1 -- Complex Numbers
3 module PreludeComplex where
5 import Cls
6 import Core
8 import IDouble -- instances
9 import IChar
10 import IFloat
11 import IInt
12 import IInteger
13 import IList
14 import List ( (++), foldr )
15 import Prel ( (.), (&&), (||), (^), atan2 )
16 import PS ( _PackedString, _unpackPS )
17 import Text
18 import TyComplex
20 -- infix 6 :+
22 -- data (RealFloat a) => Complex a = a :+ a deriving (Eq,Binary,Text)
24 instance (Eq a) => Eq (Complex a) where
25 (x :+ y) == (x2 :+ y2) = x == x2 && y == y2
26 (x :+ y) /= (x2 :+ y2) = x /= x2 || y /= y2
28 instance (RealFloat a) => Num (Complex a) where
29 (x:+y) + (x2:+y2) = (x+x2) :+ (y+y2)
30 (x:+y) - (x2:+y2) = (x-x2) :+ (y-y2)
31 (x:+y) * (x2:+y2) = (x*x2-y*y2) :+ (x*y2+y*x2)
32 negate (x:+y) = negate x :+ negate y
33 abs z = magnitude z :+ 0
34 signum 0 = 0
35 signum z@(x:+y) = x/r :+ y/r where { r = magnitude z }
36 fromInteger n = fromInteger n :+ 0
37 fromInt n = fromInt n :+ 0
39 instance (RealFloat a) => Fractional (Complex a) where
40 (x:+y) / (x2:+y2) = (x*x3+y*y3) / d :+ (y*x3-x*y3) / d
41 where x3 = scaleFloat k x2
42 y3 = scaleFloat k y2
43 k = - max (exponent x2) (exponent y2)
44 d = x2*x3 + y2*y3
45 fromRational a = fromRational a :+ 0
46 recip a = (1 :+ 0) / a
48 instance (RealFloat a) => Floating (Complex a) where
49 pi = pi :+ 0
50 exp (x:+y) = expx * cos y :+ expx * sin y
51 where expx = exp x
52 log z = log (magnitude z) :+ phase z
54 (**) a b = exp (log a * b)
55 logBase a b = log b / log a
57 sqrt 0 = 0
58 sqrt z@(x:+y) = u :+ (if y < 0 then -v else v)
59 where (u,v) = if x < 0 then (v2,u2) else (u2,v2)
60 v2 = abs y / (u2*2)
61 u2 = sqrt ((magnitude z + abs x) / 2)
63 sin (x:+y) = sin x * cosh y :+ cos x * sinh y
64 cos (x:+y) = cos x * cosh y :+ (- sin x * sinh y)
65 tan (x:+y) = (sinx*coshy:+cosx*sinhy)/(cosx*coshy:+(-sinx*sinhy))
66 where sinx = sin x
67 cosx = cos x
68 sinhy = sinh y
69 coshy = cosh y
71 sinh (x:+y) = cos y * sinh x :+ sin y * cosh x
72 cosh (x:+y) = cos y * cosh x :+ sin y * sinh x
73 tanh (x:+y) = (cosy*sinhx:+siny*coshx)/(cosy*coshx:+siny*sinhx)
74 where siny = sin y
75 cosy = cos y
76 sinhx = sinh x
77 coshx = cosh x
79 asin z@(x:+y) = y2:+(-x2)
80 where (x2:+y2) = log (((-y):+x) + sqrt (1 - z*z))
81 acos z@(x:+y) = y3:+(-x3)
82 where (x3:+y3) = log (z + ((-y2):+x2))
83 (x2:+y2) = sqrt (1 - z*z)
84 atan z@(x:+y) = y2:+(-x2)
85 where (x2:+y2) = log (((1-y):+x) / sqrt (1+z*z))
87 asinh z = log (z + sqrt (1+z*z))
88 acosh z = log (z + (z+1) * sqrt ((z-1)/(z+1)))
89 atanh z = log ((1+z) / sqrt (1-z*z))
92 instance (Text a) => Text (Complex a) where
94 -- magic fixity wired in: infix 6 :+
97 = readParen ( p > 6 )
98 (\ r -> [ (x :+ y, s2) | (x, s0) <- readsPrec 7 r,
99 (":+", s1) <- lex s0,
100 (y, s2) <- readsPrec 7 s1 ])
101 showsPrec d (a :+ b)
102 = showParen (d > 6)
103 (showsPrec 7 a . showString " :+ " . showsPrec 7 b)
105 {-# SPECIALIZE instance Eq (Complex Double) #-}
106 {-# SPECIALIZE instance Num (Complex Double) #-}
107 {-# SPECIALIZE instance Fractional (Complex Double) #-}
108 {-# SPECIALIZE instance Floating (Complex Double) #-}
110 --NO:{-# SPECIALIZE instance Eq (Complex Float) #-}
111 --NO:{-# SPECIALIZE instance Num (Complex Float) #-}
112 --NO:{-# SPECIALIZE instance Fractional (Complex Float) #-}
113 --NO:{-# SPECIALIZE instance Floating (Complex Float) #-}
115 #if defined(__UNBOXED_INSTANCES__)
117 {-# SPECIALIZE instance Eq (Complex Double#) #-}
118 {-# SPECIALIZE instance Num (Complex Double#) #-}
119 {-# SPECIALIZE instance Fractional (Complex Double#) #-}
120 {-# SPECIALIZE instance Floating (Complex Double#) #-}
121 {-# SPECIALIZE instance Text (Complex Double#) #-}
123 #endif
125 -- ToDo: something for Binary
127 -- ToDo: Complex Double# s/a{/a{Double#,?/
129 --{-# GENERATE_SPECS realPart a{Double#} #-}
130 realPart :: Complex a -> a
131 realPart (x:+y) = x
133 --{-# GENERATE_SPECS imagPart a{Double#} #-}
134 imagPart :: Complex a -> a
135 imagPart (x:+y) = y
137 --{-# GENERATE_SPECS conjugate a{Double#,Double} #-}
138 {-# GENERATE_SPECS conjugate a{Double} #-}
139 conjugate :: (RealFloat a) => Complex a -> Complex a
140 conjugate (x:+y) = x :+ (-y)
142 --{-# GENERATE_SPECS mkPolar a{Double#,Double} #-}
143 {-# GENERATE_SPECS mkPolar a{Double} #-}
144 mkPolar :: (RealFloat a) => a -> a -> Complex a
145 mkPolar r theta = r * cos theta :+ r * sin theta
147 --{-# GENERATE_SPECS cis a{Double#,Double} #-}
148 {-# GENERATE_SPECS cis a{Double} #-}
149 cis :: (RealFloat a) => a -> Complex a
150 cis theta = cos theta :+ sin theta
152 --{-# GENERATE_SPECS polar a{Double#,Double} #-}
153 {-# GENERATE_SPECS polar a{Double} #-}
154 polar :: (RealFloat a) => Complex a -> (a,a)
155 polar z = (magnitude z, phase z)
157 --{-# GENERATE_SPECS magnitude a{Double#,Double} #-}
158 {-# GENERATE_SPECS magnitude a{Double} #-}
159 magnitude :: (RealFloat a) => Complex a -> a
160 magnitude (x:+y) = scaleFloat k
161 (sqrt ((scaleFloat mk x)^2 + (scaleFloat mk y)^2))
162 where k = max (exponent x) (exponent y)
163 mk = - k
165 --{-# GENERATE_SPECS phase a{Double#,Double} #-}
166 {-# GENERATE_SPECS phase a{Double} #-}
167 phase :: (RealFloat a) => Complex a -> a
168 phase (x:+y) = atan2 y x