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{-# LANGUAGE Trustworthy #-}
{-# LANGUAGE AutoDeriveTypeable #-}
{-# LANGUAGE StandaloneDeriving #-}
-----------------------------------------------------------------------------
-- |
-- 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
, imagPart
-- * Polar form
, mkPolar
, cis
, polar
, magnitude
, phase
-- * Conjugate
, conjugate
) where
import Prelude
import Data.Typeable
import Data.Data (Data)
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.
deriving (Eq, Show, Read, Data, Typeable)
-- -----------------------------------------------------------------------------
-- 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 => 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 :: 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 :: (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
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
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
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 = 0.5 * log ((1.0+z) / (1.0-z))
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