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 as 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
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 as 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
def apart_undetermined_coeffs(P, Q):
"""Partial fractions via method of undetermined coefficients. """
X = numbered_symbols(cls=Dummy)
partial, symbols = [], []
_, factors = Q.factor_list()
for f, k in factors:
n, q = f.degree(), Q
for i in range(1, k + 1):
coeffs, q = take(X, n), q.quo(f)
partial.append((coeffs, q, f, i))
symbols.extend(coeffs)
dom = Q.get_domain().inject(*symbols)
F = Poly(0, Q.gen, domain=dom)
for i, (coeffs, q, f, k) in enumerate(partial):
h = Poly(coeffs, Q.gen, domain=dom)
partial[i] = (h, f, k)
q = q.set_domain(dom)
F += h * q
system, result = [], S.Zero
for (k,), coeff in F.terms():
system.append(coeff - P.nth(k))
from sympy.solvers import solve
solution = solve(system, symbols)
for h, f, k in partial:
h = h.as_expr().subs(solution)
result += h / f.as_expr() ** k
return result
def apart_undetermined_coeffs(P, Q):
"""Partial fractions via method of undetermined coefficients. """
X = numbered_symbols(cls=Dummy)
partial, symbols = [], []
_, factors = Q.factor_list()
for f, k in factors:
n, q = f.degree(), Q
for i in xrange(1, k + 1):
coeffs, q = take(X, n), q.quo(f)
partial.append((coeffs, q, f, i))
symbols.extend(coeffs)
dom = Q.get_domain().inject(*symbols)
F = Poly(0, Q.gen, domain=dom)
for i, (coeffs, q, f, k) in enumerate(partial):
h = Poly(coeffs, Q.gen, domain=dom)
partial[i] = (h, f, k)
q = q.set_domain(dom)
F += h * q
system, result = [], S(0)
for (k,), coeff in F.terms():
system.append(coeff - P.nth(k))
from sympy.solvers import solve
solution = solve(system, symbols)
for h, f, k in partial:
h = h.as_expr().subs(solution)
result += h / f.as_expr() ** k
return result
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
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