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472 lines
15 KiB
472 lines
15 KiB
"""
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This module contains functions for two multivariate resultants. These
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are:
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- Dixon's resultant.
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- Macaulay's resultant.
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Multivariate resultants are used to identify whether a multivariate
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system has common roots. That is when the resultant is equal to zero.
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"""
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from sympy.core.mul import (Mul, prod)
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from sympy.matrices.dense import (Matrix, diag)
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from sympy.polys.polytools import (Poly, degree_list, rem)
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from sympy.simplify.simplify import simplify
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from sympy.tensor.indexed import IndexedBase
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from sympy.polys.monomials import itermonomials, monomial_deg
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from sympy.polys.orderings import monomial_key
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from sympy.polys.polytools import poly_from_expr, total_degree
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from sympy.functions.combinatorial.factorials import binomial
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from itertools import combinations_with_replacement
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from sympy.utilities.exceptions import sympy_deprecation_warning
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class DixonResultant():
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"""
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A class for retrieving the Dixon's resultant of a multivariate
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system.
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Examples
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========
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>>> from sympy import symbols
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>>> from sympy.polys.multivariate_resultants import DixonResultant
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>>> x, y = symbols('x, y')
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>>> p = x + y
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>>> q = x ** 2 + y ** 3
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>>> h = x ** 2 + y
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>>> dixon = DixonResultant(variables=[x, y], polynomials=[p, q, h])
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>>> poly = dixon.get_dixon_polynomial()
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>>> matrix = dixon.get_dixon_matrix(polynomial=poly)
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>>> matrix
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Matrix([
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[ 0, 0, -1, 0, -1],
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[ 0, -1, 0, -1, 0],
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[-1, 0, 1, 0, 0],
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[ 0, -1, 0, 0, 1],
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[-1, 0, 0, 1, 0]])
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>>> matrix.det()
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0
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See Also
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========
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Notebook in examples: sympy/example/notebooks.
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References
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==========
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.. [1] [Kapur1994]_
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.. [2] [Palancz08]_
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"""
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def __init__(self, polynomials, variables):
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"""
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A class that takes two lists, a list of polynomials and list of
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variables. Returns the Dixon matrix of the multivariate system.
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Parameters
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----------
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polynomials : list of polynomials
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A list of m n-degree polynomials
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variables: list
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A list of all n variables
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"""
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self.polynomials = polynomials
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self.variables = variables
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self.n = len(self.variables)
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self.m = len(self.polynomials)
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a = IndexedBase("alpha")
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# A list of n alpha variables (the replacing variables)
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self.dummy_variables = [a[i] for i in range(self.n)]
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# A list of the d_max of each variable.
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self._max_degrees = [max(degree_list(poly)[i] for poly in self.polynomials)
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for i in range(self.n)]
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@property
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def max_degrees(self):
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sympy_deprecation_warning(
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"""
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The max_degrees property of DixonResultant is deprecated.
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""",
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deprecated_since_version="1.5",
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active_deprecations_target="deprecated-dixonresultant-properties",
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)
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return self._max_degrees
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def get_dixon_polynomial(self):
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r"""
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Returns
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=======
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dixon_polynomial: polynomial
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Dixon's polynomial is calculated as:
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delta = Delta(A) / ((x_1 - a_1) ... (x_n - a_n)) where,
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A = |p_1(x_1,... x_n), ..., p_n(x_1,... x_n)|
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|p_1(a_1,... x_n), ..., p_n(a_1,... x_n)|
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|... , ..., ...|
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|p_1(a_1,... a_n), ..., p_n(a_1,... a_n)|
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"""
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if self.m != (self.n + 1):
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raise ValueError('Method invalid for given combination.')
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# First row
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rows = [self.polynomials]
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temp = list(self.variables)
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for idx in range(self.n):
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temp[idx] = self.dummy_variables[idx]
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substitution = {var: t for var, t in zip(self.variables, temp)}
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rows.append([f.subs(substitution) for f in self.polynomials])
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A = Matrix(rows)
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terms = zip(self.variables, self.dummy_variables)
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product_of_differences = Mul(*[a - b for a, b in terms])
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dixon_polynomial = (A.det() / product_of_differences).factor()
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return poly_from_expr(dixon_polynomial, self.dummy_variables)[0]
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def get_upper_degree(self):
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sympy_deprecation_warning(
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"""
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The get_upper_degree() method of DixonResultant is deprecated. Use
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get_max_degrees() instead.
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""",
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deprecated_since_version="1.5",
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active_deprecations_target="deprecated-dixonresultant-properties"
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)
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list_of_products = [self.variables[i] ** self._max_degrees[i]
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for i in range(self.n)]
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product = prod(list_of_products)
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product = Poly(product).monoms()
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return monomial_deg(*product)
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def get_max_degrees(self, polynomial):
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r"""
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Returns a list of the maximum degree of each variable appearing
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in the coefficients of the Dixon polynomial. The coefficients are
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viewed as polys in $x_1, x_2, \dots, x_n$.
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"""
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deg_lists = [degree_list(Poly(poly, self.variables))
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for poly in polynomial.coeffs()]
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max_degrees = [max(degs) for degs in zip(*deg_lists)]
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return max_degrees
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def get_dixon_matrix(self, polynomial):
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r"""
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Construct the Dixon matrix from the coefficients of polynomial
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\alpha. Each coefficient is viewed as a polynomial of x_1, ...,
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x_n.
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"""
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max_degrees = self.get_max_degrees(polynomial)
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# list of column headers of the Dixon matrix.
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monomials = itermonomials(self.variables, max_degrees)
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monomials = sorted(monomials, reverse=True,
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key=monomial_key('lex', self.variables))
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dixon_matrix = Matrix([[Poly(c, *self.variables).coeff_monomial(m)
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for m in monomials]
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for c in polynomial.coeffs()])
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# remove columns if needed
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if dixon_matrix.shape[0] != dixon_matrix.shape[1]:
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keep = [column for column in range(dixon_matrix.shape[-1])
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if any(element != 0 for element
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in dixon_matrix[:, column])]
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dixon_matrix = dixon_matrix[:, keep]
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return dixon_matrix
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def KSY_precondition(self, matrix):
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"""
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Test for the validity of the Kapur-Saxena-Yang precondition.
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The precondition requires that the column corresponding to the
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monomial 1 = x_1 ^ 0 * x_2 ^ 0 * ... * x_n ^ 0 is not a linear
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combination of the remaining ones. In SymPy notation this is
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the last column. For the precondition to hold the last non-zero
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row of the rref matrix should be of the form [0, 0, ..., 1].
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"""
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if matrix.is_zero_matrix:
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return False
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m, n = matrix.shape
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# simplify the matrix and keep only its non-zero rows
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matrix = simplify(matrix.rref()[0])
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rows = [i for i in range(m) if any(matrix[i, j] != 0 for j in range(n))]
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matrix = matrix[rows,:]
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condition = Matrix([[0]*(n-1) + [1]])
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if matrix[-1,:] == condition:
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return True
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else:
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return False
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def delete_zero_rows_and_columns(self, matrix):
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"""Remove the zero rows and columns of the matrix."""
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rows = [
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i for i in range(matrix.rows) if not matrix.row(i).is_zero_matrix]
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cols = [
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j for j in range(matrix.cols) if not matrix.col(j).is_zero_matrix]
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return matrix[rows, cols]
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def product_leading_entries(self, matrix):
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"""Calculate the product of the leading entries of the matrix."""
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res = 1
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for row in range(matrix.rows):
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for el in matrix.row(row):
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if el != 0:
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res = res * el
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break
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return res
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def get_KSY_Dixon_resultant(self, matrix):
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"""Calculate the Kapur-Saxena-Yang approach to the Dixon Resultant."""
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matrix = self.delete_zero_rows_and_columns(matrix)
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_, U, _ = matrix.LUdecomposition()
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matrix = self.delete_zero_rows_and_columns(simplify(U))
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return self.product_leading_entries(matrix)
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class MacaulayResultant():
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"""
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A class for calculating the Macaulay resultant. Note that the
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polynomials must be homogenized and their coefficients must be
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given as symbols.
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Examples
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========
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>>> from sympy import symbols
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>>> from sympy.polys.multivariate_resultants import MacaulayResultant
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>>> x, y, z = symbols('x, y, z')
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>>> a_0, a_1, a_2 = symbols('a_0, a_1, a_2')
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>>> b_0, b_1, b_2 = symbols('b_0, b_1, b_2')
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>>> c_0, c_1, c_2,c_3, c_4 = symbols('c_0, c_1, c_2, c_3, c_4')
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>>> f = a_0 * y - a_1 * x + a_2 * z
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>>> g = b_1 * x ** 2 + b_0 * y ** 2 - b_2 * z ** 2
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>>> h = c_0 * y * z ** 2 - c_1 * x ** 3 + c_2 * x ** 2 * z - c_3 * x * z ** 2 + c_4 * z ** 3
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>>> mac = MacaulayResultant(polynomials=[f, g, h], variables=[x, y, z])
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>>> mac.monomial_set
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[x**4, x**3*y, x**3*z, x**2*y**2, x**2*y*z, x**2*z**2, x*y**3,
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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]
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>>> matrix = mac.get_matrix()
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>>> submatrix = mac.get_submatrix(matrix)
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>>> submatrix
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Matrix([
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[-a_1, a_0, a_2, 0],
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[ 0, -a_1, 0, 0],
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[ 0, 0, -a_1, 0],
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[ 0, 0, 0, -a_1]])
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See Also
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========
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Notebook in examples: sympy/example/notebooks.
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References
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==========
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.. [1] [Bruce97]_
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.. [2] [Stiller96]_
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"""
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def __init__(self, polynomials, variables):
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"""
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Parameters
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==========
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variables: list
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A list of all n variables
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polynomials : list of SymPy polynomials
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A list of m n-degree polynomials
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"""
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self.polynomials = polynomials
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self.variables = variables
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self.n = len(variables)
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# A list of the d_max of each polynomial.
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self.degrees = [total_degree(poly, *self.variables) for poly
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in self.polynomials]
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self.degree_m = self._get_degree_m()
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self.monomials_size = self.get_size()
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# The set T of all possible monomials of degree degree_m
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self.monomial_set = self.get_monomials_of_certain_degree(self.degree_m)
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def _get_degree_m(self):
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r"""
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Returns
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=======
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degree_m: int
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The degree_m is calculated as 1 + \sum_1 ^ n (d_i - 1),
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where d_i is the degree of the i polynomial
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"""
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return 1 + sum(d - 1 for d in self.degrees)
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def get_size(self):
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r"""
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Returns
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=======
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size: int
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The size of set T. Set T is the set of all possible
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monomials of the n variables for degree equal to the
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degree_m
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"""
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return binomial(self.degree_m + self.n - 1, self.n - 1)
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def get_monomials_of_certain_degree(self, degree):
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"""
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Returns
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=======
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monomials: list
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A list of monomials of a certain degree.
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"""
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monomials = [Mul(*monomial) for monomial
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in combinations_with_replacement(self.variables,
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degree)]
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return sorted(monomials, reverse=True,
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key=monomial_key('lex', self.variables))
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def get_row_coefficients(self):
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"""
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Returns
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=======
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row_coefficients: list
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The row coefficients of Macaulay's matrix
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"""
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row_coefficients = []
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divisible = []
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for i in range(self.n):
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if i == 0:
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degree = self.degree_m - self.degrees[i]
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monomial = self.get_monomials_of_certain_degree(degree)
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row_coefficients.append(monomial)
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else:
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divisible.append(self.variables[i - 1] **
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self.degrees[i - 1])
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degree = self.degree_m - self.degrees[i]
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poss_rows = self.get_monomials_of_certain_degree(degree)
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for div in divisible:
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for p in poss_rows:
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if rem(p, div) == 0:
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poss_rows = [item for item in poss_rows
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if item != p]
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row_coefficients.append(poss_rows)
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return row_coefficients
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def get_matrix(self):
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"""
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Returns
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=======
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macaulay_matrix: Matrix
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The Macaulay numerator matrix
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"""
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rows = []
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row_coefficients = self.get_row_coefficients()
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for i in range(self.n):
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for multiplier in row_coefficients[i]:
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coefficients = []
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poly = Poly(self.polynomials[i] * multiplier,
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*self.variables)
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for mono in self.monomial_set:
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coefficients.append(poly.coeff_monomial(mono))
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rows.append(coefficients)
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macaulay_matrix = Matrix(rows)
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return macaulay_matrix
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def get_reduced_nonreduced(self):
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r"""
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Returns
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=======
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reduced: list
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A list of the reduced monomials
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non_reduced: list
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A list of the monomials that are not reduced
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Definition
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==========
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A polynomial is said to be reduced in x_i, if its degree (the
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maximum degree of its monomials) in x_i is less than d_i. A
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polynomial that is reduced in all variables but one is said
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simply to be reduced.
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"""
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divisible = []
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for m in self.monomial_set:
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temp = []
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for i, v in enumerate(self.variables):
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temp.append(bool(total_degree(m, v) >= self.degrees[i]))
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divisible.append(temp)
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reduced = [i for i, r in enumerate(divisible)
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if sum(r) < self.n - 1]
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non_reduced = [i for i, r in enumerate(divisible)
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if sum(r) >= self.n -1]
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return reduced, non_reduced
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def get_submatrix(self, matrix):
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r"""
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Returns
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=======
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macaulay_submatrix: Matrix
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The Macaulay denominator matrix. Columns that are non reduced are kept.
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The row which contains one of the a_{i}s is dropped. a_{i}s
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are the coefficients of x_i ^ {d_i}.
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"""
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reduced, non_reduced = self.get_reduced_nonreduced()
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# if reduced == [], then det(matrix) should be 1
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if reduced == []:
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return diag([1])
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# reduced != []
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reduction_set = [v ** self.degrees[i] for i, v
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in enumerate(self.variables)]
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ais = list([self.polynomials[i].coeff(reduction_set[i])
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for i in range(self.n)])
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reduced_matrix = matrix[:, reduced]
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keep = []
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for row in range(reduced_matrix.rows):
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check = [ai in reduced_matrix[row, :] for ai in ais]
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if True not in check:
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keep.append(row)
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return matrix[keep, non_reduced]
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