+++ /dev/null
-% -----------------------------------------------------------------------------
-% $Id: Complex.lhs,v 1.7 2001/09/19 14:06:03 simonmar Exp $
-%
-% (c) The University of Glasgow, 1994-2000
-%
-
-\section[Complex]{Module @Complex@}
-
-\begin{code}
-module 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)
- --
- -- Implementation checked wrt. Haskell 98 lib report, 1/99.
-
- ) where
-
-import Prelude
-
-infix 6 :+
-\end{code}
-
-%*********************************************************
-%* *
-\subsection{The @Complex@ type}
-%* *
-%*********************************************************
-
-\begin{code}
-data (RealFloat a) => Complex a = !a :+ !a deriving (Eq, Read, Show)
-\end{code}
-
-
-%*********************************************************
-%* *
-\subsection{Functions over @Complex@}
-%* *
-%*********************************************************
-
-\begin{code}
-realPart, imagPart :: (RealFloat a) => Complex a -> a
-realPart (x :+ _) = x
-imagPart (_ :+ y) = y
-
-{-# SPECIALISE conjugate :: Complex Double -> Complex Double #-}
-conjugate :: (RealFloat a) => Complex a -> Complex a
-conjugate (x:+y) = x :+ (-y)
-
-{-# SPECIALISE mkPolar :: Double -> Double -> Complex Double #-}
-mkPolar :: (RealFloat a) => a -> a -> Complex a
-mkPolar r theta = r * cos theta :+ r * sin theta
-
-{-# SPECIALISE cis :: Double -> Complex Double #-}
-cis :: (RealFloat a) => a -> Complex a
-cis theta = cos theta :+ sin theta
-
-{-# SPECIALISE polar :: Complex Double -> (Double,Double) #-}
-polar :: (RealFloat a) => Complex a -> (a,a)
-polar z = (magnitude z, phase z)
-
-{-# 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
-
-{-# 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
-\end{code}
-
-
-%*********************************************************
-%* *
-\subsection{Instances of @Complex@}
-%* *
-%*********************************************************
-
-\begin{code}
-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
-
-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
-
-instance (RealFloat a) => Floating (Complex a) where
- {-# SPECIALISE instance Floating (Complex Float) #-}
- {-# SPECIALISE instance Floating (Complex Double) #-}
- 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 = 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))
-\end{code}
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