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<title>The Haskell 1.3 Library Report: Complex Numbers</title>
<body bgcolor="#ffffff"> <i>The Haskell 1.4 Library Report</i><br> <a href="index.html">top</a> | <a href="ratio.html">back</a> | <a href="numeric.html">next</a> | <a href="libindex.html">contents</a> <br><hr>
<a name="lib-num"></a><a name="sect3"></a>
<h2>3<tt> </tt>Complex Numbers</h2>
<p>
<table border=2 cellpadding=3>
<tr><td>
<tt><br>
module Complex ( <br>
Complex((:+)), realPart, imagPart, conjugate, <br>
mkPolar, cis, polar, magnitude, phase ) where<br>
<br>
infix 6 :+<br>
<br>
data (RealFloat a) => Complex a = !a :+ !a<br>
<br>
realPart, imagPart :: (RealFloat a) => Complex a -> a<br>
conjugate :: (RealFloat a) => Complex a -> Complex a<br>
mkPolar :: (RealFloat a) => a -> a -> Complex a<br>
cis :: (RealFloat a) => a -> Complex a<br>
polar :: (RealFloat a) => Complex a -> (a,a)<br>
magnitude, phase :: (RealFloat a) => Complex a -> a<br>
<br>
instance (RealFloat a) => Eq (Complex a) where ...<br>
instance (RealFloat a) => Read (Complex a) where ...<br>
instance (RealFloat a) => Show (Complex a) where ...<br>
instance (RealFloat a) => Num (Complex a) where ...<br>
instance (RealFloat a) => Fractional (Complex a) where ...<br>
instance (RealFloat a) => Floating (Complex a) where ...<br>
</tt></td></tr></table>
<p>
Complex numbers are an algebraic type.
The constructor <tt>(:+)</tt> 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
<tt>mkPolar</tt>. The function <tt>cis
</tt>produces a complex number from an angle <I>t</I>.
Put another way, <tt>cis</tt> <I>t</I> is a complex value with magnitude <I>1
</I>and phase <I>t</I> (modulo <I>2</I><font face="symbol">p</font>).<p>
The function <tt>polar</tt> takes a complex number and
returns a (magnitude, phase) pair in canonical form: The magnitude is
nonnegative, and the phase, in the range (- <font face="symbol">p</font>, <font face="symbol">p</font>]; if the
magnitude is zero, then so is the phase. <p>
The functions <tt>realPart</tt> and
<tt>imagPart</tt> extract the rectangular components of a
complex number and the functions <tt>magnitude</tt> and
<tt>phase</tt> extract the polar coordinates of a complex
number. Also defined on complex numbers is
<tt>conjugate</tt>.<p>
The magnitude and sign of a complex number are defined as follows:
<tt><br>
<br>
abs z = magnitude z :+ 0<br>
signum 0 = 0<br>
signum z@(x:+y) = x/r :+ y/r where r = magnitude z<br>
<br>
</tt>That is, <tt>abs</tt> z is a number with the magnitude of z, but oriented
in the positive real direction, whereas <tt>signum</tt> z has the phase of
z, but unit magnitude. (<tt>abs</tt> for a complex number differs from
<tt>magnitude</tt> only in type.)<p>
<a name="sect3.1"></a>
<h3>3.1<tt> </tt>Library <tt>Complex</tt></h3>
<tt><br>
<br>
module Complex(Complex((:+)), realPart, imagPart, conjugate, mkPolar,<br>
cis, polar, magnitude, phase) where<br>
<br>
infix 6 :+<br>
<br>
data (RealFloat a) => Complex a = !a :+ !a deriving (Eq,Read,Show)<br>
<br>
<br>
realPart, imagPart :: (RealFloat a) => Complex a -> a<br>
realPart (x:+y) = x<br>
imagPart (x:+y) = y<br>
<br>
conjugate :: (RealFloat a) => Complex a -> Complex a<br>
conjugate (x:+y) = x :+ (-y)<br>
<br>
mkPolar :: (RealFloat a) => a -> a -> Complex a<br>
mkPolar r theta = r * cos theta :+ r * sin theta<br>
<br>
cis :: (RealFloat a) => a -> Complex a<br>
cis theta = cos theta :+ sin theta<br>
<br>
polar :: (RealFloat a) => Complex a -> (a,a)<br>
polar z = (magnitude z, phase z)<br>
<br>
magnitude, phase :: (RealFloat a) => Complex a -> a<br>
magnitude (x:+y) = scaleFloat k<br>
(sqrt ((scaleFloat mk x)^2 + (scaleFloat mk y)^2))<br>
where k = max (exponent x) (exponent y)<br>
mk = - k<br>
<br>
phase (x:+y) = atan2 y x<br>
<br>
<br>
instance (RealFloat a) => Num (Complex a) where<br>
(x:+y) + (x':+y') = (x+x') :+ (y+y')<br>
(x:+y) - (x':+y') = (x-x') :+ (y-y')<br>
(x:+y) * (x':+y') = (x*x'-y*y') :+ (x*y'+y*x')<br>
negate (x:+y) = negate x :+ negate y<br>
abs z = magnitude z :+ 0<br>
signum 0 = 0<br>
signum z@(x:+y) = x/r :+ y/r where r = magnitude z<br>
fromInteger n = fromInteger n :+ 0<br>
<br>
instance (RealFloat a) => Fractional (Complex a) where<br>
(x:+y) / (x':+y') = (x*x''+y*y'') / d :+ (y*x''-x*y'') / d<br>
where x'' = scaleFloat k x'<br>
y'' = scaleFloat k y'<br>
k = - max (exponent x') (exponent y')<br>
d = x'*x'' + y'*y''<br>
<br>
fromRational a = fromRational a :+ 0<br>
<br>
instance (RealFloat a) => Floating (Complex a) where<br>
pi = pi :+ 0<br>
exp (x:+y) = expx * cos y :+ expx * sin y<br>
where expx = exp x<br>
log z = log (magnitude z) :+ phase z<br>
<br>
sqrt 0 = 0<br>
sqrt z@(x:+y) = u :+ (if y < 0 then -v else v)<br>
where (u,v) = if x < 0 then (v',u') else (u',v')<br>
v' = abs y / (u'*2)<br>
u' = sqrt ((magnitude z + abs x) / 2)<br>
<br>
sin (x:+y) = sin x * cosh y :+ cos x * sinh y<br>
cos (x:+y) = cos x * cosh y :+ (- sin x * sinh y)<br>
tan (x:+y) = (sinx*coshy:+cosx*sinhy)/(cosx*coshy:+(-sinx*sinhy))<br>
where sinx = sin x<br>
cosx = cos x<br>
sinhy = sinh y<br>
coshy = cosh y<br>
<br>
sinh (x:+y) = cos y * sinh x :+ sin y * cosh x<br>
cosh (x:+y) = cos y * cosh x :+ sin y * sinh x<br>
tanh (x:+y) = (cosy*sinhx:+siny*coshx)/(cosy*coshx:+siny*sinhx)<br>
where siny = sin y<br>
cosy = cos y<br>
sinhx = sinh x<br>
coshx = cosh x<br>
<br>
asin z@(x:+y) = y':+(-x')<br>
where (x':+y') = log ((-y:+x) + sqrt (1 - z*z))<br>
acos z@(x:+y) = y'':+(-x'')<br>
where (x'':+y'') = log (z + ((-y'):+x'))<br>
(x':+y') = sqrt (1 - z*z)<br>
atan z@(x:+y) = y':+(-x')<br>
where (x':+y') = log (((1-y):+x) / sqrt (1+z*z))<br>
<br>
asinh z = log (z + sqrt (1+z*z))<br>
acosh z = log (z + (z+1) * sqrt ((z-1)/(z+1)))<br>
atanh z = log ((1+z) / sqrt (1-z*z))<br>
<p>
<hr><i>The Haskell 1.4 Library Report</i><br><a href="index.html">top</a> | <a href="ratio.html">back</a> | <a href="numeric.html">next</a> | <a href="libindex.html">contents</a> <br><font size=2>April 4, 1997</font>
<p>
</tt>
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