-----------------------------------------------------------------------------
module Data.Complex
- ( Complex((:+))
-
+ (
+ -- * Rectangular form
+ Complex((:+))
+
, realPart -- :: (RealFloat a) => Complex a -> a
, imagPart -- :: (RealFloat a) => Complex a -> a
- , conjugate -- :: (RealFloat a) => Complex a -> Complex 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
+
-- Complex instances:
--
-- (RealFloat a) => Eq (Complex a)
import Prelude
-#ifndef __NHC__
import Data.Typeable
+#ifdef __GLASGOW_HASKELL__
+import Data.Generics.Basics( Data )
#endif
#ifdef __HUGS__
-- -----------------------------------------------------------------------------
-- The Complex type
-data (RealFloat a) => Complex a = !a :+ !a deriving (Eq, Read, Show)
-
+-- | 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 (RealFloat a) => Complex a
+ = !a :+ !a -- ^ forms a complex number from its real and imaginary
+ -- rectangular components.
+# if __GLASGOW_HASKELL__
+ deriving (Eq, Show, Read, Data)
+# else
+ deriving (Eq, Show, Read)
+# endif
-- -----------------------------------------------------------------------------
-- Functions over Complex
-realPart, imagPart :: (RealFloat a) => Complex a -> a
+-- | Extracts the real part of a complex number.
+realPart :: (RealFloat a) => Complex a -> a
realPart (x :+ _) = x
+
+-- | Extracts the imaginary part of a complex number.
+imagPart :: (RealFloat a) => Complex a -> a
imagPart (_ :+ y) = y
+-- | The conjugate of a complex number.
{-# SPECIALISE conjugate :: Complex Double -> Complex Double #-}
conjugate :: (RealFloat 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 :: (RealFloat 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 :: (RealFloat 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
where k = max (exponent x) (exponent y)
mk = - k
+-- | 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
-- -----------------------------------------------------------------------------
-- Instances of Complex
-#ifndef __NHC__
#include "Typeable.h"
INSTANCE_TYPEABLE1(Complex,complexTc,"Complex")
-#endif
instance (RealFloat a) => Num (Complex a) where
{-# SPECIALISE instance Num (Complex Float) #-}
(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 (0:+0) = 0
signum z@(x:+y) = x/r :+ y/r where r = magnitude z
fromInteger n = fromInteger n :+ 0
#ifdef __HUGS__
where expx = exp x
log z = log (magnitude z) :+ phase z
- sqrt 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)