def prde_no_cancel_b_large(b, Q, n, DE):
"""
Parametric Poly Risch Differential Equation - No cancellation: deg(b) large enough.
Given a derivation D on k[t], n in ZZ, and b, q1, ..., qm in k[t] with
b != 0 and either D == d/dt or deg(b) > max(0, deg(D) - 1), returns
h1, ..., hr in k[r] and a matrix A with coefficients in Const(k) such that
if c1, ..., cm in Const(k) and q in k[t] satisfy deg(q) <= n and
Dq + b*Q == Sum(ci*qi, (i, 1, m)), then q = Sum(dj*hj, (j, 1, r)), where
d1, ..., dr in Const(k) and A*Matrix([[c1, ..., cm, d1, ..., dr]]).T == 0.
"""
db = b.degree(DE.t)
m = len(Q)
H = [Poly(0, DE.t)]*m
for N in range(n, -1, -1): # [n, ..., 0]
for i in range(m):
si = Q[i].nth(N + db)/b.LC()
sitn = Poly(si*DE.t**N, DE.t)
H[i] = H[i] + sitn
Q[i] = Q[i] - derivation(sitn, DE) - b*sitn
if all(qi.is_zero for qi in Q):
dc = -1
M = zeros(0, 2)
else:
dc = max([qi.degree(t) for qi in Q])
M = Matrix(dc + 1, m, lambda i, j: Q[j].nth(i))
A, u = constant_system(M, zeros(dc + 1, 1), DE)
c = eye(m)
A = A.row_join(zeros(A.rows, m)).col_join(c.row_join(-c))
return H, A
def RGS_generalized(m):
"""
Computes the m + 1 generalized unrestricted growth strings
and returns them as rows in matrix.
Examples
========
>>> from diofant.combinatorics.partitions import RGS_generalized
>>> RGS_generalized(6)
Matrix([
[ 1, 1, 1, 1, 1, 1, 1],
[ 1, 2, 3, 4, 5, 6, 0],
[ 2, 5, 10, 17, 26, 0, 0],
[ 5, 15, 37, 77, 0, 0, 0],
[ 15, 52, 151, 0, 0, 0, 0],
[ 52, 203, 0, 0, 0, 0, 0],
[203, 0, 0, 0, 0, 0, 0]])
"""
d = zeros(m + 1)
for i in range(0, m + 1):
d[0, i] = 1
for i in range(1, m + 1):
for j in range(m):
if j <= m - i:
d[i, j] = j * d[i - 1, j] + d[i - 1, j + 1]
else:
d[i, j] = 0
return d
def limited_integrate(fa, fd, G, DE):
"""
Solves the limited integration problem: f = Dv + Sum(ci*wi, (i, 1, n))
"""
fa, fd = fa*Poly(1/fd.LC(), DE.t), fd.monic()
A, B, h, N, g, V = limited_integrate_reduce(fa, fd, G, DE)
V = [g] + V
g = A.gcd(B)
A, B, V = A.quo(g), B.quo(g), [via.cancel(vid*g, include=True) for
via, vid in V]
Q, M = prde_linear_constraints(A, B, V, DE)
M, _ = constant_system(M, zeros(M.rows, 1), DE)
l = M.nullspace()
if M == Matrix() or len(l) > 1:
# Continue with param_rischDE()
raise NotImplementedError("param_rischDE() is required to solve this "
"integral.")
elif len(l) == 0:
raise NonElementaryIntegralException
elif len(l) == 1:
# The c1 == 1. In this case, we can assume a normal Risch DE
if l[0][0].is_zero:
raise NonElementaryIntegralException
else:
l[0] *= 1/l[0][0]
C = sum(Poly(i, DE.t)*q for (i, q) in zip(l[0], Q))
# Custom version of rischDE() that uses the already computed
# denominator and degree bound from above.
B, C, m, alpha, beta = spde(A, B, C, N, DE)
y = solve_poly_rde(B, C, m, DE)
return (alpha*y + beta, h), list(l[0][1:])
else:
raise NotImplementedError
def eqs_to_matrix(eqs, ring):
"""Transform from equations to matrix form. """
m = zeros(len(eqs), len(ring.gens) + 1, cls=RawMatrix)
for j, e_j in enumerate(eqs):
for i, x_i in enumerate(ring.gens):
m[j, i] = e_j.coeff(x_i)
m[j, -1] = -e_j.coeff(1)
return m
def prde_no_cancel_b_small(b, Q, n, DE):
"""
Parametric Poly Risch Differential Equation - No cancellation: deg(b) small enough.
Given a derivation D on k[t], n in ZZ, and b, q1, ..., qm in k[t] with
deg(b) < deg(D) - 1 and either D == d/dt or deg(D) >= 2, returns
h1, ..., hr in k[t] and a matrix A with coefficients in Const(k) such that
if c1, ..., cm in Const(k) and q in k[t] satisfy deg(q) <= n and
Dq + b*q == Sum(ci*qi, (i, 1, m)) then q = Sum(dj*hj, (j, 1, r)) where
d1, ..., dr in Const(k) and A*Matrix([[c1, ..., cm, d1, ..., dr]]).T == 0.
"""
m = len(Q)
H = [Poly(0, DE.t)]*m
for N in range(n, 0, -1): # [n, ..., 1]
for i in range(m):
si = Q[i].nth(N + DE.d.degree(DE.t) - 1)/(N*DE.d.LC())
sitn = Poly(si*DE.t**N, DE.t)
H[i] = H[i] + sitn
Q[i] = Q[i] - derivation(sitn, DE) - b*sitn
if b.degree(DE.t) > 0:
for i in range(m):
si = Poly(Q[i].nth(b.degree(DE.t))/b.LC(), DE.t)
H[i] = H[i] + si
Q[i] = Q[i] - derivation(si, DE) - b*si
if all(qi.is_zero for qi in Q):
dc = -1
M = Matrix()
else:
dc = max([qi.degree(DE.t) for qi in Q])
M = Matrix(dc + 1, m, lambda i, j: Q[j].nth(i))
A, u = constant_system(M, zeros(dc + 1, 1), DE)
c = eye(m)
A = A.row_join(zeros(A.rows, m)).col_join(c.row_join(-c))
return H, A
else:
# TODO: implement this (requires recursive param_rischDE() call)
raise NotImplementedError
def param_rischDE(fa, fd, G, DE):
"""
Solve a Parametric Risch Differential Equation: Dy + f*y == Sum(ci*Gi, (i, 1, m)).
"""
_, (fa, fd) = weak_normalizer(fa, fd, DE)
a, (ba, bd), G, hn = prde_normal_denom(ga, gd, G, DE)
A, B, G, hs = prde_special_denom(a, ba, bd, G, DE)
g = gcd(A, B)
A, B, G = A.quo(g), B.quo(g), [gia.cancel(gid*g, include=True) for
gia, gid in G]
Q, M = prde_linear_constraints(A, B, G, DE)
M, _ = constant_system(M, zeros(M.rows, 1), DE)
# Reduce number of constants at this point
try:
# Similar to rischDE(), we try oo, even though it might lead to
# non-termination when there is no solution. At least for prde_spde,
# it will always terminate no matter what n is.
n = bound_degree(A, B, G, DE, parametric=True)
except NotImplementedError:
debug("param_rischDE: Proceeding with n = oo; may cause "
"non-termination.")
n = oo
A, B, Q, R, n1 = prde_spde(A, B, Q, n, DE)
def test_sparse_zeros_sparse_eye():
assert SparseMatrix.eye(3) == eye(3, cls=SparseMatrix)
assert len(SparseMatrix.eye(3)._smat) == 3
assert SparseMatrix.zeros(3) == zeros(3, cls=SparseMatrix)
assert len(SparseMatrix.zeros(3)._smat) == 0
def test_eq():
A = SparseMatrix(((1, 2), (3, 4)))
assert A != 1
assert A != zeros(2, 1)
def test_sparse_zeros_sparse_eye():
assert SparseMatrix.eye(3) == eye(3, cls=SparseMatrix)
assert len(SparseMatrix.eye(3)._smat) == 3
assert SparseMatrix.zeros(3) == zeros(3, cls=SparseMatrix)
assert len(SparseMatrix.zeros(3)._smat) == 0
def test_eq():
A = SparseMatrix(((1, 2), (3, 4)))
assert A != 1
assert A != zeros(2, 1)
