/*
BVOrthogonalizeCGS1 - Compute |v'| (estimated), |v| and one step of CGS with
only one global synchronization
*/
PetscErrorCode BVOrthogonalizeCGS1(BV bv,PetscInt j,Vec v,PetscScalar *H,PetscReal *onorm,PetscReal *norm)
{
PetscErrorCode ierr;
PetscInt i;
PetscReal sum,nrm,beta;
Vec w=v;
PetscFunctionBegin;
/* h = W^* v ; alpha = (v, v) */
bv->k = j;
if (onorm || norm) {
if (!v) {
bv->k++;
ierr = BVGetColumn(bv,j,&w);CHKERRQ(ierr);
}
ierr = BVDotVec(bv,w,H);CHKERRQ(ierr);
if (!v) {
ierr = BVRestoreColumn(bv,j,&w);CHKERRQ(ierr);
bv->k--;
beta = PetscSqrtReal(PetscRealPart(H[bv->nc+j]));
} else {
ierr = BVNormVec(bv,w,NORM_2,&beta);CHKERRQ(ierr);
}
} else {
if (!v) { ierr = BVDotColumn(bv,j,H);CHKERRQ(ierr); }
else { ierr = BVDotVec(bv,w,H);CHKERRQ(ierr); }
}
/* q = v - V h */
if (bv->indef) {
for (i=0;i<bv->nc+j;i++) H[i] /= bv->omega[i]; /* apply inverse of signature */
}
if (!v) { ierr = BVMultColumn(bv,-1.0,1.0,j,H);CHKERRQ(ierr); }
else { ierr = BVMultVec(bv,-1.0,1.0,w,H);CHKERRQ(ierr); }
if (bv->indef) {
for (i=0;i<bv->nc+j;i++) H[i] *= bv->omega[i]; /* revert signature */
}
/* compute |v| */
if (onorm) *onorm = beta;
if (bv->indef) {
if (!v) { ierr = BVNormColumn(bv,j,NORM_2,&nrm);CHKERRQ(ierr); }
else { ierr = BVNormVec(bv,w,NORM_2,&nrm);CHKERRQ(ierr); }
if (norm) *norm = nrm;
bv->omega[bv->nc+j] = (nrm<0.0)? -1.0: 1.0;
} else if (norm) {
/* estimate |v'| from |v| */
sum = 0.0;
for (i=0;i<bv->nc+j;i++) sum += PetscRealPart(H[i]*PetscConj(H[i]));
*norm = beta*beta-sum;
if (*norm <= 0.0) {
if (!v) { ierr = BVNormColumn(bv,j,NORM_2,norm);CHKERRQ(ierr); }
else { ierr = BVNormVec(bv,w,NORM_2,norm);CHKERRQ(ierr); }
} else *norm = PetscSqrtReal(*norm);
}
PetscFunctionReturn(0);
}
/*
EPSSelectiveLanczos - Selective reorthogonalization.
*/
static PetscErrorCode EPSSelectiveLanczos(EPS eps,PetscReal *alpha,PetscReal *beta,PetscInt k,PetscInt *M,PetscBool *breakdown,PetscReal anorm)
{
PetscErrorCode ierr;
EPS_LANCZOS *lanczos = (EPS_LANCZOS*)eps->data;
PetscInt i,j,m = *M,n,nritz=0,nritzo;
Vec vj,vj1,av;
PetscReal *d,*e,*ritz,norm;
PetscScalar *Y,*hwork;
PetscBool *which;
PetscFunctionBegin;
ierr = PetscCalloc6(m+1,&d,m,&e,m,&ritz,m*m,&Y,m,&which,m,&hwork);CHKERRQ(ierr);
for (i=0;i<k;i++) which[i] = PETSC_TRUE;
for (j=k;j<m;j++) {
ierr = BVSetActiveColumns(eps->V,0,m);CHKERRQ(ierr);
/* Lanczos step */
ierr = BVGetColumn(eps->V,j,&vj);CHKERRQ(ierr);
ierr = BVGetColumn(eps->V,j+1,&vj1);CHKERRQ(ierr);
ierr = STApply(eps->st,vj,vj1);CHKERRQ(ierr);
ierr = BVRestoreColumn(eps->V,j,&vj);CHKERRQ(ierr);
ierr = BVRestoreColumn(eps->V,j+1,&vj1);CHKERRQ(ierr);
which[j] = PETSC_TRUE;
if (j-2>=k) which[j-2] = PETSC_FALSE;
ierr = BVOrthogonalizeSomeColumn(eps->V,j+1,which,hwork,&norm,breakdown);CHKERRQ(ierr);
alpha[j] = PetscRealPart(hwork[j]);
beta[j] = norm;
if (*breakdown) {
*M = j+1;
break;
}
/* Compute eigenvalues and eigenvectors Y of the tridiagonal block */
n = j-k+1;
for (i=0;i<n;i++) {
d[i] = alpha[i+k];
e[i] = beta[i+k];
}
ierr = DenseTridiagonal(n,d,e,ritz,Y);CHKERRQ(ierr);
/* Estimate ||A|| */
for (i=0;i<n;i++)
if (PetscAbsReal(ritz[i]) > anorm) anorm = PetscAbsReal(ritz[i]);
/* Compute nearly converged Ritz vectors */
nritzo = 0;
for (i=0;i<n;i++) {
if (norm*PetscAbsScalar(Y[i*n+n-1]) < PETSC_SQRT_MACHINE_EPSILON*anorm) nritzo++;
}
if (nritzo>nritz) {
nritz = 0;
for (i=0;i<n;i++) {
if (norm*PetscAbsScalar(Y[i*n+n-1]) < PETSC_SQRT_MACHINE_EPSILON*anorm) {
ierr = BVSetActiveColumns(eps->V,k,k+n);CHKERRQ(ierr);
ierr = BVGetColumn(lanczos->AV,nritz,&av);CHKERRQ(ierr);
ierr = BVMultVec(eps->V,1.0,0.0,av,Y+i*n);CHKERRQ(ierr);
ierr = BVRestoreColumn(lanczos->AV,nritz,&av);CHKERRQ(ierr);
nritz++;
}
}
}
if (nritz > 0) {
ierr = BVGetColumn(eps->V,j+1,&vj1);CHKERRQ(ierr);
ierr = BVSetActiveColumns(lanczos->AV,0,nritz);CHKERRQ(ierr);
ierr = BVOrthogonalizeVec(lanczos->AV,vj1,hwork,&norm,breakdown);CHKERRQ(ierr);
ierr = BVRestoreColumn(eps->V,j+1,&vj1);CHKERRQ(ierr);
if (*breakdown) {
*M = j+1;
break;
}
}
ierr = BVScaleColumn(eps->V,j+1,1.0/norm);CHKERRQ(ierr);
}
ierr = PetscFree6(d,e,ritz,Y,which,hwork);CHKERRQ(ierr);
PetscFunctionReturn(0);
}
int main(int argc,char **argv)
{
PetscErrorCode ierr;
Vec t,v;
Mat Q,M;
BV X,Y;
PetscInt i,j,n=10,kx=6,lx=3,ky=5,ly=2;
PetscScalar *q,*z;
PetscReal nrm;
PetscViewer view;
PetscBool verbose,trans;
SlepcInitialize(&argc,&argv,(char*)0,help);
ierr = PetscOptionsGetInt(NULL,"-n",&n,NULL);CHKERRQ(ierr);
ierr = PetscOptionsGetInt(NULL,"-kx",&kx,NULL);CHKERRQ(ierr);
ierr = PetscOptionsGetInt(NULL,"-lx",&lx,NULL);CHKERRQ(ierr);
ierr = PetscOptionsGetInt(NULL,"-ky",&ky,NULL);CHKERRQ(ierr);
ierr = PetscOptionsGetInt(NULL,"-ly",&ly,NULL);CHKERRQ(ierr);
ierr = PetscOptionsHasName(NULL,"-verbose",&verbose);CHKERRQ(ierr);
ierr = PetscPrintf(PETSC_COMM_WORLD,"First BV with %D active columns (%D leading columns) of dimension %D.\n",kx,lx,n);CHKERRQ(ierr);
ierr = PetscPrintf(PETSC_COMM_WORLD,"Second BV with %D active columns (%D leading columns) of dimension %D.\n",ky,ly,n);CHKERRQ(ierr);
/* Create template vector */
ierr = VecCreate(PETSC_COMM_WORLD,&t);CHKERRQ(ierr);
ierr = VecSetSizes(t,PETSC_DECIDE,n);CHKERRQ(ierr);
ierr = VecSetFromOptions(t);CHKERRQ(ierr);
/* Create BV object X */
ierr = BVCreate(PETSC_COMM_WORLD,&X);CHKERRQ(ierr);
ierr = PetscObjectSetName((PetscObject)X,"X");CHKERRQ(ierr);
ierr = BVSetSizesFromVec(X,t,kx+2);CHKERRQ(ierr); /* two extra columns to test active columns */
ierr = BVSetFromOptions(X);CHKERRQ(ierr);
ierr = BVSetActiveColumns(X,lx,kx);CHKERRQ(ierr);
/* Set up viewer */
ierr = PetscViewerASCIIGetStdout(PETSC_COMM_WORLD,&view);CHKERRQ(ierr);
if (verbose) {
ierr = PetscViewerPushFormat(view,PETSC_VIEWER_ASCII_MATLAB);CHKERRQ(ierr);
}
/* Fill X entries */
for (j=0;j<kx+2;j++) {
ierr = BVGetColumn(X,j,&v);CHKERRQ(ierr);
ierr = VecZeroEntries(v);CHKERRQ(ierr);
for (i=0;i<4;i++) {
if (i+j<n) {
ierr = VecSetValue(v,i+j,(PetscScalar)(3*i+j-2),INSERT_VALUES);CHKERRQ(ierr);
}
}
ierr = VecAssemblyBegin(v);CHKERRQ(ierr);
ierr = VecAssemblyEnd(v);CHKERRQ(ierr);
ierr = BVRestoreColumn(X,j,&v);CHKERRQ(ierr);
}
if (verbose) {
ierr = BVView(X,view);CHKERRQ(ierr);
}
/* Create BV object Y */
ierr = BVCreate(PETSC_COMM_WORLD,&Y);CHKERRQ(ierr);
ierr = PetscObjectSetName((PetscObject)Y,"Y");CHKERRQ(ierr);
ierr = BVSetSizesFromVec(Y,t,ky+1);CHKERRQ(ierr);
ierr = BVSetFromOptions(Y);CHKERRQ(ierr);
ierr = BVSetActiveColumns(Y,ly,ky);CHKERRQ(ierr);
/* Fill Y entries */
for (j=0;j<ky+1;j++) {
ierr = BVGetColumn(Y,j,&v);CHKERRQ(ierr);
ierr = VecSet(v,(PetscScalar)(j+1)/4.0);CHKERRQ(ierr);
ierr = BVRestoreColumn(Y,j,&v);CHKERRQ(ierr);
}
if (verbose) {
ierr = BVView(Y,view);CHKERRQ(ierr);
}
/* Create Mat */
ierr = MatCreateSeqDense(PETSC_COMM_SELF,kx,ky,NULL,&Q);CHKERRQ(ierr);
ierr = PetscObjectSetName((PetscObject)Q,"Q");CHKERRQ(ierr);
ierr = MatDenseGetArray(Q,&q);CHKERRQ(ierr);
for (i=0;i<kx;i++)
for (j=0;j<ky;j++)
q[i+j*kx] = (i<j)? 2.0: -0.5;
ierr = MatDenseRestoreArray(Q,&q);CHKERRQ(ierr);
if (verbose) {
ierr = MatView(Q,NULL);CHKERRQ(ierr);
}
/* Test BVMult */
ierr = BVMult(Y,2.0,1.0,X,Q);CHKERRQ(ierr);
if (verbose) {
ierr = PetscPrintf(PETSC_COMM_WORLD,"After BVMult - - - - - - - - -\n");CHKERRQ(ierr);
ierr = BVView(Y,view);CHKERRQ(ierr);
}
/* Test BVMultVec */
ierr = BVGetColumn(Y,0,&v);CHKERRQ(ierr);
ierr = PetscMalloc1(kx-lx,&z);CHKERRQ(ierr);
z[0] = 2.0;
for (i=1;i<kx-lx;i++) z[i] = -0.5*z[i-1];
ierr = BVMultVec(X,-1.0,1.0,v,z);CHKERRQ(ierr);
ierr = PetscFree(z);CHKERRQ(ierr);
ierr = BVRestoreColumn(Y,0,&v);CHKERRQ(ierr);
if (verbose) {
ierr = PetscPrintf(PETSC_COMM_WORLD,"After BVMultVec - - - - - - -\n");CHKERRQ(ierr);
ierr = BVView(Y,view);CHKERRQ(ierr);
}
/* Test BVDot */
ierr = MatCreateSeqDense(PETSC_COMM_SELF,ky,kx,NULL,&M);CHKERRQ(ierr);
ierr = PetscObjectSetName((PetscObject)M,"M");CHKERRQ(ierr);
ierr = BVDot(X,Y,M);CHKERRQ(ierr);
if (verbose) {
ierr = PetscPrintf(PETSC_COMM_WORLD,"After BVDot - - - - - - - - -\n");CHKERRQ(ierr);
ierr = MatView(M,NULL);CHKERRQ(ierr);
}
/* Test BVDotVec */
ierr = BVGetColumn(Y,0,&v);CHKERRQ(ierr);
ierr = PetscMalloc1(kx-lx,&z);CHKERRQ(ierr);
ierr = BVDotVec(X,v,z);CHKERRQ(ierr);
ierr = BVRestoreColumn(Y,0,&v);CHKERRQ(ierr);
if (verbose) {
ierr = PetscPrintf(PETSC_COMM_WORLD,"After BVDotVec - - - - - - -\n");CHKERRQ(ierr);
ierr = VecCreateSeqWithArray(PETSC_COMM_SELF,1,kx-lx,z,&v);CHKERRQ(ierr);
ierr = PetscObjectSetName((PetscObject)v,"z");CHKERRQ(ierr);
ierr = VecView(v,view);CHKERRQ(ierr);
ierr = VecDestroy(&v);CHKERRQ(ierr);
}
ierr = PetscFree(z);CHKERRQ(ierr);
/* Test BVMultInPlace and BVScale */
ierr = PetscOptionsHasName(NULL,"-trans",&trans);CHKERRQ(ierr);
if (trans) {
Mat Qt;
ierr = MatTranspose(Q,MAT_INITIAL_MATRIX,&Qt);CHKERRQ(ierr);
ierr = BVMultInPlaceTranspose(X,Qt,lx+1,ky);CHKERRQ(ierr);
ierr = MatDestroy(&Qt);CHKERRQ(ierr);
} else {
ierr = BVMultInPlace(X,Q,lx+1,ky);CHKERRQ(ierr);
}
ierr = BVScale(X,2.0);CHKERRQ(ierr);
if (verbose) {
ierr = PetscPrintf(PETSC_COMM_WORLD,"After BVMultInPlace - - - - -\n");CHKERRQ(ierr);
ierr = BVView(X,view);CHKERRQ(ierr);
}
/* Test BVNorm */
ierr = BVNormColumn(X,lx,NORM_2,&nrm);CHKERRQ(ierr);
ierr = PetscPrintf(PETSC_COMM_WORLD,"2-Norm or X[%D] = %g\n",lx,(double)nrm);CHKERRQ(ierr);
ierr = BVNorm(X,NORM_FROBENIUS,&nrm);CHKERRQ(ierr);
ierr = PetscPrintf(PETSC_COMM_WORLD,"Frobenius Norm or X = %g\n",(double)nrm);CHKERRQ(ierr);
ierr = BVDestroy(&X);CHKERRQ(ierr);
ierr = BVDestroy(&Y);CHKERRQ(ierr);
ierr = MatDestroy(&Q);CHKERRQ(ierr);
ierr = MatDestroy(&M);CHKERRQ(ierr);
ierr = VecDestroy(&t);CHKERRQ(ierr);
ierr = SlepcFinalize();
return 0;
}
PetscErrorCode NEPSolve_NArnoldi(NEP nep)
{
PetscErrorCode ierr;
Mat T=nep->function,Tsigma;
Vec f,r=nep->work[0],x=nep->work[1],w=nep->work[2];
PetscScalar *X,lambda;
PetscReal beta,resnorm=0.0,nrm;
PetscInt n;
PetscBool breakdown;
KSPConvergedReason kspreason;
PetscFunctionBegin;
/* get initial space and shift */
ierr = NEPGetDefaultShift(nep,&lambda);CHKERRQ(ierr);
if (!nep->nini) {
ierr = BVSetRandomColumn(nep->V,0,nep->rand);CHKERRQ(ierr);
ierr = BVNormColumn(nep->V,0,NORM_2,&nrm);CHKERRQ(ierr);
ierr = BVScaleColumn(nep->V,0,1.0/nrm);CHKERRQ(ierr);
n = 1;
} else n = nep->nini;
/* build projected matrices for initial space */
ierr = DSSetDimensions(nep->ds,n,0,0,0);CHKERRQ(ierr);
ierr = NEPProjectOperator(nep,0,n);CHKERRQ(ierr);
/* prepare linear solver */
ierr = NEPComputeFunction(nep,lambda,T,T);CHKERRQ(ierr);
ierr = MatDuplicate(T,MAT_COPY_VALUES,&Tsigma);CHKERRQ(ierr);
ierr = KSPSetOperators(nep->ksp,Tsigma,Tsigma);CHKERRQ(ierr);
/* Restart loop */
while (nep->reason == NEP_CONVERGED_ITERATING) {
nep->its++;
/* solve projected problem */
ierr = DSSetDimensions(nep->ds,n,0,0,0);CHKERRQ(ierr);
ierr = DSSetState(nep->ds,DS_STATE_RAW);CHKERRQ(ierr);
ierr = DSSolve(nep->ds,nep->eigr,NULL);CHKERRQ(ierr);
lambda = nep->eigr[0];
/* compute Ritz vector, x = V*s */
ierr = DSGetArray(nep->ds,DS_MAT_X,&X);CHKERRQ(ierr);
ierr = BVSetActiveColumns(nep->V,0,n);CHKERRQ(ierr);
ierr = BVMultVec(nep->V,1.0,0.0,x,X);CHKERRQ(ierr);
ierr = DSRestoreArray(nep->ds,DS_MAT_X,&X);CHKERRQ(ierr);
/* compute the residual, r = T(lambda)*x */
ierr = NEPApplyFunction(nep,lambda,x,w,r,NULL,NULL);CHKERRQ(ierr);
/* convergence test */
ierr = VecNorm(r,NORM_2,&resnorm);CHKERRQ(ierr);
nep->errest[nep->nconv] = resnorm;
if (resnorm<=nep->rtol) {
ierr = BVInsertVec(nep->V,nep->nconv,x);CHKERRQ(ierr);
nep->nconv = nep->nconv + 1;
nep->reason = NEP_CONVERGED_FNORM_RELATIVE;
}
ierr = NEPMonitor(nep,nep->its,nep->nconv,nep->eigr,nep->errest,1);CHKERRQ(ierr);
if (nep->reason == NEP_CONVERGED_ITERATING) {
/* continuation vector: f = T(sigma)\r */
ierr = BVGetColumn(nep->V,n,&f);CHKERRQ(ierr);
ierr = NEP_KSPSolve(nep,r,f);CHKERRQ(ierr);
ierr = BVRestoreColumn(nep->V,n,&f);CHKERRQ(ierr);
ierr = KSPGetConvergedReason(nep->ksp,&kspreason);CHKERRQ(ierr);
if (kspreason<0) {
ierr = PetscInfo1(nep,"iter=%D, linear solve failed, stopping solve\n",nep->its);CHKERRQ(ierr);
nep->reason = NEP_DIVERGED_LINEAR_SOLVE;
break;
}
/* orthonormalize */
ierr = BVOrthogonalizeColumn(nep->V,n,NULL,&beta,&breakdown);CHKERRQ(ierr);
if (breakdown || beta==0.0) {
ierr = PetscInfo1(nep,"iter=%D, orthogonalization failed, stopping solve\n",nep->its);CHKERRQ(ierr);
nep->reason = NEP_DIVERGED_BREAKDOWN;
break;
}
ierr = BVScaleColumn(nep->V,n,1.0/beta);CHKERRQ(ierr);
/* update projected matrices */
ierr = DSSetDimensions(nep->ds,n+1,0,0,0);CHKERRQ(ierr);
ierr = NEPProjectOperator(nep,n,n+1);CHKERRQ(ierr);
n++;
}
if (nep->its >= nep->max_it) nep->reason = NEP_DIVERGED_MAX_IT;
}
ierr = MatDestroy(&Tsigma);CHKERRQ(ierr);
PetscFunctionReturn(0);
}