a Ям>> 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|И_|И_tИjГИ_tИjГИ_tdГЙЗfddДtИjГDГИ_ЗfddДtИjГDГИ_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|]}И|СqSйrй┌.0┌i)┌ar·s/Users/vegardjervell/Documents/master/model/venv/lib/python3.9/site-packages/sympy/polys/multivariate_resultants.py┌ Wєz+DixonResultant.__init__..cs$g|]ЙtЗfddДИjDГГСqS)c3s|]}t|ГИVqdSйN)rйr┌polyйrrr┌ Zrz5DixonResultant.__init__...)┌max┌polynomialsйrй┌selfr rrZs N) r#┌ variables┌len┌n┌mr ┌range┌dummy_variables┌_max_degreesйr&r#r'r)rr&r┌__init__Cs zDixonResultant.__init__cCstddddН|jS)NzS The max_degrees property of DixonResultant is deprecated. ·1.5·$deprecated-dixonresultant-propertiesйZdeprecated_since_versionZactive_deprecations_target)rr-r%rrr┌max_degrees]s√zDixonResultant.max_degreescs└|j|jdkrtdГВ|jg}t|jГ}t|jГD]B}|j|||<ddДt|j|ГDГЙ|а ЗfddД|jDГбq4t |Г}t|j|jГ}tddД|DГО}|аб|а б}t||jГdS) 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)| щz%Method invalid for given combination.cSsi|]\}}||УqSrr)r┌var┌trrr┌ Бrz7DixonResultant.get_dixon_polynomial..csg|]}|аИбСqSr)┌subs)r┌fйZsubstitutionrrrВrz7DixonResultant.get_dixon_polynomial..cSsg|]\}}||СqSrr)rr┌brrrrЗrr)r*r)┌ ValueErrorr#┌listr'r+r,┌zip┌appendrrZdetZfactorr)r&┌rows┌temp┌idx┌AZtermsZproduct_of_differencesZdixon_polynomialrr:r┌get_dixon_polynomialhs z#DixonResultant.get_dixon_polynomialcsBtddddНЗfddДtИjГDГ}t|Г}t|Габ}t|ОS)NzГ The get_upper_degree() method of DixonResultant is deprecated. Use get_max_degrees() instead. r0r1r2cs g|]}Иj|Иj|СqSr)r'r-rr%rrrХs z3DixonResultant.get_upper_degree..)rr+r)rrZmonomsr)r&Zlist_of_products┌productrr%r┌get_upper_degreeМs· zDixonResultant.get_upper_degreecs,ЗfddД|абDГ}ddДt|ОDГ}|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|ИjГГСqSr)rrr'rr%rrrвs z2DixonResultant.get_max_degrees..cSsg|]}t|ГСqSr)r")rZdegsrrrrеr)┌coeffsr>)r&┌ polynomialZ deg_listsr3rr%r┌get_max_degreesЬs zDixonResultant.get_max_degreescsМИа|б}tИj|ГЙtИdtdИjГdНЙtЗЗfddД|абDГГЙИjdИjdkrИЗfddДtИjd ГDГ}Иd d Е|fЙИS)zе Construct the Dixon matrix from the coefficients of polynomial \alpha. Each coefficient is viewed as a polynomial of x_1, ..., x_n. T┌lexй┌reverse┌keycs g|]ЙЗЗfddДИDГСqS)cs$g|]}tИgИjвRОа|бСqSr)rr'┌coeff_monomial)rr*)┌cr&rrr╖s z>DixonResultant.get_dixon_matrix...rr$)┌ monomialsr&)rOrr╖s■ z3DixonResultant.get_dixon_matrix..rr4cs.g|]&}tddДИddЕ|fDГГr|СqS)css|]}|dkVqdSйrNr)r┌elementrrrr!╛rz=DixonResultant.get_dixon_matrix...N)┌any)r┌column)┌dixon_matrixrrr╜s щ N) rIrr'┌sortedr rrG┌shaper+)r&rHr3┌keepr)rUrPr&r┌get_dixon_matrixйs ■zDixonResultant.get_dixon_matrixcsЖИjr dSИj\}ЙtИабdГЙЗЗfddДt|ГDГ}И|ddЕfЙtdgИddggГ}ИdddЕf|kr~dSdSdS) aп 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|]$ЙtЗЗfddДtИГDГГrИСqS)c3s|]}ИИ|fdkVqdSrQrйr┌j)r┌matrixrrr!╓rz=DixonResultant.KSY_precondition...)rSr+r$йr]r)r rr╓rz3DixonResultant.KSY_precondition..Nr4rVT)┌is_zero_matrixrXr Zrrefr+r)r&r]r*r@┌ conditionrr^r┌KSY_precondition┼s zDixonResultant.KSY_preconditioncs<ЗfddДtИjГDГ}ЗfddДtИjГDГ}И||fS)z/Remove the zero rows and columns of the matrix.csg|]}Иа|бjs|СqSr)┌rowr_rйr]rrrтsz?DixonResultant.delete_zero_rows_and_columns..csg|]}Иа|бjs|СqSr)┌colr_r[rcrrrфs)r+r@┌cols)r&r]r@rerrcr┌delete_zero_rows_and_columnsрs z+DixonResultant.delete_zero_rows_and_columnscCs<d}t|jГD](}|а|бD]}|dkr||}qqq|S)z;Calculate the product of the leading entries of the matrix.r4r)r+r@rb)r&r]┌resrb┌elrrr┌product_leading_entriesщsz&DixonResultant.product_leading_entriescCs0|а|б}|аб\}}}|аt|Гб}|а|бS)z@Calculate the Kapur-Saxena-Yang approach to the Dixon Resultant.)rfZLUdecompositionr ri)r&r]┌_┌Urrr┌get_KSY_Dixon_resultantєs z&DixonResultant.get_KSY_Dixon_resultantN)┌__name__┌ __module__┌__qualname__┌__doc__r/┌propertyr3rDrFrIrZrarfrirlrrrrrs* $ rc@sPeZdZdZddДZddДZddДZdd ДZd dДZdd ДZ ddДZ ddДZdS)┌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|И_|И_t|ГИ_ЗfddДИjDГИ_ИабИ_ИабИ_Иа ИjбИ_ 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|gИjвRОСqSr)rr'rr%rrr9rz.MacaulayResultant.__init__..N)r#r'r(r)┌degrees┌ _get_degree_m┌degree_m┌get_sizeZmonomials_size┌get_monomials_of_certain_degree┌monomial_setr.rr%rr/*s zMacaulayResultant.__init__cCsdtddД|jDГГS)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)r┌drrrr!Krz2MacaulayResultant._get_degree_m..)┌sumrsr%rrrrtBs zMacaulayResultant._get_degree_mcCst|j|jd|jdГS)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)rrur)r%rrrrvMs zMacaulayResultant.get_sizecCs,ddДt|j|ГDГ}t|dtd|jГdНS)zw Returns ======= monomials: list A list of monomials of a certain degree. cSsg|]}t|ОСqSr)r)r┌monomialrrrrarzEMacaulayResultant.get_monomials_of_certain_degree..TrJrK)rr'rWr )r&┌degreerPrrrrwYs z1MacaulayResultant.get_monomials_of_certain_degreecs─g}g}t|jГD]м}|dkrD|j|j|}|а|б}|а|бq|а|j|d|j|dб|j|j|}|а|б}|D].}|D]$ЙtИ|ГdkrМЗfddД|DГ}qМqД|а|бq|S)z Returns ======= row_coefficients: list The row coefficients of Macaulay's matrix rr4csg|]}|Иkr|СqSrr)r┌itemй┌prrrs z:MacaulayResultant.get_row_coefficients..)r+r)rursrwr?r'r)r&┌row_coefficients┌ divisiblerr|r{Z poss_rows┌divrr~r┌get_row_coefficientshs$ z&MacaulayResultant.get_row_coefficientsc Cs|g}|аб}t|jГD]X}||D]J}g}t|j||g|jвRО}|jD]}|а|а|ббqL|а|бq"qt |Г}|S)zt Returns ======= macaulay_matrix: Matrix The Macaulay numerator matrix ) rГr+r)rr#r'rxr?rNr) r&r@rАr┌ multiplierZcoefficientsrZmonoZmacaulay_matrixrrr┌ get_matrixДs zMacaulayResultant.get_matrixcsДg}ИjD]D}g}tИjГD]&\}}|аtt||ГИj|kГбq|а|бq ЗfddДt|ГDГ}ЗfddДt|ГDГ}||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|ГИjdkr|СqSйr4йrzr)йrr┌rr%rrr│s z.cs&g|]\}}t|ГИjdkr|СqSrЖrЗrИr%rrr╡s )rx┌ enumerater'r?┌boolrrs)r&rБr*rAr┌v┌reduced┌non_reducedrr%r┌get_reduced_nonreducedЫs z(MacaulayResultant.get_reduced_nonreducedcsкИаб\}}|gkrtdgГSЗfddДtИjГDГЙtЗЗfddДtИjГDГГ}|ddЕ|fЙg}tИjГD]*ЙЗЗfddД|DГ}d|vrr|аИбqr|||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|]\}}|Иj|СqSr)rs)rrrМr%rrr╦rz3MacaulayResultant.get_submatrix..cs g|]}Иj|аИ|бСqSr)r#Zcoeffr)┌ reduction_setr&rrr╬s Ncs g|]}|ИИddЕfvСqSrr)rZai)┌reduced_matrixrbrrr╘rT) rПrrКr'r=r+r)r@r?)r&r]rНrОZaisrY┌checkr)rСrРrbr&r┌ get_submatrix║s zMacaulayResultant.get_submatrixN)rmrnrorpr/rtrvrwrГrЕrПrУrrrrrr√s.rrN)rpZsympy.core.mulrrZsympy.matrices.denserrZsympy.polys.polytoolsrrrZsympy.simplify.simplifyr Zsympy.tensor.indexedr Zsympy.polys.monomialsrrZsympy.polys.orderingsr rrZ(sympy.functions.combinatorial.factorialsr┌ itertoolsrZsympy.utilities.exceptionsrrrrrrrr┌sd