content: {-# 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 Data.Typeable
  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.
  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))
  
  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
  