# 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)# 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.# 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 mathimport sys twopi = math.pi*2.0halfpi = 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 __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 __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 __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): 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, try: result = eval(expr) except: result = sys.exc_type 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()
# 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)
# 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.
# 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 __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 __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 __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):
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,
try:
result = eval(expr)
except:
result = sys.exc_type
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()