o 8VaC:@sdZddlmZmZmZmZddlmZmZmZm Z m Z ddl m Z m Z ddlmZddlmZmZddlmZddlmZdd lmZGd d d ZGd d d ZdS)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)remprod degree_listdiagsimplify) itermonomials monomial_deg) monomial_key)poly_from_expr total_degree)binomial)combinations_with_replacement)SymPyDeprecationWarningc@sdeZdZdZddZeddZddZdd Zd d Z d d Z ddZ ddZ ddZ ddZdS)DixonResultantaL 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]_ 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 Zalphacsg|]}|qSr.0i)arE/usr/lib/python3/dist-packages/sympy/polys/multivariate_resultants.py Tz+DixonResultant.__init__..cs$g|]tfddjDqS)c3s|] }t|VqdSN)rrpolyrrr Wsz5DixonResultant.__init__...)max polynomialsrselfrrrWN) r" variableslennmrrangedummy_variables _max_degreesr%r"r'r)rr%r__init__@s     zDixonResultant.__init__cCstdddd|jS)N max_degreescE1.5)featureissuedeprecated_since_version)rwarnr-r$rrrr0Zs zDixonResultant.max_degreescs|j|jdkr td|jg}t|j}t|jD]!}|j|||<ddt|j|D| fdd|jDqt |}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 zz7DixonResultant.get_dixon_polynomial..csg|]}|qSr)Zsubs)rfZ substitutionrrr{r;z7DixonResultant.get_dixon_polynomial..cSsg|]\}}||qSrr)rrbrrrrsr)r*r) ValueErrorr"listr'r+r,zipappendrrZdetZfactorr)r%rowstempidxAZtermsZproduct_of_differencesZdixon_polynomialrr=rget_dixon_polynomialas z#DixonResultant.get_dixon_polynomialcsHtdddddfddtjD}t|}t|}t|S)Nget_upper_degreeget_max_degreesr1r2)r3Z useinsteadr4r5cs g|] }j|j|qSr)r'r-rr$rrrz3DixonResultant.get_upper_degree..)rr6r+r)rrZmonomsr )r%Zlist_of_productsproductrr$rrHs  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_n. csg|] }tt|jqSr)rrr'rr$rrrsz2DixonResultant.get_max_degrees..cSsg|]}t|qSr)r!)rZdegsrrrrr)coeffsrA)r% polynomialZ deg_listsr0rr$rrIs zDixonResultant.get_max_degreescs|}tj|tdtdjdtfdd|DjdjdkrDfddtjd 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|]}tgjR|qSr)rr'coeff_monomial)rr*)cr%rrrr&>DixonResultant.get_dixon_matrix...rr#) monomialsr%)rSrrs   z3DixonResultant.get_dixon_matrix..rr7cs.g|]}tdddd|fDr|qS)cSsg|]}|dkqS)rr)relementrrrrrrTN)any)rcolumn) dixon_matrixrrrs  N) rIr r'sortedr rrLshaper+)r%rMr0keepr)rYrUr%rget_dixon_matrixs   zDixonResultant.get_dixon_matrixcsjrdSj\}tdfddt|D}|ddftdgddgg}dddf|kr?dSdS) 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 |] }|fdkVqdS)rNrrj)rmatrixrrr sz=DixonResultant.KSY_precondition...)rWr+r#rar)rrrs,z3DixonResultant.KSY_precondition..Nr7rZT)is_zero_matrixr\r Zrrefr+r)r%rar*rCZ conditionrrbrKSY_preconditions  zDixonResultant.KSY_preconditioncs<fddtjD}fddtjD}||fS)z/Remove the zero rows and columns of the matrix.cg|] }|js|qSr)rowrcrrarrr  z?DixonResultant.delete_zero_rows_and_columns..crer)colrcr_rgrrrrh)r+rCcols)r%rarCrjrrgrdelete_zero_rows_and_columnss   z+DixonResultant.delete_zero_rows_and_columnscCs<d}t|jD]}||D] }|dkr||}nqq|S)z;Calculate the product of the leading entries of the matrix.r7r)r+rCrf)r%raresrfelrrrproduct_leading_entriessz&DixonResultant.product_leading_entriescCs0||}|\}}}|t|}||S)z@Calculate the Kapur-Saxena-Yang approach to the Dixon Resultant.)rkZLUdecompositionr rn)r%ra_Urrrget_KSY_Dixon_resultants  z&DixonResultant.get_KSY_Dixon_resultantN)__name__ __module__ __qualname____doc__r/propertyr0rGrHrIr^rdrkrnrqrrrrrs* $   rc@sPeZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ dS)MacaulayResultanta2 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]_ 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|gjRqSr)rr'rr$rrr/z.MacaulayResultant.__init__..N) r"r'r(r)degrees _get_degree_mdegree_mget_sizeZmonomials_sizeget_monomials_of_certain_degree monomial_setr.rr$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 r7css|]}|dVqdS)r7Nr)rdrrrr Asz2MacaulayResultant._get_degree_m..)sumryr$rrrrz8s zMacaulayResultant._get_degree_mcCst|j|jd|jdS)z Returns ======= size: int The size of set T. Set T is the set of all possible monomials of the n variables for degree equal to the degree_m r7)rr{r)r$rrrr|Cs zMacaulayResultant.get_sizecCs,ddt|j|D}t|dtd|jdS)zw Returns ======= monomials: list A list of monomials of a certain degree. cSsg|]}t|qSr)r)rmonomialrrrrWrzEMacaulayResultant.get_monomials_of_certain_degree..TrNrO)rr'r[r )r%degreerUrrrr}Os z1MacaulayResultant.get_monomials_of_certain_degreecsg}g}t|jD]V}|dkr"|j|j|}||}||q ||j|d|j|d|j|j|}||}|D]}|D]t|dkrXfdd|D}qFqB||q |S)z Returns ======= row_coefficients: list The row coefficients of Macaulay's matrix rr7csg|]}|kr|qSrr)ritemprrrus z:MacaulayResultant.get_row_coefficients..)r+r)r{ryr}rBr'r)r%row_coefficients divisiblerrrZ poss_rowsZdivrrrget_row_coefficients^s(     z&MacaulayResultant.get_row_coefficientsc Cs|g}|}t|jD],}||D]%}g}t|j||g|jR}|jD] }|||q&||qq t |}|S)zt Returns ======= macaulay_matrix: Matrix The Macaulay numerator matrix ) rr+r)rr"r'r~rBrRr) r%rCrrZ multiplierZ coefficientsrZmonoZmacaulay_matrixrrr get_matrixzs    zMacaulayResultant.get_matrixcsg}jD]"}g}tjD]\}}|tt||j|kq||qfddt|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|qSr7rr)rrrr$rrr  z.cs&g|]\}}t|jdkr|qSrrrr$rrrr)r~ enumerater'rBboolrry)r%rr*rDrvreduced non_reducedrr$rget_reduced_nonreduceds   z(MacaulayResultant.get_reduced_nonreducedcs\}}|gkrtdgSfddtjDtfddtjD}|dd|fg}tjD]fdd|D}d|vrN|q9|||fS)a Returns ======= macaulay_submatrix: Matrix The Macaulay denominator matrix. Columns that are non reduced are kept. The row which contains one of the a_{i}s is dropped. a_{i}s are the coefficients of x_i ^ {d_i}. r7csg|] \}}|j|qSr)ry)rrrr$rrrrxz3MacaulayResultant.get_submatrix..cs g|] }j||qSr)r"Zcoeffr) reduction_setr%rrrrJNcs g|] }|ddfvqSrr)rZai)reduced_matrixrfrrrs T) rr rr'r@r+r)rCrB)r%rarrZaisr]checkr)rrrfr%r get_submatrixs"     zMacaulayResultant.get_submatrixN) rrrsrtrur/rzr|r}rrrrrrrrrws.   rwN)ruZsympyrrrrrrrr r Zsympy.polys.monomialsr r Zsympy.polys.orderingsr Zsympy.polys.polytoolsrrZ(sympy.functions.combinatorial.factorialsr itertoolsrZsympy.utilities.exceptionsrrrwrrrrs     ]