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numeric-prelude 0.4.0.1 → 0.4.0.2

raw patch · 32 files changed

+1435/−2185 lines, 32 filesdep +numeric-preludedep ~QuickCheckdep ~basedep ~random

Dependencies added: numeric-prelude

Dependency ranges changed: QuickCheck, base, random, utility-ht

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+ gaussian/Gaussian.hs view
@@ -0,0 +1,6 @@+module Main where++import qualified MathObj.Gaussian.Example as Example++main :: IO ()+main = Example.polyApprox
+ gaussian/MathObj/Gaussian/Bell.hs view
@@ -0,0 +1,324 @@+{-# 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+-}
+ gaussian/MathObj/Gaussian/Example.hs view
@@ -0,0 +1,231 @@+{-# 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)]
+ gaussian/MathObj/Gaussian/Polynomial.hs view
@@ -0,0 +1,480 @@+{-# 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) $+   zipWith h+      (laurentAbscissas unit_ y)+      (LPoly.coeffs y)++laurentAbscissas :: Ring.C a => a -> LPoly.T c -> [a]+laurentAbscissas unit_ =+   map (\d -> fromIntegral d * unit_) .+   iterate (1+) . LPoly.expon+++{- No Ring instance for Gaussians+instance (Ring.C a) => Differential.C (T a) where+   differentiate = differentiate+-}++{- laws+differentiate (f*g) =+   (differentiate f) * g + f * (differentiate g)+-}
+ gaussian/MathObj/Gaussian/Variance.hs view
@@ -0,0 +1,206 @@+{-# LANGUAGE NoImplicitPrelude #-}+{-+We represent a Gaussian bell curve in terms of the reciprocal of its variance+and its value at the origin.++We could do some projective geometry in the exponent+in order to also have zero variance,+which corresponds to the dirac impulse.++The Gaussians form a nice multiplicative commutative monoid.+Maybe we should have such a structure.+It would also be useful for the Root data type+and a new Exponential data type.+-}+module MathObj.Gaussian.Variance where++import qualified MathObj.Polynomial as Poly+import qualified Number.Root as Root++import qualified Algebra.Transcendental as Trans+import qualified Algebra.Algebraic      as Algebraic+import qualified Algebra.Field          as Field+import qualified Algebra.Absolute       as Absolute+import qualified Algebra.Ring           as Ring+import qualified Algebra.Additive       as Additive++{-+import Algebra.Transcendental (pi, )+import Algebra.Ring ((*), (^), )+import Algebra.Additive ((+))+-}+import Test.QuickCheck (Arbitrary, arbitrary, )+import Control.Monad (liftM2, )++-- import Prelude (($))+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+-}
numeric-prelude.cabal view
@@ -1,6 +1,6 @@ Name:           numeric-prelude-Version:        0.4.0.1-License:        GPL+Version:        0.4.0.2+License:        BSD3 License-File:   LICENSE Author:         Dylan Thurston <dpt@math.harvard.edu>, Henning Thielemann <numericprelude@henning-thielemann.de>, Mikael Johansson Maintainer:     Henning Thielemann <numericprelude@henning-thielemann.de>@@ -9,7 +9,7 @@ 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, GHC==7.6.3-Cabal-Version:  >=1.6+Cabal-Version:  >=1.8 Build-Type:     Simple Synopsis:       An experimental alternative hierarchy of numeric type classes Description:@@ -79,7 +79,7 @@   .   Write modules in the following style:   .-  > [-# NoImplicitPrelude #-]+  > [-# LANGUAGE NoImplicitPrelude #-]   > module MyModule where   >   > ... various specific imports ...@@ -146,15 +146,12 @@   docs/README   src/Algebra/GenerateRules.hs -Flag splitBase-  description: Choose the new smaller, split-up base package.- Flag buildTests   description: Build test executables   default:     False  Source-Repository this-  Tag:         0.4.0.1+  Tag:         0.4.0.2   Type:        darcs   Location:    http://code.haskell.org/numeric-prelude/ @@ -170,15 +167,14 @@     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 +  -- splitBase+  Build-Depends:+    array >=0.1 && <0.5,+    containers >=0.1 && <0.6,+    random >=1.0 && <1.1,+    base >= 2 && <5+   If impl(ghc>=7.0)     CPP-Options: -DNoImplicitPrelude=RebindableSyntax     Extensions: CPP@@ -282,29 +278,29 @@     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+  Hs-Source-Dirs: test   GHC-Options:    -Wall   Main-Is: Test.hs -  If !flag(buildTests)-    Buildable:         False+  If flag(buildTests)+    Build-Depends:+      numeric-prelude,+      base+  Else+    Buildable: False    If impl(ghc>=7.0)     CPP-Options: -DNoImplicitPrelude=RebindableSyntax     Extensions: CPP  Executable testsuite-  Hs-Source-Dirs: src, test+  Hs-Source-Dirs: test, gaussian   GHC-Options:    -Wall   Other-modules:     Test.NumericPrelude.Utility@@ -321,10 +317,17 @@     Test.MathObj.Gaussian.Variance     Test.MathObj.Gaussian.Bell     Test.MathObj.Gaussian.Polynomial+    Number.ComplexSquareRoot   Main-Is: Test/Run.hs    If flag(buildTests)-    Build-Depends: HUnit >=1 && <2+    Build-Depends:+      HUnit >=1 && <2,+      numeric-prelude,+      QuickCheck,+      utility-ht,+      random,+      base   Else     Buildable: False @@ -333,14 +336,22 @@     Extensions: CPP  Executable test-gaussian-  Hs-Source-Dirs: src, test+  Hs-Source-Dirs: gaussian   Main-Is: Gaussian.hs   Other-Modules:     MathObj.Gaussian.Example+    MathObj.Gaussian.Variance+    MathObj.Gaussian.Bell+    MathObj.Gaussian.Polynomial+   If flag(buildTests)     Build-Depends:       gnuplot >=0.5 && <0.6,-      HTam >=0.0.2 && <0.1+      HTam >=0.0.2 && <0.1,+      numeric-prelude,+      QuickCheck,+      utility-ht,+      base   Else     Buildable: False
src/Algebra/DimensionTerm.hs view
@@ -1,12 +1,4 @@ {- |-Copyright   :  (c) Henning Thielemann 2008-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable-- We already have the dynamically checked physical units provided by "Number.Physical" and the statically checked ones of the @dimensional@ package of Buckwalter,
src/Algebra/NormedSpace/Euclidean.hs view
@@ -3,13 +3,6 @@ {-# LANGUAGE FlexibleInstances #-}  {- |-Copyright   :  (c) Henning Thielemann 2005-2010-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  requires multi-parameter type classes- Abstraction of normed vector spaces -}
src/Algebra/NormedSpace/Maximum.hs view
@@ -3,13 +3,6 @@ {-# LANGUAGE FlexibleInstances #-}  {- |-Copyright   :  (c) Henning Thielemann 2005-2010-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  requires multi-parameter type classes- Abstraction of normed vector spaces -}
src/Algebra/NormedSpace/Sum.hs view
@@ -3,13 +3,6 @@ {-# LANGUAGE FlexibleInstances #-}  {- |-Copyright   :  (c) Henning Thielemann 2005-2010-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  requires multi-parameter type classes- Abstraction of normed vector spaces -}
− src/MathObj/Gaussian/Bell.hs 
@@ -1,324 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{--Complex translated and modulated Gaussian bell curve.--It could be extended to chirps-using a complex valued quadratic term with (real c >= 0).-This allows for a new test:-Express the Fourier transform in terms of a convolution with a chirp.--}-module MathObj.Gaussian.Bell where--import qualified MathObj.Polynomial as Poly-import qualified Number.Complex as Complex--import qualified Algebra.Transcendental as Trans-import qualified Algebra.Field          as Field-import qualified Algebra.Absolute       as Absolute-import qualified Algebra.Ring           as Ring-import qualified Algebra.Additive       as Additive--import Number.Complex ((+:), )--import Test.QuickCheck (Arbitrary, arbitrary, )-import Control.Monad (liftM4, )---- import Prelude (($))-import NumericPrelude.Numeric-import NumericPrelude.Base hiding (reverse, )---data T a = Cons {amp :: a, c0, c1 :: Complex.T a, c2 :: a}-   deriving (Eq, Show)--instance (Absolute.C a, Arbitrary a) => Arbitrary (T a) where-   arbitrary =-      liftM4-         (\k a b c -> Cons (abs k) a b (1 + abs c))-         arbitrary arbitrary arbitrary arbitrary---constant :: Ring.C a => T a-constant = Cons one zero zero zero--{- |-eigenfunction of 'fourier'--}-unit :: Ring.C a => T a-unit = Cons one zero zero one--{-# INLINE evaluate #-}-evaluate :: (Trans.C a) =>-   T a -> a -> Complex.T a-evaluate f x =-   Complex.scale-     (sqrt (amp f))-     (Complex.exp $ Complex.scale (-pi) $-      c0 f + Complex.scale x (c1 f) + Complex.fromReal (c2 f * x^2))--evaluateSqRt :: (Trans.C a) =>-   T a -> a -> Complex.T a-evaluateSqRt f x0 =-   Complex.scale-     (sqrt (amp f))-     (let x = sqrt pi * x0-      in  Complex.exp $ negate $-          c0 f + Complex.scale x (c1 f) + Complex.fromReal (c2 f * x^2))--exponentPolynomial :: (Additive.C a) =>-   T a -> Poly.T (Complex.T a)-exponentPolynomial f =-   Poly.fromCoeffs [c0 f, c1 f, Complex.fromReal (c2 f)]---{--norm functions depend on interpretation-and would have to return both a rational and transcendental part-expressed as @exp a@.--}--variance :: (Trans.C a) =>-   T a -> a-variance f =-   recip $ c2 f * 2*pi--multiply :: (Ring.C a) =>-   T a -> T a -> T a-multiply f g =-   Cons-      (amp f * amp g)-      (c0 f + c0 g) (c1 f + c1 g) (c2 f + c2 g)--powerRing :: (Trans.C a) =>-   Integer -> T a -> T a-powerRing p f =-   let pa = fromInteger p-   in  Cons-          (amp f ^ p)-          (pa * c0 f) (pa * c1 f) (fromInteger p * c2 f)--{--powerField does not makes sense,-since the reciprocal of a Gaussian diverges.--}--powerAlgebraic :: (Trans.C a) =>-   Rational -> T a -> T a-powerAlgebraic p f =-   let pa = fromRational' p-   in  Cons-          (amp f ^/ p)-          (pa * c0 f) (pa * c1 f) (fromRational' p * c2 f)--powerTranscendental :: (Trans.C a) =>-   a -> T a -> T a-powerTranscendental p f =-   Cons-      (amp f ^? p)-      (Complex.scale p $ c0 f) (Complex.scale p $ c1 f) (p * c2 f)---{--let x=Cons 2 (1+:3) (4+:5) (7::Rational); y=Cons 7 (1+:4) (3+:2) (5::Rational)--}-convolve :: (Field.C a) =>-   T a -> T a -> T a-convolve f g =-   let s = c2 f + c2 g-       {--       fd = f1/(2*f2)-       gd = g1/(2*g2)-       c = f2*g2/(f2+g2)--       c*(fd+gd) = (f1*g2+f2*g1)/(2*(f2+g2)) = b/2--       c*(fd+gd)^2 - fd^2*f2 - gd^2*g2-         = f2*g2*(fd+gd)^2/(f2 + g2) - (fd^2*f2 + gd^2*g2)-         = (f2*g2*(fd+gd)^2 - (f2+g2)*(fd^2*f2+gd^2*g2)) / (f2 + g2)-         = (2*f2*g2*fd*gd - (fd^2*f2^2+gd^2*g2^2)) / (f2 + g2)-         = (2*f1*g1 - (f1^2+g1^2)) / (4*(f2 + g2))-         = -(f1 - g1)^2/(4*(f2 + g2))-       -}-   in  Cons-          (amp f * amp g / s)-          (c0 f + c0 g-              - Complex.scale (recip (4*s)) ((c1 f - c1 g)^2))-          (Complex.scale (c2 g / s) (c1 f) +-           Complex.scale (c2 f / s) (c1 g))-          (c2 f * c2 g / s)-            -- recip $ recip (c2 f) + recip (c2 g)-{--   Cons-      (c0 f + c0 g) (c1 f + c1 g)-      (recip $ recip (c2 f) + recip (c2 g))--}--convolveByTranslation :: (Field.C a) =>-   T a -> T a -> T a-convolveByTranslation f0 g0 =-   let fd = Complex.scale (recip (2 * c2 f0)) $ c1 f0-       gd = Complex.scale (recip (2 * c2 g0)) $ c1 g0-       f1 = translateComplex fd f0-       g1 = translateComplex gd g0-       s = c2 f1 + c2 g1-   in  translateComplex (negate $ fd + gd) $-       Cons-          (amp f1 * amp g1 / s)-          (c0 f1 + c0 g1) zero-          (c2 f1 * c2 g1 / s)--convolveByFourier :: (Field.C a) =>-   T a -> T a -> T a-convolveByFourier f g =-   reverse $ fourier $ multiply (fourier f) (fourier g)--fourier :: (Field.C a) =>-   T a -> T a-fourier f =-   let a = c0 f-       b = c1 f-       rc = recip $ c2 f-   in  Cons-          (amp f * rc)-          (Complex.scale (rc/4) (-b^2) + a)-          (Complex.scale rc $ Complex.quarterRight b)-          rc--fourierByTranslation :: (Field.C a) =>-   T a -> T a-fourierByTranslation f =-   translateComplex (Complex.scale (1/2) $ Complex.quarterLeft $ c1 f) $-   Cons (amp f / c2 f) (c0 f) zero (recip $ c2 f)--{--a + b*x + c*x^2- = c*(a/c + b/c*x + x^2)- = c*((x-b/(2*c))^2 + (4*a*c+b^2)/(4*c^2))- = c*(x-b/(2*c))^2 + (4*a*c+b^2)/(4*c)--fourier ->-   x^2/c - i*b/c*x + (4*a*c+b^2)/(4*c)--fourier (x -> exp(-pi*c*(x-t)^2))- = fourier $ translate t $ shrink (sqrt c) $ x -> exp(-pi*x^2)- = modulate t $ dilate (sqrt c) $ fourier $ x -> exp(-pi*x^2)- = modulate t $ dilate (sqrt c) $ x -> exp(-pi*x^2)- = modulate t $ x -> exp(-pi*x^2/c)- = x -> exp(-pi*x^2/c) * exp(-2*pi*i*x*t)- = x -> exp(-pi*(x^2/c - 2*i*x*t))--}--{--b*x + c*x^2- = c*(b/c*x + x^2)- = c*((x-br/(2*c))^2 + i*x*bi/c - br^2/(4*c^2))- = c*(x-br/(2*c))^2 + i*x*bi - br^2/(4*c)--fourier ->-   (x+bi/2)^2/c - i*br/c*(x+bi/2) - br^2/(4*c)- = (1/c) * ((x+bi/2)^2 - i*br*(x+bi/2) + (br/2)^2)- = (1/c) * (x^2 - i*b*x + -(br/2)^2 + (bi/2)^2 - i*br*bi/2)- = (1/c) * (x^2 - i*b*x - (br^2-bi^2+2*br*bi*i)^2 /4)- = (1/c) * (x^2 - i*b*x - b^2 / 4)- = (1/c) * (x^2 - i*b*x + (i*b/2)^2)- = (1/c) * (x - i*b/2)^2--Example:-  (x-b)^2 = b^2 - 2*b*x + x^2-    ->  (- i*2*b*x + x^2)---fourier (x -> exp(-pi*(c*(x-t)^2 + 2*i*m*x)))- = fourier $ modulate m $ translate t $ shrink (sqrt c) $ x -> exp(-pi*x^2)- = translate (-m) $ modulate t $ dilate (sqrt c) $ fourier $ x -> exp(-pi*x^2)- = translate (-m) $ modulate t $ dilate (sqrt c) $ x -> exp(-pi*x^2)- = translate (-m) $ modulate t $ x -> exp(-pi*x^2/c)- = translate (-m) $ x -> exp(-pi*x^2/c) * exp(-2*pi*i*x*t)- = x -> exp(-pi*(x+m)^2/c) * exp(-2*pi*i*(x+m)*t)- = x -> exp(-pi*((x+m)^2/c - 2*i*(x+m)*t))--}--{--fourier (Cons a 0 0) =-  Cons a 0 infinity--fourier (Cons 0 0 c) =-  Cons 0 0 (recip c)--fourier (Cons 0 b 1) =-  Cons 0 (i*b) 1--}--translate :: Ring.C a => a -> T a -> T a-translate d f =-   let a = c0 f-       b = c1 f-       c = c2 f-   in  Cons-          (amp f)-          (Complex.fromReal (c*d^2) - Complex.scale d b + a)-          (Complex.fromReal (-2*c*d) + b)-          c--translateComplex :: Ring.C a => Complex.T a -> T a -> T a-translateComplex d f =-   let a = c0 f-       b = c1 f-       c = c2 f-   in  Cons-          (amp f)-          (Complex.scale c (d^2) - b*d + a)-          (Complex.scale (-2*c) d + b)-          c--modulate :: Ring.C a => a -> T a -> T a-modulate d f =-   Cons-      (amp f)-      (c0 f)-      (c1 f + (zero +: 2*d))-      (c2 f)--turn :: Ring.C a => a -> T a -> T a-turn d f =-   Cons-      (amp f)-      (c0 f + (zero +: 2*d))-      (c1 f)-      (c2 f)--reverse :: Additive.C a => T a -> T a-reverse f =-   f{c1 = negate $ c1 f}---dilate :: Field.C a => a -> T a -> T a-dilate k f =-   Cons-      (amp f)-      (c0 f)-      (Complex.scale (recip k) $ c1 f)-      (c2 f / k^2)--shrink :: Ring.C a => a -> T a -> T a-shrink k f =-   Cons-      (amp f)-      (c0 f)-      (Complex.scale k $ c1 f)-      (c2 f * k^2)--amplify :: (Ring.C a) => a -> T a -> T a-amplify k f =-   Cons-      (k^2 * amp f)-      (c0 f)-      (c1 f)-      (c2 f)---{- laws-fourier (convolve f g) = fourier f * fourier g--fourier (fourier f) = reverse f--}
− src/MathObj/Gaussian/Example.hs 
@@ -1,231 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{--Reciprocal of variance of a Gaussian bell curve.-We describe the curve only in terms of its variance-thus we represent a bell curve at the coordinate origin-neglecting its amplitude.--We could also define the amplitude as @root 4 c@,-thus preserving L2 norm being one,-but then @dilate@ and @shrink@ also include an amplification.--We could do some projective geometry in the exponent-in order to also have zero variance,-which corresponds to the dirac impulse.--}-module MathObj.Gaussian.Example where--import qualified MathObj.Gaussian.Polynomial as PolyBell-import qualified MathObj.Gaussian.Bell as Bell-import qualified MathObj.Gaussian.Variance as Var--import qualified MathObj.Polynomial as Poly--import qualified Algebra.Transcendental as Trans-import qualified Algebra.Algebraic      as Algebraic-import qualified Algebra.Field          as Field--- import qualified Algebra.Absolute           as Absolute-import qualified Algebra.Ring           as Ring--- import qualified Algebra.Additive       as Additive--import qualified Number.Complex as Complex-import qualified Number.Root as Root--import Algebra.Transcendental (pi, )-import Algebra.Algebraic (root, )-import Algebra.Ring ((*), (^), )--import Number.Complex ((+:), )--import qualified Numerics.Function as Func-import qualified Numerics.Fourier as Fourier-import qualified Numerics.Integration as Integ-import qualified Numerics.Differentiation as Diff--import qualified Graphics.Gnuplot.Simple as GP--import Control.Applicative (liftA2, )---- import System.Exit (ExitCode, )---- import Prelude (($))-import NumericPrelude.Numeric-import NumericPrelude.Base-import qualified Prelude as P---curve0 :: Var.T Double-curve0 = curve0a--curve0a :: Var.T Double-curve0a = Var.Cons 1.4 3.3--curve0b :: Var.T Double-curve0b = Var.Cons 2.2 1.7--variance0 :: (Double, Double)-variance0 =-   (Var.variance curve0,-    (Integ.rectangular 1000 (-2,2) $ liftA2 (*) (^2) (Var.evaluate curve0)) /-    (Integ.rectangular 1000 (-2,2) $ Var.evaluate curve0))--norm10 :: (Double, Double, Double)-norm10 =-   (Integ.rectangular 1000 (-2,2) $ Var.evaluate curve0,-    Var.norm1 curve0,-    Root.toNumber (Var.norm1Root curve0))--norm20 :: (Double, Double, Double)-norm20 =-   (sqrt $ Integ.rectangular 1000 (-2,2) $ (^2) . Var.evaluate curve0,-    Var.norm2 curve0,-    Root.toNumber (Var.norm2Root curve0))--norm30 :: (Double, Double, Double)-norm30 =-   (root 3 $ Integ.rectangular 1000 (-2,2) $ (^3) . Var.evaluate curve0,-    Var.normP 3 curve0,-    Root.toNumber (Var.normPRoot 3 curve0))--fourier0 :: IO ()-fourier0 =-   GP.plotFuncs []-      (GP.linearScale 100 (-2,2))-      [Var.evaluate $ Var.fourier curve0,-       Fourier.analysisTransformOneReal 100 (-2,2) $ Var.evaluate curve0]--multiply0 :: IO ()-multiply0 =-   GP.plotFuncs []-      (GP.linearScale 100 (-1,1))-      [Var.evaluate $ Var.multiply curve0a curve0b,-       liftA2 (*) (Var.evaluate curve0a) (Var.evaluate curve0b)]--convolve0 :: IO ()-convolve0 =-   GP.plotFuncs []-      (GP.linearScale 100 (-2,2))-      [Var.evaluate $ Var.convolve curve0a curve0b,-       Integ.convolve 1000 (-3,3) (Var.evaluate curve0a) (Var.evaluate curve0b)]---curve1 :: Bell.T Double-curve1 = curve1a--curve1a :: Bell.T Double-curve1a = Bell.Cons 1.4 (0.1+:0.3) ((-0.2)+:1.4) 2.3--curve1b :: Bell.T Double-curve1b = Bell.Cons 2.2 ((-0.3)+:2.1) (0.2+:(-0.4)) 1.7--variance1 :: (Double, Double)-variance1 =-   (Bell.variance curve1,-    (Integ.rectangular 1000 (-2,2) $-        liftA2 (*) (^2)-           (Complex.magnitudeSqr .-            Func.translateRight-               (Complex.real (Bell.c1 curve1) / (2 * Bell.c2 curve1))-               (Bell.evaluate curve1))) /-    (Integ.rectangular 1000 (-2,2) $ Complex.magnitude . Bell.evaluate curve1))--{- the norm depends on too much things-norm0vs1 :: (Double, Double)-norm0vs1 =-   ((Integ.rectangular 1000 (-5,5) $ Var.evaluate curve0)-         * exp (- Complex.real (Bell.c0 curve1)),-    Integ.rectangular 1000 (-5,5) $ Complex.magnitude . Bell.evaluate curve1)--}--fourier1 :: IO ()-fourier1 =-   GP.plotFuncs []-      (GP.linearScale 100 (-5,5))-      [Complex.real . (Bell.evaluate $ Bell.fourier curve1),-       fourierAnalysisReal 100 (-2,2) $ Bell.evaluate curve1]---curve2 :: PolyBell.T Double-curve2 =-   PolyBell.Cons---      Bell.unit---      (Bell.Cons 1.4 (0.1+:0.3) 0 1.2)---      (Bell.Cons 1.4 (0.1+:0.3) ((-0.2)+:1.4) 1)-      curve1---      (Poly.fromCoeffs [one])---      (Poly.fromCoeffs [zero,one])---      (Poly.fromCoeffs [zero,zero,one])---      (Poly.fromCoeffs [0,Complex.imaginaryUnit])-      (Poly.fromCoeffs [1.4+:(-0.1),0.8+:(0.1),(-1.1)+:0.3])--differentiate2 :: IO ()-differentiate2 =-   GP.plotFuncs []-      (GP.linearScale 100 (-2,2))-      [Complex.real . (PolyBell.evaluateSqRt $ PolyBell.differentiate curve2),-       ((/ sqrt pi) . ) $ Diff.diff (1e-5) $ Complex.real . PolyBell.evaluateSqRt curve2]--fourier2 :: IO ()-fourier2 =-   GP.plotFuncs []-      (GP.linearScale 100 (-5,5))-      [Complex.real . (PolyBell.evaluateSqRt $ PolyBell.fourier curve2),-       fourierAnalysisReal 100 (-2,2) $ PolyBell.evaluateSqRt curve2]----fourierAnalysisReal ::-   (P.Floating a) =>-   Integer -> (a, a) -> (a -> Complex.T a) -> a -> a-fourierAnalysisReal n rng f =-   liftA2 (P.-)-      (Fourier.analysisTransformOneReal n rng (Complex.real . f))-      (Fourier.analysisTransformOneImag n rng (Complex.imag . f))---{- |-Try to approximate @\x -> exp (-x^2) * x@-by a difference of translated Gaussian bells.--exp(-x^2) * x-  ==  exp(-(a+b*x+c*x^2)) - exp(-(a-b*x+c*x^2))-  ==  exp(-(a+c*x^2)) * (exp(-b*x) - exp(b*x))-  ==  exp(-(a+c*x^2)) * 2*sinh (b*x)--It holds-  lim (\b x -> sinh (b*x) / b)  =  id--}-diffApprox :: IO ()-diffApprox =-   let amp = (2*b)^- (-2)-       a = 0-       {--       amp = 1-       a = log (2 * abs b)-       -}-       b = -0.1-       c = 1-       ac = Complex.fromReal a-       bc = Complex.fromReal b-   in  GP.plotFuncs []-          (GP.linearScale 100 (-2,2::Double))-          [Complex.real .-           (PolyBell.evaluateSqRt $-              PolyBell.Cons Bell.unit (Poly.fromCoeffs [zero,one])),-           Complex.real .-           liftA2 (-)-             (PolyBell.evaluateSqRt $-                PolyBell.Cons (Bell.Cons amp ac bc c) (Poly.fromCoeffs [one]))-             (PolyBell.evaluateSqRt $-                PolyBell.Cons (Bell.Cons amp ac (-bc) c) (Poly.fromCoeffs [one]))]---polyApprox :: IO ()-polyApprox =-   GP.plotFuncs []-      (GP.linearScale 100 (-2,2::Double))-      [Complex.real .-         PolyBell.evaluateSqRt curve2,-       Complex.real . sum .-         mapM (\(amp,b) -> \x -> amp * Bell.evaluateSqRt b x)-         (PolyBell.approximateByBells 0.1 curve2)]
− src/MathObj/Gaussian/Polynomial.hs 
@@ -1,480 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{--Complex Gaussian bell multiplied with a polynomial.--In order to make this free of @pi@ factors,-we have to choose @recip (sqrt pi)@-as unit for translations and modulations,-for linear factors and in the differentiation.--}-{--ToDo:--* In order to avoid the weird @sqrt pi@ factor,-  use a polynomial expression in @pi@.--* sum of multiple bells using Data.Map from exponent polynomial to coefficient polynomial-  use of Algebra object.--* Discrete Fourier Transform and its eigenvectors--* Use projective geometry in order to support Dirac impulse.-  There are many open questions:-  1. What shall be the product of two Dirac impulses --     whether they are at the same location or not.-  2. How to organize coefficients-     such that the constant function can be modulated-     and the Dirac impulse can be translated.--}-module MathObj.Gaussian.Polynomial where--import qualified MathObj.Gaussian.Bell as Bell--import qualified MathObj.LaurentPolynomial as LPoly-import qualified MathObj.Polynomial.Core   as PolyCore-import qualified MathObj.Polynomial        as Poly-import qualified Number.Complex     as Complex--import qualified Algebra.ZeroTestable   as ZeroTestable-import qualified Algebra.Differential   as Differential-import qualified Algebra.Transcendental as Trans-import qualified Algebra.Field          as Field-import qualified Algebra.Absolute       as Absolute-import qualified Algebra.Ring           as Ring-import qualified Algebra.Additive       as Additive--import qualified Data.Record.HT as Rec-import qualified Data.List as List-import Data.Function.HT (nest, )-import Data.Eq.HT (equating, )-import Data.List.HT (mapAdjacent, )-import Data.Tuple.HT (forcePair, )--import Test.QuickCheck (Arbitrary, arbitrary, )-import Control.Monad (liftM2, )--import NumericPrelude.Numeric-import NumericPrelude.Base hiding (reverse, )--- import Prelude ()---data T a = Cons {bell :: Bell.T a, polynomial :: Poly.T (Complex.T a)}-   deriving (Show)--instance (Absolute.C a, ZeroTestable.C a, Eq a) => Eq (T a) where-   (==) = equal---{--Helper data type for 'equal',-that allows to call the (not quite trivial) polynomial equality check.-@RootProduct r a@ represents @sqrt r * a@.-The test using 'signum' works for real numbers,-and I do not know, whether it is correct for other mathematical objects.-However I cannot imagine other mathematical objects,-that make sense at all, here.-Maybe elements of a finite field.--}-data RootProduct a = RootProduct a a--instance (Absolute.C a, ZeroTestable.C a, Eq a) => Eq (RootProduct a) where-   (RootProduct xr xa) == (RootProduct yr ya)  =-      let xp = xr*xa^2-          yp = yr*ya^2-      in  xp==yp &&-          (isZero xp || signum xa == signum ya)--instance (ZeroTestable.C a) => ZeroTestable.C (RootProduct a) where-   isZero (RootProduct r a) = isZero r || isZero a---{--The derived Eq is not correct.-We have to combine the amplitude of the bell with the polynomial,-respecting signs and the square root of the bell amplitude.--}-equal :: (Absolute.C a, ZeroTestable.C a, Eq a) => T a -> T a -> Bool-equal x y =-   let bx = bell x-       by = bell y-       scaleSqr b =-          (\p ->-              (fmap (RootProduct (Bell.amp b) . Complex.real) p,-               fmap (RootProduct (Bell.amp b) . Complex.imag) p))-           . polynomial-   in  Rec.equal-          (equating Bell.c0 :-           equating Bell.c1 :-           equating Bell.c2 :-           [])-          bx by-       &&-       scaleSqr bx x == scaleSqr by y---instance (Absolute.C a, ZeroTestable.C a, Arbitrary a) => Arbitrary (T a) where-   arbitrary =---      liftM2 Cons arbitrary arbitrary-      liftM2 Cons-         arbitrary-         -- we have to restrict the number of polynomial coefficients,-         -- since with the quadratic time algorithms like fourier and convolve,-         -- in connection with Rational slow down tests too much.-         (fmap (Poly.fromCoeffs . take 5 . Poly.coeffs) arbitrary)----{-# INLINE evaluateSqRt #-}-evaluateSqRt :: (Trans.C a) =>-   T a -> a -> Complex.T a-evaluateSqRt f x =-   Bell.evaluateSqRt (bell f) x *-   Poly.evaluate (polynomial f) (Complex.fromReal $ sqrt pi * x)-{- ToDo: evaluating a complex polynomial for a real argument can be optimized -}---constant :: (Ring.C a) => T a-constant =-   Cons Bell.constant (Poly.const one)--scale :: (Ring.C a) => a -> T a -> T a-scale x f =-   f{polynomial = fmap (Complex.scale x) $ polynomial f}--scaleComplex :: (Ring.C a) => Complex.T a -> T a -> T a-scaleComplex x f =-   f{polynomial = fmap (x*) $ polynomial f}---unit :: (Ring.C a) => T a-unit = eigenfunction0--eigenfunction :: (Field.C a) => Int -> T a-eigenfunction =-   eigenfunctionDifferential--eigenfunction0 :: (Ring.C a) => T a-eigenfunction0 =-   Cons Bell.unit (Poly.fromCoeffs [one])--eigenfunction1 :: (Ring.C a) => T a-eigenfunction1 =-   Cons Bell.unit (Poly.fromCoeffs [zero, one])--eigenfunction2 :: (Field.C a) => T a-eigenfunction2 =-   Cons Bell.unit (Poly.fromCoeffs [-(1/4), zero, one])--eigenfunction3 :: (Field.C a) => T a-eigenfunction3 =-   Cons Bell.unit (Poly.fromCoeffs [zero, -(3/4), zero, one])---eigenfunctionDifferential :: (Field.C a) => Int -> T a-eigenfunctionDifferential n =-   (\f -> f{bell = Bell.unit}) $-   nest n (scale (-1/4) . differentiate) $-   Cons (Bell.Cons one zero zero 2) one--eigenfunctionIterative ::-   (Field.C a, Absolute.C a, ZeroTestable.C a, Eq a) => Int -> T a-eigenfunctionIterative n =-   fst . head . dropWhile (uncurry (/=)) . mapAdjacent (,) $-   eigenfunctionIteration $-   Cons-      Bell.unit-      (Poly.fromCoeffs $ replicate n zero ++ [one])--eigenfunctionIteration :: (Field.C a) => T a -> [T a]-eigenfunctionIteration =-   iterate (\x ->-      let y = fourier x-          px = polynomial x-          py = polynomial y-          c = last (Poly.coeffs px) / last (Poly.coeffs py)-      in  y{polynomial = fmap (0.5*) (px + fmap (c*) py)})---multiply :: (Ring.C a) =>-   T a -> T a -> T a-multiply f g =-   Cons-      (Bell.multiply (bell f) (bell g))-      (polynomial f * polynomial g)--convolve, {- convolveByDifferentiation, -} convolveByFourier :: (Field.C a) =>-   T a -> T a -> T a-convolve = convolveByFourier--{--f <*> g =-   let (foff,fint) = integrate f-   in  fint <*> differentiate g + makeGaussPoly foff * g--In principle this would work,-but (makeGaussPoly foff * g) contains a lot of-convolutions of Gaussian with Gaussian-polynomial-product,-where the Gaussians have different parameters.--convolveByDifferentiation f g =-   case polynomial f of-      fpoly ->-         if null $ Poly.coeffs fpoly-           then ...-           else ...--}--convolveByFourier f g =-   reverse $ fourier $ multiply (fourier f) (fourier g)--{--We use a Horner like scheme-in order to translate multiplications with @id@-to differentations on the Fourier side.-Quadratic runtime.--fourier (Cons bell (Poly.const a + Poly.shift f))-  = fourier (Cons bell (Poly.const a)) + fourier (Cons bell (Poly.shift f))-  = fourier (Cons bell (Poly.const a)) + differentiate (fourier (Cons bell f))--We can certainly speed this up considerably-by decomposing the polynomial into four polynomials,-one for each of the four eigenvalues 1, i, -1, -i.--}-fourier :: (Field.C a) =>-   T a -> T a-fourier f =-   foldr-      (\c p ->-          let q = differentiate p-          in  q{polynomial =-                   Poly.const c +-                   fmap (Complex.scale (1/2) . Complex.quarterLeft) (polynomial q)})-      (Cons (Bell.fourier $ bell f) zero) $-   Poly.coeffs $ polynomial f--{- |-Differentiate and divide by @sqrt pi@ in order to stay in a ring.-This way, we do not need to fiddle with pi factors.--}-differentiate :: (Ring.C a) => T a -> T a-differentiate f =-   f{polynomial =-        Differential.differentiate (polynomial f)-        - Differential.differentiate (Bell.exponentPolynomial (bell f))-           * polynomial f}--{--snd $ integrate $ differentiate (Cons Bell.unit (Poly.fromCoeffs [7,7,7,7]) :: T Double)--g = (bell f * poly f)'-  = bell f * ((poly f)' - (exppoly (bell f))' * poly f)-poly g = (poly f)' - (exppoly (bell f))' * poly f--Integration means we have g and ask for f.--poly f = ((poly f)' - poly g) / (exppoly (bell f))'--However must start with the highest term of 'poly f',-and thus we need to perform the division on reversed polynomials.--}-integrate ::-   (Field.C a, ZeroTestable.C a) =>-   T a -> (Complex.T a, T a)-integrate f =-   let fs = Poly.coeffs $ polynomial f-       (ys,~[r]) =-          PolyCore.divModRev-             {--             We need the shortening convention of 'zipWith'-             in order to limit the result list,-             we cannot use list instance for (-).-             -}-             (zipWith (-)-                (0 : 0 : diffRev ys)-                (List.reverse fs))-             (List.reverse $ Poly.coeffs $-              Differential.differentiate $-              Bell.exponentPolynomial $ bell f)-   in  forcePair $-       if null fs-         then (zero, f)-         else (r, f{polynomial = Poly.fromCoeffs $ List.reverse ys})--diffRev :: Ring.C a => [a] -> [a]-diffRev xs =-   zipWith (*) xs-      (drop 1 (iterate (subtract 1) (fromIntegral $ length xs)))--{--integrateDefinite-   (maybe rename integrate to antiderivative and call this one integrate)--int(x^(2*n)*exp(-x^2),x=-infinity..infinity)- = 2 * int(x^(2*n)*exp(-x^2),x=0..infinity)-     substitute t=x^2, dt = dx * 2 * sqrt t- = int(t^(n-1/2)*exp(-t),x=0..infinity)- = Gamma(n+1/2)- = (2n-1)!!/2^n * sqrt pi--int(pi^n*x^(2*n)*exp(-pi*x^2),x=-infinity..infinity)- = (2n-1)!!/2^n---The remainder value of 'integrate'-is the coefficient of the error function-and this is the only part that does not vanish when approaching the limit.---In order to stay in a field,-we have to return a rational number-and a transcendental part written es @exp a@.--It would be interesting to see how integral inequalities-translate to scalar inequalities containing exponential functions.--}---translate :: Ring.C a => a -> T a -> T a-translate d =-   translateComplex (Complex.fromReal d)--translateComplex :: Ring.C a => Complex.T a -> T a -> T a-translateComplex d f =-   Cons-      (Bell.translateComplex d $ bell f)-      (Poly.translate d $ polynomial f)--modulate :: Ring.C a => a -> T a -> T a-modulate d f =-   Cons-      (Bell.modulate d $ bell f)-      (polynomial f)--turn :: Ring.C a => a -> T a -> T a-turn d f =-   Cons-      (Bell.turn d $ bell f)-      (polynomial f)--reverse :: Additive.C a => T a -> T a-reverse f =-   Cons-      (Bell.reverse $ bell f)-      (Poly.reverse $ polynomial f)--dilate :: Field.C a => a -> T a -> T a-dilate k f =-   Cons-      (Bell.dilate k $ bell f)-      (Poly.dilate (Complex.fromReal k) $ polynomial f)--shrink :: Ring.C a => a -> T a -> T a-shrink k f =-   Cons-      (Bell.shrink k $ bell f)-      (Poly.shrink (Complex.fromReal k) $ polynomial f)--{--We could also amplify the polynomial coefficients.--}-amplify :: Ring.C a => a -> T a -> T a-amplify k f =-   Cons-      (Bell.amplify k $ bell f)-      (polynomial f)---{- |-Approximate a @T a@ using a linear combination of translated @Bell.T a@.-The smaller the unit (e.g. 0.1, 0.01, 0.001)-the better the approximation but the worse the numeric properties.--We cannot put all information into @amp@ of @Bell@,-since @amp@ must be real, but is complex here by construction.-We really need at least signed amplitudes at this place,-since we want to represent differences of Gaussians.--}-approximateByBells ::-   Field.C a =>-   a -> T a -> [(Complex.T a, Bell.T a)]-approximateByBells unit_ f =-   let b = bell f-       amps =-          -- approximateByBellsByTranslation-          approximateByBellsAtOnce-             unit_-             (Complex.scale (recip (2 * Bell.c2 b)) (Bell.c1 b))-             (recip (2*unit_*Bell.c2 b))-             (polynomial f)-   in  zip (LPoly.coeffs amps) $-       map-          (\d -> Bell.translate d b)-          (laurentAbscissas (unit_/2) amps)--approximateByBellsAtOnce ::-   Field.C a =>-   a -> Complex.T a -> a -> Poly.T (Complex.T a) -> LPoly.T (Complex.T a)-approximateByBellsAtOnce unit_ d s p =-   foldr-      (\x amps0 ->-         {--         Decompose (bell t * (t-d)) = bell t * t - bell t * d-         -}-         let y = fmap (Complex.scale s) amps0-         in  -- \t -> bell t * t-             --    ~   (translate unit_ bell - translate (-unit_) bell) / unit_-             LPoly.shift 1 y --             LPoly.shift (-1) y +-             -- bell t * d-             zipWithAbscissas-                (\t z -> (Complex.fromReal t - d) * z)-                (unit_/2) amps0 +-             LPoly.const x)-      (LPoly.fromCoeffs [])-      (Poly.coeffs p)--approximateByBellsByTranslation ::-   Field.C a =>-   a -> Complex.T a -> a -> Poly.T (Complex.T a) -> LPoly.T (Complex.T a)-approximateByBellsByTranslation unit_ d s p =-   foldr-      (\x amps0 ->-         {--         Decompose (bell t * (t-d)) = bell t * t - bell t * d-         -}-         let y = fmap (Complex.scale s) amps0-         in  -- \t -> bell t * t-             --    ~   (translate unit_ bell - translate (-unit_) bell) / unit_-             LPoly.shift 1 y --             LPoly.shift (-1) y +-             -- bell t * d-             zipWithAbscissas Complex.scale (unit_/2) amps0 +-             LPoly.const x)-      (LPoly.fromCoeffs [])-      (Poly.coeffs $ Poly.translate d p)--zipWithAbscissas ::-   (Ring.C a) =>-   (a -> b -> c) -> a -> LPoly.T b -> LPoly.T c-zipWithAbscissas h unit_ y =-   LPoly.fromShiftCoeffs (LPoly.expon y) $-   zipWith h-      (laurentAbscissas unit_ y)-      (LPoly.coeffs y)--laurentAbscissas :: Ring.C a => a -> LPoly.T c -> [a]-laurentAbscissas unit_ =-   map (\d -> fromIntegral d * unit_) .-   iterate (1+) . LPoly.expon---{- No Ring instance for Gaussians-instance (Ring.C a) => Differential.C (T a) where-   differentiate = differentiate--}--{- laws-differentiate (f*g) =-   (differentiate f) * g + f * (differentiate g)--}
− src/MathObj/Gaussian/Variance.hs 
@@ -1,206 +0,0 @@-{-# LANGUAGE NoImplicitPrelude #-}-{--We represent a Gaussian bell curve in terms of the reciprocal of its variance-and its value at the origin.--We could do some projective geometry in the exponent-in order to also have zero variance,-which corresponds to the dirac impulse.--The Gaussians form a nice multiplicative commutative monoid.-Maybe we should have such a structure.-It would also be useful for the Root data type-and a new Exponential data type.--}-module MathObj.Gaussian.Variance where--import qualified MathObj.Polynomial as Poly-import qualified Number.Root as Root--import qualified Algebra.Transcendental as Trans-import qualified Algebra.Algebraic      as Algebraic-import qualified Algebra.Field          as Field-import qualified Algebra.Absolute       as Absolute-import qualified Algebra.Ring           as Ring-import qualified Algebra.Additive       as Additive--{--import Algebra.Transcendental (pi, )-import Algebra.Ring ((*), (^), )-import Algebra.Additive ((+))--}-import Test.QuickCheck (Arbitrary, arbitrary, )-import Control.Monad (liftM2, )---- import Prelude (($))-import NumericPrelude.Numeric-import NumericPrelude.Base---{- |-Since @amp@ is the square of the actual amplitude it must be non-negative.--}-data T a = Cons {amp, c :: a}-   deriving (Eq, Show)--instance (Absolute.C a, Arbitrary a) => Arbitrary (T a) where-   arbitrary =-      liftM2 Cons-         (fmap abs arbitrary)-         (fmap ((1+) . abs) arbitrary)---constant :: Ring.C a => T a-constant = Cons one zero--{- |-eigenfunction of 'fourier'--}-unit :: Ring.C a => T a-unit = Cons one one--{-# INLINE evaluate #-}-evaluate :: (Trans.C a) =>-   T a -> a -> a-evaluate f x =-   sqrt (amp f) * exp (-pi * c f * x^2)--exponentPolynomial :: (Additive.C a) =>-   T a -> Poly.T a-exponentPolynomial f =-   Poly.fromCoeffs [zero, zero, c f]---integrateRoot :: (Field.C a) => T a -> Root.T a-integrateRoot f =-   Root.sqrt $ Root.fromNumber $ amp f / c f--scalarProductRoot :: (Field.C a) => T a -> T a -> Root.T a-scalarProductRoot f g =-   integrateRoot (multiply f g)---norm1Root :: (Field.C a) => T a -> Root.T a-norm1Root = integrateRoot--norm2Root :: (Field.C a) => T a -> Root.T a-norm2Root f =-   Root.sqrt $-      Root.fromNumber (amp f)-      `Root.div`-      Root.sqrt (Root.fromNumber $ 2 * c f)--normInfRoot :: (Field.C a) => T a -> Root.T a-normInfRoot f =-   Root.sqrt $ Root.fromNumber $ amp f--normPRoot :: (Field.C a) => Rational -> T a -> Root.T a-normPRoot p f =-   Root.sqrt (Root.fromNumber (amp f))-   `Root.div`-   Root.rationalPower (recip (2*p)) (Root.fromNumber (fromRational' p * c f))----- ToDo: implement NormedSpace.Sum et.al.-norm1 :: (Algebraic.C a) => T a -> a-norm1 f =-   sqrt $ amp f / c f--norm2 :: (Algebraic.C a) => T a -> a-norm2 f =-   sqrt $ amp f / (sqrt $ 2 * c f)--normInf :: (Algebraic.C a) => T a -> a-normInf f =-   sqrt (amp f)--normP :: (Trans.C a) => a -> T a -> a-normP p f =-   sqrt (amp f) * (p * c f) ^? (- recip (2*p))---variance :: (Trans.C a) =>-   T a -> a-variance f =-   recip $ c f * 2*pi--multiply :: (Ring.C a) =>-   T a -> T a -> T a-multiply f g =-   Cons (amp f * amp g) (c f + c g)--powerRing :: (Trans.C a) =>-   Integer -> T a -> T a-powerRing p f =-   Cons (amp f ^ p) (fromInteger p * c f)--{--powerField does not makes sense,-since the reciprocal of a Gaussian diverges.--}--powerAlgebraic :: (Trans.C a) =>-   Rational -> T a -> T a-powerAlgebraic p f =-   Cons (amp f ^/ p) (fromRational' p * c f)--powerTranscendental :: (Trans.C a) =>-   a -> T a -> T a-powerTranscendental p f =-   Cons (amp f ^? p) (p * c f)--{- |-> convolve x y t =->    integrate $ \s -> x s * y(t-s)--Convergence only for @c f + c g > 0@.--}-convolve :: (Field.C a) =>-   T a -> T a -> T a-convolve f g =-   let s = c f + c g-   in  Cons-          (amp f * amp g / s)-          (c f * c g / s)--{- |-> fourier x f =->    integrate $ \t -> x t * cis (-2*pi*t*f)--Convergence only for @c f > 0@.--}-fourier :: (Field.C a) =>-   T a -> T a-fourier f =-   Cons (amp f / c f) (recip $ c f)-{--fourier (t -> exp(-(a*t)^2))--}--dilate :: (Field.C a) => a -> T a -> T a-dilate k f =-   Cons (amp f) $ c f / k^2--shrink :: (Ring.C a) => a -> T a -> T a-shrink k f =-   Cons (amp f) $ c f * k^2--{- |-@amplify k@ scales by @abs k@!--}-amplify :: (Ring.C a) => a -> T a -> T a-amplify k f =-   Cons (k^2 * amp f) $ c f---{- laws-fourier (convolve f g) = multiply (fourier f) (fourier g)--dilate k (dilate m f) = dilate (k*m) f--dilate k (shrink k f) = f--variance (dilate k f) = k^2 * variance f--variance (convolve f g) = variance f + variance g--}
− src/Number/ComplexSquareRoot.hs 
@@ -1,117 +0,0 @@-module Number.ComplexSquareRoot where---- import qualified Algebra.Algebraic as Algebraic-import qualified Algebra.RealField as RealField-import qualified Algebra.RealRing as RealRing--- import qualified Algebra.Field as Field-import qualified Algebra.Ring as Ring-import qualified Algebra.Additive as Additive-import qualified Algebra.ZeroTestable as ZeroTestable--import qualified Number.Complex as Complex--import Test.QuickCheck (Arbitrary, arbitrary, )--import Control.Monad (liftM2, )--import qualified NumericPrelude.Numeric as NP-import NumericPrelude.Numeric hiding (recip, )-import NumericPrelude.Base-import Prelude ()--{- |-Represent the square root of a complex number-without actually having to compute a square root.-If the Bool is False,-then the square root is represented with positive real part-or zero real part and positive imaginary part.-If the Bool is True the square root is negated.--}-data T a = Cons Bool (Complex.T a)-   deriving (Show)--{- |-You must use @fmap@ only for number type conversion.--}-instance Functor T where-   fmap f (Cons n x) = Cons n (fmap f x)--instance (ZeroTestable.C a) => ZeroTestable.C (T a) where-   isZero (Cons _b s) = isZero s--instance (ZeroTestable.C a, Eq a) => Eq (T a) where-   (Cons xb xs) == (Cons yb ys) =-      isZero xs && isZero ys  ||-      xb==yb && xs==ys--instance (Arbitrary a) => Arbitrary (T a) where-   arbitrary = liftM2 Cons arbitrary arbitrary---fromNumber :: (RealRing.C a) => Complex.T a -> T a-fromNumber x =-   Cons-      (case compare zero (Complex.real x) of-         LT -> False-         GT -> True-         EQ -> Complex.imag x < zero)-      (x^2)---- htam:Wavelet.DyadicResultant.parityFlip-toNumber :: (RealRing.C a, Complex.Power a) => T a -> Complex.T a-toNumber (Cons n x) =-   case sqrt x of y -> if n then NP.negate y else y---one :: (Ring.C a) => T a-one = Cons False NP.one--inUpperHalfplane :: (Additive.C a, Ord a) => Complex.T a -> Bool-inUpperHalfplane x =-   case compare (Complex.imag x) zero of-      GT -> True-      LT -> False-      EQ -> Complex.real x < zero--mul, mulAlt, mulAlt2 :: (RealRing.C a) => T a -> T a -> T a-mul (Cons xb xs) (Cons yb ys) =-   let zs = xs*ys-   in  Cons-          ((xb /= yb) /=-             case (inUpperHalfplane xs,-                   inUpperHalfplane ys,-                   inUpperHalfplane zs) of-                (True,True,False) -> True-                (False,False,True) -> True-                _ -> False)-          zs--mulAlt (Cons xb xs) (Cons yb ys) =-   let zs = xs*ys-   in  Cons-          ((xb /= yb) /=-             let xi = Complex.imag xs-                 yi = Complex.imag ys-                 zi = Complex.imag zs-             in  (xi>=zero) /= (yi>=zero) &&-                 (xi>=zero) /= (zi>=zero))-          zs--mulAlt2 (Cons xb xs) (Cons yb ys) =-   let zs = xs*ys-   in  Cons-          ((xb /= yb) /=-             let xi = Complex.imag xs-                 yi = Complex.imag ys-                 zi = Complex.imag zs-             in  xi*yi<zero && xi*zi<zero)-          zs--div :: (RealField.C a) => T a -> T a -> T a-div x y = mul x (recip y)--recip :: (RealField.C a) => T a -> T a-recip (Cons b s) =-   Cons-      (b /= (Complex.imag s == zero && Complex.real s < zero))-      (NP.recip s)
src/Number/DimensionTerm.hs view
@@ -1,14 +1,6 @@ {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} {- |-Copyright   :  (c) Henning Thielemann 2008-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable-- See "Algebra.DimensionTerm". -}
src/Number/DimensionTerm/SI.hs view
@@ -1,12 +1,5 @@ {-# LANGUAGE NoImplicitPrelude #-} {- |-Copyright   :  (c) Henning Thielemann 2003-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable- Special physical units: SI unit system -}
src/Number/OccasionallyScalarExpression.hs view
@@ -2,13 +2,6 @@ {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} {- |-Copyright   :  (c) Henning Thielemann 2004-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  multi-type parameter classes (vector space)- Physical expressions track the operations made on physical values so we are able to give detailed information on how to resolve unit violations.
src/Number/Physical.hs view
@@ -2,13 +2,6 @@ {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} {- |-Copyright   :  (c) Henning Thielemann 2003-2006-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  generic instances- Numeric values combined with abstract Physical Units -}
src/Number/Physical/Read.hs view
@@ -1,12 +1,5 @@ {-# LANGUAGE NoImplicitPrelude #-} {- |-Copyright   :  (c) Henning Thielemann 2004-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  multi-parameter type classes (VectorSpace.hs)- Convert a human readable string to a physical value. -}
src/Number/Physical/Show.hs view
@@ -1,12 +1,5 @@ {-# LANGUAGE NoImplicitPrelude #-} {- |-Copyright   :  (c) Henning Thielemann 2004-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  multi-parameter type classes (VectorSpace.hs, Normalization.hs)- Convert a physical value to a human readable string. -}
src/Number/Physical/Unit.hs view
@@ -1,12 +1,5 @@ {-# LANGUAGE NoImplicitPrelude #-} {- |-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
src/Number/Physical/UnitDatabase.hs view
@@ -1,12 +1,5 @@ {-# LANGUAGE NoImplicitPrelude #-} {- |-Copyright   :  (c) Henning Thielemann 2003-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable- Tools for creating a data base of physical units and for extracting data from it -}
src/Number/Positional.hs view
@@ -1,12 +1,5 @@ {-# LANGUAGE NoImplicitPrelude #-} {- |-Copyright   :  (c) Henning Thielemann 2006-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-- Exact Real Arithmetic - Computable reals. Inspired by ''The most unreliable technique for computing pi.'' See also <http://www.haskell.org/haskellwiki/Exact_real_arithmetic> .
src/Number/Positional/Check.hs view
@@ -1,12 +1,5 @@ {-# LANGUAGE NoImplicitPrelude #-} {- |-Copyright   :  (c) Henning Thielemann 2006-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-- Interface to "Number.Positional" which dynamically checks for equal bases. -} module Number.Positional.Check where
src/Number/SI.hs view
@@ -3,13 +3,6 @@ {-# LANGUAGE FlexibleInstances #-} {-# LANGUAGE GeneralizedNewtypeDeriving #-} {- |-Copyright   :  (c) Henning Thielemann 2003-2012-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. -}
src/Number/SI/Unit.hs view
@@ -1,12 +1,5 @@ {-# LANGUAGE NoImplicitPrelude #-} {- |-Copyright   :  (c) Henning Thielemann 2003-License     :  GPL--Maintainer  :  numericprelude@henning-thielemann.de-Stability   :  provisional-Portability :  portable- Special physical units: SI unit system -}
− test/Gaussian.hs 
@@ -1,6 +0,0 @@-module Main where--import qualified MathObj.Gaussian.Example as Example--main :: IO ()-main = Example.polyApprox
+ test/Number/ComplexSquareRoot.hs view
@@ -0,0 +1,117 @@+module Number.ComplexSquareRoot where++-- import qualified Algebra.Algebraic as Algebraic+import qualified Algebra.RealField as RealField+import qualified Algebra.RealRing as RealRing+-- import qualified Algebra.Field as Field+import qualified Algebra.Ring as Ring+import qualified Algebra.Additive as Additive+import qualified Algebra.ZeroTestable as ZeroTestable++import qualified Number.Complex as Complex++import Test.QuickCheck (Arbitrary, arbitrary, )++import Control.Monad (liftM2, )++import qualified NumericPrelude.Numeric as NP+import NumericPrelude.Numeric hiding (recip, )+import NumericPrelude.Base+import Prelude ()++{- |+Represent the square root of a complex number+without actually having to compute a square root.+If the Bool is False,+then the square root is represented with positive real part+or zero real part and positive imaginary part.+If the Bool is True the square root is negated.+-}+data T a = Cons Bool (Complex.T a)+   deriving (Show)++{- |+You must use @fmap@ only for number type conversion.+-}+instance Functor T where+   fmap f (Cons n x) = Cons n (fmap f x)++instance (ZeroTestable.C a) => ZeroTestable.C (T a) where+   isZero (Cons _b s) = isZero s++instance (ZeroTestable.C a, Eq a) => Eq (T a) where+   (Cons xb xs) == (Cons yb ys) =+      isZero xs && isZero ys  ||+      xb==yb && xs==ys++instance (Arbitrary a) => Arbitrary (T a) where+   arbitrary = liftM2 Cons arbitrary arbitrary+++fromNumber :: (RealRing.C a) => Complex.T a -> T a+fromNumber x =+   Cons+      (case compare zero (Complex.real x) of+         LT -> False+         GT -> True+         EQ -> Complex.imag x < zero)+      (x^2)++-- htam:Wavelet.DyadicResultant.parityFlip+toNumber :: (RealRing.C a, Complex.Power a) => T a -> Complex.T a+toNumber (Cons n x) =+   case sqrt x of y -> if n then NP.negate y else y+++one :: (Ring.C a) => T a+one = Cons False NP.one++inUpperHalfplane :: (Additive.C a, Ord a) => Complex.T a -> Bool+inUpperHalfplane x =+   case compare (Complex.imag x) zero of+      GT -> True+      LT -> False+      EQ -> Complex.real x < zero++mul, mulAlt, mulAlt2 :: (RealRing.C a) => T a -> T a -> T a+mul (Cons xb xs) (Cons yb ys) =+   let zs = xs*ys+   in  Cons+          ((xb /= yb) /=+             case (inUpperHalfplane xs,+                   inUpperHalfplane ys,+                   inUpperHalfplane zs) of+                (True,True,False) -> True+                (False,False,True) -> True+                _ -> False)+          zs++mulAlt (Cons xb xs) (Cons yb ys) =+   let zs = xs*ys+   in  Cons+          ((xb /= yb) /=+             let xi = Complex.imag xs+                 yi = Complex.imag ys+                 zi = Complex.imag zs+             in  (xi>=zero) /= (yi>=zero) &&+                 (xi>=zero) /= (zi>=zero))+          zs++mulAlt2 (Cons xb xs) (Cons yb ys) =+   let zs = xs*ys+   in  Cons+          ((xb /= yb) /=+             let xi = Complex.imag xs+                 yi = Complex.imag ys+                 zi = Complex.imag zs+             in  xi*yi<zero && xi*zi<zero)+          zs++div :: (RealField.C a) => T a -> T a -> T a+div x y = mul x (recip y)++recip :: (RealField.C a) => T a -> T a+recip (Cons b s) =+   Cons+      (b /= (Complex.imag s == zero && Complex.real s < zero))+      (NP.recip s)
test/Test.hs view
@@ -12,7 +12,7 @@  import qualified Number.NonNegativeChunky as Chunky import qualified Number.NonNegative       as NonNegW-import qualified Number.Positional.Check  as Absolute+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@@ -79,13 +79,13 @@ {- * Reals -}  testReal :: String-testReal = Absolute.defltShow (sqrt 2 + log 2 * pi)+testReal = Real.defltShow (sqrt 2 + log 2 * pi) -testComplexReal :: Complex.T Absolute.T+testComplexReal :: Complex.T Real.T testComplexReal = exp (0 +: pi) + exp (0 -: pi) -showReal :: Absolute.T -> String-showReal = Absolute.defltShow+showReal :: Real.T -> String+showReal = Real.defltShow   {- * Fixed point numbers -}
test/Test/MathObj/PartialFraction.hs view
@@ -16,11 +16,12 @@ 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 qualified Control.Monad.HT as M  import NumericPrelude.Base as P import NumericPrelude.Numeric as NP