def qrp(A, rank = None, extra_prec = 16):
m = A.nrows()
n = A.ncols()
s = min(m,n)
if rank is not None:
#s = min(s, rank + 1);
pass;
base_prec = A.base_ring().precision();
base_field = A.base_ring()
if is_ComplexField(base_field):
field = ComplexField(base_prec + extra_prec + m);
else:
field = RealField(base_prec + extra_prec + m);
# field = base_field;
R = copy(A);
A = A.change_ring(field);
P = [ j for j in xrange(n) ];
Q = identity_matrix(field,m)
# print A.base_ring()
# print Q.base_ring()
normbound = field(2)**(-0.5*field.precision());
colnorms2 = [ float_sum([ R[k,j].norm() for k in range(m)]) for j in range(n) ];
for j in xrange(s):
# find column with max norm
maxnorm2 = colnorms2[j];
p = j;
for k in xrange(j, n):
if colnorms2[k] > maxnorm2:
p = k;
maxnorm2 = colnorms2[k];
#it is worth it to recompute the maxnorm
xnorm2 = float_sum([ R[i,p].norm() for i in range(j, m)])
xnorm = sqrt(xnorm2);
# swap j and p in A, P and colnorms
if j != p:
P[j], P[p] = P[p], P[j]
colnorms2[j], colnorms2[p] = colnorms2[p], colnorms2[j]
for i in xrange(0, m):
R[i, j], R[i, p] = R[i, p], R[i, j];
# compute householder vector
u = [ R[k,j] for k in xrange(j, m) ];
alpha = R[j,j];
# alphanorm = alpha.abs();
if alpha.real() < 0:
u[0] -= xnorm
z = -1
else:
u[0] += xnorm
z = 1
beta = xnorm*(xnorm + alpha * z);
if beta.abs() < normbound:
beta = float_sum( [u[i].conjugate() * R[i+j,j] for i in range(m - j)])
if beta == field(0):
break;
beta = 1/beta;
# H = I - beta * u * u^*
# Apply householder matrix
#update R
R[j,j] = -z * xnorm;
for i in xrange(j+1,m):
R[i,j] = 0
for k in xrange(j+1, n):
lambdR = beta * float_sum( [u[l - j].conjugate() * R[l, k] for l in xrange(j, m)] );
for i in xrange(j, m):
R[i, k] -= lambdR * u[i - j];
#update Q
for k in xrange(m):
lambdQ = float_sum( [u[l - j].conjugate() * Q[l, k] for l in xrange(j, m)] );
lambdQ *= beta
for i in xrange(j,m):
Q[i, k] -= lambdQ * u[i - j];
for k in xrange(j + 1, n):
square = R[j,k].norm();
colnorms2[k] -= square;
# sometimes is better to just recompute the norm
if True: #colnorms2[k] < normbound or colnorms2[k] < square*normbound:
colnorms2[k] = float_sum([ R[i,k].norm() for i in range(j + 1, m)])
# compute P matrix
Pmatrix = Matrix(ZZ, n);
for i in xrange(n):
Pmatrix[ P[i], i] = 1;
return Q.conjugate_transpose().change_ring(base_field), R.change_ring(base_field), Pmatrix;
