+{-# LANGUAGE CPP, DeriveDataTypeable #-}
+#ifdef __GLASGOW_HASKELL__
+{-# LANGUAGE StandaloneDeriving #-}
+#endif
+
-----------------------------------------------------------------------------
-- |
-- Module : Data.Complex
-----------------------------------------------------------------------------
module Data.Complex
- ( Complex((:+))
-
- , realPart -- :: (RealFloat a) => Complex a -> a
- , imagPart -- :: (RealFloat a) => Complex a -> a
- , conjugate -- :: (RealFloat a) => Complex a -> Complex a
- , 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
-
- -- Complex instances:
- --
- -- (RealFloat a) => Eq (Complex a)
- -- (RealFloat a) => Read (Complex a)
- -- (RealFloat a) => Show (Complex a)
- -- (RealFloat a) => Num (Complex a)
- -- (RealFloat a) => Fractional (Complex a)
- -- (RealFloat a) => Floating (Complex a)
- --
+ (
+ -- * 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
+
+ -- Complex instances:
+ --
+ -- (RealFloat a) => Eq (Complex a)
+ -- (RealFloat a) => Read (Complex a)
+ -- (RealFloat a) => Show (Complex a)
+ -- (RealFloat a) => Num (Complex a)
+ -- (RealFloat a) => Fractional (Complex a)
+ -- (RealFloat a) => Floating (Complex a)
+ --
-- Implementation checked wrt. Haskell 98 lib report, 1/99.
) where
import Prelude
-#ifndef __NHC__
-import Data.Dynamic
+import Data.Typeable
+#ifdef __GLASGOW_HASKELL__
+import Data.Data (Data)
+#endif
+
+#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 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 :: (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
+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
+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)
+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
+ (sqrt (sqr (scaleFloat mk x) + sqr (scaleFloat mk y)))
+ where k = max (exponent x) (exponent y)
+ mk = - k
+ sqr z = z * z
+-- | 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
+phase (0 :+ 0) = 0 -- SLPJ July 97 from John Peterson
+phase (x:+y) = atan2 y x
-- -----------------------------------------------------------------------------
-- Instances of Complex
-#ifndef __NHC__
-#include "Dynamic.h"
+#include "Typeable.h"
INSTANCE_TYPEABLE1(Complex,complexTc,"Complex")
-#endif
instance (RealFloat a) => Num (Complex a) where
{-# SPECIALISE instance Num (Complex Float) #-}
{-# SPECIALISE instance Num (Complex Double) #-}
- (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
+ (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) = 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) #-}
{-# SPECIALISE instance Fractional (Complex Double) #-}
- (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
+ (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
+#ifdef __HUGS__
+ fromDouble a = fromDouble a :+ 0
+#endif
-instance (RealFloat a) => Floating (Complex a) where
+instance (RealFloat a) => Floating (Complex a) where
{-# SPECIALISE instance Floating (Complex Float) #-}
{-# SPECIALISE instance Floating (Complex Double) #-}
pi = pi :+ 0
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)
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))
+ atanh z = 0.5 * log ((1.0+z) / (1.0-z))
projects / ghc-base.git / blobdiff