venv/lib/python3.12/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, XFAIL
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) == C0*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 == C0*(2*n - 3)/(n**2 - 1)/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) == 0

    assert rsolve_hyper([-n - 1, -1, 1], 0, n) == 0

    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) == 0


@XFAIL
def test_rsolve_ratio_missed():
    # this arises during computation
    # assert rsolve_hyper([-1, 1], 3*(n + n**2), n).expand() == C0 + n**3 - n
    assert rsolve_ratio([-n, n + 2], n, n) is not 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) and len(c) <= 3:
                assert rsolve_hyper(c, q, n).subs(zip(symbols('C:3'), [0, 0, 0])).expand() == p


def test_rsolve_0_sol_homogeneous():
    # fixed by cherry-pick from
    # https://github.com/diofant/diofant/commit/e1d2e52125199eb3df59f12e8944f8a5f24b00a5
    assert rsolve_hyper([n**2 - n + 12, 1], n*(n**2 - n + 12) + n + 1, n) == n


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) - b*(-sqrt(a))**n

    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))
    assert str(sol) == 'C0*((-sqrt(1 - a**2) - 1)/a)**n + C1*((sqrt(1 - a**2) - 1)/a)**n'

    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 + 1)*C1*n + 2**(-n)*C0
    assert rsolve(f, y(n), {y(0): 0, y(1): 1}) == 2**(1 - 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*C1 + C0 + C2*n
    assert rsolve(y(n+3)+3*y(n+2)+3*y(n+1)+y(n), y(n)).expand() == (-1)**n*C0 - (-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


def test_issue_8697():
    a = Function('a')
    eq = a(n + 3) - a(n + 2) - a(n + 1) + a(n)
    assert rsolve(eq, a(n)) == (-1)**n*C1 + C0 + C2*n
    eq2 = a(n + 3) + 3*a(n + 2) + 3*a(n + 1) + a(n)
    assert (rsolve(eq2, a(n)) ==
            (-1)**n*C0 + (-1)**(n + 1)*C1*n + (-1)**(n + 1)*C2*n**2)

    assert rsolve(a(n) - 2*a(n - 3) + 5*a(n - 2) - 4*a(n - 1),
                  a(n), {a(0): 1, a(1): 3, a(2): 8}) == 3*2**n - n - 2

    # From issue thread (but fixed by https://github.com/diofant/diofant/commit/da9789c6cd7d0c2ceeea19fbf59645987125b289):
    assert rsolve(a(n) - 2*a(n - 1) - n, a(n), {a(0): 1}) == 3*2**n - n - 2


def test_diofantissue_294():
    f = y(n) - y(n - 1) - 2*y(n - 2) - 2*n
    assert rsolve(f, y(n)) == (-1)**n*C0 + 2**n*C1 - n - Rational(5, 2)
    # issue sympy/sympy#11261
    assert rsolve(f, y(n), {y(0): -1, y(1): 1}) == (-(-1)**n/2 + 2*2**n -
                                                    n - Rational(5, 2))
    # issue sympy/sympy#7055
    assert rsolve(-2*y(n) + y(n + 1) + n - 1, y(n)) == 2**n*C0 + n


def test_issue_15553():
    f = Function("f")
    assert rsolve(Eq(f(n), 2*f(n - 1) + n), f(n)) == 2**n*C0 - n - 2
    assert rsolve(Eq(f(n + 1), 2*f(n) + n**2 + 1), f(n)) == 2**n*C0 - n**2 - 2*n - 4
    assert rsolve(Eq(f(n + 1), 2*f(n) + n**2 + 1), f(n), {f(1): 0}) == 7*2**n/2 - n**2 - 2*n - 4
    assert rsolve(Eq(f(n), 2*f(n - 1) + 3*n**2), f(n)) == 2**n*C0 - 3*n**2 - 12*n - 18
    assert rsolve(Eq(f(n), 2*f(n - 1) + n**2), f(n)) == 2**n*C0 - n**2 - 4*n - 6
    assert rsolve(Eq(f(n), 2*f(n - 1) + n), f(n), {f(0): 1}) == 3*2**n - n - 2