/* Finds eigenvector s of T and returns residual norm. */
double
Tevec (
double *alpha, /* vector of Lanczos scalars */
double *beta, /* vector of Lanczos scalars */
int j, /* number of Lanczos iterations taken */
double ritz, /* approximate eigenvalue of T */
double *s /* approximate eigenvector of T */
)
{
extern double SRESTOL; /* limit on relative residual tol for evec of T */
extern double DOUBLE_MAX; /* maximum double precision value */
int i; /* index */
double residual=0.0; /* how well recurrence gives eigenvector */
double temp; /* used to compute residual */
double *work; /* temporary work vector allocated within if used */
double w[MAXDIMS + 1]; /* holds eigenvalue for tinvit */
long index[MAXDIMS +1];/* index vector for tinvit */
long ierr; /* error flag for tinvit */
long nevals; /* number of evals sought */
long long_j; /* long copy of j for tinvit interface */
double hurdle; /* hurdle for local maximum in recurrence */
double prev_resid; /* stores residual from previous computation */
int tinvit_(); /* eispack's tinvit for evecs of symmetric T */
double *mkvec(); /* allocates double vectors */
void frvec(); /* frees double vectors */
double bidir(); /* bidirectional recurrence for evec of T */
void cpvec(); /* vector copy routine */
s[1] = 1.0;
if (j == 1) {
residual = fabs(alpha[1] - ritz);
}
if (j >= 2) {
/*Bidirectional recurrence - corrected and modified from Parlett and Reid,
"Tracking the Progress of the Lanczos Algorithm ..., IMA JNA 1, 1981 */
hurdle = 1.0;
residual = bidir(alpha,beta,j,ritz,s,hurdle);
}
if (residual > SRESTOL) {
/* Try again with Eispack's Tinvit iteration */
SRES_SWITCHES++;
index[1] = 1;
work = mkvec(1, 7*j); /* lump things to save mallocs */
w[1] = ritz;
work[1] = 0;
for (i = 2; i <= j; i++) {
work[i] = beta[i] * beta[i];
}
nevals = 1;
long_j = j;
/* save the previously computed evec in case it's better */
cpvec(&(work[6*j]),1,j,s);
prev_resid = residual;
tinvit_(&long_j, &long_j, &(alpha[1]), &(beta[1]), &(work[1]), &nevals,
&(w[1]), &(index[1]), &(s[1]), &ierr, &(work[j+1]), &(work[(2*j)+1]),
&(work[(3*j)+1]), &(work[(4*j)+1]), &(work[(5*j)+1]));
/* fix up sign if needed */
if (s[j] < 0) {
for(i=1; i<=j; i++) {
s[i] = - s[i];
}
}
if (ierr != 0) {
residual = DOUBLE_MAX;
/* ... don't want to use evec since it is set to zero */
}
else {
temp = (alpha[1] - ritz) * s[1] + beta[2] * s[2];
residual = temp * temp;
for (i = 2; i < j; i++) {
temp = beta[i] * s[i - 1] + (alpha[i] - ritz) * s[i]
+ beta[i + 1] * s[i + 1];
residual += temp * temp;
}
temp = beta[j] * s[j - 1] + (alpha[j] - ritz) * s[j];
residual += temp * temp;
residual = sqrt(residual);
/* tinvit normalizes, so we don't need to. */
}
/* restore previous evec if it had a better residual */
if (prev_resid < residual) {
residual = prev_resid;
cpvec(s,1,j,&(work[6*j]));
SRES_SWITCHES++; /* count since switching back as well */
}
frvec(work, 1);
}
return (residual);
}
/* Benchmark certain kernel operations */
void
time_kernels (
struct vtx_data **A, /* matrix/graph being analyzed */
int n, /* number of rows/columns in matrix */
double *vwsqrt /* square roots of vertex weights */
)
{
extern int DEBUG_PERTURB; /* debug flag for matrix perturbation */
extern int PERTURB; /* randomly perturb to break symmetry? */
extern int NPERTURB; /* number of edges to perturb */
extern int DEBUG_TRACE; /* trace main execution path */
extern double PERTURB_MAX; /* maximum size of perturbation */
int i, beg, end;
double *dvec1, *dvec2, *dvec3;
float *svec1, *svec2, *svec3, *vwsqrt_float;
double norm_dvec, norm_svec;
double dot_dvec, dot_svec;
double time, time_dvec, time_svec;
double diff;
double factor, fac;
float factor_float, fac_float;
int loops;
double min_time, target_time;
double *mkvec();
float *mkvec_float();
void frvec(), frvec_float();
void vecran();
double ch_norm(), dot();
double norm_float(), dot_float();
double seconds();
void scadd(), scadd_float(), update(), update_float();
void splarax(), splarax_float();
void perturb_init(), perturb_clear();
if (DEBUG_TRACE > 0) {
printf("<Entering time_kernels>\n");
}
beg = 1;
end = n;
dvec1 = mkvec(beg, end);
dvec2 = mkvec(beg, end);
dvec3 = mkvec(beg - 1, end);
svec1 = mkvec_float(beg, end);
svec2 = mkvec_float(beg, end);
svec3 = mkvec_float(beg - 1, end);
if (vwsqrt == NULL) {
vwsqrt_float = NULL;
}
else {
vwsqrt_float = mkvec_float(beg - 1, end);
for (i = beg - 1; i <= end; i++) {
vwsqrt_float[i] = vwsqrt[i];
}
}
vecran(dvec1, beg, end);
vecran(dvec2, beg, end);
vecran(dvec3, beg, end);
for (i = beg; i <= end; i++) {
svec1[i] = dvec1[i];
svec2[i] = dvec2[i];
svec3[i] = dvec3[i];
}
/* Set number of loops so that ch_norm() takes about one second. This should
insulate against inaccurate timings on faster machines. */
loops = 1;
time_dvec = 0;
min_time = 0.5;
target_time = 1.0;
while (time_dvec < min_time) {
time = seconds();
for (i = loops; i; i--) {
norm_dvec = ch_norm(dvec1, beg, end);
}
time_dvec = seconds() - time;
if (time_dvec < min_time) {
loops = 10 * loops;
}
}
loops = (target_time / time_dvec) * loops;
if (loops < 1)
loops = 1;
printf(" Kernel benchmarking\n");
printf("Time (in seconds) for %d loops of each operation:\n\n", loops);
printf("Routine Double Float Discrepancy Description\n");
printf("------- ------ ----- ----------- -----------\n");
/* Norm operation */
time = seconds();
for (i = loops; i; i--) {
norm_dvec = ch_norm(dvec1, beg, end);
}
time_dvec = seconds() - time;
time = seconds();
for (i = loops; i; i--) {
norm_svec = norm_float(svec1, beg, end);
}
time_svec = seconds() - time;
diff = norm_dvec - norm_svec;
printf("norm %6.2f %6.2f %14.5e", time_dvec, time_svec, diff);
printf(" 2 norm\n");
/* Dot operation */
time = seconds();
for (i = loops; i; i--) {
dot_dvec = dot(dvec1, beg, end, dvec2);
}
time_dvec = seconds() - time;
time = seconds();
for (i = loops; i; i--) {
dot_svec = dot_float(svec1, beg, end, svec2);
}
time_svec = seconds() - time;
diff = dot_dvec - dot_svec;
printf("dot %6.2f %6.2f %14.5e", time_dvec, time_svec, diff);
printf(" scalar product\n");
/* Scadd operation */
factor = 1.01;
factor_float = factor;
fac = factor;
time = seconds();
for (i = loops; i; i--) {
scadd(dvec1, beg, end, fac, dvec2);
fac = -fac; /* to keep things in scale */
}
time_dvec = seconds() - time;
fac_float = factor_float;
time = seconds();
for (i = loops; i; i--) {
scadd_float(svec1, beg, end, fac_float, svec2);
fac_float = -fac_float; /* to keep things in scale */
}
time_svec = seconds() - time;
diff = checkvec(dvec1, beg, end, svec1);
printf("scadd %6.2f %6.2f %14.5e", time_dvec, time_svec, diff);
printf(" vec1 <- vec1 + alpha*vec2\n");
/* Update operation */
time = seconds();
for (i = loops; i; i--) {
update(dvec1, beg, end, dvec2, factor, dvec3);
}
time_dvec = seconds() - time;
time = seconds();
for (i = loops; i; i--) {
update_float(svec1, beg, end, svec2, factor_float, svec3);
}
time_svec = seconds() - time;
diff = checkvec(dvec1, beg, end, svec1);
printf("update %6.2f %6.2f %14.2g", time_dvec, time_svec, diff);
printf(" vec1 <- vec2 + alpha*vec3\n");
/* splarax operation */
if (PERTURB) {
if (NPERTURB > 0 && PERTURB_MAX > 0.0) {
perturb_init(n);
if (DEBUG_PERTURB > 0) {
printf("Matrix being perturbed with scale %e\n", PERTURB_MAX);
}
}
else if (DEBUG_PERTURB > 0) {
printf("Matrix not being perturbed\n");
}
}
time = seconds();
for (i = loops; i; i--) {
splarax(dvec1, A, n, dvec2, vwsqrt, dvec3);
}
time_dvec = seconds() - time;
time = seconds();
for (i = loops; i; i--) {
splarax_float(svec1, A, n, svec2, vwsqrt_float, svec3);
}
time_svec = seconds() - time;
diff = checkvec(dvec1, beg, end, svec1);
printf("splarax %6.2f %6.2f %14.5e", time_dvec, time_svec, diff);
printf(" sparse matrix vector multiply\n");
if (PERTURB && NPERTURB > 0 && PERTURB_MAX > 0.0) {
perturb_clear();
}
printf("\n");
/* Free memory */
frvec(dvec1, 1);
frvec(dvec2, 1);
frvec(dvec3, 0);
frvec_float(svec1, 1);
frvec_float(svec2, 1);
frvec_float(svec3, 0);
if (vwsqrt_float != NULL) {
frvec_float(vwsqrt_float, beg - 1);
}
}
void
lanczos_FO (
struct vtx_data **A, /* graph data structure */
int n, /* number of rows/colums in matrix */
int d, /* problem dimension = # evecs 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 root of vertex weights */
double maxdeg, /* maximum degree of graph */
int version /* 1 = standard mode, 2 = inverse operator mode */
)
{
extern FILE *Output_File; /* output file or NULL */
extern int DEBUG_EVECS; /* print debugging output? */
extern int DEBUG_TRACE; /* trace main execution path */
extern int WARNING_EVECS; /* print warning messages? */
extern int LANCZOS_MAXITNS; /* maximum Lanczos iterations allowed */
extern double BISECTION_SAFETY; /* safety factor for bisection algorithm */
extern double SRESTOL; /* resid tol for T evec comp */
extern double DOUBLE_MAX; /* Warning on inaccurate computation of evec of T */
extern double splarax_time; /* time matvecs */
extern double orthog_time; /* time orthogonalization work */
extern double tevec_time; /* time tridiagonal eigvec work */
extern double evec_time; /* time to generate eigenvectors */
extern double ql_time; /* time tridiagonal eigval work */
extern double blas_time; /* time for blas (not assembly coded) */
extern double init_time; /* time for allocating memory, etc. */
extern double scan_time; /* time for scanning bounds list */
extern double debug_time; /* time for debug computations and output */
int i, j; /* indicies */
int maxj; /* maximum number of Lanczos iterations */
double *u, *r; /* Lanczos vectors */
double *Aq; /* sparse matrix-vector product vector */
double *alpha, *beta; /* the Lanczos scalars from each step */
double *ritz; /* copy of alpha for tqli */
double *workj; /* work vector (eg. for tqli) */
double *workn; /* work vector (eg. for checkeig) */
double *s; /* eigenvector of T */
double **q; /* columns of q = Lanczos basis vectors */
double *bj; /* beta(j)*(last element of evecs of T) */
double bis_safety; /* real safety factor for bisection algorithm */
double Sres; /* how well Tevec calculated eigvecs */
double Sres_max; /* Maximum value of Sres */
int inc_bis_safety; /* need to increase bisection safety */
double *Ares; /* how well Lanczos calculated each eigpair */
double *inv_lambda; /* eigenvalues of inverse operator */
int *index; /* the Ritz index of an eigenpair */
struct orthlink *orthlist = NULL; /* vectors to orthogonalize against in Lanczos */
struct orthlink *orthlist2 = NULL; /* vectors to orthogonalize against in Symmlq */
struct orthlink *temp; /* for expanding orthogonalization list */
double *ritzvec=NULL; /* ritz vector for current iteration */
double *zeros=NULL; /* vector of all zeros */
double *ones=NULL; /* vector of all ones */
struct scanlink *scanlist; /* list of fields for min ritz vals */
struct scanlink *curlnk; /* for traversing the scanlist */
double bji_tol; /* tol on bji estimate of A e-residual */
int converged; /* has the iteration converged? */
double time; /* current clock time */
double shift, rtol; /* symmlq input */
long precon, goodb, nout; /* symmlq input */
long checka, intlim; /* symmlq input */
double anorm, acond; /* symmlq output */
double rnorm, ynorm; /* symmlq output */
long istop, itn; /* symmlq output */
double macheps; /* machine precision calculated by symmlq */
double normxlim; /* a stopping criteria for symmlq */
long itnmin; /* enforce minimum number of iterations */
int symmlqitns; /* # symmlq itns */
double *wv1=NULL, *wv2=NULL, *wv3=NULL; /* Symmlq work space */
double *wv4=NULL, *wv5=NULL, *wv6=NULL; /* Symmlq work space */
long long_n; /* long int copy of n for symmlq */
int ritzval_flag = 0; /* status flag for ql() */
double Anorm; /* Norm estimate of the Laplacian matrix */
int left, right; /* ranges on the search for ritzvals */
int memory_ok; /* TRUE as long as don't run out of memory */
double *mkvec(); /* allocates space for a vector */
double *mkvec_ret(); /* mkvec() which returns error code */
double dot(); /* standard dot product routine */
struct orthlink *makeorthlnk(); /* make space for entry in orthog. set */
double ch_norm(); /* vector norm */
double Tevec(); /* calc evec of T by linear recurrence */
struct scanlink *mkscanlist(); /* make scan list for min ritz vecs */
double lanc_seconds(); /* current clock timer */
int symmlq_(), get_ritzvals();
void setvec(), vecscale(), update(), vecran(), strout();
void splarax(), scanmin(), scanmax(), frvec(), orthogonalize();
void orthog1(), orthogvec(), bail(), warnings(), mkeigvecs();
if (DEBUG_TRACE > 0) {
printf("<Entering lanczos_FO>\n");
}
if (DEBUG_EVECS > 0) {
if (version == 1) {
printf("Full orthogonalization Lanczos, matrix size = %d\n", n);
}
else {
printf("Full orthogonalization Lanczos, inverted operator, matrix size = %d\n", 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 Lanczos space. */
maxj = LANCZOS_MAXITNS;
u = mkvec(1, n);
r = mkvec(1, n);
Aq = mkvec(1, n);
ritzvec = mkvec(1, n);
zeros = mkvec(1, n);
setvec(zeros, 1, n, 0.0);
workn = mkvec(1, n);
Ares = mkvec(1, d);
inv_lambda = mkvec(1, d);
index = smalloc((d + 1) * sizeof(int));
alpha = mkvec(1, maxj);
beta = mkvec(1, maxj + 1);
ritz = mkvec(1, maxj);
s = mkvec(1, maxj);
bj = mkvec(1, maxj);
workj = mkvec(1, maxj + 1);
q = smalloc((maxj + 1) * sizeof(double *));
scanlist = mkscanlist(d);
if (version == 2) {
/* Allocate Symmlq space all in one chunk. */
wv1 = smalloc(6 * (n + 1) * sizeof(double));
wv2 = &wv1[(n + 1)];
wv3 = &wv1[2 * (n + 1)];
wv4 = &wv1[3 * (n + 1)];
wv5 = &wv1[4 * (n + 1)];
wv6 = &wv1[5 * (n + 1)];
/* Set invariant symmlq parameters */
precon = FALSE; /* FALSE until we figure out a good way */
goodb = FALSE; /* should be FALSE for this application */
checka = FALSE; /* if don't know by now, too bad */
intlim = n; /* set to enforce a maximum number of Symmlq itns */
itnmin = 0; /* set to enforce a minimum number of Symmlq itns */
shift = 0.0; /* since just solving rather than doing RQI */
symmlqitns = 0; /* total number of Symmlq iterations */
nout = 0; /* Effectively disabled - see notes in symmlq.f */
rtol = 1.0e-5; /* requested residual tolerance */
normxlim = DOUBLE_MAX; /* Effectively disables ||x|| termination criterion */
long_n = n; /* copy to long for linting */
}
/* Initialize. */
vecran(r, 1, n);
if (vwsqrt == NULL) {
/* whack one's direction from initial vector */
orthog1(r, 1, n);
/* list the ones direction for later use in Symmlq */
if (version == 2) {
orthlist2 = makeorthlnk();
ones = mkvec(1, n);
setvec(ones, 1, n, 1.0);
orthlist2->vec = ones;
orthlist2->pntr = NULL;
}
}
else {
/* whack vwsqrt direction from initial vector */
orthogvec(r, 1, n, vwsqrt);
if (version == 2) {
/* list the vwsqrt direction for later use in Symmlq */
orthlist2 = makeorthlnk();
orthlist2->vec = vwsqrt;
orthlist2->pntr = NULL;
}
}
beta[1] = ch_norm(r, 1, n);
q[0] = zeros;
bji_tol = eigtol;
orthlist = NULL;
Sres_max = 0.0;
Anorm = 2 * maxdeg; /* Gershgorin estimate for ||A|| */
bis_safety = BISECTION_SAFETY;
inc_bis_safety = FALSE;
init_time += lanc_seconds() - time;
/* Main Lanczos loop. */
j = 1;
converged = FALSE;
memory_ok = TRUE;
while ((j <= maxj) && (converged == FALSE) && memory_ok) {
time = lanc_seconds();
/* Allocate next Lanczos vector. If fail, back up one step and compute approx. eigvec. */
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 (j <= 2) {
bail("ERROR: Sorry, can't salvage Lanczos.",1);
/* ... save yourselves, men. */
}
j--;
}
vecscale(q[j], 1, n, 1.0 / beta[j], r);
blas_time += lanc_seconds() - time;
time = lanc_seconds();
if (version == 1) {
splarax(Aq, A, n, q[j], vwsqrt, workn);
}
else {
symmlq_(&long_n, &(q[j][1]), &wv1[1], &wv2[1], &wv3[1], &wv4[1], &Aq[1], &wv5[1],
&wv6[1], &checka, &goodb, &precon, &shift, &nout,
&intlim, &rtol, &istop, &itn, &anorm, &acond,
&rnorm, &ynorm, (double *) A, vwsqrt, (double *) orthlist2,
&macheps, &normxlim, &itnmin);
symmlqitns += itn;
if (DEBUG_EVECS > 2) {
printf("Symmlq report: rtol %g\n", rtol);
printf(" system norm %g, solution norm %g\n", anorm, ynorm);
printf(" system condition %g, residual %g\n", acond, rnorm);
printf(" termination condition %2ld, iterations %3ld\n", istop, itn);
}
}
splarax_time += lanc_seconds() - time;
time = lanc_seconds();
update(u, 1, n, Aq, -beta[j], q[j - 1]);
alpha[j] = dot(u, 1, n, q[j]);
update(r, 1, n, u, -alpha[j], q[j]);
blas_time += lanc_seconds() - time;
time = lanc_seconds();
if (vwsqrt == NULL) {
orthog1(r, 1, n);
}
else {
orthogvec(r, 1, n, vwsqrt);
}
orthogonalize(r, n, orthlist);
temp = orthlist;
orthlist = makeorthlnk();
orthlist->vec = q[j];
orthlist->pntr = temp;
beta[j + 1] = ch_norm(r, 1, n);
orthog_time += lanc_seconds() - time;
time = lanc_seconds();
left = j/2;
right = j - left + 1;
if (inc_bis_safety) {
bis_safety *= 10;
inc_bis_safety = FALSE;
}
ritzval_flag = get_ritzvals(alpha, beta+1, j, Anorm, workj+1,
ritz, d, left, right, eigtol, bis_safety);
/* ... have to off-set beta and workj since full orthogonalization
indexes these from 1 to maxj+1 whereas selective orthog.
indexes them from 0 to maxj */
if (ritzval_flag != 0) {
bail("ERROR: Both Sturm bisection and QL failed.",1);
/* ... give up. */
}
ql_time += lanc_seconds() - time;
/* Convergence check using Paige bji estimates. */
time = lanc_seconds();
for (i = 1; i <= j; i++) {
Sres = Tevec(alpha, beta, j, ritz[i], s);
if (Sres > Sres_max) {
Sres_max = Sres;
}
if (Sres > SRESTOL) {
inc_bis_safety = TRUE;
}
bj[i] = s[j] * beta[j + 1];
}
tevec_time += lanc_seconds() - time;
time = lanc_seconds();
if (version == 1) {
scanmin(ritz, 1, j, &scanlist);
}
else {
scanmax(ritz, 1, j, &scanlist);
}
converged = TRUE;
if (j < d)
converged = FALSE;
else {
curlnk = scanlist;
while (curlnk != NULL) {
if (bj[curlnk->indx] > bji_tol) {
converged = FALSE;
}
curlnk = curlnk->pntr;
}
}
scan_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+1,j,s,y,n,q);
evec_time += lanc_seconds() - time;
/* Analyze computation for and report additional problems */
time = lanc_seconds();
if (DEBUG_EVECS>0 && version == 2) {
printf("\nTotal Symmlq iterations %3d\n", symmlqitns);
}
if (version == 2) {
for (i = 1; i <= d; i++) {
lambda[i] = 1.0/lambda[i];
}
}
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 any memory allocated in this routine. */
time = lanc_seconds();
frvec(u, 1);
frvec(r, 1);
frvec(Aq, 1);
frvec(ritzvec, 1);
frvec(zeros, 1);
if (vwsqrt == NULL && version == 2) {
frvec(ones, 1);
}
frvec(workn, 1);
frvec(Ares, 1);
frvec(inv_lambda, 1);
sfree(index);
frvec(alpha, 1);
frvec(beta, 1);
frvec(ritz, 1);
frvec(s, 1);
frvec(bj, 1);
frvec(workj, 1);
if (version == 2) {
frvec(wv1, 0);
}
while (scanlist != NULL) {
curlnk = scanlist->pntr;
sfree(scanlist);
scanlist = curlnk;
}
for (i = 1; i <= j; i++) {
frvec(q[i], 1);
}
while (orthlist != NULL) {
temp = orthlist->pntr;
sfree(orthlist);
orthlist = temp;
}
while (version == 2 && orthlist2 != NULL) {
temp = orthlist2->pntr;
sfree(orthlist2);
orthlist2 = temp;
}
sfree(q);
init_time += lanc_seconds() - time;
}
/* Finds needed eigenvalues of tridiagonal T using either the QL algorithm
or Sturm sequence bisection, whichever is predicted to be faster based
on a simple complexity model. If one fails (which is rare), the other
is tried. The return value is 0 if one of the routines succeeds. If they
both fail, the return value is 1, and Lanczos should compute the best
approximation it can based on previous iterations. */
int get_ritzvals(double *alpha, /* vector of Lanczos scalars */
double *beta, /* vector of Lanczos scalars */
int j, /* number of Lanczos iterations taken */
double Anorm, /* Gershgorin estimate */
double *workj, /* work vector for Sturm sequence */
double *ritz, /* array holding evals */
int d, /* problem dimension = num. eigenpairs needed */
int left_goodlim, /* number of ritz pairs checked on left end */
int right_goodlim, /* number of ritz pairs checked on right end */
double eigtol, /* tolerance on eigenpair */
double bis_safety /* bisection tolerance function divisor */
)
{
extern int DEBUG_EVECS; /* debug flag for eigen computation */
extern int WARNING_EVECS; /* warning flag for eigen computation */
int nvals_left; /* numb. evals to find on left end of spectrum */
int nvals_right; /* numb. evals to find on right end of spectrum */
double bisection_tol; /* width of interval bisection should converge to */
int pred_steps; /* predicts # of required bisection steps per eval */
int tot_pred_steps; /* predicts total # of required bisection steps */
double * ritz_sav = NULL; /* copy of ritzvals for debugging */
int bisect_flag; /* return status of bisect() */
int ql_flag; /* return status of ql() */
int local_debug; /* whether to check bisection results with ql */
int bisect(); /* locates eigvals using bisection on Sturm seq. */
int ql(); /* computes eigenvalues of T using eispack algorithm */
void shell_sort(); /* sorts vector of eigenvalues */
double * mkvec(); /* to allocate a vector */
void frvec(); /* free vector */
void cpvec(); /* vector copy */
void bail(); /* our exit routine */
void strout(); /* string out to screen and output file */
/* Determine number of ritzvals to find on left and right ends */
nvals_left = max(d, left_goodlim);
nvals_right = min(j - nvals_left, right_goodlim);
/* Estimate work for bisection vs. ql assuming bisection takes 5j flops per
step, ql takes 30j^2 flops per call. (Ignore sorts, copies, addressing.) */
bisection_tol = eigtol * eigtol / bis_safety;
pred_steps = (log10(Anorm / bisection_tol) / log10(2.0)) + 1;
tot_pred_steps = (nvals_left + nvals_right) * pred_steps;
bisect_flag = ql_flag = 0;
if (5 * tot_pred_steps < 30 * j) {
if (DEBUG_EVECS > 2)
printf(" tridiagonal solver: bisection\n");
/* Set local_debug = TRUE for a table checking bisection against QL. */
local_debug = FALSE;
if (local_debug) {
ritz_sav = mkvec(1, j);
cpvec(ritz_sav, 1, j, alpha);
cpvec(workj, 0, j, beta);
ql_flag = ql(ritz_sav, workj, j);
if (ql_flag != 0) {
bail("Aborting debugging procedure in get_ritzvals().\n", 1);
}
shell_sort(j, &ritz_sav[1]);
}
bisect_flag = bisect(alpha, beta, j, Anorm, workj, ritz, nvals_left, nvals_right, bisection_tol,
ritz_sav, pred_steps + 10);
if (local_debug)
frvec(ritz_sav, 1);
}
else {
if (DEBUG_EVECS > 2)
printf(" tridiagonal solver: ql\n");
cpvec(ritz, 1, j, alpha);
cpvec(workj, 0, j, beta);
ql_flag = ql(ritz, workj, j);
shell_sort(j, &ritz[1]);
}
if (bisect_flag != 0 && ql_flag == 0) {
if (DEBUG_EVECS > 0 || WARNING_EVECS > 0) {
strout("WARNING: Sturm bisection of T failed; switching to QL.\n");
}
if (DEBUG_EVECS > 1 || WARNING_EVECS > 1) {
if (bisect_flag == 1)
strout(" - failure detected in sturmcnt().\n");
if (bisect_flag == 2)
strout(" - maximum number of bisection steps reached.\n");
}
cpvec(ritz, 1, j, alpha);
cpvec(workj, 0, j, beta);
ql_flag = ql(ritz, workj, j);
shell_sort(j, &ritz[1]);
}
if (ql_flag != 0 && bisect_flag == 0) {
if (DEBUG_EVECS > 0 || WARNING_EVECS > 0) {
strout("WARNING: QL failed for T; switching to Sturm bisection.\n");
}
bisect_flag = bisect(alpha, beta, j, Anorm, workj, ritz, nvals_left, nvals_right, bisection_tol,
ritz_sav, pred_steps + 3);
}
if (bisect_flag != 0 && ql_flag != 0) {
if (DEBUG_EVECS > 0 || WARNING_EVECS > 0) {
return (1); /* can't recover; bail out with error code */
}
}
return (0); /* ... things seem ok. */
}
int
lanczos_ext_float (
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 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 */
double *gvec, /* the rhs n-vector in the extended eigen problem */
double sigma /* specifies the norm constraint on extended
eigenvector */
)
{
extern FILE *Output_File; /* output file or null */
extern int LANCZOS_SO_INTERVAL; /* interval between orthogonalizations */
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 T bisection */
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, intialize 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 */
int i, j, k; /* indicies */
int maxj; /* maximum number of Lanczos iterations */
float *u, *r; /* Lanczos vectors */
double *u_double; /* double version of u */
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 */
float *workn; /* work vector, e.g. product Av for checkeig */
double *workn_double; /* work vector, e.g. product Av for checkeig */
double *s; /* eigenvector of T */
float **q; /* columns of q are Lanczos basis vectors */
double *bj; /* beta(j)*(last el. of corr. eigvec s of T) */
double bis_safety; /* real safety factor for T bisection */
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_float **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 lastpause; /* Most recent step with good ritz vecs */
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 */
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 */
double *extvec; /* n-vector solving the extended A eigenproblem */
double *v; /* j-vector solving the extended T eigenproblem */
double extval=0.0; /* computed extended eigenvalue (of both A and T) */
double *work1, *work2; /* work vectors */
double check; /* to check an orthogonality condition */
double numerical_zero; /* used for zero in presense of round-off */
int ritzval_flag; /* status flag for get_ritzvals() */
double resid; /* residual */
int memory_ok; /* TRUE until memory runs out */
float *vwsqrt_float = NULL; /* float version of vwsqrt */
struct orthlink_float *makeorthlnk_float(); /* makes space for new entry in orthog. set */
struct scanlink *mkscanlist(); /* init scan list for min ritz vecs */
double *mkvec(); /* allocates space for a vector */
float *mkvec_float(); /* allocates space for a vector */
float *mkvec_ret_float(); /* mkvec() which returns error code */
double dot_float(); /* standard dot product routine */
double ch_norm(); /* vector norm */
double norm_float(); /* vector norm */
double Tevec(); /* calc eigenvector of T by linear recurrence */
double lanc_seconds(); /* switcheable timer */
/* free allocated memory safely */
int lanpause_float(); /* figure when to pause Lanczos iteration */
int get_ritzvals(); /* compute eigenvalues of T */
void setvec(); /* initialize a vector */
void setvec_float(); /* initialize a vector */
void vecscale_float(); /* scale a vector */
void splarax(); /* matrix vector multiply */
void splarax_float(); /* matrix vector multiply */
void update_float(); /* add scalar multiple of a vector to another */
void sorthog_float(); /* orthogonalize vector against list of others */
void bail(); /* our exit routine */
void scanmin(); /* store small values of vector in linked list */
void frvec(); /* free vector */
void frvec_float(); /* free vector */
void scadd(); /* add scalar multiple of vector to another */
void scadd_float(); /* add scalar multiple of vector to another */
void scadd_mixed(); /* add scalar multiple of vector to another */
void orthog1_float(); /* efficiently orthog. against vector of ones */
void solistout_float(); /* print out orthogonalization list */
void doubleout(); /* print a double precision number */
void orthogvec_float(); /* orthogonalize one vector against another */
void double_to_float(); /* copy a double vector to a float vector */
void get_extval(); /* find extended Ritz values */
void scale_diag(); /* scale vector by diagonal matrix */
void scale_diag_float(); /* scale vector by diagonal matrix */
void strout(); /* print string to screen and file */
if (DEBUG_TRACE > 0) {
printf("<Entering lanczos_ext_float>\n");
}
if (DEBUG_EVECS > 0) {
printf("Selective orthogonalization Lanczos for extended eigenproblem, matrix size = %d.\n", n);
}
/* Initialize time. */
time = lanc_seconds();
if (d != 1) {
bail("ERROR: Extended Lanczos only available for bisection.",1);
/* ... something must be wrong upstream. */
}
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_float(1, n);
u_double = mkvec(1, n);
r = mkvec_float(1, n);
workn = mkvec_float(1, n);
workn_double = 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(float *));
solist = smalloc((maxj + 1) * sizeof(struct orthlink_float *));
scanlist = mkscanlist(d);
extvec = mkvec(1, n);
v = mkvec(1, maxj);
work1 = mkvec(1, maxj);
work2 = mkvec(1, maxj);
/* Set some constants governing orthogonalization */
ngood = 0;
maxngood = 0;
bji_tol = eigtol;
Anorm = 2 * maxdeg; /* Gershgorin estimate for ||A|| */
goodtol = Anorm * sqrt(DOUBLE_EPSILON); /* Parlett & Scott's bound, p.224 */
interval = 2 + (int) min(LANCZOS_SO_INTERVAL - 2, n / (2 * LANCZOS_SO_INTERVAL));
bis_safety = BISECTION_SAFETY;
numerical_zero = 1.0e-6;
if (DEBUG_EVECS > 0) {
printf(" maxdeg %g\n", maxdeg);
printf(" goodtol %g\n", goodtol);
printf(" interval %d\n", interval);
printf(" maxj %d\n", maxj);
}
/* Make a float copy of vwsqrt */
if (vwsqrt != NULL) {
vwsqrt_float = mkvec_float(0,n);
double_to_float(vwsqrt_float,1,n,vwsqrt);
}
/* Initialize space. */
double_to_float(r,1,n,gvec);
if (vwsqrt_float != NULL) {
scale_diag_float(r,1,n,vwsqrt_float);
}
check = norm_float(r,1,n);
if (vwsqrt_float == NULL) {
orthog1_float(r, 1, n);
}
else {
orthogvec_float(r, 1, n, vwsqrt_float);
}
check = fabs(check - norm_float(r,1,n));
if (check > 10*numerical_zero && WARNING_EVECS > 0) {
strout("WARNING: In terminal propagation, rhs should have no component in the");
printf(" nullspace of the Laplacian, so check val %g should be zero.\n", check);
if (Output_File != NULL) {
fprintf(Output_File,
" nullspace of the Laplacian, so check val %g should be zero.\n",
check);
}
}
beta[0] = norm_float(r, 1, n);
q[0] = mkvec_float(1, n);
setvec_float(q[0], 1, n, 0.0);
setvec(bj, 1, maxj, DOUBLE_MAX);
if (beta[0] < numerical_zero) {
/* The rhs vector, Dg, of the transformed problem is numerically zero or is
in the null space of the Laplacian, so this is not a well posed extended
eigenproblem. Set maxj to zero to force a quick exit but still clean-up
memory and return(1) to indicate to eigensolve that it should call the
default eigensolver routine for the standard eigenproblem. */
maxj = 0;
}
/* Main Lanczos loop. */
j = 1;
lastpause = 0;
pausemode = 1;
left_ngood = 0;
left_goodlim = 0;
converged = FALSE;
Sres_max = 0.0;
inc_bis_safety = FALSE;
nopauses = TRUE;
memory_ok = TRUE;
init_time += lanc_seconds() - time;
while ((j <= maxj) && (!converged) && memory_ok) {
time = lanc_seconds();
/* Allocate next Lanczos vector. If fail, back up to last pause. */
q[j] = mkvec_ret_float(1, n);
if (q[j] == NULL) {
memory_ok = FALSE;
if (DEBUG_EVECS > 0 || WARNING_EVECS > 0) {
strout("WARNING: Lanczos_ext out of memory; computing best approximation available.\n");
}
if (nopauses) {
bail("ERROR: Sorry, can't salvage Lanczos_ext.",1);
/* ... save yourselves, men. */
}
for (i = lastpause+1; i <= j-1; i++) {
frvec_float(q[i], 1);
}
j = lastpause;
}
/* Basic Lanczos iteration */
vecscale_float(q[j], 1, n, (float)(1.0 / beta[j - 1]), r);
blas_time += lanc_seconds() - time;
time = lanc_seconds();
splarax_float(u, A, n, q[j], vwsqrt_float, workn);
splarax_time += lanc_seconds() - time;
time = lanc_seconds();
update_float(r, 1, n, u, (float)(-beta[j - 1]), q[j - 1]);
alpha[j] = dot_float(r, 1, n, q[j]);
update_float(r, 1, n, r, (float)(-alpha[j]), q[j]);
blas_time += lanc_seconds() - time;
/* Selective orthogonalization */
time = lanc_seconds();
if (vwsqrt_float == NULL) {
orthog1_float(r, 1, n);
}
else {
orthogvec_float(r, 1, n, vwsqrt_float);
}
if ((j == (lastpause + 1)) || (j == (lastpause + 2))) {
sorthog_float(r, n, solist, ngood);
}
orthog_time += lanc_seconds() - time;
beta[j] = norm_float(r, 1, n);
time = lanc_seconds();
pause = lanpause_float(j, lastpause, interval, q, n, &pausemode, version, beta[j]);
pause_time += lanc_seconds() - time;
if (pause) {
nopauses = FALSE;
lastpause = j;
/* Compute limits for checking Ritz pair convergence. */
if (version == 2) {
if (left_ngood + 2 > left_goodlim) {
left_goodlim = left_ngood + 2;
}
}
/* 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 > j) {
left_goodlim = min(left_goodlim, j);
}
/* Find Ritz vals using faster of Sturm bisection or ql. */
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, 0, eigtol, bis_safety);
ql_time += lanc_seconds() - time;
if (ritzval_flag != 0) {
bail("ERROR: Lanczos_ext failed in computing eigenvalues of T.",1);
/* ... we recover from this in lanczos_SO, but don't worry here. */
}
/* 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];
}
ritz_time += lanc_seconds() - time;
/* Show the portion of the spectrum checked for convergence. */
if (DEBUG_EVECS > 2) {
time = lanc_seconds();
printf("\nindex Ritz vals bji bounds\n");
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)) {
printf(" %3d", temp);
doubleout(ritz[temp], 1);
doubleout(bj[temp], 1);
printf("\n");
}
curlnk = curlnk->pntr;
}
printf(" -------------------\n");
printf(" goodtol: %19.16f\n\n", goodtol);
debug_time += lanc_seconds() - time;
}
get_extval(alpha, beta, j, ritz[1], s, eigtol, beta[0], sigma, &extval,
v, work1, work2);
/* check convergence of Ritz pairs */
time = lanc_seconds();
converged = TRUE;
if (j < d)
converged = FALSE;
else {
curlnk = scanlist;
while (curlnk != NULL) {
if (bj[curlnk->indx] > bji_tol) {
converged = FALSE;
}
curlnk = curlnk->pntr;
}
}
scan_time += lanc_seconds() - time;
if (!converged) {
ngood = 0;
left_ngood = 0; /* for setting left_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_float();
(solist[ngood])->vec = mkvec_float(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_float((solist[ngood])->vec, 1, n, 0.0);
for (k = 1; k <= j; k++) {
scadd_float((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, 0);
solistout_float(solist, n, ngood, j);
printf("---------------------end of iteration---------------------\n\n");
debug_time += lanc_seconds() - time;
}
}
}
j++;
}
j--;
if (DEBUG_EVECS > 0) {
time = lanc_seconds();
if (maxj == 0) {
printf("Not extended eigenproblem -- calling ordinary eigensolver.\n");
}
else {
printf(" Lanczos_ext itns: %d\n",j);
printf(" eigenvalue: %g\n",ritz[1]);
printf(" extended eigenvalue: %g\n",extval);
}
debug_time += lanc_seconds() - time;
}
if (maxj != 0) {
/* Compute (scaled) extended eigenvector. */
time = lanc_seconds();
setvec(y[1], 1, n, 0.0);
for (k = 1; k <= j; k++) {
scadd_mixed(y[1], 1, n, v[k], q[k]);
}
evec_time += lanc_seconds() - time;
/* Note: assign() will scale this y vector back to x (since y = Dx) */
/* Compute and check residual directly. Use the Ay = extval*y + Dg version of
the problem for convenience. Note that u and v are used here as workspace */
time = lanc_seconds();
splarax(workn_double, A, n, y[1], vwsqrt, u_double);
scadd(workn_double, 1, n, -extval, y[1]);
scale_diag(gvec,1,n,vwsqrt);
scadd(workn_double, 1, n, -1.0, gvec);
resid = ch_norm(workn_double, 1, n);
if (DEBUG_EVECS > 0) {
printf(" extended residual: %g\n",resid);
if (Output_File != NULL) {
fprintf(Output_File, " extended residual: %g\n",resid);
}
}
if (WARNING_EVECS > 0 && resid > eigtol) {
printf("WARNING: Extended residual (%g) greater than tolerance (%g).\n",
resid, eigtol);
if (Output_File != NULL) {
fprintf(Output_File,
"WARNING: Extended residual (%g) greater than tolerance (%g).\n",
resid, eigtol);
}
}
debug_time += lanc_seconds() - time;
}
/* free up memory */
time = lanc_seconds();
frvec_float(u, 1);
frvec(u_double, 1);
frvec_float(r, 1);
frvec_float(workn, 1);
frvec(workn_double, 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_float(q[i], 1);
}
sfree(q);
while (scanlist != NULL) {
curlnk = scanlist->pntr;
sfree(scanlist);
scanlist = curlnk;
}
for (i = 1; i <= maxngood; i++) {
frvec_float((solist[i])->vec, 1);
sfree(solist[i]);
}
sfree(solist);
frvec(extvec, 1);
frvec(v, 1);
frvec(work1, 1);
frvec(work2, 1);
if (vwsqrt != NULL)
frvec_float(vwsqrt_float, 1);
init_time += lanc_seconds() - time;
if (maxj == 0) return(1); /* see note on beta[0] and maxj above */
else return(0);
}
