Data/Complex.hs

{-# LANGUAGE DeriveDataTypeable #-}
{-# LANGUAGE Trustworthy #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE DeriveGeneric #-}
{-# LANGUAGE DeriveTraversable #-}

-----------------------------------------------------------------------------
-- |
-- 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 GHC.Base (Applicative (..))
import GHC.Generics (Generic, Generic1)
import GHC.Float (Floating(..))
import Data.Data (Data)
import Foreign (Storable, castPtr, peek, poke, pokeElemOff, peekElemOff, sizeOf,
 alignment)

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.
--
-- The 'Foldable' and 'Traversable' instances traverse the real part first.
--
-- Note that `Complex`'s instances inherit the deficiencies from the type
-- parameter's. For example, @Complex Float@'s 'Ord' instance has similar
-- problems to `Float`'s.
data Complex a
 = !a :+ !a -- ^ forms a complex number from its real and imaginary
 -- rectangular components.
 deriving ( Eq -- ^ @since 2.01
 , Show -- ^ @since 2.01
 , Read -- ^ @since 2.01
 , Data -- ^ @since 2.01
 , Generic -- ^ @since 4.9.0.0
 , Generic1 -- ^ @since 4.9.0.0
 , Functor -- ^ @since 4.9.0.0
 , Foldable -- ^ @since 4.9.0.0
 , Traversable -- ^ @since 4.9.0.0
 )

-- -----------------------------------------------------------------------------
-- 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

-- | @since 2.01
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

-- | @since 2.01
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

-- | @since 2.01
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

 x ** y = case (x,y) of
 (_ , (0:+0)) -> 1 :+ 0
 ((0:+0), (exp_re:+_)) -> case compare exp_re 0 of
 GT -> 0 :+ 0
 LT -> inf :+ 0
 EQ -> nan :+ nan
 ((re:+im), (exp_re:+_))
 | (isInfinite re || isInfinite im) -> case compare exp_re 0 of
 GT -> inf :+ 0
 LT -> 0 :+ 0
 EQ -> nan :+ nan
 | otherwise -> exp (log x * y)
 where
 inf = 1/0
 nan = 0/0

 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))
 -- Take care to allow (-1)::Complex, fixing #8532
 acosh z = log (z + (sqrt $ z+1) * (sqrt $ z-1))
 atanh z = 0.5 * log ((1.0+z) / (1.0-z))

 log1p x@(a :+ b)
 | abs a < 0.5 && abs b < 0.5
 , u <- 2*a + a*a + b*b = log1p (u/(1 + sqrt(u+1))) :+ atan2 (1 + a) b
 | otherwise = log (1 + x)
 {-# INLINE log1p #-}

 expm1 x@(a :+ b)
 | a*a + b*b < 1
 , u <- expm1 a
 , v <- sin (b/2)
 , w <- -2*v*v = (u*w + u + w) :+ (u+1)*sin b
 | otherwise = exp x - 1
 {-# INLINE expm1 #-}

-- | @since 4.8.0.0
instance Storable a => Storable (Complex a) where
 sizeOf a = 2 * sizeOf (realPart a)
 alignment a = alignment (realPart a)
 peek p = do
 q <- return $ castPtr p
 r <- peek q
 i <- peekElemOff q 1
 return (r :+ i)
 poke p (r :+ i) = do
 q <-return $ (castPtr p)
 poke q r
 pokeElemOff q 1 i

-- | @since 4.9.0.0
instance Applicative Complex where
 pure a = a :+ a
 f :+ g <*> a :+ b = f a :+ g b
 liftA2 f (x :+ y) (a :+ b) = f x a :+ f y b

-- | @since 4.9.0.0
instance Monad Complex where
 a :+ b >>= f = realPart (f a) :+ imagPart (f b)

-- -----------------------------------------------------------------------------
-- Rules on Complex

{-# RULES

"realToFrac/a->Complex Double"
 realToFrac = \x -> realToFrac x :+ (0 :: Double)

"realToFrac/a->Complex Float"
 realToFrac = \x -> realToFrac x :+ (0 :: Float)

 #-}