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polyfuncs.py
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polyfuncs.py
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"""High-level polynomials manipulation functions. """
from sympy.polys.polytools import (
poly_from_expr, parallel_poly_from_expr, Poly)
from sympy.polys.polyoptions import allowed_flags
from sympy.polys.specialpolys import (
symmetric_poly, interpolating_poly)
from sympy.polys.polyerrors import (
PolificationFailed, ComputationFailed,
MultivariatePolynomialError)
from sympy.utilities import numbered_symbols, take
from sympy.core import S, Basic, Add, Mul
def symmetrize(F, *gens, **args):
"""
Rewrite a polynomial in terms of elementary symmetric polynomials.
**Examples**
>>> from sympy.polys.polyfuncs import symmetrize
>>> from sympy.abc import x, y
>>> symmetrize(x**2 + y**2)
(-2*x*y + (x + y)**2, 0)
>>> symmetrize(x**2 + y**2, formal=True)
(s1**2 - 2*s2, 0, [(s1, x + y), (s2, x*y)])
>>> symmetrize(x**2 - y**2)
(-2*x*y + (x + y)**2, -2*y**2)
>>> symmetrize(x**2 - y**2, formal=True)
(s1**2 - 2*s2, -2*y**2, [(s1, x + y), (s2, x*y)])
"""
allowed_flags(args, ['formal', 'symbols'])
iterable = True
if not hasattr(F, '__iter__'):
iterable = False
F = [F]
try:
F, opt = parallel_poly_from_expr(F, *gens, **args)
except PolificationFailed, exc:
result = []
for expr in exc.exprs:
if expr.is_Number:
result.append((expr, S.Zero))
else:
raise ComputationFailed('symmetrize', len(F), exc)
else:
if not iterable:
result, = result
if not exc.opt.formal:
return result
else:
if iterable:
return result, []
else:
return result + ([],)
polys, symbols = [], opt.symbols
gens, dom = opt.gens, opt.domain
for i in xrange(0, len(gens)):
poly = symmetric_poly(i+1, gens, polys=True)
polys.append((symbols.next(), poly.set_domain(dom)))
indices = range(0, len(gens) - 1)
weights = range(len(gens), 0, -1)
result = []
for f in F:
symmetric = []
if not f.is_homogeneous:
symmetric.append(f.TC())
f -= f.TC()
while f:
_height, _monom, _coeff = -1, None, None
for i, (monom, coeff) in enumerate(f.terms()):
if all(monom[i] >= monom[i+1] for i in indices):
height = max([ n*m for n, m in zip(weights, monom) ])
if height > _height:
_height, _monom, _coeff = height, monom, coeff
if _height != -1:
monom, coeff = _monom, _coeff
else:
break
exponents = []
for m1, m2 in zip(monom, monom[1:] + (0,)):
exponents.append(m1 - m2)
term = [ s**n for (s, _), n in zip(polys, exponents) ]
poly = [ p**n for (_, p), n in zip(polys, exponents) ]
symmetric.append(Mul(coeff, *term))
product = poly[0].mul(coeff)
for p in poly[1:]:
product = product.mul(p)
f -= product
result.append((Add(*symmetric), f.as_expr()))
polys = [ (s, p.as_expr()) for s, p in polys ]
if not opt.formal:
for i, (sym, non_sym) in enumerate(result):
result[i] = (sym.subs(polys), non_sym)
if not iterable:
result, = result
if not opt.formal:
return result
else:
if iterable:
return result, polys
else:
return result + (polys,)
def horner(f, *gens, **args):
"""
Rewrite a polynomial in Horner form.
**Examples**
>>> from sympy.polys.polyfuncs import horner
>>> from sympy.abc import x, y, a, b, c, d, e
>>> horner(9*x**4 + 8*x**3 + 7*x**2 + 6*x + 5)
x*(x*(x*(9*x + 8) + 7) + 6) + 5
>>> horner(a*x**4 + b*x**3 + c*x**2 + d*x + e)
e + x*(d + x*(c + x*(a*x + b)))
>>> f = 4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y
>>> horner(f, wrt=x)
x*(x*y*(4*y + 2) + y*(2*y + 1))
>>> horner(f, wrt=y)
y*(x*y*(4*x + 2) + x*(2*x + 1))
"""
allowed_flags(args, [])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed, exc:
return exc.expr
form, gen = S.Zero, F.gen
if F.is_univariate:
for coeff in F.all_coeffs():
form = form*gen + coeff
else:
F, gens = Poly(F, gen), gens[1:]
for coeff in F.all_coeffs():
form = form*gen + horner(coeff, *gens, **args)
return form
def interpolate(data, x):
"""
Construct an interpolating polynomial for the data points.
**Examples**
>>> from sympy.polys.polyfuncs import interpolate
>>> from sympy.abc import x
>>> interpolate([1, 4, 9, 16], x)
x**2
>>> interpolate([(1, 1), (2, 4), (3, 9)], x)
x**2
>>> interpolate([(1, 2), (2, 5), (3, 10)], x)
x**2 + 1
>>> interpolate({1: 2, 2: 5, 3: 10}, x)
x**2 + 1
"""
n = len(data)
if isinstance(data, dict):
X, Y = zip(*data.items())
else:
if isinstance(data[0], tuple):
X, Y = zip(*data)
else:
X = range(1, n+1)
Y = list(data)
poly = interpolating_poly(n, x, X, Y)
return poly.expand()
def viete(f, roots=None, *gens, **args):
"""
Generate Viete's formulas for ``f``.
**Examples**
>>> from sympy.polys.polyfuncs import viete
>>> from sympy import symbols
>>> x, a, b, c, r1, r2 = symbols('x,a:c,r1:3')
>>> viete(a*x**2 + b*x + c, [r1, r2], x)
[(r1 + r2, -b/a), (r1*r2, c/a)]
"""
allowed_flags(args, [])
if isinstance(roots, Basic):
gens, roots = (roots,) + gens, None
try:
f, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed, exc:
raise ComputationFailed('viete', 1, exc)
if f.is_multivariate:
raise MultivariatePolynomialError("multivariate polynomials are not allowed")
n = f.degree()
if n < 1:
raise ValueError("can't derive Viete's formulas for a constant polynomial")
if roots is None:
roots = numbered_symbols('r', start=1)
roots = take(roots, n)
if n != len(roots):
raise ValueError("required %s roots, got %s" % (n, len(roots)))
lc, coeffs = f.LC(), f.all_coeffs()
result, sign = [], -1
for i, coeff in enumerate(coeffs[1:]):
poly = symmetric_poly(i+1, roots)
coeff = sign*(coeff/lc)
result.append((poly, coeff))
sign = -sign
return result