B [9.@sdZddlmZmZmZmZddlmZmZmZm Z ddl m Z ddl m Z ddl mZddlmZddlmZdd lmZdd lmZGd d d ZGd ddZdS)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. ) IndexedBaseMatrixMulPoly)remprodfraction total_degree)range) monomial_deg) itermonomials) monomial_key)poly_from_expr)binomial)combinations_with_replacementc@s0eZdZdZddZddZddZdd Zd S) DixonResultanta? 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 Reference ========== 1. [Kapur1994]_ 2. [Palancz08]_ See Also ======== Notebook in examples: sympy/example/notebooks. cs`|_|_tj_tj_tdfddtjD_fddjD_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 Zalphacsg|] }|qSr).0i)arBlib/python3.7/site-packages/sympy/polys/multivariate_resultants.py Rsz+DixonResultant.__init__..csg|]}t|fjqSr)r variables)rpoly)selfrrrUsN) polynomialsrlennmrr dummy_variables max_degrees)rrrr)rrr__init__>s    zDixonResultant.__init__cs|j|jdkrtd|jg}t|j}xPt|jD]B}|j|||<ddt|j|D| fdd|jDq6Wt |}t|j|j}t dd|D}| | }t||jdS) a Returns ------- dixon_polynomial: polynomial Dixon's polynomial is calculated as: delta = Delta(A) / ((x_1 - a_1) ... (x_n - a_n)) where, A = |p_1(x_1,... x_n), ..., p_n(x_1,... x_n)| |p_1(a_1,... x_n), ..., p_n(a_1,... x_n)| |... , ..., ...| |p_1(a_1,... a_n), ..., p_n(a_1,... a_n)| z%Method invalid for given combination.cSsi|]\}}||qSrr)rvartrrr qsz7DixonResultant.get_dixon_polynomial..csg|]}|qSr)Zsubs)rf) substitutionrrrrsz7DixonResultant.get_dixon_polynomial..cSsg|]\}}||qSrr)rrbrrrrwsr)rr ValueErrorrlistrr rzipappendrrZdetZfactorr)rrowstempidxAZtermsZproduct_of_differencesZdixon_polynomialr)r'rget_dixon_polynomialXs z#DixonResultant.get_dixon_polynomialcs4fddtjD}t|}t|}t|S)Ncs,g|]$}j||dj|dqS)r")rr )rr)rrrr}sz3DixonResultant.get_upper_degree..)r rrrZmonomsr )rZlist_of_productsproductr)rrget_upper_degree|s   zDixonResultant.get_upper_degreecsv|}ttjtdtdjdtfdd|Dfddtj dD}dd|fS) z Construct the Dixon matrix from the coefficients of polynomial \alpha. Each coefficient is viewed as a polynomial of x_1, ..., x_n. Tlex)reversekeycs g|]fddDqS)cs"g|]}tfj|qSr)rrcoeff_monomial)rr)crrrrsz>DixonResultant.get_dixon_matrix...r)r) monomialsr)r8rrsz3DixonResultant.get_dixon_matrix..cs.g|]&}tdddd|fDr|qS)cSsg|] }|dkqS)rr)relementrrrrsz>DixonResultant.get_dixon_matrix...N)any)rcolumn) dixon_matrixrrrsN) Zcoeffsr*r rr3sortedr rr shape)rZ polynomial coefficientskeepr)r=r9rrget_dixon_matrixs   zDixonResultant.get_dixon_matrixN)__name__ __module__ __qualname____doc__r!r1r3rCrrrrrs %$rc@sXeZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ dS)MacaulayResultanta~ A class for calculating the Macaulay resultant. Note that the coefficients of the polynomials 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 - 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.get_monomials_set() >>> 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]]) Reference ========== 1. [Bruce97]_ 2. [Stiller96]_ See Also ======== Notebook in examples: sympy/example/notebooks. csD|_|_t|_fddjD___dS)z Parameters ---------- variables: list A list of all n variables polynomials : list of sympy polynomials A list of m n-degree polynomials csg|]}t|fjqSr)r r)rr)rrrrsz.MacaulayResultant.__init__..N) rrrrdegrees _get_degree_mdegree_mget_sizeZmonomials_size)rrrr)rrr!s     zMacaulayResultant.__init__cCsdtdd|jDS)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"css|]}|dVqdS)r"Nr)rdrrr sz2MacaulayResultant._get_degree_m..)sumrI)rrrrrJszMacaulayResultant._get_degree_mcCst|j|jd|jdS)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")rrKr)rrrrrLs zMacaulayResultant.get_sizecCs,ddt|j|D}t|dtd|jdS)zv Returns ------- monomials: list A list of monomials of a certain degree. cSsg|] }t|qSr)r)rmonomialrrrrszEMacaulayResultant.get_monomials_of_certain_degree..Tr4)r5r6)rrr?r )rdegreer9rrrget_monomials_of_certain_degrees  z1MacaulayResultant.get_monomials_of_certain_degreecCs||j}||_dS)z Returns ------- self.monomial_set: set The set T. Set of all possible monomials of degree degree_m N)rRrK monomial_set)rrSrrrget_monomials_sets z#MacaulayResultant.get_monomials_setcsg}g}xt|jD]}|dkrF|j|j|}||}||q||j|d|j|d|j|j|}||}x:|D]2}x,|D]$t|dkrfdd|D}qWqW||qW|S)z~ Returns ------- row_coefficients: list The row coefficients of Macaulay's matrix rr"csg|]}|kr|qSrr)ritem)prrrsz:MacaulayResultant.get_row_coefficients..)r rrKrIrRr,rr)rrow_coefficients divisiblerrQrPZ poss_rowsZdivr)rVrget_row_coefficientss"     z&MacaulayResultant.get_row_coefficientsc Csg}|}xlt|jD]^}xX||D]L}g}t|j||f|j}x|jD]}|||qPW||q&WqWt |}|S)zk Returns ------- macaulay_matrix: Matrix The Macaulay's matrix ) rYr rrrrrSr,r7r) rr-rWrZ multiplierrArZmonoZmacaulay_matrixrrr get_matrix!s  zMacaulayResultant.get_matrixcsg}xRjD]H}g}x4tjD]&\}}|tt||j|kq W||q Wfddt|D}fddt|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. cs&g|]\}}t|jdkr|qS)r")rOr)rrr)rrrrMsz.cs&g|]\}}t|jdkr|qS)r")rOr)rrr[)rrrrOs)rS enumeraterr,boolr rI)rrXrr.rvreduced non_reducedr)rrget_reduced_nonreduced7s "z(MacaulayResultant.get_reduced_nonreducedcs\}}fddtjDtfddtjD}|dd|fg}x8tjD]*fdd|D}d|krb|qbW|||fS)a Returns ------- macaulay_submatrix: Matrix The Macaulay's matrix. Columns that are non reduced are kept. The row which contain one if the a_{i}s is dropped. a_{i}s are the coefficients of x_i ^ {d_i}. csg|]\}}|j|qSr)rI)rrr^)rrrr_sz3MacaulayResultant.get_submatrix..cs g|]}j||qSr)rZcoeff)rr) reduction_setrrrrbsNcs g|]}|ddfkqS)Nr)rZai)reduced_matrixrowrrrhsT)rar\rr*r rr-r,)rZmatrixr_r`ZaisrBZcheckr)rcrbrdrr get_submatrixTs  zMacaulayResultant.get_submatrixN) rDrErFrGr!rJrLrRrTrYrZrarerrrrrHs'   rHN)rGZsympyrrrrrrrr Zsympy.core.compatibilityr Zsympy.polys.monomialsr r Zsympy.polys.orderingsr Zsympy.polys.polytoolsrZ(sympy.functions.combinatorial.factorialsr itertoolsrrrHrrrr s