diff --git a/LICENSE b/LICENSE
--- a/LICENSE
+++ b/LICENSE
@@ -1,674 +1,26 @@
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-copy of the Program in return for a fee.
-
- END OF TERMS AND CONDITIONS
-
- How to Apply These Terms to Your New Programs
-
- If you develop a new program, and you want it to be of the greatest
-possible use to the public, the best way to achieve this is to make it
-free software which everyone can redistribute and change under these terms.
-
- To do so, attach the following notices to the program. It is safest
-to attach them to the start of each source file to most effectively
-state the exclusion of warranty; and each file should have at least
-the "copyright" line and a pointer to where the full notice is found.
-
-
- Copyright (C)
-
- This program is free software: you can redistribute it and/or modify
- it under the terms of the GNU General Public License as published by
- the Free Software Foundation, either version 3 of the License, or
- (at your option) any later version.
-
- This program is distributed in the hope that it will be useful,
- but WITHOUT ANY WARRANTY; without even the implied warranty of
- MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- GNU General Public License for more details.
-
- You should have received a copy of the GNU General Public License
- along with this program. If not, see .
-
-Also add information on how to contact you by electronic and paper mail.
-
- If the program does terminal interaction, make it output a short
-notice like this when it starts in an interactive mode:
-
- Copyright (C)
- This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
- This is free software, and you are welcome to redistribute it
- under certain conditions; type `show c' for details.
-
-The hypothetical commands `show w' and `show c' should show the appropriate
-parts of the General Public License. Of course, your program's commands
-might be different; for a GUI interface, you would use an "about box".
+Copyright (c) 2013 Henning Thielemann, Dylan Thurston, Mikael Johansson
+All rights reserved.
- You should also get your employer (if you work as a programmer) or school,
-if any, to sign a "copyright disclaimer" for the program, if necessary.
-For more information on this, and how to apply and follow the GNU GPL, see
-.
+Redistribution and use in source and binary forms, with or without
+modification, are permitted provided that the following conditions
+are met:
+1. Redistributions of source code must retain the above copyright
+ notice, this list of conditions and the following disclaimer.
+2. Redistributions in binary form must reproduce the above copyright
+ notice, this list of conditions and the following disclaimer in the
+ documentation and/or other materials provided with the distribution.
+3. Neither the name of the University nor the names of its contributors
+ may be used to endorse or promote products derived from this software
+ without specific prior written permission.
- The GNU General Public License does not permit incorporating your program
-into proprietary programs. If your program is a subroutine library, you
-may consider it more useful to permit linking proprietary applications with
-the library. If this is what you want to do, use the GNU Lesser General
-Public License instead of this License. But first, please read
-.
+THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
+ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
+FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
+DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
+OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
+HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
+LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
+OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
+SUCH DAMAGE.
diff --git a/Makefile b/Makefile
--- a/Makefile
+++ b/Makefile
@@ -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 $< --output=$@
diff --git a/README.md b/README.md
new file mode 100644
--- /dev/null
+++ b/README.md
@@ -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`.
diff --git a/docs/NOTES b/docs/NOTES
--- a/docs/NOTES
+++ b/docs/NOTES
@@ -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,
diff --git a/gaussian/Gaussian.hs b/gaussian/Gaussian.hs
new file mode 100644
--- /dev/null
+++ b/gaussian/Gaussian.hs
@@ -0,0 +1,6 @@
+module Main where
+
+import qualified MathObj.Gaussian.Example as Example
+
+main :: IO ()
+main = Example.polyApprox
diff --git a/gaussian/MathObj/Gaussian/Bell.hs b/gaussian/MathObj/Gaussian/Bell.hs
new file mode 100644
--- /dev/null
+++ b/gaussian/MathObj/Gaussian/Bell.hs
@@ -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)
diff --git a/gaussian/MathObj/Gaussian/Example.hs b/gaussian/MathObj/Gaussian/Example.hs
new file mode 100644
--- /dev/null
+++ b/gaussian/MathObj/Gaussian/Example.hs
@@ -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)]
diff --git a/gaussian/MathObj/Gaussian/ExponentTuple.hs b/gaussian/MathObj/Gaussian/ExponentTuple.hs
new file mode 100644
--- /dev/null
+++ b/gaussian/MathObj/Gaussian/ExponentTuple.hs
@@ -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 aa.
+ 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 aa.
+ 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
diff --git a/gaussian/MathObj/Gaussian/Polynomial.hs b/gaussian/MathObj/Gaussian/Polynomial.hs
new file mode 100644
--- /dev/null
+++ b/gaussian/MathObj/Gaussian/Polynomial.hs
@@ -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
+-}
diff --git a/gaussian/MathObj/Gaussian/Variance.hs b/gaussian/MathObj/Gaussian/Variance.hs
new file mode 100644
--- /dev/null
+++ b/gaussian/MathObj/Gaussian/Variance.hs
@@ -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
diff --git a/numeric-prelude.cabal b/numeric-prelude.cabal
--- a/numeric-prelude.cabal
+++ b/numeric-prelude.cabal
@@ -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 , Henning Thielemann , Mikael Johansson
Maintainer: Henning Thielemann
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
diff --git a/playground/Number/ComplexSquareRoot.hs b/playground/Number/ComplexSquareRoot.hs
new file mode 100644
--- /dev/null
+++ b/playground/Number/ComplexSquareRoot.hs
@@ -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 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)
diff --git a/src/Algebra/Absolute.hs b/src/Algebra/Absolute.hs
--- a/src/Algebra/Absolute.hs
+++ b/src/Algebra/Absolute.hs
@@ -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, )
diff --git a/src/Algebra/Additive.hs b/src/Algebra/Additive.hs
--- a/src/Algebra/Additive.hs
+++ b/src/Algebra/Additive.hs
@@ -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
diff --git a/src/Algebra/Algebraic.hs b/src/Algebra/Algebraic.hs
--- a/src/Algebra/Algebraic.hs
+++ b/src/Algebra/Algebraic.hs
@@ -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)
diff --git a/src/Algebra/Differential.hs b/src/Algebra/Differential.hs
--- a/src/Algebra/Differential.hs
+++ b/src/Algebra/Differential.hs
@@ -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
diff --git a/src/Algebra/DimensionTerm.hs b/src/Algebra/DimensionTerm.hs
--- a/src/Algebra/DimensionTerm.hs
+++ b/src/Algebra/DimensionTerm.hs
@@ -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,
diff --git a/src/Algebra/DivisibleSpace.hs b/src/Algebra/DivisibleSpace.hs
--- a/src/Algebra/DivisibleSpace.hs
+++ b/src/Algebra/DivisibleSpace.hs
@@ -1,4 +1,4 @@
-{-# LANGUAGE NoImplicitPrelude #-}
+{-# LANGUAGE RebindableSyntax #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FlexibleInstances #-}
module Algebra.DivisibleSpace where
diff --git a/src/Algebra/Field.hs b/src/Algebra/Field.hs
--- a/src/Algebra/Field.hs
+++ b/src/Algebra/Field.hs
@@ -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./)
diff --git a/src/Algebra/FloatingPoint.hs b/src/Algebra/FloatingPoint.hs
new file mode 100644
--- /dev/null
+++ b/src/Algebra/FloatingPoint.hs
@@ -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
diff --git a/src/Algebra/IntegralDomain.hs b/src/Algebra/IntegralDomain.hs
--- a/src/Algebra/IntegralDomain.hs
+++ b/src/Algebra/IntegralDomain.hs
@@ -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
diff --git a/src/Algebra/Lattice.hs b/src/Algebra/Lattice.hs
--- a/src/Algebra/Lattice.hs
+++ b/src/Algebra/Lattice.hs
@@ -1,4 +1,4 @@
-{-# LANGUAGE NoImplicitPrelude #-}
+{-# LANGUAGE RebindableSyntax #-}
module Algebra.Lattice (
C(up, dn)
, max, min, abs
diff --git a/src/Algebra/Module.hs b/src/Algebra/Module.hs
--- a/src/Algebra/Module.hs
+++ b/src/Algebra/Module.hs
@@ -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 -}
diff --git a/src/Algebra/ModuleBasis.hs b/src/Algebra/ModuleBasis.hs
--- a/src/Algebra/ModuleBasis.hs
+++ b/src/Algebra/ModuleBasis.hs
@@ -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:
diff --git a/src/Algebra/Monoid.hs b/src/Algebra/Monoid.hs
--- a/src/Algebra/Monoid.hs
+++ b/src/Algebra/Monoid.hs
@@ -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.
diff --git a/src/Algebra/NonNegative.hs b/src/Algebra/NonNegative.hs
--- a/src/Algebra/NonNegative.hs
+++ b/src/Algebra/NonNegative.hs
@@ -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, )
diff --git a/src/Algebra/NormedSpace/Euclidean.hs b/src/Algebra/NormedSpace/Euclidean.hs
--- a/src/Algebra/NormedSpace/Euclidean.hs
+++ b/src/Algebra/NormedSpace/Euclidean.hs
@@ -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
diff --git a/src/Algebra/NormedSpace/Maximum.hs b/src/Algebra/NormedSpace/Maximum.hs
--- a/src/Algebra/NormedSpace/Maximum.hs
+++ b/src/Algebra/NormedSpace/Maximum.hs
@@ -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)
diff --git a/src/Algebra/NormedSpace/Sum.hs b/src/Algebra/NormedSpace/Sum.hs
--- a/src/Algebra/NormedSpace/Sum.hs
+++ b/src/Algebra/NormedSpace/Sum.hs
@@ -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
diff --git a/src/Algebra/OccasionallyScalar.hs b/src/Algebra/OccasionallyScalar.hs
--- a/src/Algebra/OccasionallyScalar.hs
+++ b/src/Algebra/OccasionallyScalar.hs
@@ -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)
diff --git a/src/Algebra/PrincipalIdealDomain.hs b/src/Algebra/PrincipalIdealDomain.hs
--- a/src/Algebra/PrincipalIdealDomain.hs
+++ b/src/Algebra/PrincipalIdealDomain.hs
@@ -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 =
diff --git a/src/Algebra/RealField.hs b/src/Algebra/RealField.hs
--- a/src/Algebra/RealField.hs
+++ b/src/Algebra/RealField.hs
@@ -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, )
{- |
diff --git a/src/Algebra/RealIntegral.hs b/src/Algebra/RealIntegral.hs
--- a/src/Algebra/RealIntegral.hs
+++ b/src/Algebra/RealIntegral.hs
@@ -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, )
diff --git a/src/Algebra/RealRing.hs b/src/Algebra/RealRing.hs
--- a/src/Algebra/RealRing.hs
+++ b/src/Algebra/RealRing.hs
@@ -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
diff --git a/src/Algebra/RealTranscendental.hs b/src/Algebra/RealTranscendental.hs
--- a/src/Algebra/RealTranscendental.hs
+++ b/src/Algebra/RealTranscendental.hs
@@ -1,4 +1,4 @@
-{-# LANGUAGE NoImplicitPrelude #-}
+{-# LANGUAGE RebindableSyntax #-}
module Algebra.RealTranscendental where
import qualified Algebra.Transcendental as Trans
diff --git a/src/Algebra/RightModule.hs b/src/Algebra/RightModule.hs
--- a/src/Algebra/RightModule.hs
+++ b/src/Algebra/RightModule.hs
@@ -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?
diff --git a/src/Algebra/Ring.hs b/src/Algebra/Ring.hs
--- a/src/Algebra/Ring.hs
+++ b/src/Algebra/Ring.hs
@@ -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.*)
diff --git a/src/Algebra/ToInteger.hs b/src/Algebra/ToInteger.hs
--- a/src/Algebra/ToInteger.hs
+++ b/src/Algebra/ToInteger.hs
@@ -49,6 +49,7 @@
toInteger :: a -> Integer
+{-# NOINLINE [2] fromIntegral #-}
fromIntegral :: (C a, Ring.C b) => a -> b
fromIntegral = fromInteger . toInteger
diff --git a/src/Algebra/ToRational.hs b/src/Algebra/ToRational.hs
--- a/src/Algebra/ToRational.hs
+++ b/src/Algebra/ToRational.hs
@@ -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
diff --git a/src/Algebra/Transcendental.hs b/src/Algebra/Transcendental.hs
--- a/src/Algebra/Transcendental.hs
+++ b/src/Algebra/Transcendental.hs
@@ -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
diff --git a/src/Algebra/Units.hs b/src/Algebra/Units.hs
--- a/src/Algebra/Units.hs
+++ b/src/Algebra/Units.hs
@@ -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
diff --git a/src/Algebra/Vector.hs b/src/Algebra/Vector.hs
--- a/src/Algebra/Vector.hs
+++ b/src/Algebra/Vector.hs
@@ -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
diff --git a/src/Algebra/VectorSpace.hs b/src/Algebra/VectorSpace.hs
--- a/src/Algebra/VectorSpace.hs
+++ b/src/Algebra/VectorSpace.hs
@@ -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)
diff --git a/src/Algebra/ZeroTestable.hs b/src/Algebra/ZeroTestable.hs
--- a/src/Algebra/ZeroTestable.hs
+++ b/src/Algebra/ZeroTestable.hs
@@ -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
diff --git a/src/MathObj/Algebra.hs b/src/MathObj/Algebra.hs
--- a/src/MathObj/Algebra.hs
+++ b/src/MathObj/Algebra.hs
@@ -1,4 +1,4 @@
-{-# LANGUAGE NoImplicitPrelude #-}
+{-# LANGUAGE RebindableSyntax #-}
{- |
Copyright : (c) Mikael Johansson 2006
Maintainer : mik@math.uni-jena.de
diff --git a/src/MathObj/DiscreteMap.hs b/src/MathObj/DiscreteMap.hs
--- a/src/MathObj/DiscreteMap.hs
+++ b/src/MathObj/DiscreteMap.hs
@@ -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?
diff --git a/src/MathObj/Gaussian/Bell.hs b/src/MathObj/Gaussian/Bell.hs
deleted file mode 100644
--- a/src/MathObj/Gaussian/Bell.hs
+++ /dev/null
@@ -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
--}
diff --git a/src/MathObj/Gaussian/Example.hs b/src/MathObj/Gaussian/Example.hs
deleted file mode 100644
--- a/src/MathObj/Gaussian/Example.hs
+++ /dev/null
@@ -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)]
diff --git a/src/MathObj/Gaussian/Polynomial.hs b/src/MathObj/Gaussian/Polynomial.hs
deleted file mode 100644
--- a/src/MathObj/Gaussian/Polynomial.hs
+++ /dev/null
@@ -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)
--}
diff --git a/src/MathObj/Gaussian/Variance.hs b/src/MathObj/Gaussian/Variance.hs
deleted file mode 100644
--- a/src/MathObj/Gaussian/Variance.hs
+++ /dev/null
@@ -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
--}
diff --git a/src/MathObj/LaurentPolynomial.hs b/src/MathObj/LaurentPolynomial.hs
--- a/src/MathObj/LaurentPolynomial.hs
+++ b/src/MathObj/LaurentPolynomial.hs
@@ -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, )
diff --git a/src/MathObj/Matrix.hs b/src/MathObj/Matrix.hs
--- a/src/MathObj/Matrix.hs
+++ b/src/MathObj/Matrix.hs
@@ -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) $
diff --git a/src/MathObj/Monoid.hs b/src/MathObj/Monoid.hs
--- a/src/MathObj/Monoid.hs
+++ b/src/MathObj/Monoid.hs
@@ -1,4 +1,4 @@
-{-# LANGUAGE NoImplicitPrelude #-}
+{-# LANGUAGE RebindableSyntax #-}
module MathObj.Monoid where
import qualified Algebra.PrincipalIdealDomain as PID
diff --git a/src/MathObj/PartialFraction.hs b/src/MathObj/PartialFraction.hs
--- a/src/MathObj/PartialFraction.hs
+++ b/src/MathObj/PartialFraction.hs
@@ -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
diff --git a/src/MathObj/Permutation.hs b/src/MathObj/Permutation.hs
--- a/src/MathObj/Permutation.hs
+++ b/src/MathObj/Permutation.hs
@@ -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
{- |
diff --git a/src/MathObj/Permutation/CycleList.hs b/src/MathObj/Permutation/CycleList.hs
--- a/src/MathObj/Permutation/CycleList.hs
+++ b/src/MathObj/Permutation/CycleList.hs
@@ -1,4 +1,4 @@
-{-# LANGUAGE NoImplicitPrelude #-}
+{-# LANGUAGE RebindableSyntax #-}
{- |
Copyright : (c) Mikael Johansson 2006
Maintainer : mik@math.uni-jena.de
diff --git a/src/MathObj/Permutation/CycleList/Check.hs b/src/MathObj/Permutation/CycleList/Check.hs
--- a/src/MathObj/Permutation/CycleList/Check.hs
+++ b/src/MathObj/Permutation/CycleList/Check.hs
@@ -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)
{- |
diff --git a/src/MathObj/Permutation/Table.hs b/src/MathObj/Permutation/Table.hs
--- a/src/MathObj/Permutation/Table.hs
+++ b/src/MathObj/Permutation/Table.hs
@@ -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)
diff --git a/src/MathObj/Polynomial.hs b/src/MathObj/Polynomial.hs
--- a/src/MathObj/Polynomial.hs
+++ b/src/MathObj/Polynomial.hs
@@ -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 "(/)"
diff --git a/src/MathObj/Polynomial/Core.hs b/src/MathObj/Polynomial/Core.hs
--- a/src/MathObj/Polynomial/Core.hs
+++ b/src/MathObj/Polynomial/Core.hs
@@ -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
{-
diff --git a/src/MathObj/PowerSeries.hs b/src/MathObj/PowerSeries.hs
--- a/src/MathObj/PowerSeries.hs
+++ b/src/MathObj/PowerSeries.hs
@@ -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
diff --git a/src/MathObj/PowerSeries/Core.hs b/src/MathObj/PowerSeries/Core.hs
--- a/src/MathObj/PowerSeries/Core.hs
+++ b/src/MathObj/PowerSeries/Core.hs
@@ -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
diff --git a/src/MathObj/PowerSeries/DifferentialEquation.hs b/src/MathObj/PowerSeries/DifferentialEquation.hs
--- a/src/MathObj/PowerSeries/DifferentialEquation.hs
+++ b/src/MathObj/PowerSeries/DifferentialEquation.hs
@@ -1,4 +1,4 @@
-{-# LANGUAGE NoImplicitPrelude #-}
+{-# LANGUAGE RebindableSyntax #-}
{- |
Lazy evaluation allows for the solution
of differential equations in terms of power series.
diff --git a/src/MathObj/PowerSeries/Example.hs b/src/MathObj/PowerSeries/Example.hs
--- a/src/MathObj/PowerSeries/Example.hs
+++ b/src/MathObj/PowerSeries/Example.hs
@@ -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)
diff --git a/src/MathObj/PowerSeries/Mean.hs b/src/MathObj/PowerSeries/Mean.hs
--- a/src/MathObj/PowerSeries/Mean.hs
+++ b/src/MathObj/PowerSeries/Mean.hs
@@ -1,4 +1,4 @@
-{-# LANGUAGE NoImplicitPrelude #-}
+{-# LANGUAGE RebindableSyntax #-}
{- |
This module computes power series for
representing some means as generalized $f$-means.
diff --git a/src/MathObj/PowerSeries2.hs b/src/MathObj/PowerSeries2.hs
--- a/src/MathObj/PowerSeries2.hs
+++ b/src/MathObj/PowerSeries2.hs
@@ -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
diff --git a/src/MathObj/PowerSeries2/Core.hs b/src/MathObj/PowerSeries2/Core.hs
--- a/src/MathObj/PowerSeries2/Core.hs
+++ b/src/MathObj/PowerSeries2/Core.hs
@@ -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
diff --git a/src/MathObj/PowerSum.hs b/src/MathObj/PowerSum.hs
--- a/src/MathObj/PowerSum.hs
+++ b/src/MathObj/PowerSum.hs
@@ -1,4 +1,4 @@
-{-# LANGUAGE NoImplicitPrelude #-}
+{-# LANGUAGE RebindableSyntax #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FlexibleInstances #-}
{- |
diff --git a/src/MathObj/RefinementMask2.hs b/src/MathObj/RefinementMask2.hs
--- a/src/MathObj/RefinementMask2.hs
+++ b/src/MathObj/RefinementMask2.hs
@@ -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
diff --git a/src/MathObj/RootSet.hs b/src/MathObj/RootSet.hs
--- a/src/MathObj/RootSet.hs
+++ b/src/MathObj/RootSet.hs
@@ -1,4 +1,4 @@
-{-# LANGUAGE NoImplicitPrelude #-}
+{-# LANGUAGE RebindableSyntax #-}
{- |
Copyright : (c) Henning Thielemann 2004-2005
diff --git a/src/MathObj/Wrapper/Haskell98.hs b/src/MathObj/Wrapper/Haskell98.hs
--- a/src/MathObj/Wrapper/Haskell98.hs
+++ b/src/MathObj/Wrapper/Haskell98.hs
@@ -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
diff --git a/src/MathObj/Wrapper/NumericPrelude.hs b/src/MathObj/Wrapper/NumericPrelude.hs
--- a/src/MathObj/Wrapper/NumericPrelude.hs
+++ b/src/MathObj/Wrapper/NumericPrelude.hs
@@ -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
diff --git a/src/Number/Complex.hs b/src/Number/Complex.hs
--- a/src/Number/Complex.hs
+++ b/src/Number/Complex.hs
@@ -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./)
diff --git a/src/Number/ComplexSquareRoot.hs b/src/Number/ComplexSquareRoot.hs
deleted file mode 100644
--- a/src/Number/ComplexSquareRoot.hs
+++ /dev/null
@@ -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 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)
diff --git a/src/Number/DimensionTerm.hs b/src/Number/DimensionTerm.hs
--- a/src/Number/DimensionTerm.hs
+++ b/src/Number/DimensionTerm.hs
@@ -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".
-}
diff --git a/src/Number/DimensionTerm/SI.hs b/src/Number/DimensionTerm/SI.hs
--- a/src/Number/DimensionTerm/SI.hs
+++ b/src/Number/DimensionTerm/SI.hs
@@ -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
diff --git a/src/Number/FixedPoint.hs b/src/Number/FixedPoint.hs
--- a/src/Number/FixedPoint.hs
+++ b/src/Number/FixedPoint.hs
@@ -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
diff --git a/src/Number/FixedPoint/Check.hs b/src/Number/FixedPoint/Check.hs
--- a/src/Number/FixedPoint/Check.hs
+++ b/src/Number/FixedPoint/Check.hs
@@ -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
+ (/) = (/)
diff --git a/src/Number/GaloisField2p32m5.hs b/src/Number/GaloisField2p32m5.hs
--- a/src/Number/GaloisField2p32m5.hs
+++ b/src/Number/GaloisField2p32m5.hs
@@ -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
diff --git a/src/Number/NonNegative.hs b/src/Number/NonNegative.hs
--- a/src/Number/NonNegative.hs
+++ b/src/Number/NonNegative.hs
@@ -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
diff --git a/src/Number/NonNegativeChunky.hs b/src/Number/NonNegativeChunky.hs
--- a/src/Number/NonNegativeChunky.hs
+++ b/src/Number/NonNegativeChunky.hs
@@ -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 []
diff --git a/src/Number/OccasionallyScalarExpression.hs b/src/Number/OccasionallyScalarExpression.hs
--- a/src/Number/OccasionallyScalarExpression.hs
+++ b/src/Number/OccasionallyScalarExpression.hs
@@ -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.
diff --git a/src/Number/PartiallyTranscendental.hs b/src/Number/PartiallyTranscendental.hs
--- a/src/Number/PartiallyTranscendental.hs
+++ b/src/Number/PartiallyTranscendental.hs
@@ -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./)
diff --git a/src/Number/Peano.hs b/src/Number/Peano.hs
--- a/src/Number/Peano.hs
+++ b/src/Number/Peano.hs
@@ -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
diff --git a/src/Number/Physical.hs b/src/Number/Physical.hs
--- a/src/Number/Physical.hs
+++ b/src/Number/Physical.hs
@@ -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
diff --git a/src/Number/Physical/Read.hs b/src/Number/Physical/Read.hs
--- a/src/Number/Physical/Read.hs
+++ b/src/Number/Physical/Read.hs
@@ -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)
diff --git a/src/Number/Physical/Show.hs b/src/Number/Physical/Show.hs
--- a/src/Number/Physical/Show.hs
+++ b/src/Number/Physical/Show.hs
@@ -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.
-}
diff --git a/src/Number/Physical/Unit.hs b/src/Number/Physical/Unit.hs
--- a/src/Number/Physical/Unit.hs
+++ b/src/Number/Physical/Unit.hs
@@ -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
-}
diff --git a/src/Number/Physical/UnitDatabase.hs b/src/Number/Physical/UnitDatabase.hs
--- a/src/Number/Physical/UnitDatabase.hs
+++ b/src/Number/Physical/UnitDatabase.hs
@@ -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)
diff --git a/src/Number/Positional.hs b/src/Number/Positional.hs
--- a/src/Number/Positional.hs
+++ b/src/Number/Positional.hs
@@ -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 .
@@ -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)
diff --git a/src/Number/Positional/Check.hs b/src/Number/Positional/Check.hs
--- a/src/Number/Positional/Check.hs
+++ b/src/Number/Positional/Check.hs
@@ -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
+ (/) = (/)
{-
diff --git a/src/Number/Quaternion.hs b/src/Number/Quaternion.hs
--- a/src/Number/Quaternion.hs
+++ b/src/Number/Quaternion.hs
@@ -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)
diff --git a/src/Number/Ratio.hs b/src/Number/Ratio.hs
--- a/src/Number/Ratio.hs
+++ b/src/Number/Ratio.hs
@@ -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
+-}
diff --git a/src/Number/ResidueClass.hs b/src/Number/ResidueClass.hs
--- a/src/Number/ResidueClass.hs
+++ b/src/Number/ResidueClass.hs
@@ -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)
diff --git a/src/Number/ResidueClass/Check.hs b/src/Number/ResidueClass/Check.hs
--- a/src/Number/ResidueClass/Check.hs
+++ b/src/Number/ResidueClass/Check.hs
@@ -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"
diff --git a/src/Number/ResidueClass/Func.hs b/src/Number/ResidueClass/Func.hs
--- a/src/Number/ResidueClass/Func.hs
+++ b/src/Number/ResidueClass/Func.hs
@@ -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"
diff --git a/src/Number/ResidueClass/Maybe.hs b/src/Number/ResidueClass/Maybe.hs
--- a/src/Number/ResidueClass/Maybe.hs
+++ b/src/Number/ResidueClass/Maybe.hs
@@ -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
diff --git a/src/Number/ResidueClass/Reader.hs b/src/Number/ResidueClass/Reader.hs
--- a/src/Number/ResidueClass/Reader.hs
+++ b/src/Number/ResidueClass/Reader.hs
@@ -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
diff --git a/src/Number/SI.hs b/src/Number/SI.hs
--- a/src/Number/SI.hs
+++ b/src/Number/SI.hs
@@ -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./)
diff --git a/src/Number/SI/Unit.hs b/src/Number/SI/Unit.hs
--- a/src/Number/SI/Unit.hs
+++ b/src/Number/SI/Unit.hs
@@ -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
-}
diff --git a/src/NumericPrelude/Base.hs b/src/NumericPrelude/Base.hs
--- a/src/NumericPrelude/Base.hs
+++ b/src/NumericPrelude/Base.hs
@@ -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
diff --git a/src/NumericPrelude/List.hs b/src/NumericPrelude/List.hs
--- a/src/NumericPrelude/List.hs
+++ b/src/NumericPrelude/List.hs
@@ -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
diff --git a/src/NumericPrelude/List/Checked.hs b/src/NumericPrelude/List/Checked.hs
--- a/src/NumericPrelude/List/Checked.hs
+++ b/src/NumericPrelude/List/Checked.hs
@@ -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, (-), )
diff --git a/src/NumericPrelude/List/Generic.hs b/src/NumericPrelude/List/Generic.hs
--- a/src/NumericPrelude/List/Generic.hs
+++ b/src/NumericPrelude/List/Generic.hs
@@ -1,4 +1,4 @@
-{-# LANGUAGE NoImplicitPrelude #-}
+{-# LANGUAGE RebindableSyntax #-}
{- |
Functions that are counterparts of the @generic@ functions in "Data.List"
using NumericPrelude.Numeric type classes.
diff --git a/src/NumericPrelude/Numeric.hs b/src/NumericPrelude/Numeric.hs
--- a/src/NumericPrelude/Numeric.hs
+++ b/src/NumericPrelude/Numeric.hs
@@ -1,4 +1,4 @@
-{-# LANGUAGE NoImplicitPrelude #-}
+{-# LANGUAGE RebindableSyntax #-}
module NumericPrelude.Numeric (
{- Additive -} (+), (-), negate, zero, subtract, sum, sum1,
{- ZeroTestable -} isZero,
diff --git a/test/Demo.hs b/test/Demo.hs
new file mode 100644
--- /dev/null
+++ b/test/Demo.hs
@@ -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
diff --git a/test/Gaussian.hs b/test/Gaussian.hs
deleted file mode 100644
--- a/test/Gaussian.hs
+++ /dev/null
@@ -1,6 +0,0 @@
-module Main where
-
-import qualified MathObj.Gaussian.Example as Example
-
-main :: IO ()
-main = Example.polyApprox
diff --git a/test/Test.hs b/test/Test.hs
deleted file mode 100644
--- a/test/Test.hs
+++ /dev/null
@@ -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
diff --git a/test/Test/Algebra/Additive.hs b/test/Test/Algebra/Additive.hs
--- a/test/Test/Algebra/Additive.hs
+++ b/test/Test/Algebra/Additive.hs
@@ -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))
diff --git a/test/Test/Algebra/IntegralDomain.hs b/test/Test/Algebra/IntegralDomain.hs
--- a/test/Test/Algebra/IntegralDomain.hs
+++ b/test/Test/Algebra/IntegralDomain.hs
@@ -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))
diff --git a/test/Test/Algebra/PrincipalIdealDomain.hs b/test/Test/Algebra/PrincipalIdealDomain.hs
new file mode 100644
--- /dev/null
+++ b/test/Test/Algebra/PrincipalIdealDomain.hs
@@ -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)"]]
diff --git a/test/Test/Algebra/RealRing.hs b/test/Test/Algebra/RealRing.hs
--- a/test/Test/Algebra/RealRing.hs
+++ b/test/Test/Algebra/RealRing.hs
@@ -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)))
diff --git a/test/Test/MathObj/Gaussian/Bell.hs b/test/Test/MathObj/Gaussian/Bell.hs
--- a/test/Test/MathObj/Gaussian/Bell.hs
+++ b/test/Test/MathObj/Gaussian/Bell.hs
@@ -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)
diff --git a/test/Test/MathObj/Gaussian/ExponentTuple.hs b/test/Test/MathObj/Gaussian/ExponentTuple.hs
new file mode 100644
--- /dev/null
+++ b/test/Test/MathObj/Gaussian/ExponentTuple.hs
@@ -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)
diff --git a/test/Test/MathObj/Gaussian/Polynomial.hs b/test/Test/MathObj/Gaussian/Polynomial.hs
--- a/test/Test/MathObj/Gaussian/Polynomial.hs
+++ b/test/Test/MathObj/Gaussian/Polynomial.hs
@@ -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)
diff --git a/test/Test/MathObj/Gaussian/Variance.hs b/test/Test/MathObj/Gaussian/Variance.hs
--- a/test/Test/MathObj/Gaussian/Variance.hs
+++ b/test/Test/MathObj/Gaussian/Variance.hs
@@ -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 aa.
- 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 aa.
- 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)
diff --git a/test/Test/MathObj/Matrix.hs b/test/Test/MathObj/Matrix.hs
--- a/test/Test/MathObj/Matrix.hs
+++ b/test/Test/MathObj/Matrix.hs
@@ -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))
diff --git a/test/Test/MathObj/PartialFraction.hs b/test/Test/MathObj/PartialFraction.hs
--- a/test/Test/MathObj/PartialFraction.hs
+++ b/test/Test/MathObj/PartialFraction.hs
@@ -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)
diff --git a/test/Test/MathObj/Polynomial.hs b/test/Test/MathObj/Polynomial.hs
--- a/test/Test/MathObj/Polynomial.hs
+++ b/test/Test/MathObj/Polynomial.hs
@@ -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)
diff --git a/test/Test/MathObj/Polynomial/Core.hs b/test/Test/MathObj/Polynomial/Core.hs
new file mode 100644
--- /dev/null
+++ b/test/Test/MathObj/Polynomial/Core.hs
@@ -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)
diff --git a/test/Test/MathObj/PowerSeries.hs b/test/Test/MathObj/PowerSeries.hs
--- a/test/Test/MathObj/PowerSeries.hs
+++ b/test/Test/MathObj/PowerSeries.hs
@@ -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))
diff --git a/test/Test/MathObj/PowerSeries/Core.hs b/test/Test/MathObj/PowerSeries/Core.hs
new file mode 100644
--- /dev/null
+++ b/test/Test/MathObj/PowerSeries/Core.hs
@@ -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)
diff --git a/test/Test/MathObj/PowerSeries/Example.hs b/test/Test/MathObj/PowerSeries/Example.hs
new file mode 100644
--- /dev/null
+++ b/test/Test/MathObj/PowerSeries/Example.hs
@@ -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))
diff --git a/test/Test/MathObj/RefinementMask2.hs b/test/Test/MathObj/RefinementMask2.hs
--- a/test/Test/MathObj/RefinementMask2.hs
+++ b/test/Test/MathObj/RefinementMask2.hs
@@ -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]"]]
diff --git a/test/Test/Number/ComplexSquareRoot.hs b/test/Test/Number/ComplexSquareRoot.hs
--- a/test/Test/Number/ComplexSquareRoot.hs
+++ b/test/Test/Number/ComplexSquareRoot.hs
@@ -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))
diff --git a/test/Test/Number/GaloisField2p32m5.hs b/test/Test/Number/GaloisField2p32m5.hs
--- a/test/Test/Number/GaloisField2p32m5.hs
+++ b/test/Test/Number/GaloisField2p32m5.hs
@@ -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)
diff --git a/test/Test/NumericPrelude/Utility.hs b/test/Test/NumericPrelude/Utility.hs
--- a/test/Test/NumericPrelude/Utility.hs
+++ b/test/Test/NumericPrelude/Utility.hs
@@ -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
diff --git a/test/Test/Run.hs b/test/Test/Run.hs
--- a/test/Test/Run.hs
+++ b/test/Test/Run.hs
@@ -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