# Complex numbers
# ---------------
# [Now that Python has a complex data type built-in, this is not very
# useful, but it's still a nice example class]
# 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)
# ToComplex(z) -> a complex number equal to z; z itself if IsComplex(z) is true
# if z is a tuple(re, im) it will also be converted
# PolarToComplex([r [,phi [,fullcircle]]]) ->
# the complex number z for which r == z.radius() and phi == z.angle(fullcircle)
# (r and phi default to 0)
# exp(z) -> returns the complex exponential of z. Equivalent to pow(math.e,z).
#
# 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)
# 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.
# The cmath modules should be used instead.
#
# 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 math
import sys
twopi = math.pi*2.0
halfpi = math.pi/2.0
def IsComplex(obj):
return hasattr(obj, 're') and hasattr(obj, 'im')
def ToComplex(obj):
if IsComplex(obj):
return obj
elif isinstance(obj, tuple):
return Complex(*obj)
else:
return Complex(obj)
def PolarToComplex(r = 0, phi = 0, fullcircle = twopi):
phi = phi * (twopi / fullcircle)
return Complex(math.cos(phi)*r, math.sin(phi)*r)
def Re(obj):
if IsComplex(obj):
return obj.re
return obj
def Im(obj):
if IsComplex(obj):
return obj.im
return 0
class Complex:
def __init__(self, re=0, im=0):
_re = 0
_im = 0
if IsComplex(re):
_re = re.re
_im = re.im
else:
_re = re
if IsComplex(im):
_re = _re - im.im
_im = _im + im.re
else:
_im = _im + im
# this class is immutable, so setting self.re directly is
# not possible.
self.__dict__['re'] = _re
self.__dict__['im'] = _im
def __setattr__(self, name, value):
raise TypeError('Complex numbers are immutable')
def __hash__(self):
if not self.im:
return hash(self.re)
return hash((self.re, self.im))
def __repr__(self):
if not self.im:
return 'Complex(%r)' % (self.re,)
else:
return 'Complex(%r, %r)' % (self.re, self.im)
def __str__(self):
if not self.im:
return repr(self.re)
else:
return 'Complex(%r, %r)' % (self.re, self.im)
def __neg__(self):
return Complex(-self.re, -self.im)
def __pos__(self):
return self
def __abs__(self):
return math.hypot(self.re, self.im)
def __int__(self):
if self.im:
raise ValueError("can't convert Complex with nonzero im to int")
return int(self.re)
def __float__(self):
if self.im:
raise ValueError("can't convert Complex with nonzero im to float")
return float(self.re)
def __cmp__(self, other):
other = ToComplex(other)
return cmp((self.re, self.im), (other.re, other.im))
def __rcmp__(self, other):
other = ToComplex(other)
return cmp(other, self)
def __bool__(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):
other = ToComplex(other)
return Complex(self.re + other.re, self.im + other.im)
__radd__ = __add__
def __sub__(self, other):
other = ToComplex(other)
return Complex(self.re - other.re, self.im - other.im)
def __rsub__(self, other):
other = ToComplex(other)
return other - self
def __mul__(self, other):
other = ToComplex(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):
other = ToComplex(other)
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):
other = ToComplex(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:
if self.im: raise TypeError('Complex to the Complex power')
else: return exp(math.log(self.re)*n)
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):
base = ToComplex(base)
return pow(base, self)
def exp(z):
r = math.exp(z.re)
return Complex(math.cos(z.im)*r,math.sin(z.im)*r)
def checkop(expr, a, b, value, fuzz = 1e-6):
print(' ', a, 'and', b, end=' ')
try:
result = eval(expr)
except:
result = sys.exc_info()[0]
print('->', result)
if isinstance(result, str) or isinstance(value, str):
ok = (result == value)
else:
ok = abs(result - value) <= fuzz
if not ok:
print('!!\t!!\t!! should be', value, 'diff', abs(result - value))
def test():
print('test constructors')
constructor_test = (
# "expect" is an array [re,im] "got" the Complex.
( (0,0), Complex() ),
( (0,0), Complex() ),
( (1,0), Complex(1) ),
( (0,1), Complex(0,1) ),
( (1,2), Complex(Complex(1,2)) ),
( (1,3), Complex(Complex(1,2),1) ),
( (0,0), Complex(0,Complex(0,0)) ),
( (3,4), Complex(3,Complex(4)) ),
( (-1,3), Complex(1,Complex(3,2)) ),
( (-7,6), Complex(Complex(1,2),Complex(4,8)) ) )
cnt = [0,0]
for t in constructor_test:
cnt[0] += 1
if ((t[0][0]!=t[1].re)or(t[0][1]!=t[1].im)):
print(" expected", t[0], "got", t[1])
cnt[1] += 1
print(" ", cnt[1], "of", cnt[0], "tests failed")
# test operators
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), 1),
(Complex(0,10), 1, Complex(0,10)),
(Complex(0,10), Complex(1), Complex(0,10)),
(Complex(1), Complex(0,10), 1),
(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),
],
}
for expr in sorted(testsuite):
print(expr + ':')
t = (expr,)
for item in testsuite[expr]:
checkop(*(t+item))
if __name__ == '__main__':
test()
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