/*
* Copyright(C) 1999-2020, 2022, 2023 National Technology & Engineering Solutions
* of Sandia, LLC (NTESS). Under the terms of Contract DE-NA0003525 with
* NTESS, the U.S. Government retains certain rights in this software.
*
* See packages/seacas/LICENSE for details
*/
#include "defs.h" // for FALSE, TRUE, min, max
#include "smalloc.h" // for sfree, smalloc
#include "structs.h" // for orthlink, scanlink
#include <math.h> // for sqrt
#include <stdio.h> // for printf, NULL, FILE
/* These comments are for version 1 of selective orthogonalization:
This version of selective orthogonalization largely follows that described in
Pareltt and Scott, "The Lanczos Algorithm with Selective Orthogonalization",
Math Comp v33 #145, 1979. Different heuristics are used to control loss of
orthogonality. Specifically, the pauses to check for convergence of Ritz pairs
at first come at a small but increasing interval as the computation builds up.
Later they come at a regular, pre-set interval, e.g. every 10 steps. Hence this
is similar to Grear's periodic reorthogonalization scheme, but with an adaptive
period. A small number of ritz pairs at both ends of the spectrum are monitored
for convergence at each pause. The number monitored gradually increases as more
Ritz pairs converge so that the number monitored at each pause is always at least
2 greater than the number that have previously converged. This is a fairly
conservative strategy, at least in the context of the typical Laplacian graph
matrices studied in connection with load balancing. But it may occasionally
result in premature loss of orthogonality. If the loss of orthogonality is
severe, Lanczos may fail to meet the set eigen tolerance or mis-converge to
the wrong eigenpair. These conditions will usually be detected by the algorithm
and a warning issued. If the graph is small, full orthogonalization is
then a fall back option. If the graph is large, one of the multi-level schemes
should be tried. The algorithm orthogonalizes the starting vector and each
residual vector against the vector of all ones since we know that is an
eigenvector of the Laplacian (which we don't want to see). This is
accomplished through calls to orthog1. These can be removed, but then
we must compute an extra eigenvalue and the vector of all ones shows up as
a Ritz pair we need to orthogonalize against in order to maintain the basis.
This is more expensive then simply taking it out of the residual on each step,
and doing the latter also reduces the number of iterations. The Ritz values
being monitored at each step are computed using the QL algorithm or bisection
on the Sturm sequence, whichever is faster based on a simple complexity model.
In practice the time spent computing Ritz values is only a small portion of
the total Lanczos time, so more complex schemes based on interpolating the
bottom pivot function represent only a very marginal savings in execution
time at the expense of either some robustness or substantially increased
code complexity. */
/* These comments are for version 2 of selective orthogonalization:
These comments are cumulative from version 1. Modified the heuristics to
make it faster. Now only checks left end of spectrum for converging Ritz pairs.
Only checking the left end of the spectrum for converging Ritz pairs is in principle
not as good as checking both ends, but in practice seems to be no worse and usually
better than checking both ends. Since the distribution of eigenvalues for Laplacian
graphs of interest seems to generally result in much more rapid convergence on the
right end of the spectrum, ignoring the converging Ritz pairs there saves a lot of
work. */
/* These comments are for version 3 of selective orthogonalization:
This option hasn't performed that well, so I've rescinded it from the menu.
Uses Paige suggestion of monitoring dot product of current Lanczos vector
against first to sense loss of orthogonality. This can save time by avoiding
calls to the QL or bisection routines for finding Ritz values. */
/* These comments explain some of indexing conventions used:
Note that the indexing of the tridiagonal matrix is consistent with Parlett and
Scott's Math Comp '79 paper "The Lanczos Algorithm With Selective Orthogonalization"
and with Golub and Van Loan's (2nd ed.) treatment in Chapter 9. So on iteration j
the tridiagonal matrix T has diagonal = alpha[1], alpha[2] ... alpha[j] and
off-diagonal = beta[1], beta[2] ... beta[j-1]. beta[0] is the norm of the initial
residual and beta[j] is used in monitoring convergence. */
/* These comments pertain to the calculation of the eigenvalues of T:
There are several safety features in the code which computes the Ritz values.
Computation is first attempted using either the QL algorithm or Sturm sequence
bisection, whichever is predicted to be cheaper based on simple complexity analysis
(normally bisection is cheaper in this context). The classic bisection algorithm
can fail due to overflow, so a re-scaling heuristic is used to guard against this.
The number of bisection and QL iterations is monitored and if a reasonable maximum
is observed these routines abort. If either routine fails, the other is applied
to the problem. If both routines fail, Lanczos is backed up to the last pause point
and an approximate eigenvector is computed based on the available information. */
/* These comments pertain to the calculation of eigenvectors of T:
In the first instance the eigenvectors are computed using a bidirectional
recurrence which amounts to one fancy step of inverse iteration. If this
fails to achieve the requested tolerance, a more expensive version of inverse
iteration (Tinvit from Eispack) is tried. The better of the two answers is
then used. See the routine Tevec for more comments and details. */
void lanczos_SO(struct vtx_data **A, /* sparse matrix in row linked list format */
int n, /* problem size */
int d, /* problem dimension = number of eigvecs to find */
double **y, /* columns of y are eigenvectors of A */
double *lambda, /* ritz approximation to eigenvals of A */
double *bound, /* on ritz pair approximations to eig pairs of A */
double eigtol, /* tolerance on eigenvectors */
double *vwsqrt, /* square roots of vertex weights */
double maxdeg, /* maximum degree of graph */
int version, /* flags which version of sel. orth. to use */
int cube_or_mesh, /* 0 => hypercube, d => d-dimensional mesh */
int nsets, /* number of sets to divide into */
int *assignment, /* set number of each vtx (length n+1) */
int *active, /* space for nvtxs integers */
int mediantype, /* which partitioning strategy to use */
double *goal, /* desired set sizes */
int vwgt_max /* largest vertex weight */
)
{
extern FILE *Output_File; /* output file or null */
extern int LANCZOS_SO_INTERVAL; /* interval between orthogonalizations */
extern int LANCZOS_CONVERGENCE_MODE; /* type of Lanczos convergence test */
extern int LANCZOS_MAXITNS; /* maximum Lanczos iterations allowed */
extern int DEBUG_EVECS; /* print debugging output? */
extern int DEBUG_TRACE; /* trace main execution path */
extern int WARNING_EVECS; /* print warning messages? */
extern double BISECTION_SAFETY; /* safety factor for bisection alg. */
extern double SRESTOL; /* resid tol for T evec comp */
extern double DOUBLE_EPSILON; /* machine precision */
extern double DOUBLE_MAX; /* largest double value */
extern double splarax_time; /* time matvec */
extern double orthog_time; /* time orthogonalization work */
extern double evec_time; /* time to generate eigenvectors */
extern double ql_time; /* time tridiagonal eigenvalue work */
extern double blas_time; /* time for blas. linear algebra */
extern double init_time; /* time to allocate, initialize variables */
extern double scan_time; /* time for scanning eval and bound lists */
extern double debug_time; /* time for (some of) debug computations */
extern double ritz_time; /* time to generate ritz vectors */
extern double pause_time; /* time to compute whether to pause */
double bis_safety; /* real safety factor for bisection alg. */
int i, j, k; /* indices */
int maxj; /* maximum number of Lanczos iterations */
double *u, *r; /* Lanczos vectors */
double *alpha, *beta; /* the Lanczos scalars from each step */
double *ritz; /* copy of alpha for ql */
double *workj; /* work vector, e.g. copy of beta for ql */
double *workn; /* work vector, e.g. product Av for checkeig */
double *s; /* eigenvector of T */
double **q; /* columns of q are Lanczos basis vectors */
double *bj; /* beta(j)*(last el. of corr. eigvec s of T) */
double Sres; /* how well Tevec calculated eigvec s */
double Sres_max; /* Max value of Sres */
int inc_bis_safety; /* need to increase bisection safety */
double *Ares; /* how well Lanczos calc. eigpair lambda,y */
int *index; /* the Ritz index of an eigenpair */
struct orthlink **solist; /* vec. of structs with vecs. to orthog. against */
struct scanlink *scanlist; /* linked list of fields to do with min ritz vals */
struct scanlink *curlnk; /* for traversing the scanlist */
double bji_tol; /* tol on bji est. of eigen residual of A */
int converged; /* has the iteration converged? */
double goodtol; /* error tolerance for a good Ritz vector */
int ngood; /* total number of good Ritz pairs at current step */
int maxngood; /* biggest val of ngood through current step */
int left_ngood; /* number of good Ritz pairs on left end */
int right_ngood; /* number of good Ritz pairs on right end */
int lastpause; /* Most recent step with good ritz vecs */
int firstpause; /* Is this the first pause? */
int nopauses; /* Have there been any pauses? */
int interval; /* number of steps between pauses */
double time; /* Current clock time */
int left_goodlim; /* number of ritz pairs checked on left end */
int right_goodlim; /* number of ritz pairs checked on right end */
double Anorm; /* Norm estimate of the Laplacian matrix */
int pausemode; /* which Lanczos pausing criterion to use */
int pause; /* whether to pause */
int temp; /* used to prevent redundant index computations */
int *old_assignment = NULL; /* set # of each vtx on previous pause, length n+1 */
int *assgn_pntr; /* pntr to assignment vector */
int *old_assgn_pntr; /* pntr to previous assignment vector */
int assigndiff; /* # of differences between old and new assignment */
int assigntol; /* tolerance on convergence of assignment vector */
int ritzval_flag; /* status flag for get_ritzvals() */
int memory_ok; /* True until lanczos runs out of memory */
if (DEBUG_TRACE > 0) {
printf("<Entering lanczos_so>\n");
}
if (DEBUG_EVECS > 0) {
printf("Selective orthogonalization Lanczos (v. %d), matrix size = %d.\n", version, n);
}
/* Initialize time. */
time = lanc_seconds();
if (n < d + 1) {
bail("ERROR: System too small for number of eigenvalues requested.", 1);
/* d+1 since don't use zero eigenvalue pair */
}
/* Allocate space. */
maxj = LANCZOS_MAXITNS;
u = mkvec(1, n);
r = mkvec(1, n);
workn = mkvec(1, n);
Ares = mkvec(0, d);
index = smalloc((d + 1) * sizeof(int));
alpha = mkvec(1, maxj);
beta = mkvec(0, maxj);
ritz = mkvec(1, maxj);
s = mkvec(1, maxj);
bj = mkvec(1, maxj);
workj = mkvec(0, maxj);
q = smalloc((maxj + 1) * sizeof(double *));
solist = smalloc((maxj + 1) * sizeof(struct orthlink *));
scanlist = mkscanlist(d);
if (LANCZOS_CONVERGENCE_MODE == 1) {
old_assignment = smalloc((n + 1) * sizeof(int));
}
/* Set some constants governing the orthogonalization heuristic. */
ngood = 0;
maxngood = 0;
bji_tol = eigtol;
assigntol = eigtol * n;
Anorm = 2 * maxdeg; /* Gershgorin estimate for ||A|| */
goodtol = Anorm * sqrt(DOUBLE_EPSILON); /* Parlett & Scott's bound, p.224 */
interval = 2 + min(LANCZOS_SO_INTERVAL - 2, n / (2 * LANCZOS_SO_INTERVAL));
bis_safety = BISECTION_SAFETY;
if (DEBUG_EVECS > 0) {
printf(" eigtol %g\n", eigtol);
printf(" maxdeg %g\n", maxdeg);
printf(" goodtol %g\n", goodtol);
printf(" interval %d\n", interval);
printf(" maxj %d\n", maxj);
printf(" srestol %g\n", SRESTOL);
if (LANCZOS_CONVERGENCE_MODE == 1) {
printf(" assigntol %d\n", assigntol);
}
}
/* Initialize space. */
vecran(r, 1, n);
if (vwsqrt == NULL) {
orthog1(r, 1, n);
}
else {
orthogvec(r, 1, n, vwsqrt);
}
beta[0] = ch_norm(r, 1, n);
q[0] = mkvec(1, n);
setvec(q[0], 1, n, 0.0);
setvec(bj, 1, maxj, DOUBLE_MAX);
/* Main Lanczos loop. */
j = 1;
lastpause = 0;
pausemode = 1;
left_ngood = 0;
right_ngood = 0;
left_goodlim = 0;
right_goodlim = 0;
converged = FALSE;
Sres_max = 0.0;
inc_bis_safety = FALSE;
ritzval_flag = 0;
memory_ok = TRUE;
firstpause = FALSE;
nopauses = TRUE;
init_time += lanc_seconds() - time;
while ((j <= maxj) && (!converged) && (ritzval_flag == 0) && memory_ok) {
time = lanc_seconds();
/* Allocate next Lanczos vector. If fail, back up to last pause. */
q[j] = mkvec_ret(1, n);
if (q[j] == NULL) {
memory_ok = FALSE;
if (DEBUG_EVECS > 0 || WARNING_EVECS > 0) {
strout("WARNING: Lanczos out of memory; computing best approximation available.\n");
}
if (nopauses) {
bail("ERROR: Sorry, can't salvage Lanczos.", 1);
/* ... save yourselves, men. */
}
for (i = lastpause + 1; i <= j - 1; i++) {
frvec(q[i], 1);
}
j = lastpause;
}
/* Basic Lanczos iteration */
vecscale(q[j], 1, n, 1.0 / beta[j - 1], r);
blas_time += lanc_seconds() - time;
time = lanc_seconds();
splarax(u, A, n, q[j], vwsqrt, workn);
splarax_time += lanc_seconds() - time;
time = lanc_seconds();
update(r, 1, n, u, -beta[j - 1], q[j - 1]);
alpha[j] = dot(r, 1, n, q[j]);
update(r, 1, n, r, -alpha[j], q[j]);
blas_time += lanc_seconds() - time;
/* Selective orthogonalization */
time = lanc_seconds();
if (vwsqrt == NULL) {
orthog1(r, 1, n);
}
else {
orthogvec(r, 1, n, vwsqrt);
}
if ((j == (lastpause + 1)) || (j == (lastpause + 2))) {
sorthog(r, n, solist, ngood);
}
orthog_time += lanc_seconds() - time;
beta[j] = ch_norm(r, 1, n);
time = lanc_seconds();
pause = lanpause(j, lastpause, interval, q, n, &pausemode, version, beta[j]);
pause_time += lanc_seconds() - time;
if (pause) {
nopauses = FALSE;
if (lastpause == 0) {
firstpause = TRUE;
}
else {
firstpause = FALSE;
}
lastpause = j;
/* Compute limits for checking Ritz pair convergence. */
if (version == 1) {
if (left_ngood + 2 > left_goodlim) {
left_goodlim = left_ngood + 2;
}
if (right_ngood + 3 > right_goodlim) {
right_goodlim = right_ngood + 3;
}
}
if (version == 2) {
if (left_ngood + 2 > left_goodlim) {
left_goodlim = left_ngood + 2;
}
right_goodlim = 0;
}
/* Special case: need at least d Ritz vals on left. */
left_goodlim = max(left_goodlim, d);
/* Special case: can't find more than j total Ritz vals. */
if (left_goodlim + right_goodlim > j) {
left_goodlim = min(left_goodlim, j);
right_goodlim = j - left_goodlim;
}
/* Find Ritz vals using faster of Sturm bisection or QL. */
time = lanc_seconds();
blas_time += lanc_seconds() - time;
time = lanc_seconds();
if (inc_bis_safety) {
bis_safety *= 10;
inc_bis_safety = FALSE;
}
ritzval_flag = get_ritzvals(alpha, beta, j, Anorm, workj, ritz, d, left_goodlim,
right_goodlim, eigtol, bis_safety);
ql_time += lanc_seconds() - time;
/* If get_ritzvals() fails, back up to last pause point and exit main loop. */
if (ritzval_flag != 0) {
if (DEBUG_EVECS > 0 || WARNING_EVECS > 0) {
strout("ERROR: Lanczos failed in computing eigenvalues of T; computing");
strout(" best readily available approximation to eigenvector.\n");
}
if (firstpause) {
bail("ERROR: Sorry, can't salvage Lanczos.", 1);
/* ... save yourselves, men. */
}
for (i = lastpause + 1; i <= j; i++) {
frvec(q[i], 1);
}
j = lastpause;
get_ritzvals(alpha, beta, j, Anorm, workj, ritz, d, left_goodlim, right_goodlim, eigtol,
bis_safety);
}
/* Scan for minimum evals of tridiagonal. */
time = lanc_seconds();
scanmin(ritz, 1, j, &scanlist);
scan_time += lanc_seconds() - time;
/* Compute Ritz pair bounds at left end. */
time = lanc_seconds();
setvec(bj, 1, j, 0.0);
for (i = 1; i <= left_goodlim; i++) {
Sres = Tevec(alpha, beta - 1, j, ritz[i], s);
if (Sres > Sres_max) {
Sres_max = Sres;
}
if (Sres > SRESTOL) {
inc_bis_safety = TRUE;
}
bj[i] = s[j] * beta[j];
/* bj[i] = fabs(s[j] * beta[j]); if don't enforce in Tevec */
}
/* Compute Ritz pair bounds at right end. */
for (i = j; i > j - right_goodlim; i--) {
Sres = Tevec(alpha, beta - 1, j, ritz[i], s);
if (Sres > Sres_max) {
Sres_max = Sres;
}
if (Sres > SRESTOL) {
inc_bis_safety = TRUE;
}
bj[i] = s[j] * beta[j];
/* bj[i] = fabs(s[j] * beta[j]); if don't enforce in Tevec */
}
ritz_time += lanc_seconds() - time;
/* Show the portion of the spectrum checked for convergence. */
if (DEBUG_EVECS > 2) {
time = lanc_seconds();
printf("index Ritz vals bji bounds (j = %d)\n", j);
for (i = 1; i <= left_goodlim; i++) {
printf(" %3d", i);
doubleout(ritz[i], 1);
doubleout(bj[i], 1);
printf("\n");
}
printf("\n");
curlnk = scanlist;
while (curlnk != NULL) {
temp = curlnk->indx;
if ((temp > left_goodlim) && (temp < j - right_goodlim)) {
printf(" %3d", temp);
doubleout(ritz[temp], 1);
doubleout(bj[temp], 1);
printf("\n");
}
curlnk = curlnk->pntr;
}
printf("\n");
for (i = j - right_goodlim + 1; i <= j; i++) {
printf(" %3d", i);
doubleout(ritz[i], 1);
doubleout(bj[i], 1);
printf("\n");
}
printf(" -------------------\n");
printf(" goodtol: %19.16f\n\n", goodtol);
debug_time += lanc_seconds() - time;
}
/* Check for convergence. */
time = lanc_seconds();
if (LANCZOS_CONVERGENCE_MODE != 1 || d > 1) {
/* check convergence of residual bound */
converged = TRUE;
if (j < d) {
converged = FALSE;
}
else {
curlnk = scanlist;
while (curlnk != NULL) {
if (bj[curlnk->indx] > bji_tol) {
converged = FALSE;
}
curlnk = curlnk->pntr;
}
}
}
if (LANCZOS_CONVERGENCE_MODE == 1 && d == 1) {
/* check change in partition */
if (firstpause) {
converged = TRUE;
if (j < d) {
converged = FALSE;
}
else {
curlnk = scanlist;
while (curlnk != NULL) {
if (bj[curlnk->indx] > bji_tol) {
converged = FALSE;
}
curlnk = curlnk->pntr;
}
}
if (!converged) {
/* compute current approx. to eigenvectors */
mkeigvecs(scanlist, lambda, bound, index, bj, d, &Sres_max, alpha, beta, j, s, y, n, q);
/* compute first assignment */
assign(A, y, n, d, cube_or_mesh, nsets, vwsqrt, assignment, active, mediantype, goal,
vwgt_max);
}
}
else {
/* copy assignment to old_assignment */
assgn_pntr = assignment;
old_assgn_pntr = old_assignment;
for (i = n + 1; i; i--) {
*old_assgn_pntr++ = *assgn_pntr++;
}
/* compute current approx. to eigenvectors */
mkeigvecs(scanlist, lambda, bound, index, bj, d, &Sres_max, alpha, beta, j, s, y, n, q);
/* write new assignment */
assign(A, y, n, d, cube_or_mesh, nsets, vwsqrt, assignment, active, mediantype, goal,
vwgt_max);
assigndiff = 0;
assgn_pntr = assignment;
old_assgn_pntr = old_assignment;
for (i = n + 1; i; i--) {
if (*old_assgn_pntr++ != *assgn_pntr++) {
assigndiff++;
}
}
assigndiff = min(assigndiff, n - assigndiff);
if (DEBUG_EVECS > 1) {
printf(" j %d, change from last assignment %d\n\n", j, assigndiff);
}
if (assigndiff <= assigntol) {
converged = TRUE;
}
else {
converged = FALSE;
}
}
}
scan_time += lanc_seconds() - time;
/* Show current estimates of evals and bounds (for help in tuning) */
if (DEBUG_EVECS > 2 && !converged) {
time = lanc_seconds();
/* compute current approx. to eigenvectors */
mkeigvecs(scanlist, lambda, bound, index, bj, d, &Sres_max, alpha, beta, j, s, y, n, q);
/* Compute residuals and display associated info. */
printf("j %4d; lambda Ares est. Ares index\n", j);
for (i = 1; i <= d; i++) {
Ares[i] = checkeig(workn, A, y[i], n, lambda[i], vwsqrt, u);
printf("%2d.", i);
doubleout(lambda[i], 1);
doubleout(bound[i], 1);
doubleout(Ares[i], 1);
printf(" %3d\n", index[i]);
}
printf("\n");
debug_time += lanc_seconds() - time;
}
if (!converged) {
ngood = 0;
left_ngood = 0; /* for setting left_goodlim on next loop */
right_ngood = 0; /* for setting right_goodlim on next loop */
/* Compute converged Ritz pairs on left end */
time = lanc_seconds();
for (i = 1; i <= left_goodlim; i++) {
if (bj[i] <= goodtol) {
ngood += 1;
left_ngood += 1;
if (ngood > maxngood) {
maxngood = ngood;
solist[ngood] = makeorthlnk();
(solist[ngood])->vec = mkvec(1, n);
}
(solist[ngood])->index = i;
Sres = Tevec(alpha, beta - 1, j, ritz[i], s);
if (Sres > Sres_max) {
Sres_max = Sres;
}
if (Sres > SRESTOL) {
inc_bis_safety = TRUE;
}
setvec((solist[ngood])->vec, 1, n, 0.0);
for (k = 1; k <= j; k++) {
scadd((solist[ngood])->vec, 1, n, s[k], q[k]);
}
}
}
/* Compute converged Ritz pairs on right end */
for (i = j; i > j - right_goodlim; i--) {
if (bj[i] <= goodtol) {
ngood += 1;
right_ngood += 1;
if (ngood > maxngood) {
maxngood = ngood;
solist[ngood] = makeorthlnk();
(solist[ngood])->vec = mkvec(1, n);
}
(solist[ngood])->index = i;
Sres = Tevec(alpha, beta - 1, j, ritz[i], s);
if (Sres > Sres_max) {
Sres_max = Sres;
}
if (Sres > SRESTOL) {
inc_bis_safety = TRUE;
}
setvec((solist[ngood])->vec, 1, n, 0.0);
for (k = 1; k <= j; k++) {
scadd((solist[ngood])->vec, 1, n, s[k], q[k]);
}
}
}
ritz_time += lanc_seconds() - time;
if (DEBUG_EVECS > 2) {
time = lanc_seconds();
printf(" j %3d; goodlim lft %2d, rgt %2d; list ", j, left_goodlim, right_goodlim);
solistout(solist, ngood, j);
printf("---------------------end of iteration---------------------\n\n");
debug_time += lanc_seconds() - time;
}
}
}
j++;
}
j--;
/* Collect eigenvalue and bound information. */
time = lanc_seconds();
mkeigvecs(scanlist, lambda, bound, index, bj, d, &Sres_max, alpha, beta, j, s, y, n, q);
evec_time += lanc_seconds() - time;
/* Analyze computation for and report additional problems */
time = lanc_seconds();
warnings(workn, A, y, n, lambda, vwsqrt, Ares, bound, index, d, j, maxj, Sres_max, eigtol, u,
Anorm, Output_File);
debug_time += lanc_seconds() - time;
/* free up memory */
time = lanc_seconds();
frvec(u, 1);
frvec(r, 1);
frvec(workn, 1);
frvec(Ares, 0);
sfree(index);
frvec(alpha, 1);
frvec(beta, 0);
frvec(ritz, 1);
frvec(s, 1);
frvec(bj, 1);
frvec(workj, 0);
for (i = 0; i <= j; i++) {
frvec(q[i], 1);
}
sfree(q);
while (scanlist != NULL) {
curlnk = scanlist->pntr;
sfree(scanlist);
scanlist = curlnk;
}
for (i = 1; i <= maxngood; i++) {
frvec((solist[i])->vec, 1);
sfree(solist[i]);
}
sfree(solist);
if (LANCZOS_CONVERGENCE_MODE == 1) {
sfree(old_assignment);
}
init_time += lanc_seconds() - time;
}