441 lines
13 KiB
Python
441 lines
13 KiB
Python
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"""Solvers of systems of polynomial equations. """
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import itertools
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from sympy.core import S
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from sympy.core.sorting import default_sort_key
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from sympy.polys import Poly, groebner, roots
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from sympy.polys.polytools import parallel_poly_from_expr
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from sympy.polys.polyerrors import (ComputationFailed,
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PolificationFailed, CoercionFailed)
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from sympy.simplify import rcollect
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from sympy.utilities import postfixes
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from sympy.utilities.misc import filldedent
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class SolveFailed(Exception):
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"""Raised when solver's conditions were not met. """
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def solve_poly_system(seq, *gens, strict=False, **args):
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"""
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Return a list of solutions for the system of polynomial equations
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or else None.
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Parameters
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==========
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seq: a list/tuple/set
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Listing all the equations that are needed to be solved
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gens: generators
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generators of the equations in seq for which we want the
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solutions
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strict: a boolean (default is False)
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if strict is True, NotImplementedError will be raised if
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the solution is known to be incomplete (which can occur if
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not all solutions are expressible in radicals)
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args: Keyword arguments
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Special options for solving the equations.
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Returns
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=======
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List[Tuple]
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a list of tuples with elements being solutions for the
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symbols in the order they were passed as gens
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None
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None is returned when the computed basis contains only the ground.
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Examples
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========
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>>> from sympy import solve_poly_system
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>>> from sympy.abc import x, y
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>>> solve_poly_system([x*y - 2*y, 2*y**2 - x**2], x, y)
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[(0, 0), (2, -sqrt(2)), (2, sqrt(2))]
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>>> solve_poly_system([x**5 - x + y**3, y**2 - 1], x, y, strict=True)
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Traceback (most recent call last):
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...
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UnsolvableFactorError
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"""
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try:
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polys, opt = parallel_poly_from_expr(seq, *gens, **args)
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except PolificationFailed as exc:
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raise ComputationFailed('solve_poly_system', len(seq), exc)
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if len(polys) == len(opt.gens) == 2:
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f, g = polys
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if all(i <= 2 for i in f.degree_list() + g.degree_list()):
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try:
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return solve_biquadratic(f, g, opt)
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except SolveFailed:
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pass
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return solve_generic(polys, opt, strict=strict)
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def solve_biquadratic(f, g, opt):
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"""Solve a system of two bivariate quadratic polynomial equations.
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Parameters
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==========
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f: a single Expr or Poly
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First equation
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g: a single Expr or Poly
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Second Equation
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opt: an Options object
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For specifying keyword arguments and generators
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Returns
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=======
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List[Tuple]
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a list of tuples with elements being solutions for the
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symbols in the order they were passed as gens
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None
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None is returned when the computed basis contains only the ground.
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Examples
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========
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>>> from sympy import Options, Poly
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>>> from sympy.abc import x, y
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>>> from sympy.solvers.polysys import solve_biquadratic
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>>> NewOption = Options((x, y), {'domain': 'ZZ'})
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>>> a = Poly(y**2 - 4 + x, y, x, domain='ZZ')
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>>> b = Poly(y*2 + 3*x - 7, y, x, domain='ZZ')
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>>> solve_biquadratic(a, b, NewOption)
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[(1/3, 3), (41/27, 11/9)]
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>>> a = Poly(y + x**2 - 3, y, x, domain='ZZ')
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>>> b = Poly(-y + x - 4, y, x, domain='ZZ')
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>>> solve_biquadratic(a, b, NewOption)
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[(7/2 - sqrt(29)/2, -sqrt(29)/2 - 1/2), (sqrt(29)/2 + 7/2, -1/2 + \
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sqrt(29)/2)]
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"""
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G = groebner([f, g])
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if len(G) == 1 and G[0].is_ground:
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return None
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if len(G) != 2:
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raise SolveFailed
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x, y = opt.gens
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p, q = G
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if not p.gcd(q).is_ground:
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# not 0-dimensional
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raise SolveFailed
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p = Poly(p, x, expand=False)
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p_roots = [rcollect(expr, y) for expr in roots(p).keys()]
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q = q.ltrim(-1)
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q_roots = list(roots(q).keys())
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solutions = [(p_root.subs(y, q_root), q_root) for q_root, p_root in
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itertools.product(q_roots, p_roots)]
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return sorted(solutions, key=default_sort_key)
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def solve_generic(polys, opt, strict=False):
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"""
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Solve a generic system of polynomial equations.
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Returns all possible solutions over C[x_1, x_2, ..., x_m] of a
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set F = { f_1, f_2, ..., f_n } of polynomial equations, using
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Groebner basis approach. For now only zero-dimensional systems
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are supported, which means F can have at most a finite number
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of solutions. If the basis contains only the ground, None is
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returned.
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The algorithm works by the fact that, supposing G is the basis
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of F with respect to an elimination order (here lexicographic
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order is used), G and F generate the same ideal, they have the
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same set of solutions. By the elimination property, if G is a
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reduced, zero-dimensional Groebner basis, then there exists an
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univariate polynomial in G (in its last variable). This can be
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solved by computing its roots. Substituting all computed roots
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for the last (eliminated) variable in other elements of G, new
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polynomial system is generated. Applying the above procedure
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recursively, a finite number of solutions can be found.
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The ability of finding all solutions by this procedure depends
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on the root finding algorithms. If no solutions were found, it
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means only that roots() failed, but the system is solvable. To
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overcome this difficulty use numerical algorithms instead.
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Parameters
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==========
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polys: a list/tuple/set
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Listing all the polynomial equations that are needed to be solved
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opt: an Options object
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For specifying keyword arguments and generators
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strict: a boolean
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If strict is True, NotImplementedError will be raised if the solution
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is known to be incomplete
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Returns
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=======
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List[Tuple]
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a list of tuples with elements being solutions for the
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symbols in the order they were passed as gens
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None
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None is returned when the computed basis contains only the ground.
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References
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==========
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.. [Buchberger01] B. Buchberger, Groebner Bases: A Short
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Introduction for Systems Theorists, In: R. Moreno-Diaz,
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B. Buchberger, J.L. Freire, Proceedings of EUROCAST'01,
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February, 2001
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.. [Cox97] D. Cox, J. Little, D. O'Shea, Ideals, Varieties
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and Algorithms, Springer, Second Edition, 1997, pp. 112
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Raises
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========
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NotImplementedError
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If the system is not zero-dimensional (does not have a finite
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number of solutions)
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UnsolvableFactorError
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If ``strict`` is True and not all solution components are
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expressible in radicals
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Examples
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========
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>>> from sympy import Poly, Options
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>>> from sympy.solvers.polysys import solve_generic
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>>> from sympy.abc import x, y
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>>> NewOption = Options((x, y), {'domain': 'ZZ'})
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>>> a = Poly(x - y + 5, x, y, domain='ZZ')
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>>> b = Poly(x + y - 3, x, y, domain='ZZ')
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>>> solve_generic([a, b], NewOption)
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[(-1, 4)]
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>>> a = Poly(x - 2*y + 5, x, y, domain='ZZ')
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>>> b = Poly(2*x - y - 3, x, y, domain='ZZ')
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>>> solve_generic([a, b], NewOption)
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[(11/3, 13/3)]
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>>> a = Poly(x**2 + y, x, y, domain='ZZ')
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>>> b = Poly(x + y*4, x, y, domain='ZZ')
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>>> solve_generic([a, b], NewOption)
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[(0, 0), (1/4, -1/16)]
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>>> a = Poly(x**5 - x + y**3, x, y, domain='ZZ')
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>>> b = Poly(y**2 - 1, x, y, domain='ZZ')
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>>> solve_generic([a, b], NewOption, strict=True)
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Traceback (most recent call last):
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...
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UnsolvableFactorError
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"""
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def _is_univariate(f):
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"""Returns True if 'f' is univariate in its last variable. """
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for monom in f.monoms():
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if any(monom[:-1]):
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return False
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return True
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def _subs_root(f, gen, zero):
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"""Replace generator with a root so that the result is nice. """
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p = f.as_expr({gen: zero})
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if f.degree(gen) >= 2:
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p = p.expand(deep=False)
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return p
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def _solve_reduced_system(system, gens, entry=False):
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"""Recursively solves reduced polynomial systems. """
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if len(system) == len(gens) == 1:
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# the below line will produce UnsolvableFactorError if
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# strict=True and the solution from `roots` is incomplete
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zeros = list(roots(system[0], gens[-1], strict=strict).keys())
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return [(zero,) for zero in zeros]
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basis = groebner(system, gens, polys=True)
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if len(basis) == 1 and basis[0].is_ground:
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if not entry:
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return []
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else:
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return None
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univariate = list(filter(_is_univariate, basis))
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if len(basis) < len(gens):
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raise NotImplementedError(filldedent('''
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only zero-dimensional systems supported
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(finite number of solutions)
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'''))
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if len(univariate) == 1:
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f = univariate.pop()
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else:
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raise NotImplementedError(filldedent('''
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only zero-dimensional systems supported
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(finite number of solutions)
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'''))
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gens = f.gens
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gen = gens[-1]
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# the below line will produce UnsolvableFactorError if
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# strict=True and the solution from `roots` is incomplete
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zeros = list(roots(f.ltrim(gen), strict=strict).keys())
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if not zeros:
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return []
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if len(basis) == 1:
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return [(zero,) for zero in zeros]
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solutions = []
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for zero in zeros:
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new_system = []
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new_gens = gens[:-1]
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for b in basis[:-1]:
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eq = _subs_root(b, gen, zero)
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if eq is not S.Zero:
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new_system.append(eq)
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for solution in _solve_reduced_system(new_system, new_gens):
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solutions.append(solution + (zero,))
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if solutions and len(solutions[0]) != len(gens):
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raise NotImplementedError(filldedent('''
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only zero-dimensional systems supported
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(finite number of solutions)
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'''))
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return solutions
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try:
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result = _solve_reduced_system(polys, opt.gens, entry=True)
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except CoercionFailed:
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raise NotImplementedError
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if result is not None:
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return sorted(result, key=default_sort_key)
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def solve_triangulated(polys, *gens, **args):
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"""
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Solve a polynomial system using Gianni-Kalkbrenner algorithm.
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The algorithm proceeds by computing one Groebner basis in the ground
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domain and then by iteratively computing polynomial factorizations in
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appropriately constructed algebraic extensions of the ground domain.
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Parameters
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==========
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polys: a list/tuple/set
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Listing all the equations that are needed to be solved
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gens: generators
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generators of the equations in polys for which we want the
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solutions
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args: Keyword arguments
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Special options for solving the equations
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Returns
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=======
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List[Tuple]
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A List of tuples. Solutions for symbols that satisfy the
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equations listed in polys
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Examples
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========
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>>> from sympy import solve_triangulated
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>>> from sympy.abc import x, y, z
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>>> F = [x**2 + y + z - 1, x + y**2 + z - 1, x + y + z**2 - 1]
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>>> solve_triangulated(F, x, y, z)
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[(0, 0, 1), (0, 1, 0), (1, 0, 0)]
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References
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==========
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1. Patrizia Gianni, Teo Mora, Algebraic Solution of System of
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Polynomial Equations using Groebner Bases, AAECC-5 on Applied Algebra,
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Algebraic Algorithms and Error-Correcting Codes, LNCS 356 247--257, 1989
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"""
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G = groebner(polys, gens, polys=True)
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G = list(reversed(G))
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domain = args.get('domain')
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if domain is not None:
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for i, g in enumerate(G):
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|
G[i] = g.set_domain(domain)
|
||
|
|
|
||
|
|
f, G = G[0].ltrim(-1), G[1:]
|
||
|
|
dom = f.get_domain()
|
||
|
|
|
||
|
|
zeros = f.ground_roots()
|
||
|
|
solutions = set()
|
||
|
|
|
||
|
|
for zero in zeros:
|
||
|
|
solutions.add(((zero,), dom))
|
||
|
|
|
||
|
|
var_seq = reversed(gens[:-1])
|
||
|
|
vars_seq = postfixes(gens[1:])
|
||
|
|
|
||
|
|
for var, vars in zip(var_seq, vars_seq):
|
||
|
|
_solutions = set()
|
||
|
|
|
||
|
|
for values, dom in solutions:
|
||
|
|
H, mapping = [], list(zip(vars, values))
|
||
|
|
|
||
|
|
for g in G:
|
||
|
|
_vars = (var,) + vars
|
||
|
|
|
||
|
|
if g.has_only_gens(*_vars) and g.degree(var) != 0:
|
||
|
|
h = g.ltrim(var).eval(dict(mapping))
|
||
|
|
|
||
|
|
if g.degree(var) == h.degree():
|
||
|
|
H.append(h)
|
||
|
|
|
||
|
|
p = min(H, key=lambda h: h.degree())
|
||
|
|
zeros = p.ground_roots()
|
||
|
|
|
||
|
|
for zero in zeros:
|
||
|
|
if not zero.is_Rational:
|
||
|
|
dom_zero = dom.algebraic_field(zero)
|
||
|
|
else:
|
||
|
|
dom_zero = dom
|
||
|
|
|
||
|
|
_solutions.add(((zero,) + values, dom_zero))
|
||
|
|
|
||
|
|
solutions = _solutions
|
||
|
|
|
||
|
|
solutions = list(solutions)
|
||
|
|
|
||
|
|
for i, (solution, _) in enumerate(solutions):
|
||
|
|
solutions[i] = solution
|
||
|
|
|
||
|
|
return sorted(solutions, key=default_sort_key)
|