Python sympy.rf函数代码示例

def test_special_products():
    # Wallis product
    assert product((4*k)**2 / (4*k**2 - 1), (k, 1, n)) == \
        4**n*factorial(n)**2/rf(Rational(1, 2), n)/rf(Rational(3, 2), n)

    # Euler's product formula for sin
    assert product(1 + a/k**2, (k, 1, n)) == \
        rf(1 - sqrt(-a), n)*rf(1 + sqrt(-a), n)/factorial(n)**2

def test_rf_ff_eval_hiprec():
    maple = Float('6.9109401292234329956525265438452')
    us = ff(18, S(2)/3).evalf(32)
    assert abs(us - maple)/us < 1e-31

    maple = Float('6.8261540131125511557924466355367')
    us = rf(18, S(2)/3).evalf(32)
    assert abs(us - maple)/us < 1e-31

    maple = Float('34.007346127440197150854651814225')
    us = rf(Float('4.4', 32), Float('2.2', 32));
    assert abs(us - maple)/us < 1e-31

def test_rf_lambdify_mpmath():
    from sympy import lambdify
    x, y = symbols('x,y')
    f = lambdify((x,y), rf(x, y), 'mpmath')
    maple = Float('34.007346127440197')
    us = f(4.4, 2.2)
    assert abs(us - maple)/us < 1e-15

def test_issue_9699():
    n, k = symbols('n k', real=True)
    x, y = symbols('x, y')
    assert combsimp((n + 1)*factorial(n)) == factorial(n + 1)
    assert combsimp((x + 1)*factorial(x)/gamma(y)) == gamma(x + 2)/gamma(y)
    assert combsimp(factorial(n)/n) == factorial(n - 1)
    assert combsimp(rf(x + n, k)*binomial(n, k)) == binomial(n, k)*gamma(k + n + x)/gamma(n + x)

def test_simple_products():
    assert product(2, (k, a, n)) == 2**(n - a + 1)
    assert product(k, (k, 1, n)) == factorial(n)
    assert product(k**3, (k, 1, n)) == factorial(n)**3

    assert product(k + 1, (k, 0, n - 1)) == factorial(n)
    assert product(k + 1, (k, a, n - 1)) == rf(1 + a, n - a)

    assert product(cos(k), (k, 0, 5)) == cos(1)*cos(2)*cos(3)*cos(4)*cos(5)
    assert product(cos(k), (k, 3, 5)) == cos(3)*cos(4)*cos(5)
    assert product(cos(k), (k, 1, Rational(5, 2))) != cos(1)*cos(2)

    assert isinstance(product(k**k, (k, 1, n)), Product)

    assert Product(x**k, (k, 1, n)).variables == [k]

    raises(ValueError, lambda: Product(n))
    raises(ValueError, lambda: Product(n, k))
    raises(ValueError, lambda: Product(n, k, 1))
    raises(ValueError, lambda: Product(n, k, 1, 10))
    raises(ValueError, lambda: Product(n, (k, 1)))

    assert product(1, (n, 1, oo)) == 1  # issue 8301
    assert product(2, (n, 1, oo)) == oo
    assert product(-1, (n, 1, oo)).func is Product

def test_ff_eval_apply():
    x, y = symbols('x,y')
    n, k = symbols('n k', integer=True)
    m = Symbol('m', integer=True, nonnegative=True)

    assert ff(nan, y) == nan
    assert ff(x, nan) == nan

    assert ff(x, y) == ff(x, y)

    assert ff(oo, 0) == 1
    assert ff(-oo, 0) == 1

    assert ff(oo, 6) == oo
    assert ff(-oo, 7) == -oo

    assert ff(oo, -6) == oo
    assert ff(-oo, -7) == oo

    assert ff(x, 0) == 1
    assert ff(x, 1) == x
    assert ff(x, 2) == x*(x - 1)
    assert ff(x, 3) == x*(x - 1)*(x - 2)
    assert ff(x, 5) == x*(x - 1)*(x - 2)*(x - 3)*(x - 4)

    assert ff(x, -1) == 1/(x + 1)
    assert ff(x, -2) == 1/((x + 1)*(x + 2))
    assert ff(x, -3) == 1/((x + 1)*(x + 2)*(x + 3))

    assert ff(100, 100) == factorial(100)

    assert ff(2*x**2 - 5*x, 2) == (2*x**2  - 5*x)*(2*x**2 - 5*x - 1)
    assert isinstance(ff(2*x**2 - 5*x, 2), Mul)
    assert ff(x**2 + 3*x, -2) == 1/((x**2 + 3*x + 1)*(x**2 + 3*x + 2))

    assert ff(Poly(2*x**2 - 5*x, x), 2) == Poly(4*x**4 - 28*x**3 + 59*x**2 - 35*x, x)
    assert isinstance(ff(Poly(2*x**2 - 5*x, x), 2), Poly)
    raises(ValueError, lambda: ff(Poly(2*x**2 - 5*x, x, y), 2))
    assert ff(Poly(x**2 + 3*x, x), -2) == 1/(x**4 + 12*x**3 + 49*x**2 + 78*x + 40)
    raises(ValueError, lambda: ff(Poly(x**2 + 3*x, x, y), -2))


    assert ff(x, m).is_integer is None
    assert ff(n, k).is_integer is None
    assert ff(n, m).is_integer is True
    assert ff(n, k + pi).is_integer is False
    assert ff(n, m + pi).is_integer is False
    assert ff(pi, m).is_integer is False

    assert isinstance(ff(x, x), ff)
    assert ff(n, n) == factorial(n)

    assert ff(x, k).rewrite(rf) == rf(x - k + 1, k)
    assert ff(x, k).rewrite(gamma) == (-1)**k*gamma(k - x) / gamma(-x)
    assert ff(n, k).rewrite(factorial) == factorial(n) / factorial(n - k)
    assert ff(x, k).rewrite(binomial) == factorial(k) * binomial(x, k)

def test_simple_products():
    assert product(2, (k, a, n)) == 2**(n-a+1)
    assert product(k, (k, 1, n)) == factorial(n)
    assert product(k**3, (k, 1, n)) == factorial(n)**3

    assert product(k+1, (k, 0, n-1)) == factorial(n)
    assert product(k+1, (k, a, n-1)) == rf(1+a, n-a)

    assert product(cos(k), (k, 0, 5)) == cos(1)*cos(2)*cos(3)*cos(4)*cos(5)
    assert product(cos(k), (k, 3, 5)) == cos(3)*cos(4)*cos(5)
    assert product(cos(k), (k, 1, Rational(5, 2))) == cos(1)*cos(2)

    assert isinstance(product(k**k, (k, 1, n)), Product)

def test_rsolve_hyper():
    assert rsolve_hyper([-1, -1, 1], 0, n) in [
        C0*(S.Half - S.Half*sqrt(5))**n + C1*(S.Half + S.Half*sqrt(5))**n,
        C1*(S.Half - S.Half*sqrt(5))**n + C0*(S.Half + S.Half*sqrt(5))**n,
    ]

    assert rsolve_hyper([n**2 - 2, -2*n - 1, 1], 0, n) in [
        C0*rf(sqrt(2), n) + C1*rf(-sqrt(2), n),
        C1*rf(sqrt(2), n) + C0*rf(-sqrt(2), n),
    ]

    assert rsolve_hyper([n**2 - k, -2*n - 1, 1], 0, n) in [
        C0*rf(sqrt(k), n) + C1*rf(-sqrt(k), n),
        C1*rf(sqrt(k), n) + C0*rf(-sqrt(k), n),
    ]

    assert rsolve_hyper(
        [2*n*(n + 1), -n**2 - 3*n + 2, n - 1], 0, n) == C1*factorial(n) + C0*2**n

    assert rsolve_hyper(
        [n + 2, -(2*n + 3)*(17*n**2 + 51*n + 39), n + 1], 0, n) == None

    assert rsolve_hyper([-n - 1, -1, 1], 0, n) == None

    assert rsolve_hyper([-1, 1], n, n).expand() == C0 + n**2/2 - n/2

    assert rsolve_hyper([-1, 1], 1 + n, n).expand() == C0 + n**2/2 + n/2

    assert rsolve_hyper([-1, 1], 3*(n + n**2), n).expand() == C0 + n**3 - n

    assert rsolve_hyper([-a, 1],0,n).expand() == C0*a**n

    assert rsolve_hyper([-a, 0, 1], 0, n).expand() == (-1)**n*C1*a**(n/2) + C0*a**(n/2)

    assert rsolve_hyper([1, 1, 1], 0, n).expand() == \
        C0*(-S(1)/2 - sqrt(3)*I/2)**n + C1*(-S(1)/2 + sqrt(3)*I/2)**n

    assert rsolve_hyper([1, -2*n/a - 2/a, 1], 0, n) == None

def test_rf_eval_apply():
    x, y = symbols('x,y')
    n, k = symbols('n k', integer=True)
    m = Symbol('m', integer=True, nonnegative=True)

    assert rf(nan, y) == nan
    assert rf(x, nan) == nan

    assert rf(x, y) == rf(x, y)

    assert rf(oo, 0) == 1
    assert rf(-oo, 0) == 1

    assert rf(oo, 6) == oo
    assert rf(-oo, 7) == -oo

    assert rf(oo, -6) == oo
    assert rf(-oo, -7) == oo

    assert rf(x, 0) == 1
    assert rf(x, 1) == x
    assert rf(x, 2) == x*(x + 1)
    assert rf(x, 3) == x*(x + 1)*(x + 2)
    assert rf(x, 5) == x*(x + 1)*(x + 2)*(x + 3)*(x + 4)

    assert rf(x, -1) == 1/(x - 1)
    assert rf(x, -2) == 1/((x - 1)*(x - 2))
    assert rf(x, -3) == 1/((x - 1)*(x - 2)*(x - 3))

    assert rf(1, 100) == factorial(100)

    assert rf(x**2 + 3*x, 2) == (x**2 + 3*x)*(x**2 + 3*x + 1)
    assert isinstance(rf(x**2 + 3*x, 2), Mul)
    assert rf(x**3 + x, -2) == 1/((x**3 + x - 1)*(x**3 + x - 2))

    assert rf(Poly(x**2 + 3*x, x), 2) == Poly(x**4 + 8*x**3 + 19*x**2 + 12*x, x)
    assert isinstance(rf(Poly(x**2 + 3*x, x), 2), Poly)
    raises(ValueError, lambda: rf(Poly(x**2 + 3*x, x, y), 2))
    assert rf(Poly(x**3 + x, x), -2) == 1/(x**6 - 9*x**5 + 35*x**4 - 75*x**3 + 94*x**2 - 66*x + 20)
    raises(ValueError, lambda: rf(Poly(x**3 + x, x, y), -2))

    assert rf(x, m).is_integer is None
    assert rf(n, k).is_integer is None
    assert rf(n, m).is_integer is True
    assert rf(n, k + pi).is_integer is False
    assert rf(n, m + pi).is_integer is False
    assert rf(pi, m).is_integer is False

    assert rf(x, k).rewrite(ff) == ff(x + k - 1, k)
    assert rf(x, k).rewrite(binomial) == factorial(k)*binomial(x + k - 1, k)
    assert rf(n, k).rewrite(factorial) == \
        factorial(n + k - 1) / factorial(n - 1)

def test_rf_eval_apply():
    x, y = symbols('x,y')

    assert rf(nan, y) == nan

    assert rf(x, y) == rf(x, y)

    assert rf(oo, 0) == 1
    assert rf(-oo, 0) == 1

    assert rf(oo, 6) == oo
    assert rf(-oo, 7) == -oo

    assert rf(oo, -6) == oo
    assert rf(-oo, -7) == oo

    assert rf(x, 0) == 1
    assert rf(x, 1) == x
    assert rf(x, 2) == x*(x + 1)
    assert rf(x, 3) == x*(x + 1)*(x + 2)
    assert rf(x, 5) == x*(x + 1)*(x + 2)*(x + 3)*(x + 4)

    assert rf(x, -1) == 1/(x - 1)
    assert rf(x, -2) == 1/((x - 1)*(x - 2))
    assert rf(x, -3) == 1/((x - 1)*(x - 2)*(x - 3))

    assert rf(1, 100) == factorial(100)

def test_rf_eval_apply():
    x, y = symbols('x,y')

    assert rf(nan, y) == nan

    assert rf(x, y) == rf(x, y)

    assert rf(oo, 0) == 1
    assert rf(-oo, 0) == 1

    assert rf(oo, 6) == oo
    assert rf(-oo, 7) == -oo

    assert rf(oo, -6) == oo
    assert rf(-oo, -7) == oo

    assert rf(x, 0) == 1
    assert rf(x, 1) == x
    assert rf(x, 2) == x*(x + 1)
    assert rf(x, 3) == x*(x + 1)*(x + 2)
    assert rf(x, 5) == x*(x + 1)*(x + 2)*(x + 3)*(x + 4)

    assert rf(x, -1) == 1/(x - 1)
    assert rf(x, -2) == 1/((x - 1)*(x - 2))
    assert rf(x, -3) == 1/((x - 1)*(x - 2)*(x - 3))

    assert rf(1, 100) == factorial(100)

    n = Symbol('n', integer=True)
    k = Symbol('k', integer=True)
    m = Symbol('m', integer=True, nonnegative=True)
    assert rf(x, m).is_integer is None
    assert rf(n, k).is_integer is None
    assert rf(n, m).is_integer is True
    assert rf(n, k + pi).is_integer is False
    assert rf(n, m + pi).is_integer is False
    assert rf(pi, m).is_integer is False

    def to_sequence(self):
        """
        Finds the recurrence relation in power series expansion
        of the function.

        Examples
        ========

        >>> from sympy.holonomic.holonomic import HolonomicFunction, DifferentialOperators
        >>> from sympy.polys.domains import ZZ, QQ
        >>> from sympy import symbols
        >>> x = symbols('x')
        >>> R, Dx = DifferentialOperators(QQ.old_poly_ring(x),'Dx')

        >>> HolonomicFunction(Dx - 1, x, 0, [1]).to_sequence()
        HolonomicSequence((-1) + (n + 1)Sn, n), u(0) = 1

        See Also
        ========

        HolonomicFunction.series

        References
        ==========

        hal.inria.fr/inria-00070025/document
        """

        dict1 = {}
        n = symbols('n', integer=True)
        dom = self.annihilator.parent.base.dom
        R, _ = RecurrenceOperators(dom.old_poly_ring(n), 'Sn')

        for i, j in enumerate(self.annihilator.listofpoly):
            listofdmp = j.all_coeffs()
            degree = len(listofdmp) - 1
            for k in range(degree + 1):
                coeff = listofdmp[degree - k]
                if coeff == 0:
                    continue
                if i - k in dict1:
                    dict1[i - k] += (coeff * rf(n - k + 1, i))
                else:
                    dict1[i - k] = (coeff * rf(n - k + 1, i))

        sol = []
        lower = min(dict1.keys())
        upper = max(dict1.keys())

        for j in range(lower, upper + 1):
            if j in dict1.keys():
                sol.append(dict1[j].subs(n, n - lower))
            else:
                sol.append(S(0))
        # recurrence relation
        sol = RecurrenceOperator(sol, R)

        if not self._have_init_cond:
            return HolonomicSequence(sol)
        if self.x0 != 0:
            return HolonomicSequence(sol)
        # computing the initial conditions for recurrence
        order = sol.order
        all_roots = roots(sol.listofpoly[-1].rep, filter='Z')
        all_roots = all_roots.keys()

        if all_roots:
            max_root = max(all_roots)
            if max_root >= 0:
                order += max_root + 1

        y0 = _extend_y0(self, order)
        u0 = []
        # u(n) = y^n(0)/factorial(n)
        for i, j in enumerate(y0):
            u0.append(j / factorial(i))

        return HolonomicSequence(sol, u0)

def test_rf_eval_apply():
    x, y = symbols("x,y")
    n, k = symbols("n k", integer=True)
    m = Symbol("m", integer=True, nonnegative=True)

    assert rf(nan, y) == nan
    assert rf(x, nan) == nan

    assert rf(x, y) == rf(x, y)

    assert rf(oo, 0) == 1
    assert rf(-oo, 0) == 1

    assert rf(oo, 6) == oo
    assert rf(-oo, 7) == -oo

    assert rf(oo, -6) == oo
    assert rf(-oo, -7) == oo

    assert rf(x, 0) == 1
    assert rf(x, 1) == x
    assert rf(x, 2) == x * (x + 1)
    assert rf(x, 3) == x * (x + 1) * (x + 2)
    assert rf(x, 5) == x * (x + 1) * (x + 2) * (x + 3) * (x + 4)

    assert rf(x, -1) == 1 / (x - 1)
    assert rf(x, -2) == 1 / ((x - 1) * (x - 2))
    assert rf(x, -3) == 1 / ((x - 1) * (x - 2) * (x - 3))

    assert rf(1, 100) == factorial(100)

    assert rf(x ** 2 + 3 * x, 2) == x ** 4 + 8 * x ** 3 + 19 * x ** 2 + 12 * x
    assert rf(x ** 3 + x, -2) == 1 / (x ** 6 - 9 * x ** 5 + 35 * x ** 4 - 75 * x ** 3 + 94 * x ** 2 - 66 * x + 20)

    assert rf(x, m).is_integer is None
    assert rf(n, k).is_integer is None
    assert rf(n, m).is_integer is True
    assert rf(n, k + pi).is_integer is False
    assert rf(n, m + pi).is_integer is False
    assert rf(pi, m).is_integer is False

    assert rf(x, k).rewrite(ff) == ff(x + k - 1, k)
    assert rf(x, k).rewrite(binomial) == factorial(k) * binomial(x + k - 1, k)
    assert rf(n, k).rewrite(factorial) == factorial(n + k - 1) / factorial(n - 1)

def test_F1():
    assert rf(x, 3) == x*(1 + x)*(2 + x)

def test_rational_products():
    assert product(1 + 1/k, (k, 1, n)) == rf(2, n)/factorial(n)