def test_quo_z(): x = var('x') p = x**8 + x**6 - 3 * x**4 - 3 * x**3 + 8 * x**2 + 2 * x - 5 q = 3 * x**6 + 5 * x**4 - 4 * x**2 - 9 * x + 21 assert quo_z(p, -q, x) != pquo(p, -q, x) y = var('y') q = 3 * x**6 + 5 * y**4 - 4 * x**2 - 9 * x + 21 assert quo_z(p, -q, x) == pquo(p, -q, x)
def test_sturm_q(): x = var('x') p = x**3 - 7 * x + 7 q = 3 * x**2 - 7 assert sturm_q(p, q, x) == sturm(p) assert sturm_q(-p, -q, x) != sturm(-p)
def test_bezout(): x = var('x') p = -2 * x**5 + 7 * x**3 + 9 * x**2 - 3 * x + 1 q = -10 * x**4 + 21 * x**2 + 18 * x - 3 assert bezout(p, q, x, 'bz').det() == sylvester(p, q, x, 2).det() assert bezout(p, q, x, 'bz').det() != sylvester(p, q, x, 1).det() assert bezout( p, q, x, 'prs') == backward_eye(5) * bezout(p, q, x, 'bz') * backward_eye(5)
def test_euclid_amv(): x = var('x') p = x**3 - 7 * x + 7 q = 3 * x**2 - 7 assert euclid_amv(p, q, x)[-1] == sylvester(p, q, x).det() assert euclid_amv(p, q, x) == subresultants_amv(p, q, x) p = x**8 + x**6 - 3 * x**4 - 3 * x**3 + 8 * x**2 + 2 * x - 5 q = 3 * x**6 + 5 * x**4 - 4 * x**2 - 9 * x + 21 assert euclid_amv(p, q, x)[-1] != sylvester(p, q, x, 2).det() sam_factors = [1, 1, -1, -1, 1, 1] assert euclid_amv( p, q, x) == [i * j for i, j in zip(sam_factors, sturm_amv(p, q, x))]
def test_modified_subresultants_sylv(): x = var('x') p = x**8 + x**6 - 3 * x**4 - 3 * x**3 + 8 * x**2 + 2 * x - 5 q = 3 * x**6 + 5 * x**4 - 4 * x**2 - 9 * x + 21 amv_factors = [1, 1, -1, 1, -1, 1] assert modified_subresultants_sylv(p, q, x) == [ i * j for i, j in zip(amv_factors, subresultants_amv(p, q, x)) ] assert modified_subresultants_sylv(p, q, x)[-1] != res_q(p + x**8, q, x) assert modified_subresultants_sylv(p, q, x) != sturm_amv(p, q, x) p = x**3 - 7 * x + 7 q = 3 * x**2 - 7 assert modified_subresultants_sylv(p, q, x) == sturm_amv(p, q, x) assert modified_subresultants_sylv(-p, q, x) != sturm_amv(-p, q, x)
def test_subresultants_sylv(): x = var('x') p = x**8 + x**6 - 3 * x**4 - 3 * x**3 + 8 * x**2 + 2 * x - 5 q = 3 * x**6 + 5 * x**4 - 4 * x**2 - 9 * x + 21 assert subresultants_sylv(p, q, x) == subresultants(p, q, x) assert subresultants_sylv(p, q, x)[-1] == res(p, q, x) assert subresultants_sylv(p, q, x) != euclid_amv(p, q, x) amv_factors = [1, 1, -1, 1, -1, 1] assert subresultants_sylv(p, q, x) == [ i * j for i, j in zip(amv_factors, modified_subresultants_amv(p, q, x)) ] p = x**3 - 7 * x + 7 q = 3 * x**2 - 7 assert subresultants_sylv(p, q, x) == euclid_amv(p, q, x)
def test_sturm_amv(): x = var('x') p = x**8 + x**6 - 3 * x**4 - 3 * x**3 + 8 * x**2 + 2 * x - 5 q = 3 * x**6 + 5 * x**4 - 4 * x**2 - 9 * x + 21 assert sturm_amv(p, q, x)[-1] != sylvester(p, q, x, 2).det() sam_factors = [1, 1, -1, -1, 1, 1] assert sturm_amv( p, q, x) == [i * j for i, j in zip(sam_factors, euclid_amv(p, q, x))] p = -9 * x**5 - 5 * x**3 - 9 q = -45 * x**4 - 15 * x**2 assert sturm_amv(p, q, x, 1)[-1] == sylvester(p, q, x, 1).det() assert sturm_amv(p, q, x)[-1] != sylvester(p, q, x, 2).det() assert sturm_amv(-p, q, x)[-1] == sylvester(-p, q, x, 2).det() assert sturm_pg(-p, q, x) == modified_subresultants_pg(-p, q, x)
def test_sylvester(): x = var('x') assert sylvester(x**3 - 7, 0, x) == sylvester(x**3 - 7, 0, x, 1) == Matrix( [[0]]) assert sylvester(0, x**3 - 7, x) == sylvester(0, x**3 - 7, x, 1) == Matrix( [[0]]) assert sylvester(x**3 - 7, 0, x, 2) == Matrix([[0]]) assert sylvester(0, x**3 - 7, x, 2) == Matrix([[0]]) assert sylvester(x**3 - 7, 7, x).det() == sylvester(x**3 - 7, 7, x, 1).det() == 343 assert sylvester(7, x**3 - 7, x).det() == sylvester(7, x**3 - 7, x, 1).det() == 343 assert sylvester(x**3 - 7, 7, x, 2).det() == -343 assert sylvester(7, x**3 - 7, x, 2).det() == 343 assert sylvester(3, 7, x).det() == sylvester( 3, 7, x, 1).det() == sylvester(3, 7, x, 2).det() == 1 assert sylvester(3, 0, x).det() == sylvester( 3, 0, x, 1).det() == sylvester(3, 0, x, 2).det() == 1 assert sylvester(x - 3, x - 8, x) == sylvester(x - 3, x - 8, x, 1) == sylvester( x - 3, x - 8, x, 2) == Matrix([[1, -3], [1, -8]]) assert sylvester(x**3 - 7 * x + 7, 3 * x**2 - 7, x) == sylvester( x**3 - 7 * x + 7, 3 * x**2 - 7, x, 1) == Matrix( [[1, 0, -7, 7, 0], [0, 1, 0, -7, 7], [3, 0, -7, 0, 0], [0, 3, 0, -7, 0], [0, 0, 3, 0, -7]]) assert sylvester(x**3 - 7 * x + 7, 3 * x**2 - 7, x, 2) == Matrix([[1, 0, -7, 7, 0, 0], [0, 3, 0, -7, 0, 0], [0, 1, 0, -7, 7, 0], [0, 0, 3, 0, -7, 0], [0, 0, 1, 0, -7, 7], [0, 0, 0, 3, 0, -7]])
def test_res_z(): x = var('x') assert res_z(3, 5, x) == 1 assert res(3, 5, x) == res_q(3, 5, x) == res_z(3, 5, x)
def test_res_q(): x = var('x') assert res_q(3, 5, x) == 1
def test_rem_z(): x = var('x') p = x**8 + x**6 - 3 * x**4 - 3 * x**3 + 8 * x**2 + 2 * x - 5 q = 3 * x**6 + 5 * x**4 - 4 * x**2 - 9 * x + 21 assert rem_z(p, -q, x) != prem(p, -q, x)
def test_euclid_q(): x = var('x') p = x**3 - 7 * x + 7 q = 3 * x**2 - 7 assert euclid_q(p, q, x)[-1] == -sturm(p)[-1]