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/* This function computes the solution to the least squares system
phi = [ A x = b , lambda D x = 0 ]^2
where A is an M by N matrix, D is an N by N diagonal matrix, lambda
is a scalar parameter and b is a vector of length M.
The function requires the factorization of A into A = Q R P^T,
where Q is an orthogonal matrix, R is an upper triangular matrix
with diagonal elements of non-increasing magnitude and P is a
permuation matrix. The system above is then equivalent to
[ R z = Q^T b, P^T (lambda D) P z = 0 ]
where x = P z. If this system does not have full rank then a least
squares solution is obtained. On output the function also provides
an upper triangular matrix S such that
P^T (A^T A + lambda^2 D^T D) P = S^T S
Parameters,
r: On input, contains the full upper triangle of R. On output the
strict lower triangle contains the transpose of the strict upper
triangle of S, and the diagonal of S is stored in sdiag. The full
upper triangle of R is not modified.
p: the encoded form of the permutation matrix P. column j of P is
column p[j] of the identity matrix.
lambda, diag: contains the scalar lambda and the diagonal elements
of the matrix D
qtb: contains the product Q^T b
x: on output contains the least squares solution of the system
wa: is a workspace of length N
*/
static int
qrsolv (gsl_matrix * r, const gsl_permutation * p, const double lambda,
const gsl_vector * diag, const gsl_vector * qtb,
gsl_vector * x, gsl_vector * sdiag, gsl_vector * wa)
{
size_t n = r->size2;
size_t i, j, k, nsing;
/* Copy r and qtb to preserve input and initialise s. In particular,
save the diagonal elements of r in x */
for (j = 0; j < n; j++)
{
double rjj = gsl_matrix_get (r, j, j);
double qtbj = gsl_vector_get (qtb, j);
for (i = j + 1; i < n; i++)
{
double rji = gsl_matrix_get (r, j, i);
gsl_matrix_set (r, i, j, rji);
}
gsl_vector_set (x, j, rjj);
gsl_vector_set (wa, j, qtbj);
}
/* Eliminate the diagonal matrix d using a Givens rotation */
for (j = 0; j < n; j++)
{
double qtbpj;
size_t pj = gsl_permutation_get (p, j);
double diagpj = lambda * gsl_vector_get (diag, pj);
if (diagpj == 0)
{
continue;
}
gsl_vector_set (sdiag, j, diagpj);
for (k = j + 1; k < n; k++)
{
gsl_vector_set (sdiag, k, 0.0);
}
/* The transformations to eliminate the row of d modify only a
single element of qtb beyond the first n, which is initially
zero */
qtbpj = 0;
for (k = j; k < n; k++)
{
/* Determine a Givens rotation which eliminates the
appropriate element in the current row of d */
double sine, cosine;
double wak = gsl_vector_get (wa, k);
double rkk = gsl_matrix_get (r, k, k);
double sdiagk = gsl_vector_get (sdiag, k);
if (sdiagk == 0)
{
continue;
}
if (fabs (rkk) < fabs (sdiagk))
{
double cotangent = rkk / sdiagk;
sine = 0.5 / sqrt (0.25 + 0.25 * cotangent * cotangent);
cosine = sine * cotangent;
}
else
{
double tangent = sdiagk / rkk;
cosine = 0.5 / sqrt (0.25 + 0.25 * tangent * tangent);
sine = cosine * tangent;
}
/* Compute the modified diagonal element of r and the
modified element of [qtb,0] */
{
double new_rkk = cosine * rkk + sine * sdiagk;
double new_wak = cosine * wak + sine * qtbpj;
qtbpj = -sine * wak + cosine * qtbpj;
gsl_matrix_set(r, k, k, new_rkk);
gsl_vector_set(wa, k, new_wak);
}
/* Accumulate the transformation in the row of s */
for (i = k + 1; i < n; i++)
{
double rik = gsl_matrix_get (r, i, k);
double sdiagi = gsl_vector_get (sdiag, i);
double new_rik = cosine * rik + sine * sdiagi;
double new_sdiagi = -sine * rik + cosine * sdiagi;
gsl_matrix_set(r, i, k, new_rik);
gsl_vector_set(sdiag, i, new_sdiagi);
}
}
/* Store the corresponding diagonal element of s and restore the
corresponding diagonal element of r */
{
double rjj = gsl_matrix_get (r, j, j);
double xj = gsl_vector_get(x, j);
gsl_vector_set (sdiag, j, rjj);
gsl_matrix_set (r, j, j, xj);
}
}
/* Solve the triangular system for z. If the system is singular then
obtain a least squares solution */
nsing = n;
for (j = 0; j < n; j++)
{
double sdiagj = gsl_vector_get (sdiag, j);
if (sdiagj == 0)
{
nsing = j;
break;
}
}
for (j = nsing; j < n; j++)
{
gsl_vector_set (wa, j, 0.0);
}
for (k = 0; k < nsing; k++)
{
double sum = 0;
j = (nsing - 1) - k;
for (i = j + 1; i < nsing; i++)
{
sum += gsl_matrix_get(r, i, j) * gsl_vector_get(wa, i);
}
{
double waj = gsl_vector_get (wa, j);
double sdiagj = gsl_vector_get (sdiag, j);
gsl_vector_set (wa, j, (waj - sum) / sdiagj);
}
}
/* Permute the components of z back to the components of x */
for (j = 0; j < n; j++)
{
size_t pj = gsl_permutation_get (p, j);
double waj = gsl_vector_get (wa, j);
gsl_vector_set (x, pj, waj);
}
return GSL_SUCCESS;
}
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