2 % (c) The AQUA Project, Glasgow University, 1994-1999
5 \section[Complex]{Module @Complex@}
11 , realPart -- :: (RealFloat a) => Complex a -> a
12 , imagPart -- :: (RealFloat a) => Complex a -> a
13 , conjugate -- :: (RealFloat a) => Complex a -> Complex a
14 , mkPolar -- :: (RealFloat a) => a -> a -> Complex a
15 , cis -- :: (RealFloat a) => a -> Complex a
16 , polar -- :: (RealFloat a) => Complex a -> (a,a)
17 , magnitude -- :: (RealFloat a) => Complex a -> a
18 , phase -- :: (RealFloat a) => Complex a -> a
22 -- (RealFloat a) => Eq (Complex a)
23 -- (RealFloat a) => Read (Complex a)
24 -- (RealFloat a) => Show (Complex a)
25 -- (RealFloat a) => Num (Complex a)
26 -- (RealFloat a) => Fractional (Complex a)
27 -- (RealFloat a) => Floating (Complex a)
29 -- Implementation checked wrt. Haskell 98 lib report, 1/99.
38 %*********************************************************
40 \subsection{The @Complex@ type}
42 %*********************************************************
45 data (RealFloat a) => Complex a = !a :+ !a deriving (Eq, Read, Show)
49 %*********************************************************
51 \subsection{Functions over @Complex@}
53 %*********************************************************
56 realPart, imagPart :: (RealFloat a) => Complex a -> a
60 conjugate :: (RealFloat a) => Complex a -> Complex a
61 conjugate (x:+y) = x :+ (-y)
63 mkPolar :: (RealFloat a) => a -> a -> Complex a
64 mkPolar r theta = r * cos theta :+ r * sin theta
66 cis :: (RealFloat a) => a -> Complex a
67 cis theta = cos theta :+ sin theta
69 polar :: (RealFloat a) => Complex a -> (a,a)
70 polar z = (magnitude z, phase z)
72 magnitude :: (RealFloat a) => Complex a -> a
73 magnitude (x:+y) = scaleFloat k
74 (sqrt ((scaleFloat mk x)^2 + (scaleFloat mk y)^2))
75 where k = max (exponent x) (exponent y)
78 phase :: (RealFloat a) => Complex a -> a
79 phase (0 :+ 0) = 0 -- SLPJ July 97 from John Peterson
80 phase (x:+y) = atan2 y x
84 %*********************************************************
86 \subsection{Instances of @Complex@}
88 %*********************************************************
91 instance (RealFloat a) => Num (Complex a) where
92 (x:+y) + (x':+y') = (x+x') :+ (y+y')
93 (x:+y) - (x':+y') = (x-x') :+ (y-y')
94 (x:+y) * (x':+y') = (x*x'-y*y') :+ (x*y'+y*x')
95 negate (x:+y) = negate x :+ negate y
96 abs z = magnitude z :+ 0
98 signum z@(x:+y) = x/r :+ y/r where r = magnitude z
99 fromInteger n = fromInteger n :+ 0
101 instance (RealFloat a) => Fractional (Complex a) where
102 (x:+y) / (x':+y') = (x*x''+y*y'') / d :+ (y*x''-x*y'') / d
103 where x'' = scaleFloat k x'
104 y'' = scaleFloat k y'
105 k = - max (exponent x') (exponent y')
108 fromRational a = fromRational a :+ 0
110 instance (RealFloat a) => Floating (Complex a) where
112 exp (x:+y) = expx * cos y :+ expx * sin y
114 log z = log (magnitude z) :+ phase z
117 sqrt z@(x:+y) = u :+ (if y < 0 then -v else v)
118 where (u,v) = if x < 0 then (v',u') else (u',v')
120 u' = sqrt ((magnitude z + abs x) / 2)
122 sin (x:+y) = sin x * cosh y :+ cos x * sinh y
123 cos (x:+y) = cos x * cosh y :+ (- sin x * sinh y)
124 tan (x:+y) = (sinx*coshy:+cosx*sinhy)/(cosx*coshy:+(-sinx*sinhy))
130 sinh (x:+y) = cos y * sinh x :+ sin y * cosh x
131 cosh (x:+y) = cos y * cosh x :+ sin y * sinh x
132 tanh (x:+y) = (cosy*sinhx:+siny*coshx)/(cosy*coshx:+siny*sinhx)
138 asin z@(x:+y) = y':+(-x')
139 where (x':+y') = log (((-y):+x) + sqrt (1 - z*z))
141 where (x'':+y'') = log (z + ((-y'):+x'))
142 (x':+y') = sqrt (1 - z*z)
143 atan z@(x:+y) = y':+(-x')
144 where (x':+y') = log (((1-y):+x) / sqrt (1+z*z))
146 asinh z = log (z + sqrt (1+z*z))
147 acosh z = log (z + (z+1) * sqrt ((z-1)/(z+1)))
148 atanh z = log ((1+z) / sqrt (1-z*z))