Data/Complex.hs

   1 -----------------------------------------------------------------------------
   2 -- |
   3 -- Module      :  Data.Complex
   4 -- Copyright   :  (c) The University of Glasgow 2001
   5 -- License     :  BSD-style (see the file libraries/base/LICENSE)
   6 --
   7 -- Maintainer  :  libraries@haskell.org
   8 -- Stability   :  provisional
   9 -- Portability :  portable
  10 --
  11 -- Complex numbers.
  12 --
  13 -----------------------------------------------------------------------------
  14
  15 module Data.Complex
  16         (
  17         -- * Rectangular form
  18           Complex((:+))
  19
  20         , realPart      -- :: (RealFloat a) => Complex a -> a
  21         , imagPart      -- :: (RealFloat a) => Complex a -> a
  22         -- * Polar form
  23         , mkPolar       -- :: (RealFloat a) => a -> a -> Complex a
  24         , cis           -- :: (RealFloat a) => a -> Complex a
  25         , polar         -- :: (RealFloat a) => Complex a -> (a,a)
  26         , magnitude     -- :: (RealFloat a) => Complex a -> a
  27         , phase         -- :: (RealFloat a) => Complex a -> a
  28         -- * Conjugate
  29         , conjugate     -- :: (RealFloat a) => Complex a -> Complex a
  30
  31         -- Complex instances:
  32         --
  33         --  (RealFloat a) => Eq         (Complex a)
  34         --  (RealFloat a) => Read       (Complex a)
  35         --  (RealFloat a) => Show       (Complex a)
  36         --  (RealFloat a) => Num        (Complex a)
  37         --  (RealFloat a) => Fractional (Complex a)
  38         --  (RealFloat a) => Floating   (Complex a)
  39         --
  40         -- Implementation checked wrt. Haskell 98 lib report, 1/99.
  41
  42         )  where
  43
  44 import Prelude
  45
  46 import Data.Typeable
  47
  48 #ifdef __HUGS__
  49 import Hugs.Prelude(Num(fromInt), Fractional(fromDouble))
  50 #endif
  51
  52 infix  6  :+
  53
  54 -- -----------------------------------------------------------------------------
  55 -- The Complex type
  56
  57 -- | Complex numbers are an algebraic type.
  58 --
  59 -- For a complex number @z@, @'abs' z@ is a number with the magnitude of @z@,
  60 -- but oriented in the positive real direction, whereas @'signum' z@
  61 -- has the phase of @z@, but unit magnitude.
  62 data (RealFloat a) => Complex a
  63   = !a :+ !a    -- ^ forms a complex number from its real and imaginary
  64                 -- rectangular components.
  65   deriving (Eq, Read, Show)
  66
  67 -- -----------------------------------------------------------------------------
  68 -- Functions over Complex
  69
  70 -- | Extracts the real part of a complex number.
  71 realPart :: (RealFloat a) => Complex a -> a
  72 realPart (x :+ _) =  x
  73
  74 -- | Extracts the imaginary part of a complex number.
  75 imagPart :: (RealFloat a) => Complex a -> a
  76 imagPart (_ :+ y) =  y
  77
  78 -- | The conjugate of a complex number.
  79 {-# SPECIALISE conjugate :: Complex Double -> Complex Double #-}
  80 conjugate        :: (RealFloat a) => Complex a -> Complex a
  81 conjugate (x:+y) =  x :+ (-y)
  82
  83 -- | Form a complex number from polar components of magnitude and phase.
  84 {-# SPECIALISE mkPolar :: Double -> Double -> Complex Double #-}
  85 mkPolar          :: (RealFloat a) => a -> a -> Complex a
  86 mkPolar r theta  =  r * cos theta :+ r * sin theta
  87
  88 -- | @'cis' t@ is a complex value with magnitude @1@
  89 -- and phase @t@ (modulo @2*'pi'@).
  90 {-# SPECIALISE cis :: Double -> Complex Double #-}
  91 cis              :: (RealFloat a) => a -> Complex a
  92 cis theta        =  cos theta :+ sin theta
  93
  94 -- | The function 'polar' takes a complex number and
  95 -- returns a (magnitude, phase) pair in canonical form:
  96 -- the magnitude is nonnegative, and the phase in the range @(-'pi', 'pi']@;
  97 -- if the magnitude is zero, then so is the phase.
  98 {-# SPECIALISE polar :: Complex Double -> (Double,Double) #-}
  99 polar            :: (RealFloat a) => Complex a -> (a,a)
 100 polar z          =  (magnitude z, phase z)
 101
 102 -- | The nonnegative magnitude of a complex number.
 103 {-# SPECIALISE magnitude :: Complex Double -> Double #-}
 104 magnitude :: (RealFloat a) => Complex a -> a
 105 magnitude (x:+y) =  scaleFloat k
 106                      (sqrt ((scaleFloat mk x)^(2::Int) + (scaleFloat mk y)^(2::Int)))
 107                     where k  = max (exponent x) (exponent y)
 108                           mk = - k
 109
 110 -- | The phase of a complex number, in the range @(-'pi', 'pi']@.
 111 -- If the magnitude is zero, then so is the phase.
 112 {-# SPECIALISE phase :: Complex Double -> Double #-}
 113 phase :: (RealFloat a) => Complex a -> a
 114 phase (0 :+ 0)   = 0            -- SLPJ July 97 from John Peterson
 115 phase (x:+y)     = atan2 y x
 116
 117
 118 -- -----------------------------------------------------------------------------
 119 -- Instances of Complex
 120
 121 #include "Typeable.h"
 122 INSTANCE_TYPEABLE1(Complex,complexTc,"Complex")
 123
 124 instance  (RealFloat a) => Num (Complex a)  where
 125     {-# SPECIALISE instance Num (Complex Float) #-}
 126     {-# SPECIALISE instance Num (Complex Double) #-}
 127     (x:+y) + (x':+y')   =  (x+x') :+ (y+y')
 128     (x:+y) - (x':+y')   =  (x-x') :+ (y-y')
 129     (x:+y) * (x':+y')   =  (x*x'-y*y') :+ (x*y'+y*x')
 130     negate (x:+y)       =  negate x :+ negate y
 131     abs z               =  magnitude z :+ 0
 132     signum 0            =  0
 133     signum z@(x:+y)     =  x/r :+ y/r  where r = magnitude z
 134     fromInteger n       =  fromInteger n :+ 0
 135 #ifdef __HUGS__
 136     fromInt n           =  fromInt n :+ 0
 137 #endif
 138
 139 instance  (RealFloat a) => Fractional (Complex a)  where
 140     {-# SPECIALISE instance Fractional (Complex Float) #-}
 141     {-# SPECIALISE instance Fractional (Complex Double) #-}
 142     (x:+y) / (x':+y')   =  (x*x''+y*y'') / d :+ (y*x''-x*y'') / d
 143                            where x'' = scaleFloat k x'
 144                                  y'' = scaleFloat k y'
 145                                  k   = - max (exponent x') (exponent y')
 146                                  d   = x'*x'' + y'*y''
 147
 148     fromRational a      =  fromRational a :+ 0
 149 #ifdef __HUGS__
 150     fromDouble a        =  fromDouble a :+ 0
 151 #endif
 152
 153 instance  (RealFloat a) => Floating (Complex a) where
 154     {-# SPECIALISE instance Floating (Complex Float) #-}
 155     {-# SPECIALISE instance Floating (Complex Double) #-}
 156     pi             =  pi :+ 0
 157     exp (x:+y)     =  expx * cos y :+ expx * sin y
 158                       where expx = exp x
 159     log z          =  log (magnitude z) :+ phase z
 160
 161     sqrt 0         =  0
 162     sqrt z@(x:+y)  =  u :+ (if y < 0 then -v else v)
 163                       where (u,v) = if x < 0 then (v',u') else (u',v')
 164                             v'    = abs y / (u'*2)
 165                             u'    = sqrt ((magnitude z + abs x) / 2)
 166
 167     sin (x:+y)     =  sin x * cosh y :+ cos x * sinh y
 168     cos (x:+y)     =  cos x * cosh y :+ (- sin x * sinh y)
 169     tan (x:+y)     =  (sinx*coshy:+cosx*sinhy)/(cosx*coshy:+(-sinx*sinhy))
 170                       where sinx  = sin x
 171                             cosx  = cos x
 172                             sinhy = sinh y
 173                             coshy = cosh y
 174
 175     sinh (x:+y)    =  cos y * sinh x :+ sin  y * cosh x
 176     cosh (x:+y)    =  cos y * cosh x :+ sin y * sinh x
 177     tanh (x:+y)    =  (cosy*sinhx:+siny*coshx)/(cosy*coshx:+siny*sinhx)
 178                       where siny  = sin y
 179                             cosy  = cos y
 180                             sinhx = sinh x
 181                             coshx = cosh x
 182
 183     asin z@(x:+y)  =  y':+(-x')
 184                       where  (x':+y') = log (((-y):+x) + sqrt (1 - z*z))
 185     acos z         =  y'':+(-x'')
 186                       where (x'':+y'') = log (z + ((-y'):+x'))
 187                             (x':+y')   = sqrt (1 - z*z)
 188     atan z@(x:+y)  =  y':+(-x')
 189                       where (x':+y') = log (((1-y):+x) / sqrt (1+z*z))
 190
 191     asinh z        =  log (z + sqrt (1+z*z))
 192     acosh z        =  log (z + (z+1) * sqrt ((z-1)/(z+1)))
 193     atanh z        =  log ((1+z) / sqrt (1-z*z))