ó ¡¼™\c@sàdZddlmZmZmZmZddlmZmZmZm Z ddl m Z ddl m Z mZddlmZddlmZmZddlmZdd lmZd dd „ƒYZd dd „ƒYZdS(s This module contains functions for two multivariate resultants. These are: - Dixon's resultant. - Macaulay's resultant. Multivariate resultants are used to identify whether a multivariate system has common roots. That is when the resultant is equal to zero. iÿÿÿÿ(t IndexedBasetMatrixtMultPoly(tremtprodt degree_listtdiag(trange(t monomial_degt itermonomials(t monomial_key(tpoly_from_exprt total_degree(tbinomial(tcombinations_with_replacementtDixonResultantcBs2eZdZd„Zd„Zd„Zd„ZRS(sL A class for retrieving the Dixon's resultant of a multivariate system. Examples ======== >>> from sympy.core import symbols >>> from sympy.polys.multivariate_resultants import DixonResultant >>> x, y = symbols('x, y') >>> p = x + y >>> q = x ** 2 + y ** 3 >>> h = x ** 2 + y >>> dixon = DixonResultant(variables=[x, y], polynomials=[p, q, h]) >>> poly = dixon.get_dixon_polynomial() >>> matrix = dixon.get_dixon_matrix(polynomial=poly) >>> matrix Matrix([ [ 0, 0, -1, 0, -1], [ 0, -1, 0, -1, 0], [-1, 0, 1, 0, 0], [ 0, -1, 0, 0, 1], [-1, 0, 0, 1, 0]]) >>> matrix.det() 0 See Also ======== Notebook in examples: sympy/example/notebooks. References ========== .. [1] [Kapur1994]_ .. [2] [Palancz08]_ cs­||_||_t|jƒ|_t|jƒ|_tdƒ}gt|jƒD]‰|ˆ^qR|_gt|jƒD]%‰t‡fd†|jDƒƒ^q{|_ dS(sV A class that takes two lists, a list of polynomials and list of variables. Returns the Dixon matrix of the multivariate system. Parameters ---------- polynomials : list of polynomials A list of m n-degree polynomials variables: list A list of all n variables talphac3s|]}t|ƒˆVqdS(N(R(t.0tpoly(ti(sBlib/python2.7/site-packages/sympy/polys/multivariate_resultants.pys ZsN( t polynomialst variablestlentntmRRtdummy_variablestmaxt max_degrees(tselfRRta((RsBlib/python2.7/site-packages/sympy/polys/multivariate_resultants.pyt__init__Bs   )c Cs'|j|jdkr%tdƒ‚n|jg}t|jƒ}xpt|jƒD]_}|j|||vs i(RRt ValueErrorRtlistRRRtziptappendtsubsRRtdettfactorR ( Rtrowsttemptidxt substitutiontftAttermsRtbtproduct_of_differencestdixon_polynomial((sBlib/python2.7/site-packages/sympy/polys/multivariate_resultants.pytget_dixon_polynomial]s 0 )cCs\gt|jƒD]}|j||j|^q}t|ƒ}t|ƒjƒ}t|ŒS(N(RRRRRRtmonomsR (RRtlist_of_productstproduct((sBlib/python2.7/site-packages/sympy/polys/multivariate_resultants.pytget_upper_degrees 1 c Cs|jƒ}tt|j|jƒƒƒ}t|dtdtd|jƒƒ}tg|D]4}g|D]!}t ||jŒj |ƒ^qe^qXƒ}gt |j dƒD]A}t g|dd…|fD]}|dk^qɃr©|^q©} |dd…| fS(s¥ Construct the Dixon matrix from the coefficients of polynomial \alpha. Each coefficient is viewed as a polynomial of x_1, ..., x_n. treversetkeytlexiÿÿÿÿNi(tcoeffsR#R RR7tsortedtTrueR RRtcoeff_monomialRtshapetany( Rt polynomialt coefficientst monomialstcRt dixon_matrixtcolumntelementtkeep((sBlib/python2.7/site-packages/sympy/polys/multivariate_resultants.pytget_dixon_matrix‰s  A;(t__name__t __module__t__doc__RR3R7RI(((sBlib/python2.7/site-packages/sympy/polys/multivariate_resultants.pyRs )  $ tMacaulayResultantcBsVeZdZd„Zd„Zd„Zd„Zd„Zd„Zd„Z d„Z RS( s2 A class for calculating the Macaulay resultant. Note that the polynomials must be homogenized and their coefficients must be given as symbols. Examples ======== >>> from sympy.core import symbols >>> from sympy.polys.multivariate_resultants import MacaulayResultant >>> x, y, z = symbols('x, y, z') >>> a_0, a_1, a_2 = symbols('a_0, a_1, a_2') >>> b_0, b_1, b_2 = symbols('b_0, b_1, b_2') >>> c_0, c_1, c_2,c_3, c_4 = symbols('c_0, c_1, c_2, c_3, c_4') >>> f = a_0 * y - a_1 * x + a_2 * z >>> g = b_1 * x ** 2 + b_0 * y ** 2 - b_2 * z ** 2 >>> h = c_0 * y * z ** 2 - c_1 * x ** 3 + c_2 * x ** 2 * z - c_3 * x * z ** 2 + c_4 * z ** 3 >>> mac = MacaulayResultant(polynomials=[f, g, h], variables=[x, y, z]) >>> mac.monomial_set [x**4, x**3*y, x**3*z, x**2*y**2, x**2*y*z, x**2*z**2, x*y**3, x*y**2*z, x*y*z**2, x*z**3, y**4, y**3*z, y**2*z**2, y*z**3, z**4] >>> matrix = mac.get_matrix() >>> submatrix = mac.get_submatrix(matrix) >>> submatrix Matrix([ [-a_1, a_0, a_2, 0], [ 0, -a_1, 0, 0], [ 0, 0, -a_1, 0], [ 0, 0, 0, -a_1]]) See Also ======== Notebook in examples: sympy/example/notebooks. References ========== .. [1] [Bruce97]_ .. [2] [Stiller96]_ cCsƒ||_||_t|ƒ|_g|jD]}t||jŒ^q+|_|jƒ|_|jƒ|_ |j |jƒ|_ dS(sÌ Parameters ========== variables: list A list of all n variables polynomials : list of sympy polynomials A list of m n-degree polynomials N( RRRRR tdegreest _get_degree_mtdegree_mtget_sizetmonomials_sizetget_monomials_of_certain_degreet monomial_set(RRRR((sBlib/python2.7/site-packages/sympy/polys/multivariate_resultants.pyRÑs  (cCsdtd„|jDƒƒS(s½ Returns ======= degree_m: int The degree_m is calculated as 1 + \sum_1 ^ n (d_i - 1), where d_i is the degree of the i polynomial icss|]}|dVqdS(iN((Rtd((sBlib/python2.7/site-packages/sympy/polys/multivariate_resultants.pys òs(tsumRN(R((sBlib/python2.7/site-packages/sympy/polys/multivariate_resultants.pyROés cCs"t|j|jd|jdƒS(sÒ Returns ======= size: int The size of set T. Set T is the set of all possible monomials of the n variables for degree equal to the degree_m i(RRPR(R((sBlib/python2.7/site-packages/sympy/polys/multivariate_resultants.pyRQôs cCsMgt|j|ƒD]}t|Œ^q}t|dtdtd|jƒƒS(sw Returns ======= monomials: list A list of monomials of a certain degree. R8R9R:(RRRR<R=R (RtdegreetmonomialRC((sBlib/python2.7/site-packages/sympy/polys/multivariate_resultants.pyRSs  c Csg}g}x t|jƒD]û}|dkra|j|j|}|j|ƒ}|j|ƒq|j|j|d|j|dƒ|j|j|}|j|ƒ}x\|D]T}xK|D]C}t||ƒdkr¿g|D]} | |krá| ^qá}q¿q¿Wq²W|j|ƒqW|S(s Returns ======= row_coefficients: list The row coefficients of Macaulay's matrix ii(RRRPRNRSR%RR( Rtrow_coefficientst divisibleRRWRXt poss_rowstdivtptitem((sBlib/python2.7/site-packages/sympy/polys/multivariate_resultants.pytget_row_coefficientss$    #c Cs«g}|jƒ}x†t|jƒD]u}xl||D]`}g}t|j|||jŒ}x'|jD]}|j|j|ƒƒqfW|j|ƒq3Wq"Wt |ƒ}|S(st Returns ======= macaulay_matrix: Matrix The Macaulay numerator matrix ( R_RRRRRRTR%R>R( RR)RYRt multiplierRBRtmonotmacaulay_matrix((sBlib/python2.7/site-packages/sympy/polys/multivariate_resultants.pyt get_matrix+s   c Csüg}xm|jD]b}g}xFt|jƒD]5\}}|jtt||ƒ|j|kƒƒq,W|j|ƒqWgt|ƒD]+\}}t|ƒ|jdkrƒ|^qƒ}gt|ƒD]+\}}t|ƒ|jdkrÁ|^qÁ}||fS(sÓ Returns ======= reduced: list A list of the reduced monomials non_reduced: list A list of the monomials that are not reduced Definition ========== A polynomial is said to be reduced in x_i, if its degree (the maximum degree of its monomials) in x_i is less than d_i. A polynomial that is reduced in all variables but one is said simply to be reduced. i( RTt enumerateRR%tboolR RNRVR( RRZRR*Rtvtrtreducedt non_reduced((sBlib/python2.7/site-packages/sympy/polys/multivariate_resultants.pytget_reduced_nonreducedBs-%%c Cs)|jƒ\}}|gkr+tdgƒSgt|jƒD]\}}||j|^q;}tgt|jƒD] }|j|j ||ƒ^qqƒ}|dd…|f}g} xbt|j ƒD]Q} g|D]"} | || dd…fk^qÓ} t | krÆ| j | ƒqÆqÆW|| |fS(s Returns ======= macaulay_submatrix: Matrix The Macaulay denominator matrix. Columns that are non reduced are kept. The row which contains one of the a_{i}s is dropped. a_{i}s are the coefficients of x_i ^ {d_i}. iN( RjRRdRRNR#RRRtcoeffR)R=R%( RtmatrixRhRiRRft reduction_settaistreduced_matrixRHtrowtaitcheck((sBlib/python2.7/site-packages/sympy/polys/multivariate_resultants.pyt get_submatrixas   06/ ( RJRKRLRRORQRSR_RcRjRs(((sBlib/python2.7/site-packages/sympy/polys/multivariate_resultants.pyRM¢s.     N(((RLtsympyRRRRRRRRtsympy.core.compatibilityRtsympy.polys.monomialsR R tsympy.polys.orderingsR tsympy.polys.polytoolsR R t(sympy.functions.combinatorial.factorialsRt itertoolsRRRM(((sBlib/python2.7/site-packages/sympy/polys/multivariate_resultants.pyt s""‹