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2 -- Standard Library: Complex numbers
4 -- Suitable for use with Hugs 98
5 -----------------------------------------------------------------------------
7 module Complex(Complex((:+)), realPart, imagPart, conjugate, mkPolar,
8 cis, polar, magnitude, phase) where
12 data (RealFloat a) => Complex a = !a :+ !a
13 deriving (Eq,Read,Show)
15 realPart, imagPart :: (RealFloat a) => Complex a -> a
19 conjugate :: (RealFloat a) => Complex a -> Complex a
20 conjugate (x:+y) = x :+ (-y)
22 mkPolar :: (RealFloat a) => a -> a -> Complex a
23 mkPolar r theta = r * cos theta :+ r * sin theta
25 cis :: (RealFloat a) => a -> Complex a
26 cis theta = cos theta :+ sin theta
28 polar :: (RealFloat a) => Complex a -> (a,a)
29 polar z = (magnitude z, phase z)
31 magnitude, phase :: (RealFloat a) => Complex a -> a
32 magnitude (x:+y) = scaleFloat k
33 (sqrt ((scaleFloat mk x)^2 + (scaleFloat mk y)^2))
34 where k = max (exponent x) (exponent y)
37 phase (x:+y) = atan2 y x
39 instance (RealFloat a) => Num (Complex a) where
40 (x:+y) + (x':+y') = (x+x') :+ (y+y')
41 (x:+y) - (x':+y') = (x-x') :+ (y-y')
42 (x:+y) * (x':+y') = (x*x'-y*y') :+ (x*y'+y*x')
43 negate (x:+y) = negate x :+ negate y
44 abs z = magnitude z :+ 0
46 signum z@(x:+y) = x/r :+ y/r where r = magnitude z
47 fromInteger n = fromInteger n :+ 0
48 fromInt n = fromInt n :+ 0
50 instance (RealFloat a) => Fractional (Complex a) where
51 (x:+y) / (x':+y') = (x*x''+y*y'') / d :+ (y*x''-x*y'') / d
52 where x'' = scaleFloat k x'
54 k = - max (exponent x') (exponent y')
56 fromRational a = fromRational a :+ 0
57 fromDouble a = fromDouble a :+ 0
59 instance (RealFloat a) => Floating (Complex a) where
61 exp (x:+y) = expx * cos y :+ expx * sin y
63 log z = log (magnitude z) :+ phase z
65 sqrt z@(x:+y) = u :+ (if y < 0 then -v else v)
66 where (u,v) = if x < 0 then (v',u') else (u',v')
68 u' = sqrt ((magnitude z + abs x) / 2)
69 sin (x:+y) = sin x * cosh y :+ cos x * sinh y
70 cos (x:+y) = cos x * cosh y :+ (- sin x * sinh y)
71 tan (x:+y) = (sinx*coshy:+cosx*sinhy)/(cosx*coshy:+(-sinx*sinhy))
76 sinh (x:+y) = cos y * sinh x :+ sin y * cosh x
77 cosh (x:+y) = cos y * cosh x :+ sin y * sinh x
78 tanh (x:+y) = (cosy*sinhx:+siny*coshx)/(cosy*coshx:+siny*sinhx)
83 asin z@(x:+y) = y':+(-x')
84 where (x':+y') = log (((-y):+x) + sqrt (1 - z*z))
85 acos z@(x:+y) = y'':+(-x'')
86 where (x'':+y'') = log (z + ((-y'):+x'))
87 (x':+y') = sqrt (1 - z*z)
88 atan z@(x:+y) = y':+(-x')
89 where (x':+y') = log (((1-y):+x) / sqrt (1+z*z))
90 asinh z = log (z + sqrt (1+z*z))
91 acosh z = log (z + (z+1) * sqrt ((z-1)/(z+1)))
92 atanh z = log ((1+z) / sqrt (1-z*z))
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