ó 9­\c@ sîdZddlmZmZddlmZddlmZmZm Z ddl m Z ddl m Z mZmZmZddlmZddlmZmZdd lmZd efd „ƒYZd „Zd „Zd„Zd„ZdS(s,Solvers of systems of polynomial equations. iÿÿÿÿ(tprint_functiontdivision(tS(tPolytgroebnertroots(tparallel_poly_from_expr(tComputationFailedtPolificationFailedtCoercionFailedtPolynomialError(trcollect(tdefault_sort_keyt postfixes(t filldedentt SolveFailedcB seZdZRS(s-Raised when solver's conditions weren't met. (t__name__t __module__t__doc__(((s4lib/python2.7/site-packages/sympy/solvers/polysys.pyRscO sâyt|||Ž\}}Wn+tk rI}tdt|ƒ|ƒ‚nXt|ƒt|jƒkopdknrÕ|\}}td„|jƒ|jƒDƒƒrÕyt|||ƒSWqÒtk rÎqÒXqÕnt ||ƒS(s Solve a system of polynomial equations. 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))] tsolve_poly_systemics s|]}|dkVqdS(iN((t.0ti((s4lib/python2.7/site-packages/sympy/solvers/polysys.pys )s( RRRtlentgenstallt degree_listtsolve_biquadraticRt solve_generic(tseqRtargstpolystopttexctftg((s4lib/python2.7/site-packages/sympy/solvers/polysys.pyRs+ &  cC sMt||gƒ}t|ƒdkr5|djr5dSt|ƒdkrPt‚n|j\}}|\}}|j|ƒjs†t‚nt||dtƒ}gt |ƒj ƒD]}t ||ƒ^q®} |j dƒ}t t |ƒj ƒƒ} g} xD| D]<} x3| D]+} | j|| ƒ| f}| j|ƒq WqýWt| dtƒS(s¹Solve a system of two bivariate quadratic polynomial equations. 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)] iiitexpandiÿÿÿÿtkeyN(RRt is_groundtNoneRRtgcdRtFalseRtkeysR tltrimtlisttsubstappendtsortedR (R!R"RtGtxtytptqtexprtp_rootstq_rootst solutionstq_roottp_roottsolution((s4lib/python2.7/site-packages/sympy/solvers/polysys.pyR2s&   .  c s„d„‰d„‰t‡‡‡fd†‰yˆ||jdtƒ}Wntk r_t‚nX|dk r|t|dtƒSdSdS(s 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. 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)] cS s9x2|jƒD]$}td„|d Dƒƒr tSq WtS(s8Returns True if 'f' is univariate in its last variable. cs s|] }|VqdS(N((Rtm((s4lib/python2.7/site-packages/sympy/solvers/polysys.pys ¥siÿÿÿÿ(tmonomstanyR(tTrue(R!tmonom((s4lib/python2.7/site-packages/sympy/solvers/polysys.pyt_is_univariate¢scS sD|ji||6ƒ}|j|ƒdkr@|jdtƒ}n|S(s:Replace generator with a root so that the result is nice. itdeep(tas_exprtdegreeR#R((R!tgentzeroR2((s4lib/python2.7/site-packages/sympy/solvers/polysys.pyt _subs_rootªsc s2t|ƒt|ƒko#dknrett|d|dƒjƒƒ}g|D]}|f^qRSt||dtƒ}t|ƒdkrª|djrª|s£gSdSnttˆ|ƒƒ}t|ƒdkrà|j ƒ}nt t dƒƒ‚|j }|d}tt|j |ƒƒjƒƒ}|s0gSt|ƒdkr\g|D]}|f^qISg} x’|D]Š}g} |d } xC|d D]7} ˆ| ||ƒ} | tjk rŠ| j| ƒqŠqŠWx+ˆ| | ƒD]}| j||fƒqÕWqiW| r.t| dƒt|ƒkr.t t dƒƒ‚n| S(s/Recursively solves reduced polynomial systems. iiiÿÿÿÿRsv only zero-dimensional systems supported (finite number of solutions) N(RR+RR)RR>R%R&tfiltertpoptNotImplementedErrorRRR*RtZeroR-(tsystemRtentrytzerosREtbasist univariateR!RDR7t new_systemtnew_genstbteqR:(R@t_solve_reduced_systemRF(s4lib/python2.7/site-packages/sympy/solvers/polysys.pyRT³sD(#   !  "RLR$N(R(RR>R RIR&R.R (RRtresult((R@RTRFs4lib/python2.7/site-packages/sympy/solvers/polysys.pyRfs<  9   cO s§t||dtƒ}tt|ƒƒ}|jdƒ}|dk rux0t|ƒD]\}}|j|ƒ||>> 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 RtdomainiiÿÿÿÿiR$cS s |jƒS(N(RC(th((s4lib/python2.7/site-packages/sympy/solvers/polysys.pyt6tN(RR>R+treversedtgetR&t enumeratet set_domainR*t get_domaint ground_rootstsettaddR tzipt has_only_gensRCtevaltdictR-tmint is_Rationaltalgebraic_fieldR.R (RRRR/RVRR"R!tdomRMR7REtvar_seqtvars_seqtvartvarst _solutionstvaluestHtmappingt_varsRWR2tdom_zeroR:t_((s4lib/python2.7/site-packages/sympy/solvers/polysys.pytsolve_triangulated÷sH        $   "  N(Rt __future__RRt sympy.coreRt sympy.polysRRRtsympy.polys.polytoolsRtsympy.polys.polyerrorsRRR R tsympy.simplifyR tsympy.utilitiesR R tsympy.utilities.miscRt ExceptionRRRRRu(((s4lib/python2.7/site-packages/sympy/solvers/polysys.pyts"  4 ‘