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# Complex numbers
# ---------------
# This module represents complex numbers as instances of the class Complex.
# A Complex instance z has two data attribues, z.re (the real part) and z.im
# (the imaginary part). In fact, z.re and z.im can have any value -- all
# arithmetic operators work regardless of the type of z.re and z.im (as long
# as they support numerical operations).
#
# The following functions exist (Complex is actually a class):
# Complex([re [,im]) -> creates a complex number from a real and an imaginary part
# IsComplex(z) -> true iff z is a complex number (== has .re and .im attributes)
# Polar([r [,phi [,fullcircle]]]) ->
# the complex number z for which r == z.radius() and phi == z.angle(fullcircle)
# (r and phi default to 0)
#
# Complex numbers have the following methods:
# z.abs() -> absolute value of z
# z.radius() == z.abs()
# z.angle([fullcircle]) -> angle from positive X axis; fullcircle gives units
# z.phi([fullcircle]) == z.angle(fullcircle)
#
# These standard functions and unary operators accept complex arguments:
# abs(z)
# -z
# +z
# not z
# repr(z) == `z`
# str(z)
# hash(z) -> a combination of hash(z.re) and hash(z.im) such that if z.im is zero
# the result equals hash(z.re)
# Note that hex(z) and oct(z) are not defined.
#
# These conversions accept complex arguments only if their imaginary part is zero:
# int(z)
# long(z)
# float(z)
#
# The following operators accept two complex numbers, or one complex number
# and one real number (int, long or float):
# z1 + z2
# z1 - z2
# z1 * z2
# z1 / z2
# pow(z1, z2)
# cmp(z1, z2)
# Note that z1 % z2 and divmod(z1, z2) are not defined,
# nor are shift and mask operations.
#
# The standard module math does not support complex numbers.
# (I suppose it would be easy to implement a cmath module.)
#
# Idea:
# add a class Polar(r, phi) and mixed-mode arithmetic which
# chooses the most appropriate type for the result:
# Complex for +,-,cmp
# Polar for *,/,pow
import types, math
if not hasattr(math, 'hypot'):
def hypot(x, y):
# XXX I know there's a way to compute this without possibly causing
# overflow, but I can't remember what it is right now...
return math.sqrt(x*x + y*y)
math.hypot = hypot
twopi = math.pi*2.0
halfpi = math.pi/2.0
def IsComplex(obj):
return hasattr(obj, 're') and hasattr(obj, 'im')
def Polar(r = 0, phi = 0, fullcircle = twopi):
phi = phi * (twopi / fullcircle)
return Complex(math.cos(phi)*r, math.sin(phi)*r)
class Complex:
def __init__(self, re=0, im=0):
if IsComplex(re):
im = im + re.im
re = re.re
if IsComplex(im):
re = re - im.im
im = im.re
self.re = re
self.im = im
def __setattr__(self, name, value):
if hasattr(self, name):
raise TypeError, "Complex numbers have set-once attributes"
self.__dict__[name] = value
def __repr__(self):
if not self.im:
return 'Complex(%s)' % `self.re`
else:
return 'Complex(%s, %s)' % (`self.re`, `self.im`)
def __str__(self):
if not self.im:
return `self.re`
else:
return 'Complex(%s, %s)' % (`self.re`, `self.im`)
def __coerce__(self, other):
if IsComplex(other):
return self, other
return self, Complex(other) # May fail
def __cmp__(self, other):
return cmp(self.re, other.re) or cmp(self.im, other.im)
def __hash__(self):
if not self.im: return hash(self.re)
mod = sys.maxint + 1L
return int((hash(self.re) + 2L*hash(self.im) + mod) % (2L*mod) - mod)
def __neg__(self):
return Complex(-self.re, -self.im)
def __pos__(self):
return self
def __abs__(self):
return math.hypot(self.re, self.im)
##return math.sqrt(self.re*self.re + self.im*self.im)
def __int__(self):
if self.im:
raise ValueError, "can't convert Complex with nonzero im to int"
return int(self.re)
def __long__(self):
if self.im:
raise ValueError, "can't convert Complex with nonzero im to long"
return long(self.re)
def __float__(self):
if self.im:
raise ValueError, "can't convert Complex with nonzero im to float"
return float(self.re)
def __nonzero__(self):
return not (self.re == self.im == 0)
abs = radius = __abs__
def angle(self, fullcircle = twopi):
return (fullcircle/twopi) * ((halfpi - math.atan2(self.re, self.im)) % twopi)
phi = angle
def __add__(self, other):
return Complex(self.re + other.re, self.im + other.im)
__radd__ = __add__
def __sub__(self, other):
return Complex(self.re - other.re, self.im - other.im)
def __rsub__(self, other):
return Complex(other.re - self.re, other.im - self.im)
def __mul__(self, other):
return Complex(self.re*other.re - self.im*other.im,
self.re*other.im + self.im*other.re)
__rmul__ = __mul__
def __div__(self, other):
# Deviating from the general principle of not forcing re or im
# to be floats, we cast to float here, otherwise division
# of Complex numbers with integer re and im parts would use
# the (truncating) integer division
d = float(other.re*other.re + other.im*other.im)
if not d: raise ZeroDivisionError, 'Complex division'
return Complex((self.re*other.re + self.im*other.im) / d,
(self.im*other.re - self.re*other.im) / d)
def __rdiv__(self, other):
return other / self
def __pow__(self, n, z=None):
if z is not None:
raise TypeError, 'Complex does not support ternary pow()'
if IsComplex(n):
if n.im: raise TypeError, 'Complex to the Complex power'
n = n.re
r = pow(self.abs(), n)
phi = n*self.angle()
return Complex(math.cos(phi)*r, math.sin(phi)*r)
def __rpow__(self, base):
return pow(base, self)
# Everything below this point is part of the test suite
def checkop(expr, a, b, value, fuzz = 1e-6):
import sys
print ' ', a, 'and', b,
try:
result = eval(expr)
except:
result = sys.exc_type
print '->', result
if (type(result) == type('') or type(value) == type('')):
ok = result == value
else:
ok = abs(result - value) <= fuzz
if not ok:
print '!!\t!!\t!! should be', value, 'diff', abs(result - value)
def test():
testsuite = {
'a+b': [
(1, 10, 11),
(1, Complex(0,10), Complex(1,10)),
(Complex(0,10), 1, Complex(1,10)),
(Complex(0,10), Complex(1), Complex(1,10)),
(Complex(1), Complex(0,10), Complex(1,10)),
],
'a-b': [
(1, 10, -9),
(1, Complex(0,10), Complex(1,-10)),
(Complex(0,10), 1, Complex(-1,10)),
(Complex(0,10), Complex(1), Complex(-1,10)),
(Complex(1), Complex(0,10), Complex(1,-10)),
],
'a*b': [
(1, 10, 10),
(1, Complex(0,10), Complex(0, 10)),
(Complex(0,10), 1, Complex(0,10)),
(Complex(0,10), Complex(1), Complex(0,10)),
(Complex(1), Complex(0,10), Complex(0,10)),
],
'a/b': [
(1., 10, 0.1),
(1, Complex(0,10), Complex(0, -0.1)),
(Complex(0, 10), 1, Complex(0, 10)),
(Complex(0, 10), Complex(1), Complex(0, 10)),
(Complex(1), Complex(0,10), Complex(0, -0.1)),
],
'pow(a,b)': [
(1, 10, 1),
(1, Complex(0,10), 'TypeError'),
(Complex(0,10), 1, Complex(0,10)),
(Complex(0,10), Complex(1), Complex(0,10)),
(Complex(1), Complex(0,10), 'TypeError'),
(2, Complex(4,0), 16),
],
'cmp(a,b)': [
(1, 10, -1),
(1, Complex(0,10), 1),
(Complex(0,10), 1, -1),
(Complex(0,10), Complex(1), -1),
(Complex(1), Complex(0,10), 1),
],
}
exprs = testsuite.keys()
exprs.sort()
for expr in exprs:
print expr + ':'
t = (expr,)
for item in testsuite[expr]:
apply(checkop, t+item)
if __name__ == '__main__':
test()
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