#ifdef __GLASGOW_HASKELL__
#endif
module Data.Complex
(
Complex((:+))
, realPart
, imagPart
, mkPolar
, cis
, polar
, magnitude
, phase
, conjugate
) where
import Prelude
import Data.Typeable
#ifdef __GLASGOW_HASKELL__
import Data.Data (Data)
#endif
#ifdef __HUGS__
import Hugs.Prelude(Num(fromInt), Fractional(fromDouble))
#endif
infix 6 :+
data Complex a
= !a :+ !a
# if __GLASGOW_HASKELL__
deriving (Eq, Show, Read, Data)
# else
deriving (Eq, Show, Read)
# endif
realPart :: (RealFloat a) => Complex a -> a
realPart (x :+ _) = x
imagPart :: (RealFloat a) => Complex a -> a
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 (sqr (scaleFloat mk x) + sqr (scaleFloat mk y)))
where k = max (exponent x) (exponent y)
mk = k
sqr z = z * z
phase :: (RealFloat a) => Complex a -> a
phase (0 :+ 0) = 0
phase (x:+y) = atan2 y x
#include "Typeable.h"
INSTANCE_TYPEABLE1(Complex,complexTc,"Complex")
instance (RealFloat a) => Num (Complex a) where
(x:+y) + (x':+y') = (x+x') :+ (y+y')
(x:+y) (x':+y') = (xx') :+ (yy')
(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
(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
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) = 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 (((1y):+x) / sqrt (1+z*z))
asinh z = log (z + sqrt (1+z*z))
acosh z = log (z + (z+1) * sqrt ((z1)/(z+1)))
atanh z = 0.5 * log ((1.0+z) / (1.0z))
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