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numeric-prelude 0.3.0.2 → 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
@@ -1,81 +1,32 @@--OBJECT_DIR    := build/$(shell uname -s)-$(shell uname -m)-INTERFACE_DIR := build/Interface--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) \-          $(wildcard src/MathObj/PowerSeries/*.hs)--GHC_OPTIONS = -Wall -odir$(OBJECT_DIR) -hidir$(INTERFACE_DIR)---# names of literate modules after removing literary information-UNLIT_MODULES = $(patsubst %.lhs, %.hs, $(patsubst %.hs, , $(MODULES)))--# names of all modules without literary information-HS_MODULES = $(patsubst %.lhs, %.hs, $(MODULES))--STDINTERFACES = base/base.haddock parsec/parsec.haddock--HADDOCK_INCL = $(patsubst %, -i /usr/local/share/ghc-6.2/html/libraries/%, \-                    $(STDINTERFACES))--HC = ghc--HCI = ghci----.INTERMEDIATE:	$(UNLIT_MODULES)--.PHONY:	all doc clean build test ghci publish+HCI6 = ghci+HCI7 = ghci -XCPP -DNoImplicitPrelude=RebindableSyntax -all:	build+.PHONY: ghci ghci6 ghci7 ghci-gauss ghci-compile -clean:-	-rm `find $(OBJECT_DIR) -name "*.o"`-	-rm `find $(INTERFACE_DIR) -name "*.hi"`+ghci:	ghci7 -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+ghci6:+	$(HCI6) -Wall -i:src:test +RTS -M256m -c30 -RTS test/Demo.hs -ghci:-	$(HCI) -Wall -i:src:test +RTS -M256m -c30 -RTS test/Test.hs+ghci7:+	$(HCI7) -Wall -i:src:test +RTS -M256m -c30 -RTS test/Demo.hs  ghci-gauss:-	$(HCI) -Wall -i:src:test +RTS -M256m -c30 -RTS test/Test/MathObj/Gaussian/Variance.hs+	$(HCI7) -Wall -i:src:test:gaussian +RTS -M256m -c30 -RTS test/Test/MathObj/Gaussian/Variance.hs  ghci-compile:-	$(HCI) -Wall -i:src:test +RTS -M256m -c30 -RTS -fobject-code -O -hidir=dist/build -odir=dist/build test/Test.hs+	$(HCI7) -Wall -i:src:test +RTS -M256m -c30 -RTS -fobject-code -O -hidir=dist/build -odir=dist/build test/Demo.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)+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 -%.hs:	%.lhs-	unlit $< $@+update-test:+	doctest-extract-0.1 -i src/ -i gaussian/ -i playground/ -o test/ --executable-main=Test/Run.hs $$(cat test-module.list) -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`+%.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 +1,36 @@+* 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,
+ 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
@@ -1,188 +1,60 @@+Cabal-Version:  2.2 Name:           numeric-prelude-Version:        0.3.0.2-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-Tested-With:    GHC==6.4.1, GHC==6.8.2, GHC==6.10.4, GHC==6.12.3-Tested-With:    GHC==7.2.2, GHC==7.4.1-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 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.-  .-  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.+  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/GenerateRules.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:         0.3.0.2+  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,-    deepseq >=1.1 && <1.4-  If flag(splitBase)-    Build-Depends:-      base >= 2 && <5,-      array >=0.1 && <0.5,-      containers >=0.1 && <0.6,-      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 -  If impl(ghc>=7.0)-    CPP-Options: -DNoImplicitPrelude=RebindableSyntax-    Extensions: CPP+  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   Hs-source-dirs: src   Exposed-modules:@@ -193,6 +65,7 @@     Algebra.DimensionTerm     Algebra.DivisibleSpace     Algebra.Field+    Algebra.FloatingPoint     Algebra.Indexable     Algebra.IntegralDomain     Algebra.NonNegative@@ -281,68 +154,86 @@     NumericPrelude.List     Algebra.AffineSpace     Algebra.RealRing98-    MathObj.Gaussian.Variance-    MathObj.Gaussian.Bell-    MathObj.Gaussian.Polynomial-    Number.ComplexSquareRoot     -- 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-  Hs-Source-Dirs: src, test+Executable numeric-prelude-demo+  Hs-Source-Dirs: test   GHC-Options:    -Wall-  Main-Is: Test.hs--  If !flag(buildTests)-    Buildable:         False+  Default-Language: Haskell98+  Main-Is: Demo.hs -  If impl(ghc>=7.0)-    CPP-Options: -DNoImplicitPrelude=RebindableSyntax-    Extensions: CPP+  If flag(buildExamples)+    Build-Depends:+      numeric-prelude,+      base+  Else+    Buildable: False -Executable testsuite-  Hs-Source-Dirs: src, test+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--  If impl(ghc>=7.0)-    CPP-Options: -DNoImplicitPrelude=RebindableSyntax-    Extensions: CPP+  Build-Depends:+    doctest-exitcode-stdio >=0.0 && <0.1,+    doctest-lib >=0.1 && <0.1.1,+    numeric-prelude,+    QuickCheck,+    utility-ht,+    random,+    base -Executable test-gaussian-  Hs-Source-Dirs: src, test+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--  If impl(ghc>=7.0)-    CPP-Options: -DNoImplicitPrelude=RebindableSyntax-    Extensions: CPP
+ 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/Algebra/Absolute.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module Algebra.Absolute (    C(abs, signum),    absOrd, signumOrd,@@ -7,7 +7,7 @@ import qualified Algebra.Ring         as Ring import qualified Algebra.Additive     as Additive -import Algebra.Ring (one, ) -- fromInteger+import Algebra.Ring (one, ) import Algebra.Additive (zero, negate,)  import Data.Int  (Int,  Int8,  Int16,  Int32,  Int64,  )
src/Algebra/Additive.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module Algebra.Additive (     -- * Class     C,@@ -32,12 +32,19 @@ 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  +, -  {- |@@ -57,6 +64,7 @@ -}  class C a where+    {-# MINIMAL zero, (+), ((-) | negate) #-}     -- | zero element of the vector space     zero     :: a     -- | add and subtract elements@@ -95,6 +103,9 @@ 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 (+)@@ -106,19 +117,23 @@ 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@@ -363,6 +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@@ -405,7 +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
@@ -1,8 +1,7 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} 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
@@ -1,10 +1,8 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module Algebra.Differential where  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/DivisibleSpace.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} module Algebra.DivisibleSpace where
src/Algebra/Field.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module Algebra.Field (     {- * Class -}     C,@@ -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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module Algebra.IntegralDomain (     {- * Class -}     C,@@ -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
src/Algebra/Lattice.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module Algebra.Lattice (       C(up, dn)     , max, min, abs
src/Algebra/Module.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} {- |@@ -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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} {- |@@ -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
@@ -1,15 +1,8 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# 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 -} @@ -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
@@ -1,15 +1,8 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# 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 -} @@ -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
@@ -1,15 +1,8 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# 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 -} @@ -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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} @@ -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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module Algebra.PrincipalIdealDomain (     {- * Class -}     C,@@ -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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module Algebra.RealField (    C,    ) where@@ -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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {- | Generally before using 'quot' and 'rem', think twice. In most cases 'divMod' and friends are the right choice,@@ -19,8 +19,6 @@ 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, )
src/Algebra/RealRing.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module Algebra.RealRing where  import qualified Algebra.RealRing98 as RealRing98@@ -35,6 +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'@@ -115,8 +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@@ -156,6 +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@@ -169,6 +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 #-}@@ -183,20 +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 #-}@@ -417,6 +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 =@@ -424,30 +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/RealTranscendental.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module Algebra.RealTranscendental where  import qualified Algebra.Transcendental      as Trans
src/Algebra/RightModule.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} module Algebra.RightModule where@@ -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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module Algebra.Ring (     {- * Class -}     C,@@ -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
@@ -49,6 +49,7 @@    toInteger :: a -> Integer  +{-# NOINLINE [2] fromIntegral #-} fromIntegral :: (C a, Ring.C b) => a -> b fromIntegral = fromInteger . toInteger
src/Algebra/ToRational.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module Algebra.ToRational where  import qualified Algebra.ZeroTestable as ZeroTestable@@ -68,6 +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
@@ -1,9 +1,7 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} 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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module Algebra.Units (     {- * Class -}     C,@@ -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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {- | Copyright   :  (c) Henning Thielemann 2004-2005 @@ -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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} module Algebra.VectorSpace where@@ -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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module Algebra.ZeroTestable where  import qualified Algebra.Additive as Additive@@ -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/Algebra.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {- | Copyright    :   (c) Mikael Johansson 2006 Maintainer   :   mik@math.uni-jena.de
src/MathObj/DiscreteMap.hs view
@@ -1,5 +1,5 @@ {-# OPTIONS_GHC -fno-warn-orphans #-}-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} @@ -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 
@@ -1,324 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{--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 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)]---{--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--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 
@@ -1,231 +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 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 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, 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)]
− src/MathObj/Gaussian/Polynomial.hs 
@@ -1,480 +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.--* 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, )--- import Prelude ()---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--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, 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)})---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))--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.--}-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)))--{--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.--}---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 
@@ -1,206 +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.--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 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--{- |-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--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))----- 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--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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} {- |@@ -26,7 +26,6 @@  import qualified Number.Complex as Complex --- import qualified NumericPrelude.Base as P import qualified NumericPrelude.Numeric as NP  import NumericPrelude.Base    hiding (const, reverse, )
src/MathObj/Matrix.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} {- |@@ -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/Monoid.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module MathObj.Monoid where  import qualified Algebra.PrincipalIdealDomain as PID
src/MathObj/PartialFraction.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {- | Copyright    :   (c) Henning Thielemann 2007 Maintainer   :   numericprelude@henning-thielemann.de@@ -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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {- | Copyright    :   (c) Henning Thielemann 2006 Maintainer   :   numericprelude@henning-thielemann.de@@ -16,8 +16,6 @@  import Data.Array(Ix) --- import NumericPrelude.Numeric (Integer)--- import NumericPrelude.Base   {- |
src/MathObj/Permutation/CycleList.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {- | Copyright    :   (c) Mikael Johansson 2006 Maintainer   :   mik@math.uni-jena.de
src/MathObj/Permutation/CycleList/Check.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {- | Copyright    :   (c) Henning Thielemann 2006 Maintainer   :   numericprelude@henning-thielemann.de@@ -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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {- | Copyright    :   (c) Henning Thielemann 2006 Maintainer   :   numericprelude@henning-thielemann.de@@ -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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} @@ -73,15 +73,46 @@  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@@ -268,18 +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.@@ -288,19 +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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {- | This module implements polynomial functions on plain lists. We use such functions in order to implement methods of other datatypes.@@ -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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} @@ -27,6 +27,17 @@ 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 #-}@@ -126,6 +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)@@ -189,3 +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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module MathObj.PowerSeries.Core where  import qualified MathObj.Polynomial.Core as Poly@@ -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/DifferentialEquation.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {- | Lazy evaluation allows for the solution  of differential equations in terms of power series.
src/MathObj/PowerSeries/Example.hs view
@@ -1,11 +1,10 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} 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 @@ -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/PowerSeries/Mean.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {- | This module computes power series for representing some means as generalized $f$-means.
src/MathObj/PowerSeries2.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} @@ -19,11 +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 @@ -86,6 +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) @@ -124,5 +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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module MathObj.PowerSeries2.Core where  import qualified MathObj.PowerSeries as PS@@ -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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} {- |
src/MathObj/RefinementMask2.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module MathObj.RefinementMask2 (    T, coeffs, fromCoeffs,    fromPolynomial,@@ -29,6 +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]}  @@ -85,6 +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@@ -115,6 +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)@@ -131,10 +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,@@ -162,17 +203,18 @@                 (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)-{--> 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]--}  convolve ::    (Ring.C a) => T a -> T a -> T a
src/MathObj/RootSet.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {- | Copyright   :  (c) Henning Thielemann 2004-2005
src/MathObj/Wrapper/Haskell98.hs view
@@ -10,6 +10,7 @@ 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@@ -43,7 +44,7 @@ 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 a+newtype T a = Cons {decons :: a}    deriving       (Show, Eq, Ord, Ix, Bounded, Enum,        Num, Integral, Fractional, Floating,@@ -59,6 +60,15 @@ 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)@@ -155,6 +165,21 @@  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
src/MathObj/Wrapper/NumericPrelude.hs view
@@ -11,6 +11,7 @@ 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@@ -51,14 +52,14 @@ then @T (Polynomial (MathObj.Wrapper.Haskell98.T a))@ is in 'Num' for all types @a@ that are in 'Num'. -}-newtype T a = Cons a+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,+       ToInteger.C, ToRational.C, Float.C,        Differential.C)  {-# INLINE lift1 #-}@@ -151,21 +152,21 @@    truncate (Cons a) = fromInteger (RealRing.truncate a)    round (Cons a) = fromInteger (RealRing.round a) -instance (Trans.C a, RealRing.C a, ToRational.C a, Absolute.C a, Ord a, Show a) => RealFloat (T a) where-   atan2 = atan2-   floatRadix = unimplemented "floatRadix"-   floatDigits = unimplemented "floatDigits"-   floatRange = unimplemented "floatRange"-   decodeFloat = unimplemented "decodeFloat"-   encodeFloat = unimplemented "encodeFloat"-   exponent = unimplemented "exponent"-   significand = unimplemented "significand"-   scaleFloat = unimplemented "scaleFloat"-   isNaN = unimplemented "isNaN"-   isInfinite = unimplemented "isInfinite"-   isDenormalized = unimplemented "isDenormalized"-   isNegativeZero = unimplemented "isNegativeZero"-   isIEEE = unimplemented "isIEEE"+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
src/Number/Complex.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} {- Rules should be processed -}@@ -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@@ -81,8 +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, )@@ -90,7 +92,6 @@ import Text.Read.HT (readsInfixPrec, )  --- import qualified Data.Typeable as Ty  infix  6  +:, `Cons` @@ -359,7 +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'.@@ -545,29 +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 
@@ -1,117 +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 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". -}
src/Number/DimensionTerm/SI.hs view
@@ -1,12 +1,5 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# 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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {- | Copyright   :  (c) Henning Thielemann 2006 @@ -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
src/Number/FixedPoint/Check.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module Number.FixedPoint.Check where  import qualified Number.FixedPoint as FP@@ -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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# LANGUAGE MultiParamTypeClasses #-} {- | This number type is intended for tests of functions over fields,@@ -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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# OPTIONS_GHC -fno-warn-orphans #-}  {-@@ -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
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@@ -36,6 +35,7 @@  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, )@@ -44,9 +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.@@ -283,27 +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
@@ -1,14 +1,7 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# 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.
src/Number/PartiallyTranscendental.hs view
@@ -1,10 +1,10 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {- | 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 NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {- |-Copyright    :   (c) Henning Thielemann 2007+Copyright    :   (c) Henning Thielemann 2007-2012 Maintainer   :   numericprelude@henning-thielemann.de Stability    :   provisional Portability  :   portable@@ -32,14 +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 @@ -403,9 +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@@ -413,8 +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
@@ -1,14 +1,7 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# 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 -} @@ -23,7 +16,7 @@ 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@@ -32,7 +25,8 @@  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)@@ -226,9 +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 NoImplicitPrelude #-}+{-# 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 NoImplicitPrelude #-}+{-# 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 NoImplicitPrelude #-}+{-# 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 NoImplicitPrelude #-}+{-# 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 NoImplicitPrelude #-}+{-# 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 NoImplicitPrelude #-}+{-# 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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} {- |@@ -55,11 +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, )@@ -103,34 +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) @@ -140,11 +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@@ -265,9 +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 NoImplicitPrelude #-}+{-# 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
@@ -1,10 +1,8 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} 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)
src/Number/ResidueClass/Check.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module Number.ResidueClass.Check where  import qualified Number.ResidueClass as Res@@ -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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module Number.ResidueClass.Func where  import qualified Number.ResidueClass as Res@@ -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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module Number.ResidueClass.Maybe where  import qualified Number.ResidueClass as Res@@ -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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module Number.ResidueClass.Reader where  import qualified Number.ResidueClass as Res@@ -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/SI.hs view
@@ -1,14 +1,8 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# 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. -}@@ -40,6 +34,8 @@  import Data.Tuple.HT (mapFst, ) +import qualified MathObj.Wrapper.Haskell98 as W98+ import qualified Prelude as P  import NumericPrelude.Numeric@@ -47,9 +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 @@ -249,21 +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 NoImplicitPrelude #-}+{-# 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
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {- | Some functions that are counterparts of functions from "Data.List" using NumericPrelude.Numeric type classes.@@ -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, (-), )
src/NumericPrelude/List/Generic.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} {- | Functions that are counterparts of the @generic@ functions in "Data.List" using NumericPrelude.Numeric type classes.
src/NumericPrelude/Numeric.hs view
@@ -1,4 +1,4 @@-{-# LANGUAGE NoImplicitPrelude #-}+{-# LANGUAGE RebindableSyntax #-} module NumericPrelude.Numeric (     {- Additive -} (+), (-), negate, zero, subtract, sum, sum1,     {- ZeroTestable -} isZero,
+ 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 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/Test/Algebra/Additive.hs view
@@ -1,35 +1,28 @@-{-# LANGUAGE NoImplicitPrelude #-}-module Test.Algebra.Additive where--import qualified Algebra.Additive as A--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/Additive.hs+{-# LINE 42 "src/Algebra/Additive.hs" #-} -test ::-   (Testable t) =>-   ([Integer] -> t) -> IO ()-test = quickCheck+module Test.Algebra.Additive where +import qualified Test.DocTest.Driver as DocTest -tests :: HUnit.Test-tests =-   HUnit.TestLabel "additive group" $-   HUnit.TestList $-   map testUnit $-   testList+{-# LINE 43 "src/Algebra/Additive.hs" #-}+import     qualified Algebra.Additive as A+import     qualified Test.QuickCheck as QC -testList :: [(String, IO ())]-testList =-   ("sum1", test $ \ns n ->-      A.sum (n:ns) == A.sum1 (n:ns)) :-   ("sumNestedAssociative", test $ \ns ->-      A.sum ns == A.sumNestedAssociative ns) :-   ("sumNestedCommutative", test $ \ns ->-      A.sum ns == A.sumNestedCommutative ns) :-   []+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 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-+-- 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 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, )+-- 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
@@ -1,103 +1,157 @@-{-# LANGUAGE NoImplicitPrelude #-}-{-# LANGUAGE MultiParamTypeClasses #-}-{-# LANGUAGE FlexibleInstances #-}-module Test.MathObj.Gaussian.Bell where--import qualified MathObj.Gaussian.Bell as G--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 = quickCheck+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) :-      ("convolution by constant function",-          {--          using a G.norm1 we could exactly compute the amplitude-          of the resulting constant function.-          -}-          simple $ \x ->-             case G.convolve x (G.constant) of-                G.Cons _amp _a b c -> isZero b && isZero c) :-      ("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 by translation",-          simple $ \x -> G.fourier x == G.fourierByTranslation x) :-      ("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
@@ -1,165 +1,215 @@-{-# 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+-- Do not edit! Automatically created with doctest-extract from gaussian/MathObj/Gaussian/Polynomial.hs+{-# LINE 60 "gaussian/MathObj/Gaussian/Polynomial.hs" #-} -import qualified Number.NonNegative as NonNeg-import Data.Function.HT (nest, )-import Data.Tuple.HT (mapSnd, )+{-# OPTIONS_GHC -XRebindableSyntax #-} --- import Debug.Trace (trace, )+module Test.MathObj.Gaussian.Polynomial where -import NumericPrelude.Base as P-import NumericPrelude.Numeric as NP+import Test.DocTest.Base+import qualified Test.DocTest.Driver as DocTest +{-# 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) -simple ::-   (Testable t) =>-   (G.T Rational -> t) -> IO ()-simple f =-   quickCheck (\x -> f (x :: G.T Rational))+asRational     :: Id (G.T Rational)+asRational     = id -tests :: HUnit.Test-tests =-   HUnit.TestLabel "polynomial" $-   HUnit.TestList $-   map testUnit $-   testList+withRational     :: Id (G.T Rational -> a)+withRational     = id -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) :-      []+mulLinear2i     :: Id (G.T Rational)+mulLinear2i     x =+       x{G.polynomial = Poly.fromCoeffs [0, 0+:2] * G.polynomial x} -{--inequalities:+rotateQuarter     :: Int -> Id (G.T Rational)+rotateQuarter     n =+       G.scaleComplex (negate Complex.imaginaryUnit ^ fromIntegral n) -Heisenberg's uncertainty relation-   needs integrals and thus needs product of exponential numbers and roots--}+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
@@ -1,227 +1,142 @@-{-# 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--{--This is simpler, but may yield exponents smaller than 1.--instance Arbitrary YoungConjugates where-   arbitrary = liftM3-      (\(PositiveInteger a0) (PositiveInteger b0) (PositiveInteger 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))-      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 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+-- 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 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) :-      []+-- 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 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)-+-- 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 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 (\(label, len, x, y) -> (label, equalInfLists len [x,y]))---+-- Do not edit! Automatically created with doctest-extract from src/MathObj/PowerSeries.hs+{-# LINE 30 "src/MathObj/PowerSeries.hs" #-} -powerMult :: Rational -> Rational -> Bool-powerMult exp0 exp1 =-   PS.mul (PSE.pow exp0) (PSE.pow exp1)  ==  PSE.pow (exp0+exp1)+module Test.MathObj.PowerSeries where -powerExplODE :: Rational -> Bool-powerExplODE expon =-   PSE.powODE expon == PSE.powExpl expon+import qualified Test.DocTest.Driver as DocTest +{-# LINE 31 "src/MathObj/PowerSeries.hs" #-}+import     qualified MathObj.PowerSeries.Core as PS+import     qualified MathObj.PowerSeries as PST+import     qualified Test.QuickCheck as QC+import     Test.NumericPrelude.Utility (equalTrunc, (/\))+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 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--+-- 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, ZeroTestable.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, ZeroTestable.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 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+-- 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 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-+-- 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
@@ -1,36 +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.Algebra.Additive as Additive-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 :-         Additive.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