Ed;dZddlmZddlmZddlmZmZddlm Z m Z m Z ddl m Z ddlmZddlmZmZdd lmZdd lmZmZdd lmZdd lmZdd lmZGddZGddZdS)a 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. )prodMul)Matrixdiag)Poly degree_listrem)simplify) IndexedBase) itermonomials monomial_deg) monomial_key)poly_from_expr total_degree)binomial)combinations_with_replacement)sympy_deprecation_warningc^eZdZdZdZedZdZdZdZ dZ dZ d Z d Z d Zd S) DixonResultantaG A class for retrieving the Dixon's resultant of a multivariate system. Examples ======== >>> from sympy 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]_ c<|_|_tj_tj_t dfdt jD_fdt jD_dS)aV 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 alphac g|] }| Sr).0ias Clib/python3.11/site-packages/sympy/polys/multivariate_resultants.py z+DixonResultant.__init__..Xs<<<!<<<cRg|]"tfdjD#S)c3BK|]}t|VdSN)r )rpolyrs r z5DixonResultant.__init__...[s0 S S$T!2!21!5 S S S S S Sr )max polynomialsrrselfs @rrz+DixonResultant.__init__..[sJ$$$! S S S S$BR S S SSS$$$r N) r' variableslennmr rangedummy_variables _max_degrees)r)r'r*rs` @r__init__zDixonResultant.__init__Ds'"T^$$T%&&  <<<z7DixonResultant.get_dixon_polynomial..sKKKvsACKKKr c:g|]}|Sr)subs)rf substitutions rrz7DixonResultant.get_dixon_polynomial..s%HHH! --HHHr cg|] \}}||z  Srr)rrbs rrz7DixonResultant.get_dixon_polynomial..s &?&?&?Aq1u&?&?&?r r)r-r, ValueErrorr'listr*r.r/zipappendrrdetfactorr) r)rowstempidxAtermsproduct_of_differencesdixon_polynomialrCs @rget_dixon_polynomialz#DixonResultant.get_dixon_polynomialis. 6dfqj ! FDEE E !DN##== J JC,S1DIKKT^T1J1JKKKL KKHHHHt7GHHH I I I I 4LLDND$899!$&?&?&?&?&?!@EEGG&<<DDFF.0DEEaHHr ctdddfdtjD}t|}t |}t |S)Nz The get_upper_degree() method of DixonResultant is deprecated. Use get_max_degrees() instead. r3r4r5cFg|]}j|j|zSr)r*r0r(s rrz3DixonResultant.get_upper_degree..s=444 !!N1-1B11EE444r )rr.r,rrmonomsr)r)list_of_productsproducts` rget_upper_degreezDixonResultant.get_upper_degrees! &+'M     4444%*46]]444'((w--&&((W%%r cnfd|D}dt|D}|S)z Returns a list of the maximum degree of each variable appearing in the coefficients of the Dixon polynomial. The coefficients are viewed as polys in $x_1, x_2, \dots, x_n$. cTg|]$}tt|j%Sr)r rr*rr$r)s rrz2DixonResultant.get_max_degrees..s=666!dDN!;!;<<666r c,g|]}t|Sr)r&)rdegss rrz2DixonResultant.get_max_degrees..s===Ts4yy===r )coeffsrH)r) polynomial deg_listsr9s` rget_max_degreeszDixonResultant.get_max_degreessV 6666!+!2!2!4!4666 >=S)_=== r c|}tj|tdt djt fd|Djdjdkr2fdtjdD}d d |fS) z Construct the Dixon matrix from the coefficients of polynomial \alpha. Each coefficient is viewed as a polynomial of x_1, ..., x_n. Tlexreversekeyc0g|]fdDS)cZg|]'}tgjR|(Sr)rr*coeff_monomial)rr-cr)s rrz>DixonResultant.get_dixon_matrix...sI 4 4 4$%!%Q 8 8 8 8 G G J J 4 4 4r r)rrk monomialsr)s @rrz3DixonResultant.get_dixon_matrix..sP>>>$% 4 4 4 4 4)2 4 4 4>>>r rr;cZg|]'}tddd|fD%|(S)c3"K|] }|dkV dSrNr)relements rr%z=DixonResultant.get_dixon_matrix...s644G7a<444444r N)any)rcolumn dixon_matrixs rrz3DixonResultant.get_dixon_matrix..s`555v44'6 2444445F555r N) rbr r*sortedrrr_shaper.)r)r`r9keeprsrls` @@rget_dixon_matrixzDixonResultant.get_dixon_matrixs**:66 "$.+>> 9d+E4>BBDDD >>>>>)3):):)<)<>>>??  a L$6q$9 9 15555|/A"/E)F)F555D(40Lr c(jrdSj\}tdfdt |D}|ddft dgdz zdgzg}dddf|krdSdS)a Test for the validity of the Kapur-Saxena-Yang precondition. The precondition requires that the column corresponding to the monomial 1 = x_1 ^ 0 * x_2 ^ 0 * ... * x_n ^ 0 is not a linear combination of the remaining ones. In SymPy notation this is the last column. For the precondition to hold the last non-zero row of the rref matrix should be of the form [0, 0, ..., 1]. Frchg|]-tfdtD+.S)c34K|]}|fdkVdSror)rjrmatrixs rr%z=DixonResultant.KSY_precondition...s0*O*O6!Q$<1+<*O*O*O*O*O*Or )rqr.)rrr}r,s @rrz3DixonResultant.KSY_precondition..sEPPPas*O*O*O*O*OeAhh*O*O*O'O'OPPPPr Nr;rtT)is_zero_matrixrvr rrefr.r)r)r}r-rL conditionr,s ` @rKSY_preconditionzDixonResultant.KSY_preconditions   5|1&++--*++PPPPP588PPPQQQQC1IO,-- "QQQ$<9 $ 45r cfdtjD}fdtjD}||fS)z/Remove the zero rows and columns of the matrix.cHg|]}|j|Sr)rowr~)rrr}s rrz?DixonResultant.delete_zero_rows_and_columns..BOOOA1MO OOOr cHg|]}|j|Sr)colr~)rr|r}s rrz?DixonResultant.delete_zero_rows_and_columns..rr )r.rLcols)r)r}rLrs ` rdelete_zero_rows_and_columnsz+DixonResultant.delete_zero_rows_and_columnssOOOOV[))OOOOOOOV[))OOOdDj!!r cd}t|jD]'}||D]}|dkr||z}n(|S)z;Calculate the product of the leading entries of the matrix.r;r)r.rLr)r)r}resrels rproduct_leading_entriesz&DixonResultant.product_leading_entriess`%%  Cjjoo  7(CE r c||}|\}}}|t|}||S)z@Calculate the Kapur-Saxena-Yang approach to the Dixon Resultant.)rLUdecompositionr r)r)r}_Us rget_KSY_Dixon_resultantz&DixonResultant.get_KSY_Dixon_resultantsY226::((**1a228A;;??++F333r N)__name__ __module__ __qualname____doc__r1propertyr9rSrYrbrxrrrrrr rrrs((T$$$4!!X!"I"I"IH&&&   86"""44444r rcBeZdZdZdZdZdZdZdZdZ dZ d Z d S) MacaulayResultanta- 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 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]_ c|_|_t|_fdjD___ j_ dS)z Parameters ========== variables: list A list of all n variables polynomials : list of SymPy polynomials A list of m n-degree polynomials c4g|]}t|gjRSr)rr*r\s rrz.MacaulayResultant.__init__..:s7--- T;DN;;;---r N) r'r*r+r,degrees _get_degree_mdegree_mget_sizemonomials_sizeget_monomials_of_certain_degree monomial_set)r)r'r*s` rr1zMacaulayResultant.__init__+s'"Y----+--- **,, "mmoo!@@OOr cDdtd|jDzS)z 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 r;c3 K|] }|dz V dS)r;Nr)rds rr%z2MacaulayResultant._get_degree_m..Ls&33q1u333333r )sumrr8s rrzMacaulayResultant._get_degree_mCs(333dl3333333r cRt|j|jzdz |jdz S)z 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 r;)rrr,r8s rrzMacaulayResultant.get_sizeNs( .2DFQJ???r cdt|j|D}t|dtd|jS)zw Returns ======= monomials: list A list of monomials of a certain degree. c g|] }t| Srr)rmonomials rrzEMacaulayResultant.get_monomials_of_certain_degree..bs(???S(^???r Trdre)rr*rur)r)degreerls rrz1MacaulayResultant.get_monomials_of_certain_degreeZs]??5dn6<>>??? i&udn==??? ?r cg}g}t|jD]}|dkr@|j|j|z }||}||H||j|dz |j|dz z|j|j|z }||}|D])}|D]$t|dkrfd|D}%*|||S)z Returns ======= row_coefficients: list The row coefficients of Macaulay's matrix rr;c g|] }|k| Srr)ritemps rrz:MacaulayResultant.get_row_coefficients..s1)7)7)7$,0AI)7)7)7)7r )r.r,rrrrIr*r ) r)row_coefficients divisiblerrr poss_rowsdivrs @rget_row_coefficientsz&MacaulayResultant.get_row_coefficientsisK tv 3 3AAv 3a8??GG ''1111  A!6!%a!e!4"5666a8 @@HH $77C&77q#;;!+7)7)7)7)7))7)7)7I7!'' 2222r cfg}|}t|jD]v}||D]k}g}t|j||zg|jR}|jD]*}|||+||lwt|}|S)zt Returns ======= macaulay_matrix: Matrix The Macaulay numerator matrix ) rr.r,rr'r*rrIrjr) r)rLrr multiplier coefficientsr$monomacaulay_matrixs r get_matrixzMacaulayResultant.get_matrixs4466tv * *A.q1 * * ! D,Q/*<-!^---!-CCD ''(;(;D(A(ABBBB L)))) *!,,r c pg}jD]r}g}tjD]D\}}|t t ||j|kE||sfdt|D}fdt|D}||fS)a 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. cPg|]"\}}t|jdz k |#Sr;rr,rrrr)s rrz.sB+++A!fftvz)+1+++r cPg|]"\}}t|jdz k |#Srrrs rrz.sB///TQa&&DFAI-/q///r )r enumerater*rIboolrr)r)rr-rMrvreduced non_reduceds` rget_reduced_nonreducedz(MacaulayResultant.get_reduced_nonreduceds$ " # #AD!$.11 I I1 Da!3!3t|A!FGGHHHH   T " " " "++++9!5!5+++////Yy%9%9///  ##r c \}}|gkrtdgSfdtjDfdt jD}|dd|fg}t jD]*  fd|D}d|vr| +|||fS)a Returns ======= macaulay_submatrix: Matrix The Macaulay denominator matrix. Columns that are non reduced are kept. The row which contains one of the a_{i}s is dropped. a_{i}s are the coefficients of x_i ^ {d_i}. r;c6g|]\}}|j|zSr)r)rrrr)s rrz3MacaulayResultant.get_submatrix..s4777$!Qdl1o-777r c\g|](}j||)Sr)r'coeff)rr reduction_setr)s rrz3MacaulayResultant.get_submatrix..sD'''"((q)9::'''r Nc,g|]}|ddfvSr#r)raireduced_matrixrs rrz3MacaulayResultant.get_submatrix..s+@@@bR>#qqq&11@@@r T)rrrr*r.r,rLrI) r)r}rraisrwcheckrrrs ` @@@r get_submatrixzMacaulayResultant.get_submatrixs  $::<< b= 99 7777%dn55777 '''''df ''' 7 +,-- ! !C@@@@@C@@@E5  ! C   dK'((r N) rrrrr1rrrrrrrrr rrrs--\PPP0 4 4 4 @ @ @ ? ? ?   8.$$$>)))))r rN)rmathrsympy.core.mulrsympy.matrices.denserrsympy.polys.polytoolsrr r sympy.simplify.simplifyr sympy.tensor.indexedr sympy.polys.monomialsr rsympy.polys.orderingsrrr(sympy.functions.combinatorial.factorialsr itertoolsrsympy.utilities.exceptionsrrrrr rrst  ////////::::::::::,,,,,,,,,,,,========......>>>>>>>>======333333@@@@@@a4a4a4a4a4a4a4a4F])])])])])])])])])])r