"""Tests for Dixon's and Macaulay's classes. """ from sympy.matrices.dense import Matrix from sympy.polys.polytools import factor from sympy.core import symbols from sympy.tensor.indexed import IndexedBase from sympy.polys.multivariate_resultants import (DixonResultant, MacaulayResultant) c, d = symbols("a, b") x, y = symbols("x, y") p = c * x + y q = x + d * y dixon = DixonResultant(polynomials=[p, q], variables=[x, y]) macaulay = MacaulayResultant(polynomials=[p, q], variables=[x, y]) def test_dixon_resultant_init(): """Test init method of DixonResultant.""" a = IndexedBase("alpha") assert dixon.polynomials == [p, q] assert dixon.variables == [x, y] assert dixon.n == 2 assert dixon.m == 2 assert dixon.dummy_variables == [a[0], a[1]] def test_get_dixon_polynomial_numerical(): """Test Dixon's polynomial for a numerical example.""" a = IndexedBase("alpha") p = x + y q = x ** 2 + y **3 h = x ** 2 + y dixon = DixonResultant([p, q, h], [x, y]) polynomial = -x * y ** 2 * a[0] - x * y ** 2 * a[1] - x * y * a[0] \ * a[1] - x * y * a[1] ** 2 - x * a[0] * a[1] ** 2 + x * a[0] - \ y ** 2 * a[0] * a[1] + y ** 2 * a[1] - y * a[0] * a[1] ** 2 + y * \ a[1] ** 2 assert dixon.get_dixon_polynomial().as_expr().expand() == polynomial def test_get_max_degrees(): """Tests max degrees function.""" p = x + y q = x ** 2 + y **3 h = x ** 2 + y dixon = DixonResultant(polynomials=[p, q, h], variables=[x, y]) dixon_polynomial = dixon.get_dixon_polynomial() assert dixon.get_max_degrees(dixon_polynomial) == [1, 2] def test_get_dixon_matrix(): """Test Dixon's resultant for a numerical example.""" x, y = symbols('x, y') p = x + y q = x ** 2 + y ** 3 h = x ** 2 + y dixon = DixonResultant([p, q, h], [x, y]) polynomial = dixon.get_dixon_polynomial() assert dixon.get_dixon_matrix(polynomial).det() == 0 def test_get_dixon_matrix_example_two(): """Test Dixon's matrix for example from [Palancz08]_.""" x, y, z = symbols('x, y, z') f = x ** 2 + y ** 2 - 1 + z * 0 g = x ** 2 + z ** 2 - 1 + y * 0 h = y ** 2 + z ** 2 - 1 example_two = DixonResultant([f, g, h], [y, z]) poly = example_two.get_dixon_polynomial() matrix = example_two.get_dixon_matrix(poly) expr = 1 - 8 * x ** 2 + 24 * x ** 4 - 32 * x ** 6 + 16 * x ** 8 assert (matrix.det() - expr).expand() == 0 def test_KSY_precondition(): """Tests precondition for KSY Resultant.""" A, B, C = symbols('A, B, C') m1 = Matrix([[1, 2, 3], [4, 5, 12], [6, 7, 18]]) m2 = Matrix([[0, C**2], [-2 * C, -C ** 2]]) m3 = Matrix([[1, 0], [0, 1]]) m4 = Matrix([[A**2, 0, 1], [A, 1, 1 / A]]) m5 = Matrix([[5, 1], [2, B], [0, 1], [0, 0]]) assert dixon.KSY_precondition(m1) == False assert dixon.KSY_precondition(m2) == True assert dixon.KSY_precondition(m3) == True assert dixon.KSY_precondition(m4) == False assert dixon.KSY_precondition(m5) == True def test_delete_zero_rows_and_columns(): """Tests method for deleting rows and columns containing only zeros.""" A, B, C = symbols('A, B, C') m1 = Matrix([[0, 0], [0, 0], [1, 2]]) m2 = Matrix([[0, 1, 2], [0, 3, 4], [0, 5, 6]]) m3 = Matrix([[0, 0, 0, 0], [0, 1, 2, 0], [0, 3, 4, 0], [0, 0, 0, 0]]) m4 = Matrix([[1, 0, 2], [0, 0, 0], [3, 0, 4]]) m5 = Matrix([[0, 0, 0, 1], [0, 0, 0, 2], [0, 0, 0, 3], [0, 0, 0, 4]]) m6 = Matrix([[0, 0, A], [B, 0, 0], [0, 0, C]]) assert dixon.delete_zero_rows_and_columns(m1) == Matrix([[1, 2]]) assert dixon.delete_zero_rows_and_columns(m2) == Matrix([[1, 2], [3, 4], [5, 6]]) assert dixon.delete_zero_rows_and_columns(m3) == Matrix([[1, 2], [3, 4]]) assert dixon.delete_zero_rows_and_columns(m4) == Matrix([[1, 2], [3, 4]]) assert dixon.delete_zero_rows_and_columns(m5) == Matrix([[1], [2], [3], [4]]) assert dixon.delete_zero_rows_and_columns(m6) == Matrix([[0, A], [B, 0], [0, C]]) def test_product_leading_entries(): """Tests product of leading entries method.""" A, B = symbols('A, B') m1 = Matrix([[1, 2, 3], [0, 4, 5], [0, 0, 6]]) m2 = Matrix([[0, 0, 1], [2, 0, 3]]) m3 = Matrix([[0, 0, 0], [1, 2, 3], [0, 0, 0]]) m4 = Matrix([[0, 0, A], [1, 2, 3], [B, 0, 0]]) assert dixon.product_leading_entries(m1) == 24 assert dixon.product_leading_entries(m2) == 2 assert dixon.product_leading_entries(m3) == 1 assert dixon.product_leading_entries(m4) == A * B def test_get_KSY_Dixon_resultant_example_one(): """Tests the KSY Dixon resultant for example one""" x, y, z = symbols('x, y, z') p = x * y * z q = x**2 - z**2 h = x + y + z dixon = DixonResultant([p, q, h], [x, y]) dixon_poly = dixon.get_dixon_polynomial() dixon_matrix = dixon.get_dixon_matrix(dixon_poly) D = dixon.get_KSY_Dixon_resultant(dixon_matrix) assert D == -z**3 def test_get_KSY_Dixon_resultant_example_two(): """Tests the KSY Dixon resultant for example two""" x, y, A = symbols('x, y, A') p = x * y + x * A + x - A**2 - A + y**2 + y q = x**2 + x * A - x + x * y + y * A - y h = x**2 + x * y + 2 * x - x * A - y * A - 2 * A dixon = DixonResultant([p, q, h], [x, y]) dixon_poly = dixon.get_dixon_polynomial() dixon_matrix = dixon.get_dixon_matrix(dixon_poly) D = factor(dixon.get_KSY_Dixon_resultant(dixon_matrix)) assert D == -8*A*(A - 1)*(A + 2)*(2*A - 1)**2 def test_macaulay_resultant_init(): """Test init method of MacaulayResultant.""" assert macaulay.polynomials == [p, q] assert macaulay.variables == [x, y] assert macaulay.n == 2 assert macaulay.degrees == [1, 1] assert macaulay.degree_m == 1 assert macaulay.monomials_size == 2 def test_get_degree_m(): assert macaulay._get_degree_m() == 1 def test_get_size(): assert macaulay.get_size() == 2 def test_macaulay_example_one(): """Tests the Macaulay for example from [Bruce97]_""" x, y, z = symbols('x, y, z') a_1_1, a_1_2, a_1_3 = symbols('a_1_1, a_1_2, a_1_3') a_2_2, a_2_3, a_3_3 = symbols('a_2_2, a_2_3, a_3_3') b_1_1, b_1_2, b_1_3 = symbols('b_1_1, b_1_2, b_1_3') b_2_2, b_2_3, b_3_3 = symbols('b_2_2, b_2_3, b_3_3') c_1, c_2, c_3 = symbols('c_1, c_2, c_3') f_1 = a_1_1 * x ** 2 + a_1_2 * x * y + a_1_3 * x * z + \ a_2_2 * y ** 2 + a_2_3 * y * z + a_3_3 * z ** 2 f_2 = b_1_1 * x ** 2 + b_1_2 * x * y + b_1_3 * x * z + \ b_2_2 * y ** 2 + b_2_3 * y * z + b_3_3 * z ** 2 f_3 = c_1 * x + c_2 * y + c_3 * z mac = MacaulayResultant([f_1, f_2, f_3], [x, y, z]) assert mac.degrees == [2, 2, 1] assert mac.degree_m == 3 assert mac.monomial_set == [x ** 3, x ** 2 * y, x ** 2 * z, x * y ** 2, x * y * z, x * z ** 2, y ** 3, y ** 2 *z, y * z ** 2, z ** 3] assert mac.monomials_size == 10 assert mac.get_row_coefficients() == [[x, y, z], [x, y, z], [x * y, x * z, y * z, z ** 2]] matrix = mac.get_matrix() assert matrix.shape == (mac.monomials_size, mac.monomials_size) assert mac.get_submatrix(matrix) == Matrix([[a_1_1, a_2_2], [b_1_1, b_2_2]]) def test_macaulay_example_two(): """Tests the Macaulay formulation for example from [Stiller96]_.""" 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([f, g, h], [x, y, z]) assert mac.degrees == [1, 2, 3] assert mac.degree_m == 4 assert mac.monomials_size == 15 assert len(mac.get_row_coefficients()) == mac.n matrix = mac.get_matrix() assert matrix.shape == (mac.monomials_size, mac.monomials_size) assert mac.get_submatrix(matrix) == Matrix([[-a_1, a_0, a_2, 0], [0, -a_1, 0, 0], [0, 0, -a_1, 0], [0, 0, 0, -a_1]])