int main(int argc,char **argv)
{
PetscErrorCode ierr;
DS ds;
SlepcSC sc;
PetscReal *T,*s,re,im;
PetscScalar *eigr,*eigi;
PetscInt i,n=10,l=2,k=5,ld;
PetscViewer viewer;
PetscBool verbose;
SlepcInitialize(&argc,&argv,(char*)0,help);
ierr = PetscOptionsGetInt(NULL,"-n",&n,NULL);CHKERRQ(ierr);
ierr = PetscPrintf(PETSC_COMM_WORLD,"Solve a Dense System of type GHIEP with compact storage - dimension %D.\n",n);CHKERRQ(ierr);
ierr = PetscOptionsGetInt(NULL,"-l",&l,NULL);CHKERRQ(ierr);
ierr = PetscOptionsGetInt(NULL,"-k",&k,NULL);CHKERRQ(ierr);
if (l>n || k>n || l>k) SETERRQ(PETSC_COMM_WORLD,1,"Wrong value of dimensions");
ierr = PetscOptionsHasName(NULL,"-verbose",&verbose);CHKERRQ(ierr);
/* Create DS object */
ierr = DSCreate(PETSC_COMM_WORLD,&ds);CHKERRQ(ierr);
ierr = DSSetType(ds,DSGHIEP);CHKERRQ(ierr);
ierr = DSSetFromOptions(ds);CHKERRQ(ierr);
ld = n+2; /* test leading dimension larger than n */
ierr = DSAllocate(ds,ld);CHKERRQ(ierr);
ierr = DSSetDimensions(ds,n,0,l,k);CHKERRQ(ierr);
ierr = DSSetCompact(ds,PETSC_TRUE);CHKERRQ(ierr);
/* Set up viewer */
ierr = PetscViewerASCIIGetStdout(PETSC_COMM_WORLD,&viewer);CHKERRQ(ierr);
ierr = PetscViewerPushFormat(viewer,PETSC_VIEWER_ASCII_INFO_DETAIL);CHKERRQ(ierr);
ierr = DSView(ds,viewer);CHKERRQ(ierr);
ierr = PetscViewerPopFormat(viewer);CHKERRQ(ierr);
if (verbose) {
ierr = PetscViewerPushFormat(viewer,PETSC_VIEWER_ASCII_MATLAB);CHKERRQ(ierr);
}
/* Fill arrow-tridiagonal matrix */
ierr = DSGetArrayReal(ds,DS_MAT_T,&T);CHKERRQ(ierr);
ierr = DSGetArrayReal(ds,DS_MAT_D,&s);CHKERRQ(ierr);
for (i=0;i<n;i++) T[i] = (PetscReal)(i+1);
for (i=k;i<n-1;i++) T[i+ld] = 1.0;
for (i=l;i<k;i++) T[i+2*ld] = 1.0;
T[2*ld+l+1] = -7; T[ld+k+1] = -7;
/* Signature matrix */
for (i=0;i<n;i++) s[i] = 1.0;
s[l+1] = -1.0;
s[k+1] = -1.0;
ierr = DSRestoreArrayReal(ds,DS_MAT_T,&T);CHKERRQ(ierr);
ierr = DSRestoreArrayReal(ds,DS_MAT_D,&s);CHKERRQ(ierr);
if (l==0 && k==0) {
ierr = DSSetState(ds,DS_STATE_INTERMEDIATE);CHKERRQ(ierr);
} else {
ierr = DSSetState(ds,DS_STATE_RAW);CHKERRQ(ierr);
}
if (verbose) {
ierr = PetscPrintf(PETSC_COMM_WORLD,"Initial - - - - - - - - -\n");CHKERRQ(ierr);
ierr = DSView(ds,viewer);CHKERRQ(ierr);
}
/* Solve */
ierr = PetscCalloc2(n,&eigr,n,&eigi);CHKERRQ(ierr);
ierr = DSGetSlepcSC(ds,&sc);CHKERRQ(ierr);
sc->comparison = SlepcCompareLargestMagnitude;
sc->comparisonctx = NULL;
sc->map = NULL;
sc->mapobj = NULL;
ierr = DSSolve(ds,eigr,eigi);CHKERRQ(ierr);
ierr = DSSort(ds,eigr,eigi,NULL,NULL,NULL);CHKERRQ(ierr);
if (verbose) {
ierr = PetscPrintf(PETSC_COMM_WORLD,"After solve - - - - - - - - -\n");CHKERRQ(ierr);
ierr = DSView(ds,viewer);CHKERRQ(ierr);
}
/* Print eigenvalues */
ierr = PetscPrintf(PETSC_COMM_WORLD,"Computed eigenvalues =\n",n);CHKERRQ(ierr);
for (i=0;i<n;i++) {
#if defined(PETSC_USE_COMPLEX)
re = PetscRealPart(eigr[i]);
im = PetscImaginaryPart(eigr[i]);
#else
re = eigr[i];
im = eigi[i];
#endif
if (PetscAbs(im)<1e-10) {
ierr = PetscViewerASCIIPrintf(viewer," %.5f\n",(double)re);CHKERRQ(ierr);
} else {
ierr = PetscViewerASCIIPrintf(viewer," %.5f%+.5fi\n",(double)re,(double)im);CHKERRQ(ierr);
}
}
ierr = PetscFree2(eigr,eigi);CHKERRQ(ierr);
ierr = DSDestroy(&ds);CHKERRQ(ierr);
ierr = SlepcFinalize();
return 0;
}
PetscErrorCode EPSSolve_Lanczos(EPS eps)
{
EPS_LANCZOS *lanczos = (EPS_LANCZOS*)eps->data;
PetscErrorCode ierr;
PetscInt nconv,i,j,k,l,x,n,*perm,restart,ncv=eps->ncv,r,ld;
Vec vi,vj,w;
Mat U;
PetscScalar *Y,*ritz,stmp;
PetscReal *d,*e,*bnd,anorm,beta,norm,rtmp,resnorm;
PetscBool breakdown;
char *conv,ctmp;
PetscFunctionBegin;
ierr = DSGetLeadingDimension(eps->ds,&ld);CHKERRQ(ierr);
ierr = PetscMalloc4(ncv,&ritz,ncv,&bnd,ncv,&perm,ncv,&conv);CHKERRQ(ierr);
/* The first Lanczos vector is the normalized initial vector */
ierr = EPSGetStartVector(eps,0,NULL);CHKERRQ(ierr);
anorm = -1.0;
nconv = 0;
/* Restart loop */
while (eps->reason == EPS_CONVERGED_ITERATING) {
eps->its++;
/* Compute an ncv-step Lanczos factorization */
n = PetscMin(nconv+eps->mpd,ncv);
ierr = DSGetArrayReal(eps->ds,DS_MAT_T,&d);CHKERRQ(ierr);
e = d + ld;
ierr = EPSBasicLanczos(eps,d,e,nconv,&n,&breakdown,anorm);CHKERRQ(ierr);
beta = e[n-1];
ierr = DSRestoreArrayReal(eps->ds,DS_MAT_T,&d);CHKERRQ(ierr);
ierr = DSSetDimensions(eps->ds,n,0,nconv,0);CHKERRQ(ierr);
ierr = DSSetState(eps->ds,DS_STATE_INTERMEDIATE);CHKERRQ(ierr);
ierr = BVSetActiveColumns(eps->V,nconv,n);CHKERRQ(ierr);
/* Solve projected problem */
ierr = DSSolve(eps->ds,ritz,NULL);CHKERRQ(ierr);
ierr = DSSort(eps->ds,ritz,NULL,NULL,NULL,NULL);CHKERRQ(ierr);
/* Estimate ||A|| */
for (i=nconv;i<n;i++)
anorm = PetscMax(anorm,PetscAbsReal(PetscRealPart(ritz[i])));
/* Compute residual norm estimates as beta*abs(Y(m,:)) + eps*||A|| */
ierr = DSGetArray(eps->ds,DS_MAT_Q,&Y);CHKERRQ(ierr);
for (i=nconv;i<n;i++) {
resnorm = beta*PetscAbsScalar(Y[n-1+i*ld]) + PETSC_MACHINE_EPSILON*anorm;
ierr = (*eps->converged)(eps,ritz[i],eps->eigi[i],resnorm,&bnd[i],eps->convergedctx);CHKERRQ(ierr);
if (bnd[i]<eps->tol) conv[i] = 'C';
else conv[i] = 'N';
}
ierr = DSRestoreArray(eps->ds,DS_MAT_Q,&Y);CHKERRQ(ierr);
/* purge repeated ritz values */
if (lanczos->reorthog == EPS_LANCZOS_REORTHOG_LOCAL) {
for (i=nconv+1;i<n;i++) {
if (conv[i] == 'C' && PetscAbsScalar((ritz[i]-ritz[i-1])/ritz[i]) < eps->tol) conv[i] = 'R';
}
}
/* Compute restart vector */
if (breakdown) {
ierr = PetscInfo2(eps,"Breakdown in Lanczos method (it=%D norm=%g)\n",eps->its,(double)beta);CHKERRQ(ierr);
} else {
restart = nconv;
while (restart<n && conv[restart] != 'N') restart++;
if (restart >= n) {
breakdown = PETSC_TRUE;
} else {
for (i=restart+1;i<n;i++) {
if (conv[i] == 'N') {
ierr = SlepcSCCompare(eps->sc,ritz[restart],0.0,ritz[i],0.0,&r);CHKERRQ(ierr);
if (r>0) restart = i;
}
}
ierr = DSGetArray(eps->ds,DS_MAT_Q,&Y);CHKERRQ(ierr);
ierr = BVMultColumn(eps->V,1.0,0.0,n,Y+restart*ld+nconv);CHKERRQ(ierr);
ierr = DSRestoreArray(eps->ds,DS_MAT_Q,&Y);CHKERRQ(ierr);
}
}
/* Count and put converged eigenvalues first */
for (i=nconv;i<n;i++) perm[i] = i;
for (k=nconv;k<n;k++) {
if (conv[perm[k]] != 'C') {
j = k + 1;
while (j<n && conv[perm[j]] != 'C') j++;
if (j>=n) break;
l = perm[k]; perm[k] = perm[j]; perm[j] = l;
}
}
/* Sort eigenvectors according to permutation */
ierr = DSGetArray(eps->ds,DS_MAT_Q,&Y);CHKERRQ(ierr);
for (i=nconv;i<k;i++) {
x = perm[i];
if (x != i) {
j = i + 1;
while (perm[j] != i) j++;
/* swap eigenvalues i and j */
stmp = ritz[x]; ritz[x] = ritz[i]; ritz[i] = stmp;
rtmp = bnd[x]; bnd[x] = bnd[i]; bnd[i] = rtmp;
ctmp = conv[x]; conv[x] = conv[i]; conv[i] = ctmp;
perm[j] = x; perm[i] = i;
/* swap eigenvectors i and j */
for (l=0;l<n;l++) {
stmp = Y[l+x*ld]; Y[l+x*ld] = Y[l+i*ld]; Y[l+i*ld] = stmp;
}
}
}
ierr = DSRestoreArray(eps->ds,DS_MAT_Q,&Y);CHKERRQ(ierr);
/* compute converged eigenvectors */
ierr = DSGetMat(eps->ds,DS_MAT_Q,&U);CHKERRQ(ierr);
ierr = BVMultInPlace(eps->V,U,nconv,k);CHKERRQ(ierr);
ierr = MatDestroy(&U);CHKERRQ(ierr);
/* purge spurious ritz values */
if (lanczos->reorthog == EPS_LANCZOS_REORTHOG_LOCAL) {
for (i=nconv;i<k;i++) {
ierr = BVGetColumn(eps->V,i,&vi);CHKERRQ(ierr);
ierr = VecNorm(vi,NORM_2,&norm);CHKERRQ(ierr);
ierr = VecScale(vi,1.0/norm);CHKERRQ(ierr);
w = eps->work[0];
ierr = STApply(eps->st,vi,w);CHKERRQ(ierr);
ierr = VecAXPY(w,-ritz[i],vi);CHKERRQ(ierr);
ierr = BVRestoreColumn(eps->V,i,&vi);CHKERRQ(ierr);
ierr = VecNorm(w,NORM_2,&norm);CHKERRQ(ierr);
ierr = (*eps->converged)(eps,ritz[i],eps->eigi[i],norm,&bnd[i],eps->convergedctx);CHKERRQ(ierr);
if (bnd[i]>=eps->tol) conv[i] = 'S';
}
for (i=nconv;i<k;i++) {
if (conv[i] != 'C') {
j = i + 1;
while (j<k && conv[j] != 'C') j++;
if (j>=k) break;
/* swap eigenvalues i and j */
stmp = ritz[j]; ritz[j] = ritz[i]; ritz[i] = stmp;
rtmp = bnd[j]; bnd[j] = bnd[i]; bnd[i] = rtmp;
ctmp = conv[j]; conv[j] = conv[i]; conv[i] = ctmp;
/* swap eigenvectors i and j */
ierr = BVGetColumn(eps->V,i,&vi);CHKERRQ(ierr);
ierr = BVGetColumn(eps->V,j,&vj);CHKERRQ(ierr);
ierr = VecSwap(vi,vj);CHKERRQ(ierr);
ierr = BVRestoreColumn(eps->V,i,&vi);CHKERRQ(ierr);
ierr = BVRestoreColumn(eps->V,j,&vj);CHKERRQ(ierr);
}
}
k = i;
}
/* store ritz values and estimated errors */
for (i=nconv;i<n;i++) {
eps->eigr[i] = ritz[i];
eps->errest[i] = bnd[i];
}
ierr = EPSMonitor(eps,eps->its,nconv,eps->eigr,eps->eigi,eps->errest,n);CHKERRQ(ierr);
nconv = k;
if (eps->its >= eps->max_it) eps->reason = EPS_DIVERGED_ITS;
if (nconv >= eps->nev) eps->reason = EPS_CONVERGED_TOL;
if (eps->reason == EPS_CONVERGED_ITERATING) { /* copy restart vector */
ierr = BVCopyColumn(eps->V,n,nconv);CHKERRQ(ierr);
if (lanczos->reorthog == EPS_LANCZOS_REORTHOG_LOCAL && !breakdown) {
/* Reorthonormalize restart vector */
ierr = BVOrthogonalizeColumn(eps->V,nconv,NULL,&norm,&breakdown);CHKERRQ(ierr);
ierr = BVScaleColumn(eps->V,nconv,1.0/norm);CHKERRQ(ierr);
}
if (breakdown) {
/* Use random vector for restarting */
ierr = PetscInfo(eps,"Using random vector for restart\n");CHKERRQ(ierr);
ierr = EPSGetStartVector(eps,nconv,&breakdown);CHKERRQ(ierr);
}
if (breakdown) { /* give up */
eps->reason = EPS_DIVERGED_BREAKDOWN;
ierr = PetscInfo(eps,"Unable to generate more start vectors\n");CHKERRQ(ierr);
}
}
}
eps->nconv = nconv;
ierr = PetscFree4(ritz,bnd,perm,conv);CHKERRQ(ierr);
PetscFunctionReturn(0);
}
PetscErrorCode SVDSolve_TRLanczos(SVD svd)
{
PetscErrorCode ierr;
SVD_TRLANCZOS *lanczos = (SVD_TRLANCZOS*)svd->data;
PetscReal *alpha,*beta,lastbeta,norm;
PetscScalar *Q,*swork=NULL,*w;
PetscInt i,k,l,nv,ld;
Mat U,VT;
PetscBool conv;
BVOrthogType orthog;
PetscFunctionBegin;
/* allocate working space */
ierr = DSGetLeadingDimension(svd->ds,&ld);CHKERRQ(ierr);
ierr = BVGetOrthogonalization(svd->V,&orthog,NULL,NULL);CHKERRQ(ierr);
ierr = PetscMalloc1(ld,&w);CHKERRQ(ierr);
if (lanczos->oneside && orthog == BV_ORTHOG_CGS) {
ierr = PetscMalloc1(svd->ncv+1,&swork);CHKERRQ(ierr);
}
/* normalize start vector */
if (!svd->nini) {
ierr = BVSetRandomColumn(svd->V,0,svd->rand);CHKERRQ(ierr);
ierr = BVNormColumn(svd->V,0,NORM_2,&norm);CHKERRQ(ierr);
ierr = BVScaleColumn(svd->V,0,1.0/norm);CHKERRQ(ierr);
}
l = 0;
while (svd->reason == SVD_CONVERGED_ITERATING) {
svd->its++;
/* inner loop */
nv = PetscMin(svd->nconv+svd->mpd,svd->ncv);
ierr = BVSetActiveColumns(svd->V,svd->nconv,nv);CHKERRQ(ierr);
ierr = BVSetActiveColumns(svd->U,svd->nconv,nv);CHKERRQ(ierr);
ierr = DSGetArrayReal(svd->ds,DS_MAT_T,&alpha);CHKERRQ(ierr);
beta = alpha + ld;
if (lanczos->oneside) {
if (orthog == BV_ORTHOG_MGS) {
ierr = SVDOneSideTRLanczosMGS(svd,alpha,beta,svd->V,svd->U,svd->nconv,l,nv);CHKERRQ(ierr);
} else {
ierr = SVDOneSideTRLanczosCGS(svd,alpha,beta,svd->V,svd->U,svd->nconv,l,nv,swork);CHKERRQ(ierr);
}
} else {
ierr = SVDTwoSideLanczos(svd,alpha,beta,svd->V,svd->U,svd->nconv+l,nv);CHKERRQ(ierr);
}
lastbeta = beta[nv-1];
ierr = DSRestoreArrayReal(svd->ds,DS_MAT_T,&alpha);CHKERRQ(ierr);
ierr = BVScaleColumn(svd->V,nv,1.0/lastbeta);CHKERRQ(ierr);
/* compute SVD of general matrix */
ierr = DSSetDimensions(svd->ds,nv,nv,svd->nconv,svd->nconv+l);CHKERRQ(ierr);
if (l==0) {
ierr = DSSetState(svd->ds,DS_STATE_INTERMEDIATE);CHKERRQ(ierr);
} else {
ierr = DSSetState(svd->ds,DS_STATE_RAW);CHKERRQ(ierr);
}
ierr = DSSolve(svd->ds,w,NULL);CHKERRQ(ierr);
ierr = DSSort(svd->ds,w,NULL,NULL,NULL,NULL);CHKERRQ(ierr);
/* compute error estimates */
k = 0;
conv = PETSC_TRUE;
ierr = DSGetArray(svd->ds,DS_MAT_U,&Q);CHKERRQ(ierr);
ierr = DSGetArrayReal(svd->ds,DS_MAT_T,&alpha);CHKERRQ(ierr);
beta = alpha + ld;
for (i=svd->nconv;i<nv;i++) {
svd->sigma[i] = PetscRealPart(w[i]);
beta[i] = PetscRealPart(Q[nv-1+i*ld])*lastbeta;
svd->errest[i] = PetscAbsScalar(beta[i]);
if (svd->sigma[i] > svd->tol) svd->errest[i] /= svd->sigma[i];
if (conv) {
if (svd->errest[i] < svd->tol) k++;
else conv = PETSC_FALSE;
}
}
ierr = DSRestoreArrayReal(svd->ds,DS_MAT_T,&alpha);CHKERRQ(ierr);
ierr = DSRestoreArray(svd->ds,DS_MAT_U,&Q);CHKERRQ(ierr);
/* check convergence and update l */
if (svd->its >= svd->max_it) svd->reason = SVD_DIVERGED_ITS;
if (svd->nconv+k >= svd->nsv) svd->reason = SVD_CONVERGED_TOL;
if (svd->reason != SVD_CONVERGED_ITERATING) l = 0;
else l = PetscMax((nv-svd->nconv-k)/2,0);
/* compute converged singular vectors and restart vectors */
ierr = DSGetMat(svd->ds,DS_MAT_VT,&VT);CHKERRQ(ierr);
ierr = BVMultInPlaceTranspose(svd->V,VT,svd->nconv,svd->nconv+k+l);CHKERRQ(ierr);
ierr = MatDestroy(&VT);CHKERRQ(ierr);
ierr = DSGetMat(svd->ds,DS_MAT_U,&U);CHKERRQ(ierr);
ierr = BVMultInPlace(svd->U,U,svd->nconv,svd->nconv+k+l);CHKERRQ(ierr);
ierr = MatDestroy(&U);CHKERRQ(ierr);
/* copy the last vector to be the next initial vector */
if (svd->reason == SVD_CONVERGED_ITERATING) {
ierr = BVCopyColumn(svd->V,nv,svd->nconv+k+l);CHKERRQ(ierr);
}
svd->nconv += k;
ierr = SVDMonitor(svd,svd->its,svd->nconv,svd->sigma,svd->errest,nv);CHKERRQ(ierr);
}
/* orthonormalize U columns in one side method */
if (lanczos->oneside) {
for (i=0;i<svd->nconv;i++) {
ierr = BVOrthogonalizeColumn(svd->U,i,NULL,&norm,NULL);CHKERRQ(ierr);
ierr = BVScaleColumn(svd->U,i,1.0/norm);CHKERRQ(ierr);
}
}
/* free working space */
ierr = PetscFree(w);CHKERRQ(ierr);
if (swork) { ierr = PetscFree(swork);CHKERRQ(ierr); }
PetscFunctionReturn(0);
}
