shady-gen-0.5.1: src/Shady/Complex.hs
{-# LANGUAGE TypeOperators, CPP, DeriveDataTypeable, TypeFamilies #-}
{-# OPTIONS_GHC -Wall #-}
-----------------------------------------------------------------------------
-- |
-- Module : Data.Complex
-- Copyright : (c) The University of Glasgow 2001, Conal Elliott 2009
-- License : BSD-style
--
-- Maintainer : conal@conal.net
-- Stability : provisional
-- Portability : portable
--
-- Complex numbers. This version is modified from Data.Complex in base.
-- It eliminates the RealFloat requirement by using a more naive
-- definition of 'magnitude'. Also, defines instances for vector-space classes.
--
-----------------------------------------------------------------------------
module Shady.Complex
(
-- * Rectangular form
Complex((:+))
, realPart -- :: Complex a -> a
, imagPart -- :: Complex a -> a
-- * Polar form
, mkPolar -- :: a -> a -> Complex a
, cis -- :: a -> Complex a
, polar -- :: Complex a -> (a,a)
-- , magnitude -- :: Complex a -> a
, phase -- :: Complex a -> a
-- * Conjugate
, conjugate -- :: Complex a -> Complex a
-- Complex instances: (Eq,Read,Show,Num,Fractional,Floating)
-- Complex instances: (AdditiveGroup, VectorSpace, InnerSpace)
-- * Misc interface additions
, onRI, onRI2
) where
import Prelude
import Data.Typeable
#ifdef __GLASGOW_HASKELL__
import Data.Data (Data)
#endif
#ifdef __HUGS__
import Hugs.Prelude(Num(fromInt), Fractional(fromDouble))
#endif
import Data.VectorSpace
import Shady.Misc (Unop,Binop,FMod(..),Frac(..))
import Text.PrettyPrint.Leijen.DocExpr
infix 6 :+
-- -----------------------------------------------------------------------------
-- The Complex type
-- | Complex numbers are an algebraic type.
--
-- For a complex number @z@, @'abs' z@ is a number with the magnitude of @z@,
-- but oriented in the positive real direction, whereas @'signum' z@
-- has the phase of @z@, but unit magnitude.
data Complex a
= !a :+ !a -- ^ forms a complex number from its real and imaginary
-- rectangular components.
# if __GLASGOW_HASKELL__
deriving (Eq, Show, Read, Data)
# else
deriving (Eq, Show, Read)
# endif
-- -----------------------------------------------------------------------------
-- Functions over Complex
-- | Extracts the real part of a complex number.
realPart :: Complex a -> a
realPart (x :+ _) = x
-- | Extracts the imaginary part of a complex number.
imagPart :: Complex a -> a
imagPart (_ :+ y) = y
-- | The conjugate of a complex number.
{-# SPECIALISE conjugate :: Complex Double -> Complex Double #-}
conjugate :: Num a => Unop (Complex a)
conjugate (x:+y) = x :+ (-y)
-- | Form a complex number from polar components of magnitude and phase.
{-# SPECIALISE mkPolar :: Double -> Double -> Complex Double #-}
mkPolar :: Floating a => a -> a -> Complex a
mkPolar r theta = r * cos theta :+ r * sin theta
-- | @'cis' t@ is a complex value with magnitude @1@
-- and phase @t@ (modulo @2*'pi'@).
{-# SPECIALISE cis :: Double -> Complex Double #-}
cis :: Floating a => a -> Complex a
cis theta = cos theta :+ sin theta
-- | The function 'polar' takes a complex number and
-- returns a (magnitude, phase) pair in canonical form:
-- the magnitude is nonnegative, and the phase in the range @(-'pi', 'pi']@;
-- if the magnitude is zero, then so is the phase.
{-# SPECIALISE polar :: Complex Double -> (Double,Double) #-}
polar :: Floating a => Complex a -> (a,a)
polar z = (magnitude z, phase z)
-- | Operate on the real & imaginary components
onRI :: Unop a -> Unop (Complex a)
onRI f (x :+ y) = f x :+ f y
-- | Operate on the real & imaginary components
onRI2 :: Binop a -> Binop (Complex a)
onRI2 f (x :+ y) (x' :+ y') = f x x' :+ f y y'
instance Floating a => AdditiveGroup (Complex a) where
{ zeroV = 0 ; negateV = negate ; (^+^) = (+) }
instance Floating a => VectorSpace (Complex a) where
type Scalar (Complex a) = a
-- s *^ (x :+ y) = s * x :+ s * y
(*^) s = onRI (s *)
instance Floating a => InnerSpace (Complex a) where
(x :+ y) <.> (x' :+ y') = x*x' + y*y'
{-
-- | The nonnegative magnitude of a complex number.
{-# SPECIALISE magnitude :: Complex Double -> Double #-}
magnitude :: Floating a => Complex a -> a
magnitude = sqrt . magSq
magnitudeSq :: Floating a => Complex a -> a
-- magnitude (x:+y) = scaleFloat k
-- (sqrt (sqr (scaleFloat mk x) + sqr (scaleFloat mk y)))
-- where k = max (exponent x) (exponent y)
-- mk = - k
-- sqr z = z * z
-}
-- | The phase of a complex number, in the range @(-'pi', 'pi']@.
-- If the magnitude is zero, then so is the phase.
{-# SPECIALISE phase :: Complex Double -> Double #-}
phase :: Floating a => Complex a -> a
-- The zero case requires a real EQ instance
-- phase (0 :+ 0) = 0 -- SLPJ July 97 from John Peterson
phase (x:+y) = atan2' y x
-- To avoid reliance on 'RealFloat'.
atan2' :: (Floating a) => a -> a -> a
atan2' y x = atan (y/x)
-- -----------------------------------------------------------------------------
-- Instances of Complex
#include "Typeable.h"
INSTANCE_TYPEABLE1(Complex,complexTc,"Complex")
instance Floating a => Num (Complex a) where
{-# SPECIALISE instance Num (Complex Float) #-}
{-# SPECIALISE instance Num (Complex Double) #-}
(x:+y) + (x':+y') = (x+x') :+ (y+y')
(x:+y) - (x':+y') = (x-x') :+ (y-y')
(x:+y) * (x':+y') = (x*x'-y*y') :+ (x*y'+y*x')
negate (x:+y) = negate x :+ negate y
abs z = magnitude z :+ 0
signum (0:+0) = 0
signum z@(x:+y) = x/r :+ y/r where r = magnitude z
fromInteger n = fromInteger n :+ 0
#ifdef __HUGS__
fromInt n = fromInt n :+ 0
#endif
instance Floating a => Fractional (Complex a) where
{-# SPECIALISE instance Fractional (Complex Float) #-}
{-# SPECIALISE instance Fractional (Complex Double) #-}
(x:+y) / v@(x':+y') = ((x*x'+y*y') :+ (y*x'-x*y')) ^/ magnitudeSq v
-- (x:+y) / (x':+y') = (x*x''+y*y'') / d :+ (y*x''-x*y'') / d
-- where x'' = scaleFloat k x'
-- y'' = scaleFloat k y'
-- k = - max (exponent x') (exponent y')
-- d = x'*x'' + y'*y''
fromRational a = fromRational a :+ 0
#ifdef __HUGS__
fromDouble a = fromDouble a :+ 0
#endif
instance Floating a => Floating (Complex a) where
{-# SPECIALISE instance Floating (Complex Float) #-}
{-# SPECIALISE instance Floating (Complex Double) #-}
pi = pi :+ 0
exp (x:+y) = expx * cos y :+ expx * sin y
where expx = exp x
log z = log (magnitude z) :+ phase z
-- x ** y = exp (log x * y)
-- sqrt = (** 0.5)
-- Use default sqrt (** 0.5)
-- sqrt (0:+0) = 0
-- sqrt z@(x:+y) = u :+ (if y < 0 then -v else v)
-- where (u,v) = if x < 0 then (v',u') else (u',v')
-- v' = abs y / (u'*2)
-- u' = sqrt ((magnitude z + abs x) / 2)
sin (x:+y) = sin x * cosh y :+ cos x * sinh y
cos (x:+y) = cos x * cosh y :+ (- sin x * sinh y)
tan (x:+y) = (sinx*coshy:+cosx*sinhy)/(cosx*coshy:+(-sinx*sinhy))
where sinx = sin x
cosx = cos x
sinhy = sinh y
coshy = cosh y
sinh (x:+y) = cos y * sinh x :+ sin y * cosh x
cosh (x:+y) = cos y * cosh x :+ sin y * sinh x
tanh (x:+y) = (cosy*sinhx:+siny*coshx)/(cosy*coshx:+siny*sinhx)
where siny = sin y
cosy = cos y
sinhx = sinh x
coshx = cosh x
asin z@(x:+y) = y':+(-x')
where (x':+y') = log (((-y):+x) + sqrt (1 - z*z))
acos z = y'':+(-x'')
where (x'':+y'') = log (z + ((-y'):+x'))
(x':+y') = sqrt (1 - z*z)
atan z@(x:+y) = y':+(-x')
where (x':+y') = log (((1-y):+x) / sqrt (1+z*z))
asinh z = log (z + sqrt (1+z*z))
acosh z = log (z + (z+1) * sqrt ((z-1)/(z+1)))
atanh z = log ((1+z) / sqrt (1-z*z))
{--------------------------------------------------------------------
Pretty printing
--------------------------------------------------------------------}
-- infix 6 :+
instance HasExpr a => HasExpr (Complex a) where
expr (x :+ y) = op Infix 6 ":+" (expr x) (expr y)
{--------------------------------------------------------------------
Misc
--------------------------------------------------------------------}
instance Frac s => Frac (Complex s) where frac = onRI frac
instance FMod s => FMod (Complex s) where fmod = onRI2 fmod