The Haskell 98 Library Report: Complex Numbers

The Haskell 98 Report
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## 13`` Complex Numbers

`module Complex ( Complex((:+)), realPart, imagPart, conjugate, mkPolar, cis, polar, magnitude, phase ) where infix 6 :+ data (RealFloat a) => Complex a = !a :+ !a realPart, imagPart :: (RealFloat a) => Complex a -> a conjugate :: (RealFloat a) => Complex a -> Complex a mkPolar :: (RealFloat a) => a -> a -> Complex a cis :: (RealFloat a) => a -> Complex a polar :: (RealFloat a) => Complex a -> (a,a) magnitude, phase :: (RealFloat a) => Complex a -> a instance (RealFloat a) => Eq (Complex a) where ... instance (RealFloat a) => Read (Complex a) where ... instance (RealFloat a) => Show (Complex a) where ... instance (RealFloat a) => Num (Complex a) where ... instance (RealFloat a) => Fractional (Complex a) where ... instance (RealFloat a) => Floating (Complex a) where ...`

Complex numbers are an algebraic type. The constructor `(:+)` forms a complex number from its real and imaginary rectangular components. This constructor is strict: if either the real part or the imaginary part of the number is _|_, the entire number is _|_. A complex number may also be formed from polar components of magnitude and phase by the function `mkPolar`. The function `cis` produces a complex number from an angle t. Put another way, `cis` t is a complex value with magnitude 1 and phase t (modulo 2p).

The function `polar` takes a complex number and returns a (magnitude, phase) pair in canonical form: The magnitude is nonnegative, and the phase, in the range (- p, p]; if the magnitude is zero, then so is the phase.

The functions `realPart` and `imagPart` extract the rectangular components of a complex number and the functions `magnitude` and `phase` extract the polar components of a complex number. The function `conjugate` computes the conjugate of a complex number in the usual way.

The magnitude and sign of a complex number are defined as follows: `
abs z = magnitude z :+ 0
signum 0 = 0
signum z@(x:+y) = x/r :+ y/r where r = magnitude z
` That is, `abs` z is a number with the magnitude of z, but oriented in the positive real direction, whereas `signum` z has the phase of z, but unit magnitude.

### 13.1`` Library `Complex`

`
module Complex(Complex((:+)), realPart, imagPart, conjugate, mkPolar,
cis, polar, magnitude, phase) where
infix 6 :+
data (RealFloat a) => Complex a = !a :+ !a deriving (Eq,Read,Show)
realPart, imagPart :: (RealFloat a) => Complex a -> a
realPart (x:+y) = x
imagPart (x:+y) = y
conjugate :: (RealFloat a) => Complex a -> Complex a
conjugate (x:+y) = x :+ (-y)
mkPolar :: (RealFloat a) => a -> a -> Complex a
mkPolar r theta = r * cos theta :+ r * sin theta
cis :: (RealFloat a) => a -> Complex a
cis theta = cos theta :+ sin theta
polar :: (RealFloat a) => Complex a -> (a,a)
polar z = (magnitude z, phase z)
magnitude :: (RealFloat a) => Complex a -> a
magnitude (x:+y) = scaleFloat k
(sqrt ((scaleFloat mk x)^2 + (scaleFloat mk y)^2))
where k = max (exponent x) (exponent y)
mk = - k
phase :: (RealFloat a) => Complex a -> a
phase (0 :+ 0) = 0
phase (x :+ y) = atan2 y x
instance (RealFloat a) => Num (Complex a) where
(x:+y) + (x':+y') = (x+x') :+ (y+y')
(x:+y) - (x':+y') = (x-x') :+ (y-y')
(x:+y) * (x':+y') = (x*x'-y*y') :+ (x*y'+y*x')
negate (x:+y) = negate x :+ negate y
abs z = magnitude z :+ 0
signum 0 = 0
signum z@(x:+y) = x/r :+ y/r where r = magnitude z
fromInteger n = fromInteger n :+ 0
instance (RealFloat a) => Fractional (Complex a) where
(x:+y) / (x':+y') = (x*x''+y*y'') / d :+ (y*x''-x*y'') / d
where x'' = scaleFloat k x'
y'' = scaleFloat k y'
k = - max (exponent x') (exponent y')
d = x'*x'' + y'*y''
fromRational a = fromRational a :+ 0
instance (RealFloat a) => Floating (Complex a) where
pi = pi :+ 0
exp (x:+y) = expx * cos y :+ expx * sin y
where expx = exp x
log z = log (magnitude z) :+ phase z
sqrt 0 = 0
sqrt z@(x:+y) = u :+ (if y < 0 then -v else v)
where (u,v) = if x < 0 then (v',u') else (u',v')
v' = abs y / (u'*2)
u' = sqrt ((magnitude z + abs x) / 2)
sin (x:+y) = sin x * cosh y :+ cos x * sinh y
cos (x:+y) = cos x * cosh y :+ (- sin x * sinh y)
tan (x:+y) = (sinx*coshy:+cosx*sinhy)/(cosx*coshy:+(-sinx*sinhy))
where sinx = sin x
cosx = cos x
sinhy = sinh y
coshy = cosh y
sinh (x:+y) = cos y * sinh x :+ sin y * cosh x
cosh (x:+y) = cos y * cosh x :+ sin y * sinh x
tanh (x:+y) = (cosy*sinhx:+siny*coshx)/(cosy*coshx:+siny*sinhx)
where siny = sin y
cosy = cos y
sinhx = sinh x
coshx = cosh x
asin z@(x:+y) = y':+(-x')
where (x':+y') = log (((-y):+x) + sqrt (1 - z*z))
acos z@(x:+y) = y'':+(-x'')
where (x'':+y'') = log (z + ((-y'):+x'))
(x':+y') = sqrt (1 - z*z)
atan z@(x:+y) = y':+(-x')
where (x':+y') = log (((1-y):+x) / sqrt (1+z*z))
asinh z = log (z + sqrt (1+z*z))
acosh z = log (z + (z+1) * sqrt ((z-1)/(z+1)))
atanh z = log ((1+z) / sqrt (1-z*z))

The Haskell 98 Report
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December 2002

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