// abs(x'*y)/\sqrt(x'*x)/sqrt(y'*y)
double dvec_dcos_abs(int n, const double *x, const double *y){
double norm_x,norm_y,dcos,dot;
norm_x=dvec_norm2(n,x);
norm_y=dvec_norm2(n,y);
dot=dvec_dot(n,x,y);
dcos=fabs(dot)/(norm_x*norm_y);
return dcos;
}
// x=x/sqrt(x'*x) where x[i]>0, abs(x[i]) is max
void dvec_normalize_sgn(int n, double *x)
{
int k;
double a=0;
a=dvec_norm2(n,x);
a=1.0/a;
dvec_max_abs_index(n,x,&k);
if(x[k]<0) a=-a;
dvec_scale(n,x,a);
}
int main(int argc, char *argv[])
{
struct sparse_matrix_t *sparseA = NULL;
struct vector_t *b = NULL;
struct vector_t *x;
struct mesh_t *mesh;
char *xml_output;
long int *compress2fat = NULL;
struct vector_t *solution;
struct vector_t *std_error_sol;
long int fat_sol_nb_col;
lsqr_input *input;
lsqr_output *output;
lsqr_work *work; /* zone temoraire de travail */
lsqr_func *func; /* func->mat_vec_prod -> APROD */
/* cmd line arg */
char *mesh_filename = NULL;
char *importfilename = NULL;
char *output_filename = NULL;
char *sol_error_filename = NULL;
char *log_filename = NULL;
char *output_type = NULL;
int max_iter;
double damping, grad_damping;
int use_ach = 0; /* ACH : tele-seismic inversion tomography */
int check_sparse = 0; /* check sparse matrix disable by default */
/* velocity model */
char *vmodel = NULL;
struct velocity_model_t *vm = NULL;
struct mesh_t **imported_mesh = NULL;
char **xmlfilelist = NULL;
int nb_xmlfile = 0;
int i, j;
int nb_irm = 0;
struct irm_t **irm = NULL;
int *nb_metacell = NULL;
FILE *logfd;
/*************************************************************/
parse_command_line(argc, argv,
&mesh_filename,
&vmodel,
&importfilename,
&log_filename,
&output_filename, &output_type,
&max_iter,
&damping, &grad_damping, &use_ach, &check_sparse);
if (use_ach) {
fprintf(stderr, "Using ACH tomographic inversion\n");
} else {
fprintf(stderr, "Using STANDARD tomographic inversion\n");
}
/* load the velocity model */
if (vmodel) {
char *myfile;
vm = load_velocity_model(vmodel);
if (!vm) {
fprintf(stderr, "Can not initialize velocity model '%s'\n",
vmodel);
exit(1);
}
myfile = strdup(vmodel);
fprintf(stderr, "Velocity model '%s' loaded\n", basename(myfile));
free(myfile);
} else {
vm = NULL;
}
/* Open log file */
if (!log_filename) {
logfd = stdout;
} else {
if (!(logfd = fopen(log_filename, "w"))) {
perror(log_filename);
exit(1);
}
}
/*check_write_access (output_filename); */
/**************************************/
/* test if we can open file to import */
/**************************************/
if (importfilename) {
xmlfilelist = parse_separated_list(importfilename, ",");
nb_xmlfile = 0;
while (xmlfilelist[nb_xmlfile]) {
if (access(xmlfilelist[nb_xmlfile], R_OK) == -1) {
perror(xmlfilelist[nb_xmlfile]);
exit(1);
}
nb_xmlfile++;
}
} else {
fprintf(stderr, "No file to import ... exiting\n");
exit(0);
}
/****************************/
/* main mesh initialization */
/****************************/
mesh = mesh_init_from_file(mesh_filename);
if (!mesh) {
fprintf(stderr, "Error decoding %s.\n", mesh_filename);
exit(1);
}
fprintf(stderr, "read %s ok\n", mesh_filename);
/*****************************************/
/* check and initialize slice xml files */
/*****************************************/
if (nb_xmlfile) {
int nb_sparse = 0;
int nb_res = 0;
int f;
imported_mesh =
(struct mesh_t **) malloc(sizeof(struct mesh_t *) *
nb_xmlfile);
assert(imported_mesh);
for (i = 0; i < nb_xmlfile; i++) {
imported_mesh[i] = mesh_init_from_file(xmlfilelist[i]);
if (!imported_mesh[i]) {
fprintf(stderr, "Error decoding %s.\n", mesh_filename);
exit(1);
}
for (f = 0; f < NB_MESH_FILE_FORMAT; f++) {
/* mandatory field : res, sparse, and irm if provided */
if (f == RES || f == SPARSE || f == IRM) {
check_files_access(f, imported_mesh[i]->data[f],
xmlfilelist[i]);
}
}
if (imported_mesh[i]->data[SPARSE]) {
nb_sparse += imported_mesh[i]->data[SPARSE]->ndatafile;
}
if (imported_mesh[i]->data[RES]) {
nb_res += imported_mesh[i]->data[RES]->ndatafile;
}
if (imported_mesh[i]->data[IRM]) {
nb_irm += imported_mesh[i]->data[IRM]->ndatafile;
}
}
if (!nb_sparse || !nb_res) {
fprintf(stderr, "Error no sparse or res file available !\n");
exit(0);
}
}
/*********************************************/
/* read and import the sparse(s) matrix(ces) */
/*********************************************/
for (i = 0; i < nb_xmlfile; i++) {
if (!imported_mesh[i]->data[SPARSE]) {
continue;
}
for (j = 0; j < imported_mesh[i]->data[SPARSE]->ndatafile; j++) {
sparseA = import_sparse_matrix(sparseA,
imported_mesh[i]->data[SPARSE]->
filename[j]);
}
}
if (check_sparse) {
if (check_sparse_matrix(sparseA)) {
exit(1);
}
}
/*sparse_compute_length(sparseA, "length1.txt"); */
fat_sol_nb_col = sparseA->nb_col;
show_sparse_stats(sparseA);
/*********************************************/
/* read and import the residual time vector */
/*********************************************/
for (i = 0; i < nb_xmlfile; i++) {
if (!imported_mesh[i]->data[RES]) {
continue;
}
for (j = 0; j < imported_mesh[i]->data[RES]->ndatafile; j++) {
b = import_vector(b, imported_mesh[i]->data[RES]->filename[j]);
}
}
/*************************************************/
/* check compatibility between matrix and vector */
/*************************************************/
if (sparseA->nb_line != b->length) {
fprintf(stderr,
"Error, check your matrix/vector sizes (%ld/%ld)\n",
sparseA->nb_line, b->length);
exit(1);
}
/********************/
/* show memory used */
/********************/
#ifdef __APPLE__
{
struct mstats memusage;
memusage = mstats();
fprintf(stderr, "Memory used: %.2f MBytes\n",
(float) (memusage.bytes_used) / (1024. * 1024));
}
#else
{
struct mallinfo m_info;
m_info = mallinfo();
fprintf(stderr, "Memory used: %.2f MBytes\n",
(float) (m_info.uordblks +
m_info.usmblks) / (1024. * 1024.));
}
#endif
/**************************************/
/* relative traveltime mode */
/**************************************/
if (use_ach) {
int nb_evt_imported = 0;
for (i = 0; i < nb_xmlfile; i++) {
if (!imported_mesh[i]->data[EVT]) {
continue;
}
for (j = 0; j < imported_mesh[i]->data[EVT]->ndatafile; j++) {
relative_tt(sparseA, b,
imported_mesh[i]->data[EVT]->filename[j]);
nb_evt_imported++;
}
}
if (!nb_evt_imported) {
fprintf(stderr,
"Error in ACH mode, can not import any .evt file !\n");
exit(1);
}
}
/************************************************/
/* read the irregular mesh definition if needed */
/* one by layer */
/************************************************/
if (nb_irm) {
int cpt = 0;
struct mesh_offset_t **offset;
int l;
irm = (struct irm_t **) malloc(nb_irm * sizeof(struct irm_t *));
assert(irm);
nb_metacell = (int *) calloc(nb_irm, sizeof(int));
assert(nb_metacell);
make_mesh(mesh);
for (i = 0; i < nb_xmlfile; i++) {
if (!imported_mesh[i]->data[IRM]) {
continue;
}
/* offset between meshes */
offset = compute_mesh_offset(mesh, imported_mesh[i]);
for (l = 0; l < mesh->nlayers; l++) {
if (!offset[l])
continue;
fprintf(stderr,
"\t%s, [%s] offset[layer=%d] : lat=%d lon=%d z=%d\n",
xmlfilelist[i], MESH_FILE_FORMAT[IRM], l,
offset[l]->lat, offset[l]->lon, offset[l]->z);
}
for (j = 0; j < imported_mesh[i]->data[IRM]->ndatafile; j++) {
/* FIXME: read only once the irm file */
irm[cpt] =
read_irm(imported_mesh[i]->data[IRM]->filename[j],
&(nb_metacell[cpt]));
import2mesh_irm_file(mesh,
imported_mesh[i]->data[IRM]->
filename[j], offset);
cpt++;
}
for (l = 0; l < mesh->nlayers; l++) {
if (offset[l])
free(offset[l]);
}
free(offset);
}
metacell_find_neighbourhood(mesh);
}
/*sparse_compute_length(sparseA, "length1.txt"); */
fat_sol_nb_col = sparseA->nb_col;
show_sparse_stats(sparseA);
/***********************/
/* remove empty column */
/***********************/
fprintf(stderr, "starting compression ...\n");
sparse_compress_column(mesh, sparseA, &compress2fat);
if (check_sparse) {
if (check_sparse_matrix(sparseA)) {
exit(1);
}
}
show_sparse_stats(sparseA);
/***************************************/
/* add gradient damping regularisation */
/***************************************/
if (fabs(grad_damping) > 1.e-6) {
int nb_faces = 6; /* 1 cell may have 6 neighbours */
long int nb_lines = 0;
char *regul_name;
fprintf(stdout, "using gradient damping : %f\n", grad_damping);
/* tmp file name */
regul_name = tempnam("/tmp", "regul");
if (!regul_name) {
perror("lsqrsolve: ");
exit(1);
}
if (nb_irm) {
create_regul_DtD_irm(sparseA, compress2fat,
mesh, regul_name, nb_faces,
grad_damping, &nb_lines);
} else {
create_regul_DtD(sparseA, compress2fat,
mesh, regul_name, nb_faces,
grad_damping, &nb_lines);
}
sparse_matrix_resize(sparseA,
sparseA->nb_line + sparseA->nb_col,
sparseA->nb_col);
sparseA = import_sparse_matrix(sparseA, regul_name);
if (check_sparse) {
if (check_sparse_matrix(sparseA)) {
exit(1);
}
}
vector_resize(b, sparseA->nb_line);
unlink(regul_name);
show_sparse_stats(sparseA);
}
/*********************************/
/* the real mesh is no more used */
/* keep only the light mesh */
/*********************************/
fprintf(stdout,
"Time to free the real mesh and keep only the light structure\n");
free_mesh(mesh);
mesh = mesh_init_from_file(mesh_filename);
if (!mesh) {
fprintf(stderr, "Error decoding %s.\n", mesh_filename);
exit(1);
}
fprintf(stderr, "read %s ok\n", mesh_filename);
/********************************/
/* init vector solution to zero */
/********************************/
x = new_vector(sparseA->nb_col);
/*************************************************************/
/* solve A.x = B */
/* A = ray length in the cells */
/* B = residual travel time observed - computed */
/* x solution to satisfy the lsqr problem */
/*************************************************************/
/* LSQR alloc */
alloc_lsqr_mem(&input, &output, &work, &func, sparseA->nb_line,
sparseA->nb_col);
fprintf(stderr, "alloc_lsqr_mem : ok\n");
/* defines the routine Mat.Vect to use */
func->mat_vec_prod = sparseMATRIXxVECTOR;
/* Set the input parameters for LSQR */
input->num_rows = sparseA->nb_line;
input->num_cols = sparseA->nb_col;
input->rel_mat_err = 1.0e-3; /* in km */
input->rel_rhs_err = 1.0e-2; /* in seconde */
/*input->rel_mat_err = 0.;
input->rel_rhs_err = 0.; */
input->cond_lim = .0;
input->lsqr_fp_out = logfd;
/* input->rhs_vec = (dvec *) b; */
dvec_copy((dvec *) b, input->rhs_vec);
input->sol_vec = (dvec *) x; /* initial guess */
input->damp_val = damping;
if (max_iter == -1) {
input->max_iter = 4 * (sparseA->nb_col);
} else {
input->max_iter = max_iter;
}
/* catch Ctrl-C signal */
signal(SIGINT, emergency_halt);
/******************************/
/* resolution du systeme Ax=b */
/******************************/
lsqr(input, output, work, func, sparseA);
fprintf(stderr, "*** lsqr ended (%ld iter) : %s\n",
output->num_iters, lsqr_msg[output->term_flag]);
if (output->term_flag == 0) { /* solution x=x0 */
exit(0);
}
/* uncompress the solution */
solution = uncompress_column((struct vector_t *) output->sol_vec,
compress2fat, fat_sol_nb_col);
/* uncompress the standard error on solution */
std_error_sol =
uncompress_column((struct vector_t *) output->std_err_vec,
compress2fat, fat_sol_nb_col);
/* if irm file was provided, set the right value to each cell
* from a given metacell
*/
if (irm) {
irm_update(solution, irm, nb_metacell, nb_irm, mesh);
free_irm(irm, nb_irm);
free(nb_metacell);
}
/* write solution */
if (strchr(output_type, 'm')) {
export2matlab(solution, output_filename, mesh, vm,
output->num_iters,
input->damp_val, grad_damping, use_ach);
}
if (strchr(output_type, 's')) {
export2sco(solution, output_filename, mesh, vm,
output->num_iters,
input->damp_val, grad_damping, use_ach);
}
if (strchr(output_type, 'g')) {
/* solution */
export2gmt(solution, output_filename, mesh, vm,
output->num_iters,
input->damp_val, grad_damping, use_ach);
/* error on solution */
sol_error_filename = (char *)
malloc(sizeof(char) *
(strlen(output_filename) + strlen(".err") + 1));
sprintf(sol_error_filename, "%s.err", output_filename);
export2gmt(std_error_sol, sol_error_filename, mesh, vm,
output->num_iters,
input->damp_val, grad_damping, use_ach);
free(sol_error_filename);
}
/* save the xml enrichied with sections */
xml_output = (char *)
malloc((strlen(output_filename) + strlen(".xml") +
1) * sizeof(char));
assert(xml_output);
sprintf(xml_output, "%s.xml", output_filename);
mesh2xml(mesh, xml_output);
free(xml_output);
/******************************************************/
/* variance reduction, ie how the model fits the data */
/* X = the final solution */
/* */
/* ||b-AX||² */
/* VR= 1 - -------- */
/* ||b||² */
/* */
/******************************************************/
{
double norm_b;
double norm_b_AX;
double VR; /* variance reduction */
struct vector_t *rhs; /* right hand side */
rhs = new_vector(sparseA->nb_line);
/* use copy */
dvec_copy((dvec *) b, (dvec *) rhs);
norm_b = dvec_norm2((dvec *) rhs);
/* does rhs = rhs + sparseA . output->sol_vec */
/* here rhs is overwritten */
dvec_scale((-1.0), (dvec *) rhs);
sparseMATRIXxVECTOR(0, output->sol_vec, (dvec *) rhs, sparseA);
dvec_scale((-1.0), (dvec *) rhs);
norm_b_AX = dvec_norm2((dvec *) rhs);
VR = 1 - (norm_b_AX * norm_b_AX) / (norm_b * norm_b);
fprintf(stdout, "Variance reduction = %.2f%%\n", VR * 100);
free_vector(rhs);
}
/********/
/* free */
/********/
if (vm) {
free_velocity_model(vm);
}
free_mesh(mesh);
free_sparse_matrix(sparseA);
free_lsqr_mem(input, output, work, func);
free_vector(solution);
free_vector(std_error_sol);
free(compress2fat);
for (i = 0; i < nb_xmlfile; i++) {
free(xmlfilelist[i]);
free_mesh(imported_mesh[i]);
}
free(xmlfilelist);
free(imported_mesh);
return (0);
}
// x=x/sqrt(x'*x)
void dvec_normalize(int n, double *x)
{
double norm=0;
norm=dvec_norm2(n,x);
dvec_scale(n,x,(1.0/norm));
}
/*
*------------------------------------------------------------------------------
*
* LSQR finds a solution x to the following problems:
*
* 1. Unsymmetric equations -- solve A*x = b
*
* 2. Linear least squares -- solve A*x = b
* in the least-squares sense
*
* 3. Damped least squares -- solve ( A )*x = ( b )
* ( damp*I ) ( 0 )
* in the least-squares sense
*
* where 'A' is a matrix with 'm' rows and 'n' columns, 'b' is an
* 'm'-vector, and 'damp' is a scalar. (All quantities are real.)
* The matrix 'A' is intended to be large and sparse.
*
*
* Notation
* --------
*
* The following quantities are used in discussing the subroutine
* parameters:
*
* 'Abar' = ( A ), 'bbar' = ( b )
* ( damp*I ) ( 0 )
*
* 'r' = b - A*x, 'rbar' = bbar - Abar*x
*
* 'rnorm' = sqrt( norm(r)**2 + damp**2 * norm(x)**2 )
* = norm( rbar )
*
* 'rel_prec' = the relative precision of floating-point arithmetic
* on the machine being used. Typically 2.22e-16
* with 64-bit arithmetic.
*
* LSQR minimizes the function 'rnorm' with respect to 'x'.
*
*
* References
* ----------
*
* C.C. Paige and M.A. Saunders, LSQR: An algorithm for sparse
* linear equations and sparse least squares,
* ACM Transactions on Mathematical Software 8, 1 (March 1982),
* pp. 43-71.
*
* C.C. Paige and M.A. Saunders, Algorithm 583, LSQR: Sparse
* linear equations and least-squares problems,
* ACM Transactions on Mathematical Software 8, 2 (June 1982),
* pp. 195-209.
*
* C.L. Lawson, R.J. Hanson, D.R. Kincaid and F.T. Krogh,
* Basic linear algebra subprograms for Fortran usage,
* ACM Transactions on Mathematical Software 5, 3 (Sept 1979),
* pp. 308-323 and 324-325.
*
*------------------------------------------------------------------------------
*/
void lsqr( lsqr_input *input, lsqr_output *output, lsqr_work *work,
lsqr_func *func,
void *prod,
int (*per_iteration_callback)(lsqr_input*, lsqr_output*, void* token),
void* token)
{
double dvec_norm2( dvec * );
long indx,
term_iter,
term_iter_max;
double alpha,
beta,
rhobar,
phibar,
bnorm,
bbnorm,
cs1,
sn1,
psi,
rho,
cs,
sn,
theta,
phi,
tau,
ddnorm,
delta,
gammabar,
zetabar,
gamma,
cs2,
sn2,
zeta,
xxnorm,
res,
resid_tol,
cond_tol,
resid_tol_mach,
temp,
stop_crit_1,
stop_crit_2,
stop_crit_3;
static char term_msg[8][80] =
{
"The exact solution is x = x0",
"The residual Ax - b is small enough, given ATOL and BTOL",
"The least squares error is small enough, given ATOL",
"The estimated condition number has exceeded CONLIM",
"The residual Ax - b is small enough, given machine precision",
"The least squares error is small enough, given machine precision",
"The estimated condition number has exceeded machine precision",
"The iteration limit has been reached"
};
if( input->lsqr_fp_out != NULL )
fprintf( input->lsqr_fp_out, " Least Squares Solution of A*x = b\n"
" The matrix A has %7li rows and %7li columns\n"
" The damping parameter is\tDAMP = %10.2e\n"
" ATOL = %10.2e\t\tCONDLIM = %10.2e\n"
" BTOL = %10.2e\t\tITERLIM = %10li\n\n",
input->num_rows, input->num_cols, input->damp_val, input->rel_mat_err,
input->cond_lim, input->rel_rhs_err, input->max_iter );
output->term_flag = 0;
term_iter = 0;
output->num_iters = 0;
output->frob_mat_norm = 0.0;
output->mat_cond_num = 0.0;
output->sol_norm = 0.0;
for(indx = 0; indx < input->num_cols; indx++)
{
work->bidiag_wrk_vec->elements[indx] = 0.0;
work->srch_dir_vec->elements[indx] = 0.0;
output->std_err_vec->elements[indx] = 0.0;
output->sol_vec->elements[indx] = 0.0;
}
bbnorm = 0.0;
ddnorm = 0.0;
xxnorm = 0.0;
cs2 = -1.0;
sn2 = 0.0;
zeta = 0.0;
res = 0.0;
if( input->cond_lim > 0.0 )
cond_tol = 1.0 / input->cond_lim;
else
cond_tol = DBL_EPSILON;
alpha = 0.0;
beta = 0.0;
/*
* Set up the initial vectors u and v for bidiagonalization. These satisfy
* the relations
* BETA*u = b - A*x0
* ALPHA*v = A^T*u
*/
/* Compute b - A*x0 and store in vector u which initially held vector b */
dvec_scale( (-1.0), input->rhs_vec );
func->mat_vec_prod( 0, input->sol_vec, input->rhs_vec, prod );
dvec_scale( (-1.0), input->rhs_vec );
/* compute Euclidean length of u and store as BETA */
beta = dvec_norm2( input->rhs_vec );
if( beta > 0.0 )
{
/* scale vector u by the inverse of BETA */
dvec_scale( (1.0 / beta), input->rhs_vec );
/* Compute matrix-vector product A^T*u and store it in vector v */
func->mat_vec_prod( 1, work->bidiag_wrk_vec, input->rhs_vec, prod );
/* compute Euclidean length of v and store as ALPHA */
alpha = dvec_norm2( work->bidiag_wrk_vec );
}
if( alpha > 0.0 )
{
/* scale vector v by the inverse of ALPHA */
dvec_scale( (1.0 / alpha), work->bidiag_wrk_vec );
/* copy vector v to vector w */
dvec_copy( work->bidiag_wrk_vec, work->srch_dir_vec );
}
output->mat_resid_norm = alpha * beta;
output->resid_norm = beta;
bnorm = beta;
/*
* If the norm || A^T r || is zero, then the initial guess is the exact
* solution. Exit and report this.
*/
if( (output->mat_resid_norm == 0.0) && (input->lsqr_fp_out != NULL) )
{
fprintf( input->lsqr_fp_out, "\tISTOP = %3li\t\t\tITER = %9li\n"
" || A ||_F = %13.5e\tcond( A ) = %13.5e\n"
" || r ||_2 = %13.5e\t|| A^T r ||_2 = %13.5e\n"
" || b ||_2 = %13.5e\t|| x - x0 ||_2 = %13.5e\n\n",
output->term_flag, output->num_iters, output->frob_mat_norm,
output->mat_cond_num, output->resid_norm, output->mat_resid_norm,
bnorm, output->sol_norm );
fprintf( input->lsqr_fp_out, " %s\n\n", term_msg[output->term_flag]);
return;
}
rhobar = alpha;
phibar = beta;
/*
* If statistics are printed at each iteration, print a header and the initial
* values for each quantity.
*/
if( input->lsqr_fp_out != NULL )
{
fprintf( input->lsqr_fp_out,
" ITER || r || Compatible "
"||A^T r|| / ||A|| ||r|| || A || cond( A )\n\n" );
stop_crit_1 = 1.0;
stop_crit_2 = alpha / beta;
fprintf( input->lsqr_fp_out,
"%6li %13.5e %10.2e \t%10.2e \t%10.2e %10.2e\n",
output->num_iters, output->resid_norm, stop_crit_1, stop_crit_2,
output->frob_mat_norm, output->mat_cond_num);
}
/*
* The main iteration loop is continued as long as no stopping criteria
* are satisfied and the number of total iterations is less than some upper
* bound.
*/
while( output->term_flag == 0 )
{
output->num_iters++;
/*
* Perform the next step of the bidiagonalization to obtain
* the next vectors u and v, and the scalars ALPHA and BETA.
* These satisfy the relations
* BETA*u = A*v - ALPHA*u,
* ALFA*v = A^T*u - BETA*v.
*/
/* scale vector u by the negative of ALPHA */
dvec_scale( (-alpha), input->rhs_vec );
/* compute A*v - ALPHA*u and store in vector u */
func->mat_vec_prod( 0, work->bidiag_wrk_vec, input->rhs_vec, prod );
/* compute Euclidean length of u and store as BETA */
beta = dvec_norm2( input->rhs_vec );
/* accumulate this quantity to estimate Frobenius norm of matrix A */
/* bbnorm += sqr(alpha) + sqr(beta) + sqr(input->damp_val);*/
bbnorm += alpha*alpha + beta*beta
+ input->damp_val*input->damp_val;
if( beta > 0.0 )
{
/* scale vector u by the inverse of BETA */
dvec_scale( (1.0 / beta), input->rhs_vec );
/* scale vector v by the negative of BETA */
dvec_scale( (-beta), work->bidiag_wrk_vec );
/* compute A^T*u - BETA*v and store in vector v */
func->mat_vec_prod( 1, work->bidiag_wrk_vec, input->rhs_vec, prod );
/* compute Euclidean length of v and store as ALPHA */
alpha = dvec_norm2( work->bidiag_wrk_vec );
if( alpha > 0.0 )
/* scale vector v by the inverse of ALPHA */
dvec_scale( (1.0 / alpha), work->bidiag_wrk_vec );
}
/*
* Use a plane rotation to eliminate the damping parameter.
* This alters the diagonal (RHOBAR) of the lower-bidiagonal matrix.
*/
cs1 = rhobar / sqrt( lsqr_sqr(rhobar) + lsqr_sqr(input->damp_val) );
sn1 = input->damp_val
/ sqrt( lsqr_sqr(rhobar) + lsqr_sqr(input->damp_val) );
psi = sn1 * phibar;
phibar = cs1 * phibar;
/*
* Use a plane rotation to eliminate the subdiagonal element (BETA)
* of the lower-bidiagonal matrix, giving an upper-bidiagonal matrix.
*/
rho = sqrt( lsqr_sqr(rhobar) + lsqr_sqr(input->damp_val)
+ lsqr_sqr(beta) );
cs = sqrt( lsqr_sqr(rhobar) + lsqr_sqr(input->damp_val) ) / rho;
sn = beta / rho;
theta = sn * alpha;
rhobar = -cs * alpha;
phi = cs * phibar;
phibar = sn * phibar;
tau = sn * phi;
/*
* Update the solution vector x, the search direction vector w, and the
* standard error estimates vector se.
*/
for(indx = 0; indx < input->num_cols; indx++)
{
/* update the solution vector x */
output->sol_vec->elements[indx] += (phi / rho) *
work->srch_dir_vec->elements[indx];
/* update the standard error estimates vector se */
output->std_err_vec->elements[indx] += lsqr_sqr( (1.0 / rho) *
work->srch_dir_vec->elements[indx] );
/* accumulate this quantity to estimate condition number of A
*/
ddnorm += lsqr_sqr( (1.0 / rho) *
work->srch_dir_vec->elements[indx] );
/* update the search direction vector w */
work->srch_dir_vec->elements[indx] =
work->bidiag_wrk_vec->elements[indx] -
(theta / rho) * work->srch_dir_vec->elements[indx];
}
/*
* Use a plane rotation on the right to eliminate the super-diagonal element
* (THETA) of the upper-bidiagonal matrix. Then use the result to estimate
* the solution norm || x ||.
*/
delta = sn2 * rho;
gammabar = -cs2 * rho;
zetabar = (phi - delta * zeta) / gammabar;
/* compute an estimate of the solution norm || x || */
output->sol_norm = sqrt( xxnorm + lsqr_sqr(zetabar) );
gamma = sqrt( lsqr_sqr(gammabar) + lsqr_sqr(theta) );
cs2 = gammabar / gamma;
sn2 = theta / gamma;
zeta = (phi - delta * zeta) / gamma;
/* accumulate this quantity to estimate solution norm || x || */
xxnorm += lsqr_sqr(zeta);
/*
* Estimate the Frobenius norm and condition of the matrix A, and the
* Euclidean norms of the vectors r and A^T*r.
*/
output->frob_mat_norm = sqrt( bbnorm );
output->mat_cond_num = output->frob_mat_norm * sqrt( ddnorm );
res += lsqr_sqr(psi);
output->resid_norm = sqrt( lsqr_sqr(phibar) + res );
output->mat_resid_norm = alpha * fabs( tau );
/*
* Use these norms to estimate the values of the three stopping criteria.
*/
stop_crit_1 = output->resid_norm / bnorm;
stop_crit_2 = 0.0;
if( output->resid_norm > 0.0 )
stop_crit_2 = output->mat_resid_norm / ( output->frob_mat_norm *
output->resid_norm );
stop_crit_3 = 1.0 / output->mat_cond_num;
/* 05 Jul 2007: Bug reported by Joel Erickson <*****@*****.**>.
*/
resid_tol = input->rel_rhs_err + input->rel_mat_err *
output->frob_mat_norm * /* (not output->mat_resid_norm *) */
output->sol_norm / bnorm;
resid_tol_mach = DBL_EPSILON + DBL_EPSILON *
output->frob_mat_norm * /* (not output->mat_resid_norm *) */
output->sol_norm / bnorm;
/*
* Check to see if any of the stopping criteria are satisfied.
* First compare the computed criteria to the machine precision.
* Second compare the computed criteria to the the user specified precision.
*/
/* iteration limit reached */
if( output->num_iters >= input->max_iter )
output->term_flag = 7;
/* condition number greater than machine precision */
if( stop_crit_3 <= DBL_EPSILON )
output->term_flag = 6;
/* least squares error less than machine precision */
if( stop_crit_2 <= DBL_EPSILON )
output->term_flag = 5;
/* residual less than a function of machine precision */
if( stop_crit_1 <= resid_tol_mach )
output->term_flag = 4;
/* condition number greater than CONLIM */
if( stop_crit_3 <= cond_tol )
output->term_flag = 3;
/* least squares error less than ATOL */
if( stop_crit_2 <= input->rel_mat_err )
output->term_flag = 2;
/* residual less than a function of ATOL and BTOL */
if( stop_crit_1 <= resid_tol )
output->term_flag = 1;
/*
* If statistics are printed at each iteration, print a header and the initial
* values for each quantity.
*/
if( input->lsqr_fp_out != NULL )
{
fprintf( input->lsqr_fp_out,
"%6li %13.5e %10.2e \t%10.2e \t%10.2e %10.2e\n",
output->num_iters, output->resid_norm, stop_crit_1,
stop_crit_2,
output->frob_mat_norm, output->mat_cond_num);
}
/*
* The convergence criteria are required to be met on NCONV consecutive
* iterations, where NCONV is set below. Suggested values are 1, 2, or 3.
*/
if( output->term_flag == 0 )
term_iter = -1;
term_iter_max = 1;
term_iter++;
if( (term_iter < term_iter_max) &&
(output->num_iters < input->max_iter) )
output->term_flag = 0;
if (per_iteration_callback)
per_iteration_callback(input, output, token);
} /* end while loop */
/*
* Finish computing the standard error estimates vector se.
*/
temp = 1.0;
if( input->num_rows > input->num_cols )
temp = ( double ) ( input->num_rows - input->num_cols );
if( lsqr_sqr(input->damp_val) > 0.0 )
temp = ( double ) ( input->num_rows );
temp = output->resid_norm / sqrt( temp );
for(indx = 0; indx < input->num_cols; indx++)
/* update the standard error estimates vector se */
output->std_err_vec->elements[indx] = temp *
sqrt( output->std_err_vec->elements[indx] );
/*
* If statistics are printed at each iteration, print the statistics for the
* stopping condition.
*/
if( input->lsqr_fp_out != NULL )
{
fprintf( input->lsqr_fp_out, "\n\tISTOP = %3li\t\t\tITER = %9li\n"
" || A ||_F = %13.5e\tcond( A ) = %13.5e\n"
" || r ||_2 = %13.5e\t|| A^T r ||_2 = %13.5e\n"
" || b ||_2 = %13.5e\t|| x - x0 ||_2 = %13.5e\n\n",
output->term_flag, output->num_iters, output->frob_mat_norm,
output->mat_cond_num, output->resid_norm, output->mat_resid_norm,
bnorm, output->sol_norm );
fprintf( input->lsqr_fp_out, " %s\n\n", term_msg[output->term_flag]);
}
return;
}
