void QR_update (Matrix& Q, Matrix& R,
VectorT& w, const VectorT& v)
{
Assert(Q.m == R.m && Q.n == R.m);
Assert(w.n == R.m);
Assert(v.n == R.n);
int j, k;
T w0;
/* Apply Given's rotations to reduce w to (|w|, 0, 0, ... , 0)
J_1^T .... J_(n-1)^T w = +/- |w| e_1
simultaneously applied to R, H = J_1^T ... J^T_(n-1) R
so that H is upper Hessenberg. (12.5.2) */
for (k = R.m - 1; k > 0; k--) {
double c, s;
double wk = w(k);
double wkm1 = w(k-1);
create_givens (wkm1, wk, &c, &s);
apply_givens_vec (w, k - 1, k, c, s);
apply_givens_qr (R.m, R.n, Q, R, k - 1, k, c, s);
}
w0 = w(0);
/* Add in w v^T (Equation 12.5.3) */
for (j = 0; j < R.n; j++)
R(0,j) += w0 * v(j);
/* Apply Givens transformations R' = G_(n-1)^T ... G_1^T H
Equation 12.5.4 */
for (k=1; k<Min(R.m,R.n+1); k++) {
double c, s;
double diag = R(k-1,k-1);
double offdiag = R(k,k-1);
create_givens (diag, offdiag, &c, &s);
apply_givens_qr (R.m, R.n, Q, R, k - 1, k, c, s);
R(k,k-1)=0;
}
}
int
gsl_linalg_PTLQ_update (gsl_matrix * Q, gsl_matrix * L,
const gsl_permutation * p,
const gsl_vector * v, gsl_vector * w)
{
if (Q->size1 != Q->size2 || L->size1 != L->size2)
{
return GSL_ENOTSQR;
}
else if (L->size1 != Q->size2 || v->size != Q->size2 || w->size != Q->size2)
{
return GSL_EBADLEN;
}
else
{
size_t j, k;
const size_t N = Q->size1;
const size_t M = Q->size2;
double w0;
/* Apply Given's rotations to reduce w to (|w|, 0, 0, ... , 0)
J_1^T .... J_(n-1)^T w = +/- |w| e_1
simultaneously applied to L, H = J_1^T ... J^T_(n-1) L
so that H is upper Hessenberg. (12.5.2) */
for (k = M - 1; k > 0; k--)
{
double c, s;
double wk = gsl_vector_get (w, k);
double wkm1 = gsl_vector_get (w, k - 1);
create_givens (wkm1, wk, &c, &s);
apply_givens_vec (w, k - 1, k, c, s);
apply_givens_lq (M, N, Q, L, k - 1, k, c, s);
}
w0 = gsl_vector_get (w, 0);
/* Add in v w^T (Equation 12.5.3) */
for (j = 0; j < N; j++)
{
double lj0 = gsl_matrix_get (L, j, 0);
size_t p_j = gsl_permutation_get (p, j);
double vj = gsl_vector_get (v, p_j);
gsl_matrix_set (L, j, 0, lj0 + w0 * vj);
}
/* Apply Givens transformations L' = G_(n-1)^T ... G_1^T H
Equation 12.5.4 */
for (k = 1; k < N; k++)
{
double c, s;
double diag = gsl_matrix_get (L, k - 1, k - 1);
double offdiag = gsl_matrix_get (L, k - 1, k );
create_givens (diag, offdiag, &c, &s);
apply_givens_lq (M, N, Q, L, k - 1, k, c, s);
}
return GSL_SUCCESS;
}
}
int
gsl_linalg_LQ_update (gsl_matrix * Q, gsl_matrix * L,
const gsl_vector * v, gsl_vector * w)
{
const size_t N = L->size1;
const size_t M = L->size2;
if (Q->size1 != M || Q->size2 != M)
{
GSL_ERROR ("Q matrix must be N x N if L is M x N", GSL_ENOTSQR);
}
else if (w->size != M)
{
GSL_ERROR ("w must be length N if L is M x N", GSL_EBADLEN);
}
else if (v->size != N)
{
GSL_ERROR ("v must be length M if L is M x N", GSL_EBADLEN);
}
else
{
size_t j, k;
double w0;
/* Apply Given's rotations to reduce w to (|w|, 0, 0, ... , 0)
J_1^T .... J_(n-1)^T w = +/- |w| e_1
simultaneously applied to L, H = J_1^T ... J^T_(n-1) L
so that H is upper Hessenberg. (12.5.2) */
for (k = M - 1; k > 0; k--) /* loop from k = M-1 to 1 */
{
double c, s;
double wk = gsl_vector_get (w, k);
double wkm1 = gsl_vector_get (w, k - 1);
create_givens (wkm1, wk, &c, &s);
apply_givens_vec (w, k - 1, k, c, s);
apply_givens_lq (M, N, Q, L, k - 1, k, c, s);
}
w0 = gsl_vector_get (w, 0);
/* Add in v w^T (Equation 12.5.3) */
for (j = 0; j < N; j++)
{
double lj0 = gsl_matrix_get (L, j, 0);
double vj = gsl_vector_get (v, j);
gsl_matrix_set (L, j, 0, lj0 + w0 * vj);
}
/* Apply Givens transformations L' = G_(n-1)^T ... G_1^T H
Equation 12.5.4 */
for (k = 1; k < GSL_MIN(M,N+1); k++)
{
double c, s;
double diag = gsl_matrix_get (L, k - 1, k - 1);
double offdiag = gsl_matrix_get (L, k - 1 , k);
create_givens (diag, offdiag, &c, &s);
apply_givens_lq (M, N, Q, L, k - 1, k, c, s);
gsl_matrix_set (L, k - 1, k, 0.0); /* exact zero of G^T */
}
return GSL_SUCCESS;
}
}
int
gsl_linalg_QRPT_update (gsl_matrix * Q, gsl_matrix * R,
const gsl_permutation * p,
gsl_vector * w, const gsl_vector * v)
{
const size_t M = R->size1;
const size_t N = R->size2;
if (Q->size1 != M || Q->size2 != M)
{
GSL_ERROR ("Q matrix must be M x M if R is M x N", GSL_ENOTSQR);
}
else if (w->size != M)
{
GSL_ERROR ("w must be length M if R is M x N", GSL_EBADLEN);
}
else if (v->size != N)
{
GSL_ERROR ("v must be length N if R is M x N", GSL_EBADLEN);
}
else
{
size_t j, k;
double w0;
/* Apply Given's rotations to reduce w to (|w|, 0, 0, ... , 0)
J_1^T .... J_(n-1)^T w = +/- |w| e_1
simultaneously applied to R, H = J_1^T ... J^T_(n-1) R
so that H is upper Hessenberg. (12.5.2) */
for (k = M - 1; k > 0; k--)
{
double c, s;
double wk = gsl_vector_get (w, k);
double wkm1 = gsl_vector_get (w, k - 1);
create_givens (wkm1, wk, &c, &s);
apply_givens_vec (w, k - 1, k, c, s);
apply_givens_qr (M, N, Q, R, k - 1, k, c, s);
}
w0 = gsl_vector_get (w, 0);
/* Add in w v^T (Equation 12.5.3) */
for (j = 0; j < N; j++)
{
double r0j = gsl_matrix_get (R, 0, j);
size_t p_j = gsl_permutation_get (p, j);
double vj = gsl_vector_get (v, p_j);
gsl_matrix_set (R, 0, j, r0j + w0 * vj);
}
/* Apply Givens transformations R' = G_(n-1)^T ... G_1^T H
Equation 12.5.4 */
for (k = 1; k < GSL_MIN(M,N+1); k++)
{
double c, s;
double diag = gsl_matrix_get (R, k - 1, k - 1);
double offdiag = gsl_matrix_get (R, k, k - 1);
create_givens (diag, offdiag, &c, &s);
apply_givens_qr (M, N, Q, R, k - 1, k, c, s);
gsl_matrix_set (R, k, k - 1, 0.0); /* exact zero of G^T */
}
return GSL_SUCCESS;
}
}
