1 % -----------------------------------------------------------------------------
2 % $Id: Complex.lhs,v 1.7 2001/09/19 14:06:03 simonmar Exp $
4 % (c) The University of Glasgow, 1994-2000
7 \section[Complex]{Module @Complex@}
13 , realPart -- :: (RealFloat a) => Complex a -> a
14 , imagPart -- :: (RealFloat a) => Complex a -> a
15 , conjugate -- :: (RealFloat a) => Complex a -> Complex a
16 , mkPolar -- :: (RealFloat a) => a -> a -> Complex a
17 , cis -- :: (RealFloat a) => a -> Complex a
18 , polar -- :: (RealFloat a) => Complex a -> (a,a)
19 , magnitude -- :: (RealFloat a) => Complex a -> a
20 , phase -- :: (RealFloat a) => Complex a -> a
24 -- (RealFloat a) => Eq (Complex a)
25 -- (RealFloat a) => Read (Complex a)
26 -- (RealFloat a) => Show (Complex a)
27 -- (RealFloat a) => Num (Complex a)
28 -- (RealFloat a) => Fractional (Complex a)
29 -- (RealFloat a) => Floating (Complex a)
31 -- Implementation checked wrt. Haskell 98 lib report, 1/99.
40 %*********************************************************
42 \subsection{The @Complex@ type}
44 %*********************************************************
47 data (RealFloat a) => Complex a = !a :+ !a deriving (Eq, Read, Show)
51 %*********************************************************
53 \subsection{Functions over @Complex@}
55 %*********************************************************
58 realPart, imagPart :: (RealFloat a) => Complex a -> a
62 {-# SPECIALISE conjugate :: Complex Double -> Complex Double #-}
63 conjugate :: (RealFloat a) => Complex a -> Complex a
64 conjugate (x:+y) = x :+ (-y)
66 {-# SPECIALISE mkPolar :: Double -> Double -> Complex Double #-}
67 mkPolar :: (RealFloat a) => a -> a -> Complex a
68 mkPolar r theta = r * cos theta :+ r * sin theta
70 {-# SPECIALISE cis :: Double -> Complex Double #-}
71 cis :: (RealFloat a) => a -> Complex a
72 cis theta = cos theta :+ sin theta
74 {-# SPECIALISE polar :: Complex Double -> (Double,Double) #-}
75 polar :: (RealFloat a) => Complex a -> (a,a)
76 polar z = (magnitude z, phase z)
78 {-# SPECIALISE magnitude :: Complex Double -> Double #-}
79 magnitude :: (RealFloat a) => Complex a -> a
80 magnitude (x:+y) = scaleFloat k
81 (sqrt ((scaleFloat mk x)^(2::Int) + (scaleFloat mk y)^(2::Int)))
82 where k = max (exponent x) (exponent y)
85 {-# SPECIALISE phase :: Complex Double -> Double #-}
86 phase :: (RealFloat a) => Complex a -> a
87 phase (0 :+ 0) = 0 -- SLPJ July 97 from John Peterson
88 phase (x:+y) = atan2 y x
92 %*********************************************************
94 \subsection{Instances of @Complex@}
96 %*********************************************************
99 instance (RealFloat a) => Num (Complex a) where
100 {-# SPECIALISE instance Num (Complex Float) #-}
101 {-# SPECIALISE instance Num (Complex Double) #-}
102 (x:+y) + (x':+y') = (x+x') :+ (y+y')
103 (x:+y) - (x':+y') = (x-x') :+ (y-y')
104 (x:+y) * (x':+y') = (x*x'-y*y') :+ (x*y'+y*x')
105 negate (x:+y) = negate x :+ negate y
106 abs z = magnitude z :+ 0
108 signum z@(x:+y) = x/r :+ y/r where r = magnitude z
109 fromInteger n = fromInteger n :+ 0
111 instance (RealFloat a) => Fractional (Complex a) where
112 {-# SPECIALISE instance Fractional (Complex Float) #-}
113 {-# SPECIALISE instance Fractional (Complex Double) #-}
114 (x:+y) / (x':+y') = (x*x''+y*y'') / d :+ (y*x''-x*y'') / d
115 where x'' = scaleFloat k x'
116 y'' = scaleFloat k y'
117 k = - max (exponent x') (exponent y')
120 fromRational a = fromRational a :+ 0
122 instance (RealFloat a) => Floating (Complex a) where
123 {-# SPECIALISE instance Floating (Complex Float) #-}
124 {-# SPECIALISE instance Floating (Complex Double) #-}
126 exp (x:+y) = expx * cos y :+ expx * sin y
128 log z = log (magnitude z) :+ phase z
131 sqrt z@(x:+y) = u :+ (if y < 0 then -v else v)
132 where (u,v) = if x < 0 then (v',u') else (u',v')
134 u' = sqrt ((magnitude z + abs x) / 2)
136 sin (x:+y) = sin x * cosh y :+ cos x * sinh y
137 cos (x:+y) = cos x * cosh y :+ (- sin x * sinh y)
138 tan (x:+y) = (sinx*coshy:+cosx*sinhy)/(cosx*coshy:+(-sinx*sinhy))
144 sinh (x:+y) = cos y * sinh x :+ sin y * cosh x
145 cosh (x:+y) = cos y * cosh x :+ sin y * sinh x
146 tanh (x:+y) = (cosy*sinhx:+siny*coshx)/(cosy*coshx:+siny*sinhx)
152 asin z@(x:+y) = y':+(-x')
153 where (x':+y') = log (((-y):+x) + sqrt (1 - z*z))
155 where (x'':+y'') = log (z + ((-y'):+x'))
156 (x':+y') = sqrt (1 - z*z)
157 atan z@(x:+y) = y':+(-x')
158 where (x':+y') = log (((1-y):+x) / sqrt (1+z*z))
160 asinh z = log (z + sqrt (1+z*z))
161 acosh z = log (z + (z+1) * sqrt ((z-1)/(z+1)))
162 atanh z = log ((1+z) / sqrt (1-z*z))