-----------------------------------------------------------------------------
---
+-- |
-- Module : Data.Complex
-- Copyright : (c) The University of Glasgow 2001
--- License : BSD-style (see the file libraries/core/LICENSE)
+-- License : BSD-style (see the file libraries/base/LICENSE)
--
-- Maintainer : libraries@haskell.org
-- Stability : provisional
-- Portability : portable
--
--- $Id: Complex.hs,v 1.1 2001/06/28 14:15:02 simonmar Exp $
---
-- Complex numbers.
--
-----------------------------------------------------------------------------
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
-import Data.Dynamic
+import Data.Typeable
+import Data.Generics.Basics( Data )
+
+#ifdef __HUGS__
+import Hugs.Prelude(Num(fromInt), Fractional(fromDouble))
+#endif
infix 6 :+
-- -----------------------------------------------------------------------------
-- 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
(sqrt ((scaleFloat mk x)^(2::Int) + (scaleFloat mk y)^(2::Int)))
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
phase (x:+y) = atan2 y x
-- -----------------------------------------------------------------------------
-- Instances of Complex
-#include "Dynamic.h"
+#include "Typeable.h"
INSTANCE_TYPEABLE1(Complex,complexTc,"Complex")
instance (RealFloat a) => Num (Complex a) where
signum 0 = 0
signum z@(x:+y) = x/r :+ y/r where r = magnitude z
fromInteger n = fromInteger n :+ 0
+#ifdef __HUGS__
+ fromInt n = fromInt n :+ 0
+#endif
instance (RealFloat a) => Fractional (Complex a) where
{-# SPECIALISE instance Fractional (Complex Float) #-}
d = x'*x'' + y'*y''
fromRational a = fromRational a :+ 0
+#ifdef __HUGS__
+ fromDouble a = fromDouble a :+ 0
+#endif
instance (RealFloat a) => Floating (Complex a) where
{-# SPECIALISE instance Floating (Complex Float) #-}