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)
    import Control.Monad.Fix (MonadFix(..))
    import Control.Monad.Zip (MonadZip(..))

    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)

    -- | @since 4.15.0.0
    instance MonadZip Complex where
      mzipWith = liftA2

    -- | @since 4.15.0.0
    instance MonadFix Complex where
      mfix f = (let a :+ _ = f a in a) :+ (let _ :+ a = f a in a)

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

    {-# RULES

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

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

      #-}