Data/Complex.hs

    {-# LANGUAGE CPP, PackageImports #-}
    #if __GLASGOW_HASKELL__ >= 701
    {-# LANGUAGE Safe #-}
    #endif

    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

            -- * Specification

            -- $code
      ) where
    import "base" Data.Complex

    {- $code
    > module Data.Complex(Complex((:+)), realPart, imagPart, conjugate, mkPolar,
    >                     cis, polar, magnitude, phase)  where
    >
    > infix  6  :+
    >
    > data  (RealFloat a)     => Complex a = !a :+ !a  deriving (Eq,Read,Show)
    >
    >
    > realPart, imagPart :: (RealFloat a) => Complex a -> a
    > realPart (x:+y)        =  x
    > imagPart (x:+y)        =  y
    >
    > conjugate      :: (RealFloat a) => Complex a -> Complex a
    > conjugate (x:+y) =  x :+ (-y)
    >
    > mkPolar                :: (RealFloat a) => a -> a -> Complex a
    > mkPolar r theta        =  r * cos theta :+ r * sin theta
    >
    > cis            :: (RealFloat a) => a -> Complex a
    > cis theta      =  cos theta :+ sin theta
    >
    > polar          :: (RealFloat a) => Complex a -> (a,a)
    > polar z                =  (magnitude z, phase z)
    >
    > magnitude :: (RealFloat a) => Complex a -> a
    > magnitude (x:+y) =  scaleFloat k
    >                    (sqrt ((scaleFloat mk x)^2 + (scaleFloat mk y)^2))
    >                   where k  = max (exponent x) (exponent y)
    >                         mk = - k
    >
    > phase :: (RealFloat a) => Complex a -> a
    > phase (0 :+ 0) = 0
    > phase (x :+ y) = atan2 y x
    >
    >
    > instance  (RealFloat a) => Num (Complex a)  where
    >     (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
    >     signum z@(x:+y)   =  x/r :+ y/r  where r = magnitude z
    >     fromInteger n     =  fromInteger n :+ 0
    >
    > instance  (RealFloat a) => Fractional (Complex a)  where
    >     (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
    >     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
    >     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@(x:+y)  =  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))
    > -}