MTtranslateService/Lib/site-packages/sympy/solvers/tests/test_recurr.py

from sympy.core.function import (Function, Lambda, expand)
from sympy.core.numbers import (I, Rational)
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.combinatorial.factorials import (rf, binomial, factorial)
from sympy.functions.elementary.complexes import Abs
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (cos, sin)
from sympy.polys.polytools import factor
from sympy.solvers.recurr import rsolve, rsolve_hyper, rsolve_poly, rsolve_ratio
from sympy.testing.pytest import raises, slow
from sympy.abc import a, b

y = Function('y')
n, k = symbols('n,k', integer=True)
C0, C1, C2 = symbols('C0,C1,C2')


def test_rsolve_poly():
    assert rsolve_poly([-1, -1, 1], 0, n) == 0
    assert rsolve_poly([-1, -1, 1], 1, n) == -1

    assert rsolve_poly([-1, n + 1], n, n) == 1
    assert rsolve_poly([-1, 1], n, n) == C0 + (n**2 - n)/2
    assert rsolve_poly([-n - 1, n], 1, n) == C1*n - 1
    assert rsolve_poly([-4*n - 2, 1], 4*n + 1, n) == -1

    assert rsolve_poly([-1, 1], n**5 + n**3, n) == \
        C0 - n**3 / 2 - n**5 / 2 + n**2 / 6 + n**6 / 6 + 2*n**4 / 3


def test_rsolve_ratio():
    solution = rsolve_ratio([-2*n**3 + n**2 + 2*n - 1, 2*n**3 + n**2 - 6*n,
        -2*n**3 - 11*n**2 - 18*n - 9, 2*n**3 + 13*n**2 + 22*n + 8], 0, n)

    assert solution in [
        C1*((-2*n + 3)/(n**2 - 1))/3,
        (S.Half)*(C1*(-3 + 2*n)/(-1 + n**2)),
        (S.Half)*(C1*( 3 - 2*n)/( 1 - n**2)),
        (S.Half)*(C2*(-3 + 2*n)/(-1 + n**2)),
        (S.Half)*(C2*( 3 - 2*n)/( 1 - n**2)),
    ]


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*(Rational(-1, 2) - sqrt(3)*I/2)**n + C1*(Rational(-1, 2) + sqrt(3)*I/2)**n

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


def recurrence_term(c, f):
    """Compute RHS of recurrence in f(n) with coefficients in c."""
    return sum(c[i]*f.subs(n, n + i) for i in range(len(c)))


def test_rsolve_bulk():
    """Some bulk-generated tests."""
    funcs = [ n, n + 1, n**2, n**3, n**4, n + n**2, 27*n + 52*n**2 - 3*
        n**3 + 12*n**4 - 52*n**5 ]
    coeffs = [ [-2, 1], [-2, -1, 1], [-1, 1, 1, -1, 1], [-n, 1], [n**2 -
        n + 12, 1] ]
    for p in funcs:
        # compute difference
        for c in coeffs:
            q = recurrence_term(c, p)
            if p.is_polynomial(n):
                assert rsolve_poly(c, q, n) == p
            # See issue 3956:
            #if p.is_hypergeometric(n):
            #    assert rsolve_hyper(c, q, n) == p


def test_rsolve():
    f = y(n + 2) - y(n + 1) - y(n)
    h = sqrt(5)*(S.Half + S.Half*sqrt(5))**n \
        - sqrt(5)*(S.Half - S.Half*sqrt(5))**n

    assert rsolve(f, y(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(f, y(n), [0, 5]) == h
    assert rsolve(f, y(n), {0: 0, 1: 5}) == h
    assert rsolve(f, y(n), {y(0): 0, y(1): 5}) == h
    assert rsolve(y(n) - y(n - 1) - y(n - 2), y(n), [0, 5]) == h
    assert rsolve(Eq(y(n), y(n - 1) + y(n - 2)), y(n), [0, 5]) == h

    assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0

    f = (n - 1)*y(n + 2) - (n**2 + 3*n - 2)*y(n + 1) + 2*n*(n + 1)*y(n)
    g = C1*factorial(n) + C0*2**n
    h = -3*factorial(n) + 3*2**n

    assert rsolve(f, y(n)) == g
    assert rsolve(f, y(n), []) == g
    assert rsolve(f, y(n), {}) == g

    assert rsolve(f, y(n), [0, 3]) == h
    assert rsolve(f, y(n), {0: 0, 1: 3}) == h
    assert rsolve(f, y(n), {y(0): 0, y(1): 3}) == h

    assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0

    f = y(n) - y(n - 1) - 2

    assert rsolve(f, y(n), {y(0): 0}) == 2*n
    assert rsolve(f, y(n), {y(0): 1}) == 2*n + 1
    assert rsolve(f, y(n), {y(0): 0, y(1): 1}) is None

    assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0

    f = 3*y(n - 1) - y(n) - 1

    assert rsolve(f, y(n), {y(0): 0}) == -3**n/2 + S.Half
    assert rsolve(f, y(n), {y(0): 1}) == 3**n/2 + S.Half
    assert rsolve(f, y(n), {y(0): 2}) == 3*3**n/2 + S.Half

    assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0

    f = y(n) - 1/n*y(n - 1)
    assert rsolve(f, y(n)) == C0/factorial(n)
    assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0

    f = y(n) - 1/n*y(n - 1) - 1
    assert rsolve(f, y(n)) is None

    f = 2*y(n - 1) + (1 - n)*y(n)/n

    assert rsolve(f, y(n), {y(1): 1}) == 2**(n - 1)*n
    assert rsolve(f, y(n), {y(1): 2}) == 2**(n - 1)*n*2
    assert rsolve(f, y(n), {y(1): 3}) == 2**(n - 1)*n*3

    assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0

    f = (n - 1)*(n - 2)*y(n + 2) - (n + 1)*(n + 2)*y(n)

    assert rsolve(f, y(n), {y(3): 6, y(4): 24}) == n*(n - 1)*(n - 2)
    assert rsolve(
        f, y(n), {y(3): 6, y(4): -24}) == -n*(n - 1)*(n - 2)*(-1)**(n)

    assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0

    assert rsolve(Eq(y(n + 1), a*y(n)), y(n), {y(1): a}).simplify() == a**n

    assert rsolve(y(n) - a*y(n-2),y(n), \
            {y(1): sqrt(a)*(a + b), y(2): a*(a - b)}).simplify() == \
            a**(n/2)*(-(-1)**n*b + a)

    f = (-16*n**2 + 32*n - 12)*y(n - 1) + (4*n**2 - 12*n + 9)*y(n)

    yn = rsolve(f, y(n), {y(1): binomial(2*n + 1, 3)})
    sol = 2**(2*n)*n*(2*n - 1)**2*(2*n + 1)/12
    assert factor(expand(yn, func=True)) == sol

    sol = rsolve(y(n) + a*(y(n + 1) + y(n - 1))/2, y(n))
    Y = lambda i: sol.subs(n, i)
    assert (Y(n) + a*(Y(n + 1) + Y(n - 1))/2).expand().cancel() == 0

    assert rsolve((k + 1)*y(k), y(k)) is None
    assert (rsolve((k + 1)*y(k) + (k + 3)*y(k + 1) + (k + 5)*y(k + 2), y(k))
            is None)

    assert rsolve(y(n) + y(n + 1) + 2**n + 3**n, y(n)) == (-1)**n*C0 - 2**n/3 - 3**n/4


def test_rsolve_raises():
    x = Function('x')
    raises(ValueError, lambda: rsolve(y(n) - y(k + 1), y(n)))
    raises(ValueError, lambda: rsolve(y(n) - y(n + 1), x(n)))
    raises(ValueError, lambda: rsolve(y(n) - x(n + 1), y(n)))
    raises(ValueError, lambda: rsolve(y(n) - sqrt(n)*y(n + 1), y(n)))
    raises(ValueError, lambda: rsolve(y(n) - y(n + 1), y(n), {x(0): 0}))
    raises(ValueError, lambda: rsolve(y(n) + y(n + 1) + 2**n + cos(n), y(n)))


def test_issue_6844():
    f = y(n + 2) - y(n + 1) + y(n)/4
    assert rsolve(f, y(n)) == 2**(-n)*(C0 + C1*n)
    assert rsolve(f, y(n), {y(0): 0, y(1): 1}) == 2*2**(-n)*n


def test_issue_18751():
    r = Symbol('r', positive=True)
    theta = Symbol('theta', real=True)
    f = y(n) - 2 * r * cos(theta) * y(n - 1) + r**2 * y(n - 2)
    assert rsolve(f, y(n)) == \
        C0*(r*(cos(theta) - I*Abs(sin(theta))))**n + C1*(r*(cos(theta) + I*Abs(sin(theta))))**n

def test_constant_naming():
    #issue 8697
    assert rsolve(y(n+3) - y(n+2) - y(n+1) + y(n), y(n)) == (-1)**n*C0+C1+C2*n
    assert rsolve(y(n+3)+3*y(n+2)+3*y(n+1)+y(n), y(n)).expand() == C0*(-1)**n + (-1)**n*C1*n + (-1)**n*C2*n**2
    assert rsolve(y(n) - 2*y(n - 3) + 5*y(n - 2) - 4*y(n - 1),y(n),[1,3,8]) == 3*2**n - n - 2

    #issue 19630
    assert rsolve(y(n+3) - 3*y(n+1) + 2*y(n), y(n), {y(1):0, y(2):8, y(3):-2}) == (-2)**n + 2*n

@slow
def test_issue_15751():
    f = y(n) + 21*y(n + 1) - 273*y(n + 2) - 1092*y(n + 3) + 1820*y(n + 4) + 1092*y(n + 5) - 273*y(n + 6) - 21*y(n + 7) + y(n + 8)
    assert rsolve(f, y(n)) is not None


def test_issue_17990():
    f = -10*y(n) + 4*y(n + 1) + 6*y(n + 2) + 46*y(n + 3)
    sol = rsolve(f, y(n))
    expected = C0*((86*18**(S(1)/3)/69 + (-12 + (-1 + sqrt(3)*I)*(290412 +
        3036*sqrt(9165))**(S(1)/3))*(1 - sqrt(3)*I)*(24201 + 253*sqrt(9165))**
        (S(1)/3)/276)/((1 - sqrt(3)*I)*(24201 + 253*sqrt(9165))**(S(1)/3))
        )**n + C1*((86*18**(S(1)/3)/69 + (-12 + (-1 - sqrt(3)*I)*(290412 + 3036
        *sqrt(9165))**(S(1)/3))*(1 + sqrt(3)*I)*(24201 + 253*sqrt(9165))**
        (S(1)/3)/276)/((1 + sqrt(3)*I)*(24201 + 253*sqrt(9165))**(S(1)/3))
        )**n + C2*(-43*18**(S(1)/3)/(69*(24201 + 253*sqrt(9165))**(S(1)/3)) -
        S(1)/23 + (290412 + 3036*sqrt(9165))**(S(1)/3)/138)**n
    assert sol == expected
    e = sol.subs({C0: 1, C1: 1, C2: 1, n: 1}).evalf()
    assert abs(e + 0.130434782608696) < 1e-13