Python QQ.old_poly_ring方法代码示例

# 需要导入模块: from sympy import QQ [as 别名]
# 或者: from sympy.QQ import old_poly_ring [as 别名]
def test_operations():
    F = QQ.old_poly_ring(x).free_module(2)
    G = QQ.old_poly_ring(x).free_module(3)
    f = F.identity_hom()
    g = homomorphism(F, F, [0, [1, x]])
    h = homomorphism(F, F, [[1, 0], 0])
    i = homomorphism(F, G, [[1, 0, 0], [0, 1, 0]])

    assert f == f
    assert f != g
    assert f != i
    assert (f != F.identity_hom()) is False
    assert 2*f == f*2 == homomorphism(F, F, [[2, 0], [0, 2]])
    assert f/2 == homomorphism(F, F, [[S(1)/2, 0], [0, S(1)/2]])
    assert f + g == homomorphism(F, F, [[1, 0], [1, x + 1]])
    assert f - g == homomorphism(F, F, [[1, 0], [-1, 1 - x]])
    assert f*g == g == g*f
    assert h*g == homomorphism(F, F, [0, [1, 0]])
    assert g*h == homomorphism(F, F, [0, 0])
    assert i*f == i
    assert f([1, 2]) == [1, 2]
    assert g([1, 2]) == [2, 2*x]

    assert i.restrict_domain(F.submodule([x, x]))([x, x]) == i([x, x])
    h1 = h.quotient_domain(F.submodule([0, 1]))
    assert h1([1, 0]) == h([1, 0])
    assert h1.restrict_domain(h1.domain.submodule([x, 0]))([x, 0]) == h([x, 0])

    raises(TypeError, lambda: f/g)
    raises(TypeError, lambda: f + 1)
    raises(TypeError, lambda: f + i)
    raises(TypeError, lambda: f - 1)
    raises(TypeError, lambda: f*i)

# 需要导入模块: from sympy import QQ [as 别名]
# 或者: from sympy.QQ import old_poly_ring [as 别名]
def test_to_Sequence_Initial_Coniditons():
    x = symbols('x')
    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
    n = symbols('n', integer=True)
    _, Sn = RecurrenceOperators(QQ.old_poly_ring(n), 'Sn')
    p = HolonomicFunction(Dx - 1, x, 0, [1]).to_sequence()
    q = [(HolonomicSequence(-1 + (n + 1)*Sn, 1), 0)]
    assert p == q
    p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]).to_sequence()
    q = [(HolonomicSequence(1 + (n**2 + 3*n + 2)*Sn**2, [0, 1]), 0)]
    assert p == q
    p = HolonomicFunction(Dx**2 + 1 + x**3*Dx, x, 0, [2, 3]).to_sequence()
    q = [(HolonomicSequence(n + Sn**2 + (n**2 + 7*n + 12)*Sn**4, [2, 3, -1, -1/2, 1/12]), 1)]
    assert p == q
    p = HolonomicFunction(x**3*Dx**5 + 1 + Dx, x).to_sequence()
    q = [(HolonomicSequence(1 + (n + 1)*Sn + (n**5 - 5*n**3 + 4*n)*Sn**2), 0, 3)]
    assert p == q
    C_0, C_1, C_2, C_3 = symbols('C_0, C_1, C_2, C_3')
    p = expr_to_holonomic(log(1+x**2))
    q = [(HolonomicSequence(n**2 + (n**2 + 2*n)*Sn**2, [0, 0, C_2]), 0, 1)]
    assert p.to_sequence() == q
    p = p.diff()
    q = [(HolonomicSequence((n + 2) + (n + 2)*Sn**2, [C_0, 0]), 1, 0)]
    assert p.to_sequence() == q
    p = expr_to_holonomic(erf(x) + x).to_sequence()
    q = [(HolonomicSequence((2*n**2 - 2*n) + (n**3 + 2*n**2 - n - 2)*Sn**2, [0, 1 + 2/sqrt(pi), 0, C_3]), 0, 2)]
    assert p == q

# 需要导入模块: from sympy import QQ [as 别名]
# 或者: from sympy.QQ import old_poly_ring [as 别名]
def test_negative_power():
    x = symbols("x")
    _, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
    h1 = HolonomicFunction((-1) + (x)*Dx, x) ** -2
    h2 = HolonomicFunction((2) + (x)*Dx, x)

    assert h1 == h2

# 需要导入模块: from sympy import QQ [as 别名]
# 或者: from sympy.QQ import old_poly_ring [as 别名]
def test_multiplication_initial_condition():
    x = symbols('x')
    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
    p = HolonomicFunction(Dx**2 + x*Dx - 1, x, 0, [3, 1])
    q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1])
    r = HolonomicFunction((x**4 + 14*x**2 + 60) + 4*x*Dx + (x**4 + 9*x**2 + 20)*Dx**2 + \
        (2*x**3 + 18*x)*Dx**3 + (x**2 + 10)*Dx**4, x, 0, [3, 4, 2, 3])
    assert p * q == r
    p = HolonomicFunction(Dx**2 + x, x, 0, [1, 0])
    q = HolonomicFunction(Dx**3 - x**2, x, 0, [3, 3, 3])
    r = HolonomicFunction((x**8 - 37*x**7/27 - 10*x**6/27 - 164*x**5/9 - 184*x**4/9 + \
        160*x**3/27 + 404*x**2/9 + 8*x + 40/3) + (6*x**7 - 128*x**6/9 - 98*x**5/9 - 28*x**4/9 + \
        8*x**3/9 + 28*x**2 + 40*x/9 - 40)*Dx + (3*x**6 - 82*x**5/9 + 76*x**4/9 + 4*x**3/3 + \
        220*x**2/9 - 80*x/3)*Dx**2 + (-2*x**6 + 128*x**5/27 - 2*x**4/3 -80*x**2/9 + 200/9)*Dx**3 + \
        (3*x**5 - 64*x**4/9 - 28*x**3/9 + 6*x**2 - 20*x/9 - 20/3)*Dx**4 + (-4*x**3 + 64*x**2/9 + \
            8*x/3)*Dx**5 + (x**4 - 64*x**3/27 - 4*x**2/3 + 20/9)*Dx**6, x, 0, [3, 3, 3, -3, -12, -24])
    assert p * q == r
    p = HolonomicFunction(Dx - 1, x, 0, [2])
    q = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1])
    r = HolonomicFunction(2 -2*Dx + Dx**2, x, 0, [0, 2])
    assert p * q == r
    q = HolonomicFunction(x*Dx**2 + 1 + 2*Dx, x, 0,[0, 1])
    r = HolonomicFunction((x - 1) + (-2*x + 2)*Dx + x*Dx**2, x, 0, [0, 2])
    assert p * q == r
    p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 3])
    q = HolonomicFunction(Dx**3 + 1, x, 0, [1, 2, 1])
    r = HolonomicFunction(6*Dx + 3*Dx**2 + 2*Dx**3 - 3*Dx**4 + Dx**6, x, 0, [1, 5, 14, 17, 17, 2])
    assert p * q == r
    p = expr_to_holonomic(sin(x))
    q = expr_to_holonomic(1/x)
    r = HolonomicFunction(x + 2*Dx + x*Dx**2, x, 1, [sin(1), -sin(1) + cos(1)])
    assert p * q == r

# 需要导入模块: from sympy import QQ [as 别名]
# 或者: from sympy.QQ import old_poly_ring [as 别名]
def test_multiplication_initial_condition():
    x = symbols('x')
    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
    p = HolonomicFunction(Dx**2 + x*Dx - 1, x, 0, [3, 1])
    q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1])
    r = HolonomicFunction((x**4 + 14*x**2 + 60) + 4*x*Dx + (x**4 + 9*x**2 + 20)*Dx**2 + \
        (2*x**3 + 18*x)*Dx**3 + (x**2 + 10)*Dx**4, x, 0, [3, 4, 2, 3])
    assert p * q == r
    p = HolonomicFunction(Dx**2 + x, x, 0, [1, 0])
    q = HolonomicFunction(Dx**3 - x**2, x, 0, [3, 3, 3])
    r = HolonomicFunction((27*x**8 - 37*x**7 - 10*x**6 - 492*x**5 - 552*x**4 + 160*x**3 + \
        1212*x**2 + 216*x + 360) + (162*x**7 - 384*x**6 - 294*x**5 - 84*x**4 + 24*x**3 + \
        756*x**2 + 120*x - 1080)*Dx + (81*x**6 - 246*x**5 + 228*x**4 + 36*x**3 + \
        660*x**2 - 720*x)*Dx**2 + (-54*x**6 + 128*x**5 - 18*x**4 - 240*x**2 + 600)*Dx**3 + \
        (81*x**5 - 192*x**4 - 84*x**3 + 162*x**2 - 60*x - 180)*Dx**4 + (-108*x**3 + \
        192*x**2 + 72*x)*Dx**5 + (27*x**4 - 64*x**3 - 36*x**2 + 60)*Dx**6, x, 0, [3, 3, 3, -3, -12, -24])
    assert p * q == r
    p = HolonomicFunction(Dx - 1, x, 0, [2])
    q = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1])
    r = HolonomicFunction(2 -2*Dx + Dx**2, x, 0, [0, 2])
    assert p * q == r
    q = HolonomicFunction(x*Dx**2 + 1 + 2*Dx, x, 0,[0, 1])
    r = HolonomicFunction((x - 1) + (-2*x + 2)*Dx + x*Dx**2, x, 0, [0, 2])
    assert p * q == r
    p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 3])
    q = HolonomicFunction(Dx**3 + 1, x, 0, [1, 2, 1])
    r = HolonomicFunction(6*Dx + 3*Dx**2 + 2*Dx**3 - 3*Dx**4 + Dx**6, x, 0, [1, 5, 14, 17, 17, 2])
    assert p * q == r

# 需要导入模块: from sympy import QQ [as 别名]
# 或者: from sympy.QQ import old_poly_ring [as 别名]
def test_from_sympy():
    x = symbols("x")
    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), "Dx")
    p = from_sympy((sin(x) / x) ** 2)
    q = HolonomicFunction(
        8 * x + (4 * x ** 2 + 6) * Dx + 6 * x * Dx ** 2 + x ** 2 * Dx ** 3,
        x,
        1,
        [sin(1) ** 2, -2 * sin(1) ** 2 + 2 * sin(1) * cos(1), -8 * sin(1) * cos(1) + 2 * cos(1) ** 2 + 4 * sin(1) ** 2],
    )
    assert p == q
    p = from_sympy(1 / (1 + x ** 2) ** 2)
    q = HolonomicFunction(4 * x + (x ** 2 + 1) * Dx, x, 0, 1)
    assert p == q
    p = from_sympy(exp(x) * sin(x) + x * log(1 + x))
    q = HolonomicFunction(
        (4 * x ** 3 + 20 * x ** 2 + 40 * x + 36)
        + (-4 * x ** 4 - 20 * x ** 3 - 40 * x ** 2 - 36 * x) * Dx
        + (4 * x ** 5 + 12 * x ** 4 + 14 * x ** 3 + 16 * x ** 2 + 20 * x - 8) * Dx ** 2
        + (-4 * x ** 5 - 10 * x ** 4 - 4 * x ** 3 + 4 * x ** 2 - 2 * x + 8) * Dx ** 3
        + (2 * x ** 5 + 4 * x ** 4 - 2 * x ** 3 - 7 * x ** 2 + 2 * x + 5) * Dx ** 4,
        x,
        0,
        [0, 1, 4, -1],
    )
    assert p == q
    p = from_sympy(x * exp(x) + cos(x) + 1)
    q = HolonomicFunction(
        (-x - 3) * Dx + (x + 2) * Dx ** 2 + (-x - 3) * Dx ** 3 + (x + 2) * Dx ** 4, x, 0, [2, 1, 1, 3]
    )
    assert p == q
    assert (x * exp(x) + cos(x) + 1).series(n=10) == p.series(n=10)
    p = from_sympy(log(1 + x) ** 2 + 1)
    q = HolonomicFunction(Dx + (3 * x + 3) * Dx ** 2 + (x ** 2 + 2 * x + 1) * Dx ** 3, x, 0, [1, 0, 2])
    assert p == q
    p = from_sympy(erf(x) ** 2 + x)
    q = HolonomicFunction(
        (32 * x ** 4 - 8 * x ** 2 + 8) * Dx ** 2 + (24 * x ** 3 - 2 * x) * Dx ** 3 + (4 * x ** 2 + 1) * Dx ** 4,
        x,
        0,
        [0, 1, 8 / pi, 0],
    )
    assert p == q
    p = from_sympy(cosh(x) * x)
    q = HolonomicFunction((-x ** 2 + 2) - 2 * x * Dx + x ** 2 * Dx ** 2, x, 0, [0, 1])
    assert p == q
    p = from_sympy(besselj(2, x))
    q = HolonomicFunction((x ** 2 - 4) + x * Dx + x ** 2 * Dx ** 2, x, 0, [0, 0])
    assert p == q
    p = from_sympy(besselj(0, x) + exp(x))
    q = HolonomicFunction(
        (-2 * x ** 2 - x + 1)
        + (2 * x ** 2 - x - 3) * Dx
        + (-2 * x ** 2 + x + 2) * Dx ** 2
        + (2 * x ** 2 + x) * Dx ** 3,
        x,
        0,
        [2, 1, 1 / 2],
    )
    assert p == q

# 需要导入模块: from sympy import QQ [as 别名]
# 或者: from sympy.QQ import old_poly_ring [as 别名]
def test_evalf():
    x = symbols('x')
    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
    # a straight line on real axis
    r = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1]
    p = HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1])
    s = '0.699525841805253'
    assert sstr(p.evalf(r)[-1]) == s
    # a traingle with vertices (0, 1+i, 2)
    r = [0.1 + 0.1*I]
    for i in range(9):
        r.append(r[-1]+0.1+0.1*I)
    for i in range(10):
        r.append(r[-1]+0.1-0.1*I)
    s = '1.07530466271334 - 0.0251200594793912*I'
    assert sstr(p.evalf(r)[-1]) == s
    p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1])
    s = '0.905546532085401 - 6.93889390390723e-18*I'
    assert sstr(p.evalf(r)[-1]) == s
    # a rectangular path (0 -> i -> 2+i -> 2)
    r = [0.1*I]
    for i in range(9):
        r.append(r[-1]+0.1*I)
    for i in range(20):
        r.append(r[-1]+0.1)
    for i in range(10):
        r.append(r[-1]-0.1*I)
    p = HolonomicFunction(Dx**2 + 1, x, 0, [1,1]).evalf(r)
    s = '0.501421652861245 - 3.88578058618805e-16*I'
    assert sstr(p[-1]) == s

# 需要导入模块: from sympy import QQ [as 别名]
# 或者: from sympy.QQ import old_poly_ring [as 别名]
def test_to_hyper():
    x = symbols('x')
    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
    p = HolonomicFunction(Dx - 2, x, 0, [3]).to_hyper()
    q = 3 * hyper([], [], 2*x)
    assert p == q
    p = hyperexpand(HolonomicFunction((1 + x) * Dx - 3, x, 0, [2]).to_hyper()).expand()
    q = 2*x**3 + 6*x**2 + 6*x + 2
    assert p == q
    p = HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1]).to_hyper()
    q = -x**2*hyper((2, 2, 1), (2, 3), -x)/2 + x
    assert p == q
    p = HolonomicFunction(2*x*Dx + Dx**2, x, 0, [0, 2/sqrt(pi)]).to_hyper()
    q = 2*x*hyper((1/2,), (3/2,), -x**2)/sqrt(pi)
    assert p == q
    p = hyperexpand(HolonomicFunction(2*x*Dx + Dx**2, x, 0, [1, -2/sqrt(pi)]).to_hyper())
    q = erfc(x)
    assert p.rewrite(erfc) == q
    p =  hyperexpand(HolonomicFunction((x**2 - 1) + x*Dx + x**2*Dx**2,
        x, 0, [0, S(1)/2]).to_hyper())
    q = besselj(1, x)
    assert p == q
    p = hyperexpand(HolonomicFunction(x*Dx**2 + Dx + x, x, 0, [1, 0]).to_hyper())
    q = besselj(0, x)
    assert p == q

# 需要导入模块: from sympy import QQ [as 别名]
# 或者: from sympy.QQ import old_poly_ring [as 别名]
def test_addition_initial_condition():
    x = symbols('x')
    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
    p = HolonomicFunction(Dx-1, x, 0, 3)
    q = HolonomicFunction(Dx**2+1, x, 0, [1, 0])
    r = HolonomicFunction(-1 + Dx - Dx**2 + Dx**3, x, 0, [4, 3, 2])
    assert p + q == r
    p = HolonomicFunction(Dx - x + Dx**2, x, 0, [1, 2])
    q = HolonomicFunction(Dx**2 + x, x, 0, [1, 0])
    r = HolonomicFunction((-x**4 - x**3/4 - x**2 + 1/4) + (x**3 + x**2/4 + 3*x/4 + 1)*Dx + \
        (-3*x/2 + 7/4)*Dx**2 + (x**2 - 7*x/4 + 1/4)*Dx**3 + (x**2 + x/4 + 1/2)*Dx**4, x, 0, [2, 2, -2, 2])
    assert p + q == r
    p = HolonomicFunction(Dx**2 + 4*x*Dx + x**2, x, 0, [3, 4])
    q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1])
    r = HolonomicFunction((x**6 + 2*x**4 - 5*x**2 - 6) + (4*x**5 + 36*x**3 - 32*x)*Dx + \
         (x**6 + 3*x**4 + 5*x**2 - 9)*Dx**2 + (4*x**5 + 36*x**3 - 32*x)*Dx**3 + (x**4 + \
            10*x**2 - 3)*Dx**4, x, 0, [4, 5, -1, -17])
    assert p + q == r
    q = HolonomicFunction(Dx**3 + x, x, 2, [3, 0, 1])
    p = HolonomicFunction(Dx - 1, x, 2, [1])
    r = HolonomicFunction((-x**2 - x + 1) + (x**2 + x)*Dx + (-x - 2)*Dx**3 + \
        (x + 1)*Dx**4, x, 2, [4, 1, 2, -5 ])
    assert p + q == r
    p = from_sympy(sin(x))
    q = from_sympy(1/x)
    r = HolonomicFunction((x**2 + 6) + (x**3 + 2*x)*Dx + (x**2 + 6)*Dx**2 + (x**3 + 2*x)*Dx**3, \
        x, 1, [sin(1) + 1, -1 + cos(1), -sin(1) + 2])
    assert p + q == r

# 需要导入模块: from sympy import QQ [as 别名]
# 或者: from sympy.QQ import old_poly_ring [as 别名]
def test_addition_initial_condition():
    x = symbols('x')
    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
    p = HolonomicFunction(Dx-1, x, 0, [3])
    q = HolonomicFunction(Dx**2+1, x, 0, [1, 0])
    r = HolonomicFunction(-1 + Dx - Dx**2 + Dx**3, x, 0, [4, 3, 2])
    assert p + q == r
    p = HolonomicFunction(Dx - x + Dx**2, x, 0, [1, 2])
    q = HolonomicFunction(Dx**2 + x, x, 0, [1, 0])
    r = HolonomicFunction((-x**4 - x**3/4 - x**2 + 1/4) + (x**3 + x**2/4 + 3*x/4 + 1)*Dx + \
        (-3*x/2 + 7/4)*Dx**2 + (x**2 - 7*x/4 + 1/4)*Dx**3 + (x**2 + x/4 + 1/2)*Dx**4, x, 0, [2, 2, -2, 2])
    assert p + q == r
    p = HolonomicFunction(Dx**2 + 4*x*Dx + x**2, x, 0, [3, 4])
    q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1])
    r = HolonomicFunction((x**6 + 2*x**4 - 5*x**2 - 6) + (4*x**5 + 36*x**3 - 32*x)*Dx + \
         (x**6 + 3*x**4 + 5*x**2 - 9)*Dx**2 + (4*x**5 + 36*x**3 - 32*x)*Dx**3 + (x**4 + \
            10*x**2 - 3)*Dx**4, x, 0, [4, 5, -1, -17])
    assert p + q == r
    q = HolonomicFunction(Dx**3 + x, x, 2, [3, 0, 1])
    p = HolonomicFunction(Dx - 1, x, 2, [1])
    r = HolonomicFunction((-x**2 - x + 1) + (x**2 + x)*Dx + (-x - 2)*Dx**3 + \
        (x + 1)*Dx**4, x, 2, [4, 1, 2, -5 ])
    assert p + q == r
    p = expr_to_holonomic(sin(x))
    q = expr_to_holonomic(1/x, x0=1)
    r = HolonomicFunction((x**2 + 6) + (x**3 + 2*x)*Dx + (x**2 + 6)*Dx**2 + (x**3 + 2*x)*Dx**3, \
        x, 1, [sin(1) + 1, -1 + cos(1), -sin(1) + 2])
    assert p + q == r
    C_1 = symbols('C_1')
    p = expr_to_holonomic(sqrt(x))
    q = expr_to_holonomic(sqrt(x**2-x))
    r = (p + q).to_expr().subs(C_1, -I/2).expand()
    assert r == I*sqrt(x)*sqrt(-x + 1) + sqrt(x)

# 需要导入模块: from sympy import QQ [as 别名]
# 或者: from sympy.QQ import old_poly_ring [as 别名]
def test_to_Sequence_Initial_Coniditons():
    x = symbols('x')
    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
    n = symbols('n', integer=True)
    _, Sn = RecurrenceOperators(QQ.old_poly_ring(n), 'Sn')
    p = HolonomicFunction(Dx - 1, x, 0, 1).to_sequence()
    q = HolonomicSequence(-1 + (n + 1)*Sn, 1)
    assert p == q
    p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]).to_sequence()
    q = HolonomicSequence(1 + (n**2 + 3*n + 2)*Sn**2, [0, 1])
    assert p == q
    p = HolonomicFunction(Dx**2 + 1 + x**3*Dx, x, 0, [2, 3]).to_sequence()
    q = HolonomicSequence(n + Sn**2 + (n**2 + 7*n + 12)*Sn**4, [2, 3, -1, -1/2])
    assert p == q
    p = HolonomicFunction(x**3*Dx**5 + 1 + Dx, x).to_sequence()
    q = HolonomicSequence(1 + (n + 1)*Sn + (n**5 - 5*n**3 + 4*n)*Sn**2)
    assert p == q

# 需要导入模块: from sympy import QQ [as 别名]
# 或者: from sympy.QQ import old_poly_ring [as 别名]
def test_properties():
    R = QQ.old_poly_ring(x, y)
    F = R.free_module(2)
    h = homomorphism(F, F, [[x, 0], [y, 0]])
    assert h.kernel() == F.submodule([-y, x])
    assert h.image() == F.submodule([x, 0], [y, 0])
    assert not h.is_injective()
    assert not h.is_surjective()
    assert h.restrict_codomain(h.image()).is_surjective()
    assert h.restrict_domain(F.submodule([1, 0])).is_injective()
    assert h.quotient_domain(
        h.kernel()).restrict_codomain(h.image()).is_isomorphism()

    R2 = QQ.old_poly_ring(x, y, order=(("lex", x), ("ilex", y))) / [x**2 + 1]
    F = R2.free_module(2)
    h = homomorphism(F, F, [[x, 0], [y, y + 1]])
    assert h.is_isomorphism()

# 需要导入模块: from sympy import QQ [as 别名]
# 或者: from sympy.QQ import old_poly_ring [as 别名]
def test_expr_to_holonomic():
    x = symbols('x')
    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
    p = expr_to_holonomic((sin(x)/x)**2)
    q = HolonomicFunction(8*x + (4*x**2 + 6)*Dx + 6*x*Dx**2 + x**2*Dx**3, x, 0, \
        [1, 0, -2/3])
    assert p == q
    p = expr_to_holonomic(1/(1+x**2)**2)
    q = HolonomicFunction(4*x + (x**2 + 1)*Dx, x, 0, 1)
    assert p == q
    p = expr_to_holonomic(exp(x)*sin(x)+x*log(1+x))
    q = HolonomicFunction((2*x**3 + 10*x**2 + 20*x + 18) + (-2*x**4 - 10*x**3 - 20*x**2 \
        - 18*x)*Dx + (2*x**5 + 6*x**4 + 7*x**3 + 8*x**2 + 10*x - 4)*Dx**2 + \
        (-2*x**5 - 5*x**4 - 2*x**3 + 2*x**2 - x + 4)*Dx**3 + (x**5 + 2*x**4 - x**3 - \
        7*x**2/2 + x + 5/2)*Dx**4, x, 0, [0, 1, 4, -1])
    assert p == q
    p = expr_to_holonomic(x*exp(x)+cos(x)+1)
    q = HolonomicFunction((-x - 3)*Dx + (x + 2)*Dx**2 + (-x - 3)*Dx**3 + (x + 2)*Dx**4, x, \
        0, [2, 1, 1, 3])
    assert p == q
    assert (x*exp(x)+cos(x)+1).series(n=10) == p.series(n=10)
    p = expr_to_holonomic(log(1 + x)**2 + 1)
    q = HolonomicFunction(Dx + (3*x + 3)*Dx**2 + (x**2 + 2*x + 1)*Dx**3, x, 0, [1, 0, 2])
    assert p == q
    p = expr_to_holonomic(erf(x)**2 + x)
    q = HolonomicFunction((8*x**4 - 2*x**2 + 2)*Dx**2 + (6*x**3 - x/2)*Dx**3 + \
        (x**2+ 1/4)*Dx**4, x, 0, [0, 1, 8/pi, 0])
    assert p == q
    p = expr_to_holonomic(cosh(x)*x)
    q = HolonomicFunction((-x**2 + 2) -2*x*Dx + x**2*Dx**2, x, 0, [0, 1])
    assert p == q
    p = expr_to_holonomic(besselj(2, x))
    q = HolonomicFunction((x**2 - 4) + x*Dx + x**2*Dx**2, x, 0, [0, 0])
    assert p == q
    p = expr_to_holonomic(besselj(0, x) + exp(x))
    q = HolonomicFunction((-x**2 - x/2 + 1/2) + (x**2 - x/2 - 3/2)*Dx + (-x**2 + x/2 + 1)*Dx**2 +\
        (x**2 + x/2)*Dx**3, x, 0, [2, 1, 1/2])
    assert p == q
    p = expr_to_holonomic(sin(x)**2/x)
    q = HolonomicFunction(4 + 4*x*Dx + 3*Dx**2 + x*Dx**3, x, 0, [0, 1, 0])
    assert p == q
    p = expr_to_holonomic(sin(x)**2/x, x0=2)
    q = HolonomicFunction((4) + (4*x)*Dx + (3)*Dx**2 + (x)*Dx**3, x, 2, [sin(2)**2/2,
        sin(2)*cos(2) - sin(2)**2/4, -3*sin(2)**2/4 + cos(2)**2 - sin(2)*cos(2)])
    assert p == q
    p = expr_to_holonomic(log(x)/2 - Ci(2*x)/2 + Ci(2)/2)
    q = HolonomicFunction(4*Dx + 4*x*Dx**2 + 3*Dx**3 + x*Dx**4, x, 0, \
        [-log(2)/2 - EulerGamma/2 + Ci(2)/2, 0, 1, 0])
    assert p == q
    p = p.to_expr()
    q = log(x)/2 - Ci(2*x)/2 + Ci(2)/2
    assert p == q
    p = expr_to_holonomic(x**(S(1)/2), x0=1)
    q = HolonomicFunction(x*Dx - 1/2, x, 1, 1)
    assert p == q
    p = expr_to_holonomic(sqrt(1 + x**2))
    q = HolonomicFunction((-x) + (x**2 + 1)*Dx, x, 0, 1)
    assert p == q

# 需要导入模块: from sympy import QQ [as 别名]
# 或者: from sympy.QQ import old_poly_ring [as 别名]
def test_RecurrenceOperator():
    n = symbols('n', integer=True)
    R, Sn = RecurrenceOperators(QQ.old_poly_ring(n), 'Sn')
    assert Sn*n == (n + 1)*Sn
    assert Sn*n**2 == (n**2+1+2*n)*Sn
    assert Sn**2*n**2 == (n**2 + 4*n + 4)*Sn**2
    p = (Sn**3*n**2 + Sn*n)**2
    q = (n**2 + 3*n + 2)*Sn**2 + (2*n**3 + 19*n**2 + 57*n + 52)*Sn**4 + (n**4 + 18*n**3 + \
        117*n**2 + 324*n + 324)*Sn**6
    assert p == q

# 需要导入模块: from sympy import QQ [as 别名]
# 或者: from sympy.QQ import old_poly_ring [as 别名]
def test_printing():
    R = QQ.old_poly_ring(x)

    assert str(homomorphism(R.free_module(1), R.free_module(1), [0])) == \
        'Matrix([[0]]) : QQ[x]**1 -> QQ[x]**1'
    assert str(homomorphism(R.free_module(2), R.free_module(2), [0, 0])) == \
        'Matrix([                       \n[0, 0], : QQ[x]**2 -> QQ[x]**2\n[0, 0]])                       '
    assert str(homomorphism(R.free_module(1), R.free_module(1) / [[x]], [0])) == \
        'Matrix([[0]]) : QQ[x]**1 -> QQ[x]**1/<[x]>'
    assert str(R.free_module(0).identity_hom()) == 'Matrix(0, 0, []) : QQ[x]**0 -> QQ[x]**0'