra/PrincipalIdealDomain.hs" #-}+ DocTest.example+{-# LINE 298 "src/Algebra/PrincipalIdealDomain.hs" #-}+   (PID.chineseRemainderMulti [(100,21), (10000,2021::Integer)])+  [ExpectedLine [LineChunk "Just (10000,2021)"]]+ DocTest.printPrefix "Algebra.PrincipalIdealDomain:300: "+{-# LINE 300 "src/Algebra/PrincipalIdealDomain.hs" #-}+ DocTest.example+{-# LINE 300 "src/Algebra/PrincipalIdealDomain.hs" #-}+   (PID.chineseRemainderMulti [(97,90),(99,10),(100,0::Integer)])+  [ExpectedLine [LineChunk "Just (960300,100000)"]]+ DocTest.printPrefix "Algebra.PrincipalIdealDomain:302: "+{-# LINE 302 "src/Algebra/PrincipalIdealDomain.hs" #-}+ DocTest.example+{-# LINE 302 "src/Algebra/PrincipalIdealDomain.hs" #-}+   (PID.chineseRemainderMulti [(95,30),(97,27),(98,8),(99,1::Integer)])+  [ExpectedLine [LineChunk "Just (89403930,1000000)"]]
test/Test/Algebra/RealRing.hs view
@@ -1,40 +1,126 @@-{-# LANGUAGE RebindableSyntax #-}-module Test.Algebra.RealRing where--import qualified Algebra.RealRing as RealRing--import Test.NumericPrelude.Utility (testUnit, )-import Test.QuickCheck (quickCheck, )-import qualified Test.HUnit as HUnit--import Data.Tuple.HT (mapFst, )+-- Do not edit! Automatically created with doctest-extract from src/Algebra/RealRing.hs+{-# LINE 38 "src/Algebra/RealRing.hs" #-} -import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP+module Test.Algebra.RealRing where +import qualified Test.DocTest.Driver as DocTest -test :: (Eq a) => (Double -> a) -> (Double -> a) -> IO ()-test f g =-   quickCheck (\x -> f x == g x)+{-# LINE 39 "src/Algebra/RealRing.hs" #-}+import     qualified Algebra.RealRing as RealRing+import     Data.Tuple.HT (mapFst)+import     NumericPrelude.Numeric as NP+import     NumericPrelude.Base+import     Prelude () +infix     4 =~= -tests :: HUnit.Test-tests =-   HUnit.TestLabel "rounding functions" $-   HUnit.TestList $-   map testUnit $-      ("round",         test RealRing.genericRound    (NP.round :: Double -> Integer)) :-      ("truncate",      test RealRing.genericTruncate (NP.truncate :: Double -> Integer)) :-      ("ceiling",       test RealRing.genericCeiling  (NP.ceiling :: Double -> Integer)) :-      ("floor",         test RealRing.genericFloor    (NP.floor :: Double -> Integer)) :-      ("fraction",      test RealRing.genericFraction (NP.fraction :: Double -> Double)) :-      ("splitFraction", test RealRing.genericSplitFraction (NP.splitFraction :: Double -> (Integer, Double))) :+(=~=)     :: (Eq b) => (a -> b) -> (a -> b) -> a -> Bool+(f     =~= g) x = f x == g x -{--      ("splitFractionId", quickCheck (\x -> (x::Double) == (uncurry (+) $ mapFst fromInteger $ splitFraction x))) :--}-      ("splitFractionId", quickCheck (\x ->  uncurry (==) $ mapFst (((x::Double)-) . fromInteger) $ splitFraction x)) :-      ("splitFractionFloorFraction", quickCheck (\x -> (floor (x::Double) :: Integer, fraction x) == splitFraction x)) :-      ("fractionBound", quickCheck (\x -> let y = fraction (x::Double) in 0<=y && y<1)) :-      ("floorCeiling", quickCheck (\x -> negate (floor (x::Double) :: Integer) == ceiling (-x))) :-      []+test :: DocTest.T ()+test = do+ DocTest.printPrefix "Algebra.RealRing:134: "+{-# LINE 134 "src/Algebra/RealRing.hs" #-}+ DocTest.property+{-# LINE 134 "src/Algebra/RealRing.hs" #-}+         (\x -> (x::Rational) == (uncurry (+) $ mapFst fromInteger $ splitFraction x))+ DocTest.printPrefix "Algebra.RealRing:135: "+{-# LINE 135 "src/Algebra/RealRing.hs" #-}+ DocTest.property+{-# LINE 135 "src/Algebra/RealRing.hs" #-}+         (\x -> uncurry (==) $ mapFst (((x::Double)-) . fromInteger) $ splitFraction x)+ DocTest.printPrefix "Algebra.RealRing:136: "+{-# LINE 136 "src/Algebra/RealRing.hs" #-}+ DocTest.property+{-# LINE 136 "src/Algebra/RealRing.hs" #-}+         (\x -> uncurry (==) $ mapFst (((x::Rational)-) . fromInteger) $ splitFraction x)+ DocTest.printPrefix "Algebra.RealRing:137: "+{-# LINE 137 "src/Algebra/RealRing.hs" #-}+ DocTest.property+{-# LINE 137 "src/Algebra/RealRing.hs" #-}+         (\x -> splitFraction x == (floor (x::Double) :: Integer, fraction x))+ DocTest.printPrefix "Algebra.RealRing:138: "+{-# LINE 138 "src/Algebra/RealRing.hs" #-}+ DocTest.property+{-# LINE 138 "src/Algebra/RealRing.hs" #-}+         (\x -> splitFraction x == (floor (x::Rational) :: Integer, fraction x))+ DocTest.printPrefix "Algebra.RealRing:142: "+{-# LINE 142 "src/Algebra/RealRing.hs" #-}+ DocTest.property+{-# LINE 142 "src/Algebra/RealRing.hs" #-}+         (\x -> let y = fraction (x::Double) in 0<=y && y<1)+ DocTest.printPrefix "Algebra.RealRing:143: "+{-# LINE 143 "src/Algebra/RealRing.hs" #-}+ DocTest.property+{-# LINE 143 "src/Algebra/RealRing.hs" #-}+         (\x -> let y = fraction (x::Rational) in 0<=y && y<1)+ DocTest.printPrefix "Algebra.RealRing:147: "+{-# LINE 147 "src/Algebra/RealRing.hs" #-}+ DocTest.property+{-# LINE 147 "src/Algebra/RealRing.hs" #-}+         (\x -> ceiling (-x) == negate (floor (x::Double) :: Integer))+ DocTest.printPrefix "Algebra.RealRing:148: "+{-# LINE 148 "src/Algebra/RealRing.hs" #-}+ DocTest.property+{-# LINE 148 "src/Algebra/RealRing.hs" #-}+         (\x -> ceiling (-x) == negate (floor (x::Rational) :: Integer))+ DocTest.printPrefix "Algebra.RealRing:564: "+{-# LINE 564 "src/Algebra/RealRing.hs" #-}+ DocTest.property+{-# LINE 564 "src/Algebra/RealRing.hs" #-}+     (RealRing.genericFloor =~= (NP.floor :: Double -> Integer))+ DocTest.printPrefix "Algebra.RealRing:565: "+{-# LINE 565 "src/Algebra/RealRing.hs" #-}+ DocTest.property+{-# LINE 565 "src/Algebra/RealRing.hs" #-}+     (RealRing.genericFloor =~= (NP.floor :: Rational -> Integer))+ DocTest.printPrefix "Algebra.RealRing:574: "+{-# LINE 574 "src/Algebra/RealRing.hs" #-}+ DocTest.property+{-# LINE 574 "src/Algebra/RealRing.hs" #-}+     (RealRing.genericCeiling =~= (NP.ceiling :: Double -> Integer))+ DocTest.printPrefix "Algebra.RealRing:575: "+{-# LINE 575 "src/Algebra/RealRing.hs" #-}+ DocTest.property+{-# LINE 575 "src/Algebra/RealRing.hs" #-}+     (RealRing.genericCeiling =~= (NP.ceiling :: Rational -> Integer))+ DocTest.printPrefix "Algebra.RealRing:584: "+{-# LINE 584 "src/Algebra/RealRing.hs" #-}+ DocTest.property+{-# LINE 584 "src/Algebra/RealRing.hs" #-}+     (RealRing.genericTruncate =~= (NP.truncate :: Double -> Integer))+ DocTest.printPrefix "Algebra.RealRing:585: "+{-# LINE 585 "src/Algebra/RealRing.hs" #-}+ DocTest.property+{-# LINE 585 "src/Algebra/RealRing.hs" #-}+     (RealRing.genericTruncate =~= (NP.truncate :: Rational -> Integer))+ DocTest.printPrefix "Algebra.RealRing:594: "+{-# LINE 594 "src/Algebra/RealRing.hs" #-}+ DocTest.property+{-# LINE 594 "src/Algebra/RealRing.hs" #-}+     (RealRing.genericRound =~= (NP.round :: Double -> Integer))+ DocTest.printPrefix "Algebra.RealRing:595: "+{-# LINE 595 "src/Algebra/RealRing.hs" #-}+ DocTest.property+{-# LINE 595 "src/Algebra/RealRing.hs" #-}+     (RealRing.genericRound =~= (NP.round :: Rational -> Integer))+ DocTest.printPrefix "Algebra.RealRing:604: "+{-# LINE 604 "src/Algebra/RealRing.hs" #-}+ DocTest.property+{-# LINE 604 "src/Algebra/RealRing.hs" #-}+     (RealRing.genericFraction =~= (NP.fraction :: Double -> Double))+ DocTest.printPrefix "Algebra.RealRing:605: "+{-# LINE 605 "src/Algebra/RealRing.hs" #-}+ DocTest.property+{-# LINE 605 "src/Algebra/RealRing.hs" #-}+     (RealRing.genericFraction =~= (NP.fraction :: Rational -> Rational))+ DocTest.printPrefix "Algebra.RealRing:614: "+{-# LINE 614 "src/Algebra/RealRing.hs" #-}+ DocTest.property+{-# LINE 614 "src/Algebra/RealRing.hs" #-}+     (RealRing.genericSplitFraction =~= (NP.splitFraction :: Double -> (Integer,Double)))+ DocTest.printPrefix "Algebra.RealRing:615: "+{-# LINE 615 "src/Algebra/RealRing.hs" #-}+ DocTest.property+{-# LINE 615 "src/Algebra/RealRing.hs" #-}+     (RealRing.genericSplitFraction =~= (NP.splitFraction :: Rational -> (Integer,Rational)))
test/Test/MathObj/Gaussian/Bell.hs view
@@ -1,96 +1,157 @@-{-# LANGUAGE RebindableSyntax #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-module Test.MathObj.Gaussian.Bell where--import qualified MathObj.Gaussian.Bell as G---- import qualified Algebra.Ring           as Ring--import qualified Algebra.Laws as Laws+-- Do not edit! Automatically created with doctest-extract from gaussian/MathObj/Gaussian/Bell.hs+{-# LINE 30 "gaussian/MathObj/Gaussian/Bell.hs" #-} -import qualified Number.Complex as Complex+module Test.MathObj.Gaussian.Bell where -import Test.NumericPrelude.Utility (testUnit)-import Test.QuickCheck (Testable, quickCheck, (==>))-import qualified Test.HUnit as HUnit+import Test.DocTest.Base+import qualified Test.DocTest.Driver as DocTest -import Data.Function.HT (nest, )+{-# LINE 31 "gaussian/MathObj/Gaussian/Bell.hs" #-}+import     qualified MathObj.Gaussian.Bell as G+import     qualified Algebra.ZeroTestable as ZeroTestable+import     qualified Algebra.Laws as Laws+import     qualified Number.Complex as Complex+import     Number.Complex ((+:))+import     NumericPrelude.Base as P+import     NumericPrelude.Numeric as NP+import     Prelude ()+import     qualified Test.QuickCheck as QC+import     Data.Function.HT (Id, nest) -import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP+asRational     :: Id (G.T Rational)+asRational     = id +withRational     :: Id (G.T Rational -> a)+withRational     = id -simple ::-   (Testable t) =>-   (G.T Rational -> t) -> IO ()-simple f =-   quickCheck (\x -> f (x :: G.T Rational))+isConstant     :: ZeroTestable.C a => G.T a -> Bool+isConstant     (G.Cons _amp _a b c) = isZero b && isZero c -tests :: HUnit.Test-tests =-   HUnit.TestLabel "polynomial" $-   HUnit.TestList $-   map testUnit $-{--      ("convolution, dirac",-          simple $ Laws.identity (+) zero) :--}-      ("convolution, commutative",-          simple $ Laws.commutative G.convolve) :-      ("convolution, associative",-          simple $ Laws.associative G.convolve) :-      ("multiplication, one",-          simple $ Laws.identity G.multiply G.constant) :-      ("multiplication, commutative",-          simple $ Laws.commutative G.multiply) :-      ("multiplication, associative",-          simple $ Laws.associative G.multiply) :-      ("convolution, multplication, fourier",-          simple $ \x y ->-             G.fourier (G.convolve x y)-              == G.multiply (G.fourier x) (G.fourier y)) :-      ("convolution via translation",-          simple $ \x y ->-             G.convolve x y-              == G.convolveByTranslation x y) :-      ("convolution via fourier",-          simple $ \x y ->-             G.convolve x y-              == G.convolveByFourier x y) :-      ("fourier reverse",-          simple $ \x -> nest 2 G.fourier x == G.reverse x) :-      ("reverse identity",-          simple $ \x -> nest 2 G.reverse x == x) :-      ("fourier unit",-          quickCheck $ G.fourier G.unit == (G.unit :: G.T Rational)) :-      ("translate additive",-          simple $ \x a b ->-             G.translate a (G.translate b x) == G.translate (a+b) x) :-      ("translateComplex additive",-          simple $ \x a b ->-             G.translateComplex a (G.translateComplex b x) == G.translateComplex (a+b) x) :-      ("translateComplex real",-          simple $ \x a ->-             G.translateComplex (Complex.fromReal a) x == G.translate a x) :-      ("modulate additive",-          simple $ \x a b ->-             G.modulate a (G.modulate b x) == G.modulate (a+b) x) :-      ("commute translate modulate",-          simple $ \x a b ->-             G.modulate b (G.translate a x)-              == G.turn (a*b) (G.translate a (G.modulate b x))) :-      ("fourier translate",-          simple $ \x a ->-             G.fourier (G.translate a x)-              == G.modulate a (G.fourier x)) :-      ("dilate multiplicative",-          simple $ \x a b -> a>0 && b>0 ==>-             G.dilate a (G.dilate b x) == G.dilate (a*b) x) :-      ("dilate by shrink",-          simple $ \x a -> a>0 ==>-             G.shrink a x == G.dilate (recip a) x) :-      ("fourier dilate",-          simple $ \x a -> a>0 ==>-             G.fourier (G.dilate a x) == G.amplify a (G.shrink a (G.fourier x))) :-      []+test :: DocTest.T ()+test = do+ DocTest.printPrefix "MathObj.Gaussian.Bell:108: "+{-# LINE 108 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 108 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (Laws.identity G.multiply G.constant . asRational)+ DocTest.printPrefix "MathObj.Gaussian.Bell:109: "+{-# LINE 109 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 109 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (Laws.commutative G.multiply . asRational)+ DocTest.printPrefix "MathObj.Gaussian.Bell:110: "+{-# LINE 110 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 110 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (Laws.associative G.multiply . asRational)+ DocTest.printPrefix "MathObj.Gaussian.Bell:152: "+{-# LINE 152 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 152 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (Laws.commutative G.convolve . asRational)+ DocTest.printPrefix "MathObj.Gaussian.Bell:153: "+{-# LINE 153 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 153 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (Laws.associative G.convolve . asRational)+ DocTest.printPrefix "MathObj.Gaussian.Bell:161: "+{-# LINE 161 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 161 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (isConstant . G.convolve G.constant . asRational)+ DocTest.printPrefix "MathObj.Gaussian.Bell:149: "+{-# LINE 149 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.example+{-# LINE 149 "gaussian/MathObj/Gaussian/Bell.hs" #-}+   (let x=G.Cons 2 (1+:3) (4+:5) (7::Rational); y=G.Cons 7 (1+:4) (3+:2) (5::Rational) in G.convolve x y)+  [ExpectedLine [LineChunk "Cons {amp = 7 % 6, c0 = 13 % 6 +: 55 % 8, c1 = 41 % 12 +: 13 % 4, c2 = 35 % 12}"]]+ DocTest.printPrefix "MathObj.Gaussian.Bell:200: "+{-# LINE 200 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 200 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (withRational $ \x y -> G.convolve x y == G.convolveByTranslation x y)+ DocTest.printPrefix "MathObj.Gaussian.Bell:217: "+{-# LINE 217 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 217 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (withRational $ \x y -> G.convolve x y == G.convolveByFourier x y)+ DocTest.printPrefix "MathObj.Gaussian.Bell:225: "+{-# LINE 225 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 225 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (withRational $ \x y -> G.fourier (G.convolve x y) == G.multiply (G.fourier x) (G.fourier y))+ DocTest.printPrefix "MathObj.Gaussian.Bell:226: "+{-# LINE 226 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 226 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (withRational $ \x -> nest 2 G.fourier x == G.reverse x)+ DocTest.printPrefix "MathObj.Gaussian.Bell:227: "+{-# LINE 227 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 227 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (G.fourier G.unit == (asRational G.unit))+ DocTest.printPrefix "MathObj.Gaussian.Bell:228: "+{-# LINE 228 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 228 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (withRational $ \x a -> G.fourier (G.translate a x) == G.modulate a (G.fourier x))+ DocTest.printPrefix "MathObj.Gaussian.Bell:229: "+{-# LINE 229 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 229 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (withRational $ \x (QC.Positive a) -> G.fourier (G.dilate a x) == G.amplify a (G.shrink a (G.fourier x)))+ DocTest.printPrefix "MathObj.Gaussian.Bell:244: "+{-# LINE 244 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 244 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (withRational $ \x -> G.fourier x == G.fourierByTranslation x)+ DocTest.printPrefix "MathObj.Gaussian.Bell:312: "+{-# LINE 312 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 312 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (withRational $ \x a b -> G.translate a (G.translate b x) == G.translate (a+b) x)+ DocTest.printPrefix "MathObj.Gaussian.Bell:326: "+{-# LINE 326 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 326 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (withRational $ \x a b -> G.translateComplex a (G.translateComplex b x) == G.translateComplex (a+b) x)+ DocTest.printPrefix "MathObj.Gaussian.Bell:327: "+{-# LINE 327 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 327 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (withRational $ \x a -> G.translateComplex (Complex.fromReal a) x == G.translate a x)+ DocTest.printPrefix "MathObj.Gaussian.Bell:341: "+{-# LINE 341 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 341 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (withRational $ \x a b -> G.modulate a (G.modulate b x) == G.modulate (a+b) x)+ DocTest.printPrefix "MathObj.Gaussian.Bell:342: "+{-# LINE 342 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 342 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (withRational $ \x a b -> G.modulate b (G.translate a x) == G.turn (a*b) (G.translate a (G.modulate b x)))+ DocTest.printPrefix "MathObj.Gaussian.Bell:361: "+{-# LINE 361 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 361 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (withRational $ \x -> nest 2 G.reverse x == x)+ DocTest.printPrefix "MathObj.Gaussian.Bell:369: "+{-# LINE 369 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 369 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (withRational $ \x (QC.Positive a) (QC.Positive b) -> G.dilate a (G.dilate b x) == G.dilate (a*b) x)+ DocTest.printPrefix "MathObj.Gaussian.Bell:370: "+{-# LINE 370 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 370 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (withRational $ \x (QC.Positive a) -> G.shrink a x == G.dilate (recip a) x)+ DocTest.printPrefix "MathObj.Gaussian.Bell:381: "+{-# LINE 381 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 381 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (withRational $ \x (QC.Positive a) -> G.dilate a (G.shrink a x) == x)+ DocTest.printPrefix "MathObj.Gaussian.Bell:382: "+{-# LINE 382 "gaussian/MathObj/Gaussian/Bell.hs" #-}+ DocTest.property+{-# LINE 382 "gaussian/MathObj/Gaussian/Bell.hs" #-}+     (withRational $ \x (QC.Positive a) -> G.shrink a (G.dilate a x) == x)
+ test/Test/MathObj/Gaussian/ExponentTuple.hs view
@@ -0,0 +1,26 @@+-- Do not edit! Automatically created with doctest-extract from gaussian/MathObj/Gaussian/ExponentTuple.hs+{-# LINE 14 "gaussian/MathObj/Gaussian/ExponentTuple.hs" #-}++module Test.MathObj.Gaussian.ExponentTuple where++import qualified Test.DocTest.Driver as DocTest++{-# LINE 15 "gaussian/MathObj/Gaussian/ExponentTuple.hs" #-}+import     MathObj.Gaussian.ExponentTuple (HoelderConjugates(HoelderConjugates))+import     MathObj.Gaussian.ExponentTuple (YoungConjugates(YoungConjugates))+import     NumericPrelude.Base as P+import     NumericPrelude.Numeric as NP+import     Prelude ()++test :: DocTest.T ()+test = do+ DocTest.printPrefix "MathObj.Gaussian.ExponentTuple:26: "+{-# LINE 26 "gaussian/MathObj/Gaussian/ExponentTuple.hs" #-}+ DocTest.property+{-# LINE 26 "gaussian/MathObj/Gaussian/ExponentTuple.hs" #-}+     (\(HoelderConjugates p q)  ->  p>=1 && q>=1 && 1/p + 1/q == 1)+ DocTest.printPrefix "MathObj.Gaussian.ExponentTuple:53: "+{-# LINE 53 "gaussian/MathObj/Gaussian/ExponentTuple.hs" #-}+ DocTest.property+{-# LINE 53 "gaussian/MathObj/Gaussian/ExponentTuple.hs" #-}+     (\(YoungConjugates p q r)  ->  p>=1 && q>=1 && r>=1 && 1/p + 1/q == 1/r + 1)
test/Test/MathObj/Gaussian/Polynomial.hs view
@@ -1,158 +1,215 @@-{-# LANGUAGE RebindableSyntax #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-module Test.MathObj.Gaussian.Polynomial where--import qualified MathObj.Gaussian.Polynomial as G-import qualified MathObj.Gaussian.Bell as B--import qualified MathObj.Polynomial as Poly---- import qualified Algebra.Ring           as Ring--import qualified Algebra.Laws as Laws+-- Do not edit! Automatically created with doctest-extract from gaussian/MathObj/Gaussian/Polynomial.hs+{-# LINE 60 "gaussian/MathObj/Gaussian/Polynomial.hs" #-} -import qualified Number.Complex as Complex+{-# OPTIONS_GHC -XRebindableSyntax #-} -import Test.NumericPrelude.Utility (testUnit)-import Test.QuickCheck (Testable, quickCheck, (==>))-import qualified Test.HUnit as HUnit+module Test.MathObj.Gaussian.Polynomial where -import qualified Number.NonNegative as NonNeg-import Data.Function.HT (nest, )-import Data.Tuple.HT (mapSnd, )+import Test.DocTest.Base+import qualified Test.DocTest.Driver as DocTest --- import Debug.Trace (trace, )+{-# LINE 63 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+import     qualified MathObj.Gaussian.Polynomial as G+import     qualified MathObj.Gaussian.Bell as Bell+import     qualified MathObj.Polynomial as Poly+import     qualified Algebra.Laws as Laws+import     qualified Number.Complex as Complex+import     Number.Complex ((+:))+import     NumericPrelude.Base as P+import     NumericPrelude.Numeric as NP+import     qualified Test.QuickCheck as QC+import     Data.Function.HT (Id, nest)+import     Data.Tuple.HT (mapSnd) -import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP+asRational     :: Id (G.T Rational)+asRational     = id +withRational     :: Id (G.T Rational -> a)+withRational     = id -simple ::-   (Testable t) =>-   (G.T Rational -> t) -> IO ()-simple f =-   quickCheck (\x -> f (x :: G.T Rational))+mulLinear2i     :: Id (G.T Rational)+mulLinear2i     x =+       x{G.polynomial = Poly.fromCoeffs [0, 0+:2] * G.polynomial x} -tests :: HUnit.Test-tests =-   HUnit.TestLabel "polynomial" $-   HUnit.TestList $-   map testUnit $-   testList+rotateQuarter     :: Int -> Id (G.T Rational)+rotateQuarter     n =+       G.scaleComplex (negate Complex.imaginaryUnit ^ fromIntegral n) -testList :: [(String, IO ())]-testList =-{--      ("convolution, dirac",-          simple $ Laws.identity (+) zero) :--}-      ("convolution, commutative",-          simple $ Laws.commutative G.convolve) :---          simple $ \x -> Laws.commutative G.convolve (trace (show x) x)) :-      ("convolution, associative",-          simple $ Laws.associative G.convolve) :-{--      ("convolution by differentiation vs. fourier",-          simple $ \x y ->-             G.convolveByDifferentiation x y-              == G.convolveByFourier x y) :--}-      ("multiplication, one",-          simple $ Laws.identity G.multiply G.constant) :-      ("multiplication, commutative",-          simple $ Laws.commutative G.multiply) :-      ("multiplication, associative",-          simple $ Laws.associative G.multiply) :-      ("convolution, multplication, fourier",-          simple $ \x y ->-             G.fourier (G.convolve x y)-              == G.multiply (G.fourier x) (G.fourier y)) :-      ("fourier reverse",-          simple $ \x -> nest 2 G.fourier x == G.reverse x) :-      ("reverse identity",-          simple $ \x -> nest 2 G.reverse x == x) :-      ("fourier eigenfunction differential",-          quickCheck $ \m ->-             m <= 15 ==>-                let n = NonNeg.toNumber m-                    x = G.eigenfunctionDifferential n :: G.T Rational-                    k = Complex.conjugate Complex.imaginaryUnit ^ fromIntegral n-                in  G.fourier x  ==  G.scaleComplex k x) :-      ("fourier eigenfunction iterative",-          quickCheck $ \m ->-             m <= 15 ==>-                let n = NonNeg.toNumber m-                    x = G.eigenfunctionIterative n :: G.T Rational-                    k = Complex.conjugate Complex.imaginaryUnit ^ fromIntegral n-                in  G.fourier x  ==  G.scaleComplex k x) :-{- this does not hold, both functions compute different eigenbases-      ("fourier eigenfunction diff vs. iterative",-          quickCheck $ \n ->-             n <= 15 ==>-                G.eigenfunctionDifferential n ==-                (G.eigenfunctionIterative n :: G.T Rational)) :--}-      ("translate additive",-          simple $ \x a b ->-             G.translate a (G.translate b x) == G.translate (a+b) x) :-      ("translateComplex additive",-          simple $ \x a b ->-             G.translateComplex a (G.translateComplex b x) == G.translateComplex (a+b) x) :-      ("translateComplex real",-          simple $ \x a ->-             G.translateComplex (Complex.fromReal a) x == G.translate a x) :-      ("modulate additive",-          simple $ \x a b ->-             G.modulate a (G.modulate b x) == G.modulate (a+b) x) :-      ("commute translate modulate",-          simple $ \x a b ->-             G.modulate b (G.translate a x)-              == G.turn (a*b) (G.translate a (G.modulate b x))) :-      ("fourier translate",-          simple $ \x a ->-             G.fourier (G.translate a x)-              == G.modulate a (G.fourier x)) :-      ("dilate multiplicative",-          simple $ \x a b -> a>0 && b>0 ==>-             G.dilate a (G.dilate b x) == G.dilate (a*b) x) :-      ("dilate by shrink",-          simple $ \x a -> a>0 ==>-             G.shrink a x == G.dilate (recip a) x) :-      ("fourier dilate",-          simple $ \x a -> a>0 ==>-             G.fourier (G.dilate a x) == G.amplify a (G.shrink a (G.fourier x))) :-      ("integrate differentiate",-          simple $ \x ->-             G.integrate (G.differentiate x) == (zero, x)) :-      ("differentiate integrate",-          simple $ \x@(G.Cons b p) ->-             let (xoff,xint) = G.integrate x-             in  G.differentiate xint == G.Cons b (p + Poly.const xoff)) :-      ("fourier differentiate",-          simple $ \x ->-             G.fourier (G.differentiate x) ==-              let y = G.fourier x-              in  y{G.polynomial =-                      Poly.fromCoeffs [0, 0 Complex.+: 2] * G.polynomial y}) :-      ("differentiate convolve",-          simple $ \x y ->-             G.convolve (G.differentiate x) y ==-             G.convolve x (G.differentiate y)) :-      ("approximate by bells, translate",-          simple $ \x unit d -> unit/=0 ==>-             G.approximateByBells unit (G.translateComplex d x) ==-             map (mapSnd (B.translateComplex d)) (G.approximateByBells unit x)) :-      ("approximate by bells, dilate",-          simple $ \x unit d -> unit/=0 && d/=0 ==>-             G.approximateByBells unit (G.dilate d x) ==-             map (mapSnd (B.dilate d)) (G.approximateByBells (unit/d) x)) :-      ("approximate by bells, shrink",-          simple $ \x unit d -> unit/=0 && d/=0 ==>-             G.approximateByBells unit (G.shrink d x) ==-             map (mapSnd (B.shrink d)) (G.approximateByBells (unit*d) x)) :-      ("approximate by bells, different implementations",-          quickCheck $ \unit d s p -> unit/=0 ==>-             G.approximateByBellsAtOnce unit d s (p::Poly.T (Complex.T Rational)) ==-             G.approximateByBellsByTranslation unit d s p) :-      []+test :: DocTest.T ()+test = do+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:185: "+{-# LINE 185 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 185 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (QC.forAll (QC.choose (0,3)) $ \n -> G.eigenfunctionDifferential n == asRational (G.eigenfunctionIterative n))+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:193: "+{-# LINE 193 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 193 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+          (G.eigenfunction0  ==  asRational (G.eigenfunctionDifferential 0))+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:198: "+{-# LINE 198 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 198 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+          (G.eigenfunction1  ==  asRational (G.eigenfunctionDifferential 1))+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:203: "+{-# LINE 203 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 203 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+          (G.eigenfunction2  ==  asRational (G.eigenfunctionDifferential 2))+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:208: "+{-# LINE 208 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 208 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+          (G.eigenfunction3  ==  asRational (G.eigenfunctionDifferential 3))+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:215: "+{-# LINE 215 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 215 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (QC.forAll (QC.choose (0,15)) $ \n -> let x = G.eigenfunctionDifferential n in G.fourier x  ==  rotateQuarter n x)+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:224: "+{-# LINE 224 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 224 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (QC.forAll (QC.choose (0,15)) $ \n -> let x = G.eigenfunctionIterative n in G.fourier x  ==  rotateQuarter n x)+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:246: "+{-# LINE 246 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 246 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ Laws.identity G.multiply G.constant)+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:247: "+{-# LINE 247 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 247 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ Laws.commutative G.multiply)+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:248: "+{-# LINE 248 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 248 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ Laws.associative G.multiply)+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:258: "+{-# LINE 258 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 258 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ Laws.commutative G.convolve)+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:259: "+{-# LINE 259 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 259 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ Laws.associative G.convolve)+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:301: "+{-# LINE 301 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 301 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ \x y -> G.fourier (G.convolve x y) == G.multiply (G.fourier x) (G.fourier y))+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:302: "+{-# LINE 302 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 302 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ \x -> nest 2 G.fourier x == G.reverse x)+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:303: "+{-# LINE 303 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 303 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ \x a -> G.fourier (G.translate a x) == G.modulate a (G.fourier x))+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:304: "+{-# LINE 304 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 304 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ \x (QC.Positive a) -> G.fourier (G.dilate a x) == G.amplify a (G.shrink a (G.fourier x)))+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:305: "+{-# LINE 305 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 305 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ \x -> G.fourier (G.differentiate x) == mulLinear2i (G.fourier x))+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:323: "+{-# LINE 323 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 323 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ \x y -> G.convolve (G.differentiate x) y == G.convolve x (G.differentiate y))+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:348: "+{-# LINE 348 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 348 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ \x -> G.integrate (G.differentiate x) == (zero, x))+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:349: "+{-# LINE 349 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 349 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ \x@(G.Cons b p) -> let (xoff,xint) = G.integrate x in G.differentiate xint == G.Cons b (p + Poly.const xoff))+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:345: "+{-# LINE 345 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.example+{-# LINE 345 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+   (snd $ G.integrate $ G.differentiate $ G.Cons Bell.unit (Poly.fromCoeffs [7,7,7,7 :: Complex.T Rational]))+  [ExpectedLine [LineChunk "Cons {bell = Cons {amp = 1 % 1, c0 = 0 % 1 +: 0 % 1, c1 = 0 % 1 +: 0 % 1, c2 = 1 % 1}, polynomial = Polynomial.fromCoeffs [7 % 1 +: 0 % 1,7 % 1 +: 0 % 1,7 % 1 +: 0 % 1,7 % 1 +: 0 % 1]}"]]+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:409: "+{-# LINE 409 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 409 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ \x a b -> G.translate a (G.translate b x) == G.translate (a+b) x)+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:416: "+{-# LINE 416 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 416 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ \x a b -> G.translateComplex a (G.translateComplex b x) == G.translateComplex (a+b) x)+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:417: "+{-# LINE 417 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 417 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ \x a -> G.translateComplex (Complex.fromReal a) x == G.translate a x)+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:426: "+{-# LINE 426 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 426 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ \x a b -> G.modulate a (G.modulate b x) == G.modulate (a+b) x)+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:427: "+{-# LINE 427 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 427 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ \x a b -> G.modulate b (G.translate a x) == G.turn (a*b) (G.translate a (G.modulate b x)))+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:442: "+{-# LINE 442 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 442 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ \x -> nest 2 G.reverse x == x)+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:451: "+{-# LINE 451 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 451 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ \x (QC.Positive a) (QC.Positive b) -> G.dilate a (G.dilate b x) == G.dilate (a*b) x)+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:452: "+{-# LINE 452 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 452 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ \x (QC.Positive a) -> G.shrink a x == G.dilate (recip a) x)+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:461: "+{-# LINE 461 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 461 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ \x (QC.Positive a) -> G.dilate a (G.shrink a x) == x)+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:462: "+{-# LINE 462 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 462 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ \x (QC.Positive a) -> G.shrink a (G.dilate a x) == x)+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:490: "+{-# LINE 490 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 490 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ \x (QC.NonZero unit) d -> G.approximateByBells unit (G.translateComplex d x) == map (mapSnd (Bell.translateComplex d)) (G.approximateByBells unit x))+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:491: "+{-# LINE 491 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 491 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ \x (QC.NonZero unit) (QC.NonZero d) -> G.approximateByBells unit (G.dilate d x) == map (mapSnd (Bell.dilate d)) (G.approximateByBells (unit/d) x))+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:492: "+{-# LINE 492 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 492 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (withRational $ \x (QC.NonZero unit) (QC.NonZero d) -> G.approximateByBells unit (G.shrink d x) == map (mapSnd (Bell.shrink d)) (G.approximateByBells (unit*d) x))+ DocTest.printPrefix "MathObj.Gaussian.Polynomial:512: "+{-# LINE 512 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+ DocTest.property+{-# LINE 512 "gaussian/MathObj/Gaussian/Polynomial.hs" #-}+     (\(QC.NonZero unit) d s p0 -> let p = Poly.fromCoeffs $ take 10 p0 in G.approximateByBellsAtOnce unit d s p == G.approximateByBellsByTranslation unit d (s::Rational) p)
test/Test/MathObj/Gaussian/Variance.hs view
@@ -1,210 +1,142 @@-{-# LANGUAGE RebindableSyntax #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-module Test.MathObj.Gaussian.Variance where--import qualified MathObj.Gaussian.Variance as G-import qualified Number.Root as Root---- import qualified Algebra.Ring           as Ring--import qualified Algebra.Laws as Laws--import Test.NumericPrelude.Utility (testUnit)-import Test.QuickCheck (Testable, quickCheck, (==>), Arbitrary, arbitrary, )-import qualified Test.HUnit as HUnit--import Control.Monad (liftM2, liftM3, )--import Data.Function.HT (nest, compose2, )--import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP---newtype PositiveInteger = PositiveInteger Integer-   deriving Show--instance Arbitrary PositiveInteger where-   arbitrary =-      fmap (\p -> PositiveInteger $ 1 + abs p) arbitrary---{- |-For @(HoelderConjugates p q)@ it holds--> 1/p + 1/q = 1--}-data HoelderConjugates = HoelderConjugates Rational Rational-   deriving Show--{--instance Arbitrary HoelderConjugates where-   arbitrary = liftM2-      (\(PositiveInteger p) (PositiveInteger q) ->-         let s  = 1%p + 1%q-         in  HoelderConjugates (fromInteger p * s) (fromInteger q * s))-      arbitrary arbitrary--}-instance Arbitrary HoelderConjugates where-   arbitrary = liftM2-      (\(PositiveInteger p) (PositiveInteger q) ->-         let s = p + q-         in  HoelderConjugates (s % p) (s % q))-      arbitrary arbitrary--{- |-For @(YoungConjugates p q r)@ it holds--> 1/p + 1/q = 1/r + 1--}-data YoungConjugates = YoungConjugates Rational Rational Rational-   deriving Show--{--Find positive natural numbers @a, b, c, d@ with--> a + b = c + d--and--> d >= a, d >= b, d >= c--then set--> p=d/a, q=d/b, r=d/c---a+b<=c-b+c<=a-->  2b <= 0--}-instance Arbitrary YoungConjugates where-   arbitrary = liftM3-      (\(PositiveInteger a0) (PositiveInteger b0) (PositiveInteger c0) ->-         let guardSwap cond (x,y) =-                if cond x y then (x,y) else (y,x)-             {--             If a+b<=c, then from b>0 it follows a<c and thus c+b>a.-             Swapping a and c is enough and we have not to consider more cases.-             -}-             (a1,c1) = guardSwap (\a c -> a+b0>c) (a0,c0)-             b1 = b0-             d1 = a1+b1-c1-             ((a2,b2),(c2,d2)) =-                guardSwap (compose2 (<=) snd)-                   (guardSwap (<=) (a1,b1),-                    guardSwap (<=) (c1,d1))-         in  YoungConjugates (d2%a2) (d2%b2) (d2%c2))-      arbitrary arbitrary arbitrary+-- Do not edit! Automatically created with doctest-extract from gaussian/MathObj/Gaussian/Variance.hs+{-# LINE 34 "gaussian/MathObj/Gaussian/Variance.hs" #-} +module Test.MathObj.Gaussian.Variance where -simple ::-   (Testable t) =>-   (G.T Rational -> t) -> IO ()-simple f =-   quickCheck (\x -> f (x :: G.T Rational))+import qualified Test.DocTest.Driver as DocTest -tests :: HUnit.Test-tests =-   HUnit.TestLabel "variance" $-   HUnit.TestList $-   map testUnit $-   testList+{-# LINE 35 "gaussian/MathObj/Gaussian/Variance.hs" #-}+import     qualified MathObj.Gaussian.Variance as G+import     MathObj.Gaussian.ExponentTuple (HoelderConjugates(HoelderConjugates))+import     MathObj.Gaussian.ExponentTuple (YoungConjugates(YoungConjugates))+import     qualified Algebra.Laws as Laws+import     qualified Number.Root as Root+import     NumericPrelude.Base as P+import     NumericPrelude.Numeric as NP+import     Prelude ()+import     qualified Test.QuickCheck as QC+import     Data.Function.HT (Id, nest) -testList :: [(String, IO ())]-testList =-{--      ("convolution, dirac",-          simple $ Laws.identity (+) zero) :--}-      ("convolution, commutative",-          simple $ Laws.commutative G.convolve) :-      ("convolution, associative",-          simple $ Laws.associative G.convolve) :-      ("multiplication, one",-          simple $ Laws.identity G.multiply G.constant) :-      ("multiplication, commutative",-          simple $ Laws.commutative G.multiply) :-      ("multiplication, associative",-          simple $ Laws.associative G.multiply) :-      ("convolution via fourier",-          simple $ \x y ->-             G.fourier (G.convolve x y)-              == G.multiply (G.fourier x) (G.fourier y)) :-      ("fourier identity",-          simple $ \x -> nest 4 G.fourier x == x) :-      ("dilate multiplicative",-          simple $ \x a b -> a>0 && b>0 ==>-             G.dilate a (G.dilate b x) == G.dilate (a*b) x) :-      ("dilate by shrink",-          simple $ \x a -> a>0 ==>-             G.shrink a x == G.dilate (recip a) x) :-      ("fourier dilate",-          simple $ \x a -> a>0 ==>-             G.fourier (G.dilate a x) == G.amplify a (G.shrink a (G.fourier x))) :-      ("fourier, unitary",-          simple $ \x y ->-             G.scalarProductRoot x y-              == G.scalarProductRoot (G.fourier x) (G.fourier y)) :-      ("norm1 vs. normP 1",-          simple $ \x -> G.norm1Root x == G.normPRoot 1 x) :-      ("norm2 vs. normP 2",-          simple $ \x -> G.norm2Root x == G.normPRoot 2 x) :-{--I would have liked to test for a monotony of norms.-Unfortunately, it does not hold.+asRational     :: Id (G.T Rational)+asRational     = id -Means contain a division by the size of the domain.-Norms do not have this division.-Means are monotonic with respect to the degree.-Norms are not.-We cannot turn the norms into means since the size of the domain-(the complete real axis) is infinitely large.-      ("norm monotony",-          simple $ \x p0 q0 ->-             let p = 1 + abs p0-                 q = 1 + abs q0-             in  case compare p q of-                    EQ -> G.normPRoot p x == G.normPRoot q x-                    LT -> G.normPRoot p x <= G.normPRoot q x-                    GT -> G.normPRoot p x >= G.normPRoot q x) :+withRational     :: Id (G.T Rational -> a)+withRational     = id -This should also fail,-but QuickCheck does not seem to try counterexamples.-      ("infinity norm upper bound",-          simple $ \x p0 ->-             let p = 1 + abs p0-             in  G.normPRoot p x <= G.normInfRoot x) :--}-      ("Cauchy-Schwarz inequality",-          simple $ \x y ->-             G.scalarProductRoot x y-                <= G.norm2Root x `Root.mul` G.norm2Root y) :-      ("Hoelder conjugates",-          quickCheck $ \(HoelderConjugates p q) ->-             p>=1 && q>=1 && 1/p + 1/q == 1) :-      ("Hoelder inequality with infinity norm",-          simple $ \x y ->-             G.scalarProductRoot x y-                <= G.norm1Root x `Root.mul` G.normInfRoot y) :-      ("Hoelder inequality",-          simple $ \x y (HoelderConjugates p q) ->-             G.scalarProductRoot x y-                <= G.normPRoot p x `Root.mul` G.normPRoot q y) :-      ("Young inequality with two infinity norms",-          simple $ \x y ->-             G.normInfRoot (G.convolve x y)-                <= G.norm1Root x `Root.mul` G.normInfRoot y) :-      ("Young inequality with infinity norm",-          simple $ \x y (HoelderConjugates p q) ->-             G.normInfRoot (G.convolve x y)-                <= G.normPRoot p x `Root.mul` G.normPRoot q y) :-      ("Young conjugates",-          quickCheck $ \(YoungConjugates p q r) ->-             p>=1 && q>=1 && r>=1 && 1/p + 1/q == 1/r + 1) :-      ("Young inequality",-          simple $ \x y (YoungConjugates p q r) ->-             G.normPRoot r (G.convolve x y)-                <= G.normPRoot p x `Root.mul` G.normPRoot q y) :-      []+test :: DocTest.T ()+test = do+ DocTest.printPrefix "MathObj.Gaussian.Variance:95: "+{-# LINE 95 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 95 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (withRational $ \x y -> G.scalarProductRoot x y <= G.norm2Root x `Root.mul` G.norm2Root y)+ DocTest.printPrefix "MathObj.Gaussian.Variance:99: "+{-# LINE 99 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 99 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (withRational $ \x y -> G.scalarProductRoot x y <= G.norm1Root x `Root.mul` G.normInfRoot y)+ DocTest.printPrefix "MathObj.Gaussian.Variance:100: "+{-# LINE 100 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 100 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (withRational $ \x y (HoelderConjugates p q) -> G.scalarProductRoot x y <= G.normPRoot p x `Root.mul` G.normPRoot q y)+ DocTest.printPrefix "MathObj.Gaussian.Variance:108: "+{-# LINE 108 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 108 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (withRational $ \x -> G.norm1Root x == G.normPRoot 1 x)+ DocTest.printPrefix "MathObj.Gaussian.Variance:114: "+{-# LINE 114 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 114 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (withRational $ \x -> G.norm2Root x == G.normPRoot 2 x)+ DocTest.printPrefix "MathObj.Gaussian.Variance:186: "+{-# LINE 186 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 186 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (withRational $ \x (QC.Positive a) -> G.varianceRational (G.dilate a x) == a^2 * G.varianceRational x)+ DocTest.printPrefix "MathObj.Gaussian.Variance:187: "+{-# LINE 187 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 187 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (withRational $ \x y -> G.varianceRational (G.convolve x y) == G.varianceRational x + G.varianceRational y)+ DocTest.printPrefix "MathObj.Gaussian.Variance:193: "+{-# LINE 193 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 193 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (Laws.identity G.multiply G.constant . asRational)+ DocTest.printPrefix "MathObj.Gaussian.Variance:194: "+{-# LINE 194 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 194 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (Laws.commutative G.multiply . asRational)+ DocTest.printPrefix "MathObj.Gaussian.Variance:195: "+{-# LINE 195 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 195 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (Laws.associative G.multiply . asRational)+ DocTest.printPrefix "MathObj.Gaussian.Variance:228: "+{-# LINE 228 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 228 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (Laws.commutative G.convolve . asRational)+ DocTest.printPrefix "MathObj.Gaussian.Variance:229: "+{-# LINE 229 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 229 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (Laws.associative G.convolve . asRational)+ DocTest.printPrefix "MathObj.Gaussian.Variance:233: "+{-# LINE 233 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 233 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (withRational $ \x y -> G.normInfRoot (G.convolve x y) <= G.norm1Root x `Root.mul` G.normInfRoot y)+ DocTest.printPrefix "MathObj.Gaussian.Variance:234: "+{-# LINE 234 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 234 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (withRational $ \x y (HoelderConjugates p q) -> G.normInfRoot (G.convolve x y) <= G.normPRoot p x `Root.mul` G.normPRoot q y)+ DocTest.printPrefix "MathObj.Gaussian.Variance:235: "+{-# LINE 235 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 235 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (withRational $ \x y (YoungConjugates p q r) -> G.normPRoot r (G.convolve x y) <= G.normPRoot p x `Root.mul` G.normPRoot q y)+ DocTest.printPrefix "MathObj.Gaussian.Variance:251: "+{-# LINE 251 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 251 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (withRational $ \x y -> G.fourier (G.convolve x y) == G.multiply (G.fourier x) (G.fourier y))+ DocTest.printPrefix "MathObj.Gaussian.Variance:252: "+{-# LINE 252 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 252 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (withRational $ \x -> nest 4 G.fourier x == x)+ DocTest.printPrefix "MathObj.Gaussian.Variance:253: "+{-# LINE 253 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 253 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (withRational $ \x (QC.Positive a) -> G.fourier (G.dilate a x) == G.amplify a (G.shrink a (G.fourier x)))+ DocTest.printPrefix "MathObj.Gaussian.Variance:254: "+{-# LINE 254 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 254 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (withRational $ \x y -> G.scalarProductRoot x y == G.scalarProductRoot (G.fourier x) (G.fourier y))+ DocTest.printPrefix "MathObj.Gaussian.Variance:265: "+{-# LINE 265 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 265 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (withRational $ \x (QC.Positive a) (QC.Positive b) -> G.dilate a (G.dilate b x) == G.dilate (a*b) x)+ DocTest.printPrefix "MathObj.Gaussian.Variance:266: "+{-# LINE 266 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 266 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (withRational $ \x (QC.Positive a) -> G.shrink a x == G.dilate (recip a) x)+ DocTest.printPrefix "MathObj.Gaussian.Variance:273: "+{-# LINE 273 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 273 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (withRational $ \x (QC.Positive a) -> G.dilate a (G.shrink a x) == x)+ DocTest.printPrefix "MathObj.Gaussian.Variance:274: "+{-# LINE 274 "gaussian/MathObj/Gaussian/Variance.hs" #-}+ DocTest.property+{-# LINE 274 "gaussian/MathObj/Gaussian/Variance.hs" #-}+     (withRational $ \x (QC.Positive a) -> G.shrink a (G.dilate a x) == x)
test/Test/MathObj/Matrix.hs view
@@ -1,103 +1,122 @@-{-# LANGUAGE RebindableSyntax #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-module Test.MathObj.Matrix where--import qualified MathObj.Matrix as Matrix--import qualified Algebra.Ring           as Ring--import qualified Algebra.Laws as Laws+-- Do not edit! Automatically created with doctest-extract from src/MathObj/Matrix.hs+{-# LINE 71 "src/MathObj/Matrix.hs" #-} -import qualified Number.NonNegative as NonNeg+module Test.MathObj.Matrix where -import qualified System.Random as Random+import qualified Test.DocTest.Driver as DocTest -import Data.Function.HT (nest, )+{-# LINE 72 "src/MathObj/Matrix.hs" #-}+import     qualified MathObj.Matrix as Matrix+import     qualified Algebra.Ring as Ring+import     qualified Algebra.Laws as Laws+import     Test.NumericPrelude.Utility ((/\))+import     qualified Test.QuickCheck as QC+import     NumericPrelude.Numeric as NP+import     NumericPrelude.Base as P+import     Prelude () -import Test.NumericPrelude.Utility (testUnit, )-import Test.QuickCheck (quickCheck, )-import qualified Test.HUnit as HUnit+import     Control.Monad (replicateM, join)+import     Control.Applicative (liftA2)+import     Data.Function.HT (nest) +genDimension     :: QC.Gen Int+genDimension     = QC.choose (0,20) -import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP+genMatrixFor     :: (QC.Arbitrary a) => Int -> Int -> QC.Gen (Matrix.T a)+genMatrixFor     m n =+       fmap (Matrix.fromList m n) $ replicateM (m*n) QC.arbitrary +genMatrix     :: (QC.Arbitrary a) => QC.Gen (Matrix.T a)+genMatrix     = join $ liftA2 genMatrixFor genDimension genDimension -type Seed = Int-type Dimension = NonNeg.Int+genIntMatrix     :: QC.Gen (Matrix.T Integer)+genIntMatrix     = genMatrix -random :: Dimension -> Dimension -> Seed -> Matrix.T Integer-random mn nn seed =-   fst $-   Matrix.random (NonNeg.toNumber mn) (NonNeg.toNumber nn) $-   Random.mkStdGen seed+genFactorMatrix     :: (QC.Arbitrary a) => Matrix.T a -> QC.Gen (Matrix.T a)+genFactorMatrix     a = genMatrixFor (Matrix.numColumns a) =<< genDimension +genSameMatrix     :: (QC.Arbitrary a) => Matrix.T a -> QC.Gen (Matrix.T a)+genSameMatrix     = uncurry genMatrixFor . Matrix.dimension -tests :: HUnit.Test-tests =-   HUnit.TestLabel "matrix" $-   HUnit.TestList $-   map testUnit $-      ("dimension",-          quickCheck (\m n a ->-             (NonNeg.toNumber m, NonNeg.toNumber n) == Matrix.dimension (random m n a))) :-      ("to and from rows",-          quickCheck (\m n a' ->-             let a = random m n a'-             in  a == Matrix.fromRows (NonNeg.toNumber m) (NonNeg.toNumber n) (Matrix.rows a))) :-      ("to and from columns",-          quickCheck (\m n a' ->-             let a = random m n a'-             in  a == Matrix.fromColumns (NonNeg.toNumber m) (NonNeg.toNumber n) (Matrix.columns a))) :-      ("transpose, rows, columns",-          quickCheck (\m n a' ->-             let a = random m n a'-             in  Matrix.rows a == Matrix.columns (Matrix.transpose a))) :-      ("transpose, columns, rows",-          quickCheck (\m n a' ->-             let a = random m n a'-             in  Matrix.columns a == Matrix.rows (Matrix.transpose a))) :-      ("addition, zero",-          quickCheck (\m n a ->-             Laws.identity (+) (Matrix.zero (NonNeg.toNumber m) (NonNeg.toNumber n)) (random m n a))) :-      ("addition, commutative",-          quickCheck (\m n a b ->-             Laws.commutative (+) (random m n a) (random m n b))) :-      ("addition, associative",-          quickCheck (\m n a b c ->-             Laws.associative (+) (random m n a) (random m n b) (random m n c))) :-      ("addition, transpose",-          quickCheck (\m n a b ->-             Laws.homomorphism Matrix.transpose (+) (+) (random m n a) (random m n b))) :-      ("one, diagonal",-          quickCheck (\n' ->-             let n = NonNeg.toNumber n'-             in Matrix.one n == (Matrix.diagonal $ replicate n Ring.one :: Matrix.T Integer))) :-      ("multiplication, one left",-          quickCheck (\m n a ->-             Laws.leftIdentity  (*) (Matrix.one (NonNeg.toNumber m)) (random m n a))) :-      ("multiplication, one right",-          quickCheck (\m n a ->-             Laws.rightIdentity (*) (Matrix.one (NonNeg.toNumber n)) (random m n a))) :-      ("multiplication, associative",-          quickCheck (\k l m n a b c ->-             Laws.associative (*) (random k l a) (random l m b) (random m n c))) :-      ("multiplication and addition, distributive left",-          quickCheck (\l m n a b c ->-             Laws.leftDistributive (*) (+) (random n l a) (random m n b) (random m n c))) :-      ("multiplication and addition, distributive right",-          quickCheck (\l m n a b c ->-             Laws.rightDistributive (*) (+) (random l m a) (random m n b) (random m n c))) :-      ("multiplication, transpose",-          quickCheck (\l m n a b ->-             Laws.homomorphism Matrix.transpose (*) (flip (*)) (random l m a) (random m n b))) :-      ("multiplication vs. power",-          quickCheck (\m a n0 ->-             let x = random m m a-                 n = mod n0 10-             in  x^n == nest (fromInteger n) (x*) (Matrix.one (NonNeg.toNumber m)))) :-{--      ("division",       quickCheck (\x -> Integral.propInverse (x :: Poly.T Rational))) :--}-      []+test :: DocTest.T ()+test = do+ DocTest.printPrefix "MathObj.Matrix:118: "+{-# LINE 118 "src/MathObj/Matrix.hs" #-}+ DocTest.property+{-# LINE 118 "src/MathObj/Matrix.hs" #-}+     (genIntMatrix /\ \a -> Matrix.rows a == Matrix.columns (Matrix.transpose a))+ DocTest.printPrefix "MathObj.Matrix:119: "+{-# LINE 119 "src/MathObj/Matrix.hs" #-}+ DocTest.property+{-# LINE 119 "src/MathObj/Matrix.hs" #-}+     (genIntMatrix /\ \a -> Matrix.columns a == Matrix.rows (Matrix.transpose a))+ DocTest.printPrefix "MathObj.Matrix:120: "+{-# LINE 120 "src/MathObj/Matrix.hs" #-}+ DocTest.property+{-# LINE 120 "src/MathObj/Matrix.hs" #-}+     (genIntMatrix /\ \a -> genSameMatrix a /\ \b -> Laws.homomorphism Matrix.transpose (+) (+) a b)+ DocTest.printPrefix "MathObj.Matrix:141: "+{-# LINE 141 "src/MathObj/Matrix.hs" #-}+ DocTest.property+{-# LINE 141 "src/MathObj/Matrix.hs" #-}+     (genIntMatrix /\ \a -> a == uncurry Matrix.fromRows (Matrix.dimension a) (Matrix.rows a))+ DocTest.printPrefix "MathObj.Matrix:152: "+{-# LINE 152 "src/MathObj/Matrix.hs" #-}+ DocTest.property+{-# LINE 152 "src/MathObj/Matrix.hs" #-}+     (genIntMatrix /\ \a -> a == uncurry Matrix.fromColumns (Matrix.dimension a) (Matrix.columns a))+ DocTest.printPrefix "MathObj.Matrix:195: "+{-# LINE 195 "src/MathObj/Matrix.hs" #-}+ DocTest.property+{-# LINE 195 "src/MathObj/Matrix.hs" #-}+     (genIntMatrix /\ \a -> genSameMatrix a /\ \b -> Laws.commutative (+) a b)+ DocTest.printPrefix "MathObj.Matrix:196: "+{-# LINE 196 "src/MathObj/Matrix.hs" #-}+ DocTest.property+{-# LINE 196 "src/MathObj/Matrix.hs" #-}+     (genIntMatrix /\ \a -> genSameMatrix a /\ \b -> genSameMatrix b /\ \c -> Laws.associative (+) a b c)+ DocTest.printPrefix "MathObj.Matrix:212: "+{-# LINE 212 "src/MathObj/Matrix.hs" #-}+ DocTest.property+{-# LINE 212 "src/MathObj/Matrix.hs" #-}+     (genIntMatrix /\ \a -> Laws.identity (+) (uncurry Matrix.zero $ Matrix.dimension a) a)+ DocTest.printPrefix "MathObj.Matrix:228: "+{-# LINE 228 "src/MathObj/Matrix.hs" #-}+ DocTest.property+{-# LINE 228 "src/MathObj/Matrix.hs" #-}+     (genDimension /\ \n -> Matrix.one n == Matrix.diagonal (replicate n Ring.one :: [Integer]))+ DocTest.printPrefix "MathObj.Matrix:242: "+{-# LINE 242 "src/MathObj/Matrix.hs" #-}+ DocTest.property+{-# LINE 242 "src/MathObj/Matrix.hs" #-}+     (genIntMatrix /\ \a -> Laws.leftIdentity  (*) (Matrix.one (Matrix.numRows a)) a)+ DocTest.printPrefix "MathObj.Matrix:243: "+{-# LINE 243 "src/MathObj/Matrix.hs" #-}+ DocTest.property+{-# LINE 243 "src/MathObj/Matrix.hs" #-}+     (genIntMatrix /\ \a -> Laws.rightIdentity (*) (Matrix.one (Matrix.numColumns a)) a)+ DocTest.printPrefix "MathObj.Matrix:244: "+{-# LINE 244 "src/MathObj/Matrix.hs" #-}+ DocTest.property+{-# LINE 244 "src/MathObj/Matrix.hs" #-}+     (genIntMatrix /\ \a -> genFactorMatrix a /\ \b -> Laws.homomorphism Matrix.transpose (*) (flip (*)) a b)+ DocTest.printPrefix "MathObj.Matrix:245: "+{-# LINE 245 "src/MathObj/Matrix.hs" #-}+ DocTest.property+{-# LINE 245 "src/MathObj/Matrix.hs" #-}+     (genIntMatrix /\ \a -> genFactorMatrix a /\ \b -> genFactorMatrix b /\ \c -> Laws.associative (*) a b c)+ DocTest.printPrefix "MathObj.Matrix:246: "+{-# LINE 246 "src/MathObj/Matrix.hs" #-}+ DocTest.property+{-# LINE 246 "src/MathObj/Matrix.hs" #-}+     (genIntMatrix /\ \b -> genSameMatrix b /\ \c -> genFactorMatrix b /\ \a -> Laws.leftDistributive (*) (+) a b c)+ DocTest.printPrefix "MathObj.Matrix:247: "+{-# LINE 247 "src/MathObj/Matrix.hs" #-}+ DocTest.property+{-# LINE 247 "src/MathObj/Matrix.hs" #-}+     (genIntMatrix /\ \a -> genFactorMatrix a /\ \b -> genSameMatrix b /\ \c -> Laws.rightDistributive (*) (+) a b c)+ DocTest.printPrefix "MathObj.Matrix:248: "+{-# LINE 248 "src/MathObj/Matrix.hs" #-}+ DocTest.property+{-# LINE 248 "src/MathObj/Matrix.hs" #-}+     (QC.choose (0,10) /\ \k -> genDimension /\ \n -> genMatrixFor n n /\ \a -> a^k == nest (fromInteger k) ((a::Matrix.T Integer)*) (Matrix.one n))
test/Test/MathObj/PartialFraction.hs view
@@ -1,205 +1,137 @@-{-# LANGUAGE RebindableSyntax #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-module Test.MathObj.PartialFraction where--import qualified MathObj.PartialFraction      as PartialFraction-import qualified MathObj.Polynomial           as Poly-import qualified Number.Ratio                 as Ratio--import qualified Algebra.PrincipalIdealDomain as PID--- import qualified Algebra.Ring                 as Ring-import qualified Algebra.Indexable            as Indexable-import qualified Algebra.Vector               as Vector--- import Algebra.Vector((*>))--import qualified Algebra.Laws as Laws-import qualified Test.QuickCheck as QC--import Control.Monad.HT as M-import Test.NumericPrelude.Utility (testUnit)-import Test.QuickCheck (quickCheck)-import qualified Test.HUnit as HUnit---import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP---{- * Properties for generic types -}--fractionConv :: (PID.C a, Indexable.C a) => [a] -> a -> Bool-fractionConv xs y =-   PartialFraction.toFraction (PartialFraction.fromFactoredFraction xs y) ==-   y % product xs--fractionConvAlt :: (PID.C a, Indexable.C a) => [a] -> a -> Bool-fractionConvAlt xs y =-   PartialFraction.fromFactoredFraction xs y ==-   PartialFraction.fromFactoredFractionAlt xs y--scaleInt :: (PID.C a, Indexable.C a) => a -> PartialFraction.T a -> Bool-scaleInt k a =-   PartialFraction.toFraction (PartialFraction.scaleInt k a) ==-   Ratio.scale k (PartialFraction.toFraction a)--add :: (PID.C a, Indexable.C a) => PartialFraction.T a -> PartialFraction.T a -> Bool-add = Laws.homomorphism PartialFraction.toFraction (+) (+)--sub :: (PID.C a, Indexable.C a) => PartialFraction.T a -> PartialFraction.T a -> Bool-sub = Laws.homomorphism PartialFraction.toFraction (-) (-)--mul :: (PID.C a, Indexable.C a) => PartialFraction.T a -> PartialFraction.T a -> Bool-mul = Laws.homomorphism PartialFraction.toFraction (*) (*)----{- * Properties for Integers -}--{- |-Arbitrary instance of that type generates irreducible elements for tests.-Choosing from a list of examples is a simple yet effective design.-If we would construct irreducible elements by a clever algorithm-we might obtain multiple primes only rarely.--}-newtype SmallPrime = SmallPrime {intFromSmallPrime :: Integer}--type IntFraction = ([SmallPrime],Integer)--instance QC.Arbitrary SmallPrime where-   arbitrary =-      let primes = [2,3,5,7,11,13]-      in  fmap SmallPrime $ QC.elements (primes ++ map negate primes)--instance Show SmallPrime where-   show = show . intFromSmallPrime---fractionConvInt :: [SmallPrime] -> Integer -> Bool-fractionConvInt =-   fractionConv . map intFromSmallPrime--fractionConvAltInt :: [SmallPrime] -> Integer -> Bool-fractionConvAltInt =-   fractionConvAlt . map intFromSmallPrime--fromSmallPrimes :: IntFraction -> PartialFraction.T Integer-fromSmallPrimes (xs,y) =-   PartialFraction.fromFactoredFraction (map intFromSmallPrime xs) y---scaleIntInt :: Integer -> IntFraction -> Bool-scaleIntInt k a =-   scaleInt k (fromSmallPrimes a)--addInt :: IntFraction -> IntFraction -> Bool-addInt q0 q1 =-   add-      (fromSmallPrimes q0)-      (fromSmallPrimes q1)--subInt :: IntFraction -> IntFraction -> Bool-subInt q0 q1 =-   sub-      (fromSmallPrimes q0)-      (fromSmallPrimes q1)--mulInt :: IntFraction -> IntFraction -> Bool-mulInt q0 q1 =-   mul-      (fromSmallPrimes q0)-      (fromSmallPrimes q1)---intTests :: HUnit.Test-intTests =-   HUnit.TestLabel "integer" $-   HUnit.TestList $-   map testUnit $-      ("conversion between partial and ordinary fraction", quickCheck fractionConvInt) :-      ("two conversion routines from factored fractions", quickCheck fractionConvAltInt) :-      ("integer scaling", quickCheck scaleIntInt) :-      ("addition", quickCheck addInt) :-      ("subtraction", quickCheck subInt) :-      ("multiplication", quickCheck mulInt) :-      []+-- Do not edit! Automatically created with doctest-extract from src/MathObj/PartialFraction.hs+{-# LINE 45 "src/MathObj/PartialFraction.hs" #-} +module Test.MathObj.PartialFraction where -{- * Properties for Polynomials -}+import qualified Test.DocTest.Driver as DocTest -newtype IrredPoly = IrredPoly {polyFromIrredPoly :: Poly.T Rational}+{-# LINE 46 "src/MathObj/PartialFraction.hs" #-}+import     qualified MathObj.PartialFraction as PartialFraction+import     qualified MathObj.Polynomial.Core as PolyCore+import     qualified MathObj.Polynomial as Poly+import     qualified Algebra.PrincipalIdealDomain as PID+import     qualified Algebra.Indexable as Indexable+import     qualified Algebra.Laws as Laws+import     qualified Number.Ratio as Ratio+import     Test.NumericPrelude.Utility ((/\))+import     qualified Test.QuickCheck as QC+import     NumericPrelude.Numeric as NP+import     NumericPrelude.Base as P+import     Prelude () -type RatPolynomial = Poly.T Rational-type PolyFraction = ([IrredPoly],RatPolynomial)+import     Control.Applicative (liftA2) -instance QC.Arbitrary IrredPoly where-   arbitrary =-      do poly <- QC.elements (map Poly.fromCoeffs [[2,3],[2,0,1],[3,0,1],[1,-3,0,1]])-         unit <- M.until (not. isZero) QC.arbitrary-         return (IrredPoly (unit Vector.*> poly))+{-     |+Generator     of irreducible elements for tests.+Choosing     from a list of examples is a simple yet effective design.+If     we would construct irreducible elements by a clever algorithm+we     might obtain multiple primes only rarely.+-}     --+genSmallPrime     :: QC.Gen Integer+genSmallPrime     =+       let primes = [2,3,5,7,11,13]+       in  QC.elements (primes ++ map negate primes) -instance Show IrredPoly where-   show = show . polyFromIrredPoly+genPartialFractionInt     :: QC.Gen (PartialFraction.T Integer)+genPartialFractionInt     =+       liftA2 PartialFraction.fromFactoredFraction+          (QC.listOf genSmallPrime) QC.arbitrary  -fractionConvPoly :: [IrredPoly] -> RatPolynomial -> Bool-fractionConvPoly =-   fractionConv . map polyFromIrredPoly--fractionConvAltPoly :: [IrredPoly] -> RatPolynomial -> Bool-fractionConvAltPoly =-   fractionConvAlt . map polyFromIrredPoly+genIrreduciblePolynomial     :: QC.Gen (Poly.T Rational)+genIrreduciblePolynomial     = do+       QC.NonZero unit <- QC.arbitrary+       fmap (Poly.fromCoeffs . map (unit*)) $+          QC.elements [[2,3],[2,0,1],[3,0,1],[1,-3,0,1]] -fromIrredPolys :: PolyFraction -> PartialFraction.T RatPolynomial-fromIrredPolys (xs,y) =-   PartialFraction.fromFactoredFraction (map polyFromIrredPoly xs) y+genPartialFractionPoly     :: QC.Gen (PartialFraction.T (Poly.T Rational))+genPartialFractionPoly     =+       liftA2 PartialFraction.fromFactoredFraction+          (fmap (take 3) $ QC.listOf genIrreduciblePolynomial)+          (fmap (Poly.fromCoeffs . PolyCore.normalize . take 5) QC.arbitrary)  -scaleIntPoly :: RatPolynomial -> PolyFraction -> Bool-scaleIntPoly k a =-   scaleInt k (fromIrredPolys a)--addPoly :: PolyFraction -> PolyFraction -> Bool-addPoly q0 q1 =-   add-      (fromIrredPolys q0)-      (fromIrredPolys q1)--subPoly :: PolyFraction -> PolyFraction -> Bool-subPoly q0 q1 =-   sub-      (fromIrredPolys q0)-      (fromIrredPolys q1)--mulPoly :: PolyFraction -> PolyFraction -> Bool-mulPoly q0 q1 =-   mul-      (fromIrredPolys q0)-      (fromIrredPolys q1)-+fractionConv     :: (PID.C a, Indexable.C a) => [a] -> a -> Bool+fractionConv     xs y =+       PartialFraction.toFraction (PartialFraction.fromFactoredFraction xs y) ==+       y % product xs +fractionConvAlt     :: (PID.C a, Indexable.C a) => [a] -> a -> Bool+fractionConvAlt     xs y =+       PartialFraction.fromFactoredFraction xs y ==+       PartialFraction.fromFactoredFractionAlt xs y -polyTests :: HUnit.Test-polyTests =-   HUnit.TestLabel "polynomial" $-   HUnit.TestList $-   map testUnit $-{- this test fails, because addition may result in leading zero coefficients,-      that is, polynomial addition does not contain a normalization-      if it would contain one, we would exclude computable reals -}--- wrong     ("conversion between partial and ordinary fraction", quickCheck fractionConvPoly) :--- wrong     ("two conversion routines from factored fractions", quickCheck fractionConvAltPoly) :--- too slow      ("integer scaling", quickCheck scaleIntPoly) :--- too slow      ("addition", quickCheck addPoly) :--- too slow      ("subtraction", quickCheck subPoly) :--- too slow      ("multiplication", quickCheck mulPoly) :-      []+scaleInt     :: (PID.C a, Indexable.C a) => a -> PartialFraction.T a -> Bool+scaleInt     k a =+       PartialFraction.toFraction (PartialFraction.scaleInt k a) ==+       Ratio.scale k (PartialFraction.toFraction a) +add,     sub, mul ::+       (PID.C a, Indexable.C a) =>+       PartialFraction.T a -> PartialFraction.T a -> Bool+add     = Laws.homomorphism PartialFraction.toFraction (+) (+)+sub     = Laws.homomorphism PartialFraction.toFraction (-) (-)+mul     = Laws.homomorphism PartialFraction.toFraction (*) (*) -tests :: HUnit.Test-tests =-   HUnit.TestLabel "partial fraction" $-   HUnit.TestList $-      intTests :---      polyTests :-      []+test :: DocTest.T ()+test = do+ DocTest.printPrefix "MathObj.PartialFraction:195: "+{-# LINE 195 "src/MathObj/PartialFraction.hs" #-}+ DocTest.property+{-# LINE 195 "src/MathObj/PartialFraction.hs" #-}+     (QC.listOf genSmallPrime /\ fractionConv)+ DocTest.printPrefix "MathObj.PartialFraction:196: "+{-# LINE 196 "src/MathObj/PartialFraction.hs" #-}+ DocTest.property+{-# LINE 196 "src/MathObj/PartialFraction.hs" #-}+     (fmap (take 3) (QC.listOf genIrreduciblePolynomial) /\ fractionConv)+ DocTest.printPrefix "MathObj.PartialFraction:220: "+{-# LINE 220 "src/MathObj/PartialFraction.hs" #-}+ DocTest.property+{-# LINE 220 "src/MathObj/PartialFraction.hs" #-}+     (QC.listOf genSmallPrime /\ fractionConvAlt)+ DocTest.printPrefix "MathObj.PartialFraction:221: "+{-# LINE 221 "src/MathObj/PartialFraction.hs" #-}+ DocTest.property+{-# LINE 221 "src/MathObj/PartialFraction.hs" #-}+     (fmap (take 3) (QC.listOf genIrreduciblePolynomial) /\ fractionConvAlt)+ DocTest.printPrefix "MathObj.PartialFraction:297: "+{-# LINE 297 "src/MathObj/PartialFraction.hs" #-}+ DocTest.property+{-# LINE 297 "src/MathObj/PartialFraction.hs" #-}+     (genPartialFractionInt /\ \x -> genPartialFractionInt /\ \y -> add x y)+ DocTest.printPrefix "MathObj.PartialFraction:298: "+{-# LINE 298 "src/MathObj/PartialFraction.hs" #-}+ DocTest.property+{-# LINE 298 "src/MathObj/PartialFraction.hs" #-}+     (genPartialFractionInt /\ \x -> genPartialFractionInt /\ \y -> sub x y)+ DocTest.printPrefix "MathObj.PartialFraction:300: "+{-# LINE 300 "src/MathObj/PartialFraction.hs" #-}+ DocTest.property+{-# LINE 300 "src/MathObj/PartialFraction.hs" #-}+     (genPartialFractionPoly /\ \x -> genPartialFractionPoly /\ \y -> add x y)+ DocTest.printPrefix "MathObj.PartialFraction:301: "+{-# LINE 301 "src/MathObj/PartialFraction.hs" #-}+ DocTest.property+{-# LINE 301 "src/MathObj/PartialFraction.hs" #-}+     (genPartialFractionPoly /\ \x -> genPartialFractionPoly /\ \y -> sub x y)+ DocTest.printPrefix "MathObj.PartialFraction:429: "+{-# LINE 429 "src/MathObj/PartialFraction.hs" #-}+ DocTest.property+{-# LINE 429 "src/MathObj/PartialFraction.hs" #-}+     (genPartialFractionInt /\ \x k -> scaleInt k x)+ DocTest.printPrefix "MathObj.PartialFraction:430: "+{-# LINE 430 "src/MathObj/PartialFraction.hs" #-}+ DocTest.property+{-# LINE 430 "src/MathObj/PartialFraction.hs" #-}+     (genPartialFractionPoly /\ \x k -> scaleInt k x)+ DocTest.printPrefix "MathObj.PartialFraction:449: "+{-# LINE 449 "src/MathObj/PartialFraction.hs" #-}+ DocTest.property+{-# LINE 449 "src/MathObj/PartialFraction.hs" #-}+     (genPartialFractionInt /\ \x -> genPartialFractionInt /\ \y -> mul x y)+ DocTest.printPrefix "MathObj.PartialFraction:450: "+{-# LINE 450 "src/MathObj/PartialFraction.hs" #-}+ DocTest.property+{-# LINE 450 "src/MathObj/PartialFraction.hs" #-}+     (genPartialFractionPoly /\ \x -> genPartialFractionPoly /\ \y -> mul x y)
test/Test/MathObj/Polynomial.hs view
@@ -1,56 +1,63 @@-{-# LANGUAGE RebindableSyntax #-}-module Test.MathObj.Polynomial where--import qualified MathObj.Polynomial      as Poly-import qualified MathObj.Polynomial.Core as PolyCore--import qualified Algebra.IntegralDomain as Integral-import qualified Algebra.Ring           as Ring--import qualified Algebra.ZeroTestable   as ZeroTestable-import qualified Algebra.Laws as Laws--import qualified Data.List as List--import Test.NumericPrelude.Utility (testUnit)-import Test.QuickCheck (Property, quickCheck, (==>), Testable, )-import qualified Test.HUnit as HUnit---import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP---tensorProductTranspose :: (Ring.C a, Eq a) => [a] -> [a] -> Property-tensorProductTranspose xs ys =-   not (null xs) && not (null ys) ==>-      PolyCore.tensorProduct xs ys == List.transpose (PolyCore.tensorProduct ys xs)-+-- Do not edit! Automatically created with doctest-extract from src/MathObj/Polynomial.hs+{-# LINE 84 "src/MathObj/Polynomial.hs" #-} -mul :: (Ring.C a, Eq a, ZeroTestable.C a) => [a] -> [a] -> Bool-mul xs ys  =  PolyCore.equal (PolyCore.mul xs ys) (PolyCore.mulShear xs ys)+module Test.MathObj.Polynomial where +import qualified Test.DocTest.Driver as DocTest -test :: Testable a => (Poly.T Integer -> a) -> IO ()-test = quickCheck+{-# LINE 85 "src/MathObj/Polynomial.hs" #-}+import     qualified MathObj.Polynomial as Poly+import     qualified Algebra.IntegralDomain as Integral+import     qualified Algebra.Laws as Laws+import     NumericPrelude.Numeric+import     NumericPrelude.Base+import     Prelude () -testRat :: Testable a => (Poly.T Rational -> a) -> IO ()-testRat = quickCheck+intPoly     :: Poly.T Integer -> Poly.T Integer+intPoly     = id +ratioPoly     :: Poly.T Rational -> Poly.T Rational+ratioPoly     = id -tests :: HUnit.Test-tests =-   HUnit.TestLabel "polynomial" $-   HUnit.TestList $-   map testUnit $-      ("tensor product", quickCheck (tensorProductTranspose :: [Integer] -> [Integer] -> Property)) :-      ("mul speed",      quickCheck (mul                    :: [Integer] -> [Integer] -> Bool)) :-      ("addition, zero",         test (Laws.identity (+) zero)) :-      ("addition, commutative",  test (Laws.commutative (+))) :-      ("addition, associative",  test (Laws.associative (+))) :-      ("multiplication, one",          test (Laws.identity (*) one)) :-      ("multiplication, commutative",  test (Laws.commutative (*))) :-      ("multiplication, associative",  test (Laws.associative (*))) :-      ("multiplication and addition, distributive",   test (Laws.leftDistributive (*) (+))) :-      ("division",       testRat (Integral.propInverse)) :-      []+test :: DocTest.T ()+test = do+ DocTest.printPrefix "MathObj.Polynomial:100: "+{-# LINE 100 "src/MathObj/Polynomial.hs" #-}+ DocTest.property+{-# LINE 100 "src/MathObj/Polynomial.hs" #-}+     (Laws.identity (+) zero . intPoly)+ DocTest.printPrefix "MathObj.Polynomial:101: "+{-# LINE 101 "src/MathObj/Polynomial.hs" #-}+ DocTest.property+{-# LINE 101 "src/MathObj/Polynomial.hs" #-}+     (Laws.commutative (+) . intPoly)+ DocTest.printPrefix "MathObj.Polynomial:102: "+{-# LINE 102 "src/MathObj/Polynomial.hs" #-}+ DocTest.property+{-# LINE 102 "src/MathObj/Polynomial.hs" #-}+     (Laws.associative (+) . intPoly)+ DocTest.printPrefix "MathObj.Polynomial:103: "+{-# LINE 103 "src/MathObj/Polynomial.hs" #-}+ DocTest.property+{-# LINE 103 "src/MathObj/Polynomial.hs" #-}+     (Laws.identity (*) one . intPoly)+ DocTest.printPrefix "MathObj.Polynomial:104: "+{-# LINE 104 "src/MathObj/Polynomial.hs" #-}+ DocTest.property+{-# LINE 104 "src/MathObj/Polynomial.hs" #-}+     (Laws.commutative (*) . intPoly)+ DocTest.printPrefix "MathObj.Polynomial:105: "+{-# LINE 105 "src/MathObj/Polynomial.hs" #-}+ DocTest.property+{-# LINE 105 "src/MathObj/Polynomial.hs" #-}+     (Laws.associative (*) . intPoly)+ DocTest.printPrefix "MathObj.Polynomial:106: "+{-# LINE 106 "src/MathObj/Polynomial.hs" #-}+ DocTest.property+{-# LINE 106 "src/MathObj/Polynomial.hs" #-}+     (Laws.leftDistributive (*) (+) . intPoly)+ DocTest.printPrefix "MathObj.Polynomial:107: "+{-# LINE 107 "src/MathObj/Polynomial.hs" #-}+ DocTest.property+{-# LINE 107 "src/MathObj/Polynomial.hs" #-}+     (Integral.propInverse . ratioPoly)
+ test/Test/MathObj/Polynomial/Core.hs view
@@ -0,0 +1,51 @@+-- Do not edit! Automatically created with doctest-extract from src/MathObj/Polynomial/Core.hs+{-# LINE 47 "src/MathObj/Polynomial/Core.hs" #-}++module Test.MathObj.Polynomial.Core where++import qualified Test.DocTest.Driver as DocTest++{-# LINE 48 "src/MathObj/Polynomial/Core.hs" #-}+import     qualified MathObj.Polynomial.Core as PolyCore+import     qualified MathObj.Polynomial as Poly+import     qualified Data.List as List+import     qualified Test.QuickCheck as QC+import     Test.QuickCheck ((==>))+import     Data.Tuple.HT (mapPair, mapSnd)+import     NumericPrelude.Numeric+import     NumericPrelude.Base+import     Prelude ()++intPoly     :: [Integer] -> [Integer]+intPoly     = id++ratioPoly     :: [Rational] -> [Rational]+ratioPoly     = id++test :: DocTest.T ()+test = do+ DocTest.printPrefix "MathObj.Polynomial.Core:136: "+{-# LINE 136 "src/MathObj/Polynomial/Core.hs" #-}+ DocTest.property+{-# LINE 136 "src/MathObj/Polynomial/Core.hs" #-}+     (\(QC.NonEmpty xs) (QC.NonEmpty ys) -> PolyCore.tensorProduct xs ys == List.transpose (PolyCore.tensorProduct ys (intPoly xs)))+ DocTest.printPrefix "MathObj.Polynomial.Core:161: "+{-# LINE 161 "src/MathObj/Polynomial/Core.hs" #-}+ DocTest.property+{-# LINE 161 "src/MathObj/Polynomial/Core.hs" #-}+     (\xs ys  ->  PolyCore.equal (intPoly $ PolyCore.mul xs ys) (PolyCore.mulShear xs ys))+ DocTest.printPrefix "MathObj.Polynomial.Core:173: "+{-# LINE 173 "src/MathObj/Polynomial/Core.hs" #-}+ DocTest.property+{-# LINE 173 "src/MathObj/Polynomial/Core.hs" #-}+     (\x y -> case (PolyCore.normalize x, PolyCore.normalize y) of (nx, ny) -> not (null (ratioPoly ny)) ==> mapSnd PolyCore.normalize (PolyCore.divMod nx ny) == mapPair (PolyCore.normalize, PolyCore.normalize) (PolyCore.divMod x y))+ DocTest.printPrefix "MathObj.Polynomial.Core:174: "+{-# LINE 174 "src/MathObj/Polynomial/Core.hs" #-}+ DocTest.property+{-# LINE 174 "src/MathObj/Polynomial/Core.hs" #-}+     (\x y -> not (isZero (ratioPoly y)) ==> let z = fst $ PolyCore.divMod (Poly.coeffs x) y in  PolyCore.normalize z == z)+ DocTest.printPrefix "MathObj.Polynomial.Core:175: "+{-# LINE 175 "src/MathObj/Polynomial/Core.hs" #-}+ DocTest.property+{-# LINE 175 "src/MathObj/Polynomial/Core.hs" #-}+     (\x y -> case PolyCore.normalize $ ratioPoly y of ny -> not (null ny) ==> List.length (snd $ PolyCore.divMod x y) < List.length ny)
test/Test/MathObj/PowerSeries.hs view
@@ -1,103 +1,23 @@-{-# LANGUAGE RebindableSyntax #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-module Test.MathObj.PowerSeries where--import qualified MathObj.PowerSeries.Core    as PS-import qualified MathObj.PowerSeries.Example as PSE--import Test.NumericPrelude.Utility (equalInfLists {- , testUnit -} )--- import Test.QuickCheck (Property, quickCheck, (==>))-import qualified Test.HUnit as HUnit---import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP---identitiesExplODE, identitiesSeriesFunction, identitiesInverses ::-   [(String, Int, [Rational],[Rational])]--identitiesExplODE =-   ("exp",   500, PSE.expExpl,   PSE.expODE) :-   ("sin",   500, PSE.sinExpl,   PSE.sinODE) :-   ("cos",   500, PSE.cosExpl,   PSE.cosODE) :-   ("tan",    50, PSE.tanExpl,   PSE.tanODE) :-   ("tan",    50, PSE.tanExpl,   PSE.tanExplSieve) :-   ("tan",    50, PSE.tanODE,    PSE.tanODESieve) :-   ("log",   500, PSE.logExpl,   PSE.logODE) :-   ("asin",   50, PSE.asinODE,   snd (PS.inv PSE.sinODE)) :-   ("atan",  500, PSE.atanExpl,  PSE.atanODE) :-   ("sinh",  500, PSE.sinhExpl,  PSE.sinhODE) :-   ("cosh",  500, PSE.coshExpl,  PSE.coshODE) :-   ("atanh", 500, PSE.atanhExpl, PSE.atanhODE) :-   ("sqrt",  100, PSE.sqrtExpl,  PSE.sqrtODE) :-   []--identitiesSeriesFunction =-   ("exp",   500, PSE.expExpl,  PS.exp (\0 -> 1) [0,1]) :-   ("sin",   500, PSE.sinExpl,  PS.sin (\0 -> (0,1)) [0,1]) :-   ("cos",   500, PSE.cosExpl,  PS.cos (\0 -> (0,1)) [0,1]) :-   ("tan",    50, PSE.tanExpl,  PS.tan (\0 -> (0,1)) [0,1]) :-   ("sqrt",   50, PSE.sqrtExpl, PS.sqrt (\1 -> 1) [1,1]) :-   ("power", 500, PSE.powExpl (-1/3), PS.pow (\1 -> 1) (-1/3) [1,1]) :-   ("power",  50, PSE.powExpl (-1/3), PS.exp (\0 -> 1) (PS.scale (-1/3) PSE.log)) :-   ("log",   500, PSE.logExpl, PS.log (\1 -> 0) [1,1]) :-   ("asin",   50, PSE.asin, PS.asin (\1 -> 1) (\0 -> 0) [0,1]) :- --  ("acos",  50, PSE.acos, PS.acos (\1 -> 1) (\0 -> pi/2) [0,1]) :-   ("atan",  500, PSE.atan, PS.atan (\0 -> 0) [0,1]) :-   []--identitiesInverses =-   ("exp",   100, 1:1:repeat 0, PS.exp  (\0 -> 1) PSE.log) :-   ("log",   100, 0:1:repeat 0, PS.log  (\1 -> 0) PSE.exp) :-   ("tan",    50, 0:1:repeat 0, PS.tan  (\0 -> (0,1)) PSE.atan) :-   ("atan",   50, 0:1:repeat 0, PS.atan (\0 -> 0) PSE.tan) :-   ("sin",    50, 0:1:repeat 0, PS.sin  (\0 -> (0,1)) PSE.asin) :-   ("asin",  100, 0:1:repeat 0, PS.asin (\1 -> 1) (\0 -> 0) PSE.sin) :-   ("sqrt",  500, 1:1:repeat 0, PS.sqrt (\1 -> 1) (PS.mul [1,1] [1,1])) :-   []--testSeriesIdentity :: (String, Int, [Rational], [Rational]) -> HUnit.Test-testSeriesIdentity (label, len, x, y) =-   HUnit.test (HUnit.assertBool label (equalInfLists len [x,y]))--testSeriesIdentities ::-   String -> [(String, Int, [Rational], [Rational])] -> HUnit.Test-testSeriesIdentities label ids =-   HUnit.TestLabel label $-     HUnit.TestList $ map testSeriesIdentity ids--checkSeriesIdentities ::-   [(String, Int, [Rational], [Rational])] -> [(String,Bool)]-checkSeriesIdentities =-   map (\(label, len, x, y) -> (label, equalInfLists len [x,y]))---+-- Do not edit! Automatically created with doctest-extract from src/MathObj/PowerSeries.hs+{-# LINE 30 "src/MathObj/PowerSeries.hs" #-} -powerMult :: Rational -> Rational -> Bool-powerMult exp0 exp1 =-   PS.mul (PSE.pow exp0) (PSE.pow exp1)  ==  PSE.pow (exp0+exp1)+module Test.MathObj.PowerSeries where -powerExplODE :: Rational -> Bool-powerExplODE expon =-   PSE.powODE expon == PSE.powExpl expon+import qualified Test.DocTest.Driver as DocTest +{-# LINE 31 "src/MathObj/PowerSeries.hs" #-}+import     qualified MathObj.PowerSeries.Core as PS+import     qualified MathObj.PowerSeries as PST+import     qualified Test.QuickCheck as QC+import     Test.NumericPrelude.Utility (equalTrunc, (/\))+import     NumericPrelude.Numeric as NP+import     NumericPrelude.Base as P+import     Prelude () -tests :: HUnit.Test-tests =-   HUnit.TestLabel "power series" $-   HUnit.TestList [-      testSeriesIdentities "explicit vs. ODE solution" identitiesExplODE,-      testSeriesIdentities "transcendent functions of series" identitiesSeriesFunction,-      testSeriesIdentities "inverses of some series" identitiesInverses-{--      HUnit.TestLabel "laws" $-      HUnit.TestList $-         map testUnit $-            ("products of powers",     quickCheck (powerMult)) :-            ("power explicit vs. ODE", quickCheck (powerExplODE)) :-            []--}-    ]+test :: DocTest.T ()+test = do+ DocTest.printPrefix "MathObj.PowerSeries:141: "+{-# LINE 141 "src/MathObj/PowerSeries.hs" #-}+ DocTest.property+{-# LINE 141 "src/MathObj/PowerSeries.hs" #-}+     (QC.choose (1,10) /\ \expon (QC.Positive x) xs -> let xt = x:xs in  equalTrunc 15 (PS.pow (const x) (1 % expon) (PST.coeffs (PST.fromCoeffs xt ^ expon)) ++ repeat zero) (xt ++ repeat zero))
+ test/Test/MathObj/PowerSeries/Core.hs view
@@ -0,0 +1,178 @@+-- Do not edit! Automatically created with doctest-extract from src/MathObj/PowerSeries/Core.hs+{-# LINE 23 "src/MathObj/PowerSeries/Core.hs" #-}++module Test.MathObj.PowerSeries.Core where++import qualified Test.DocTest.Driver as DocTest++{-# LINE 24 "src/MathObj/PowerSeries/Core.hs" #-}+import     qualified MathObj.PowerSeries.Core as PS+import     qualified MathObj.PowerSeries.Example as PSE+import     Test.NumericPrelude.Utility (equalTrunc, (/\))+import     qualified Test.QuickCheck as QC+import     NumericPrelude.Numeric as NP+import     NumericPrelude.Base as P+import     Prelude ()+import     Control.Applicative (liftA3)++checkHoles     ::+       Int -> ([Rational] -> [Rational]) ->+       Rational -> [Rational] -> QC.Property+checkHoles     trunc f x xs =+       QC.choose (1,10) /\ \expon ->+       equalTrunc trunc+          (f (PS.insertHoles expon (x:xs)) ++ repeat zero)+          (PS.insertHoles expon (f (x:xs)) ++ repeat zero)++genInvertible     :: QC.Gen [Rational]+genInvertible     =+       liftA3 (\x0 x1 xs -> x0:x1:xs)+          QC.arbitrary (fmap QC.getNonZero QC.arbitrary) QC.arbitrary++test :: DocTest.T ()+test = do+ DocTest.printPrefix "MathObj.PowerSeries.Core:108: "+{-# LINE 108 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 108 "src/MathObj/PowerSeries/Core.hs" #-}+     (QC.choose (1,10) /\ \m -> QC.choose (1,10) /\ \n xs -> equalTrunc 100 (PS.insertHoles m $ PS.insertHoles n xs) (PS.insertHoles (m*n) xs))+ DocTest.printPrefix "MathObj.PowerSeries.Core:190: "+{-# LINE 190 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 190 "src/MathObj/PowerSeries/Core.hs" #-}+     (equalTrunc 50 PSE.sqrtExpl (PS.sqrt (\1 -> 1) [1,1]))+ DocTest.printPrefix "MathObj.PowerSeries.Core:191: "+{-# LINE 191 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 191 "src/MathObj/PowerSeries/Core.hs" #-}+     (equalTrunc 500 (1:1:repeat 0) (PS.sqrt (\1 -> 1) (PS.mul [1,1] [1,1])))+ DocTest.printPrefix "MathObj.PowerSeries.Core:192: "+{-# LINE 192 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 192 "src/MathObj/PowerSeries/Core.hs" #-}+     (checkHoles 50 (PS.sqrt (\1 -> 1)) 1)+ DocTest.printPrefix "MathObj.PowerSeries.Core:217: "+{-# LINE 217 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 217 "src/MathObj/PowerSeries/Core.hs" #-}+     (equalTrunc 100 (PSE.powExpl (-1/3)) (PS.pow (\1 -> 1) (-1/3) [1,1]))+ DocTest.printPrefix "MathObj.PowerSeries.Core:218: "+{-# LINE 218 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 218 "src/MathObj/PowerSeries/Core.hs" #-}+     (equalTrunc 50 (PSE.powExpl (-1/3)) (PS.exp (\0 -> 1) (PS.scale (-1/3) PSE.log)))+ DocTest.printPrefix "MathObj.PowerSeries.Core:219: "+{-# LINE 219 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 219 "src/MathObj/PowerSeries/Core.hs" #-}+     (checkHoles 30 (PS.pow (\1 -> 1) (1/3)) 1)+ DocTest.printPrefix "MathObj.PowerSeries.Core:220: "+{-# LINE 220 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 220 "src/MathObj/PowerSeries/Core.hs" #-}+     (checkHoles 30 (PS.pow (\1 -> 1) (2/5)) 1)+ DocTest.printPrefix "MathObj.PowerSeries.Core:237: "+{-# LINE 237 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 237 "src/MathObj/PowerSeries/Core.hs" #-}+     (equalTrunc 500 PSE.expExpl (PS.exp (\0 -> 1) [0,1]))+ DocTest.printPrefix "MathObj.PowerSeries.Core:238: "+{-# LINE 238 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 238 "src/MathObj/PowerSeries/Core.hs" #-}+     (equalTrunc 100 (1:1:repeat 0) (PS.exp (\0 -> 1) PSE.log))+ DocTest.printPrefix "MathObj.PowerSeries.Core:239: "+{-# LINE 239 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 239 "src/MathObj/PowerSeries/Core.hs" #-}+     (checkHoles 30 (PS.exp (\0 -> 1)) 0)+ DocTest.printPrefix "MathObj.PowerSeries.Core:259: "+{-# LINE 259 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 259 "src/MathObj/PowerSeries/Core.hs" #-}+     (equalTrunc 500 PSE.sinExpl (PS.sin (\0 -> (0,1)) [0,1]))+ DocTest.printPrefix "MathObj.PowerSeries.Core:260: "+{-# LINE 260 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 260 "src/MathObj/PowerSeries/Core.hs" #-}+     (equalTrunc 50 (0:1:repeat 0) (PS.sin (\0 -> (0,1)) PSE.asin))+ DocTest.printPrefix "MathObj.PowerSeries.Core:261: "+{-# LINE 261 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 261 "src/MathObj/PowerSeries/Core.hs" #-}+     (checkHoles 20 (PS.sin (\0 -> (0,1))) 0)+ DocTest.printPrefix "MathObj.PowerSeries.Core:266: "+{-# LINE 266 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 266 "src/MathObj/PowerSeries/Core.hs" #-}+     (equalTrunc 500 PSE.cosExpl (PS.cos (\0 -> (0,1)) [0,1]))+ DocTest.printPrefix "MathObj.PowerSeries.Core:267: "+{-# LINE 267 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 267 "src/MathObj/PowerSeries/Core.hs" #-}+     (checkHoles 20 (PS.cos (\0 -> (0,1))) 0)+ DocTest.printPrefix "MathObj.PowerSeries.Core:273: "+{-# LINE 273 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 273 "src/MathObj/PowerSeries/Core.hs" #-}+     (equalTrunc 50 PSE.tanExpl (PS.tan (\0 -> (0,1)) [0,1]))+ DocTest.printPrefix "MathObj.PowerSeries.Core:274: "+{-# LINE 274 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 274 "src/MathObj/PowerSeries/Core.hs" #-}+     (equalTrunc 50 (0:1:repeat 0) (PS.tan (\0 -> (0,1)) PSE.atan))+ DocTest.printPrefix "MathObj.PowerSeries.Core:275: "+{-# LINE 275 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 275 "src/MathObj/PowerSeries/Core.hs" #-}+     (checkHoles 20 (PS.tan (\0 -> (0,1))) 0)+ DocTest.printPrefix "MathObj.PowerSeries.Core:289: "+{-# LINE 289 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 289 "src/MathObj/PowerSeries/Core.hs" #-}+     (equalTrunc 500 PSE.logExpl (PS.log (\1 -> 0) [1,1]))+ DocTest.printPrefix "MathObj.PowerSeries.Core:290: "+{-# LINE 290 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 290 "src/MathObj/PowerSeries/Core.hs" #-}+     (equalTrunc 100 (0:1:repeat 0) (PS.log (\1 -> 0) PSE.exp))+ DocTest.printPrefix "MathObj.PowerSeries.Core:291: "+{-# LINE 291 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 291 "src/MathObj/PowerSeries/Core.hs" #-}+     (checkHoles 30 (PS.log (\1 -> 0)) 1)+ DocTest.printPrefix "MathObj.PowerSeries.Core:303: "+{-# LINE 303 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 303 "src/MathObj/PowerSeries/Core.hs" #-}+     (equalTrunc 500 PSE.atan (PS.atan (\0 -> 0) [0,1]))+ DocTest.printPrefix "MathObj.PowerSeries.Core:304: "+{-# LINE 304 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 304 "src/MathObj/PowerSeries/Core.hs" #-}+     (equalTrunc 50 (0:1:repeat 0) (PS.atan (\0 -> 0) PSE.tan))+ DocTest.printPrefix "MathObj.PowerSeries.Core:305: "+{-# LINE 305 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 305 "src/MathObj/PowerSeries/Core.hs" #-}+     (checkHoles 20 (PS.atan (\0 -> 0)) 0)+ DocTest.printPrefix "MathObj.PowerSeries.Core:313: "+{-# LINE 313 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 313 "src/MathObj/PowerSeries/Core.hs" #-}+     (equalTrunc 100 (0:1:repeat 0) (PS.asin (\1 -> 1) (\0 -> 0) PSE.sin))+ DocTest.printPrefix "MathObj.PowerSeries.Core:314: "+{-# LINE 314 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 314 "src/MathObj/PowerSeries/Core.hs" #-}+     (equalTrunc 50 PSE.asin (PS.asin (\1 -> 1) (\0 -> 0) [0,1]))+ DocTest.printPrefix "MathObj.PowerSeries.Core:315: "+{-# LINE 315 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 315 "src/MathObj/PowerSeries/Core.hs" #-}+     (checkHoles 30 (PS.asin (\1 -> 1) (\0 -> 0)) 0)+ DocTest.printPrefix "MathObj.PowerSeries.Core:383: "+{-# LINE 383 "src/MathObj/PowerSeries/Core.hs" #-}+ DocTest.property+{-# LINE 383 "src/MathObj/PowerSeries/Core.hs" #-}+     (genInvertible /\ \xs -> let (y,ys) = PS.inv xs; (z,zs) = PS.invDiff xs in y==z && equalTrunc 15 ys zs)
+ test/Test/MathObj/PowerSeries/Example.hs view
@@ -0,0 +1,92 @@+-- Do not edit! Automatically created with doctest-extract from src/MathObj/PowerSeries/Example.hs+{-# LINE 21 "src/MathObj/PowerSeries/Example.hs" #-}++module Test.MathObj.PowerSeries.Example where++import qualified Test.DocTest.Driver as DocTest++{-# LINE 22 "src/MathObj/PowerSeries/Example.hs" #-}+import     qualified MathObj.PowerSeries.Core as PS+import     qualified MathObj.PowerSeries.Example as PSE+import     Test.NumericPrelude.Utility (equalTrunc)+import     NumericPrelude.Numeric as NP+import     NumericPrelude.Base as P+import     Prelude ()++test :: DocTest.T ()+test = do+ DocTest.printPrefix "MathObj.PowerSeries.Example:55: "+{-# LINE 55 "src/MathObj/PowerSeries/Example.hs" #-}+ DocTest.property+{-# LINE 55 "src/MathObj/PowerSeries/Example.hs" #-}+          (\m n -> equalTrunc 30 (PS.mul (PSE.pow m) (PSE.pow n)) (PSE.pow (m+n)))+ DocTest.printPrefix "MathObj.PowerSeries.Example:66: "+{-# LINE 66 "src/MathObj/PowerSeries/Example.hs" #-}+ DocTest.property+{-# LINE 66 "src/MathObj/PowerSeries/Example.hs" #-}+          (equalTrunc 500 PSE.expExpl PSE.expODE)+ DocTest.printPrefix "MathObj.PowerSeries.Example:69: "+{-# LINE 69 "src/MathObj/PowerSeries/Example.hs" #-}+ DocTest.property+{-# LINE 69 "src/MathObj/PowerSeries/Example.hs" #-}+          (equalTrunc 500 PSE.sinExpl PSE.sinODE)+ DocTest.printPrefix "MathObj.PowerSeries.Example:72: "+{-# LINE 72 "src/MathObj/PowerSeries/Example.hs" #-}+ DocTest.property+{-# LINE 72 "src/MathObj/PowerSeries/Example.hs" #-}+          (equalTrunc 500 PSE.cosExpl PSE.cosODE)+ DocTest.printPrefix "MathObj.PowerSeries.Example:76: "+{-# LINE 76 "src/MathObj/PowerSeries/Example.hs" #-}+ DocTest.property+{-# LINE 76 "src/MathObj/PowerSeries/Example.hs" #-}+          (equalTrunc 50 PSE.tanExpl PSE.tanODE)+ DocTest.printPrefix "MathObj.PowerSeries.Example:80: "+{-# LINE 80 "src/MathObj/PowerSeries/Example.hs" #-}+ DocTest.property+{-# LINE 80 "src/MathObj/PowerSeries/Example.hs" #-}+          (equalTrunc 50 PSE.tanExpl PSE.tanExplSieve)+ DocTest.printPrefix "MathObj.PowerSeries.Example:87: "+{-# LINE 87 "src/MathObj/PowerSeries/Example.hs" #-}+ DocTest.property+{-# LINE 87 "src/MathObj/PowerSeries/Example.hs" #-}+          (equalTrunc 500 PSE.logExpl PSE.logODE)+ DocTest.printPrefix "MathObj.PowerSeries.Example:90: "+{-# LINE 90 "src/MathObj/PowerSeries/Example.hs" #-}+ DocTest.property+{-# LINE 90 "src/MathObj/PowerSeries/Example.hs" #-}+          (equalTrunc 500 PSE.atanExpl PSE.atanODE)+ DocTest.printPrefix "MathObj.PowerSeries.Example:94: "+{-# LINE 94 "src/MathObj/PowerSeries/Example.hs" #-}+ DocTest.property+{-# LINE 94 "src/MathObj/PowerSeries/Example.hs" #-}+          (equalTrunc 500 PSE.sinhExpl PSE.sinhODE)+ DocTest.printPrefix "MathObj.PowerSeries.Example:97: "+{-# LINE 97 "src/MathObj/PowerSeries/Example.hs" #-}+ DocTest.property+{-# LINE 97 "src/MathObj/PowerSeries/Example.hs" #-}+          (equalTrunc 500 PSE.coshExpl PSE.coshODE)+ DocTest.printPrefix "MathObj.PowerSeries.Example:100: "+{-# LINE 100 "src/MathObj/PowerSeries/Example.hs" #-}+ DocTest.property+{-# LINE 100 "src/MathObj/PowerSeries/Example.hs" #-}+          (equalTrunc 500 PSE.atanhExpl PSE.atanhODE)+ DocTest.printPrefix "MathObj.PowerSeries.Example:106: "+{-# LINE 106 "src/MathObj/PowerSeries/Example.hs" #-}+ DocTest.property+{-# LINE 106 "src/MathObj/PowerSeries/Example.hs" #-}+          (\expon -> equalTrunc 50 (PSE.powODE expon) (PSE.powExpl expon))+ DocTest.printPrefix "MathObj.PowerSeries.Example:112: "+{-# LINE 112 "src/MathObj/PowerSeries/Example.hs" #-}+ DocTest.property+{-# LINE 112 "src/MathObj/PowerSeries/Example.hs" #-}+          (equalTrunc 100 PSE.sqrtExpl PSE.sqrtODE)+ DocTest.printPrefix "MathObj.PowerSeries.Example:149: "+{-# LINE 149 "src/MathObj/PowerSeries/Example.hs" #-}+ DocTest.property+{-# LINE 149 "src/MathObj/PowerSeries/Example.hs" #-}+          (equalTrunc 50 PSE.tanODE PSE.tanODESieve)+ DocTest.printPrefix "MathObj.PowerSeries.Example:165: "+{-# LINE 165 "src/MathObj/PowerSeries/Example.hs" #-}+ DocTest.property+{-# LINE 165 "src/MathObj/PowerSeries/Example.hs" #-}+          (equalTrunc 50 PSE.asinODE (snd $ PS.inv PSE.sinODE))
test/Test/MathObj/RefinementMask2.hs view
@@ -1,78 +1,72 @@-{-# LANGUAGE RebindableSyntax #-}-module Test.MathObj.RefinementMask2 where--import qualified MathObj.RefinementMask2 as Mask-import qualified Algebra.Differential    as D--import qualified MathObj.Polynomial      as Poly-import qualified MathObj.Polynomial.Core as PolyCore--import qualified Algebra.RealField      as RealField-import qualified Algebra.Ring           as Ring--import qualified Algebra.ZeroTestable   as ZeroTestable--import Data.Maybe (fromMaybe, )--import Test.NumericPrelude.Utility (testUnit)-import Test.QuickCheck (Property, quickCheck, (==>), Testable, )-import qualified Test.HUnit as HUnit---import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP--+-- Do not edit! Automatically created with doctest-extract from src/MathObj/RefinementMask2.hs+{-# LINE 32 "src/MathObj/RefinementMask2.hs" #-} -hasMultipleZero :: (Ring.C a, Eq a) => Int -> a -> Poly.T a -> Bool-hasMultipleZero n x poly =-   all (zero==) $ take n $-   map (flip Poly.evaluate x) $-   iterate D.differentiate poly+module Test.MathObj.RefinementMask2 where -inverse0 :: (RealField.C a) => Mask.T a -> Property-inverse0 mask0 =-   let (b,poly) =-          case Mask.toPolynomial mask0 of-             Just p -> (True, p)-             Nothing -> (False, error "RefinementMask2.inverse0: no admissible mask")-       mask1 = Mask.fromPolynomial poly-       maskD =-          Poly.fromCoeffs (Mask.coeffs mask1) --          Poly.fromCoeffs (Mask.coeffs mask0)-   in  b ==>-          hasMultipleZero (fromMaybe 0 $ Poly.degree poly)-             1 maskD+import Test.DocTest.Base+import qualified Test.DocTest.Driver as DocTest -truncatePolynomial :: (ZeroTestable.C a) => Int -> Poly.T a -> Poly.T a-truncatePolynomial n =-   Poly.fromCoeffs . PolyCore.normalize . take n . Poly.coeffs+{-# LINE 33 "src/MathObj/RefinementMask2.hs" #-}+import     qualified MathObj.RefinementMask2 as Mask+import     qualified MathObj.Polynomial      as Poly+import     qualified MathObj.Polynomial.Core as PolyCore -inverse1 :: (RealField.C a) => Poly.T a -> Bool-inverse1 poly0 =-   case Mask.toPolynomial (Mask.fromPolynomial poly0) of-      Just poly1 -> Poly.collinear poly0 poly1-      Nothing -> False+import     qualified Algebra.Differential as D+import     qualified Algebra.Ring as Ring+import     Test.NumericPrelude.Utility ((/\))+import     qualified Test.QuickCheck as QC+import     NumericPrelude.Numeric as NP+import     NumericPrelude.Base as P+import     Prelude () -refining :: (RealField.C a) => Poly.T a -> Bool-refining poly =-   poly == Mask.refinePolynomial (Mask.fromPolynomial poly) poly+import     Data.Function.HT (nest)+import     Data.Maybe (fromMaybe)  +hasMultipleZero     :: (Ring.C a, Eq a) => Int -> a -> Poly.T a -> Bool+hasMultipleZero     n x poly =+       all (zero==) $ take n $+       map (flip Poly.evaluate x) $+       iterate D.differentiate poly -test :: Testable a => (Poly.T Integer -> a) -> IO ()-test = quickCheck+genAdmissibleMask     :: QC.Gen (Mask.T Rational, Poly.T Rational)+genAdmissibleMask     =+       QC.suchThatMap QC.arbitrary $+          \mask -> fmap ((,) mask) $ Mask.toPolynomial mask -testRat :: Testable a => (Poly.T Rational -> a) -> IO ()-testRat = quickCheck+polyFromMask     :: Mask.T a -> Poly.T a+polyFromMask     = Poly.fromCoeffs . Mask.coeffs +genShortPolynomial     :: Int -> QC.Gen (Poly.T Rational)+genShortPolynomial     n =+       fmap (Poly.fromCoeffs . PolyCore.normalize . take n) $ QC.arbitrary -tests :: HUnit.Test-tests =-   HUnit.TestLabel "refinement mask" $-   HUnit.TestList $-   map testUnit $-      ("inverse0", quickCheck (inverse0 :: Mask.T Rational -> Property)) :-      ("inverse1", quickCheck (inverse1 . truncatePolynomial 5 :: Poly.T Rational -> Bool)) :-      ("refining", quickCheck (refining . truncatePolynomial 5 :: Poly.T Rational -> Bool)) :-      []+test :: DocTest.T ()+test = do+ DocTest.printPrefix "MathObj.RefinementMask2:127: "+{-# LINE 127 "src/MathObj/RefinementMask2.hs" #-}+ DocTest.property+{-# LINE 127 "src/MathObj/RefinementMask2.hs" #-}+     (genAdmissibleMask /\ \(mask,poly) -> hasMultipleZero (fromMaybe 0 $ Poly.degree poly) 1 (polyFromMask (Mask.fromPolynomial poly) - polyFromMask mask))+ DocTest.printPrefix "MathObj.RefinementMask2:129: "+{-# LINE 129 "src/MathObj/RefinementMask2.hs" #-}+ DocTest.property+{-# LINE 129 "src/MathObj/RefinementMask2.hs" #-}+     (genShortPolynomial 5 /\ \poly -> maybe False (Poly.collinear poly) $ Mask.toPolynomial $ Mask.fromPolynomial poly)+ DocTest.printPrefix "MathObj.RefinementMask2:161: "+{-# LINE 161 "src/MathObj/RefinementMask2.hs" #-}+ DocTest.example+{-# LINE 161 "src/MathObj/RefinementMask2.hs" #-}+   (fmap ((6::Rational) *>) $ Mask.toPolynomial (Mask.fromCoeffs [0.1, 0.02, 0.005::Rational]))+  [ExpectedLine [LineChunk "Just (Polynomial.fromCoeffs [-12732 % 109375,272 % 625,-18 % 25,1 % 1])"]]+ DocTest.printPrefix "MathObj.RefinementMask2:207: "+{-# LINE 207 "src/MathObj/RefinementMask2.hs" #-}+ DocTest.property+{-# LINE 207 "src/MathObj/RefinementMask2.hs" #-}+     (genShortPolynomial 5 /\ \poly -> poly == Mask.refinePolynomial (Mask.fromPolynomial poly) poly)+ DocTest.printPrefix "MathObj.RefinementMask2:209: "+{-# LINE 209 "src/MathObj/RefinementMask2.hs" #-}+ DocTest.example+{-# LINE 209 "src/MathObj/RefinementMask2.hs" #-}+   (fmap (round :: Double -> Integer) $ fmap (1000000*) $ nest 50 (Mask.refinePolynomial (Mask.fromCoeffs [0.1, 0.02, 0.005])) (Poly.fromCoeffs [0,0,0,1]))+  [ExpectedLine [LineChunk "Polynomial.fromCoeffs [-116407,435200,-720000,1000000]"]]
test/Test/Number/ComplexSquareRoot.hs view
@@ -1,50 +1,56 @@-{-# LANGUAGE RebindableSyntax #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-module Test.Number.ComplexSquareRoot where--import qualified Number.ComplexSquareRoot as S-import qualified Number.Complex as Complex---- import qualified Algebra.Ring           as Ring--import qualified Algebra.Laws as Laws--import Test.NumericPrelude.Utility (testUnit)-import Test.QuickCheck (Testable, quickCheck, (==>), )-import qualified Test.HUnit as HUnit+-- Do not edit! Automatically created with doctest-extract from playground/Number/ComplexSquareRoot.hs+{-# LINE 21 "playground/Number/ComplexSquareRoot.hs" #-} -import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP+module Test.Number.ComplexSquareRoot where +import qualified Test.DocTest.Driver as DocTest -simple ::-   (Testable t) =>-   (S.T Rational -> t) -> IO ()-simple = quickCheck+{-# LINE 22 "playground/Number/ComplexSquareRoot.hs" #-}+import     qualified Number.ComplexSquareRoot as SR+import     qualified Number.Complex as Complex+import     qualified Algebra.Laws as Laws+import     Test.QuickCheck ((==>))+import     NumericPrelude.Numeric+import     NumericPrelude.Base+import     Prelude () -tests :: HUnit.Test-tests =-   HUnit.TestLabel "complex square root" $-   HUnit.TestList $-   map testUnit $-   testList+sr     :: SR.T Rational -> SR.T Rational+sr     = id -testList :: [(String, IO ())]-testList =-   ("multiplication, one",-      simple $ Laws.identity S.mul S.one) :-   ("multiplication, commutative",-      simple $ Laws.commutative S.mul) :-   ("multiplication, associative",-      simple $ Laws.associative S.mul) :-   ("multiplication, homomorphism",-      quickCheck $ Laws.homomorphism S.fromNumber-         (\x y -> (x :: Complex.T Rational) * y) S.mul) :-   ("division, one",-      simple $ Laws.rightIdentity S.div S.one) :-   ("recip recip",-      simple $ \x -> not (isZero x) ==> S.recip (S.recip x) == x) :-   ("recip inverts multiplication",-      simple $ \x -> not (isZero x) ==> Laws.inverse S.mul S.recip S.one x) :-   []+test :: DocTest.T ()+test = do+ DocTest.printPrefix "Number.ComplexSquareRoot:42: "+{-# LINE 42 "playground/Number/ComplexSquareRoot.hs" #-}+ DocTest.property+{-# LINE 42 "playground/Number/ComplexSquareRoot.hs" #-}+     (Laws.identity SR.mul SR.one . sr)+ DocTest.printPrefix "Number.ComplexSquareRoot:43: "+{-# LINE 43 "playground/Number/ComplexSquareRoot.hs" #-}+ DocTest.property+{-# LINE 43 "playground/Number/ComplexSquareRoot.hs" #-}+     (Laws.commutative SR.mul . sr)+ DocTest.printPrefix "Number.ComplexSquareRoot:44: "+{-# LINE 44 "playground/Number/ComplexSquareRoot.hs" #-}+ DocTest.property+{-# LINE 44 "playground/Number/ComplexSquareRoot.hs" #-}+     (Laws.associative SR.mul . sr)+ DocTest.printPrefix "Number.ComplexSquareRoot:45: "+{-# LINE 45 "playground/Number/ComplexSquareRoot.hs" #-}+ DocTest.property+{-# LINE 45 "playground/Number/ComplexSquareRoot.hs" #-}+     (Laws.homomorphism SR.fromNumber (\x y -> x * (y :: Complex.T Rational)) SR.mul)+ DocTest.printPrefix "Number.ComplexSquareRoot:46: "+{-# LINE 46 "playground/Number/ComplexSquareRoot.hs" #-}+ DocTest.property+{-# LINE 46 "playground/Number/ComplexSquareRoot.hs" #-}+     (Laws.rightIdentity SR.div SR.one . sr)+ DocTest.printPrefix "Number.ComplexSquareRoot:47: "+{-# LINE 47 "playground/Number/ComplexSquareRoot.hs" #-}+ DocTest.property+{-# LINE 47 "playground/Number/ComplexSquareRoot.hs" #-}+     (\x -> not (isZero x) ==> SR.recip (SR.recip x) == sr x)+ DocTest.printPrefix "Number.ComplexSquareRoot:48: "+{-# LINE 48 "playground/Number/ComplexSquareRoot.hs" #-}+ DocTest.property+{-# LINE 48 "playground/Number/ComplexSquareRoot.hs" #-}+     (\x -> not (isZero x) ==> Laws.inverse SR.mul SR.recip SR.one (sr x))
test/Test/Number/GaloisField2p32m5.hs view
@@ -1,37 +1,70 @@-{-# LANGUAGE RebindableSyntax #-}-module Test.Number.GaloisField2p32m5 where--import qualified Number.GaloisField2p32m5 as GF--import qualified Algebra.Laws as Laws--import Test.NumericPrelude.Utility (testUnit)-import Test.QuickCheck (Testable, quickCheck, (==>))-import qualified Test.HUnit as HUnit-+-- Do not edit! Automatically created with doctest-extract from src/Number/GaloisField2p32m5.hs+{-# LINE 33 "src/Number/GaloisField2p32m5.hs" #-} -import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP+module Test.Number.GaloisField2p32m5 where +import qualified Test.DocTest.Driver as DocTest -test :: Testable a => (GF.T -> a) -> IO ()-test = quickCheck+{-# LINE 34 "src/Number/GaloisField2p32m5.hs" #-}+import     qualified Number.GaloisField2p32m5 as GF+import     qualified Algebra.Laws as Laws+import     Test.QuickCheck ((==>))+import     NumericPrelude.Numeric+import     NumericPrelude.Base+import     Prelude () +gf     :: GF.T -> GF.T+gf     = id -tests :: HUnit.Test-tests =-   HUnit.TestLabel "galois field 2^32-5" $-   HUnit.TestList $-   map testUnit $-      ("addition, zero",         test (Laws.identity (+) zero)) :-      ("addition, commutative",  test (Laws.commutative (+))) :-      ("addition, associative",  test (Laws.associative (+))) :-      ("addition, negate",       test (Laws.inverse (+) negate zero)) :-      ("addition, subtract",     test (\x -> Laws.inverse (+) (x-) x)) :-      ("multiplication, one",          test (Laws.identity (*) one)) :-      ("multiplication, commutative",  test (Laws.commutative (*))) :-      ("multiplication, associative",  test (Laws.associative (*))) :-      ("multiplication, recip",        test (\y -> y /= 0 ==> Laws.inverse (*) recip one y)) :-      ("multiplication, division",     test (\y x -> y /= 0 ==> Laws.inverse (*) (x/) x y)) :-      ("multiplication and addition, distributive",   test (Laws.leftDistributive (*) (+))) :-      []+test :: DocTest.T ()+test = do+ DocTest.printPrefix "Number.GaloisField2p32m5:46: "+{-# LINE 46 "src/Number/GaloisField2p32m5.hs" #-}+ DocTest.property+{-# LINE 46 "src/Number/GaloisField2p32m5.hs" #-}+     (Laws.identity (+) zero . gf)+ DocTest.printPrefix "Number.GaloisField2p32m5:47: "+{-# LINE 47 "src/Number/GaloisField2p32m5.hs" #-}+ DocTest.property+{-# LINE 47 "src/Number/GaloisField2p32m5.hs" #-}+     (Laws.commutative (+) . gf)+ DocTest.printPrefix "Number.GaloisField2p32m5:48: "+{-# LINE 48 "src/Number/GaloisField2p32m5.hs" #-}+ DocTest.property+{-# LINE 48 "src/Number/GaloisField2p32m5.hs" #-}+     (Laws.associative (+) . gf)+ DocTest.printPrefix "Number.GaloisField2p32m5:49: "+{-# LINE 49 "src/Number/GaloisField2p32m5.hs" #-}+ DocTest.property+{-# LINE 49 "src/Number/GaloisField2p32m5.hs" #-}+     (Laws.inverse (+) negate zero . gf)+ DocTest.printPrefix "Number.GaloisField2p32m5:50: "+{-# LINE 50 "src/Number/GaloisField2p32m5.hs" #-}+ DocTest.property+{-# LINE 50 "src/Number/GaloisField2p32m5.hs" #-}+     (\x -> Laws.inverse (+) (x-) (gf x))+ DocTest.printPrefix "Number.GaloisField2p32m5:51: "+{-# LINE 51 "src/Number/GaloisField2p32m5.hs" #-}+ DocTest.property+{-# LINE 51 "src/Number/GaloisField2p32m5.hs" #-}+     (Laws.identity (*) one . gf)+ DocTest.printPrefix "Number.GaloisField2p32m5:52: "+{-# LINE 52 "src/Number/GaloisField2p32m5.hs" #-}+ DocTest.property+{-# LINE 52 "src/Number/GaloisField2p32m5.hs" #-}+     (Laws.commutative (*) . gf)+ DocTest.printPrefix "Number.GaloisField2p32m5:53: "+{-# LINE 53 "src/Number/GaloisField2p32m5.hs" #-}+ DocTest.property+{-# LINE 53 "src/Number/GaloisField2p32m5.hs" #-}+     (Laws.associative (*) . gf)+ DocTest.printPrefix "Number.GaloisField2p32m5:54: "+{-# LINE 54 "src/Number/GaloisField2p32m5.hs" #-}+ DocTest.property+{-# LINE 54 "src/Number/GaloisField2p32m5.hs" #-}+     (\y -> gf y /= zero ==> Laws.inverse (*) recip one y)+ DocTest.printPrefix "Number.GaloisField2p32m5:55: "+{-# LINE 55 "src/Number/GaloisField2p32m5.hs" #-}+ DocTest.property+{-# LINE 55 "src/Number/GaloisField2p32m5.hs" #-}+     (\y x -> gf y /= zero ==> Laws.inverse (*) (x/) x y)
test/Test/NumericPrelude/Utility.hs view
@@ -1,21 +1,17 @@--- cf. utility-ht Test.Utility module Test.NumericPrelude.Utility where -import Data.List.HT (mapAdjacent, )-import qualified Data.List as List-import qualified Test.HUnit as HUnit+import qualified Test.QuickCheck as QC +import qualified NumericPrelude.Numeric as NP -testUnit :: (String, IO ()) -> HUnit.Test-testUnit (label, check) =-   HUnit.TestLabel label (HUnit.TestCase check)+import Data.Eq.HT (equating) --- compare the lists simultaneously-equalLists :: Eq a => [[a]] -> Bool-equalLists xs =-   let equalElems ys =-          and (mapAdjacent (==) ys)  &&  length xs == length ys-   in  all equalElems (List.transpose xs) -equalInfLists :: Eq a => Int -> [[a]] -> Bool-equalInfLists n xs = equalLists (map (take n) xs)+equalTrunc :: Int -> [NP.Rational] -> [NP.Rational] -> Bool+equalTrunc n = equating (take n)+++infixr 0 /\++(/\) :: (Show a, QC.Testable test) => QC.Gen a -> (a -> test) -> QC.Property+(/\) = QC.forAll
test/Test/Run.hs view
@@ -1,34 +1,44 @@+-- Do not edit! Automatically created with doctest-extract. module Main where -import qualified Test.MathObj.RefinementMask2 as RefinementMask2-import qualified Test.Algebra.RealRing as RealRing-import qualified Test.Algebra.IntegralDomain as Integral-import qualified Test.MathObj.Gaussian.Polynomial as GaussPoly-import qualified Test.MathObj.Gaussian.Variance as GaussVariance-import qualified Test.MathObj.Gaussian.Bell as GaussBell-import qualified Test.MathObj.PartialFraction as PartialFraction-import qualified Test.MathObj.Matrix  as Matrix-import qualified Test.MathObj.Polynomial  as Polynomial-import qualified Test.MathObj.PowerSeries as PowerSeries-import qualified Test.Number.ComplexSquareRoot as CSqRt-import qualified Test.Number.GaloisField2p32m5 as GF-import qualified Test.HUnit.Text as HUnitText-import qualified Test.HUnit as HUnit+import qualified Test.Algebra.Additive+import qualified Test.Algebra.IntegralDomain+import qualified Test.Algebra.PrincipalIdealDomain+import qualified Test.Algebra.RealRing+import qualified Test.MathObj.Gaussian.Bell+import qualified Test.MathObj.Gaussian.Polynomial+import qualified Test.MathObj.Gaussian.ExponentTuple+import qualified Test.MathObj.Gaussian.Variance+import qualified Test.MathObj.Matrix+import qualified Test.MathObj.PartialFraction+import qualified Test.MathObj.Polynomial+import qualified Test.MathObj.Polynomial.Core+import qualified Test.MathObj.PowerSeries+import qualified Test.MathObj.PowerSeries.Core+import qualified Test.MathObj.PowerSeries.Example+import qualified Test.MathObj.RefinementMask2+import qualified Test.Number.ComplexSquareRoot+import qualified Test.Number.GaloisField2p32m5 +import qualified Test.DocTest.Driver as DocTest+ main :: IO ()-main =-   print =<<-      HUnitText.runTestTT (HUnit.TestList $-         RefinementMask2.tests :-         RealRing.tests :-         Integral.tests :-         GaussVariance.tests :-         GaussBell.tests :-         GaussPoly.tests :-         PartialFraction.tests :-         Matrix.tests :-         Polynomial.tests :-         PowerSeries.tests :-         CSqRt.tests :-         GF.tests :-         [])+main = DocTest.run $ do+    Test.Algebra.Additive.test+    Test.Algebra.IntegralDomain.test+    Test.Algebra.PrincipalIdealDomain.test+    Test.Algebra.RealRing.test+    Test.MathObj.Gaussian.Bell.test+    Test.MathObj.Gaussian.Polynomial.test+    Test.MathObj.Gaussian.ExponentTuple.test+    Test.MathObj.Gaussian.Variance.test+    Test.MathObj.Matrix.test+    Test.MathObj.PartialFraction.test+    Test.MathObj.Polynomial.test+    Test.MathObj.Polynomial.Core.test+    Test.MathObj.PowerSeries.test+    Test.MathObj.PowerSeries.Core.test+    Test.MathObj.PowerSeries.Example.test+    Test.MathObj.RefinementMask2.test+    Test.Number.ComplexSquareRoot.test+    Test.Number.GaloisField2p32m5.test