o 8Va,@sdZddlmZddlmZmZmZddlmZddl m Z m Z m Z ddl mZddlmZmZddlmZGd d d eZd d Zd dZddZddZdS)z,Solvers of systems of polynomial equations. )S)Polygroebnerroots)parallel_poly_from_expr)ComputationFailedPolificationFailedCoercionFailedrcollect)default_sort_key postfixes) filldedentc@seZdZdZdS) SolveFailedz-Raised when solver's conditions weren't met. N)__name__ __module__ __qualname____doc__rr7/usr/lib/python3/dist-packages/sympy/solvers/polysys.pyr src Oszt|g|Ri|\}}Wnty#}ztdt||d}~wwt|t|jkr3dkrYnn$|\}}tdd||DrYzt|||WStyXYnwt ||S)a Solve a system of polynomial equations. Parameters ========== seq: a list/tuple/set Listing all the equations that are needed to be solved gens: generators generators of the equations in seq for which we want the solutions args: Keyword arguments Special options for solving the equations Returns ======= List[Tuple] A List of tuples. Solutions for symbols that satisfy the equations listed in seq Examples ======== >>> from sympy import solve_poly_system >>> from sympy.abc import x, y >>> solve_poly_system([x*y - 2*y, 2*y**2 - x**2], x, y) [(0, 0), (2, -sqrt(2)), (2, sqrt(2))] solve_poly_systemNcss|]}|dkVqdS)rNr).0irrr 9sz$solve_poly_system..) rrrlengensallZ degree_listsolve_biquadraticr solve_generic)seqrargspolysoptexcfgrrrrs "  rc st||g}t|dkr|djrdSt|dkrt|j\}|\}}||js,tt||dd}fddt|D}| d }t t|}g} |D]} |D]} | | | f} | | qUqQt | td S) a!Solve a system of two bivariate quadratic polynomial equations. Parameters ========== f: a single Expr or Poly First equation g: a single Expr or Poly Second Equation opt: an Options object For specifying keyword arguments and generators Returns ======= List[Tuple] A List of tuples. Solutions for symbols that satisfy the equations listed in seq. Examples ======== >>> from sympy.polys import Options, Poly >>> from sympy.abc import x, y >>> from sympy.solvers.polysys import solve_biquadratic >>> NewOption = Options((x, y), {'domain': 'ZZ'}) >>> a = Poly(y**2 - 4 + x, y, x, domain='ZZ') >>> b = Poly(y*2 + 3*x - 7, y, x, domain='ZZ') >>> solve_biquadratic(a, b, NewOption) [(1/3, 3), (41/27, 11/9)] >>> a = Poly(y + x**2 - 3, y, x, domain='ZZ') >>> b = Poly(-y + x - 4, y, x, domain='ZZ') >>> solve_biquadratic(a, b, NewOption) [(7/2 - sqrt(29)/2, -sqrt(29)/2 - 1/2), (sqrt(29)/2 + 7/2, -1/2 + sqrt(29)/2)] rNrF)expandcsg|]}t|qSrr )rexpryrr xsz%solve_biquadratic..key)rr is_groundrrZgcdrrkeysltrimlistZsubsappendsortedr ) r%r&r#GxpqZp_rootsZq_roots solutionsZq_rootZp_rootsolutionrr*rrBs( '      rcsbddddd fdd z ||jdd }Wn ty$tw|d ur/t|td Sd S) an Solve a generic system of polynomial equations. Returns all possible solutions over C[x_1, x_2, ..., x_m] of a set F = { f_1, f_2, ..., f_n } of polynomial equations, using Groebner basis approach. For now only zero-dimensional systems are supported, which means F can have at most a finite number of solutions. The algorithm works by the fact that, supposing G is the basis of F with respect to an elimination order (here lexicographic order is used), G and F generate the same ideal, they have the same set of solutions. By the elimination property, if G is a reduced, zero-dimensional Groebner basis, then there exists an univariate polynomial in G (in its last variable). This can be solved by computing its roots. Substituting all computed roots for the last (eliminated) variable in other elements of G, new polynomial system is generated. Applying the above procedure recursively, a finite number of solutions can be found. The ability of finding all solutions by this procedure depends on the root finding algorithms. If no solutions were found, it means only that roots() failed, but the system is solvable. To overcome this difficulty use numerical algorithms instead. Parameters ========== polys: a list/tuple/set Listing all the polynomial equations that are needed to be solved opt: an Options object For specifying keyword arguments and generators Returns ======= List[Tuple] A List of tuples. Solutions for symbols that satisfy the equations listed in seq References ========== .. [Buchberger01] B. Buchberger, Groebner Bases: A Short Introduction for Systems Theorists, In: R. Moreno-Diaz, B. Buchberger, J.L. Freire, Proceedings of EUROCAST'01, February, 2001 .. [Cox97] D. Cox, J. Little, D. O'Shea, Ideals, Varieties and Algorithms, Springer, Second Edition, 1997, pp. 112 Examples ======== >>> from sympy.polys import Poly, Options >>> from sympy.solvers.polysys import solve_generic >>> from sympy.abc import x, y >>> NewOption = Options((x, y), {'domain': 'ZZ'}) >>> a = Poly(x - y + 5, x, y, domain='ZZ') >>> b = Poly(x + y - 3, x, y, domain='ZZ') >>> solve_generic([a, b], NewOption) [(-1, 4)] >>> a = Poly(x - 2*y + 5, x, y, domain='ZZ') >>> b = Poly(2*x - y - 3, x, y, domain='ZZ') >>> solve_generic([a, b], NewOption) [(11/3, 13/3)] >>> a = Poly(x**2 + y, x, y, domain='ZZ') >>> b = Poly(x + y*4, x, y, domain='ZZ') >>> solve_generic([a, b], NewOption) [(0, 0), (1/4, -1/16)] cSs(|D] }t|ddrdSqdS)z8Returns True if 'f' is univariate in its last variable. Nr-FT)Zmonomsany)r%Zmonomrrr_is_univariates z%solve_generic.._is_univariatecSs,|||i}||dkr|jdd}|S)z:Replace generator with a root so that the result is nice. rF)Zdeep)Zas_exprdegreer()r%genzeror8rrr _subs_roots z!solve_generic.._subs_rootFcst|t|krdkr$nntt|d|d}dd|DSt||dd}t|dkr<|djr<|s:gSdStt|}t|dkrN|}ntt d |j }|d}tt| |}|sjgSt|dkrwd d|DSg}|D]3} g} |dd} |ddD]} | || } | t j ur| | q| | D] }||| fqq{|rt|dt|krtt d |S) z/Recursively solves reduced polynomial systems. r'rr-cSg|]}|fqSrrrr@rrrr,z@solve_generic.._solve_reduced_system..Tr"Nzv only zero-dimensional systems supported (finite number of solutions) cSrBrrrCrrrr,rD)rr3rr1rr0filterpopNotImplementedErrorrrr2rZZeror4)systemrentryzerosZbasisZ univariater%r?r:r@Z new_systemZnew_gensbeqr;r=_solve_reduced_systemrArrrOsD          z,solve_generic.._solve_reduced_systemT)rJNr.)F)rr rHr5r )r"r#resultrrNrrsK 9  rcOst||dd}tt|}|d}|dur&t|D] \}}||||<q|dd|dd}}|}|} t } | D] } | | f|fqAt|dd} t |dd} t | | D]n\}}t }| D]b\}}gtt ||}}|D]+}|f|}|j |r||dkr||t|}|||kr||qzt|dd d }|} | D]} | js|| }n|}| | f||fqqj|} qat| } t| D] \}\}}|| |<qt| td S) a Solve a polynomial system using Gianni-Kalkbrenner algorithm. The algorithm proceeds by computing one Groebner basis in the ground domain and then by iteratively computing polynomial factorizations in appropriately constructed algebraic extensions of the ground domain. Parameters ========== polys: a list/tuple/set Listing all the equations that are needed to be solved gens: generators generators of the equations in polys for which we want the solutions args: Keyword arguments Special options for solving the equations Returns ======= List[Tuple] A List of tuples. Solutions for symbols that satisfy the equations listed in polys Examples ======== >>> from sympy.solvers.polysys import solve_triangulated >>> from sympy.abc import x, y, z >>> F = [x**2 + y + z - 1, x + y**2 + z - 1, x + y + z**2 - 1] >>> solve_triangulated(F, x, y, z) [(0, 0, 1), (0, 1, 0), (1, 0, 0)] References ========== 1. Patrizia Gianni, Teo Mora, Algebraic Solution of System of Polynomial Equations using Groebner Bases, AAECC-5 on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, LNCS 356 247--257, 1989 TrEdomainNrr-r'cSs|S)N)r>)hrrrxsz$solve_triangulated..r.)rr3reversedget enumerateZ set_domainr2Z get_domainZ ground_rootssetaddr zipZ has_only_gensr>evaldictr4minZ is_RationalZalgebraic_fieldr5r )r"rr!r6rQrr&r%ZdomrKr:r@Zvar_seqZvars_seqvarvarsZ _solutionsvaluesHmappingZ_varsrRr8Zdom_zeror;_rrrsolve_triangulated'sL-        rcN)rZ sympy.corerZ sympy.polysrrrZsympy.polys.polytoolsrZsympy.polys.polyerrorsrrr Zsympy.simplifyr Zsympy.utilitiesr r Zsympy.utilities.miscr Exceptionrrrrrcrrrrs    1E !