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Complex.py
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# 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()