/*
SLEPc eigensolver: "subspace"
Method: Subspace Iteration
Algorithm:
Subspace iteration with Rayleigh-Ritz projection and locking,
based on the SRRIT implementation.
References:
[1] "Subspace Iteration in SLEPc", SLEPc Technical Report STR-3,
available at http://www.grycap.upv.es/slepc.
Last update: Feb 2009
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SLEPc - Scalable Library for Eigenvalue Problem Computations
Copyright (c) 2002-2011, Universitat Politecnica de Valencia, Spain
This file is part of SLEPc.
SLEPc is free software: you can redistribute it and/or modify it under the
terms of version 3 of the GNU Lesser General Public License as published by
the Free Software Foundation.
SLEPc is distributed in the hope that it will be useful, but WITHOUT ANY
WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for
more details.
You should have received a copy of the GNU Lesser General Public License
along with SLEPc. If not, see <http://www.gnu.org/licenses/>.
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*/
#include <private/epsimpl.h> /*I "slepceps.h" I*/
#include <slepcblaslapack.h>
PetscErrorCode EPSSolve_Subspace(EPS);
typedef struct {
Vec *AV;
} EPS_SUBSPACE;
#undef __FUNCT__
#define __FUNCT__ "EPSSetUp_Subspace"
PetscErrorCode EPSSetUp_Subspace(EPS eps)
{
PetscErrorCode ierr;
EPS_SUBSPACE *ctx = (EPS_SUBSPACE *)eps->data;
PetscFunctionBegin;
if (eps->ncv) { /* ncv set */
if (eps->ncv<eps->nev) SETERRQ(((PetscObject)eps)->comm,1,"The value of ncv must be at least nev");
}
else if (eps->mpd) { /* mpd set */
eps->ncv = PetscMin(eps->n,eps->nev+eps->mpd);
}
else { /* neither set: defaults depend on nev being small or large */
if (eps->nev<500) eps->ncv = PetscMin(eps->n,PetscMax(2*eps->nev,eps->nev+15));
else { eps->mpd = 500; eps->ncv = PetscMin(eps->n,eps->nev+eps->mpd); }
}
if (!eps->mpd) eps->mpd = eps->ncv;
if (!eps->max_it) eps->max_it = PetscMax(100,2*eps->n/eps->ncv);
if (!eps->which) eps->which = EPS_LARGEST_MAGNITUDE;
if (eps->which!=EPS_LARGEST_MAGNITUDE)
SETERRQ(((PetscObject)eps)->comm,1,"Wrong value of eps->which");
if (!eps->extraction) {
ierr = EPSSetExtraction(eps,EPS_RITZ);CHKERRQ(ierr);
} else if (eps->extraction!=EPS_RITZ) {
SETERRQ(((PetscObject)eps)->comm,PETSC_ERR_SUP,"Unsupported extraction type\n");
}
ierr = EPSAllocateSolution(eps);CHKERRQ(ierr);
ierr = VecDuplicateVecs(eps->t,eps->ncv,&ctx->AV);CHKERRQ(ierr);
ierr = PetscFree(eps->T);CHKERRQ(ierr);
ierr = PetscMalloc(eps->ncv*eps->ncv*sizeof(PetscScalar),&eps->T);CHKERRQ(ierr);
ierr = EPSDefaultGetWork(eps,1);CHKERRQ(ierr);
/* dispatch solve method */
if (eps->leftvecs) SETERRQ(((PetscObject)eps)->comm,PETSC_ERR_SUP,"Left vectors not supported in this solver");
eps->ops->solve = EPSSolve_Subspace;
PetscFunctionReturn(0);
}
#undef __FUNCT__
#define __FUNCT__ "EPSHessCond"
/*
EPSHessCond - Compute the inf-norm condition number of the upper
Hessenberg matrix H: cond(H) = norm(H)*norm(inv(H)).
This routine uses Gaussian elimination with partial pivoting to
compute the inverse explicitly.
*/
static PetscErrorCode EPSHessCond(PetscInt n_,PetscScalar* H,PetscInt ldh_,PetscReal* cond)
{
#if defined(PETSC_MISSING_LAPACK_GETRF) || defined(SLEPC_MISSING_LAPACK_GETRI) || defined(SLEPC_MISSING_LAPACK_LANGE) || defined(SLEPC_MISSING_LAPACK_LANHS)
PetscFunctionBegin;
SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"GETRF,GETRI - Lapack routines are unavailable.");
#else
PetscErrorCode ierr;
PetscBLASInt *ipiv,lwork,info,n=n_,ldh=ldh_;
PetscScalar *work;
PetscReal hn,hin,*rwork;
PetscFunctionBegin;
ierr = PetscLogEventBegin(EPS_Dense,0,0,0,0);CHKERRQ(ierr);
ierr = PetscMalloc(sizeof(PetscBLASInt)*n,&ipiv);CHKERRQ(ierr);
lwork = n*n;
ierr = PetscMalloc(sizeof(PetscScalar)*lwork,&work);CHKERRQ(ierr);
ierr = PetscMalloc(sizeof(PetscReal)*n,&rwork);CHKERRQ(ierr);
hn = LAPACKlanhs_("I",&n,H,&ldh,rwork);
LAPACKgetrf_(&n,&n,H,&ldh,ipiv,&info);
if (info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"Error in Lapack xGETRF %d",info);
LAPACKgetri_(&n,H,&ldh,ipiv,work,&lwork,&info);
if (info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"Error in Lapack xGETRI %d",info);
hin = LAPACKlange_("I",&n,&n,H,&ldh,rwork);
*cond = hn * hin;
ierr = PetscFree(ipiv);CHKERRQ(ierr);
ierr = PetscFree(work);CHKERRQ(ierr);
ierr = PetscFree(rwork);CHKERRQ(ierr);
ierr = PetscLogEventEnd(EPS_Dense,0,0,0,0);CHKERRQ(ierr);
PetscFunctionReturn(0);
#endif
}
#undef __FUNCT__
#define __FUNCT__ "EPSFindGroup"
/*
EPSFindGroup - Find a group of nearly equimodular eigenvalues, provided
in arrays wr and wi, according to the tolerance grptol. Also the 2-norms
of the residuals must be passed-in (rsd). Arrays are processed from index
l to index m only. The output information is:
ngrp - number of entries of the group
ctr - (w(l)+w(l+ngrp-1))/2
ae - average of wr(l),...,wr(l+ngrp-1)
arsd - average of rsd(l),...,rsd(l+ngrp-1)
*/
static PetscErrorCode EPSFindGroup(PetscInt l,PetscInt m,PetscScalar *wr,PetscScalar *wi,PetscReal *rsd,
PetscReal grptol,PetscInt *ngrp,PetscReal *ctr,PetscReal *ae,PetscReal *arsd)
{
PetscInt i;
PetscReal rmod,rmod1;
PetscFunctionBegin;
*ngrp = 0;
*ctr = 0;
rmod = SlepcAbsEigenvalue(wr[l],wi[l]);
for (i=l;i<m;) {
rmod1 = SlepcAbsEigenvalue(wr[i],wi[i]);
if (PetscAbsReal(rmod-rmod1) > grptol*(rmod+rmod1)) break;
*ctr = (rmod+rmod1)/2.0;
if (wi[i] != 0.0) {
(*ngrp)+=2;
i+=2;
} else {
(*ngrp)++;
i++;
}
}
*ae = 0;
*arsd = 0;
if (*ngrp) {
for (i=l;i<l+*ngrp;i++) {
(*ae) += PetscRealPart(wr[i]);
(*arsd) += rsd[i]*rsd[i];
}
*ae = *ae / *ngrp;
*arsd = PetscSqrtScalar(*arsd / *ngrp);
}
PetscFunctionReturn(0);
}
#undef __FUNCT__
#define __FUNCT__ "EPSSchurResidualNorms"
/*
EPSSchurResidualNorms - Computes the column norms of residual vectors
OP*V(1:n,l:m) - V*T(1:m,l:m), where, on entry, OP*V has been computed and
stored in AV. ldt is the leading dimension of T. On exit, rsd(l) to
rsd(m) contain the computed norms.
*/
static PetscErrorCode EPSSchurResidualNorms(EPS eps,Vec *V,Vec *AV,PetscScalar *T,PetscInt l,PetscInt m,PetscInt ldt,PetscReal *rsd)
{
PetscErrorCode ierr;
PetscInt i,k;
#if defined(PETSC_USE_COMPLEX)
PetscScalar t;
#endif
PetscFunctionBegin;
for (i=l;i<m;i++) {
if (i==m-1 || T[i+1+ldt*i]==0.0) k=i+1; else k=i+2;
ierr = VecCopy(AV[i],eps->work[0]);CHKERRQ(ierr);
ierr = SlepcVecMAXPBY(eps->work[0],1.0,-1.0,k,T+ldt*i,V);CHKERRQ(ierr);
#if !defined(PETSC_USE_COMPLEX)
ierr = VecDot(eps->work[0],eps->work[0],rsd+i);CHKERRQ(ierr);
#else
ierr = VecDot(eps->work[0],eps->work[0],&t);CHKERRQ(ierr);
rsd[i] = PetscRealPart(t);
#endif
}
for (i=l;i<m;i++) {
if (i == m-1) {
rsd[i] = PetscSqrtReal(rsd[i]);
} else if (T[i+1+(ldt*i)]==0.0) {
rsd[i] = PetscSqrtReal(rsd[i]);
} else {
rsd[i] = PetscSqrtReal((rsd[i]+rsd[i+1])/2.0);
rsd[i+1] = rsd[i];
i++;
}
}
PetscFunctionReturn(0);
}
#undef __FUNCT__
#define __FUNCT__ "EPSSolve_Subspace"
PetscErrorCode EPSSolve_Subspace(EPS eps)
{
PetscErrorCode ierr;
EPS_SUBSPACE *ctx = (EPS_SUBSPACE *)eps->data;
PetscInt i,k,ngrp,nogrp,*itrsd,*itrsdold,
nxtsrr,idsrr,idort,nxtort,nv,ncv = eps->ncv,its;
PetscScalar *T=eps->T,*U;
PetscReal arsd,oarsd,ctr,octr,ae,oae,*rsd,norm,tcond=1.0;
PetscBool breakdown;
/* Parameters */
PetscInt init = 5; /* Number of initial iterations */
PetscReal stpfac = 1.5, /* Max num of iter before next SRR step */
alpha = 1.0, /* Used to predict convergence of next residual */
beta = 1.1, /* Used to predict convergence of next residual */
grptol = 1e-8, /* Tolerance for EPSFindGroup */
cnvtol = 1e-6; /* Convergence criterion for cnv */
PetscInt orttol = 2; /* Number of decimal digits whose loss
can be tolerated in orthogonalization */
PetscFunctionBegin;
its = 0;
ierr = PetscMalloc(sizeof(PetscScalar)*ncv*ncv,&U);CHKERRQ(ierr);
ierr = PetscMalloc(sizeof(PetscReal)*ncv,&rsd);CHKERRQ(ierr);
ierr = PetscMalloc(sizeof(PetscInt)*ncv,&itrsd);CHKERRQ(ierr);
ierr = PetscMalloc(sizeof(PetscInt)*ncv,&itrsdold);CHKERRQ(ierr);
for (i=0;i<ncv;i++) {
rsd[i] = 0.0;
itrsd[i] = -1;
}
/* Complete the initial basis with random vectors and orthonormalize them */
k = eps->nini;
while (k<ncv) {
ierr = SlepcVecSetRandom(eps->V[k],eps->rand);CHKERRQ(ierr);
ierr = IPOrthogonalize(eps->ip,eps->nds,eps->DS,k,PETSC_NULL,eps->V,eps->V[k],PETSC_NULL,&norm,&breakdown);CHKERRQ(ierr);
if (norm>0.0 && !breakdown) {
ierr = VecScale(eps->V[k],1.0/norm);CHKERRQ(ierr);
k++;
}
}
while (eps->its<eps->max_it) {
eps->its++;
nv = PetscMin(eps->nconv+eps->mpd,ncv);
/* Find group in previously computed eigenvalues */
ierr = EPSFindGroup(eps->nconv,nv,eps->eigr,eps->eigi,rsd,grptol,&nogrp,&octr,&oae,&oarsd);CHKERRQ(ierr);
/* Compute a Rayleigh-Ritz projection step
on the active columns (idx) */
/* 1. AV(:,idx) = OP * V(:,idx) */
for (i=eps->nconv;i<nv;i++) {
ierr = STApply(eps->OP,eps->V[i],ctx->AV[i]);CHKERRQ(ierr);
}
/* 2. T(:,idx) = V' * AV(:,idx) */
for (i=eps->nconv;i<nv;i++) {
ierr = VecMDot(ctx->AV[i],nv,eps->V,T+i*ncv);CHKERRQ(ierr);
}
/* 3. Reduce projected matrix to Hessenberg form: [U,T] = hess(T) */
ierr = EPSDenseHessenberg(nv,eps->nconv,T,ncv,U);CHKERRQ(ierr);
/* 4. Reduce T to quasi-triangular (Schur) form */
ierr = EPSDenseSchur(nv,eps->nconv,T,ncv,U,eps->eigr,eps->eigi);CHKERRQ(ierr);
/* 5. Sort diagonal elements in T and accumulate rotations on U */
ierr = EPSSortDenseSchur(eps,nv,eps->nconv,T,ncv,U,eps->eigr,eps->eigi);CHKERRQ(ierr);
/* 6. AV(:,idx) = AV * U(:,idx) */
ierr = SlepcUpdateVectors(nv,ctx->AV,eps->nconv,nv,U,nv,PETSC_FALSE);CHKERRQ(ierr);
/* 7. V(:,idx) = V * U(:,idx) */
ierr = SlepcUpdateVectors(nv,eps->V,eps->nconv,nv,U,nv,PETSC_FALSE);CHKERRQ(ierr);
/* Convergence check */
ierr = EPSSchurResidualNorms(eps,eps->V,ctx->AV,T,eps->nconv,nv,ncv,rsd);CHKERRQ(ierr);
for (i=eps->nconv;i<nv;i++) {
itrsdold[i] = itrsd[i];
itrsd[i] = its;
eps->errest[i] = rsd[i];
}
for (;;) {
/* Find group in currently computed eigenvalues */
ierr = EPSFindGroup(eps->nconv,nv,eps->eigr,eps->eigi,eps->errest,grptol,&ngrp,&ctr,&ae,&arsd);CHKERRQ(ierr);
if (ngrp!=nogrp) break;
if (ngrp==0) break;
if (PetscAbsScalar(ae-oae)>ctr*cnvtol*(itrsd[eps->nconv]-itrsdold[eps->nconv])) break;
if (arsd>ctr*eps->tol) break;
eps->nconv = eps->nconv + ngrp;
if (eps->nconv>=nv) break;
}
ierr = EPSMonitor(eps,eps->its,eps->nconv,eps->eigr,eps->eigi,eps->errest,nv);CHKERRQ(ierr);
if (eps->nconv>=eps->nev) break;
/* Compute nxtsrr (iteration of next projection step) */
nxtsrr = PetscMin(eps->max_it,PetscMax((PetscInt)floor(stpfac*its),init));
if (ngrp!=nogrp || ngrp==0 || arsd>=oarsd) {
idsrr = nxtsrr - its;
} else {
idsrr = (PetscInt)floor(alpha+beta*(itrsdold[eps->nconv]-itrsd[eps->nconv])*log(arsd/eps->tol)/log(arsd/oarsd));
idsrr = PetscMax(1,idsrr);
}
nxtsrr = PetscMin(nxtsrr,its+idsrr);
/* Compute nxtort (iteration of next orthogonalization step) */
ierr = PetscMemcpy(U,T,sizeof(PetscScalar)*ncv*ncv);CHKERRQ(ierr);
ierr = EPSHessCond(nv,U,ncv,&tcond);CHKERRQ(ierr);
idort = PetscMax(1,(PetscInt)floor(orttol/PetscMax(1,log10(tcond))));
nxtort = PetscMin(its+idort,nxtsrr);
/* V(:,idx) = AV(:,idx) */
for (i=eps->nconv;i<nv;i++) {
ierr = VecCopy(ctx->AV[i],eps->V[i]);CHKERRQ(ierr);
}
its++;
/* Orthogonalization loop */
do {
while (its<nxtort) {
/* AV(:,idx) = OP * V(:,idx) */
for (i=eps->nconv;i<nv;i++) {
ierr = STApply(eps->OP,eps->V[i],ctx->AV[i]);CHKERRQ(ierr);
}
/* V(:,idx) = AV(:,idx) with normalization */
for (i=eps->nconv;i<nv;i++) {
ierr = VecCopy(ctx->AV[i],eps->V[i]);CHKERRQ(ierr);
ierr = VecNorm(eps->V[i],NORM_INFINITY,&norm);CHKERRQ(ierr);
ierr = VecScale(eps->V[i],1/norm);CHKERRQ(ierr);
}
its++;
}
/* Orthonormalize vectors */
for (i=eps->nconv;i<nv;i++) {
ierr = IPOrthogonalize(eps->ip,eps->nds,eps->DS,i,PETSC_NULL,eps->V,eps->V[i],PETSC_NULL,&norm,&breakdown);CHKERRQ(ierr);
if (breakdown) {
ierr = SlepcVecSetRandom(eps->V[i],eps->rand);CHKERRQ(ierr);
ierr = IPOrthogonalize(eps->ip,eps->nds,eps->DS,i,PETSC_NULL,eps->V,eps->V[i],PETSC_NULL,&norm,&breakdown);CHKERRQ(ierr);
}
ierr = VecScale(eps->V[i],1/norm);CHKERRQ(ierr);
}
nxtort = PetscMin(its+idort,nxtsrr);
} while (its<nxtsrr);
}
ierr = PetscFree(U);CHKERRQ(ierr);
ierr = PetscFree(rsd);CHKERRQ(ierr);
ierr = PetscFree(itrsd);CHKERRQ(ierr);
ierr = PetscFree(itrsdold);CHKERRQ(ierr);
if (eps->nconv == eps->nev) eps->reason = EPS_CONVERGED_TOL;
else eps->reason = EPS_DIVERGED_ITS;
PetscFunctionReturn(0);
}
#undef __FUNCT__
#define __FUNCT__ "EPSReset_Subspace"
PetscErrorCode EPSReset_Subspace(EPS eps)
{
PetscErrorCode ierr;
EPS_SUBSPACE *ctx = (EPS_SUBSPACE *)eps->data;
PetscFunctionBegin;
ierr = VecDestroyVecs(eps->ncv,&ctx->AV);CHKERRQ(ierr);
ierr = PetscFree(eps->T);CHKERRQ(ierr);
ierr = EPSReset_Default(eps);CHKERRQ(ierr);
PetscFunctionReturn(0);
}
#undef __FUNCT__
#define __FUNCT__ "EPSDestroy_Subspace"
PetscErrorCode EPSDestroy_Subspace(EPS eps)
{
PetscErrorCode ierr;
PetscFunctionBegin;
ierr = PetscFree(eps->data);CHKERRQ(ierr);
PetscFunctionReturn(0);
}
EXTERN_C_BEGIN
#undef __FUNCT__
#define __FUNCT__ "EPSCreate_Subspace"
PetscErrorCode EPSCreate_Subspace(EPS eps)
{
PetscErrorCode ierr;
PetscFunctionBegin;
ierr = PetscNewLog(eps,EPS_SUBSPACE,&eps->data);CHKERRQ(ierr);
eps->ops->setup = EPSSetUp_Subspace;
eps->ops->destroy = EPSDestroy_Subspace;
eps->ops->reset = EPSReset_Subspace;
eps->ops->backtransform = EPSBackTransform_Default;
eps->ops->computevectors = EPSComputeVectors_Schur;
PetscFunctionReturn(0);
}
EXTERN_C_END