322 lines
8.3 KiB
Python
322 lines
8.3 KiB
Python
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"""High-level polynomials manipulation functions. """
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from sympy.core import S, Basic, symbols, Dummy
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from sympy.polys.polyerrors import (
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PolificationFailed, ComputationFailed,
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MultivariatePolynomialError, OptionError)
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from sympy.polys.polyoptions import allowed_flags, build_options
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from sympy.polys.polytools import poly_from_expr, Poly
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from sympy.polys.specialpolys import (
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symmetric_poly, interpolating_poly)
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from sympy.polys.rings import sring
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from sympy.utilities import numbered_symbols, take, public
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@public
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def symmetrize(F, *gens, **args):
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r"""
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Rewrite a polynomial in terms of elementary symmetric polynomials.
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A symmetric polynomial is a multivariate polynomial that remains invariant
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under any variable permutation, i.e., if `f = f(x_1, x_2, \dots, x_n)`,
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then `f = f(x_{i_1}, x_{i_2}, \dots, x_{i_n})`, where
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`(i_1, i_2, \dots, i_n)` is a permutation of `(1, 2, \dots, n)` (an
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element of the group `S_n`).
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Returns a tuple of symmetric polynomials ``(f1, f2, ..., fn)`` such that
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``f = f1 + f2 + ... + fn``.
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Examples
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========
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>>> from sympy.polys.polyfuncs import symmetrize
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>>> from sympy.abc import x, y
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>>> symmetrize(x**2 + y**2)
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(-2*x*y + (x + y)**2, 0)
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>>> symmetrize(x**2 + y**2, formal=True)
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(s1**2 - 2*s2, 0, [(s1, x + y), (s2, x*y)])
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>>> symmetrize(x**2 - y**2)
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(-2*x*y + (x + y)**2, -2*y**2)
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>>> symmetrize(x**2 - y**2, formal=True)
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(s1**2 - 2*s2, -2*y**2, [(s1, x + y), (s2, x*y)])
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"""
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allowed_flags(args, ['formal', 'symbols'])
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iterable = True
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if not hasattr(F, '__iter__'):
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iterable = False
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F = [F]
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R, F = sring(F, *gens, **args)
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gens = R.symbols
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opt = build_options(gens, args)
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symbols = opt.symbols
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symbols = [next(symbols) for i in range(len(gens))]
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result = []
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for f in F:
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p, r, m = f.symmetrize()
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result.append((p.as_expr(*symbols), r.as_expr(*gens)))
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polys = [(s, g.as_expr()) for s, (_, g) in zip(symbols, m)]
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if not opt.formal:
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for i, (sym, non_sym) in enumerate(result):
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result[i] = (sym.subs(polys), non_sym)
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if not iterable:
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result, = result
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if not opt.formal:
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return result
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else:
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if iterable:
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return result, polys
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else:
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return result + (polys,)
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@public
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def horner(f, *gens, **args):
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"""
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Rewrite a polynomial in Horner form.
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Among other applications, evaluation of a polynomial at a point is optimal
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when it is applied using the Horner scheme ([1]).
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Examples
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========
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>>> from sympy.polys.polyfuncs import horner
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>>> from sympy.abc import x, y, a, b, c, d, e
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>>> horner(9*x**4 + 8*x**3 + 7*x**2 + 6*x + 5)
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x*(x*(x*(9*x + 8) + 7) + 6) + 5
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>>> horner(a*x**4 + b*x**3 + c*x**2 + d*x + e)
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e + x*(d + x*(c + x*(a*x + b)))
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>>> f = 4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y
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>>> horner(f, wrt=x)
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x*(x*y*(4*y + 2) + y*(2*y + 1))
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>>> horner(f, wrt=y)
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y*(x*y*(4*x + 2) + x*(2*x + 1))
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References
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==========
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[1] - https://en.wikipedia.org/wiki/Horner_scheme
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"""
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allowed_flags(args, [])
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try:
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F, opt = poly_from_expr(f, *gens, **args)
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except PolificationFailed as exc:
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return exc.expr
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form, gen = S.Zero, F.gen
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if F.is_univariate:
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for coeff in F.all_coeffs():
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form = form*gen + coeff
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else:
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F, gens = Poly(F, gen), gens[1:]
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for coeff in F.all_coeffs():
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form = form*gen + horner(coeff, *gens, **args)
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return form
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@public
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def interpolate(data, x):
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"""
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Construct an interpolating polynomial for the data points
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evaluated at point x (which can be symbolic or numeric).
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Examples
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========
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>>> from sympy.polys.polyfuncs import interpolate
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>>> from sympy.abc import a, b, x
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A list is interpreted as though it were paired with a range starting
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from 1:
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>>> interpolate([1, 4, 9, 16], x)
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x**2
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This can be made explicit by giving a list of coordinates:
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>>> interpolate([(1, 1), (2, 4), (3, 9)], x)
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x**2
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The (x, y) coordinates can also be given as keys and values of a
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dictionary (and the points need not be equispaced):
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>>> interpolate([(-1, 2), (1, 2), (2, 5)], x)
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x**2 + 1
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>>> interpolate({-1: 2, 1: 2, 2: 5}, x)
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x**2 + 1
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If the interpolation is going to be used only once then the
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value of interest can be passed instead of passing a symbol:
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>>> interpolate([1, 4, 9], 5)
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25
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Symbolic coordinates are also supported:
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>>> [(i,interpolate((a, b), i)) for i in range(1, 4)]
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[(1, a), (2, b), (3, -a + 2*b)]
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"""
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n = len(data)
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if isinstance(data, dict):
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if x in data:
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return S(data[x])
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X, Y = list(zip(*data.items()))
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else:
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if isinstance(data[0], tuple):
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X, Y = list(zip(*data))
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if x in X:
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return S(Y[X.index(x)])
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else:
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if x in range(1, n + 1):
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return S(data[x - 1])
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Y = list(data)
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X = list(range(1, n + 1))
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try:
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return interpolating_poly(n, x, X, Y).expand()
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except ValueError:
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d = Dummy()
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return interpolating_poly(n, d, X, Y).expand().subs(d, x)
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@public
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def rational_interpolate(data, degnum, X=symbols('x')):
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"""
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Returns a rational interpolation, where the data points are element of
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any integral domain.
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The first argument contains the data (as a list of coordinates). The
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``degnum`` argument is the degree in the numerator of the rational
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function. Setting it too high will decrease the maximal degree in the
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denominator for the same amount of data.
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Examples
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========
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>>> from sympy.polys.polyfuncs import rational_interpolate
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>>> data = [(1, -210), (2, -35), (3, 105), (4, 231), (5, 350), (6, 465)]
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>>> rational_interpolate(data, 2)
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(105*x**2 - 525)/(x + 1)
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Values do not need to be integers:
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>>> from sympy import sympify
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>>> x = [1, 2, 3, 4, 5, 6]
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>>> y = sympify("[-1, 0, 2, 22/5, 7, 68/7]")
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>>> rational_interpolate(zip(x, y), 2)
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(3*x**2 - 7*x + 2)/(x + 1)
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The symbol for the variable can be changed if needed:
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>>> from sympy import symbols
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>>> z = symbols('z')
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>>> rational_interpolate(data, 2, X=z)
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(105*z**2 - 525)/(z + 1)
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References
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==========
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.. [1] Algorithm is adapted from:
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http://axiom-wiki.newsynthesis.org/RationalInterpolation
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"""
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from sympy.matrices.dense import ones
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xdata, ydata = list(zip(*data))
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k = len(xdata) - degnum - 1
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if k < 0:
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raise OptionError("Too few values for the required degree.")
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c = ones(degnum + k + 1, degnum + k + 2)
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for j in range(max(degnum, k)):
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for i in range(degnum + k + 1):
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c[i, j + 1] = c[i, j]*xdata[i]
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for j in range(k + 1):
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for i in range(degnum + k + 1):
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c[i, degnum + k + 1 - j] = -c[i, k - j]*ydata[i]
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r = c.nullspace()[0]
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return (sum(r[i] * X**i for i in range(degnum + 1))
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/ sum(r[i + degnum + 1] * X**i for i in range(k + 1)))
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@public
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def viete(f, roots=None, *gens, **args):
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"""
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Generate Viete's formulas for ``f``.
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Examples
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========
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>>> from sympy.polys.polyfuncs import viete
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>>> from sympy import symbols
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>>> x, a, b, c, r1, r2 = symbols('x,a:c,r1:3')
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>>> viete(a*x**2 + b*x + c, [r1, r2], x)
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[(r1 + r2, -b/a), (r1*r2, c/a)]
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"""
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allowed_flags(args, [])
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if isinstance(roots, Basic):
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gens, roots = (roots,) + gens, None
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try:
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f, opt = poly_from_expr(f, *gens, **args)
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except PolificationFailed as exc:
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raise ComputationFailed('viete', 1, exc)
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if f.is_multivariate:
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raise MultivariatePolynomialError(
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"multivariate polynomials are not allowed")
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n = f.degree()
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if n < 1:
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raise ValueError(
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"Cannot derive Viete's formulas for a constant polynomial")
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if roots is None:
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roots = numbered_symbols('r', start=1)
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roots = take(roots, n)
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if n != len(roots):
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raise ValueError("required %s roots, got %s" % (n, len(roots)))
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lc, coeffs = f.LC(), f.all_coeffs()
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result, sign = [], -1
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for i, coeff in enumerate(coeffs[1:]):
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poly = symmetric_poly(i + 1, roots)
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coeff = sign*(coeff/lc)
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result.append((poly, coeff))
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sign = -sign
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return result
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