Haskell Code by HsColour

-----------------------------------------------------------------------------
-- |
-- Module : Data.Complex
-- Copyright : (c) The University of Glasgow 2001
-- License : BSD-style (see the file libraries/base/LICENSE)
-- 
-- Maintainer : libraries@haskell.org
-- Stability : provisional
-- Portability : portable
--
-- Complex numbers.
--
-----------------------------------------------------------------------------

module Data.Complex
 (
 -- * Rectangular form
 Complex((:+))

 , realPart -- :: (RealFloat a) => Complex a -> a
 , imagPart -- :: (RealFloat a) => Complex a -> a
 -- * Polar form
 , mkPolar -- :: (RealFloat a) => a -> a -> Complex a
 , cis -- :: (RealFloat a) => a -> Complex a
 , polar -- :: (RealFloat a) => Complex a -> (a,a)
 , magnitude -- :: (RealFloat a) => Complex a -> a
 , phase -- :: (RealFloat a) => Complex a -> a
 -- * Conjugate
 , conjugate -- :: (RealFloat a) => Complex a -> Complex a

 -- Complex instances:
 --
 -- (RealFloat a) => Eq (Complex a)
 -- (RealFloat a) => Read (Complex a)
 -- (RealFloat a) => Show (Complex a)
 -- (RealFloat a) => Num (Complex a)
 -- (RealFloat a) => Fractional (Complex a)
 -- (RealFloat a) => Floating (Complex a)
 -- 
 -- Implementation checked wrt. Haskell 98 lib report, 1/99.

 ) where

import Prelude

import Data.Typeable
#ifdef __GLASGOW_HASKELL__
import Data.Data (Data)
#endif

#ifdef __HUGS__
import Hugs.Prelude(Num(fromInt), Fractional(fromDouble))
#endif

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 (RealFloat a) => 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 :: (RealFloat a) => Complex a -> a
realPart (x :+ _) = x

-- | Extracts the imaginary part of a complex number.
imagPart :: (RealFloat a) => Complex a -> a
imagPart (_ :+ y) = y

-- | The conjugate of a complex number.
{-# SPECIALISE conjugate :: Complex Double -> Complex Double #-}
conjugate :: (RealFloat a) => Complex a -> 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 :: (RealFloat 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 :: (RealFloat 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 :: (RealFloat a) => Complex a -> (a,a)
polar z = (magnitude z, phase z)

-- | The nonnegative magnitude of a complex number.
{-# SPECIALISE magnitude :: Complex Double -> Double #-}
magnitude :: (RealFloat 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 :: (RealFloat a) => Complex a -> a
phase (0 :+ 0) = 0 -- SLPJ July 97 from John Peterson
phase (x:+y) = atan2 y x


-- -----------------------------------------------------------------------------
-- Instances of Complex

#include "Typeable.h"
INSTANCE_TYPEABLE1(Complex,complexTc,"Complex")

instance (RealFloat 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 (RealFloat a) => Fractional (Complex a) where
 {-# SPECIALISE instance Fractional (Complex Float) #-}
 {-# SPECIALISE instance Fractional (Complex Double) #-}
 (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 (RealFloat 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

 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))