1 % -----------------------------------------------------------------------------
2 % $Id: Complex.lhs,v 1.6 2001/09/19 14:05:01 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 {-# SPECIALISE realPart :: Complex Double -> Double #-}
59 {-# SPECIALISE imagPart :: Complex Double -> Double #-}
60 realPart, imagPart :: (RealFloat a) => Complex a -> a
64 {-# SPECIALISE conjugate :: Complex Double -> Complex Double #-}
65 conjugate :: (RealFloat a) => Complex a -> Complex a
66 conjugate (x:+y) = x :+ (-y)
68 {-# SPECIALISE mkPolar :: Double -> Double -> Complex Double #-}
69 mkPolar :: (RealFloat a) => a -> a -> Complex a
70 mkPolar r theta = r * cos theta :+ r * sin theta
72 {-# SPECIALISE cis :: Double -> Complex Double #-}
73 cis :: (RealFloat a) => a -> Complex a
74 cis theta = cos theta :+ sin theta
76 {-# SPECIALISE polar :: Complex Double -> (Double,Double) #-}
77 polar :: (RealFloat a) => Complex a -> (a,a)
78 polar z = (magnitude z, phase z)
80 {-# SPECIALISE magnitude :: Complex Double -> Double #-}
81 magnitude :: (RealFloat a) => Complex a -> a
82 magnitude (x:+y) = scaleFloat k
83 (sqrt ((scaleFloat mk x)^(2::Int) + (scaleFloat mk y)^(2::Int)))
84 where k = max (exponent x) (exponent y)
87 {-# SPECIALISE phase :: Complex Double -> Double #-}
88 phase :: (RealFloat a) => Complex a -> a
89 phase (0 :+ 0) = 0 -- SLPJ July 97 from John Peterson
90 phase (x:+y) = atan2 y x
94 %*********************************************************
96 \subsection{Instances of @Complex@}
98 %*********************************************************
101 instance (RealFloat a) => Num (Complex a) where
102 {-# SPECIALISE instance Num (Complex Float) #-}
103 {-# SPECIALISE instance Num (Complex Double) #-}
104 (x:+y) + (x':+y') = (x+x') :+ (y+y')
105 (x:+y) - (x':+y') = (x-x') :+ (y-y')
106 (x:+y) * (x':+y') = (x*x'-y*y') :+ (x*y'+y*x')
107 negate (x:+y) = negate x :+ negate y
108 abs z = magnitude z :+ 0
110 signum z@(x:+y) = x/r :+ y/r where r = magnitude z
111 fromInteger n = fromInteger n :+ 0
113 instance (RealFloat a) => Fractional (Complex a) where
114 {-# SPECIALISE instance Fractional (Complex Float) #-}
115 {-# SPECIALISE instance Fractional (Complex Double) #-}
116 (x:+y) / (x':+y') = (x*x''+y*y'') / d :+ (y*x''-x*y'') / d
117 where x'' = scaleFloat k x'
118 y'' = scaleFloat k y'
119 k = - max (exponent x') (exponent y')
122 fromRational a = fromRational a :+ 0
124 instance (RealFloat a) => Floating (Complex a) where
125 {-# SPECIALISE instance Floating (Complex Float) #-}
126 {-# SPECIALISE instance Floating (Complex Double) #-}
128 exp (x:+y) = expx * cos y :+ expx * sin y
130 log z = log (magnitude z) :+ phase z
133 sqrt z@(x:+y) = u :+ (if y < 0 then -v else v)
134 where (u,v) = if x < 0 then (v',u') else (u',v')
136 u' = sqrt ((magnitude z + abs x) / 2)
138 sin (x:+y) = sin x * cosh y :+ cos x * sinh y
139 cos (x:+y) = cos x * cosh y :+ (- sin x * sinh y)
140 tan (x:+y) = (sinx*coshy:+cosx*sinhy)/(cosx*coshy:+(-sinx*sinhy))
146 sinh (x:+y) = cos y * sinh x :+ sin y * cosh x
147 cosh (x:+y) = cos y * cosh x :+ sin y * sinh x
148 tanh (x:+y) = (cosy*sinhx:+siny*coshx)/(cosy*coshx:+siny*sinhx)
154 asin z@(x:+y) = y':+(-x')
155 where (x':+y') = log (((-y):+x) + sqrt (1 - z*z))
157 where (x'':+y'') = log (z + ((-y'):+x'))
158 (x':+y') = sqrt (1 - z*z)
159 atan z@(x:+y) = y':+(-x')
160 where (x':+y') = log (((1-y):+x) / sqrt (1+z*z))
162 asinh z = log (z + sqrt (1+z*z))
163 acosh z = log (z + (z+1) * sqrt ((z-1)/(z+1)))
164 atanh z = log ((1+z) / sqrt (1-z*z))