def test_prde_no_cancel():
# b large
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
assert prde_no_cancel_b_large(Poly(1, x), [Poly(x**2, x), Poly(1, x)], 2, DE) == \
([Poly(x**2 - 2*x + 2, x), Poly(1, x)], Matrix([[1, 0, -1, 0],
[0, 1, 0, -1]], x))
assert prde_no_cancel_b_large(Poly(1, x), [Poly(x**3, x), Poly(1, x)], 3, DE) == \
([Poly(x**3 - 3*x**2 + 6*x - 6, x), Poly(1, x)], Matrix([[1, 0, -1, 0],
[0, 1, 0, -1]], x))
assert prde_no_cancel_b_large(Poly(x, x), [Poly(x**2, x), Poly(1, x)], 1, DE) == \
([Poly(x, x, domain='ZZ'), Poly(0, x, domain='ZZ')], Matrix([[1, -1, 0, 0],
[1, 0, -1, 0],
[0, 1, 0, -1]], x))
# b small
# XXX: Is there a better example of a monomial with D.degree() > 2?
DE = DifferentialExtension(
extension={'D': [Poly(1, x), Poly(t**3 + 1, t)]})
# My original q was t**4 + t + 1, but this solution implies q == t**4
# (c1 = 4), with some of the ci for the original q equal to 0.
G = [
Poly(t**6, t),
Poly(x * t**5, t),
Poly(t**3, t),
Poly(x * t**2, t),
Poly(1 + x, t)
]
R = QQ.frac_field(x)[t]
assert prde_no_cancel_b_small(Poly(x*t, t), G, 4, DE) == \
([Poly(t**4/4 - x/12*t**3 + x**2/24*t**2 + (Rational(-11, 12) - x**3/24)*t + x/24, t),
Poly(x/3*t**3 - x**2/6*t**2 + (Rational(-1, 3) + x**3/6)*t - x/6, t), Poly(t, t),
Poly(0, t), Poly(0, t)], Matrix([[1, 0, -1, 0, 0, 0, 0, 0, 0, 0],
[0, 1, Rational(-1, 4), 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, -1, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, -1, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0, -1, 0, 0],
[0, 0, 0, 1, 0, 0, 0, 0, -1, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, -1]], ring=R))
# TODO: Add test for deg(b) <= 0 with b small
DE = DifferentialExtension(
extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
b = Poly(-1 / x**2, t, field=True) # deg(b) == 0
q = [Poly(x**i * t**j, t, field=True) for i in range(2) for j in range(3)]
h, A = prde_no_cancel_b_small(b, q, 3, DE)
V = A.nullspace()
R = QQ.frac_field(x)[t]
assert len(V) == 1
assert V[0] == Matrix([Rational(-1, 2), 0, 0, 1, 0, 0] * 3, ring=R)
assert (Matrix([h]) * V[0][6:, :])[0] == Poly(x**2 / 2, t, domain='QQ(x)')
assert (Matrix([q]) * V[0][:6, :])[0] == Poly(x - S.Half,
t,
domain='QQ(x)')
def test_prde_no_cancel():
# b large
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
assert prde_no_cancel_b_large(Poly(1, x), [Poly(x**2, x), Poly(1, x)], 2, DE) == \
([Poly(x**2 - 2*x + 2, x), Poly(1, x)], Matrix([[1, 0, -1, 0],
[0, 1, 0, -1]]))
assert prde_no_cancel_b_large(Poly(1, x), [Poly(x**3, x), Poly(1, x)], 3, DE) == \
([Poly(x**3 - 3*x**2 + 6*x - 6, x), Poly(1, x)], Matrix([[1, 0, -1, 0],
[0, 1, 0, -1]]))
# b small
# XXX: Is there a better example of a monomial with D.degree() > 2?
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**3 + 1, t)]})
# My original q was t**4 + t + 1, but this solution implies q == t**4
# (c1 = 4), with some of the ci for the original q equal to 0.
G = [Poly(t**6, t), Poly(x*t**5, t), Poly(t**3, t), Poly(x*t**2, t), Poly(1 + x, t)]
assert prde_no_cancel_b_small(Poly(x*t, t), G, 4, DE) == \
([Poly(t**4/4 - x/12*t**3 + x**2/24*t**2 + (-S(11)/12 - x**3/24)*t + x/24, t),
Poly(x/3*t**3 - x**2/6*t**2 + (-S(1)/3 + x**3/6)*t - x/6, t), Poly(t, t),
Poly(0, t), Poly(0, t)], Matrix([[1, 0, -1, 0, 0, 0, 0, 0, 0, 0],
[0, 1, -S(1)/4, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, -1, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, -1, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0, -1, 0, 0],
[0, 0, 0, 1, 0, 0, 0, 0, -1, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, -1]]))
def test_prde_no_cancel():
# b large
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
assert prde_no_cancel_b_large(Poly(1, x), [Poly(x**2, x), Poly(1, x)], 2, DE) == \
([Poly(x**2 - 2*x + 2, x), Poly(1, x)], Matrix([[1, 0, -1, 0],
[0, 1, 0, -1]]))
assert prde_no_cancel_b_large(Poly(1, x), [Poly(x**3, x), Poly(1, x)], 3, DE) == \
([Poly(x**3 - 3*x**2 + 6*x - 6, x), Poly(1, x)], Matrix([[1, 0, -1, 0],
[0, 1, 0, -1]]))
assert prde_no_cancel_b_large(Poly(x, x), [Poly(x**2, x), Poly(1, x)], 1, DE) == \
([Poly(x, x, domain='ZZ'), Poly(0, x, domain='ZZ')], Matrix([[1, -1, 0, 0],
[1, 0, -1, 0],
[0, 1, 0, -1]]))
# b small
# XXX: Is there a better example of a monomial with D.degree() > 2?
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**3 + 1, t)]})
# My original q was t**4 + t + 1, but this solution implies q == t**4
# (c1 = 4), with some of the ci for the original q equal to 0.
G = [Poly(t**6, t), Poly(x*t**5, t), Poly(t**3, t), Poly(x*t**2, t), Poly(1 + x, t)]
assert prde_no_cancel_b_small(Poly(x*t, t), G, 4, DE) == \
([Poly(t**4/4 - x/12*t**3 + x**2/24*t**2 + (-S(11)/12 - x**3/24)*t + x/24, t),
Poly(x/3*t**3 - x**2/6*t**2 + (-S(1)/3 + x**3/6)*t - x/6, t), Poly(t, t),
Poly(0, t), Poly(0, t)], Matrix([[1, 0, -1, 0, 0, 0, 0, 0, 0, 0],
[0, 1, -S(1)/4, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, -1, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, -1, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0, -1, 0, 0],
[0, 0, 0, 1, 0, 0, 0, 0, -1, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, -1]]))
# TODO: Add test for deg(b) <= 0 with b small
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
b = Poly(-1/x**2, t, field=True) # deg(b) == 0
q = [Poly(x**i*t**j, t, field=True) for i in range(2) for j in range(3)]
h, A = prde_no_cancel_b_small(b, q, 3, DE)
V = A.nullspace()
assert len(V) == 1
assert V[0] == Matrix([-S(1)/2, 0, 0, 1, 0, 0]*3)
assert (Matrix([h])*V[0][6:, :])[0] == Poly(x**2/2, t, domain='ZZ(x)')
assert (Matrix([q])*V[0][:6, :])[0] == Poly(x - S(1)/2, t, domain='QQ(x)')
def solve_poly_rde(b, cQ, n, DE, parametric=False):
"""
Solve a Polynomial Risch Differential Equation with degree bound n.
This constitutes step 4 of the outline given in the rde.py docstring.
For parametric=False, cQ is c, a Poly; for parametric=True, cQ is Q ==
[q1, ..., qm], a list of Polys.
"""
from sympy.integrals.prde import (prde_no_cancel_b_large,
prde_no_cancel_b_small)
# No cancellation
if not b.is_zero and (DE.case == 'base'
or b.degree(DE.t) > max(0,
DE.d.degree(DE.t) - 1)):
if parametric:
return prde_no_cancel_b_large(b, cQ, n, DE)
return no_cancel_b_large(b, cQ, n, DE)
elif (b.is_zero or b.degree(DE.t) < DE.d.degree(DE.t) - 1) and \
(DE.case == 'base' or DE.d.degree(DE.t) >= 2):
if parametric:
return prde_no_cancel_b_small(b, cQ, n, DE)
R = no_cancel_b_small(b, cQ, n, DE)
if isinstance(R, Poly):
return R
else:
# XXX: Might k be a field? (pg. 209)
h, b0, c0 = R
with DecrementLevel(DE):
b0, c0 = b0.as_poly(DE.t), c0.as_poly(DE.t)
assert b0 is not None # See above comment
assert c0 is not None
y = solve_poly_rde(b0, c0, n, DE).as_poly(DE.t)
return h + y
elif DE.d.degree(DE.t) >= 2 and b.degree(DE.t) == DE.d.degree(DE.t) - 1 and \
n > -b.as_poly(DE.t).LC()/DE.d.as_poly(DE.t).LC():
# TODO: Is this check necessary, and if so, what should it do if it fails?
# b comes from the first element returned from spde()
assert b.as_poly(DE.t).LC().is_number
if parametric:
raise NotImplementedError("prde_no_cancel_b_equal() is not yet "
"implemented.")
R = no_cancel_equal(b, cQ, n, DE)
if isinstance(R, Poly):
return R
else:
h, m, C = R
# XXX: Or should it be rischDE()?
y = solve_poly_rde(b, C, m, DE)
return h + y
else:
# Cancellation
if b.is_zero:
raise NotImplementedError(
"Remaining cases for Poly (P)RDE are "
"not yet implemented (is_deriv_in_field() required).")
else:
if DE.case == 'exp':
if parametric:
raise NotImplementedError(
"Parametric RDE cancellation "
"hyperexponential case is not yet implemented.")
return cancel_exp(b, cQ, n, DE)
elif DE.case == 'primitive':
if parametric:
raise NotImplementedError(
"Parametric RDE cancellation "
"primitive case is not yet implemented.")
return cancel_primitive(b, cQ, n, DE)
else:
raise NotImplementedError(
"Other Poly (P)RDE cancellation "
"cases are not yet implemented (%s)." % case)
if parametric:
raise NotImplementedError("Remaining cases for Poly PRDE not yet "
"implemented.")
raise NotImplementedError("Remaining cases for Poly RDE not yet "
"implemented.")
def solve_poly_rde(b, cQ, n, DE, parametric=False):
"""
Solve a Polynomial Risch Differential Equation with degree bound n.
This constitutes step 4 of the outline given in the rde.py docstring.
For parametric=False, cQ is c, a Poly; for parametric=True, cQ is Q ==
[q1, ..., qm], a list of Polys.
"""
from sympy.integrals.prde import (prde_no_cancel_b_large,
prde_no_cancel_b_small)
# No cancellation
if not b.is_zero and (DE.case == 'base' or
b.degree(DE.t) > max(0, DE.d.degree(DE.t) - 1)):
if parametric:
return prde_no_cancel_b_large(b, cQ, n, DE)
return no_cancel_b_large(b, cQ, n, DE)
elif (b.is_zero or b.degree(DE.t) < DE.d.degree(DE.t) - 1) and \
(DE.case == 'base' or DE.d.degree(DE.t) >= 2):
if parametric:
return prde_no_cancel_b_small(b, cQ, n, DE)
R = no_cancel_b_small(b, cQ, n, DE)
if isinstance(R, Poly):
return R
else:
# XXX: Might k be a field? (pg. 209)
h, b0, c0 = R
with DecrementLevel(DE):
b0, c0 = b0.as_poly(DE.t), c0.as_poly(DE.t)
assert b0 is not None # See above comment
assert c0 is not None
y = solve_poly_rde(b0, c0, n, DE).as_poly(DE.t)
return h + y
elif DE.d.degree(DE.t) >= 2 and b.degree(DE.t) == DE.d.degree(DE.t) - 1 and \
n > -b.as_poly(DE.t).LC()/DE.d.as_poly(DE.t).LC():
# TODO: Is this check necessary, and if so, what should it do if it fails?
# b comes from the first element returned from spde()
assert b.as_poly(DE.t).LC().is_number
if parametric:
raise NotImplementedError("prde_no_cancel_b_equal() is not yet "
"implemented.")
R = no_cancel_equal(b, cQ, n, DE)
if isinstance(R, Poly):
return R
else:
h, m, C = R
# XXX: Or should it be rischDE()?
y = solve_poly_rde(b, C, m, DE)
return h + y
else:
# Cancellation
if b.is_zero:
raise NotImplementedError("Remaining cases for Poly (P)RDE are "
"not yet implemented (is_deriv_in_field() required).")
else:
if DE.case == 'exp':
if parametric:
raise NotImplementedError("Parametric RDE cancellation "
"hyperexponential case is not yet implemented.")
return cancel_exp(b, cQ, n, DE)
elif DE.case == 'primitive':
if parametric:
raise NotImplementedError("Parametric RDE cancellation "
"primitive case is not yet implemented.")
return cancel_primitive(b, cQ, n, DE)
else:
raise NotImplementedError("Other Poly (P)RDE cancellation "
"cases are not yet implemented (%s)." % case)
if parametric:
raise NotImplementedError("Remaining cases for Poly PRDE not yet "
"implemented.")
raise NotImplementedError("Remaining cases for Poly RDE not yet "
"implemented.")