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296 lines
11 KiB
296 lines
11 KiB
from sympy.core.function import (Function, Lambda, expand)
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from sympy.core.numbers import (I, Rational)
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from sympy.core.relational import Eq
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from sympy.core.singleton import S
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from sympy.core.symbol import (Symbol, symbols)
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from sympy.functions.combinatorial.factorials import (rf, binomial, factorial)
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from sympy.functions.elementary.complexes import Abs
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from sympy.functions.elementary.miscellaneous import sqrt
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from sympy.functions.elementary.trigonometric import (cos, sin)
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from sympy.polys.polytools import factor
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from sympy.solvers.recurr import rsolve, rsolve_hyper, rsolve_poly, rsolve_ratio
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from sympy.testing.pytest import raises, slow, XFAIL
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from sympy.abc import a, b
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y = Function('y')
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n, k = symbols('n,k', integer=True)
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C0, C1, C2 = symbols('C0,C1,C2')
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def test_rsolve_poly():
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assert rsolve_poly([-1, -1, 1], 0, n) == 0
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assert rsolve_poly([-1, -1, 1], 1, n) == -1
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assert rsolve_poly([-1, n + 1], n, n) == 1
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assert rsolve_poly([-1, 1], n, n) == C0 + (n**2 - n)/2
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assert rsolve_poly([-n - 1, n], 1, n) == C0*n - 1
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assert rsolve_poly([-4*n - 2, 1], 4*n + 1, n) == -1
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assert rsolve_poly([-1, 1], n**5 + n**3, n) == \
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C0 - n**3 / 2 - n**5 / 2 + n**2 / 6 + n**6 / 6 + 2*n**4 / 3
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def test_rsolve_ratio():
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solution = rsolve_ratio([-2*n**3 + n**2 + 2*n - 1, 2*n**3 + n**2 - 6*n,
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-2*n**3 - 11*n**2 - 18*n - 9, 2*n**3 + 13*n**2 + 22*n + 8], 0, n)
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assert solution == C0*(2*n - 3)/(n**2 - 1)/2
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def test_rsolve_hyper():
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assert rsolve_hyper([-1, -1, 1], 0, n) in [
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C0*(S.Half - S.Half*sqrt(5))**n + C1*(S.Half + S.Half*sqrt(5))**n,
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C1*(S.Half - S.Half*sqrt(5))**n + C0*(S.Half + S.Half*sqrt(5))**n,
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]
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assert rsolve_hyper([n**2 - 2, -2*n - 1, 1], 0, n) in [
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C0*rf(sqrt(2), n) + C1*rf(-sqrt(2), n),
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C1*rf(sqrt(2), n) + C0*rf(-sqrt(2), n),
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]
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assert rsolve_hyper([n**2 - k, -2*n - 1, 1], 0, n) in [
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C0*rf(sqrt(k), n) + C1*rf(-sqrt(k), n),
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C1*rf(sqrt(k), n) + C0*rf(-sqrt(k), n),
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]
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assert rsolve_hyper(
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[2*n*(n + 1), -n**2 - 3*n + 2, n - 1], 0, n) == C1*factorial(n) + C0*2**n
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assert rsolve_hyper(
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[n + 2, -(2*n + 3)*(17*n**2 + 51*n + 39), n + 1], 0, n) == 0
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assert rsolve_hyper([-n - 1, -1, 1], 0, n) == 0
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assert rsolve_hyper([-1, 1], n, n).expand() == C0 + n**2/2 - n/2
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assert rsolve_hyper([-1, 1], 1 + n, n).expand() == C0 + n**2/2 + n/2
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assert rsolve_hyper([-1, 1], 3*(n + n**2), n).expand() == C0 + n**3 - n
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assert rsolve_hyper([-a, 1],0,n).expand() == C0*a**n
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assert rsolve_hyper([-a, 0, 1], 0, n).expand() == (-1)**n*C1*a**(n/2) + C0*a**(n/2)
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assert rsolve_hyper([1, 1, 1], 0, n).expand() == \
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C0*(Rational(-1, 2) - sqrt(3)*I/2)**n + C1*(Rational(-1, 2) + sqrt(3)*I/2)**n
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assert rsolve_hyper([1, -2*n/a - 2/a, 1], 0, n) == 0
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@XFAIL
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def test_rsolve_ratio_missed():
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# this arises during computation
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# assert rsolve_hyper([-1, 1], 3*(n + n**2), n).expand() == C0 + n**3 - n
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assert rsolve_ratio([-n, n + 2], n, n) is not None
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def recurrence_term(c, f):
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"""Compute RHS of recurrence in f(n) with coefficients in c."""
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return sum(c[i]*f.subs(n, n + i) for i in range(len(c)))
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def test_rsolve_bulk():
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"""Some bulk-generated tests."""
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funcs = [ n, n + 1, n**2, n**3, n**4, n + n**2, 27*n + 52*n**2 - 3*
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n**3 + 12*n**4 - 52*n**5 ]
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coeffs = [ [-2, 1], [-2, -1, 1], [-1, 1, 1, -1, 1], [-n, 1], [n**2 -
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n + 12, 1] ]
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for p in funcs:
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# compute difference
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for c in coeffs:
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q = recurrence_term(c, p)
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if p.is_polynomial(n):
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assert rsolve_poly(c, q, n) == p
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# See issue 3956:
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if p.is_hypergeometric(n) and len(c) <= 3:
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assert rsolve_hyper(c, q, n).subs(zip(symbols('C:3'), [0, 0, 0])).expand() == p
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def test_rsolve_0_sol_homogeneous():
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# fixed by cherry-pick from
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# https://github.com/diofant/diofant/commit/e1d2e52125199eb3df59f12e8944f8a5f24b00a5
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assert rsolve_hyper([n**2 - n + 12, 1], n*(n**2 - n + 12) + n + 1, n) == n
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def test_rsolve():
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f = y(n + 2) - y(n + 1) - y(n)
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h = sqrt(5)*(S.Half + S.Half*sqrt(5))**n \
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- sqrt(5)*(S.Half - S.Half*sqrt(5))**n
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assert rsolve(f, y(n)) in [
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C0*(S.Half - S.Half*sqrt(5))**n + C1*(S.Half + S.Half*sqrt(5))**n,
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C1*(S.Half - S.Half*sqrt(5))**n + C0*(S.Half + S.Half*sqrt(5))**n,
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]
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assert rsolve(f, y(n), [0, 5]) == h
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assert rsolve(f, y(n), {0: 0, 1: 5}) == h
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assert rsolve(f, y(n), {y(0): 0, y(1): 5}) == h
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assert rsolve(y(n) - y(n - 1) - y(n - 2), y(n), [0, 5]) == h
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assert rsolve(Eq(y(n), y(n - 1) + y(n - 2)), y(n), [0, 5]) == h
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assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
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f = (n - 1)*y(n + 2) - (n**2 + 3*n - 2)*y(n + 1) + 2*n*(n + 1)*y(n)
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g = C1*factorial(n) + C0*2**n
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h = -3*factorial(n) + 3*2**n
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assert rsolve(f, y(n)) == g
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assert rsolve(f, y(n), []) == g
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assert rsolve(f, y(n), {}) == g
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assert rsolve(f, y(n), [0, 3]) == h
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assert rsolve(f, y(n), {0: 0, 1: 3}) == h
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assert rsolve(f, y(n), {y(0): 0, y(1): 3}) == h
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assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
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f = y(n) - y(n - 1) - 2
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assert rsolve(f, y(n), {y(0): 0}) == 2*n
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assert rsolve(f, y(n), {y(0): 1}) == 2*n + 1
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assert rsolve(f, y(n), {y(0): 0, y(1): 1}) is None
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assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
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f = 3*y(n - 1) - y(n) - 1
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assert rsolve(f, y(n), {y(0): 0}) == -3**n/2 + S.Half
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assert rsolve(f, y(n), {y(0): 1}) == 3**n/2 + S.Half
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assert rsolve(f, y(n), {y(0): 2}) == 3*3**n/2 + S.Half
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assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
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f = y(n) - 1/n*y(n - 1)
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assert rsolve(f, y(n)) == C0/factorial(n)
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assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
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f = y(n) - 1/n*y(n - 1) - 1
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assert rsolve(f, y(n)) is None
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f = 2*y(n - 1) + (1 - n)*y(n)/n
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assert rsolve(f, y(n), {y(1): 1}) == 2**(n - 1)*n
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assert rsolve(f, y(n), {y(1): 2}) == 2**(n - 1)*n*2
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assert rsolve(f, y(n), {y(1): 3}) == 2**(n - 1)*n*3
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assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
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f = (n - 1)*(n - 2)*y(n + 2) - (n + 1)*(n + 2)*y(n)
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assert rsolve(f, y(n), {y(3): 6, y(4): 24}) == n*(n - 1)*(n - 2)
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assert rsolve(
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f, y(n), {y(3): 6, y(4): -24}) == -n*(n - 1)*(n - 2)*(-1)**(n)
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assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
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assert rsolve(Eq(y(n + 1), a*y(n)), y(n), {y(1): a}).simplify() == a**n
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assert rsolve(y(n) - a*y(n-2),y(n), \
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{y(1): sqrt(a)*(a + b), y(2): a*(a - b)}).simplify() == \
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a**(n/2 + 1) - b*(-sqrt(a))**n
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f = (-16*n**2 + 32*n - 12)*y(n - 1) + (4*n**2 - 12*n + 9)*y(n)
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yn = rsolve(f, y(n), {y(1): binomial(2*n + 1, 3)})
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sol = 2**(2*n)*n*(2*n - 1)**2*(2*n + 1)/12
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assert factor(expand(yn, func=True)) == sol
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sol = rsolve(y(n) + a*(y(n + 1) + y(n - 1))/2, y(n))
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assert str(sol) == 'C0*((-sqrt(1 - a**2) - 1)/a)**n + C1*((sqrt(1 - a**2) - 1)/a)**n'
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assert rsolve((k + 1)*y(k), y(k)) is None
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assert (rsolve((k + 1)*y(k) + (k + 3)*y(k + 1) + (k + 5)*y(k + 2), y(k))
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is None)
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assert rsolve(y(n) + y(n + 1) + 2**n + 3**n, y(n)) == (-1)**n*C0 - 2**n/3 - 3**n/4
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def test_rsolve_raises():
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x = Function('x')
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raises(ValueError, lambda: rsolve(y(n) - y(k + 1), y(n)))
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raises(ValueError, lambda: rsolve(y(n) - y(n + 1), x(n)))
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raises(ValueError, lambda: rsolve(y(n) - x(n + 1), y(n)))
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raises(ValueError, lambda: rsolve(y(n) - sqrt(n)*y(n + 1), y(n)))
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raises(ValueError, lambda: rsolve(y(n) - y(n + 1), y(n), {x(0): 0}))
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raises(ValueError, lambda: rsolve(y(n) + y(n + 1) + 2**n + cos(n), y(n)))
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def test_issue_6844():
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f = y(n + 2) - y(n + 1) + y(n)/4
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assert rsolve(f, y(n)) == 2**(-n + 1)*C1*n + 2**(-n)*C0
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assert rsolve(f, y(n), {y(0): 0, y(1): 1}) == 2**(1 - n)*n
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def test_issue_18751():
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r = Symbol('r', positive=True)
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theta = Symbol('theta', real=True)
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f = y(n) - 2 * r * cos(theta) * y(n - 1) + r**2 * y(n - 2)
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assert rsolve(f, y(n)) == \
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C0*(r*(cos(theta) - I*Abs(sin(theta))))**n + C1*(r*(cos(theta) + I*Abs(sin(theta))))**n
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def test_constant_naming():
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#issue 8697
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assert rsolve(y(n+3) - y(n+2) - y(n+1) + y(n), y(n)) == (-1)**n*C1 + C0 + C2*n
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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
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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
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#issue 19630
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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
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@slow
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def test_issue_15751():
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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)
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assert rsolve(f, y(n)) is not None
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def test_issue_17990():
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f = -10*y(n) + 4*y(n + 1) + 6*y(n + 2) + 46*y(n + 3)
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sol = rsolve(f, y(n))
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expected = C0*((86*18**(S(1)/3)/69 + (-12 + (-1 + sqrt(3)*I)*(290412 +
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3036*sqrt(9165))**(S(1)/3))*(1 - sqrt(3)*I)*(24201 + 253*sqrt(9165))**
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(S(1)/3)/276)/((1 - sqrt(3)*I)*(24201 + 253*sqrt(9165))**(S(1)/3))
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)**n + C1*((86*18**(S(1)/3)/69 + (-12 + (-1 - sqrt(3)*I)*(290412 + 3036
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*sqrt(9165))**(S(1)/3))*(1 + sqrt(3)*I)*(24201 + 253*sqrt(9165))**
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(S(1)/3)/276)/((1 + sqrt(3)*I)*(24201 + 253*sqrt(9165))**(S(1)/3))
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)**n + C2*(-43*18**(S(1)/3)/(69*(24201 + 253*sqrt(9165))**(S(1)/3)) -
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S(1)/23 + (290412 + 3036*sqrt(9165))**(S(1)/3)/138)**n
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assert sol == expected
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e = sol.subs({C0: 1, C1: 1, C2: 1, n: 1}).evalf()
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assert abs(e + 0.130434782608696) < 1e-13
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def test_issue_8697():
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a = Function('a')
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eq = a(n + 3) - a(n + 2) - a(n + 1) + a(n)
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assert rsolve(eq, a(n)) == (-1)**n*C1 + C0 + C2*n
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eq2 = a(n + 3) + 3*a(n + 2) + 3*a(n + 1) + a(n)
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assert (rsolve(eq2, a(n)) ==
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(-1)**n*C0 + (-1)**(n + 1)*C1*n + (-1)**(n + 1)*C2*n**2)
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assert rsolve(a(n) - 2*a(n - 3) + 5*a(n - 2) - 4*a(n - 1),
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a(n), {a(0): 1, a(1): 3, a(2): 8}) == 3*2**n - n - 2
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# From issue thread (but fixed by https://github.com/diofant/diofant/commit/da9789c6cd7d0c2ceeea19fbf59645987125b289):
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assert rsolve(a(n) - 2*a(n - 1) - n, a(n), {a(0): 1}) == 3*2**n - n - 2
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def test_diofantissue_294():
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f = y(n) - y(n - 1) - 2*y(n - 2) - 2*n
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assert rsolve(f, y(n)) == (-1)**n*C0 + 2**n*C1 - n - Rational(5, 2)
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# issue sympy/sympy#11261
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assert rsolve(f, y(n), {y(0): -1, y(1): 1}) == (-(-1)**n/2 + 2*2**n -
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n - Rational(5, 2))
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# issue sympy/sympy#7055
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assert rsolve(-2*y(n) + y(n + 1) + n - 1, y(n)) == 2**n*C0 + n
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def test_issue_15553():
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f = Function("f")
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assert rsolve(Eq(f(n), 2*f(n - 1) + n), f(n)) == 2**n*C0 - n - 2
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assert rsolve(Eq(f(n + 1), 2*f(n) + n**2 + 1), f(n)) == 2**n*C0 - n**2 - 2*n - 4
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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
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assert rsolve(Eq(f(n), 2*f(n - 1) + 3*n**2), f(n)) == 2**n*C0 - 3*n**2 - 12*n - 18
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assert rsolve(Eq(f(n), 2*f(n - 1) + n**2), f(n)) == 2**n*C0 - n**2 - 4*n - 6
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assert rsolve(Eq(f(n), 2*f(n - 1) + n), f(n), {f(0): 1}) == 3*2**n - n - 2
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