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This implements polynomial functions over a single variable in Python. It represents the polynomial as a list of numbers and allows most arithmetic operations, using conventional Python syntax. It does not do symbolic manipulations. Instead, you can do things like this:

``````x = SimplePolynomial()
eq = (x-1)*(x*1)
print eq     # prints 'X**2 - 1'
print eq(4)  # prints 15
``````
Python, 294 lines
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294``` ```"""\ This class implements polynomial functions over a single variable. It represents the polynomial as a list of numbers and allows most arithmetic operations, using conventional Python syntax. It does not do symbolic manipulations. Instead, you can do things like this: >>> x = SimplePolynomial() >>> quadratic = (x+1)*(x-1) >>> str(quadratic) 'X**2 - 1' >>> quadratic(4) 15 >>> for i in range(4): ... polynomial = (x+1)**i ... i, str(polynomial), polynomial(1) (0, '1', 1) (1, 'X + 1', 2) (2, 'X**2 + 2*X + 1', 4) (3, 'X**3 + 3*X**2 + 3*X + 1', 8) """ from __future__ import division, generators from operator import add try: from itertools import izip_longest except ImportError: # The izip_longest function was added in version 2.6 # If we can't find it, use an equivalent implementation. from itertools import chain, repeat class ZipExhausted(Exception): pass def izip_longest(*args, **kwds): # izip_longest('ABCD', 'xy', fillvalue='-') --> Ax By C- D- fillvalue = kwds.get('fillvalue') counter = [len(args) - 1] def sentinel(): if not counter[0]: raise ZipExhausted counter[0] -= 1 yield fillvalue fillers = repeat(fillvalue) iterators = [chain(it, sentinel(), fillers) for it in args] try: while iterators: yield tuple(map(next, iterators)) except ZipExhausted: pass try: from numbers import Number except ImportError: # The numbers module was added in version 2.6 # If we can't find it, use an equivalent implementation. Number = (int, float, long, complex) class SimplePolynomial(object): def __init__(self, terms=[0,1]): """\ >>> str(SimplePolynomial()) 'X' """ try: while terms[-1] == 0: del terms[-1] except IndexError: pass self.terms = list(terms) def __str__(self): """\ Needs some work, but adequate. """ l = len(self.terms) if l == 0: return '0' if l == 1: return str(self.terms[0]) result = [] for i, c in reversed(list(enumerate(self.terms))): if c == 0: continue if c < 0: result.append('-') c = - c else: result.append('+') if c == 1: if i == 0: result.append('1') elif i == 1: result.append('X') else: result.append('X**%g' % i) else: if i == 0: result.append('%g' % c) elif i == 1: result.append('%g*X' % c) else: result.append('%g*X**%d' % (c, i)) if len(result) == 0: return '0' if result[0] == '-': result[1] = '-'+result[1] del result[0] return ' '.join(result) def __add__(self, other): """\ >>> str(SimplePolynomial() + 1) 'X + 1' >>> str(1 + SimplePolynomial()) 'X + 1' >>> str(SimplePolynomial() + SimplePolynomial()) '2*X' """ if len(self.terms) == 0: return other if isinstance(other, Number): terms = self.terms[:] terms[0] += other return SimplePolynomial(terms) return SimplePolynomial([ add(*pair) for pair in izip_longest(self.terms, other.terms, fillvalue=0) ]) # Since addition is commutative, reuse __add__ __radd__ = __add__ def __neg__(self): """\ >>> str(- SimplePolynomial()) '-X' """ return SimplePolynomial([-c for c in self.terms]) def __sub__(self, other): """\ >>> str(SimplePolynomial() - 1) 'X - 1' >>> str(SimplePolynomial() - SimplePolynomial()) '0' """ return self + -other def __rsub__(self, other): """\ >>> str(1 - SimplePolynomial()) '-X + 1' """ return -self + other def __mul__(self, other): """\ >>> str(SimplePolynomial() * 2) '2*X' >>> str(2 * SimplePolynomial()) '2*X' >>> str(SimplePolynomial() * SimplePolynomial()) 'X**2' """ if isinstance(other, Number): return SimplePolynomial([c * other for c in self.terms]) terms = [0]*(len(self.terms)+len(other.terms)) for i1, c1 in enumerate(self.terms): for i2, c2 in enumerate(other.terms): terms[i1+i2] += c1*c2 return SimplePolynomial(terms) # Since multiplication is commutative, reuse __mul__ __rmul__ = __mul__ def __truediv__(self, other): """\ Implements some simple forms of division. See http://en.wikipedia.org/wiki/Synthetic_division for details. >>> str(SimplePolynomial() / 2) '0.5*X' >>> x = SimplePolynomial() >>> quotient, remainder = (x**3 - 12*x**2 - 42) / (x - 3) >>> str(quotient), str(remainder) ('X**2 - 9*X - 27', '-123') """ if isinstance(other, Number): return SimplePolynomial([c / other for c in self.terms]) if len(other.terms) == 1: return self/other.terms[0] assert len(other.terms) == 2 dividend = self.terms[:] divisor = other.terms[:] assert divisor[-1] == 1 xi = 0 result = [] for i in xrange(-1, -len(dividend) - 1, -1): xi = dividend[i] - xi * divisor[0] result.insert(0, xi) return SimplePolynomial(result[1:]), result[0] raise NotImplementedError('synthetic division') __div__ = __truediv__ def __eq__(self, other): """\ >>> SimplePolynomial() == SimplePolynomial() True >>> SimplePolynomial() - SimplePolynomial() == 0 True """ if isinstance(other, SimplePolynomial): return self.terms == other.terms if isinstance(other, Number): if len(self.terms) > 1: return False try: self.terms[0] == other except IndexError: return other == 0 return False def __ne__(self, other): return not self == other def copy(self): return SimplePolynomial(self.terms[:]) def __pow__(self, exponent): """\ Uses the Russian Peasant Multiplication algorithm. >>> str(SimplePolynomial() ** 2) 'X**2' >>> str(SimplePolynomial() ** 0.5) Traceback (most recent call last): ... NotImplementedError: exponent is not an integer >>> str(SimplePolynomial() ** -1) Traceback (most recent call last): ... NotImplementedError: exponent is less than zero """ if not isinstance(exponent, int): raise NotImplementedError('exponent is not an integer') if exponent < 0: raise NotImplementedError('exponent is less than zero') tmp = self.copy() result = SimplePolynomial([1]) while exponent > 0: if exponent & 1: result *= tmp tmp *= tmp exponent >>= 1 return result def __call__(self, x): """\ Evaluate the polynomial for the given value using the Horner scheme. >>> SimplePolynomial()(1) 1 """ result = 0 for c in reversed(self.terms): result = result * x + c return result def derivative(self): """\ >>> str(SimplePolynomial().derivative()) '1' """ terms = [i*c for i, c in enumerate(self.terms)] return SimplePolynomial(terms[1:]) def integrate(self, const=0): """\ >>> str(SimplePolynomial().integrate()) '0.5*X**2' """ terms = [const] terms.extend([c/(i+1) for i, c in enumerate(self.terms)]) return SimplePolynomial(terms) if __name__ == '__main__': import doctest doctest.testmod() ```

Since I posted the original, I've added doctests and the barest beginnings of polynomial division. In the doctests, I cast values to strings to avoid having to use print, so this is usable as-is in both Python 2.x and 3.x.

Gabriel Genellina 14 years, 5 months ago

Very nice recipe!!! Just a few notes:

• you might replace `sum_v` by `operator.add`

• I noticed you very carefully avoided sharing `self.terms` between instances, but forgot to do so in `copy()`.

• also, complex coefficients appear to be supported, but `integral()` uses `float(c)` and may fail in such cases. `c/float(i+1)` is enough to avoid integer division (or use `from __future__ import division`)

• To evaluate the polynomial, Ruffini's Rule usually gives better results (but is slightly more complex than your very straightforward code)

Gabriel Genellina 14 years, 5 months ago

also, itertools seems to be a leftover from earlier attempts...

Sam Denton (author) 14 years, 5 months ago

Thanks for the feedback. I've implemented all of your suggestions. In the case of `copy()`, I'd convinced myself that the sharing was harmless, but it's probably better to be safe than sorry. The `float(c)` was left over from earlier debugging; after integration my coefficients all appeared to be integers. (It turned out that I was using `'%d'` in `__str__()`.) At some point I'll probably add synthetic division.

 Created by Sam Denton on Tue, 10 Nov 2009 (MIT)