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%
% (c) The AQUA Project, Glasgow University, 1994-1999
%
\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
conjugate :: (RealFloat a) => Complex a -> Complex a
conjugate (x:+y) = x :+ (-y)
mkPolar :: (RealFloat a) => a -> a -> Complex a
mkPolar r theta = r * cos theta :+ r * sin theta
cis :: (RealFloat a) => a -> Complex a
cis theta = cos theta :+ sin theta
polar :: (RealFloat a) => Complex a -> (a,a)
polar z = (magnitude z, phase z)
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
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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