def test_sympyissue_8697():
assert rsolve(f(n + 3) - f(n + 2) - f(n + 1) + f(n)) == [{
f:
Lambda(n, (-1)**n * C1 + C0 + C2 * n)
}]
assert (rsolve(f(n + 3) + 3 * f(n + 2) + 3 * f(n + 1) + f(n)) == [{
f:
Lambda(n, (-1)**n * (C0 + C1 * n + C2 * n**2))
}])
assert rsolve(f(n) - 2 * f(n - 3) + 5 * f(n - 2) - 4 * f(n - 1),
init={
f(0): 1,
f(1): 3,
f(2): 8
}) == [{
f: Lambda(n, 3 * 2**n - n - 2)
}]
# From issue thread (but not related to the problem, fixed before):
assert rsolve(f(n) - 2 * f(n - 1) - n,
init={f(0): 1}) == [{
f: Lambda(n, 3 * 2**n - n - 2)
}]
assert (rsolve(f(n + 2) - 5 * f(n + 1) + 6 * f(n) - n) == [{
f:
Lambda(n, 2**n * C0 + 3**n * C1 + n / 2 + 3 / 4)
}])
def test_poly():
assert rsolve_poly([-1, -1, 1], 0, n) == (0, [])
assert rsolve_poly([-1, -1, 1], 1, n) == (-1, [])
assert rsolve_poly([-n**2, n, -1, 1], 1, n) is None
assert rsolve(-f(n) + (n + 1) * f(n + 1) - n) == [{
f:
Lambda(n, C0 / factorial(n) + 1)
}]
assert rsolve(-(n + 1) * f(n) + n * f(n + 1) - 1) == [{
f:
Lambda(n, C0 * n - 1)
}]
assert rsolve(-(4 * n + 2) * f(n) + f(n + 1) - 4 * n - 1) == [{
f:
Lambda(n, 4**n * C0 * gamma(n + 1 / 2) / sqrt(pi) - 1)
}]
assert (rsolve(-f(n) + f(n + 1) - n**5 - n**3) == [{
f:
Lambda(n, C0 + n**2 * (n**4 - 3 * n**3 + 4 * n**2 - 3 * n + 1) / 6)
}])
assert rsolve_poly([1, 1], sqrt(n), n) is None
assert rsolve_poly([-2, -1, 1], -2 * n**4 - (n + 1)**4 + (n + 2)**4,
n) == (n**4, [])
assert rsolve(n * f(n) - f(n + 1) - n**3 + (n + 1)**2) == [{
f:
Lambda(n, (C0 * factorial(n) + n**3) / n)
}]
def test_hyper():
assert rsolve((n**2 - 2) * f(n) - (2 * n + 1) * f(n + 1) + f(n + 2)) == [{
f:
Lambda(n,
C0 * rf(-sqrt(2), n) + C1 * rf(+sqrt(2), n))
}]
assert rsolve((n**2 - k) * f(n) - (2 * n + 1) * f(n + 1) + f(n + 2)) == [{
f:
Lambda(n,
C1 * rf(sqrt(k), n) + C0 * rf(-sqrt(k), n))
}]
assert rsolve(2 * n * (n + 1) * f(n) - (n**2 + 3 * n - 2) * f(n + 1) +
(n - 1) * f(n + 2)) == [{
f:
Lambda(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(-n - f(n) + f(n + 1)) == [{
f: Lambda(n, C0 + n * (n - 1) / 2)
}]
assert rsolve(-1 - n - f(n) + f(n + 1)) == [{
f:
Lambda(n, C0 + n * (n + 1) / 2)
}]
assert rsolve(-3 * (n + n**2) - f(n) + f(n + 1)) == [{
f:
Lambda(n, C0 + n**3 - n)
}]
assert rsolve(-n - factorial(n) - f(n) + f(n + 1)) is None
assert rsolve(-a * f(n) + f(n + 1)) == [{f: Lambda(n, C0 * a**n)}]
assert rsolve(-a * f(n) + f(n + 2)) == [{
f:
Lambda(n,
a**(n / 2) * ((-1)**n * C1 + C0))
}]
assert (rsolve(f(n) + f(n + 1) + f(n + 2)) == [{
f:
Lambda(
n,
2**-n * (C0 * (-1 - sqrt(3) * I)**n + C1 * (-1 + sqrt(3) * I)**n))
}])
assert rsolve_hyper([1, -2 * n / a - 2 / a, 1], 0, n) == (0, [])
assert rsolve_hyper([1, 1], sqrt(n), n) is None
assert rsolve_hyper([1, 1], n + sqrt(n), n) is None
def test_sympyissue_19630():
eq = f(n + 3) - 3 * f(n + 1) + 2 * f(n)
res = [{f: Lambda(n, (-2)**n * C1 + C0 + C2 * n)}]
assert rsolve(eq) == res
res0 = [{f: Lambda(n, (-2)**n + 2 * n)}]
assert rsolve(eq, init={f(1): 0, f(2): 8, f(3): -2}) == res0
def test_sympyissue_15553():
assert rsolve(Eq(f(n + 1), 2 * f(n) + n**2 + 1)) == [{
f:
Lambda(n, 2**n * C0 - n**2 - 2 * n - 4)
}]
assert rsolve(Eq(f(n + 1), 2 * f(n) + n**2 + 1),
init={f(1): 0}) == [{
f:
Lambda(n, 7 * 2**n / 2 - n**2 - 2 * n - 4)
}]
def test_sympyissue_6844():
eq = f(n + 2) - f(n + 1) + f(n) / 4
assert rsolve(eq) == [{f: Lambda(n, 2**(-n) * (C0 + C1 * n))}]
assert rsolve(eq, init={
f(0): 0,
f(1): 1
}) == [{
f: Lambda(n, 2**(1 - n) * n)
}]
def test_diofantissue_294():
eq = f(n) - f(n - 1) - 2 * f(n - 2) - 2 * n
assert rsolve(eq) == [{f: Lambda(n, (-1)**n * C0 + 2**n * C1 - n - 5 / 2)}]
# issue sympy/sympy#11261
assert rsolve(eq, init={
f(0): -1,
f(1): 1
}) == [{
f: Lambda(n, -(-1)**n / 2 + 2**(n + 1) - n - 5 / 2)
}]
# issue sympy/sympy#7055
assert rsolve(-2 * f(n) + f(n + 1) + n - 1) == [{
f: Lambda(n, 2**n * C0 + n)
}]
def test_diofantissue_13629():
assert rsolve(f(n + 1) - (f(n) + (n + 1)**2),
init={f(0): 0}) == [{
f:
Lambda(n,
n * (2 * n**2 + 3 * n + 1) / 6)
}]
def test_diofantissue_922():
assert rsolve(-2 * n / 3 + f(n) - f(n - 1) + 2 * (n - 1)**3 / 3 + 2 *
(n - 1)**2 / 3,
init={f(0): 0}) == [{
f:
Lambda(n,
n * (-3 * n**3 + 2 * n**2 + 9 * n + 4) / 18)
}]
def test_sympyissue_18751():
r = symbols('r', real=True, positive=True)
theta = symbols('theta', real=True)
eq = f(n) - 2 * r * cos(theta) * f(n - 1) + r**2 * f(n - 2)
res = [{
f:
Lambda(
n,
r**n * (C0 * (cos(theta) - I * abs(sin(theta)))**n + C1 *
(cos(theta) + I * abs(sin(theta)))**n))
}]
assert rsolve(eq) == res
def solve(lhs):
if lhs in solutions:
return solutions[lhs]
#print(' Solving for ' + lhs, ' with initial condition {}'.format(str(fs[lhs](0)) + " = " + str(initial_vals[lhs])))
if lhs in [evar for _, evar in evar_recs[lhs]]: # recurrence
eqn = fs[lhs](n) - sum(
coeff *
(solve(evar).subs({n: n - 1}) if evar != lhs else fs[lhs]
(n - 1)) for coeff, evar in evar_recs[lhs])
#print(' equation', eqn)
res = rsolve(eqn, fs[lhs](n), init={fs[lhs](0): initial_vals[lhs]})
else: # non recurrence
res = sum(coeff * solve(evar).subs({n: n - 1})
for coeff, evar in evar_recs[lhs])
solutions[lhs] = simplify(res)
#print(' ', lhs + '(n)', ' -> ', res)
return res
def test_diofantissue_451():
assert rsolve(f(n) - 2 * f(n - 1) - 3**n,
init={f(0): 1}) == [{
f: Lambda(n, -2**(n + 1) + 3**(n + 1))
}]
def test_rsolve():
eq = f(n + 2) - f(n + 1) - f(n)
res = [{
f:
Lambda(n, 2**(-n) * (C0 * (1 + sqrt(5))**n + C1 * (-sqrt(5) + 1)**n))
}]
assert rsolve(eq) == res
res = [{
k: v.subs({
C0: sqrt(5),
C1: -sqrt(5)
}).simplify()
for k, v in r.items()
} for r in res]
assert rsolve(eq, init={f(0): 0, f(1): 5}) == res
assert rsolve(f(n) - f(n - 1) - f(n - 2), init={f(0): 0, f(1): 5}) == res
assert rsolve(Eq(f(n),
f(n - 1) + f(n - 2)), init={
f(0): 0,
f(1): 5
}) == res
eq = (n - 1) * f(n + 2) - (n**2 + 3 * n -
2) * f(n + 1) + 2 * n * (n + 1) * f(n)
res = [{f: Lambda(n, C1 * factorial(n) + C0 * 2**n)}]
assert rsolve(eq) == res
res = [{f: Lambda(n, -3 * factorial(n) + 3 * 2**n)}]
assert rsolve(eq, init={f(0): 0, f(1): 3}) == res
eq = f(n) - f(n - 1) - 2
assert rsolve(eq, f(n)) == [{f: Lambda(n, C0 + 2 * n)}]
assert rsolve(eq) == [{f: Lambda(n, C0 + 2 * n)}]
assert rsolve(eq, init={f(0): 0}) == [{f: Lambda(n, 2 * n)}]
assert rsolve(eq, init={f(0): 1}) == [{f: Lambda(n, 2 * n + 1)}]
assert rsolve(eq, init={f(0): 0, f(1): 1}) is None
eq = 3 * f(n - 1) - f(n) - 1
assert rsolve(eq) == [{f: Lambda(n, 3**n * C0 + 1 / 2)}]
assert rsolve(eq, init={f(0): 0}) == [{f: Lambda(n, -3**n / 2 + 1 / 2)}]
assert rsolve(eq, init={f(0): 1}) == [{f: Lambda(n, 3**n / 2 + 1 / 2)}]
assert rsolve(eq, init={f(0): 2}) == [{f: Lambda(n, 3 * 3**n / 2 + 1 / 2)}]
assert rsolve(f(n) - 1 / n * f(n - 1), f(n)) == [{
f:
Lambda(n, C0 / factorial(n))
}]
assert rsolve(f(n) - 1 / n * f(n - 1) - 1, f(n)) is None
eq = 2 * f(n - 1) + (1 - n) * f(n) / n
assert rsolve(eq) == [{f: Lambda(n, 2**n * C0 * n)}]
assert rsolve([eq]) == [{f: Lambda(n, 2**n * C0 * n)}]
assert rsolve(eq, init={f(1): 1}) == [{f: Lambda(n, 2**(n - 1) * n)}]
assert rsolve(eq, init={f(1): 2}, simplify=False) == [{
f:
Lambda(n, 2**(n - 1) * n * 2)
}]
assert rsolve(eq, init={f(1): 2}) == [{f: Lambda(n, 2**n * n)}]
assert rsolve(eq, init={f(1): 3}) == [{f: Lambda(n, 3 * 2**n * n / 2)}]
eq = (n - 1) * (n - 2) * f(n + 2) - (n + 1) * (n + 2) * f(n)
assert rsolve(eq) == [{
f:
Lambda(n,
n * (n - 2) * (n - 1) * ((-1)**n * C1 + C0))
}]
assert rsolve(eq, init={
f(3): 6,
f(4): 24
}) == [{
f: Lambda((n),
n * (n - 1) * (n - 2))
}]
assert (rsolve(eq, init={
f(3): 6,
f(4): -24
}) == [{
f: Lambda(n, (-1)**(n + 1) * n * (n - 2) * (n - 1))
}])
assert rsolve(Eq(f(n + 1), a * f(n)), init={f(1): a}) == [{
f:
Lambda(n, a**n)
}]
assert (rsolve(f(n) - a * f(n - 2),
init={
f(1): sqrt(a) * (a + b),
f(2): a * (a - b)
}) == [{
f: Lambda(n,
a**(n / 2) * (-(-1)**n * b + a))
}])
eq = (-16 * n**2 + 32 * n - 12) * f(n - 1) + (4 * n**2 - 12 * n + 9) * f(n)
assert (rsolve(eq, init={f(1): binomial(2 * n + 1, 3)}) == [{
f:
Lambda(
n, 4**n * n * (2 * n - 1) *
gamma(n + 3 / 2) / (3 * gamma(n - 1 / 2)))
}])
assert (rsolve(f(n) + a * (f(n + 1) + f(n - 1)) / 2) == [{
f:
Lambda(
n, a**-n * (C0 * (-a * sqrt(-1 + a**-2) - 1)**n + C1 *
(a * sqrt(-1 + a**-2) - 1)**n))
}])
def test_sympyissue_17982():
assert (rsolve(f(n + 3) + 10 * f(n + 2) + 32 * f(n + 1) + 32 * f(n)) == [{
f:
Lambda(n, (-2)**n * C0 + (-4)**n * C1 + (-4)**n * C2 * n)
}])
def test_diofantissue_923():
assert rsolve(4 * f(n) + 4 * f(n + 1) + f(n + 2)) == [{
f:
Lambda(n, (-2)**n * (C0 + C1 * n))
}]
def test_rsolve_raises():
pytest.raises(ValueError, lambda: rsolve(f(n) - f(k + 1), f(n)))
pytest.raises(ValueError, lambda: rsolve(f(n) - sqrt(n) * f(n + 1)))
pytest.raises(ValueError,
lambda: rsolve(f(n) - f(n + 1), f(n), init={g(0): 0}))
pytest.raises(NotImplementedError, lambda: rsolve(f(n) - sqrt(n)))
def test_diofantissue_456():
assert rsolve(f(n) - 2 * f(n - 1) - 3**n * n,
init={f(0): 1}) == [{
f:
Lambda(n, 7 * 2**n + 3**(n + 1) * (n - 2))
}]
