Python polys.QQ类代码示例

def test_exceptions():
    I = QQ.old_poly_ring(x).ideal(x)
    J = QQ.old_poly_ring(y).ideal(1)
    raises(ValueError, lambda: I.union(x))
    raises(ValueError, lambda: I + J)
    raises(ValueError, lambda: I * J)
    raises(ValueError, lambda: I.union(J))
    assert (I == J) is False
    assert I != J

def test_intersection():
    R = QQ.old_poly_ring(x, y, z)
    # SCA, example 1.8.11
    assert R.ideal(x, y).intersect(R.ideal(y ** 2, z)) == R.ideal(y ** 2, y * z, x * z)

    assert R.ideal(x, y).intersect(R.ideal()).is_zero()

    R = QQ.old_poly_ring(x, y, z, order="ilex")
    assert R.ideal(x, y).intersect(R.ideal(y ** 2 + y ** 2 * z, z + z * x ** 3 * y)) == R.ideal(y ** 2, y * z, x * z)

def test_FreeModule():
    M1 = FreeModule(QQ[x], 2)
    assert M1 == FreeModule(QQ[x], 2)
    assert M1 != FreeModule(QQ[y], 2)
    assert M1 != FreeModule(QQ[x], 3)
    M2 = FreeModule(QQ.poly_ring(x, order="ilex"), 2)

    assert [x, 1] in M1
    assert [x] not in M1
    assert [2, y] not in M1
    assert [1/(x + 1), 2] not in M1

    e = M1.convert([x, x**2 + 1])
    X = QQ[x].convert(x)
    assert e == [X, X**2 + 1]
    assert e == [x, x**2 + 1]
    assert 2*e == [2*x, 2*x**2 + 2]
    assert e*2 == [2*x, 2*x**2 + 2]
    assert e/2 == [x/2, (x**2 + 1)/2]
    assert x*e == [x**2, x**3 + x]
    assert e*x == [x**2, x**3 + x]
    assert X*e == [x**2, x**3 + x]
    assert e*X == [x**2, x**3 + x]

    assert [x, 1] in M2
    assert [x] not in M2
    assert [2, y] not in M2
    assert [1/(x + 1), 2] in M2

    e = M2.convert([x, x**2 + 1])
    X = QQ.poly_ring(x, order="ilex").convert(x)
    assert e == [X, X**2 + 1]
    assert e == [x, x**2 + 1]
    assert 2*e == [2*x, 2*x**2 + 2]
    assert e*2 == [2*x, 2*x**2 + 2]
    assert e/2 == [x/2, (x**2 + 1)/2]
    assert x*e == [x**2, x**3 + x]
    assert e*x == [x**2, x**3 + x]
    assert e/(1 + x) == [x/(1 + x), (x**2 + 1)/(1 + x)]
    assert X*e == [x**2, x**3 + x]
    assert e*X == [x**2, x**3 + x]

    M3 = FreeModule(QQ[x, y], 2)
    assert M3.convert(e) == M3.convert([x, x**2 + 1])

    assert not M3.is_submodule(0)
    assert not M3.is_zero()

    raises(NotImplementedError, lambda: ZZ[x].free_module(2))
    raises(NotImplementedError, lambda: FreeModulePolyRing(ZZ, 2))
    raises(CoercionFailed, lambda: M1.convert(QQ[x].free_module(3)
           .convert([1, 2, 3])))
    raises(CoercionFailed, lambda: M3.convert(1))

def test_quotient():
    # SCA, example 2.8.6
    R = QQ.old_poly_ring(x, y, z)
    F = R.free_module(2)
    assert F.submodule([x*y, x*z], [y*z, x*y]).module_quotient(
        F.submodule([y, z], [z, y])) == QQ.old_poly_ring(x, y, z).ideal(x**2*y**2 - x*y*z**2)
    assert F.submodule([x, y]).module_quotient(F.submodule()).is_whole_ring()

    M = F.submodule([x**2, x**2], [y**2, y**2])
    N = F.submodule([x + y, x + y])
    q, rel = M.module_quotient(N, relations=True)
    assert q == R.ideal(y**2, x - y)
    for i, g in enumerate(q.gens):
        assert g*N.gens[0] == sum(c*x for c, x in zip(rel[i], M.gens))

def test_QuotientModule():
    R = QQ.old_poly_ring(x)
    F = R.free_module(3)
    N = F.submodule([1, x, x**2])
    M = F/N

    assert M != F
    assert M != N
    assert M == F / [(1, x, x**2)]
    assert not M.is_zero()
    assert (F / F.basis()).is_zero()

    SQ = F.submodule([1, x, x**2], [2, 0, 0]) / N
    assert SQ == M.submodule([2, x, x**2])
    assert SQ != M.submodule([2, 1, 0])
    assert SQ != M
    assert M.is_submodule(SQ)
    assert not SQ.is_full_module()

    raises(ValueError, lambda: N/F)
    raises(ValueError, lambda: F.submodule([2, 0, 0]) / N)
    raises(ValueError, lambda: R.free_module(2)/F)
    raises(CoercionFailed, lambda: F.convert(M.convert([1, x, x**2])))

    M1 = F / [[1, 1, 1]]
    M2 = M1.submodule([1, 0, 0], [0, 1, 0])
    assert M1 == M2

def test_QuotientModuleElement():
    R = QQ.old_poly_ring(x)
    F = R.free_module(3)
    N = F.submodule([1, x, x**2])
    M = F/N
    e = M.convert([x**2, 2, 0])

    assert M.convert([x + 1, x**2 + x, x**3 + x**2]) == 0
    assert e == [x**2, 2, 0] + N == F.convert([x**2, 2, 0]) + N == \
        M.convert(F.convert([x**2, 2, 0]))

    assert M.convert([x**2 + 1, 2*x + 2, x**2]) == e + [0, x, 0] == \
        e + M.convert([0, x, 0]) == e + F.convert([0, x, 0])
    assert M.convert([x**2 + 1, 2, x**2]) == e - [0, x, 0] == \
        e - M.convert([0, x, 0]) == e - F.convert([0, x, 0])
    assert M.convert([0, 2, 0]) == M.convert([x**2, 4, 0]) - e == \
        [x**2, 4, 0] - e == F.convert([x**2, 4, 0]) - e
    assert M.convert([x**3 + x**2, 2*x + 2, 0]) == (1 + x)*e == \
        R.convert(1 + x)*e == e*(1 + x) == e*R.convert(1 + x)
    assert -e == [-x**2, -2, 0]

    f = [x, x, 0] + N
    assert M.convert([1, 1, 0]) == f / x == f / R.convert(x)

    M2 = F/[(2, 2*x, 2*x**2), (0, 0, 1)]
    G = R.free_module(2)
    M3 = G/[[1, x]]
    M4 = F.submodule([1, x, x**2], [1, 0, 0]) / N
    raises(CoercionFailed, lambda: M.convert(G.convert([1, x])))
    raises(CoercionFailed, lambda: M.convert(M3.convert([1, x])))
    raises(CoercionFailed, lambda: M.convert(M2.convert([1, x, x])))
    assert M2.convert(M.convert([2, x, x**2])) == [2, x, 0]
    assert M.convert(M4.convert([2, 0, 0])) == [2, 0, 0]

def test_in_terms_of_generators():
    R = QQ.old_poly_ring(x, order="ilex")
    M = R.free_module(2).submodule([2*x, 0], [1, 2])
    assert M.in_terms_of_generators(
        [x, x]) == [R.convert(S(1)/4), R.convert(x/2)]
    raises(ValueError, lambda: M.in_terms_of_generators([1, 0]))

    M = R.free_module(2) / ([x, 0], [1, 1])
    SM = M.submodule([1, x])
    assert SM.in_terms_of_generators([2, 0]) == [R.convert(-2/(x - 1))]

    R = QQ.old_poly_ring(x, y) / [x**2 - y**2]
    M = R.free_module(2)
    SM = M.submodule([x, 0], [0, y])
    assert SM.in_terms_of_generators(
        [x**2, x**2]) == [R.convert(x), R.convert(y)]

def test_SubModulePolyRing_nontriv_global():
    R = QQ.old_poly_ring(x, y, z)
    F = R.free_module(1)

    def contains(I, f):
        return F.submodule(*[[g] for g in I]).contains([f])

    assert contains([x, y], x)
    assert contains([x, y], x + y)
    assert not contains([x, y], 1)
    assert not contains([x, y], z)
    assert contains([x**2 + y, x**2 + x], x - y)
    assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x**2)
    assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**3)
    assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**4)
    assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x*y**2)
    assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**4 + y**3 + 2*z*y*x)
    assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x*y*z)
    assert contains([x, 1 + x + y, 5 - 7*y], 1)
    assert contains(
        [x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2*y + x**2*z + y**2*z],
        x**3)
    assert not contains(
        [x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2*y + x**2*z + y**2*z],
        x**2 + y**2)

    # compare local order
    assert not contains([x*(1 + x + y), y*(1 + z)], x)
    assert not contains([x*(1 + x + y), y*(1 + z)], x + y)

def test_PrettyPoly():
    from sympy.polys.domains import QQ

    F = QQ.frac_field(x, y)
    R = QQ[x, y]
    assert sstr(F.convert(x / (x + y))) == sstr(x / (x + y))
    assert sstr(R.convert(x + y)) == sstr(x + y)

def test_SubModulePolyRing_local():
    R = QQ.poly_ring(x, y, order=ilex)
    F = R.free_module(3)
    Fd = F.submodule([1+x, 0, 0], [1+y, 2+2*y, 0], [1, 2, 3])
    M = F.submodule([x**2 + y**2, 1, 0], [x, y, 1])

    assert F == Fd
    assert Fd == F
    assert F != M
    assert M != F
    assert Fd != M
    assert M != Fd
    assert Fd == F.submodule(*F.basis())

    assert Fd.is_full_module()
    assert not M.is_full_module()
    assert not Fd.is_zero()
    assert not M.is_zero()
    assert Fd.submodule().is_zero()

    assert M.contains([x**2 + y**2 + x, 1 + y, 1])
    assert not M.contains([x**2 + y**2 + x, 1 + y, 2])
    assert M.contains([y**2, 1 - x*y, -x])

    assert F.submodule([1 + x, 0, 0]) == F.submodule([1, 0, 0])
    assert F.submodule([1, 0, 0], [0, 1, 0]).union(F.submodule([0, 0, 1 + x*y])) == F

    raises(ValueError, lambda: M.submodule([1, 0, 0]))

def test_nontriv_global():
    R = QQ.old_poly_ring(x, y, z)

    def contains(I, f):
        return R.ideal(*I).contains(f)

    assert contains([x, y], x)
    assert contains([x, y], x + y)
    assert not contains([x, y], 1)
    assert not contains([x, y], z)
    assert contains([x ** 2 + y, x ** 2 + x], x - y)
    assert not contains([x + y + z, x * y + x * z + y * z, x * y * z], x ** 2)
    assert contains([x + y + z, x * y + x * z + y * z, x * y * z], x ** 3)
    assert contains([x + y + z, x * y + x * z + y * z, x * y * z], x ** 4)
    assert not contains([x + y + z, x * y + x * z + y * z, x * y * z], x * y ** 2)
    assert contains([x + y + z, x * y + x * z + y * z, x * y * z], x ** 4 + y ** 3 + 2 * z * y * x)
    assert contains([x + y + z, x * y + x * z + y * z, x * y * z], x * y * z)
    assert contains([x, 1 + x + y, 5 - 7 * y], 1)
    assert contains([x ** 3 + y ** 3, y ** 3 + z ** 3, z ** 3 + x ** 3, x ** 2 * y + x ** 2 * z + y ** 2 * z], x ** 3)
    assert not contains(
        [x ** 3 + y ** 3, y ** 3 + z ** 3, z ** 3 + x ** 3, x ** 2 * y + x ** 2 * z + y ** 2 * z], x ** 2 + y ** 2
    )

    # compare local order
    assert not contains([x * (1 + x + y), y * (1 + z)], x)
    assert not contains([x * (1 + x + y), y * (1 + z)], x + y)

def test_PolynomialRingBase():
    assert srepr(ZZ.old_poly_ring(x)) == \
        "GlobalPolynomialRing(ZZ, Symbol('x'))"
    assert srepr(ZZ[x].old_poly_ring(y)) == \
        "GlobalPolynomialRing(ZZ[x], Symbol('y'))"
    assert srepr(QQ.frac_field(x).old_poly_ring(y)) == \
        "GlobalPolynomialRing(FractionField(FracField((Symbol('x'),), QQ, lex)), Symbol('y'))"

def test_reduction():
    from sympy.polys.distributedmodules import sdm_nf_buchberger_reduced
    R = QQ.old_poly_ring(x, y)
    I = R.ideal(x**5, y)
    e = R.convert(x**3 + y**2)
    assert I.reduce_element(e) == e
    assert I.reduce_element(e, NF=sdm_nf_buchberger_reduced) == R.convert(x**3)

def test_syzygy():
    R = QQ.old_poly_ring(x, y, z)
    M = R.free_module(1).submodule([x*y], [y*z], [x*z])
    S = R.free_module(3).submodule([0, x, -y], [z, -x, 0])
    assert M.syzygy_module() == S

    M2 = M / ([x*y*z],)
    S2 = R.free_module(3).submodule([z, 0, 0], [0, x, 0], [0, 0, y])
    assert M2.syzygy_module() == S2

    F = R.free_module(3)
    assert F.submodule(*F.basis()).syzygy_module() == F.submodule()

    R2 = QQ.old_poly_ring(x, y, z) / [x*y*z]
    M3 = R2.free_module(1).submodule([x*y], [y*z], [x*z])
    S3 = R2.free_module(3).submodule([z, 0, 0], [0, x, 0], [0, 0, y])
    assert M3.syzygy_module() == S3

def test_module_mul():
    R = QQ.old_poly_ring(x)
    M = R.free_module(2)
    S1 = M.submodule([x, 0], [0, x])
    S2 = M.submodule([x**2, 0], [0, x**2])
    I = R.ideal(x)

    assert I*M == M*I == S1 == x*M == M*x
    assert I*S1 == S2 == x*S1