def __init__(self):
self.cols = random.randint(_sage_const_1 , _sage_const_5 )
self.rows = random.randint(_sage_const_1 , _sage_const_5 )
ring = random.choice([QQ])
self.A = random_matrix(ring, self.rows, self.cols)
self.B = random_matrix(ring, self.rows, self.cols)
self.game = NormalFormGame([self.A, self.B])
def random_chain_complex(level=1):
"""
Return a random chain complex, defined by specifying a single
random matrix in a random degree, with differential of degree
either 1 or -1. The matrix is randomly sparse or dense.
:param level: measure of complexity: the larger this is, the
larger the matrix can be, and the larger its degree can be in
the chain complex.
:type level: positive integer; optional, default 1
EXAMPLES::
sage: from sage.homology.tests import random_chain_complex
sage: C = random_chain_complex()
sage: C
Chain complex with at most 2 nonzero terms over Integer Ring
sage: C.degree_of_differential() # random: either 1 or -1
1
"""
bound = 50*level
nrows = randint(0, bound)
ncols = randint(0, bound)
sparseness = bool(randint(0, 1))
mat = random_matrix(ZZ, nrows, ncols, sparse=sparseness)
dim = randint(-bound, bound)
deg = 2 * randint(0, 1) - 1 # -1 or 1
return ChainComplex({dim: mat}, degree = deg)
def det_padic(A, proof=True, stabilize=2):
"""
Return the determinant of A, computed using a p-adic/multimodular
algorithm.
INPUTS:
- ``A`` -- a square matrix
- ``proof`` -- boolean
- ``stabilize`` (default: 2) -- if proof False, number of successive primes so that
CRT det must stabilize.
EXAMPLES::
sage: import sage.matrix.matrix_integer_dense_hnf as h
sage: a = matrix(ZZ, 3, [1..9])
sage: h.det_padic(a)
0
sage: a = matrix(ZZ, 3, [1,2,5,-7,8,10,192,5,18])
sage: h.det_padic(a)
-3669
sage: a.determinant(algorithm='ntl')
-3669
"""
if not A.is_square():
raise ValueError("A must be a square matrix")
r = A.rank()
if r < A.nrows():
return ZZ(0)
v = random_matrix(ZZ, A.nrows(), 1)
d = A._solve_right_nonsingular_square(v, check_rank=False).denominator()
return det_given_divisor(A, d, proof=proof, stabilize=stabilize)
def random_inequalities(d, n):
"""
Random collections of inequalities for testing purposes.
INPUT:
- ``d`` -- integer. The dimension.
- ``n`` -- integer. The number of random inequalities to generate.
OUTPUT:
A random set of inequalites as a :class:`StandardAlgorithm` instance.
EXAMPLES::
sage: from sage.geometry.polyhedron.double_description import random_inequalities
sage: P = random_inequalities(5, 10)
sage: P.run().verify()
"""
from sage.matrix.constructor import random_matrix
while True:
A = random_matrix(QQ, n, d)
if A.rank() == min(n, d) and not any(a == 0 for a in A.rows()):
break
return StandardAlgorithm(A)
def random_lattice(dimension):
"""
Construct a random ZZ-lattice of a given dimension.
INPUT:
- ``dimension`` -- dimension of the constructed lattice.
OUTPUT:
A lattice with integer basis vectors.
EXAMPLES::
sage: random_lattice(3)
ZZ-lattice of degree 3 and rank 3
Inner product matrix:
[ 2 0 0]
[ 0 66 -22]
[ 0 -22 4246]
Basis matrix:
[ 0 1 -1]
[-8 1 1]
[14 45 45]
sage: random_lattice(100).dimension()
100
"""
basis = random_matrix(ZZ, dimension, dimension)
return Lattice(basis)
def bigraphical(self, G, A=None, K=QQ, names=None):
r"""
Return a bigraphical hyperplane arrangement.
INPUT:
- ``G`` -- graph
- ``A`` -- list, matrix, dictionary (default: ``None``
gives semiorder), or the string 'generic'
- ``K`` -- field (default: `\QQ`)
- ``names`` -- tuple of strings or ``None`` (default); the
variable names for the ambient space
OUTPUT:
The hyperplane arrangement with hyperplanes `x_i - x_j =
A[i,j]` and `x_j - x_i = A[j,i]` for each edge `v_i, v_j` of
``G``. The indices `i,j` are the indices of elements of
``G.vertices()``.
EXAMPLES::
sage: G = graphs.CycleGraph(4)
sage: G.edges()
[(0, 1, None), (0, 3, None), (1, 2, None), (2, 3, None)]
sage: G.edges(labels=False)
[(0, 1), (0, 3), (1, 2), (2, 3)]
sage: A = {0:{1:1, 3:2}, 1:{0:3, 2:0}, 2:{1:2, 3:1}, 3:{2:0, 0:2}}
sage: HA = hyperplane_arrangements.bigraphical(G, A)
sage: HA.n_regions()
63
sage: hyperplane_arrangements.bigraphical(G, 'generic').n_regions()
65
sage: hyperplane_arrangements.bigraphical(G).n_regions()
59
REFERENCES:
.. [BigraphicalArrangements] S. Hopkins, D. Perkinson.
"Bigraphical Arrangements".
:arxiv:`1212.4398`
"""
n = G.num_verts()
if A is None: # default to G-semiorder arrangement
A = matrix(K, n, lambda i, j: 1)
elif A == 'generic':
A = random_matrix(ZZ, n, x=10000)
A = matrix(K, A)
H = make_parent(K, n, names)
x = H.gens()
hyperplanes = []
for e in G.edges():
i = G.vertices().index(e[0])
j = G.vertices().index(e[1])
hyperplanes.append( x[i] - x[j] - A[i][j])
hyperplanes.append(-x[i] + x[j] - A[j][i])
return H(*hyperplanes)
def random_low_weight_bases(N,p,m,NN,weightbound):
r"""
Returns list of random integral bases of modular forms of level `N` and
(even) weight at most weightbound with coefficients reduced modulo
`(p^m,q^{NN})`.
INPUT:
- ``N`` -- positive integer (level).
- ``p`` -- prime.
- ``m``, ``NN`` -- positive integers.
- ``weightbound`` -- (even) positive integer.
OUTPUT:
- list of lists of `q`-expansions modulo `(p^m,q^{NN})`.
EXAMPLES::
sage: from sage.modular.overconvergent.hecke_series import random_low_weight_bases
sage: S = random_low_weight_bases(3,7,2,5,6); S # random
[[4 + 48*q + 46*q^2 + 48*q^3 + 42*q^4 + O(q^5)],
[3 + 5*q + 45*q^2 + 22*q^3 + 22*q^4 + O(q^5),
1 + 3*q + 27*q^2 + 27*q^3 + 23*q^4 + O(q^5)],
[2*q + 4*q^2 + 16*q^3 + 48*q^4 + O(q^5),
2 + 6*q + q^2 + 3*q^3 + 43*q^4 + O(q^5),
1 + 2*q + 6*q^2 + 14*q^3 + 4*q^4 + O(q^5)]]
sage: S[0][0].parent()
Power Series Ring in q over Ring of integers modulo 49
sage: S[0][0].prec()
5
"""
LWB = low_weight_bases(N,p,m,NN,weightbound)
# this is "approximately" row reduced (it's the mod p^n reduction of a
# matrix over ZZ in Hermite form)
RandomLWB = []
for i in xrange(len(LWB)):
n = len(LWB[i])
c = random_matrix(Zmod(p**m), n)
while c.det() % p == 0:
c = random_matrix(Zmod(p**m), n)
RandomLWB.append([ sum([c[j, k] * LWB[i][k] for k in xrange(n)]) for j in xrange(n) ])
return RandomLWB
def benchmark_hnf(nrange, bits=4):
"""
Run benchmark program.
EXAMPLES:
sage: import sage.matrix.matrix_integer_dense_hnf as hnf
sage: hnf.benchmark_hnf([50,100],32)
('sage', 50, 32, ...),
('sage', 100, 32, ...),
"""
b = 2**bits
for n in nrange:
a = random_matrix(ZZ, n, x=-b,y=b)
t = cputime()
h,_ = hnf(a, proof=False)
tm = cputime(t)
print '%s,'%(('sage', n, bits, tm),)
def hnf_with_transformation_tests(n=10, m=5, trials=10):
"""
Use this to randomly test that hnf with transformation matrix
is working.
EXAMPLES:
sage: from sage.matrix.matrix_integer_dense_hnf import hnf_with_transformation_tests
sage: hnf_with_transformation_tests(n=15,m=10, trials=10)
0 1 2 3 4 5 6 7 8 9
"""
import sys
for i in range(trials):
print i,
sys.stdout.flush()
a = random_matrix(ZZ, n, m)
w = hnf_with_transformation(a)
assert w[0] == w[1]*a
w = hnf_with_transformation(a, proof=False)
assert w[0] == w[1]*a
def hnf_with_transformation_tests(n=10, m=5, trials=10):
"""
Use this to randomly test that hnf with transformation matrix
is working.
EXAMPLES::
sage: from sage.matrix.matrix_integer_dense_hnf import hnf_with_transformation_tests
sage: hnf_with_transformation_tests(n=15,m=10, trials=10)
0 1 2 3 4 5 6 7 8 9
"""
import sys
for i in range(trials):
print(i, end=" ")
sys.stdout.flush()
A = random_matrix(ZZ, n, m)
H, U = hnf_with_transformation(A)
assert H == U * A
H, U = hnf_with_transformation(A, proof=False)
assert H == U * A
def solve_system_with_difficult_last_row(B, A):
"""
Solve the matrix equation B*Z = A when the last row of $B$
contains huge entries.
INPUT:
- B -- a square n x n nonsingular matrix with painful big bottom row.
- A -- an n x k matrix.
OUTPUT:
the unique solution to B*Z = A.
EXAMPLES::
sage: from sage.matrix.matrix_integer_dense_saturation import solve_system_with_difficult_last_row
sage: B = matrix(ZZ, 3, [1,2,3, 3,-1,2,939239082,39202803080,2939028038402834]); A = matrix(ZZ,3,2,[1,2,4,3,-1,0])
sage: X = solve_system_with_difficult_last_row(B, A); X
[ 290668794698843/226075992027744 468068726971/409557956572]
[-226078357385539/1582531944194208 1228691305937/2866905696004]
[ 2365357795/1582531944194208 -17436221/2866905696004]
sage: B*X == A
True
"""
# See the comments in the function of the same name in matrix_integer_dense_hnf.py.
# This function is just a generalization of that one to A a matrix.
C = copy(B)
while True:
C[C.nrows()-1] = random_matrix(ZZ,1,C.ncols()).row(0)
try:
X = C.solve_right(A)
except ValueError:
verbose("Try difficult solve again with different random vector")
else:
break
D = B.matrix_from_rows(range(C.nrows()-1))
N = D._rational_kernel_flint()
if N.ncols() != 1:
verbose("Difficult solve quickly failed. Using direct approach.")
return B.solve_right(A)
tm = verbose("Recover correct linear combinations")
k = N.matrix_from_columns([0])
# The sought for solution Z to B*Z = A is some linear combination
# Z = X + alpha*k
# Let w be the last row of B; then Z satisfies
# w * Z = A'
# where A' is the last row of A. Thus
# w * (X + alpha*k) = A'
# so w * X + alpha*w*k = A'
# so alpha*w*k = A' - w*X.
w = B[-1] # last row of B
A_prime = A[-1] # last row of A
lhs = w*k
rhs = A_prime - w * X
if lhs[0] == 0:
verbose("Difficult solve quickly failed. Using direct approach.")
return B.solve_right(A)
for i in range(X.ncols()):
alpha = rhs[i] / lhs[0]
X.set_column(i, (X.matrix_from_columns([i]) + alpha*k).list())
verbose("Done getting linear combinations.", tm)
return X
def solve_system_with_difficult_last_row(B, a):
"""
Solve B*x = a when the last row of $B$ contains huge entries using
a clever trick that reduces the problem to solve C*x = a where $C$
is $B$ but with the last row replaced by something small, along
with one easy null space computation. The latter are both solved
$p$-adically.
INPUT:
- B -- a square n x n nonsingular matrix with painful big bottom row.
- a -- an n x 1 column matrix
OUTPUT:
- the unique solution to B*x = a.
EXAMPLES::
sage: from sage.matrix.matrix_integer_dense_hnf import solve_system_with_difficult_last_row
sage: B = matrix(ZZ, 3, [1,2,4, 3,-4,7, 939082,2930982,132902384098234])
sage: a = matrix(ZZ,3,1, [1,2,5])
sage: z = solve_system_with_difficult_last_row(B, a)
sage: z
[ 106321906985474/132902379815497]
[132902385037291/1329023798154970]
[ -5221794/664511899077485]
sage: B*z
[1]
[2]
[5]
"""
# Here's how:
# 1. We make a copy of B but with the last *nasty* row of B replaced
# by a random very nice row.
C = copy(B)
while True:
C[C.nrows()-1] = random_matrix(ZZ,1,C.ncols()).row(0)
# 2. Then we find the unique solution to C * x = a
try:
x = C.solve_right(a)
except ValueError:
verbose("Try difficult solve again with different random vector")
else:
break
# 3. We next delete the last row of B and find a basis vector k
# for the 1-dimensional kernel.
D = B.matrix_from_rows(range(C.nrows()-1))
N = D._rational_kernel_iml()
if N.ncols() != 1:
verbose("Try difficult solve again with different random vector")
return solve_system_with_difficult_last_row(B, a)
k = N.matrix_from_columns([0])
# 4. The sought for solution z to B*z = a is some linear combination
#
# z = x + alpha*k
#
# of x and k, where k is the above fixed basis for the kernel of D.
# Setting w to be the last row of B, this column vector z satisfies
#
# w * z = a'
#
# where a' is the last entry of a. Thus
#
# w * (x + alpha*k) = a'
#
# so w * x + alpha*w*k = a'
# so alpha*w*k = a' - w*x.
w = B[-1] # last row of B
a_prime = a[-1]
lhs = w*k
rhs = a_prime - w * x
if lhs[0] == 0:
verbose("Try difficult solve again with different random vector")
return solve_system_with_difficult_last_row(B, a)
alpha = rhs[0] / lhs[0]
z = x + alpha*k
return z
def sanity_checks(times=50, n=8, m=5, proof=True, stabilize=2, check_using_magma = True):
"""
Run random sanity checks on the modular p-adic HNF with tall and wide matrices
both dense and sparse.
INPUT:
- times -- number of times to randomly try matrices with each shape
- n -- number of rows
- m -- number of columns
- proof -- test with proof true
- stabilize -- parameter to pass to hnf algorithm when proof is False
- check_using_magma -- if True use Magma instead of PARI to check
correctness of computed HNF's. Since PARI's HNF is buggy and slow (as of
2008-02-16 non-pivot entries sometimes aren't normalized to be
nonnegative) the default is Magma.
EXAMPLES::
sage: import sage.matrix.matrix_integer_dense_hnf as matrix_integer_dense_hnf
sage: matrix_integer_dense_hnf.sanity_checks(times=5, check_using_magma=False)
small 8 x 5
0 1 2 3 4 (done)
big 8 x 5
0 1 2 3 4 (done)
small 5 x 8
0 1 2 3 4 (done)
big 5 x 8
0 1 2 3 4 (done)
sparse 8 x 5
0 1 2 3 4 (done)
sparse 5 x 8
0 1 2 3 4 (done)
ill conditioned -- 1000*A -- 8 x 5
0 1 2 3 4 (done)
ill conditioned -- 1000*A but one row -- 8 x 5
0 1 2 3 4 (done)
"""
import sys
def __do_check(v):
"""
This is used internally by the sanity check code.
"""
for i,a in enumerate(v):
global sanity
sanity = a
print i,
sys.stdout.flush()
if check_using_magma:
if magma(hnf(a)[0]) != magma(a).EchelonForm():
print "bug computing hnf of a matrix"
print 'a = matrix(ZZ, %s, %s, %s)'%(a.nrows(), a.ncols(), a.list())
return
else:
if hnf(a)[0] != a.echelon_form(algorithm = 'pari'):
print "bug computing hnf of a matrix"
print 'a = matrix(ZZ, %s, %s, %s)'%(a.nrows(), a.ncols(), a.list())
return
print " (done)"
print "small %s x %s"%(n,m)
__do_check([random_matrix(ZZ, n, m, x=-1,y=1) for _ in range(times)])
print "big %s x %s"%(n,m)
__do_check([random_matrix(ZZ, n, m, x=-2^32,y=2^32) for _ in range(times)])
print "small %s x %s"%(m,n)
__do_check([random_matrix(ZZ, m, n, x=-1,y=1) for _ in range(times)])
print "big %s x %s"%(m,n)
__do_check([random_matrix(ZZ, m, n, x=-2^32,y=2^32) for _ in range(times)])
print "sparse %s x %s"%(n,m)
__do_check([random_matrix(ZZ, n, m, density=0.1) for _ in range(times)])
print "sparse %s x %s"%(m,n)
__do_check([random_matrix(ZZ, m, n, density=0.1) for _ in range(times)])
print "ill conditioned -- 1000*A -- %s x %s"%(n,m)
__do_check([1000*random_matrix(ZZ, n, m, x=-1,y=1) for _ in range(times)])
print "ill conditioned -- 1000*A but one row -- %s x %s"%(n,m)
v = []
for _ in range(times):
a = 1000*random_matrix(ZZ, n, m, x=-1,y=1)
a[a.nrows()-1] = a[a.nrows()-1]/1000
v.append(a)
__do_check(v)
def sanity_checks(times=50,
n=8,
m=5,
proof=True,
stabilize=2,
check_using_magma=True):
"""
Run random sanity checks on the modular p-adic HNF with tall and wide matrices
both dense and sparse.
INPUT:
- times -- number of times to randomly try matrices with each shape
- n -- number of rows
- m -- number of columns
- proof -- test with proof true
- stabilize -- parameter to pass to hnf algorithm when proof is False
- check_using_magma -- if True use Magma instead of PARI to check
correctness of computed HNF's. Since PARI's HNF is buggy and slow (as of
2008-02-16 non-pivot entries sometimes are not normalized to be
nonnegative) the default is Magma.
EXAMPLES::
sage: import sage.matrix.matrix_integer_dense_hnf as matrix_integer_dense_hnf
sage: matrix_integer_dense_hnf.sanity_checks(times=5, check_using_magma=False)
small 8 x 5
0 1 2 3 4 (done)
big 8 x 5
0 1 2 3 4 (done)
small 5 x 8
0 1 2 3 4 (done)
big 5 x 8
0 1 2 3 4 (done)
sparse 8 x 5
0 1 2 3 4 (done)
sparse 5 x 8
0 1 2 3 4 (done)
ill conditioned -- 1000*A -- 8 x 5
0 1 2 3 4 (done)
ill conditioned -- 1000*A but one row -- 8 x 5
0 1 2 3 4 (done)
"""
if check_using_magma:
from sage.interfaces.all import magma
def __do_check(v):
"""
This is used internally by the sanity check code.
"""
for i, a in enumerate(v):
global sanity
sanity = a
print(i, end=" ")
if check_using_magma:
if magma(hnf(a)[0]) != magma(a).EchelonForm():
print("bug computing hnf of a matrix")
print('a = matrix(ZZ, %s, %s, %s)' %
(a.nrows(), a.ncols(), a.list()))
return
else:
if hnf(a)[0] != a.echelon_form(algorithm='pari'):
print("bug computing hnf of a matrix")
print('a = matrix(ZZ, %s, %s, %s)' %
(a.nrows(), a.ncols(), a.list()))
return
print(" (done)")
print("small %s x %s" % (n, m))
__do_check([random_matrix(ZZ, n, m, x=-1, y=1) for _ in range(times)])
print("big %s x %s" % (n, m))
__do_check(
[random_matrix(ZZ, n, m, x=-2**32, y=2**32) for _ in range(times)])
print("small %s x %s" % (m, n))
__do_check([random_matrix(ZZ, m, n, x=-1, y=1) for _ in range(times)])
print("big %s x %s" % (m, n))
__do_check(
[random_matrix(ZZ, m, n, x=-2**32, y=2**32) for _ in range(times)])
print("sparse %s x %s" % (n, m))
__do_check([random_matrix(ZZ, n, m, density=0.1) for _ in range(times)])
print("sparse %s x %s" % (m, n))
__do_check([random_matrix(ZZ, m, n, density=0.1) for _ in range(times)])
print("ill conditioned -- 1000*A -- %s x %s" % (n, m))
__do_check(
[1000 * random_matrix(ZZ, n, m, x=-1, y=1) for _ in range(times)])
print("ill conditioned -- 1000*A but one row -- %s x %s" % (n, m))
v = []
for _ in range(times):
a = 1000 * random_matrix(ZZ, n, m, x=-1, y=1)
a[a.nrows() - 1] = a[a.nrows() - 1] / 1000
v.append(a)
__do_check(v)
def solve_system_with_difficult_last_row(B, A):
"""
Solve the matrix equation B*Z = A when the last row of $B$
contains huge entries.
INPUT:
- B -- a square n x n nonsingular matrix with painful big bottom row.
- A -- an n x k matrix.
OUTPUT:
the unique solution to B*Z = A.
EXAMPLES::
sage: from sage.matrix.matrix_integer_dense_saturation import solve_system_with_difficult_last_row
sage: B = matrix(ZZ, 3, [1,2,3, 3,-1,2,939239082,39202803080,2939028038402834]); A = matrix(ZZ,3,2,[1,2,4,3,-1,0])
sage: X = solve_system_with_difficult_last_row(B, A); X
[ 290668794698843/226075992027744 468068726971/409557956572]
[-226078357385539/1582531944194208 1228691305937/2866905696004]
[ 2365357795/1582531944194208 -17436221/2866905696004]
sage: B*X == A
True
"""
# See the comments in the function of the same name in matrix_integer_dense_hnf.py.
# This function is just a generalization of that one to A a matrix.
C = copy(B)
while True:
C[C.nrows() - 1] = random_matrix(ZZ, 1, C.ncols()).row(0)
try:
X = C.solve_right(A)
except ValueError:
verbose("Try difficult solve again with different random vector")
else:
break
D = B.matrix_from_rows(range(C.nrows() - 1))
N = D._rational_kernel_flint()
if N.ncols() != 1:
verbose("Difficult solve quickly failed. Using direct approach.")
return B.solve_right(A)
tm = verbose("Recover correct linear combinations")
k = N.matrix_from_columns([0])
# The sought for solution Z to B*Z = A is some linear combination
# Z = X + alpha*k
# Let w be the last row of B; then Z satisfies
# w * Z = A'
# where A' is the last row of A. Thus
# w * (X + alpha*k) = A'
# so w * X + alpha*w*k = A'
# so alpha*w*k = A' - w*X.
w = B[-1] # last row of B
A_prime = A[-1] # last row of A
lhs = w * k
rhs = A_prime - w * X
if lhs[0] == 0:
verbose("Difficult solve quickly failed. Using direct approach.")
return B.solve_right(A)
for i in range(X.ncols()):
alpha = rhs[i] / lhs[0]
X.set_column(i, (X.matrix_from_columns([i]) + alpha * k).list())
verbose("Done getting linear combinations.", tm)
return X
def solve_system_with_difficult_last_row(B, a):
"""
Solve B*x = a when the last row of $B$ contains huge entries using
a clever trick that reduces the problem to solve C*x = a where $C$
is $B$ but with the last row replaced by something small, along
with one easy null space computation. The latter are both solved
$p$-adically.
INPUT:
B -- a square n x n nonsingular matrix with painful big bottom row.
a -- an n x 1 column matrix
OUTPUT:
the unique solution to B*x = a.
EXAMPLES:
sage: from sage.matrix.matrix_integer_dense_hnf import solve_system_with_difficult_last_row
sage: B = matrix(ZZ, 3, [1,2,4, 3,-4,7, 939082,2930982,132902384098234])
sage: a = matrix(ZZ,3,1, [1,2,5])
sage: z = solve_system_with_difficult_last_row(B, a)
sage: z
[ 106321906985474/132902379815497]
[132902385037291/1329023798154970]
[ -5221794/664511899077485]
sage: B*z
[1]
[2]
[5]
"""
# Here's how:
# 1. We make a copy of B but with the last *nasty* row of B replaced
# by a random very nice row.
C = copy(B)
while True:
C[C.nrows() - 1] = random_matrix(ZZ, 1, C.ncols()).row(0)
# 2. Then we find the unique solution to C * x = a
try:
x = C.solve_right(a)
except ValueError:
verbose("Try difficult solve again with different random vector")
else:
break
# 3. We next delete the last row of B and find a basis vector k
# for the 1-dimensional kernel.
D = B.matrix_from_rows(range(C.nrows() - 1))
N = D._rational_kernel_iml()
if N.ncols() != 1:
verbose("Try difficult solve again with different random vector")
return solve_system_with_difficult_last_row(B, a)
k = N.matrix_from_columns([0])
# 4. The sought for solution z to B*z = a is some linear combination
#
# z = x + alpha*k
#
# of x and k, where k is the above fixed basis for the kernel of D.
# Setting w to be the last row of B, this column vector z satisfies
#
# w * z = a'
#
# where a' is the last entry of a. Thus
#
# w * (x + alpha*k) = a'
#
# so w * x + alpha*w*k = a'
# so alpha*w*k = a' - w*x.
w = B[-1] # last row of B
a_prime = a[-1]
lhs = w * k
rhs = a_prime - w * x
if lhs[0] == 0:
verbose("Try difficult solve again with different random vector")
return solve_system_with_difficult_last_row(B, a)
alpha = rhs[0] / lhs[0]
z = x + alpha * k
return z