int GMRFLib_gsl_spd_inverse(gsl_matrix * A)
{
/*
* replace SPD matrix A with its inverse
*/
gsl_matrix *L;
gsl_vector *x;
size_t i, n;
assert(A->size1 == A->size2);
n = A->size1;
x = gsl_vector_calloc(n);
L = GMRFLib_gsl_duplicate_matrix(A);
gsl_linalg_cholesky_decomp(L);
for (i = 0; i < n; i++) {
gsl_vector_set_basis(x, i);
gsl_linalg_cholesky_svx(L, x);
gsl_matrix_set_col(A, i, x);
}
gsl_vector_free(x);
gsl_matrix_free(L);
return GMRFLib_SUCCESS;
}
int lseShurComplement(gsl_matrix * A, gsl_matrix * C,
gsl_vector * b, gsl_vector * d,
gsl_vector * x, gsl_vector * lambda, double * sigma)
{
int i;
double xi;
gsl_vector *c0, *S, *tau;
gsl_matrix *CT, *U;
gsl_permutation *perm;
gsl_vector_view row, cp;
gsl_matrix_view R;
if (A->size2 != C->size2) return -1;
if (A->size2 != x->size) return -1;
if (A->size1 < A->size2) return -1;
if (b != NULL && A->size1 != b->size) return -1;
if (C->size1 != d->size) return -1;
if (C->size1 != lambda->size) return -1;
c0 = gsl_vector_alloc(x->size);
gsl_matrix_get_row(c0, C, 0);
/* Cholesky factorization of A^T A = R^T R via QRPT decomposition */
perm = gsl_permutation_alloc(x->size);
tau = gsl_vector_alloc(x->size);
gsl_linalg_QRPT_decomp(A, tau, perm, &i, x);
/* cp = R^{-T} P A^T b = Q^T b */
if (b != NULL) {
gsl_linalg_QR_QTvec(A, tau, b);
cp = gsl_vector_subvector(b, 0, x->size);
}
gsl_vector_free(tau);
/* C P -> C */
R = gsl_matrix_submatrix(A, 0, 0, A->size2, A->size2);
for (i = 0; i < C->size1; ++i) {
row = gsl_matrix_row(C, i);
gsl_permute_vector(perm, &row.vector);
}
/* Compute C inv(R) -> C */
gsl_blas_dtrsm(CblasRight, CblasUpper, CblasNoTrans, CblasNonUnit, 1.0,
&R.matrix, C);
/* The Schur complement D = C C^T,
Compute SVD of D = U S^2 U^T by SVD of C^T = V S U^T */
CT = gsl_matrix_alloc(C->size2, C->size1);
gsl_matrix_transpose_memcpy(CT, C);
U = gsl_matrix_alloc(CT->size2, CT->size2);
S = gsl_vector_alloc(CT->size2);
gsl_linalg_SV_decomp(CT, U, S, lambda);
/* Right hand side of the Shur complement system
d - C (A^T A)^-1 A^T b = d - C cp -> d
(with C P R^-1 -> C and R^-T P^T A^T b -> cp) */
if (b != NULL) {
gsl_blas_dgemv(CblasNoTrans, -1.0, C, &cp.vector, 1.0, d);
}
/* Calculate S U^T lambda, where -lambda is the Lagrange multiplier */
gsl_blas_dgemv(CblasTrans, 1.0, U, d, 0.0, lambda);
gsl_vector_div(lambda, S);
/* Calculate sigma = || A x ||_2 = || x ||_2 (before inv(R) x -> x) */
*sigma = gsl_blas_dnrm2(lambda);
/* Compute inv(R)^T C^T lambda = C^T lambda (with C inv(R) ->C) */
gsl_blas_dgemv(CblasNoTrans, 1.0, CT, lambda, 0.0, x);
/* x = inv(A^T A) C^T lambda = inv(R) [inv(R)^T C^T lambda] */
if (R.matrix.data[R.matrix.size1 * R.matrix.size2 - 1] != 0.0) {
gsl_blas_dtrsv(CblasUpper, CblasNoTrans, CblasNonUnit, &R.matrix, x);
}
else { /* Special case when A is singular */
gsl_vector_set_basis(x, x->size - 1);
*sigma = 0.0;
}
/* Permute back, 1-step iterative refinement on first constraint */
gsl_permute_vector_inverse(perm, x);
gsl_blas_ddot(x, c0, &xi);
gsl_vector_scale(x, d->data[0] / xi);
/* get the real lambda from S U^T lambda previously stored in lambda */
gsl_vector_div(lambda, S);
gsl_vector_memcpy(S, lambda);
gsl_blas_dgemv(CblasNoTrans, 1.0, U, S, 0.0, lambda);
gsl_vector_free(c0);
gsl_vector_free(S);
gsl_matrix_free(U);
gsl_matrix_free(CT);
gsl_permutation_free(perm);
return 0;
}
