def test_limited_integrate():
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
G = [(Poly(x, x), Poly(x + 1, x))]
assert limited_integrate(Poly(-(1 + x + 5*x**2 - 3*x**3), x),
Poly(1 - x - x**2 + x**3, x), G, DE) == \
((Poly(x**2 - x + 2, x), Poly(x - 1, x)), [2])
G = [(Poly(1, x), Poly(x, x))]
assert limited_integrate(Poly(5*x**2, x), Poly(3, x), G, DE) == \
((Poly(5*x**3/9, x), Poly(1, x)), [0])
python类Poly()的实例源码
def test_is_log_deriv_k_t_radical():
DE = DifferentialExtension(extension={'D': [Poly(1, x)], 'E_K': [], 'L_K': [],
'E_args': [], 'L_args': []})
assert is_log_deriv_k_t_radical(Poly(2*x, x), Poly(1, x), DE) is None
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(2*t1, t1), Poly(1/x, t2)],
'L_K': [2], 'E_K': [1], 'L_args': [x], 'E_args': [2*x]})
assert is_log_deriv_k_t_radical(Poly(x + t2/2, t2), Poly(1, t2), DE) == \
([(t1, 1), (x, 1)], t1*x, 2, 0)
# TODO: Add more tests
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t0, t0), Poly(1/x, t)],
'L_K': [2], 'E_K': [1], 'L_args': [x], 'E_args': [x]})
assert is_log_deriv_k_t_radical(Poly(x + t/2 + 3, t), Poly(1, t), DE) == \
([(t0, 2), (x, 1)], x*t0**2, 2, 3)
def test_parametric_log_deriv():
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
assert parametric_log_deriv_heu(Poly(5*t**2 + t - 6, t), Poly(2*x*t**2, t),
Poly(-1, t), Poly(x*t**2, t), DE) == \
(2, 6, t*x**5)
def test_ratint_logpart():
assert ratint_logpart(x, x**2 - 9, x, t) == \
[(Poly(x**2 - 9, x), Poly(-2*t + 1, t))]
assert ratint_logpart(x**2, x**3 - 5, x, t) == \
[(Poly(x**3 - 5, x), Poly(-3*t + 1, t))]
def test_log_to_atan():
f, g = (Poly(x + S(1)/2, x, domain='QQ'), Poly(sqrt(3)/2, x, domain='EX'))
fg_ans = 2*atan(2*sqrt(3)*x/3 + sqrt(3)/3)
assert log_to_atan(f, g) == fg_ans
assert log_to_atan(g, f) == -fg_ans
def test_Domain__algebraic_field():
alg = ZZ.algebraic_field(sqrt(2))
assert alg.ext.minpoly == Poly(x**2 - 2)
assert alg.dom == QQ
alg = QQ.algebraic_field(sqrt(2))
assert alg.ext.minpoly == Poly(x**2 - 2)
assert alg.dom == QQ
alg = alg.algebraic_field(sqrt(3))
assert alg.ext.minpoly == Poly(x**4 - 10*x**2 + 1)
assert alg.dom == QQ
def test_jacobi_poly():
raises(ValueError, lambda: jacobi_poly(-1, a, b, x))
assert jacobi_poly(1, a, b, x, polys=True) == Poly(
(a/2 + b/2 + 1)*x + a/2 - b/2, x, domain='ZZ(a,b)')
assert jacobi_poly(0, a, b, x) == 1
assert jacobi_poly(1, a, b, x) == a/2 - b/2 + x*(a/2 + b/2 + 1)
assert jacobi_poly(2, a, b, x) == (a**2/8 - a*b/4 - a/8 + b**2/8 - b/8 + x**2*(a**2/8 + a*b/4 + 7*a/8 +
b**2/8 + 7*b/8 + S(3)/2) + x*(a**2/4 + 3*a/4 - b**2/4 - 3*b/4) - S(1)/2)
assert jacobi_poly(1, a, b, polys=True) == Poly(
(a/2 + b/2 + 1)*x + a/2 - b/2, x, domain='ZZ(a,b)')
def test_chebyshevt_poly():
raises(ValueError, lambda: chebyshevt_poly(-1, x))
assert chebyshevt_poly(1, x, polys=True) == Poly(x)
assert chebyshevt_poly(0, x) == 1
assert chebyshevt_poly(1, x) == x
assert chebyshevt_poly(2, x) == 2*x**2 - 1
assert chebyshevt_poly(3, x) == 4*x**3 - 3*x
assert chebyshevt_poly(4, x) == 8*x**4 - 8*x**2 + 1
assert chebyshevt_poly(5, x) == 16*x**5 - 20*x**3 + 5*x
assert chebyshevt_poly(6, x) == 32*x**6 - 48*x**4 + 18*x**2 - 1
assert chebyshevt_poly(1).dummy_eq(x)
assert chebyshevt_poly(1, polys=True) == Poly(x)
def test_chebyshevu_poly():
raises(ValueError, lambda: chebyshevu_poly(-1, x))
assert chebyshevu_poly(1, x, polys=True) == Poly(2*x)
assert chebyshevu_poly(0, x) == 1
assert chebyshevu_poly(1, x) == 2*x
assert chebyshevu_poly(2, x) == 4*x**2 - 1
assert chebyshevu_poly(3, x) == 8*x**3 - 4*x
assert chebyshevu_poly(4, x) == 16*x**4 - 12*x**2 + 1
assert chebyshevu_poly(5, x) == 32*x**5 - 32*x**3 + 6*x
assert chebyshevu_poly(6, x) == 64*x**6 - 80*x**4 + 24*x**2 - 1
assert chebyshevu_poly(1).dummy_eq(2*x)
assert chebyshevu_poly(1, polys=True) == Poly(2*x)
def test_hermite_poly():
raises(ValueError, lambda: hermite_poly(-1, x))
assert hermite_poly(1, x, polys=True) == Poly(2*x)
assert hermite_poly(0, x) == 1
assert hermite_poly(1, x) == 2*x
assert hermite_poly(2, x) == 4*x**2 - 2
assert hermite_poly(3, x) == 8*x**3 - 12*x
assert hermite_poly(4, x) == 16*x**4 - 48*x**2 + 12
assert hermite_poly(5, x) == 32*x**5 - 160*x**3 + 120*x
assert hermite_poly(6, x) == 64*x**6 - 480*x**4 + 720*x**2 - 120
assert hermite_poly(1).dummy_eq(2*x)
assert hermite_poly(1, polys=True) == Poly(2*x)
def test_cyclotomic_poly():
raises(ValueError, lambda: cyclotomic_poly(0, x))
assert cyclotomic_poly(1, x, polys=True) == Poly(x - 1)
assert cyclotomic_poly(1, x) == x - 1
assert cyclotomic_poly(2, x) == x + 1
assert cyclotomic_poly(3, x) == x**2 + x + 1
assert cyclotomic_poly(4, x) == x**2 + 1
assert cyclotomic_poly(5, x) == x**4 + x**3 + x**2 + x + 1
assert cyclotomic_poly(6, x) == x**2 - x + 1
def test_symmetric_poly():
raises(ValueError, lambda: symmetric_poly(-1, x, y, z))
raises(ValueError, lambda: symmetric_poly(5, x, y, z))
assert symmetric_poly(1, x, y, z, polys=True) == Poly(x + y + z)
assert symmetric_poly(1, (x, y, z), polys=True) == Poly(x + y + z)
assert symmetric_poly(0, x, y, z) == 1
assert symmetric_poly(1, x, y, z) == x + y + z
assert symmetric_poly(2, x, y, z) == x*y + x*z + y*z
assert symmetric_poly(3, x, y, z) == x*y*z
def test_random_poly():
poly = random_poly(x, 10, -100, 100, polys=False)
assert Poly(poly).degree() == 10
assert all(-100 <= coeff <= 100 for coeff in Poly(poly).coeffs()) is True
poly = random_poly(x, 10, -100, 100, polys=True)
assert poly.degree() == 10
assert all(-100 <= coeff <= 100 for coeff in poly.coeffs()) is True
def main():
x = Symbol("x")
s = Poly(A(x), x)
num = list(reversed(s.coeffs()))[:11]
print(s.as_expr())
print(num)
def truncd(d, v, Q):
assert d <= Q.degree(v)
from sympy import Poly
cs = coefs(d+1, v, Q)
cs.reverse()
p = sum(c*v**k for k,c in enumerate(cs))
return Poly(p) if p else Poly(p, v)
def check_convergence(numer, denom, n):
"""
Returns (h, g, p) where
-- h is:
> 0 for convergence of rate 1/factorial(n)**h
< 0 for divergence of rate factorial(n)**(-h)
= 0 for geometric or polynomial convergence or divergence
-- abs(g) is:
> 1 for geometric convergence of rate 1/h**n
< 1 for geometric divergence of rate h**n
= 1 for polynomial convergence or divergence
(g < 0 indicates an alternating series)
-- p is:
> 1 for polynomial convergence of rate 1/n**h
<= 1 for polynomial divergence of rate n**(-h)
"""
from sympy import Poly
npol = Poly(numer, n)
dpol = Poly(denom, n)
p = npol.degree()
q = dpol.degree()
rate = q - p
if rate:
return rate, None, None
constant = dpol.LC() / npol.LC()
if abs(constant) != 1:
return rate, constant, None
if npol.degree() == dpol.degree() == 0:
return rate, constant, 0
pc = npol.all_coeffs()[1]
qc = dpol.all_coeffs()[1]
return rate, constant, (qc - pc)/dpol.LC()
def test_gosper_normal():
assert gosper_normal(4*n + 5, 2*(4*n + 1)*(2*n + 3), n) == \
(Poly(S(1)/4, n), Poly(n + S(3)/2), Poly(n + S(1)/4))
def test_solve_poly_system():
assert solve_poly_system([x - 1], x) == [(S.One,)]
assert solve_poly_system([y - x, y - x - 1], x, y) is None
assert solve_poly_system([y - x**2, y + x**2], x, y) == [(S.Zero, S.Zero)]
assert solve_poly_system([2*x - 3, 3*y/2 - 2*x, z - 5*y], x, y, z) == \
[(Rational(3, 2), Integer(2), Integer(10))]
assert solve_poly_system([x*y - 2*y, 2*y**2 - x**2], x, y) == \
[(0, 0), (2, -sqrt(2)), (2, sqrt(2))]
assert solve_poly_system([y - x**2, y + x**2 + 1], x, y) == \
[(-I*sqrt(S.Half), -S.Half), (I*sqrt(S.Half), -S.Half)]
f_1 = x**2 + y + z - 1
f_2 = x + y**2 + z - 1
f_3 = x + y + z**2 - 1
a, b = sqrt(2) - 1, -sqrt(2) - 1
assert solve_poly_system([f_1, f_2, f_3], x, y, z) == \
[(0, 0, 1), (0, 1, 0), (1, 0, 0), (a, a, a), (b, b, b)]
solution = [(1, -1), (1, 1)]
assert solve_poly_system([Poly(x**2 - y**2), Poly(x - 1)]) == solution
assert solve_poly_system([x**2 - y**2, x - 1], x, y) == solution
assert solve_poly_system([x**2 - y**2, x - 1]) == solution
assert solve_poly_system(
[x + x*y - 3, y + x*y - 4], x, y) == [(-3, -2), (1, 2)]
raises(NotImplementedError, lambda: solve_poly_system([x**3 - y**3], x, y))
raises(PolynomialError, lambda: solve_poly_system([1/x], x))
def test_normal_denom():
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
raises(NonElementaryIntegralException, lambda: normal_denom(Poly(1, x), Poly(1, x),
Poly(1, x), Poly(x, x), DE))
fa, fd = Poly(t**2 + 1, t), Poly(1, t)
ga, gd = Poly(1, t), Poly(t**2, t)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
assert normal_denom(fa, fd, ga, gd, DE) == \
(Poly(t, t), (Poly(t**3 - t**2 + t - 1, t), Poly(1, t)), (Poly(1, t),
Poly(1, t)), Poly(t, t))
def test_bound_degree_fail():
# Primitive
DE = DifferentialExtension(extension={'D': [Poly(1, x),
Poly(t0/x**2, t0), Poly(1/x, t)]})
assert bound_degree(Poly(t**2, t), Poly(-(1/x**2*t**2 + 1/x), t),
Poly((2*x - 1)*t**4 + (t0 + x)/x*t**3 - (t0 + 4*x**2)/2*x*t**2 + x*t,
t), DE) == 3
