phivenv/Lib/site-packages/sympy/polys/tests/test_subresultants_qq_zz.py

from sympy.core.symbol import Symbol
from sympy.polys.polytools import (pquo, prem, sturm, subresultants)
from sympy.matrices import Matrix
from sympy.polys.subresultants_qq_zz import (sylvester, res, res_q, res_z, bezout,
    subresultants_sylv,   modified_subresultants_sylv,
    subresultants_bezout, modified_subresultants_bezout,
    backward_eye,
    sturm_pg, sturm_q, sturm_amv, euclid_pg, euclid_q,
    euclid_amv, modified_subresultants_pg, subresultants_pg,
    subresultants_amv_q, quo_z, rem_z, subresultants_amv,
    modified_subresultants_amv, subresultants_rem,
    subresultants_vv, subresultants_vv_2)


def test_sylvester():
    x = Symbol('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_subresultants_sylv():
    x = Symbol('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_modified_subresultants_sylv():
    x = Symbol('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_res():
    x = Symbol('x')

    assert res(3, 5, x) == 1

def test_res_q():
    x = Symbol('x')

    assert res_q(3, 5, x) == 1

def test_res_z():
    x = Symbol('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_bezout():
    x = Symbol('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_subresultants_bezout():
    x = Symbol('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_bezout(p, q, x) == subresultants(p, q, x)
    assert subresultants_bezout(p, q, x)[-1] == sylvester(p, q, x).det()
    assert subresultants_bezout(p, q, x) != euclid_amv(p, q, x)
    amv_factors = [1, 1, -1, 1, -1, 1]
    assert subresultants_bezout(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_bezout(p, q, x) == euclid_amv(p, q, x)

def test_modified_subresultants_bezout():
    x = Symbol('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_bezout(p, q, x) == [i*j for i, j in zip(amv_factors, subresultants_amv(p, q, x))]
    assert modified_subresultants_bezout(p, q, x)[-1] != sylvester(p + x**8, q, x).det()
    assert modified_subresultants_bezout(p, q, x) != sturm_amv(p, q, x)

    p = x**3 - 7*x + 7
    q = 3*x**2 - 7
    assert modified_subresultants_bezout(p, q, x) == sturm_amv(p, q, x)
    assert modified_subresultants_bezout(-p, q, x) != sturm_amv(-p, q, x)

def test_sturm_pg():
    x = Symbol('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_pg(p, q, x)[-1] != sylvester(p, q, x, 2).det()
    sam_factors = [1, 1, -1, -1, 1, 1]
    assert sturm_pg(p, q, x) == [i*j for i,j in zip(sam_factors, euclid_pg(p, q, x))]

    p = -9*x**5 - 5*x**3 - 9
    q = -45*x**4 - 15*x**2
    assert sturm_pg(p, q, x, 1)[-1] == sylvester(p, q, x, 1).det()
    assert sturm_pg(p, q, x)[-1] != sylvester(p, q, x, 2).det()
    assert sturm_pg(-p, q, x)[-1] == sylvester(-p, q, x, 2).det()
    assert sturm_pg(-p, q, x) == modified_subresultants_pg(-p, q, x)

def test_sturm_q():
    x = Symbol('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_sturm_amv():
    x = Symbol('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_euclid_pg():
    x = Symbol('x')

    p = x**6+x**5-x**4-x**3+x**2-x+1
    q = 6*x**5+5*x**4-4*x**3-3*x**2+2*x-1
    assert euclid_pg(p, q, x)[-1] == sylvester(p, q, x).det()
    assert euclid_pg(p, q, x) == subresultants_pg(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_pg(p, q, x)[-1] != sylvester(p, q, x, 2).det()
    sam_factors = [1, 1, -1, -1, 1, 1]
    assert euclid_pg(p, q, x) == [i*j for i,j in zip(sam_factors, sturm_pg(p, q, x))]


def test_euclid_q():
    x = Symbol('x')

    p = x**3 - 7*x + 7
    q = 3*x**2 - 7
    assert euclid_q(p, q, x)[-1] == -sturm(p)[-1]


def test_euclid_amv():
    x = Symbol('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_pg():
    x = Symbol('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_pg(p, q, x) == [i*j for i, j in zip(amv_factors, subresultants_pg(p, q, x))]
    assert modified_subresultants_pg(p, q, x)[-1] != sylvester(p + x**8, q, x).det()
    assert modified_subresultants_pg(p, q, x) != sturm_pg(p, q, x)

    p = x**3 - 7*x + 7
    q = 3*x**2 - 7
    assert modified_subresultants_pg(p, q, x) == sturm_pg(p, q, x)
    assert modified_subresultants_pg(-p, q, x) != sturm_pg(-p, q, x)


def test_subresultants_pg():
    x = Symbol('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_pg(p, q, x) == subresultants(p, q, x)
    assert subresultants_pg(p, q, x)[-1] == sylvester(p, q, x).det()
    assert subresultants_pg(p, q, x) != euclid_pg(p, q, x)
    amv_factors = [1, 1, -1, 1, -1, 1]
    assert subresultants_pg(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_pg(p, q, x) == euclid_pg(p, q, x)


def test_subresultants_amv_q():
    x = Symbol('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_amv_q(p, q, x) == subresultants(p, q, x)
    assert subresultants_amv_q(p, q, x)[-1] == sylvester(p, q, x).det()
    assert subresultants_amv_q(p, q, x) != euclid_amv(p, q, x)
    amv_factors = [1, 1, -1, 1, -1, 1]
    assert subresultants_amv_q(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_amv(p, q, x) == euclid_amv(p, q, x)


def test_rem_z():
    x = Symbol('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_quo_z():
    x = Symbol('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 = Symbol('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_subresultants_amv():
    x = Symbol('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_amv(p, q, x) == subresultants(p, q, x)
    assert subresultants_amv(p, q, x)[-1] == sylvester(p, q, x).det()
    assert subresultants_amv(p, q, x) != euclid_amv(p, q, x)
    amv_factors = [1, 1, -1, 1, -1, 1]
    assert subresultants_amv(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_amv(p, q, x) == euclid_amv(p, q, x)


def test_modified_subresultants_amv():
    x = Symbol('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_amv(p, q, x) == [i*j for i, j in zip(amv_factors, subresultants_amv(p, q, x))]
    assert modified_subresultants_amv(p, q, x)[-1] != sylvester(p + x**8, q, x).det()
    assert modified_subresultants_amv(p, q, x) != sturm_amv(p, q, x)

    p = x**3 - 7*x + 7
    q = 3*x**2 - 7
    assert modified_subresultants_amv(p, q, x) == sturm_amv(p, q, x)
    assert modified_subresultants_amv(-p, q, x) != sturm_amv(-p, q, x)


def test_subresultants_rem():
    x = Symbol('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_rem(p, q, x) == subresultants(p, q, x)
    assert subresultants_rem(p, q, x)[-1] == sylvester(p, q, x).det()
    assert subresultants_rem(p, q, x) != euclid_amv(p, q, x)
    amv_factors = [1, 1, -1, 1, -1, 1]
    assert subresultants_rem(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_rem(p, q, x) == euclid_amv(p, q, x)


def test_subresultants_vv():
    x = Symbol('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_vv(p, q, x) == subresultants(p, q, x)
    assert subresultants_vv(p, q, x)[-1] == sylvester(p, q, x).det()
    assert subresultants_vv(p, q, x) != euclid_amv(p, q, x)
    amv_factors = [1, 1, -1, 1, -1, 1]
    assert subresultants_vv(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_vv(p, q, x) == euclid_amv(p, q, x)


def test_subresultants_vv_2():
    x = Symbol('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_vv_2(p, q, x) == subresultants(p, q, x)
    assert subresultants_vv_2(p, q, x)[-1] == sylvester(p, q, x).det()
    assert subresultants_vv_2(p, q, x) != euclid_amv(p, q, x)
    amv_factors = [1, 1, -1, 1, -1, 1]
    assert subresultants_vv_2(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_vv_2(p, q, x) == euclid_amv(p, q, x)