haskell2010-1.1.2.0: 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))
> -}