U Zʹj;@sdZddlmZddlmZddlmZmZddlm Z m Z m Z ddl m Z ddlmZddlmZmZdd lmZdd lmZmZdd lmZdd lmZdd lmZGdddZGdddZdS)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@sdeZdZdZddZeddZddZdd Zd d Z d d Z ddZ ddZ ddZ ddZdS)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]_ csd|_|_tj_tj_tdfddtjD_fddtjD_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 alphacsg|] }|qSr.0i)arb/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/polys/multivariate_resultants.py Xsz+DixonResultant.__init__..cs$g|]tfddjDqS)c3s|]}t|VqdSN)rrpolyrrr [sz5DixonResultant.__init__...)max polynomialsrselfr rr[sN) r# variableslennmr rangedummy_variables _max_degreesr&r#r'r)rr&r__init__Ds    zDixonResultant.__init__cCstdddd|jS)NzS The max_degrees property of DixonResultant is deprecated. 1.5$deprecated-dixonresultant-propertiesZdeprecated_since_versionZactive_deprecations_target)rr-r%rrr max_degrees^s zDixonResultant.max_degreescs|j|jdkrtd|jg}t|j}t|jD]<}|j|||<tt |j|| fdd|jDq4t |}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.csg|]}|qSr)subs)rfZ substitutionrrrsz7DixonResultant.get_dixon_polynomial..cSsg|]\}}||qSrr)rrbrrrrsr)r*r) ValueErrorr#listr'r+r,dictzipappendrrZdetfactorr)r&rowstempidxAZtermsZproduct_of_differencesZdixon_polynomialrr7rget_dixon_polynomialis z#DixonResultant.get_dixon_polynomialcsBtddddfddtjD}t|}t|}t|S)Nz The get_upper_degree() method of DixonResultant is deprecated. Use get_max_degrees() instead. r0r1r2cs g|]}j|j|qSr)r'r-rr%rrrsz3DixonResultant.get_upper_degree..)rr+r)rrZmonomsr )r&Zlist_of_productsproductrr%rget_upper_degrees  zDixonResultant.get_upper_degreecs,fdd|D}ddt|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 $x_1, x_2, \dots, x_n$. csg|]}tt|jqSr)rrr'rr%rrrsz2DixonResultant.get_max_degrees..cSsg|] }t|qSr)r")rZdegsrrrrs)coeffsr<)r& polynomialZ deg_listsr3rr%rget_max_degreess  zDixonResultant.get_max_degreescs|}tj|tdtdjdtfdd|Djdjdkrfddtjd D}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. Tlexreversekeycs g|]fddDqS)cs"g|]}tfj|qSr)rr'coeff_monomial)rr*)cr&rrrsz>DixonResultant.get_dixon_matrix...rr$) monomialsr&)rNrrs z3DixonResultant.get_dixon_matrix..rr4cs.g|]&}tdddd|fDr|qS)css|]}|dkVqdSrNr)relementrrrr!sz=DixonResultant.get_dixon_matrix...N)any)rcolumn) dixon_matrixrrrs N) rHr r'sortedrrrFshaper+)r&rGr3keepr)rTrOr&rget_dixon_matrixs   zDixonResultant.get_dixon_matrixcsjr dSj\}tdfddt|D}|ddftdgddgg}dddf|kr~dSdSdS) a Test for the validity of the Kapur-Saxena-Yang precondition. The precondition requires that the column corresponding to the monomial 1 = x_1 ^ 0 * x_2 ^ 0 * ... * x_n ^ 0 is not a linear combination of the remaining ones. In SymPy notation this is the last column. For the precondition to hold the last non-zero row of the rref matrix should be of the form [0, 0, ..., 1]. Frcs,g|]$tfddtDrqS)c3s|]}|fdkVqdSrPrrj)rmatrixrrr!sz=DixonResultant.KSY_precondition...)rRr+r$r\r)r rrsz3DixonResultant.KSY_precondition..Nr4rUT)is_zero_matrixrWr Zrrefr+r)r&r\r*r? conditionrr]rKSY_preconditions  zDixonResultant.KSY_preconditioncs<fddtjD}fddtjD}||fS)z/Remove the zero rows and columns of the matrix.csg|]}|js|qSr)rowr^rr\rrrs z?DixonResultant.delete_zero_rows_and_columns..csg|]}|js|qSr)colr^rZrbrrrs )r+r?cols)r&r\r?rdrrbrdelete_zero_rows_and_columnss  z+DixonResultant.delete_zero_rows_and_columnscCs<d}t|jD](}||D]}|dkr||}qqq|S)z;Calculate the product of the leading entries of the matrix.r4r)r+r?ra)r&r\resraelrrrproduct_leading_entriessz&DixonResultant.product_leading_entriescCs0||}|\}}}|t|}||S)z@Calculate the Kapur-Saxena-Yang approach to the Dixon Resultant.)reZLUdecompositionr rh)r&r\_Urrrget_KSY_Dixon_resultants z&DixonResultant.get_KSY_Dixon_resultantN)__name__ __module__ __qualname____doc__r/propertyr3rCrErHrYr`rerhrkrrrrrs* $   rc@sPeZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ dS)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]_ csR|_|_t|_fddjD___ j_ 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)rr'rr%rrr:sz.MacaulayResultant.__init__..N) r#r'r(r)degrees _get_degree_mdegree_mget_sizeZmonomials_sizeget_monomials_of_certain_degree monomial_setr.rr%rr/+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 r4css|]}|dVqdS)r4Nr)rdrrrr!Lsz2MacaulayResultant._get_degree_m..)sumrrr%rrrrsCs zMacaulayResultant._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 r4)rrtr)r%rrrruNs zMacaulayResultant.get_sizecCs,ddt|j|D}t|dtd|jdS)zw Returns ======= monomials: list A list of monomials of a certain degree. cSsg|] }t|qSrr)rmonomialrrrrbszEMacaulayResultant.get_monomials_of_certain_degree..TrIrJ)rr'rVr)r&degreerOrrrrvZs z1MacaulayResultant.get_monomials_of_certain_degreecsg}g}t|jD]}|dkrD|j|j|}||}||q||j|d|j|d|j|j|}||}|D].}|D]$t|dkrfdd|D}qq||q|S)z Returns ======= row_coefficients: list The row coefficients of Macaulay's matrix rr4csg|]}|kr|qSrr)ritemprrrsz:MacaulayResultant.get_row_coefficients..)r+r)rtrrrvr=r'r )r&row_coefficients divisiblerr{rzZ poss_rowsdivrr}rget_row_coefficientsis$     z&MacaulayResultant.get_row_coefficientsc Cszg}|}t|jD]V}||D]H}g}t|j||f|j}|jD]}|||qJ||q"qt |}|S)zt Returns ======= macaulay_matrix: Matrix The Macaulay numerator matrix ) rr+r)rr#r'rwr=rMr) r&r?rrZ multiplierZ coefficientsrmonoZmacaulay_matrixrrr get_matrixs  zMacaulayResultant.get_matrixcsg}jD]D}g}tjD]&\}}|tt||j|kq||q fddt|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|qSr4ryr)rrrr%rrrsz.cs&g|]\}}t|jdkr|qSrrrr%rrrs)rw enumerater'r=boolrrr)r&rr*r@rvreduced non_reducedrr%rget_reduced_nonreduceds   z(MacaulayResultant.get_reduced_nonreducedcs\}}|gkrtdgSfddtjDfddtjD}|dd|fg}tjD]*fdd|D}d|krn|qn|||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}. r4csg|]\}}|j|qSr)rr)rrrr%rrrsz3MacaulayResultant.get_submatrix..cs g|]}j||qSr)r#Zcoeffr) reduction_setr&rrrsNcs g|]}|ddfkqSrr)rZai)reduced_matrixrarrrsT)rrrr'r+r)r?r=)r&r\rrZaisrXcheckr)rrrar&r get_submatrixs     zMacaulayResultant.get_submatrixN) rlrmrnror/rsrurvrrrrrrrrrqs.  rqN)romathrZsympy.core.mulrZsympy.matrices.denserrZsympy.polys.polytoolsrrr Zsympy.simplify.simplifyr Zsympy.tensor.indexedr Zsympy.polys.monomialsr r Zsympy.polys.orderingsrrrZ(sympy.functions.combinatorial.factorialsr itertoolsrZsympy.utilities.exceptionsrrrqrrrrs        d