def test_polymatrix_getitem():
M = PolyMatrix([[1, 2], [3, 4]], x)
assert M[:, :] == M
assert M[0, :] == PolyMatrix([[1, 2]], x)
assert M[:, 0] == PolyMatrix([1, 3], x)
assert M[0, 0] == Poly(1, x, domain=QQ)
assert M[0] == Poly(1, x, domain=QQ)
assert M[:2] == [Poly(1, x, domain=QQ), Poly(2, x, domain=QQ)]
def test_polymatrix_from_Matrix():
assert PolyMatrix.from_Matrix(Matrix([1, 2]), x) == PolyMatrix([1, 2], x, ring=QQ[x])
assert PolyMatrix.from_Matrix(Matrix([1]), ring=QQ[x]) == PolyMatrix([1], x)
pmx = PolyMatrix([1, 2], x)
pmy = PolyMatrix([1, 2], y)
assert pmx != pmy
assert pmx.set_gens(y) == pmy
def test_polymatrix_rref():
M = PolyMatrix([[1, 2], [3, 4]], x)
assert M.rref() == (PolyMatrix.eye(2, x), (0, 1))
raises(ValueError, lambda: PolyMatrix([1, 2], ring=ZZ[x]).rref())
raises(ValueError, lambda: PolyMatrix([1, x], ring=QQ[x]).rref())
def test_polymatrix_ones_zeros():
assert PolyMatrix.zeros(1, 2, x) == PolyMatrix([[0, 0]], x)
assert PolyMatrix.eye(2, x) == PolyMatrix([[1, 0], [0, 1]], x)
def test_polymatrix_manipulations():
M1 = PolyMatrix([[1, 2], [3, 4]], x)
assert M1.transpose() == PolyMatrix([[1, 3], [2, 4]], x)
M2 = PolyMatrix([[5, 6], [7, 8]], x)
assert M1.row_join(M2) == PolyMatrix([[1, 2, 5, 6], [3, 4, 7, 8]], x)
assert M1.col_join(M2) == PolyMatrix([[1, 2], [3, 4], [5, 6], [7, 8]], x)
assert M1.applyfunc(lambda e: 2 * e) == PolyMatrix([[2, 4], [6, 8]], x)
def test_polymatrix_arithmetic():
M = PolyMatrix([[1, 2], [3, 4]], x)
assert M + M == PolyMatrix([[2, 4], [6, 8]], x)
assert M - M == PolyMatrix([[0, 0], [0, 0]], x)
assert -M == PolyMatrix([[-1, -2], [-3, -4]], x)
raises(TypeError, lambda: M + 1)
raises(TypeError, lambda: M - 1)
raises(TypeError, lambda: 1 + M)
raises(TypeError, lambda: 1 - M)
assert M * M == PolyMatrix([[7, 10], [15, 22]], x)
assert 2 * M == PolyMatrix([[2, 4], [6, 8]], x)
assert M * 2 == PolyMatrix([[2, 4], [6, 8]], x)
assert S(2) * M == PolyMatrix([[2, 4], [6, 8]], x)
assert M * S(2) == PolyMatrix([[2, 4], [6, 8]], x)
raises(TypeError, lambda: [] * M)
raises(TypeError, lambda: M * [])
M2 = PolyMatrix([[1, 2]], ring=ZZ[x])
assert S.Half * M2 == PolyMatrix([[S.Half, 1]], ring=QQ[x])
assert M2 * S.Half == PolyMatrix([[S.Half, 1]], ring=QQ[x])
assert M / 2 == PolyMatrix([[S(1) / 2, 1], [S(3) / 2, 2]], x)
assert M / Poly(2, x) == PolyMatrix([[S(1) / 2, 1], [S(3) / 2, 2]], x)
raises(TypeError, lambda: M / [])
def test_polymatrix_repr():
assert repr(PolyMatrix([[1, 2]], x)) == 'PolyMatrix([[1, 2]], ring=QQ[x])'
assert repr(PolyMatrix(0, 2, [], x)) == 'PolyMatrix(0, 2, [], ring=QQ[x])'
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: b is in k, deg(qi) < deg(Dt)
t = DE.t
if DE.case != 'base':
with DecrementLevel(DE):
t0 = DE.t # k = k0(t0)
ba, bd = frac_in(b, t0, field=True)
Q0 = [frac_in(qi.TC(), t0, field=True) for qi in Q]
f, B = param_rischDE(ba, bd, Q0, DE)
# f = [f1, ..., fr] in k^r and B is a matrix with
# m + r columns and entries in Const(k) = Const(k0)
# such that Dy0 + b*y0 = Sum(ci*qi, (i, 1, m)) has
# a solution y0 in k with c1, ..., cm in Const(k)
# if and only y0 = Sum(dj*fj, (j, 1, r)) where
# d1, ..., dr ar in Const(k) and
# B*Matrix([c1, ..., cm, d1, ..., dr]) == 0.
# Transform fractions (fa, fd) in f into constant
# polynomials fa/fd in k[t].
# (Is there a better way?)
f = [Poly(fa.as_expr()/fd.as_expr(), t, field=True)
for fa, fd in f]
else:
# Base case. Dy == 0 for all y in k and b == 0.
# Dy + b*y = Sum(ci*qi) is solvable if and only if
# Sum(ci*qi) == 0 in which case the solutions are
# y = d1*f1 for f1 = 1 and any d1 in Const(k) = k.
f = [Poly(1, t, field=True)] # r = 1
B = Matrix([[qi.TC() for qi in Q] + [S(0)]])
# The condition for solvability is
# B*Matrix([c1, ..., cm, d1]) == 0
# There are no constraints on d1.
# Coefficients of t^j (j > 0) in Sum(ci*qi) must be zero.
d = max([qi.degree(DE.t) for qi in Q])
if d > 0:
M = Matrix(d, m, lambda i, j: Q[j].nth(i + 1))
A, _ = constant_system(M, zeros(d, 1), DE)
else:
# No constraints on the hj.
A = Matrix(0, m, [])
# Solutions of the original equation are
# y = Sum(dj*fj, (j, 1, r) + Sum(ei*hi, (i, 1, m)),
# where ei == ci (i = 1, ..., m), when
# A*Matrix([c1, ..., cm]) == 0 and
# B*Matrix([c1, ..., cm, d1, ..., dr]) == 0
# Build combined constraint matrix with m + r + m columns.
r = len(f)
I = eye(m)
A = A.row_join(zeros(A.rows, r + m))
B = B.row_join(zeros(B.rows, m))
C = I.row_join(zeros(m, r)).row_join(-I)
return f + H, A.col_join(B).col_join(C)
def _test_polymatrix():
pm1 = PolyMatrix([[Poly(x**2, x), Poly(-x, x)],
[Poly(x**3, x), Poly(-1 + x, x)]])
v1 = PolyMatrix([[1, 0], [-1, 0]], ring='ZZ[x]')
m1 = PolyMatrix([[1, 0], [-1, 0]], ring='ZZ[x]')
A = PolyMatrix([[Poly(x**2 + x, x), Poly(0, x)], \
[Poly(x**3 - x + 1, x), Poly(0, x)]])
B = PolyMatrix([[Poly(x**2, x), Poly(-x, x)], [Poly(-x**2, x),
Poly(x, x)]])
assert A.ring == ZZ[x]
assert isinstance(pm1 * v1, PolyMatrix)
assert pm1 * v1 == A
assert pm1 * m1 == A
assert v1 * pm1 == B
pm2 = PolyMatrix([[Poly(x**2, x, domain='QQ'), Poly(0, x, domain='QQ'), Poly(-x**2, x, domain='QQ'), \
Poly(x**3, x, domain='QQ'), Poly(0, x, domain='QQ'), Poly(-x**3, x, domain='QQ')]])
assert pm2.ring == QQ[x]
v2 = PolyMatrix([1, 0, 0, 0, 0, 0], ring='ZZ[x]')
m2 = PolyMatrix([1, 0, 0, 0, 0, 0], ring='ZZ[x]')
C = PolyMatrix([[Poly(x**2, x, domain='QQ')]])
assert pm2 * v2 == C
assert pm2 * m2 == C
pm3 = PolyMatrix([[Poly(x**2, x), S.One]], ring='ZZ[x]')
v3 = S.Half * pm3
assert v3 == PolyMatrix([[Poly(S.Half * x**2, x, domain='QQ'), S.Half]],
ring='QQ[x]')
assert pm3 * S.Half == v3
assert v3.ring == QQ[x]
pm4 = PolyMatrix(
[[Poly(x**2, x, domain='ZZ'),
Poly(-x**2, x, domain='ZZ')]])
v4 = PolyMatrix([1, -1], ring='ZZ[x]')
assert pm4 * v4 == PolyMatrix([[Poly(2 * x**2, x, domain='ZZ')]])
assert len(PolyMatrix(ring=ZZ[x])) == 0
assert PolyMatrix([1, 0, 0, 1], x) / (-1) == PolyMatrix([-1, 0, 0, -1], x)
def test_polymatrix():
pm1 = PolyMatrix([[Poly(x**2, x), Poly(-x, x)],
[Poly(x**3, x), Poly(-1 + x, x)]])
v1 = PolyMatrix([[1, 0], [-1, 0]], ring='ZZ[x]')
m1 = Matrix([[1, 0], [-1, 0]], ring='ZZ[x]')
A = PolyMatrix([[Poly(x**2 + x, x), Poly(0, x)], \
[Poly(x**3 - x + 1, x), Poly(0, x)]])
B = PolyMatrix([[Poly(x**2, x), Poly(-x, x)], [Poly(-x**2, x),
Poly(x, x)]])
assert A.ring == ZZ[x]
assert isinstance(pm1 * v1, PolyMatrix)
assert pm1 * v1 == A
assert pm1 * m1 == A
assert v1 * pm1 == B
pm2 = PolyMatrix([[Poly(x**2, x, domain='QQ'), Poly(0, x, domain='QQ'), Poly(-x**2, x, domain='QQ'), \
Poly(x**3, x, domain='QQ'), Poly(0, x, domain='QQ'), Poly(-x**3, x, domain='QQ')]])
assert pm2.ring == QQ[x]
v2 = PolyMatrix([1, 0, 0, 0, 0, 0], ring='ZZ[x]')
m2 = Matrix([1, 0, 0, 0, 0, 0], ring='ZZ[x]')
C = PolyMatrix([[Poly(x**2, x, domain='QQ')]])
assert pm2 * v2 == C
assert pm2 * m2 == C
pm3 = PolyMatrix([[Poly(x**2, x), S(1)]], ring='ZZ[x]')
v3 = (S(1) / 2) * pm3
assert v3 == PolyMatrix([[Poly(S(1) / 2 * x**2, x, domain='QQ'),
S(1) / 2]],
ring='EX')
assert pm3 * (S(1) / 2) == v3
assert v3.ring == EX
pm4 = PolyMatrix(
[[Poly(x**2, x, domain='ZZ'),
Poly(-x**2, x, domain='ZZ')]])
v4 = Matrix([1, -1], ring='ZZ[x]')
assert pm4 * v4 == PolyMatrix([[Poly(2 * x**2, x, domain='ZZ')]])
assert len(PolyMatrix()) == 0
assert PolyMatrix([1, 0, 0, 1]) / (-1) == PolyMatrix([-1, 0, 0, -1])
def param_rischDE(fa, fd, G, DE):
"""
Solve a Parametric Risch Differential Equation: Dy + f*y == Sum(ci*Gi, (i, 1, m)).
Given a derivation D in k(t), f in k(t), and G
= [G1, ..., Gm] in k(t)^m, return h = [h1, ..., hr] in k(t)^r and
a matrix A with m + r columns and entries in Const(k) such that
Dy + f*y = Sum(ci*Gi, (i, 1, m)) has a solution y
in k(t) with c1, ..., cm in Const(k) if and only if y = Sum(dj*hj,
(j, 1, r)) where d1, ..., dr are in Const(k) and (c1, ..., cm,
d1, ..., dr) is a solution of Ax == 0.
Elements of k(t) are tuples (a, d) with a and d in k[t].
"""
m = len(G)
q, (fa, fd) = weak_normalizer(fa, fd, DE)
# Solutions of the weakly normalized equation Dz + f*z = q*Sum(ci*Gi)
# correspond to solutions y = z/q of the original equation.
gamma = q
G = [(q*ga).cancel(gd, include=True) for ga, gd in G]
a, (ba, bd), G, hn = prde_normal_denom(fa, fd, G, DE)
# Solutions q in k<t> of a*Dq + b*q = Sum(ci*Gi) correspond
# to solutions z = q/hn of the weakly normalized equation.
gamma *= hn
A, B, G, hs = prde_special_denom(a, ba, bd, G, DE)
# Solutions p in k[t] of A*Dp + B*p = Sum(ci*Gi) correspond
# to solutions q = p/hs of the previous equation.
gamma *= hs
g = A.gcd(B)
a, b, g = A.quo(g), B.quo(g), [gia.cancel(gid*g, include=True) for
gia, gid in G]
# a*Dp + b*p = Sum(ci*gi) may have a polynomial solution
# only if the sum is in k[t].
q, M = prde_linear_constraints(a, b, g, DE)
# q = [q1, ..., qm] where qi in k[t] is the polynomial component
# of the partial fraction expansion of gi.
# M is a matrix with m columns and entries in k.
# Sum(fi*gi, (i, 1, m)), where f1, ..., fm are elements of k,
# is a polynomial if and only if M*Matrix([f1, ..., fm]) == 0,
# in which case the sum is equal to Sum(fi*qi).
M, _ = constant_system(M, zeros(M.rows, 1), DE)
# M is a matrix with m columns and entries in Const(k).
# Sum(ci*gi) is in k[t] for c1, ..., cm in Const(k)
# if and only if M*Matrix([c1, ..., cm]) == 0,
# in which case the sum is Sum(ci*qi).
## Reduce number of constants at this point
V = M.nullspace()
# V = [v1, ..., vu] where each vj is a column matrix with
# entries aj1, ..., ajm in Const(k).
# Sum(aji*gi) is in k[t] and equal to Sum(aji*qi) (j = 1, ..., u).
# Sum(ci*gi) is in k[t] if and only is ci = Sum(dj*aji)
# (i = 1, ..., m) for some d1, ..., du in Const(k).
# In that case,
# Sum(ci*gi) = Sum(ci*qi) = Sum(dj*Sum(aji*qi)) = Sum(dj*rj)
# where rj = Sum(aji*qi) (j = 1, ..., u) in k[t].
if not V: # No non-trivial solution
return [], eye(m)
Mq = Matrix([q]) # A single row.
r = [(Mq*vj)[0] for vj in V] # [r1, ..., ru]
# Solutions of a*Dp + b*p = Sum(dj*rj) correspond to solutions
# y = p/gamma of the initial equation with ci = Sum(dj*aji).
try:
# We try n=5. At least for prde_spde, it will always
# terminate no matter what n is.
n = bound_degree(a, b, r, DE, parametric=True)
except NotImplementedError:
# A temporary bound is set. Eventually, it will be removed.
# the currently added test case takes large time
# even with n=5, and much longer with large n's.
n = 5
h, B = param_poly_rischDE(a, b, r, n, DE)
# h = [h1, ..., hv] in k[t]^v and and B is a matrix with u + v
# columns and entries in Const(k) such that
# a*Dp + b*p = Sum(dj*rj) has a solution p of degree <= n
# in k[t] if and only if p = Sum(ek*hk) where e1, ..., ev are in
# Const(k) and B*Matrix([d1, ..., du, e1, ..., ev]) == 0.
# The solutions of the original equation for ci = Sum(dj*aji)
# (i = 1, ..., m) are then y = Sum(ek*hk, (k, 1, v))/gamma.
## Build combined relation matrix with m + u + v columns.
A = -eye(m)
for vj in V:
A = A.row_join(vj)
A = A.row_join(zeros(m, len(h)))
A = A.col_join(zeros(B.rows, m).row_join(B))
## Eliminate d1, ..., du.
W = A.nullspace()
# W = [w1, ..., wt] where each wl is a column matrix with
# entries blk (k = 1, ..., m + u + v) in Const(k).
# The vectors (bl1, ..., blm) generate the space of those
# constant families (c1, ..., cm) for which a solution of
# the equation Dy + f*y == Sum(ci*Gi) exists. They generate
# the space and form a basis except possibly when Dy + f*y == 0
# is solvable in k(t}. The corresponding solutions are
# y = Sum(blk'*hk, (k, 1, v))/gamma, where k' = k + m + u.
v = len(h)
M = Matrix([wl[:m] + wl[-v:] for wl in W]) # excise dj's.
N = M.nullspace()
# N = [n1, ..., ns] where the ni in Const(k)^(m + v) are column
# vectors generating the space of linear relations between
# c1, ..., cm, e1, ..., ev.
C = Matrix([ni[:] for ni in N]) # rows n1, ..., ns.
return [hk.cancel(gamma, include=True) for hk in h], C
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: b is in k, deg(qi) < deg(Dt)
t = DE.t
if DE.case != 'base':
with DecrementLevel(DE):
t0 = DE.t # k = k0(t0)
ba, bd = frac_in(b, t0, field=True)
Q0 = [frac_in(qi.TC(), t0, field=True) for qi in Q]
f, B = param_rischDE(ba, bd, Q0, DE)
# f = [f1, ..., fr] in k^r and B is a matrix with
# m + r columns and entries in Const(k) = Const(k0)
# such that Dy0 + b*y0 = Sum(ci*qi, (i, 1, m)) has
# a solution y0 in k with c1, ..., cm in Const(k)
# if and only y0 = Sum(dj*fj, (j, 1, r)) where
# d1, ..., dr ar in Const(k) and
# B*Matrix([c1, ..., cm, d1, ..., dr]) == 0.
# Transform fractions (fa, fd) in f into constant
# polynomials fa/fd in k[t].
# (Is there a better way?)
f = [Poly(fa.as_expr()/fd.as_expr(), t, field=True)
for fa, fd in f]
else:
# Base case. Dy == 0 for all y in k and b == 0.
# Dy + b*y = Sum(ci*qi) is solvable if and only if
# Sum(ci*qi) == 0 in which case the solutions are
# y = d1*f1 for f1 = 1 and any d1 in Const(k) = k.
f = [Poly(1, t, field=True)] # r = 1
B = Matrix([[qi.TC() for qi in Q] + [S(0)]])
# The condition for solvability is
# B*Matrix([c1, ..., cm, d1]) == 0
# There are no constraints on d1.
# Coefficients of t^j (j > 0) in Sum(ci*qi) must be zero.
d = max([qi.degree(DE.t) for qi in Q])
if d > 0:
M = Matrix(d, m, lambda i, j: Q[j].nth(i + 1))
A, _ = constant_system(M, zeros(d, 1), DE)
else:
# No constraints on the hj.
A = Matrix(0, m, [])
# Solutions of the original equation are
# y = Sum(dj*fj, (j, 1, r) + Sum(ei*hi, (i, 1, m)),
# where ei == ci (i = 1, ..., m), when
# A*Matrix([c1, ..., cm]) == 0 and
# B*Matrix([c1, ..., cm, d1, ..., dr]) == 0
# Build combined constraint matrix with m + r + m columns.
r = len(f)
I = eye(m)
A = A.row_join(zeros(A.rows, r + m))
B = B.row_join(zeros(B.rows, m))
C = I.row_join(zeros(m, r)).row_join(-I)
return f + H, A.col_join(B).col_join(C)
def param_rischDE(fa, fd, G, DE):
"""
Solve a Parametric Risch Differential Equation: Dy + f*y == Sum(ci*Gi, (i, 1, m)).
Given a derivation D in k(t), f in k(t), and G
= [G1, ..., Gm] in k(t)^m, return h = [h1, ..., hr] in k(t)^r and
a matrix A with m + r columns and entries in Const(k) such that
Dy + f*y = Sum(ci*Gi, (i, 1, m)) has a solution y
in k(t) with c1, ..., cm in Const(k) if and only if y = Sum(dj*hj,
(j, 1, r)) where d1, ..., dr are in Const(k) and (c1, ..., cm,
d1, ..., dr) is a solution of Ax == 0.
Elements of k(t) are tuples (a, d) with a and d in k[t].
"""
m = len(G)
q, (fa, fd) = weak_normalizer(fa, fd, DE)
# Solutions of the weakly normalized equation Dz + f*z = q*Sum(ci*Gi)
# correspond to solutions y = z/q of the original equation.
gamma = q
G = [(q*ga).cancel(gd, include=True) for ga, gd in G]
a, (ba, bd), G, hn = prde_normal_denom(fa, fd, G, DE)
# Solutions q in k<t> of a*Dq + b*q = Sum(ci*Gi) correspond
# to solutions z = q/hn of the weakly normalized equation.
gamma *= hn
A, B, G, hs = prde_special_denom(a, ba, bd, G, DE)
# Solutions p in k[t] of A*Dp + B*p = Sum(ci*Gi) correspond
# to solutions q = p/hs of the previous equation.
gamma *= hs
g = A.gcd(B)
a, b, g = A.quo(g), B.quo(g), [gia.cancel(gid*g, include=True) for
gia, gid in G]
# a*Dp + b*p = Sum(ci*gi) may have a polynomial solution
# only if the sum is in k[t].
q, M = prde_linear_constraints(a, b, g, DE)
# q = [q1, ..., qm] where qi in k[t] is the polynomial component
# of the partial fraction expansion of gi.
# M is a matrix with m columns and entries in k.
# Sum(fi*gi, (i, 1, m)), where f1, ..., fm are elements of k,
# is a polynomial if and only if M*Matrix([f1, ..., fm]) == 0,
# in which case the sum is equal to Sum(fi*qi).
M, _ = constant_system(M, zeros(M.rows, 1), DE)
# M is a matrix with m columns and entries in Const(k).
# Sum(ci*gi) is in k[t] for c1, ..., cm in Const(k)
# if and only if M*Matrix([c1, ..., cm]) == 0,
# in which case the sum is Sum(ci*qi).
## Reduce number of constants at this point
V = M.nullspace()
# V = [v1, ..., vu] where each vj is a column matrix with
# entries aj1, ..., ajm in Const(k).
# Sum(aji*gi) is in k[t] and equal to Sum(aji*qi) (j = 1, ..., u).
# Sum(ci*gi) is in k[t] if and only is ci = Sum(dj*aji)
# (i = 1, ..., m) for some d1, ..., du in Const(k).
# In that case,
# Sum(ci*gi) = Sum(ci*qi) = Sum(dj*Sum(aji*qi)) = Sum(dj*rj)
# where rj = Sum(aji*qi) (j = 1, ..., u) in k[t].
if not V: # No non-trivial solution
return [], eye(m)
Mq = Matrix([q]) # A single row.
r = [(Mq*vj)[0] for vj in V] # [r1, ..., ru]
# Solutions of a*Dp + b*p = Sum(dj*rj) correspond to solutions
# y = p/gamma of the initial equation with ci = Sum(dj*aji).
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, r, DE, parametric=True)
except NotImplementedError:
debug("param_rischDE: Proceeding with n = oo; may cause "
"non-termination.")
n = oo
h, B = param_poly_rischDE(a, b, r, n, DE)
# h = [h1, ..., hv] in k[t]^v and and B is a matrix with u + v
# columns and entries in Const(k) such that
# a*Dp + b*p = Sum(dj*rj) has a solution p of degree <= n
# in k[t] if and only if p = Sum(ek*hk) where e1, ..., ev are in
# Const(k) and B*Matrix([d1, ..., du, e1, ..., ev]) == 0.
# The solutions of the original equation for ci = Sum(dj*aji)
# (i = 1, ..., m) are then y = Sum(ek*hk, (k, 1, v))/gamma.
## Build combined relation matrix with m + u + v columns.
A = -eye(m)
for vj in V:
A = A.row_join(vj)
A = A.row_join(zeros(m, len(h)))
A = A.col_join(zeros(B.rows, m).row_join(B))
## Eliminate d1, ..., du.
W = A.nullspace()
# W = [w1, ..., wt] where each wl is a column matrix with
# entries blk (k = 1, ..., m + u + v) in Const(k).
# The vectors (bl1, ..., blm) generate the space of those
# constant families (c1, ..., cm) for which a solution of
# the equation Dy + f*y == Sum(ci*Gi) exists. They generate
# the space and form a basis except possibly when Dy + f*y == 0
# is solvable in k(t}. The corresponding solutions are
# y = Sum(blk'*hk, (k, 1, v))/gamma, where k' = k + m + u.
v = len(h)
M = Matrix([wl[:m] + wl[-v:] for wl in W]) # excise dj's.
N = M.nullspace()
# N = [n1, ..., ns] where the ni in Const(k)^(m + v) are column
# vectors generating the space of linear relations between
# c1, ..., cm, e1, ..., ev.
C = Matrix([ni[:] for ni in N]) # rows n1, ..., ns.
return [hk.cancel(gamma, include=True) for hk in h], C
def test_polymatrix_nullspace():
M = PolyMatrix([[1, 2], [3, 6]], x)
assert M.nullspace() == [PolyMatrix([-2, 1], x)]
raises(ValueError, lambda: PolyMatrix([1, 2], ring=ZZ[x]).nullspace())
raises(ValueError, lambda: PolyMatrix([1, x], ring=QQ[x]).nullspace())
assert M.rank() == 1
def test_polymatrix_eq():
assert (PolyMatrix([x]) == PolyMatrix([x])) is True
assert (PolyMatrix([y]) == PolyMatrix([x])) is False
assert (PolyMatrix([x]) != PolyMatrix([x])) is False
assert (PolyMatrix([y]) != PolyMatrix([x])) is True
assert PolyMatrix([[x, y]]) != PolyMatrix([x, y]) == PolyMatrix([[x], [y]])
assert PolyMatrix([x], ring=QQ[x]) != PolyMatrix([x], ring=ZZ[x])
assert PolyMatrix([x]) != Matrix([x])
assert PolyMatrix([x]).to_Matrix() == Matrix([x])
assert PolyMatrix([1], x) == PolyMatrix([1], x)
assert PolyMatrix([1], x) != PolyMatrix([1], y)
def test_polymatrix_constructor():
M1 = PolyMatrix([[x, y]], ring=QQ[x, y])
assert M1.ring == QQ[x, y]
assert M1.domain == QQ
assert M1.gens == (x, y)
assert M1.shape == (1, 2)
assert M1.rows == 1
assert M1.cols == 2
assert len(M1) == 2
assert list(M1) == [Poly(x, (x, y), domain=QQ), Poly(y, (x, y), domain=QQ)]
M2 = PolyMatrix([[x, y]], ring=QQ[x][y])
assert M2.ring == QQ[x][y]
assert M2.domain == QQ[x]
assert M2.gens == (y, )
assert M2.shape == (1, 2)
assert M2.rows == 1
assert M2.cols == 2
assert len(M2) == 2
assert list(M2) == [
Poly(x, (y, ), domain=QQ[x]),
Poly(y, (y, ), domain=QQ[x])
]
assert PolyMatrix([[x, y]], y) == PolyMatrix([[x, y]],
ring=ZZ.frac_field(x)[y])
assert PolyMatrix([[x, y]], ring='ZZ[x,y]') == PolyMatrix([[x, y]],
ring=ZZ[x, y])
assert PolyMatrix([[x, y]], (x, y)) == PolyMatrix([[x, y]], ring=QQ[x, y])
assert PolyMatrix([[x, y]], x, y) == PolyMatrix([[x, y]], ring=QQ[x, y])
assert PolyMatrix([x, y]) == PolyMatrix([[x], [y]], ring=QQ[x, y])
assert PolyMatrix(1, 2, [x, y]) == PolyMatrix([[x, y]], ring=QQ[x, y])
assert PolyMatrix(1, 2, lambda i, j: [x, y][j]) == PolyMatrix([[x, y]],
ring=QQ[x,
y])
assert PolyMatrix(0, 2, [], x, y).shape == (0, 2)
assert PolyMatrix(2, 0, [], x, y).shape == (2, 0)
assert PolyMatrix([[], []], x, y).shape == (2, 0)
assert PolyMatrix(ring=QQ[x, y]) == PolyMatrix(
0, 0, [], ring=QQ[x, y]) == PolyMatrix([], ring=QQ[x, y])
raises(TypeError, lambda: PolyMatrix())
raises(TypeError, lambda: PolyMatrix(1))
assert PolyMatrix([Poly(x), Poly(y)]) == PolyMatrix([[x], [y]],
ring=ZZ[x, y])
# XXX: Maybe a bug in parallel_poly_from_expr (x lost from gens and domain):
assert PolyMatrix([Poly(y, x), 1]) == PolyMatrix([[y], [1]], ring=QQ[y])
def test_polymatrix():
pm1 = PolyMatrix([[Poly(x**2, x), Poly(-x, x)],
[Poly(x**3, x), Poly(-1 + x, x)]])
v1 = PolyMatrix([[1, 0], [-1, 0]], ring="ZZ[x]")
m1 = Matrix([[1, 0], [-1, 0]], ring="ZZ[x]")
A = PolyMatrix([[Poly(x**2 + x, x), Poly(0, x)],
[Poly(x**3 - x + 1, x), Poly(0, x)]])
B = PolyMatrix([[Poly(x**2, x), Poly(-x, x)],
[Poly(-(x**2), x), Poly(x, x)]])
assert A.ring == ZZ[x]
assert isinstance(pm1 * v1, PolyMatrix)
assert pm1 * v1 == A
assert pm1 * m1 == A
assert v1 * pm1 == B
pm2 = PolyMatrix([[
Poly(x**2, x, domain="QQ"),
Poly(0, x, domain="QQ"),
Poly(-(x**2), x, domain="QQ"),
Poly(x**3, x, domain="QQ"),
Poly(0, x, domain="QQ"),
Poly(-(x**3), x, domain="QQ"),
]])
assert pm2.ring == QQ[x]
v2 = PolyMatrix([1, 0, 0, 0, 0, 0], ring="ZZ[x]")
m2 = Matrix([1, 0, 0, 0, 0, 0], ring="ZZ[x]")
C = PolyMatrix([[Poly(x**2, x, domain="QQ")]])
assert pm2 * v2 == C
assert pm2 * m2 == C
pm3 = PolyMatrix([[Poly(x**2, x), S.One]], ring="ZZ[x]")
v3 = S.Half * pm3
assert v3 == PolyMatrix([[Poly(S.Half * x**2, x, domain="QQ"), S.Half]],
ring="EX")
assert pm3 * S.Half == v3
assert v3.ring == EX
pm4 = PolyMatrix(
[[Poly(x**2, x, domain="ZZ"),
Poly(-(x**2), x, domain="ZZ")]])
v4 = Matrix([1, -1], ring="ZZ[x]")
assert pm4 * v4 == PolyMatrix([[Poly(2 * x**2, x, domain="ZZ")]])
assert len(PolyMatrix()) == 0
assert PolyMatrix([1, 0, 0, 1]) / (-1) == PolyMatrix([-1, 0, 0, -1])
