Date: 2023-09-05T10:18:26+00:00

Actual source code: qslice.c
slepc-3.19.2 2023-09-05
Report Typos and Errors

      1: /*
      2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      4:    Copyright (c) 2002-, Universitat Politecnica de Valencia, Spain

      6:    This file is part of SLEPc.
      7:    SLEPc is distributed under a 2-clause BSD license (see LICENSE).
      8:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      9: */
     10: /*
     11:    SLEPc polynomial eigensolver: "stoar"

     13:    Method: S-TOAR with spectrum slicing for symmetric quadratic eigenproblems

     15:    Algorithm:

     17:        Symmetric Two-Level Orthogonal Arnoldi.

     19:    References:

     21:        [1] C. Campos and J.E. Roman, "Inertia-based spectrum slicing
     22:            for symmetric quadratic eigenvalue problems", Numer. Linear
     23:            Algebra Appl. 27(4):e2293, 2020.
     24: */

     26: #include <slepc/private/pepimpl.h>
     27: #include "../src/pep/impls/krylov/pepkrylov.h"
     28: #include <slepcblaslapack.h>

     30: static PetscBool  cited = PETSC_FALSE;
     31: static const char citation[] =
     32:   "@Article{slepc-slice-qep,\n"
     33:   "   author = \"C. Campos and J. E. Roman\",\n"
     34:   "   title = \"Inertia-based spectrum slicing for symmetric quadratic eigenvalue problems\",\n"
     35:   "   journal = \"Numer. Linear Algebra Appl.\",\n"
     36:   "   volume = \"27\",\n"
     37:   "   number = \"4\",\n"
     38:   "   pages = \"e2293\",\n"
     39:   "   year = \"2020,\"\n"
     40:   "   doi = \"https://doi.org/10.1002/nla.2293\"\n"
     41:   "}\n";

     43: #define SLICE_PTOL PETSC_SQRT_MACHINE_EPSILON

     45: static PetscErrorCode PEPQSliceResetSR(PEP pep)
     46: {
     47:   PEP_STOAR      *ctx=(PEP_STOAR*)pep->data;
     48:   PEP_SR         sr=ctx->sr;
     49:   PEP_shift      s;
     50:   PetscInt       i;

     52:   PetscFunctionBegin;
     53:   if (sr) {
     54:     /* Reviewing list of shifts to free memory */
     55:     s = sr->s0;
     56:     if (s) {
     57:       while (s->neighb[1]) {
     58:         s = s->neighb[1];
     59:         PetscCall(PetscFree(s->neighb[0]));
     60:       }
     61:       PetscCall(PetscFree(s));
     62:     }
     63:     PetscCall(PetscFree(sr->S));
     64:     for (i=0;i<pep->nconv;i++) PetscCall(PetscFree(sr->qinfo[i].q));
     65:     PetscCall(PetscFree(sr->qinfo));
     66:     for (i=0;i<3;i++) PetscCall(VecDestroy(&sr->v[i]));
     67:     PetscCall(EPSDestroy(&sr->eps));
     68:     PetscCall(PetscFree(sr));
     69:   }
     70:   ctx->sr = NULL;
     71:   PetscFunctionReturn(PETSC_SUCCESS);
     72: }

     74: PetscErrorCode PEPReset_STOAR_QSlice(PEP pep)
     75: {
     76:   PEP_STOAR      *ctx=(PEP_STOAR*)pep->data;

     78:   PetscFunctionBegin;
     79:   PetscCall(PEPQSliceResetSR(pep));
     80:   PetscCall(PetscFree(ctx->inertias));
     81:   PetscCall(PetscFree(ctx->shifts));
     82:   PetscFunctionReturn(PETSC_SUCCESS);
     83: }

     85: /*
     86:   PEPQSliceAllocateSolution - Allocate memory storage for common variables such
     87:   as eigenvalues and eigenvectors.
     88: */
     89: static PetscErrorCode PEPQSliceAllocateSolution(PEP pep)
     90: {
     91:   PEP_STOAR      *ctx=(PEP_STOAR*)pep->data;
     92:   PetscInt       k;
     93:   BVType         type;
     94:   Vec            t;
     95:   PEP_SR         sr = ctx->sr;

     97:   PetscFunctionBegin;
     98:   /* allocate space for eigenvalues and friends */
     99:   k = PetscMax(1,sr->numEigs);
    100:   PetscCall(PetscFree4(sr->eigr,sr->eigi,sr->errest,sr->perm));
    101:   PetscCall(PetscCalloc4(k,&sr->eigr,k,&sr->eigi,k,&sr->errest,k,&sr->perm));
    102:   PetscCall(PetscFree(sr->qinfo));
    103:   PetscCall(PetscCalloc1(k,&sr->qinfo));

    105:   /* allocate sr->V and transfer options from pep->V */
    106:   PetscCall(BVDestroy(&sr->V));
    107:   PetscCall(BVCreate(PetscObjectComm((PetscObject)pep),&sr->V));
    108:   if (!pep->V) PetscCall(PEPGetBV(pep,&pep->V));
    109:   if (!((PetscObject)(pep->V))->type_name) PetscCall(BVSetType(sr->V,BVSVEC));
    110:   else {
    111:     PetscCall(BVGetType(pep->V,&type));
    112:     PetscCall(BVSetType(sr->V,type));
    113:   }
    114:   PetscCall(STMatCreateVecsEmpty(pep->st,&t,NULL));
    115:   PetscCall(BVSetSizesFromVec(sr->V,t,k+1));
    116:   PetscCall(VecDestroy(&t));
    117:   sr->ld = k;
    118:   PetscCall(PetscFree(sr->S));
    119:   PetscCall(PetscMalloc1((k+1)*sr->ld*(pep->nmat-1),&sr->S));
    120:   PetscFunctionReturn(PETSC_SUCCESS);
    121: }

    123: /* Convergence test to compute positive Ritz values */
    124: static PetscErrorCode ConvergedPositive(EPS eps,PetscScalar eigr,PetscScalar eigi,PetscReal res,PetscReal *errest,void *ctx)
    125: {
    126:   PetscFunctionBegin;
    127:   *errest = (PetscRealPart(eigr)>0.0)?0.0:res;
    128:   PetscFunctionReturn(PETSC_SUCCESS);
    129: }

    131: static PetscErrorCode PEPQSliceMatGetInertia(PEP pep,PetscReal shift,PetscInt *inertia,PetscInt *zeros)
    132: {
    133:   KSP            ksp,kspr;
    134:   PC             pc;
    135:   Mat            F;
    136:   PetscBool      flg;

    138:   PetscFunctionBegin;
    139:   if (!pep->solvematcoeffs) PetscCall(PetscMalloc1(pep->nmat,&pep->solvematcoeffs));
    140:   if (shift==PETSC_MAX_REAL) { /* Inertia of matrix A[2] */
    141:     pep->solvematcoeffs[0] = 0.0; pep->solvematcoeffs[1] = 0.0; pep->solvematcoeffs[2] = 1.0;
    142:   } else PetscCall(PEPEvaluateBasis(pep,shift,0,pep->solvematcoeffs,NULL));
    143:   PetscCall(STMatSetUp(pep->st,pep->sfactor,pep->solvematcoeffs));
    144:   PetscCall(STGetKSP(pep->st,&ksp));
    145:   PetscCall(KSPGetPC(ksp,&pc));
    146:   PetscCall(PetscObjectTypeCompare((PetscObject)pc,PCREDUNDANT,&flg));
    147:   if (flg) {
    148:     PetscCall(PCRedundantGetKSP(pc,&kspr));
    149:     PetscCall(KSPGetPC(kspr,&pc));
    150:   }
    151:   PetscCall(PCFactorGetMatrix(pc,&F));
    152:   PetscCall(MatGetInertia(F,inertia,zeros,NULL));
    153:   PetscFunctionReturn(PETSC_SUCCESS);
    154: }

    156: static PetscErrorCode PEPQSliceGetInertia(PEP pep,PetscReal shift,PetscInt *inertia,PetscInt *zeros,PetscInt correction)
    157: {
    158:   KSP            ksp;
    159:   Mat            P;
    160:   PetscReal      nzshift=0.0,dot;
    161:   PetscRandom    rand;
    162:   PetscInt       nconv;
    163:   PEP_STOAR      *ctx=(PEP_STOAR*)pep->data;
    164:   PEP_SR         sr=ctx->sr;

    166:   PetscFunctionBegin;
    167:   if (shift >= PETSC_MAX_REAL) { /* Right-open interval */
    168:     *inertia = 0;
    169:   } else if (shift <= PETSC_MIN_REAL) {
    170:     *inertia = 0;
    171:     if (zeros) *zeros = 0;
    172:   } else {
    173:     /* If the shift is zero, perturb it to a very small positive value.
    174:        The goal is that the nonzero pattern is the same in all cases and reuse
    175:        the symbolic factorizations */
    176:     nzshift = (shift==0.0)? 10.0/PETSC_MAX_REAL: shift;
    177:     PetscCall(PEPQSliceMatGetInertia(pep,nzshift,inertia,zeros));
    178:     PetscCall(STSetShift(pep->st,nzshift));
    179:   }
    180:   if (!correction) {
    181:     if (shift >= PETSC_MAX_REAL) *inertia = 2*pep->n;
    182:     else if (shift>PETSC_MIN_REAL) {
    183:       PetscCall(STGetKSP(pep->st,&ksp));
    184:       PetscCall(KSPGetOperators(ksp,&P,NULL));
    185:       if (*inertia!=pep->n && !sr->v[0]) {
    186:         PetscCall(MatCreateVecs(P,&sr->v[0],NULL));
    187:         PetscCall(VecDuplicate(sr->v[0],&sr->v[1]));
    188:         PetscCall(VecDuplicate(sr->v[0],&sr->v[2]));
    189:         PetscCall(BVGetRandomContext(pep->V,&rand));
    190:         PetscCall(VecSetRandom(sr->v[0],rand));
    191:       }
    192:       if (*inertia<pep->n && *inertia>0) {
    193:         if (!sr->eps) {
    194:           PetscCall(EPSCreate(PetscObjectComm((PetscObject)pep),&sr->eps));
    195:           PetscCall(EPSSetProblemType(sr->eps,EPS_HEP));
    196:           PetscCall(EPSSetWhichEigenpairs(sr->eps,EPS_LARGEST_REAL));
    197:         }
    198:         PetscCall(EPSSetConvergenceTestFunction(sr->eps,ConvergedPositive,NULL,NULL));
    199:         PetscCall(EPSSetOperators(sr->eps,P,NULL));
    200:         PetscCall(EPSSolve(sr->eps));
    201:         PetscCall(EPSGetConverged(sr->eps,&nconv));
    202:         PetscCheck(nconv,((PetscObject)pep)->comm,PETSC_ERR_CONV_FAILED,"Inertia computation fails in %g",(double)nzshift);
    203:         PetscCall(EPSGetEigenpair(sr->eps,0,NULL,NULL,sr->v[0],sr->v[1]));
    204:       }
    205:       if (*inertia!=pep->n) {
    206:         PetscCall(MatMult(pep->A[1],sr->v[0],sr->v[1]));
    207:         PetscCall(MatMult(pep->A[2],sr->v[0],sr->v[2]));
    208:         PetscCall(VecAXPY(sr->v[1],2*nzshift,sr->v[2]));
    209:         PetscCall(VecDotRealPart(sr->v[1],sr->v[0],&dot));
    210:         if (dot>0.0) *inertia = 2*pep->n-*inertia;
    211:       }
    212:     }
    213:   } else if (correction<0) *inertia = 2*pep->n-*inertia;
    214:   PetscFunctionReturn(PETSC_SUCCESS);
    215: }

    217: /*
    218:    Check eigenvalue type - used only in non-hyperbolic problems.
    219:    All computed eigenvalues must have the same definite type (positive or negative).
    220:    If ini=TRUE the type is available in omega, otherwise we compute an eigenvalue
    221:    closest to shift and determine its type.
    222:  */
    223: static PetscErrorCode PEPQSliceCheckEigenvalueType(PEP pep,PetscReal shift,PetscReal omega,PetscBool ini)
    224: {
    225:   PEP            pep2;
    226:   ST             st;
    227:   PetscInt       nconv;
    228:   PetscScalar    lambda;
    229:   PetscReal      dot;
    230:   PEP_STOAR      *ctx=(PEP_STOAR*)pep->data;
    231:   PEP_SR         sr=ctx->sr;

    233:   PetscFunctionBegin;
    234:   if (!ini) {
    235:     PetscCheck(-(omega/(shift*ctx->alpha+ctx->beta))*sr->type>=0,((PetscObject)pep)->comm,PETSC_ERR_CONV_FAILED,"Different positive/negative type detected in eigenvalue %g",(double)shift);
    236:   } else {
    237:     PetscCall(PEPCreate(PetscObjectComm((PetscObject)pep),&pep2));
    238:     PetscCall(PEPSetOptionsPrefix(pep2,((PetscObject)pep)->prefix));
    239:     PetscCall(PEPAppendOptionsPrefix(pep2,"pep_eigenvalue_type_"));
    240:     PetscCall(PEPSetTolerances(pep2,PETSC_DEFAULT,pep->max_it/4));
    241:     PetscCall(PEPSetType(pep2,PEPTOAR));
    242:     PetscCall(PEPSetOperators(pep2,pep->nmat,pep->A));
    243:     PetscCall(PEPSetWhichEigenpairs(pep2,PEP_TARGET_MAGNITUDE));
    244:     PetscCall(PEPGetRG(pep2,&pep2->rg));
    245:     PetscCall(RGSetType(pep2->rg,RGINTERVAL));
    246: #if defined(PETSC_USE_COMPLEX)
    247:     PetscCall(RGIntervalSetEndpoints(pep2->rg,pep->inta,pep->intb,-PETSC_SQRT_MACHINE_EPSILON,PETSC_SQRT_MACHINE_EPSILON));
    248: #else
    249:     PetscCall(RGIntervalSetEndpoints(pep2->rg,pep->inta,pep->intb,0.0,0.0));
    250: #endif
    251:     pep2->target = shift;
    252:     st = pep2->st;
    253:     pep2->st = pep->st;
    254:     PetscCall(PEPSolve(pep2));
    255:     PetscCall(PEPGetConverged(pep2,&nconv));
    256:     if (nconv) {
    257:       PetscCall(PEPGetEigenpair(pep2,0,&lambda,NULL,pep2->work[0],NULL));
    258:       PetscCall(MatMult(pep->A[1],pep2->work[0],pep2->work[1]));
    259:       PetscCall(MatMult(pep->A[2],pep2->work[0],pep2->work[2]));
    260:       PetscCall(VecAXPY(pep2->work[1],2.0*lambda*pep->sfactor,pep2->work[2]));
    261:       PetscCall(VecDotRealPart(pep2->work[1],pep2->work[0],&dot));
    262:       PetscCall(PetscInfo(pep,"lambda=%g, %s type\n",(double)PetscRealPart(lambda),(dot>0.0)?"positive":"negative"));
    263:       if (!sr->type) sr->type = (dot>0.0)?1:-1;
    264:       else PetscCheck(sr->type*dot>=0.0,((PetscObject)pep)->comm,PETSC_ERR_CONV_FAILED,"Different positive/negative type detected in eigenvalue %g",(double)PetscRealPart(lambda));
    265:     }
    266:     pep2->st = st;
    267:     PetscCall(PEPDestroy(&pep2));
    268:   }
    269:   PetscFunctionReturn(PETSC_SUCCESS);
    270: }

    272: static inline PetscErrorCode PEPQSliceDiscriminant(PEP pep,Vec u,Vec w,PetscReal *d,PetscReal *smas,PetscReal *smenos)
    273: {
    274:   PetscReal      ap,bp,cp,dis;

    276:   PetscFunctionBegin;
    277:   PetscCall(MatMult(pep->A[0],u,w));
    278:   PetscCall(VecDotRealPart(w,u,&cp));
    279:   PetscCall(MatMult(pep->A[1],u,w));
    280:   PetscCall(VecDotRealPart(w,u,&bp));
    281:   PetscCall(MatMult(pep->A[2],u,w));
    282:   PetscCall(VecDotRealPart(w,u,&ap));
    283:   dis = bp*bp-4*ap*cp;
    284:   if (dis>=0.0 && smas) {
    285:     if (ap>0) *smas = (-bp+PetscSqrtReal(dis))/(2*ap);
    286:     else if (ap<0) *smas = (-bp-PetscSqrtReal(dis))/(2*ap);
    287:     else {
    288:       if (bp >0) *smas = -cp/bp;
    289:       else *smas = PETSC_MAX_REAL;
    290:     }
    291:   }
    292:   if (dis>=0.0 && smenos) {
    293:     if (ap>0) *smenos = (-bp-PetscSqrtReal(dis))/(2*ap);
    294:     else if (ap<0) *smenos = (-bp+PetscSqrtReal(dis))/(2*ap);
    295:     else {
    296:       if (bp<0) *smenos = -cp/bp;
    297:       else *smenos = PETSC_MAX_REAL;
    298:     }
    299:   }
    300:   if (d) *d = dis;
    301:   PetscFunctionReturn(PETSC_SUCCESS);
    302: }

    304: static inline PetscErrorCode PEPQSliceEvaluateQEP(PEP pep,PetscScalar x,Mat M,MatStructure str)
    305: {
    306:   PetscFunctionBegin;
    307:   PetscCall(MatCopy(pep->A[0],M,SAME_NONZERO_PATTERN));
    308:   PetscCall(MatAXPY(M,x,pep->A[1],str));
    309:   PetscCall(MatAXPY(M,x*x,pep->A[2],str));
    310:   PetscFunctionReturn(PETSC_SUCCESS);
    311: }

    313: /*@
    314:    PEPCheckDefiniteQEP - Determines if a symmetric/Hermitian quadratic eigenvalue problem
    315:    is definite or not.

    317:    Collective

    319:    Input Parameter:
    320: .  pep  - eigensolver context

    322:    Output Parameters:
    323: +  xi - first computed parameter
    324: .  mu - second computed parameter
    325: .  definite - flag indicating that the problem is definite
    326: -  hyperbolic - flag indicating that the problem is hyperbolic

    328:    Notes:
    329:    This function is intended for quadratic eigenvalue problems, Q(lambda)=A*lambda^2+B*lambda+C,
    330:    with symmetric (or Hermitian) coefficient matrices A,B,C.

    332:    On output, the flag 'definite' may have the values -1 (meaning that the QEP is not
    333:    definite), 1 (if the problem is definite), or 0 if the algorithm was not able to
    334:    determine whether the problem is definite or not.

    336:    If definite=1, the output flag 'hyperbolic' informs in a similar way about whether the
    337:    problem is hyperbolic or not.

    339:    If definite=1, the computed values xi and mu satisfy Q(xi)<0 and Q(mu)>0, as
    340:    obtained via the method proposed in [Niendorf and Voss, LAA 2010]. Furthermore, if
    341:    hyperbolic=1 then only xi is computed.

    343:    Level: advanced

    345: .seealso: PEPSetProblemType()
    346: @*/
    347: PetscErrorCode PEPCheckDefiniteQEP(PEP pep,PetscReal *xi,PetscReal *mu,PetscInt *definite,PetscInt *hyperbolic)
    348: {
    349:   PetscRandom    rand;
    350:   Vec            u,w;
    351:   PetscReal      d=0.0,s=0.0,sp,mut=0.0,omg=0.0,omgp;
    352:   PetscInt       k,its=10,hyp=0,check=0,nconv,inertia,n;
    353:   Mat            M=NULL;
    354:   MatStructure   str;
    355:   EPS            eps;
    356:   PetscBool      transform,ptypehyp;

    358:   PetscFunctionBegin;
    359:   PetscCheck(pep->problem_type==PEP_HERMITIAN || pep->problem_type==PEP_HYPERBOLIC,PetscObjectComm((PetscObject)pep),PETSC_ERR_SUP,"Only available for Hermitian (or hyperbolic) problems");
    360:   ptypehyp = (pep->problem_type==PEP_HYPERBOLIC)? PETSC_TRUE: PETSC_FALSE;
    361:   if (!pep->st) PetscCall(PEPGetST(pep,&pep->st));
    362:   PetscCall(PEPSetDefaultST(pep));
    363:   PetscCall(STSetMatrices(pep->st,pep->nmat,pep->A));
    364:   PetscCall(MatGetSize(pep->A[0],&n,NULL));
    365:   PetscCall(STGetTransform(pep->st,&transform));
    366:   PetscCall(STSetTransform(pep->st,PETSC_FALSE));
    367:   PetscCall(STSetUp(pep->st));
    368:   PetscCall(MatCreateVecs(pep->A[0],&u,&w));
    369:   PetscCall(PEPGetBV(pep,&pep->V));
    370:   PetscCall(BVGetRandomContext(pep->V,&rand));
    371:   PetscCall(VecSetRandom(u,rand));
    372:   PetscCall(VecNormalize(u,NULL));
    373:   PetscCall(PEPQSliceDiscriminant(pep,u,w,&d,&s,NULL));
    374:   if (d<0.0) check = -1;
    375:   if (!check) {
    376:     PetscCall(EPSCreate(PetscObjectComm((PetscObject)pep),&eps));
    377:     PetscCall(EPSSetProblemType(eps,EPS_HEP));
    378:     PetscCall(EPSSetWhichEigenpairs(eps,EPS_LARGEST_REAL));
    379:     PetscCall(EPSSetTolerances(eps,PetscSqrtReal(PETSC_SQRT_MACHINE_EPSILON),PETSC_DECIDE));
    380:     PetscCall(MatDuplicate(pep->A[0],MAT_DO_NOT_COPY_VALUES,&M));
    381:     PetscCall(STGetMatStructure(pep->st,&str));
    382:   }
    383:   for (k=0;k<its&&!check;k++) {
    384:     PetscCall(PEPQSliceEvaluateQEP(pep,s,M,str));
    385:     PetscCall(EPSSetOperators(eps,M,NULL));
    386:     PetscCall(EPSSolve(eps));
    387:     PetscCall(EPSGetConverged(eps,&nconv));
    388:     if (!nconv) break;
    389:     PetscCall(EPSGetEigenpair(eps,0,NULL,NULL,u,w));
    390:     sp = s;
    391:     PetscCall(PEPQSliceDiscriminant(pep,u,w,&d,&s,&omg));
    392:     if (d<0.0) {check = -1; break;}
    393:     if (PetscAbsReal((s-sp)/s)<100*PETSC_MACHINE_EPSILON) break;
    394:     if (s>sp) {hyp = -1;}
    395:     mut = 2*s-sp;
    396:     PetscCall(PEPQSliceMatGetInertia(pep,mut,&inertia,NULL));
    397:     if (inertia == n) {check = 1; break;}
    398:   }
    399:   for (;k<its&&!check;k++) {
    400:     mut = (s-omg)/2;
    401:     PetscCall(PEPQSliceMatGetInertia(pep,mut,&inertia,NULL));
    402:     if (inertia == n) {check = 1; break;}
    403:     if (PetscAbsReal((s-omg)/omg)<100*PETSC_MACHINE_EPSILON) break;
    404:     PetscCall(PEPQSliceEvaluateQEP(pep,omg,M,str));
    405:     PetscCall(EPSSetOperators(eps,M,NULL));
    406:     PetscCall(EPSSolve(eps));
    407:     PetscCall(EPSGetConverged(eps,&nconv));
    408:     if (!nconv) break;
    409:     PetscCall(EPSGetEigenpair(eps,0,NULL,NULL,u,w));
    410:     omgp = omg;
    411:     PetscCall(PEPQSliceDiscriminant(pep,u,w,&d,NULL,&omg));
    412:     if (d<0.0) {check = -1; break;}
    413:     if (omg<omgp) hyp = -1;
    414:   }
    415:   if (check==1) *xi = mut;
    416:   PetscCheck(hyp!=-1 || !ptypehyp,PetscObjectComm((PetscObject)pep),PETSC_ERR_PLIB,"Problem does not satisfy hyperbolic test; consider removing the hyperbolicity flag");
    417:   if (check==1 && hyp==0) {
    418:     PetscCall(PEPQSliceMatGetInertia(pep,PETSC_MAX_REAL,&inertia,NULL));
    419:     if (inertia == 0) hyp = 1;
    420:     else hyp = -1;
    421:   }
    422:   if (check==1 && hyp!=1) {
    423:     check = 0;
    424:     PetscCall(EPSSetWhichEigenpairs(eps,EPS_SMALLEST_REAL));
    425:     for (;k<its&&!check;k++) {
    426:       PetscCall(PEPQSliceEvaluateQEP(pep,s,M,str));
    427:       PetscCall(EPSSetOperators(eps,M,NULL));
    428:       PetscCall(EPSSolve(eps));
    429:       PetscCall(EPSGetConverged(eps,&nconv));
    430:       if (!nconv) break;
    431:       PetscCall(EPSGetEigenpair(eps,0,NULL,NULL,u,w));
    432:       sp = s;
    433:       PetscCall(PEPQSliceDiscriminant(pep,u,w,&d,&s,&omg));
    434:       if (d<0.0) {check = -1; break;}
    435:       if (PetscAbsReal((s-sp)/s)<100*PETSC_MACHINE_EPSILON) break;
    436:       mut = 2*s-sp;
    437:       PetscCall(PEPQSliceMatGetInertia(pep,mut,&inertia,NULL));
    438:       if (inertia == 0) {check = 1; break;}
    439:     }
    440:     for (;k<its&&!check;k++) {
    441:       mut = (s-omg)/2;
    442:       PetscCall(PEPQSliceMatGetInertia(pep,mut,&inertia,NULL));
    443:       if (inertia == 0) {check = 1; break;}
    444:       if (PetscAbsReal((s-omg)/omg)<100*PETSC_MACHINE_EPSILON) break;
    445:       PetscCall(PEPQSliceEvaluateQEP(pep,omg,M,str));
    446:       PetscCall(EPSSetOperators(eps,M,NULL));
    447:       PetscCall(EPSSolve(eps));
    448:       PetscCall(EPSGetConverged(eps,&nconv));
    449:       if (!nconv) break;
    450:       PetscCall(EPSGetEigenpair(eps,0,NULL,NULL,u,w));
    451:       PetscCall(PEPQSliceDiscriminant(pep,u,w,&d,NULL,&omg));
    452:       if (d<0.0) {check = -1; break;}
    453:     }
    454:   }
    455:   if (check==1) *mu = mut;
    456:   *definite = check;
    457:   *hyperbolic = hyp;
    458:   if (M) PetscCall(MatDestroy(&M));
    459:   PetscCall(VecDestroy(&u));
    460:   PetscCall(VecDestroy(&w));
    461:   PetscCall(EPSDestroy(&eps));
    462:   PetscCall(STSetTransform(pep->st,transform));
    463:   PetscFunctionReturn(PETSC_SUCCESS);
    464: }

    466: /*
    467:    Dummy backtransform operation
    468:  */
    469: static PetscErrorCode PEPBackTransform_Skip(PEP pep)
    470: {
    471:   PetscFunctionBegin;
    472:   PetscFunctionReturn(PETSC_SUCCESS);
    473: }

    475: PetscErrorCode PEPSetUp_STOAR_QSlice(PEP pep)
    476: {
    477:   PEP_STOAR      *ctx=(PEP_STOAR*)pep->data;
    478:   PEP_SR         sr;
    479:   PetscInt       ld,i,zeros=0;
    480:   SlepcSC        sc;
    481:   PetscReal      r;

    483:   PetscFunctionBegin;
    484:   PEPCheckSinvertCayley(pep);
    485:   PetscCheck(pep->inta<pep->intb,PetscObjectComm((PetscObject)pep),PETSC_ERR_SUP,"This solver does not support computing all eigenvalues unless you provide a computational interval with PEPSetInterval()");
    486:   PetscCheck(pep->intb<PETSC_MAX_REAL || pep->inta>PETSC_MIN_REAL,PetscObjectComm((PetscObject)pep),PETSC_ERR_ARG_WRONG,"The defined computational interval should have at least one of their sides bounded");
    487:   PEPCheckUnsupportedCondition(pep,PEP_FEATURE_STOPPING,PETSC_TRUE," (with spectrum slicing)");
    488:   if (pep->tol==(PetscReal)PETSC_DEFAULT) {
    489: #if defined(PETSC_USE_REAL_SINGLE)
    490:     pep->tol = SLEPC_DEFAULT_TOL;
    491: #else
    492:     /* use tighter tolerance */
    493:     pep->tol = SLEPC_DEFAULT_TOL*1e-2;
    494: #endif
    495:   }
    496:   if (ctx->nev==1) ctx->nev = PetscMin(20,pep->n);  /* nev not set, use default value */
    497:   PetscCheck(pep->n<=10 || ctx->nev>=10,PetscObjectComm((PetscObject)pep),PETSC_ERR_ARG_WRONG,"nev cannot be less than 10 in spectrum slicing runs");
    498:   pep->ops->backtransform = PEPBackTransform_Skip;
    499:   if (pep->max_it==PETSC_DEFAULT) pep->max_it = 100;

    501:   /* create spectrum slicing context and initialize it */
    502:   PetscCall(PEPQSliceResetSR(pep));
    503:   PetscCall(PetscNew(&sr));
    504:   ctx->sr   = sr;
    505:   sr->itsKs = 0;
    506:   sr->nleap = 0;
    507:   sr->sPres = NULL;

    509:   if (pep->solvematcoeffs) PetscCall(PetscFree(pep->solvematcoeffs));
    510:   PetscCall(PetscMalloc1(pep->nmat,&pep->solvematcoeffs));
    511:   if (!pep->st) PetscCall(PEPGetST(pep,&pep->st));
    512:   PetscCall(STSetTransform(pep->st,PETSC_FALSE));
    513:   PetscCall(STSetUp(pep->st));

    515:   ctx->hyperbolic = (pep->problem_type==PEP_HYPERBOLIC)? PETSC_TRUE: PETSC_FALSE;

    517:   /* check presence of ends and finding direction */
    518:   if (pep->inta > PETSC_MIN_REAL || pep->intb >= PETSC_MAX_REAL) {
    519:     sr->int0 = pep->inta;
    520:     sr->int1 = pep->intb;
    521:     sr->dir = 1;
    522:     if (pep->intb >= PETSC_MAX_REAL) { /* Right-open interval */
    523:       sr->hasEnd = PETSC_FALSE;
    524:     } else sr->hasEnd = PETSC_TRUE;
    525:   } else {
    526:     sr->int0 = pep->intb;
    527:     sr->int1 = pep->inta;
    528:     sr->dir = -1;
    529:     sr->hasEnd = PetscNot(pep->inta <= PETSC_MIN_REAL);
    530:   }

    532:   /* compute inertia0 */
    533:   PetscCall(PEPQSliceGetInertia(pep,sr->int0,&sr->inertia0,ctx->detect?&zeros:NULL,ctx->hyperbolic?0:1));
    534:   PetscCheck(!zeros || (sr->int0!=pep->inta && sr->int0!=pep->intb),((PetscObject)pep)->comm,PETSC_ERR_USER,"Found singular matrix for the transformed problem in the interval endpoint");
    535:   if (!ctx->hyperbolic && ctx->checket) PetscCall(PEPQSliceCheckEigenvalueType(pep,sr->int0,0.0,PETSC_TRUE));

    537:   /* compute inertia1 */
    538:   PetscCall(PEPQSliceGetInertia(pep,sr->int1,&sr->inertia1,ctx->detect?&zeros:NULL,ctx->hyperbolic?0:1));
    539:   PetscCheck(!zeros,((PetscObject)pep)->comm,PETSC_ERR_USER,"Found singular matrix for the transformed problem in an interval endpoint defined by user");
    540:   if (!ctx->hyperbolic && ctx->checket && sr->hasEnd) {
    541:     PetscCall(PEPQSliceCheckEigenvalueType(pep,sr->int1,0.0,PETSC_TRUE));
    542:     PetscCheck(sr->type || sr->inertia1==sr->inertia0,((PetscObject)pep)->comm,PETSC_ERR_CONV_FAILED,"No information of eigenvalue type in interval");
    543:     PetscCheck(!sr->type || sr->inertia1!=sr->inertia0,((PetscObject)pep)->comm,PETSC_ERR_CONV_FAILED,"Different positive/negative type detected");
    544:     if (sr->dir*(sr->inertia1-sr->inertia0)<0) {
    545:       sr->intcorr = -1;
    546:       sr->inertia0 = 2*pep->n-sr->inertia0;
    547:       sr->inertia1 = 2*pep->n-sr->inertia1;
    548:     } else sr->intcorr = 1;
    549:   } else {
    550:     if (sr->inertia0<=pep->n && sr->inertia1<=pep->n) sr->intcorr = 1;
    551:     else if (sr->inertia0>=pep->n && sr->inertia1>=pep->n) sr->intcorr = -1;
    552:   }

    554:   if (sr->hasEnd) {
    555:     sr->dir = -sr->dir; r = sr->int0; sr->int0 = sr->int1; sr->int1 = r;
    556:     i = sr->inertia0; sr->inertia0 = sr->inertia1; sr->inertia1 = i;
    557:   }

    559:   /* number of eigenvalues in interval */
    560:   sr->numEigs = (sr->dir)*(sr->inertia1 - sr->inertia0);
    561:   PetscCall(PetscInfo(pep,"QSlice setup: allocating for %" PetscInt_FMT " eigenvalues in [%g,%g]\n",sr->numEigs,(double)pep->inta,(double)pep->intb));
    562:   if (sr->numEigs) {
    563:     PetscCall(PEPQSliceAllocateSolution(pep));
    564:     PetscCall(PEPSetDimensions_Default(pep,ctx->nev,&ctx->ncv,&ctx->mpd));
    565:     pep->nev = ctx->nev; pep->ncv = ctx->ncv; pep->mpd = ctx->mpd;
    566:     ld   = ctx->ncv+2;
    567:     PetscCall(DSSetType(pep->ds,DSGHIEP));
    568:     PetscCall(DSSetCompact(pep->ds,PETSC_TRUE));
    569:     PetscCall(DSSetExtraRow(pep->ds,PETSC_TRUE));
    570:     PetscCall(DSAllocate(pep->ds,ld));
    571:     PetscCall(DSGetSlepcSC(pep->ds,&sc));
    572:     sc->rg            = NULL;
    573:     sc->comparison    = SlepcCompareLargestMagnitude;
    574:     sc->comparisonctx = NULL;
    575:     sc->map           = NULL;
    576:     sc->mapobj        = NULL;
    577:   } else {pep->ncv = 0; pep->nev = 0; pep->mpd = 0;}
    578:   PetscFunctionReturn(PETSC_SUCCESS);
    579: }

    581: /*
    582:    Fills the fields of a shift structure
    583: */
    584: static PetscErrorCode PEPCreateShift(PEP pep,PetscReal val,PEP_shift neighb0,PEP_shift neighb1)
    585: {
    586:   PEP_shift      s,*pending2;
    587:   PetscInt       i;
    588:   PEP_SR         sr;
    589:   PEP_STOAR      *ctx=(PEP_STOAR*)pep->data;

    591:   PetscFunctionBegin;
    592:   sr = ctx->sr;
    593:   PetscCall(PetscNew(&s));
    594:   s->value = val;
    595:   s->neighb[0] = neighb0;
    596:   if (neighb0) neighb0->neighb[1] = s;
    597:   s->neighb[1] = neighb1;
    598:   if (neighb1) neighb1->neighb[0] = s;
    599:   s->comp[0] = PETSC_FALSE;
    600:   s->comp[1] = PETSC_FALSE;
    601:   s->index = -1;
    602:   s->neigs = 0;
    603:   s->nconv[0] = s->nconv[1] = 0;
    604:   s->nsch[0] = s->nsch[1]=0;
    605:   /* Inserts in the stack of pending shifts */
    606:   /* If needed, the array is resized */
    607:   if (sr->nPend >= sr->maxPend) {
    608:     sr->maxPend *= 2;
    609:     PetscCall(PetscMalloc1(sr->maxPend,&pending2));
    610:     for (i=0;i<sr->nPend;i++) pending2[i] = sr->pending[i];
    611:     PetscCall(PetscFree(sr->pending));
    612:     sr->pending = pending2;
    613:   }
    614:   sr->pending[sr->nPend++]=s;
    615:   PetscFunctionReturn(PETSC_SUCCESS);
    616: }

    618: /* Provides next shift to be computed */
    619: static PetscErrorCode PEPExtractShift(PEP pep)
    620: {
    621:   PetscInt       iner,zeros=0;
    622:   PEP_STOAR      *ctx=(PEP_STOAR*)pep->data;
    623:   PEP_SR         sr;
    624:   PetscReal      newShift,aux;
    625:   PEP_shift      sPres;

    627:   PetscFunctionBegin;
    628:   sr = ctx->sr;
    629:   if (sr->nPend > 0) {
    630:     if (sr->dirch) {
    631:       aux = sr->int1; sr->int1 = sr->int0; sr->int0 = aux;
    632:       iner = sr->inertia1; sr->inertia1 = sr->inertia0; sr->inertia0 = iner;
    633:       sr->dir *= -1;
    634:       PetscCall(PetscFree(sr->s0->neighb[1]));
    635:       PetscCall(PetscFree(sr->s0));
    636:       sr->nPend--;
    637:       PetscCall(PEPCreateShift(pep,sr->int0,NULL,NULL));
    638:       sr->sPrev = NULL;
    639:       sr->sPres = sr->pending[--sr->nPend];
    640:       pep->target = sr->sPres->value;
    641:       sr->s0 = sr->sPres;
    642:       pep->reason = PEP_CONVERGED_ITERATING;
    643:     } else {
    644:       sr->sPrev = sr->sPres;
    645:       sr->sPres = sr->pending[--sr->nPend];
    646:     }
    647:     sPres = sr->sPres;
    648:     PetscCall(PEPQSliceGetInertia(pep,sPres->value,&iner,ctx->detect?&zeros:NULL,sr->intcorr));
    649:     if (zeros) {
    650:       newShift = sPres->value*(1.0+SLICE_PTOL);
    651:       if (sr->dir*(sPres->neighb[0] && newShift-sPres->neighb[0]->value) < 0) newShift = (sPres->value+sPres->neighb[0]->value)/2;
    652:       else if (sPres->neighb[1] && sr->dir*(sPres->neighb[1]->value-newShift) < 0) newShift = (sPres->value+sPres->neighb[1]->value)/2;
    653:       PetscCall(PEPQSliceGetInertia(pep,newShift,&iner,&zeros,sr->intcorr));
    654:       PetscCheck(!zeros,((PetscObject)pep)->comm,PETSC_ERR_CONV_FAILED,"Inertia computation fails in %g",(double)newShift);
    655:       sPres->value = newShift;
    656:     }
    657:     sr->sPres->inertia = iner;
    658:     pep->target = sr->sPres->value;
    659:     pep->reason = PEP_CONVERGED_ITERATING;
    660:     pep->its = 0;
    661:   } else sr->sPres = NULL;
    662:   PetscFunctionReturn(PETSC_SUCCESS);
    663: }

    665: /*
    666:   Obtains value of subsequent shift
    667: */
    668: static PetscErrorCode PEPGetNewShiftValue(PEP pep,PetscInt side,PetscReal *newS)
    669: {
    670:   PetscReal lambda,d_prev;
    671:   PetscInt  i,idxP;
    672:   PEP_SR    sr;
    673:   PEP_shift sPres,s;
    674:   PEP_STOAR *ctx=(PEP_STOAR*)pep->data;

    676:   PetscFunctionBegin;
    677:   sr = ctx->sr;
    678:   sPres = sr->sPres;
    679:   if (sPres->neighb[side]) {
    680:   /* Completing a previous interval */
    681:     if (!sPres->neighb[side]->neighb[side] && sPres->neighb[side]->nconv[side]==0) { /* One of the ends might be too far from eigenvalues */
    682:       if (side) *newS = (sPres->value + PetscRealPart(sr->eigr[sr->perm[sr->indexEig-1]]))/2;
    683:       else *newS = (sPres->value + PetscRealPart(sr->eigr[sr->perm[0]]))/2;
    684:     } else *newS=(sPres->value + sPres->neighb[side]->value)/2;
    685:   } else { /* (Only for side=1). Creating a new interval. */
    686:     if (sPres->neigs==0) {/* No value has been accepted*/
    687:       if (sPres->neighb[0]) {
    688:         /* Multiplying by 10 the previous distance */
    689:         *newS = sPres->value + 10*(sr->dir)*PetscAbsReal(sPres->value - sPres->neighb[0]->value);
    690:         sr->nleap++;
    691:         /* Stops when the interval is open and no values are found in the last 5 shifts (there might be infinite eigenvalues) */
    692:         PetscCheck(sr->hasEnd || sr->nleap<=5,PetscObjectComm((PetscObject)pep),PETSC_ERR_CONV_FAILED,"Unable to compute the wanted eigenvalues with open interval");
    693:       } else { /* First shift */
    694:         if (pep->nconv != 0) {
    695:           /* Unaccepted values give information for next shift */
    696:           idxP=0;/* Number of values left from shift */
    697:           for (i=0;i<pep->nconv;i++) {
    698:             lambda = PetscRealPart(pep->eigr[i]);
    699:             if ((sr->dir)*(lambda - sPres->value) <0) idxP++;
    700:             else break;
    701:           }
    702:           /* Avoiding subtraction of eigenvalues (might be the same).*/
    703:           if (idxP>0) {
    704:             d_prev = PetscAbsReal(sPres->value - PetscRealPart(pep->eigr[0]))/(idxP+0.3);
    705:           } else {
    706:             d_prev = PetscAbsReal(sPres->value - PetscRealPart(pep->eigr[pep->nconv-1]))/(pep->nconv+0.3);
    707:           }
    708:           *newS = sPres->value + ((sr->dir)*d_prev*pep->nev)/2;
    709:           sr->dirch = PETSC_FALSE;
    710:         } else { /* No values found, no information for next shift */
    711:           PetscCheck(!sr->dirch,PetscObjectComm((PetscObject)pep),PETSC_ERR_PLIB,"First shift renders no information");
    712:           sr->dirch = PETSC_TRUE;
    713:           *newS = sr->int1;
    714:         }
    715:       }
    716:     } else { /* Accepted values found */
    717:       sr->dirch = PETSC_FALSE;
    718:       sr->nleap = 0;
    719:       /* Average distance of values in previous subinterval */
    720:       s = sPres->neighb[0];
    721:       while (s && PetscAbs(s->inertia - sPres->inertia)==0) {
    722:         s = s->neighb[0];/* Looking for previous shifts with eigenvalues within */
    723:       }
    724:       if (s) {
    725:         d_prev = PetscAbsReal((sPres->value - s->value)/(sPres->inertia - s->inertia));
    726:       } else { /* First shift. Average distance obtained with values in this shift */
    727:         /* first shift might be too far from first wanted eigenvalue (no values found outside the interval)*/
    728:         if ((sr->dir)*(PetscRealPart(sr->eigr[0])-sPres->value)>0 && PetscAbsReal((PetscRealPart(sr->eigr[sr->indexEig-1]) - PetscRealPart(sr->eigr[0]))/PetscRealPart(sr->eigr[0])) > PetscSqrtReal(pep->tol)) {
    729:           d_prev =  PetscAbsReal((PetscRealPart(sr->eigr[sr->indexEig-1]) - PetscRealPart(sr->eigr[0])))/(sPres->neigs+0.3);
    730:         } else {
    731:           d_prev = PetscAbsReal(PetscRealPart(sr->eigr[sr->indexEig-1]) - sPres->value)/(sPres->neigs+0.3);
    732:         }
    733:       }
    734:       /* Average distance is used for next shift by adding it to value on the right or to shift */
    735:       if ((sr->dir)*(PetscRealPart(sr->eigr[sPres->index + sPres->neigs -1]) - sPres->value)>0) {
    736:         *newS = PetscRealPart(sr->eigr[sPres->index + sPres->neigs -1])+ ((sr->dir)*d_prev*(pep->nev))/2;
    737:       } else { /* Last accepted value is on the left of shift. Adding to shift */
    738:         *newS = sPres->value + ((sr->dir)*d_prev*(pep->nev))/2;
    739:       }
    740:     }
    741:     /* End of interval can not be surpassed */
    742:     if ((sr->dir)*(sr->int1 - *newS) < 0) *newS = sr->int1;
    743:   }/* of neighb[side]==null */
    744:   PetscFunctionReturn(PETSC_SUCCESS);
    745: }

    747: /*
    748:   Function for sorting an array of real values
    749: */
    750: static PetscErrorCode sortRealEigenvalues(PetscScalar *r,PetscInt *perm,PetscInt nr,PetscBool prev,PetscInt dir)
    751: {
    752:   PetscReal re;
    753:   PetscInt  i,j,tmp;

    755:   PetscFunctionBegin;
    756:   if (!prev) for (i=0;i<nr;i++) perm[i] = i;
    757:   /* Insertion sort */
    758:   for (i=1;i<nr;i++) {
    759:     re = PetscRealPart(r[perm[i]]);
    760:     j = i-1;
    761:     while (j>=0 && dir*(re - PetscRealPart(r[perm[j]])) <= 0) {
    762:       tmp = perm[j]; perm[j] = perm[j+1]; perm[j+1] = tmp; j--;
    763:     }
    764:   }
    765:   PetscFunctionReturn(PETSC_SUCCESS);
    766: }

    768: /* Stores the pairs obtained since the last shift in the global arrays */
    769: static PetscErrorCode PEPStoreEigenpairs(PEP pep)
    770: {
    771:   PEP_STOAR      *ctx=(PEP_STOAR*)pep->data;
    772:   PetscReal      lambda,err,*errest;
    773:   PetscInt       i,*aux,count=0,ndef,ld,nconv=pep->nconv,d=pep->nmat-1,idx;
    774:   PetscBool      iscayley,divide=PETSC_FALSE;
    775:   PEP_SR         sr = ctx->sr;
    776:   PEP_shift      sPres;
    777:   Vec            w,vomega;
    778:   Mat            MS;
    779:   BV             tV;
    780:   PetscScalar    *S,*eigr,*tS,*omega;

    782:   PetscFunctionBegin;
    783:   sPres = sr->sPres;
    784:   sPres->index = sr->indexEig;

    786:   if (nconv>sr->ndef0+sr->ndef1) {
    787:     /* Back-transform */
    788:     PetscCall(STBackTransform(pep->st,nconv,pep->eigr,pep->eigi));
    789:     for (i=0;i<nconv;i++) {
    790: #if defined(PETSC_USE_COMPLEX)
    791:       if (PetscImaginaryPart(pep->eigr[i])) pep->eigr[i] = sr->int0-sr->dir;
    792: #else
    793:       if (pep->eigi[i]) pep->eigr[i] = sr->int0-sr->dir;
    794: #endif
    795:     }
    796:     PetscCall(PetscObjectTypeCompare((PetscObject)pep->st,STCAYLEY,&iscayley));
    797:     /* Sort eigenvalues */
    798:     PetscCall(sortRealEigenvalues(pep->eigr,pep->perm,nconv,PETSC_FALSE,sr->dir));
    799:     PetscCall(VecCreateSeq(PETSC_COMM_SELF,nconv,&vomega));
    800:     PetscCall(BVGetSignature(ctx->V,vomega));
    801:     PetscCall(VecGetArray(vomega,&omega));
    802:     PetscCall(BVGetSizes(pep->V,NULL,NULL,&ld));
    803:     PetscCall(BVTensorGetFactors(ctx->V,NULL,&MS));
    804:     PetscCall(MatDenseGetArray(MS,&S));
    805:     /* Values stored in global array */
    806:     PetscCall(PetscCalloc4(nconv,&eigr,nconv,&errest,nconv*nconv*d,&tS,nconv,&aux));
    807:     ndef = sr->ndef0+sr->ndef1;
    808:     for (i=0;i<nconv;i++) {
    809:       lambda = PetscRealPart(pep->eigr[pep->perm[i]]);
    810:       err = pep->errest[pep->perm[i]];
    811:       if ((sr->dir)*(lambda - sPres->ext[0]) > 0 && (sr->dir)*(sPres->ext[1] - lambda) > 0) {/* Valid value */
    812:         PetscCheck(sr->indexEig+count-ndef<sr->numEigs,PetscObjectComm((PetscObject)pep),PETSC_ERR_PLIB,"Unexpected error in Spectrum Slicing");
    813:         PetscCall(PEPQSliceCheckEigenvalueType(pep,lambda,PetscRealPart(omega[pep->perm[i]]),PETSC_FALSE));
    814:         eigr[count] = lambda;
    815:         errest[count] = err;
    816:         if (((sr->dir)*(sPres->value - lambda) > 0) && ((sr->dir)*(lambda - sPres->ext[0]) > 0)) sPres->nconv[0]++;
    817:         if (((sr->dir)*(lambda - sPres->value) > 0) && ((sr->dir)*(sPres->ext[1] - lambda) > 0)) sPres->nconv[1]++;
    818:         PetscCall(PetscArraycpy(tS+count*(d*nconv),S+pep->perm[i]*(d*ld),nconv));
    819:         PetscCall(PetscArraycpy(tS+count*(d*nconv)+nconv,S+pep->perm[i]*(d*ld)+ld,nconv));
    820:         count++;
    821:       }
    822:     }
    823:     PetscCall(VecRestoreArray(vomega,&omega));
    824:     PetscCall(VecDestroy(&vomega));
    825:     for (i=0;i<count;i++) {
    826:       PetscCall(PetscArraycpy(S+i*(d*ld),tS+i*nconv*d,nconv));
    827:       PetscCall(PetscArraycpy(S+i*(d*ld)+ld,tS+i*nconv*d+nconv,nconv));
    828:     }
    829:     PetscCall(MatDenseRestoreArray(MS,&S));
    830:     PetscCall(BVTensorRestoreFactors(ctx->V,NULL,&MS));
    831:     PetscCall(BVSetActiveColumns(ctx->V,0,count));
    832:     PetscCall(BVTensorCompress(ctx->V,count));
    833:     if (sr->sPres->nconv[0] && sr->sPres->nconv[1]) {
    834:       divide = PETSC_TRUE;
    835:       PetscCall(BVTensorGetFactors(ctx->V,NULL,&MS));
    836:       PetscCall(MatDenseGetArray(MS,&S));
    837:       PetscCall(PetscArrayzero(tS,nconv*nconv*d));
    838:       for (i=0;i<count;i++) {
    839:         PetscCall(PetscArraycpy(tS+i*nconv*d,S+i*(d*ld),count));
    840:         PetscCall(PetscArraycpy(tS+i*nconv*d+nconv,S+i*(d*ld)+ld,count));
    841:       }
    842:       PetscCall(MatDenseRestoreArray(MS,&S));
    843:       PetscCall(BVTensorRestoreFactors(ctx->V,NULL,&MS));
    844:       PetscCall(BVSetActiveColumns(pep->V,0,count));
    845:       PetscCall(BVDuplicateResize(pep->V,count,&tV));
    846:       PetscCall(BVCopy(pep->V,tV));
    847:     }
    848:     if (sr->sPres->nconv[0]) {
    849:       if (divide) {
    850:         PetscCall(BVSetActiveColumns(ctx->V,0,sr->sPres->nconv[0]));
    851:         PetscCall(BVTensorCompress(ctx->V,sr->sPres->nconv[0]));
    852:       }
    853:       for (i=0;i<sr->ndef0;i++) aux[i] = sr->idxDef0[i];
    854:       for (i=sr->ndef0;i<sr->sPres->nconv[0];i++) aux[i] = sr->indexEig+i-sr->ndef0;
    855:       PetscCall(BVTensorGetFactors(ctx->V,NULL,&MS));
    856:       PetscCall(MatDenseGetArray(MS,&S));
    857:       for (i=0;i<sr->sPres->nconv[0];i++) {
    858:         sr->eigr[aux[i]] = eigr[i];
    859:         sr->errest[aux[i]] = errest[i];
    860:         PetscCall(BVGetColumn(pep->V,i,&w));
    861:         PetscCall(BVInsertVec(sr->V,aux[i],w));
    862:         PetscCall(BVRestoreColumn(pep->V,i,&w));
    863:         idx = sr->ld*d*aux[i];
    864:         PetscCall(PetscArrayzero(sr->S+idx,sr->ld*d));
    865:         PetscCall(PetscArraycpy(sr->S+idx,S+i*(ld*d),sr->sPres->nconv[0]));
    866:         PetscCall(PetscArraycpy(sr->S+idx+sr->ld,S+i*(ld*d)+ld,sr->sPres->nconv[0]));
    867:         PetscCall(PetscFree(sr->qinfo[aux[i]].q));
    868:         PetscCall(PetscMalloc1(sr->sPres->nconv[0],&sr->qinfo[aux[i]].q));
    869:         PetscCall(PetscArraycpy(sr->qinfo[aux[i]].q,aux,sr->sPres->nconv[0]));
    870:         sr->qinfo[aux[i]].nq = sr->sPres->nconv[0];
    871:       }
    872:       PetscCall(MatDenseRestoreArray(MS,&S));
    873:       PetscCall(BVTensorRestoreFactors(ctx->V,NULL,&MS));
    874:     }

    876:     if (sr->sPres->nconv[1]) {
    877:       if (divide) {
    878:         PetscCall(BVTensorGetFactors(ctx->V,NULL,&MS));
    879:         PetscCall(MatDenseGetArray(MS,&S));
    880:         for (i=0;i<sr->sPres->nconv[1];i++) {
    881:           PetscCall(PetscArraycpy(S+i*(d*ld),tS+(sr->sPres->nconv[0]+i)*nconv*d,count));
    882:           PetscCall(PetscArraycpy(S+i*(d*ld)+ld,tS+(sr->sPres->nconv[0]+i)*nconv*d+nconv,count));
    883:         }
    884:         PetscCall(MatDenseRestoreArray(MS,&S));
    885:         PetscCall(BVTensorRestoreFactors(ctx->V,NULL,&MS));
    886:         PetscCall(BVSetActiveColumns(pep->V,0,count));
    887:         PetscCall(BVCopy(tV,pep->V));
    888:         PetscCall(BVSetActiveColumns(ctx->V,0,sr->sPres->nconv[1]));
    889:         PetscCall(BVTensorCompress(ctx->V,sr->sPres->nconv[1]));
    890:       }
    891:       for (i=0;i<sr->ndef1;i++) aux[i] = sr->idxDef1[i];
    892:       for (i=sr->ndef1;i<sr->sPres->nconv[1];i++) aux[i] = sr->indexEig+sr->sPres->nconv[0]-sr->ndef0+i-sr->ndef1;
    893:       PetscCall(BVTensorGetFactors(ctx->V,NULL,&MS));
    894:       PetscCall(MatDenseGetArray(MS,&S));
    895:       for (i=0;i<sr->sPres->nconv[1];i++) {
    896:         sr->eigr[aux[i]] = eigr[sr->sPres->nconv[0]+i];
    897:         sr->errest[aux[i]] = errest[sr->sPres->nconv[0]+i];
    898:         PetscCall(BVGetColumn(pep->V,i,&w));
    899:         PetscCall(BVInsertVec(sr->V,aux[i],w));
    900:         PetscCall(BVRestoreColumn(pep->V,i,&w));
    901:         idx = sr->ld*d*aux[i];
    902:         PetscCall(PetscArrayzero(sr->S+idx,sr->ld*d));
    903:         PetscCall(PetscArraycpy(sr->S+idx,S+i*(ld*d),sr->sPres->nconv[1]));
    904:         PetscCall(PetscArraycpy(sr->S+idx+sr->ld,S+i*(ld*d)+ld,sr->sPres->nconv[1]));
    905:         PetscCall(PetscFree(sr->qinfo[aux[i]].q));
    906:         PetscCall(PetscMalloc1(sr->sPres->nconv[1],&sr->qinfo[aux[i]].q));
    907:         PetscCall(PetscArraycpy(sr->qinfo[aux[i]].q,aux,sr->sPres->nconv[1]));
    908:         sr->qinfo[aux[i]].nq = sr->sPres->nconv[1];
    909:       }
    910:       PetscCall(MatDenseRestoreArray(MS,&S));
    911:       PetscCall(BVTensorRestoreFactors(ctx->V,NULL,&MS));
    912:     }
    913:     sPres->neigs = count-sr->ndef0-sr->ndef1;
    914:     sr->indexEig += sPres->neigs;
    915:     sPres->nconv[0]-= sr->ndef0;
    916:     sPres->nconv[1]-= sr->ndef1;
    917:     PetscCall(PetscFree4(eigr,errest,tS,aux));
    918:   } else {
    919:     sPres->neigs = 0;
    920:     sPres->nconv[0]= 0;
    921:     sPres->nconv[1]= 0;
    922:   }
    923:   /* Global ordering array updating */
    924:   PetscCall(sortRealEigenvalues(sr->eigr,sr->perm,sr->indexEig,PETSC_FALSE,sr->dir));
    925:   /* Check for completion */
    926:   sPres->comp[0] = PetscNot(sPres->nconv[0] < sPres->nsch[0]);
    927:   sPres->comp[1] = PetscNot(sPres->nconv[1] < sPres->nsch[1]);
    928:   PetscCheck(sPres->nconv[0]<=sPres->nsch[0] && sPres->nconv[1]<=sPres->nsch[1],PetscObjectComm((PetscObject)pep),PETSC_ERR_PLIB,"Mismatch between number of values found and information from inertia");
    929:   if (divide) PetscCall(BVDestroy(&tV));
    930:   PetscFunctionReturn(PETSC_SUCCESS);
    931: }

    933: static PetscErrorCode PEPLookForDeflation(PEP pep)
    934: {
    935:   PetscReal val;
    936:   PetscInt  i,count0=0,count1=0;
    937:   PEP_shift sPres;
    938:   PetscInt  ini,fin;
    939:   PEP_SR    sr;
    940:   PEP_STOAR *ctx=(PEP_STOAR*)pep->data;

    942:   PetscFunctionBegin;
    943:   sr = ctx->sr;
    944:   sPres = sr->sPres;

    946:   if (sPres->neighb[0]) ini = (sr->dir)*(sPres->neighb[0]->inertia - sr->inertia0);
    947:   else ini = 0;
    948:   fin = sr->indexEig;
    949:   /* Selection of ends for searching new values */
    950:   if (!sPres->neighb[0]) sPres->ext[0] = sr->int0;/* First shift */
    951:   else sPres->ext[0] = sPres->neighb[0]->value;
    952:   if (!sPres->neighb[1]) {
    953:     if (sr->hasEnd) sPres->ext[1] = sr->int1;
    954:     else sPres->ext[1] = (sr->dir > 0)?PETSC_MAX_REAL:PETSC_MIN_REAL;
    955:   } else sPres->ext[1] = sPres->neighb[1]->value;
    956:   /* Selection of values between right and left ends */
    957:   for (i=ini;i<fin;i++) {
    958:     val=PetscRealPart(sr->eigr[sr->perm[i]]);
    959:     /* Values to the right of left shift */
    960:     if ((sr->dir)*(val - sPres->ext[1]) < 0) {
    961:       if ((sr->dir)*(val - sPres->value) < 0) count0++;
    962:       else count1++;
    963:     } else break;
    964:   }
    965:   /* The number of values on each side are found */
    966:   if (sPres->neighb[0]) {
    967:     sPres->nsch[0] = (sr->dir)*(sPres->inertia - sPres->neighb[0]->inertia)-count0;
    968:     PetscCheck(sPres->nsch[0]>=0,PetscObjectComm((PetscObject)pep),PETSC_ERR_PLIB,"Mismatch between number of values found and information from inertia");
    969:   } else sPres->nsch[0] = 0;

    971:   if (sPres->neighb[1]) {
    972:     sPres->nsch[1] = (sr->dir)*(sPres->neighb[1]->inertia - sPres->inertia) - count1;
    973:     PetscCheck(sPres->nsch[1]>=0,PetscObjectComm((PetscObject)pep),PETSC_ERR_PLIB,"Mismatch between number of values found and information from inertia");
    974:   } else sPres->nsch[1] = (sr->dir)*(sr->inertia1 - sPres->inertia);

    976:   /* Completing vector of indexes for deflation */
    977:   for (i=0;i<count0;i++) sr->idxDef0[i] = sr->perm[ini+i];
    978:   sr->ndef0 = count0;
    979:   for (i=0;i<count1;i++) sr->idxDef1[i] = sr->perm[ini+count0+i];
    980:   sr->ndef1 = count1;
    981:   PetscFunctionReturn(PETSC_SUCCESS);
    982: }

    984: /*
    985:   Compute a run of Lanczos iterations
    986: */
    987: static PetscErrorCode PEPSTOARrun_QSlice(PEP pep,PetscReal *a,PetscReal *b,PetscReal *omega,PetscInt k,PetscInt *M,PetscBool *breakdown,PetscBool *symmlost,Vec *t_)
    988: {
    989:   PEP_STOAR      *ctx = (PEP_STOAR*)pep->data;
    990:   PetscInt       i,j,m=*M,l,lock;
    991:   PetscInt       lds,d,ld,offq,nqt,ldds;
    992:   Vec            v=t_[0],t=t_[1],q=t_[2];
    993:   PetscReal      norm,sym=0.0,fro=0.0,*f;
    994:   PetscScalar    *y,*S,sigma;
    995:   PetscBLASInt   j_,one=1;
    996:   PetscBool      lindep;
    997:   Mat            MS;

    999:   PetscFunctionBegin;
    1000:   PetscCall(PetscMalloc1(*M,&y));
    1001:   PetscCall(BVGetSizes(pep->V,NULL,NULL,&ld));
    1002:   PetscCall(BVTensorGetDegree(ctx->V,&d));
    1003:   PetscCall(BVGetActiveColumns(pep->V,&lock,&nqt));
    1004:   lds = d*ld;
    1005:   offq = ld;
    1006:   PetscCall(DSGetLeadingDimension(pep->ds,&ldds));

    1008:   *breakdown = PETSC_FALSE; /* ----- */
    1009:   PetscCall(STGetShift(pep->st,&sigma));
    1010:   PetscCall(DSGetDimensions(pep->ds,NULL,&l,NULL,NULL));
    1011:   PetscCall(BVSetActiveColumns(ctx->V,0,m));
    1012:   PetscCall(BVSetActiveColumns(pep->V,0,nqt));
    1013:   for (j=k;j<m;j++) {
    1014:     /* apply operator */
    1015:     PetscCall(BVTensorGetFactors(ctx->V,NULL,&MS));
    1016:     PetscCall(MatDenseGetArray(MS,&S));
    1017:     PetscCall(BVGetColumn(pep->V,nqt,&t));
    1018:     PetscCall(BVMultVec(pep->V,1.0,0.0,v,S+j*lds));
    1019:     PetscCall(MatMult(pep->A[1],v,q));
    1020:     PetscCall(MatMult(pep->A[2],v,t));
    1021:     PetscCall(VecAXPY(q,sigma*pep->sfactor,t));
    1022:     PetscCall(VecScale(q,pep->sfactor));
    1023:     PetscCall(BVMultVec(pep->V,1.0,0.0,v,S+offq+j*lds));
    1024:     PetscCall(MatMult(pep->A[2],v,t));
    1025:     PetscCall(VecAXPY(q,pep->sfactor*pep->sfactor,t));
    1026:     PetscCall(STMatSolve(pep->st,q,t));
    1027:     PetscCall(VecScale(t,-1.0));
    1028:     PetscCall(BVRestoreColumn(pep->V,nqt,&t));

    1030:     /* orthogonalize */
    1031:     PetscCall(BVOrthogonalizeColumn(pep->V,nqt,S+(j+1)*lds,&norm,&lindep));
    1032:     if (!lindep) {
    1033:       *(S+(j+1)*lds+nqt) = norm;
    1034:       PetscCall(BVScaleColumn(pep->V,nqt,1.0/norm));
    1035:       nqt++;
    1036:     }
    1037:     for (i=0;i<nqt;i++) *(S+(j+1)*lds+offq+i) = *(S+j*lds+i)+sigma*(*(S+(j+1)*lds+i));
    1038:     PetscCall(BVSetActiveColumns(pep->V,0,nqt));
    1039:     PetscCall(MatDenseRestoreArray(MS,&S));
    1040:     PetscCall(BVTensorRestoreFactors(ctx->V,NULL,&MS));

    1042:     /* level-2 orthogonalization */
    1043:     PetscCall(BVOrthogonalizeColumn(ctx->V,j+1,y,&norm,&lindep));
    1044:     a[j] = PetscRealPart(y[j]);
    1045:     omega[j+1] = (norm > 0)?1.0:-1.0;
    1046:     PetscCall(BVScaleColumn(ctx->V,j+1,1.0/norm));
    1047:     b[j] = PetscAbsReal(norm);

    1049:     /* check symmetry */
    1050:     PetscCall(DSGetArrayReal(pep->ds,DS_MAT_T,&f));
    1051:     if (j==k) {
    1052:       for (i=l;i<j-1;i++) y[i] = PetscAbsScalar(y[i])-PetscAbsReal(f[2*ldds+i]);
    1053:       for (i=0;i<l;i++) y[i] = 0.0;
    1054:     }
    1055:     PetscCall(DSRestoreArrayReal(pep->ds,DS_MAT_T,&f));
    1056:     if (j>0) y[j-1] = PetscAbsScalar(y[j-1])-PetscAbsReal(b[j-1]);
    1057:     PetscCall(PetscBLASIntCast(j,&j_));
    1058:     sym = SlepcAbs(BLASnrm2_(&j_,y,&one),sym);
    1059:     fro = SlepcAbs(fro,SlepcAbs(a[j],b[j]));
    1060:     if (j>0) fro = SlepcAbs(fro,b[j-1]);
    1061:     if (sym/fro>PetscMax(PETSC_SQRT_MACHINE_EPSILON,10*pep->tol)) {
    1062:       *symmlost = PETSC_TRUE;
    1063:       *M=j;
    1064:       break;
    1065:     }
    1066:   }
    1067:   PetscCall(BVSetActiveColumns(pep->V,lock,nqt));
    1068:   PetscCall(BVSetActiveColumns(ctx->V,0,*M));
    1069:   PetscCall(PetscFree(y));
    1070:   PetscFunctionReturn(PETSC_SUCCESS);
    1071: }

    1073: static PetscErrorCode PEPSTOAR_QSlice(PEP pep,Mat B)
    1074: {
    1075:   PEP_STOAR      *ctx = (PEP_STOAR*)pep->data;
    1076:   PetscInt       j,k,l,nv=0,ld,ldds,t,nq=0,idx;
    1077:   PetscInt       nconv=0,deg=pep->nmat-1,count0=0,count1=0;
    1078:   PetscScalar    *om,sigma,*back,*S,*pQ;
    1079:   PetscReal      beta,norm=1.0,*omega,*a,*b,eta,lambda;
    1080:   PetscBool      breakdown,symmlost=PETSC_FALSE,sinv,falselock=PETSC_TRUE;
    1081:   Mat            MS,MQ,D;
    1082:   Vec            v,vomega;
    1083:   PEP_SR         sr;
    1084:   BVOrthogType   otype;
    1085:   BVOrthogBlockType obtype;

    1087:   PetscFunctionBegin;
    1088:   /* Resize if needed for deflating vectors  */
    1089:   sr = ctx->sr;
    1090:   sigma = sr->sPres->value;
    1091:   k = sr->ndef0+sr->ndef1;
    1092:   pep->ncv = ctx->ncv+k;
    1093:   pep->nev = ctx->nev+k;
    1094:   PetscCall(PEPAllocateSolution(pep,3));
    1095:   PetscCall(BVDestroy(&ctx->V));
    1096:   PetscCall(BVCreateTensor(pep->V,pep->nmat-1,&ctx->V));
    1097:   PetscCall(BVGetOrthogonalization(pep->V,&otype,NULL,&eta,&obtype));
    1098:   PetscCall(BVSetOrthogonalization(ctx->V,otype,BV_ORTHOG_REFINE_ALWAYS,eta,obtype));
    1099:   PetscCall(DSAllocate(pep->ds,pep->ncv+2));
    1100:   PetscCall(PetscMalloc1(pep->ncv,&back));
    1101:   PetscCall(DSGetLeadingDimension(pep->ds,&ldds));
    1102:   PetscCall(BVSetMatrix(ctx->V,B,PETSC_TRUE));
    1103:   PetscCheck(ctx->lock,PetscObjectComm((PetscObject)pep),PETSC_ERR_SUP,"A locking variant is needed for spectrum slicing");
    1104:   /* undocumented option to use a cheaper locking instead of the true locking */
    1105:   PetscCall(PetscOptionsGetBool(NULL,NULL,"-pep_stoar_falselocking",&falselock,NULL));
    1106:   PetscCall(PetscObjectTypeCompare((PetscObject)pep->st,STSINVERT,&sinv));
    1107:   PetscCall(RGPushScale(pep->rg,sinv?pep->sfactor:1.0/pep->sfactor));
    1108:   PetscCall(STScaleShift(pep->st,sinv?pep->sfactor:1.0/pep->sfactor));

    1110:   /* Get the starting Arnoldi vector */
    1111:   PetscCall(BVSetActiveColumns(pep->V,0,1));
    1112:   PetscCall(BVTensorBuildFirstColumn(ctx->V,pep->nini));
    1113:   PetscCall(BVSetActiveColumns(ctx->V,0,1));
    1114:   if (k) {
    1115:     /* Insert deflated vectors */
    1116:     PetscCall(BVSetActiveColumns(pep->V,0,0));
    1117:     idx = sr->ndef0?sr->idxDef0[0]:sr->idxDef1[0];
    1118:     for (j=0;j<k;j++) {
    1119:       PetscCall(BVGetColumn(pep->V,j,&v));
    1120:       PetscCall(BVCopyVec(sr->V,sr->qinfo[idx].q[j],v));
    1121:       PetscCall(BVRestoreColumn(pep->V,j,&v));
    1122:     }
    1123:     /* Update innerproduct matrix */
    1124:     PetscCall(BVSetActiveColumns(ctx->V,0,0));
    1125:     PetscCall(BVTensorGetFactors(ctx->V,NULL,&MS));
    1126:     PetscCall(BVSetActiveColumns(pep->V,0,k));
    1127:     PetscCall(BVTensorRestoreFactors(ctx->V,NULL,&MS));

    1129:     PetscCall(BVGetSizes(pep->V,NULL,NULL,&ld));
    1130:     PetscCall(BVTensorGetFactors(ctx->V,NULL,&MS));
    1131:     PetscCall(MatDenseGetArray(MS,&S));
    1132:     for (j=0;j<sr->ndef0;j++) {
    1133:       PetscCall(PetscArrayzero(S+j*ld*deg,ld*deg));
    1134:       PetscCall(PetscArraycpy(S+j*ld*deg,sr->S+sr->idxDef0[j]*sr->ld*deg,k));
    1135:       PetscCall(PetscArraycpy(S+j*ld*deg+ld,sr->S+sr->idxDef0[j]*sr->ld*deg+sr->ld,k));
    1136:       pep->eigr[j] = sr->eigr[sr->idxDef0[j]];
    1137:       pep->errest[j] = sr->errest[sr->idxDef0[j]];
    1138:     }
    1139:     for (j=0;j<sr->ndef1;j++) {
    1140:       PetscCall(PetscArrayzero(S+(j+sr->ndef0)*ld*deg,ld*deg));
    1141:       PetscCall(PetscArraycpy(S+(j+sr->ndef0)*ld*deg,sr->S+sr->idxDef1[j]*sr->ld*deg,k));
    1142:       PetscCall(PetscArraycpy(S+(j+sr->ndef0)*ld*deg+ld,sr->S+sr->idxDef1[j]*sr->ld*deg+sr->ld,k));
    1143:       pep->eigr[j+sr->ndef0] = sr->eigr[sr->idxDef1[j]];
    1144:       pep->errest[j+sr->ndef0] = sr->errest[sr->idxDef1[j]];
    1145:     }
    1146:     PetscCall(MatDenseRestoreArray(MS,&S));
    1147:     PetscCall(BVTensorRestoreFactors(ctx->V,NULL,&MS));
    1148:     PetscCall(BVSetActiveColumns(ctx->V,0,k+1));
    1149:     PetscCall(VecCreateSeq(PETSC_COMM_SELF,k+1,&vomega));
    1150:     PetscCall(VecGetArray(vomega,&om));
    1151:     for (j=0;j<k;j++) {
    1152:       PetscCall(BVOrthogonalizeColumn(ctx->V,j,NULL,&norm,NULL));
    1153:       PetscCall(BVScaleColumn(ctx->V,j,1/norm));
    1154:       om[j] = (norm>=0.0)?1.0:-1.0;
    1155:     }
    1156:     PetscCall(BVTensorGetFactors(ctx->V,NULL,&MS));
    1157:     PetscCall(MatDenseGetArray(MS,&S));
    1158:     for (j=0;j<deg;j++) {
    1159:       PetscCall(BVSetRandomColumn(pep->V,k+j));
    1160:       PetscCall(BVOrthogonalizeColumn(pep->V,k+j,S+k*ld*deg+j*ld,&norm,NULL));
    1161:       PetscCall(BVScaleColumn(pep->V,k+j,1.0/norm));
    1162:       S[k*ld*deg+j*ld+k+j] = norm;
    1163:     }
    1164:     PetscCall(MatDenseRestoreArray(MS,&S));
    1165:     PetscCall(BVSetActiveColumns(pep->V,0,k+deg));
    1166:     PetscCall(BVTensorRestoreFactors(ctx->V,NULL,&MS));
    1167:     PetscCall(BVOrthogonalizeColumn(ctx->V,k,NULL,&norm,NULL));
    1168:     PetscCall(BVScaleColumn(ctx->V,k,1.0/norm));
    1169:     om[k] = (norm>=0.0)?1.0:-1.0;
    1170:     PetscCall(VecRestoreArray(vomega,&om));
    1171:     PetscCall(BVSetSignature(ctx->V,vomega));
    1172:     PetscCall(DSGetArrayReal(pep->ds,DS_MAT_T,&a));
    1173:     PetscCall(VecGetArray(vomega,&om));
    1174:     for (j=0;j<k;j++) a[j] = PetscRealPart(om[j]/(pep->eigr[j]-sigma));
    1175:     PetscCall(VecRestoreArray(vomega,&om));
    1176:     PetscCall(VecDestroy(&vomega));
    1177:     PetscCall(DSRestoreArrayReal(pep->ds,DS_MAT_T,&a));
    1178:     PetscCall(DSGetArray(pep->ds,DS_MAT_Q,&pQ));
    1179:     PetscCall(PetscArrayzero(pQ,ldds*k));
    1180:     for (j=0;j<k;j++) pQ[j+j*ldds] = 1.0;
    1181:     PetscCall(DSRestoreArray(pep->ds,DS_MAT_Q,&pQ));
    1182:   }
    1183:   PetscCall(BVSetActiveColumns(ctx->V,0,k+1));
    1184:   PetscCall(DSSetDimensions(pep->ds,k+1,PETSC_DEFAULT,PETSC_DEFAULT));
    1185:   PetscCall(DSGetMatAndColumn(pep->ds,DS_MAT_D,0,&D,&vomega));
    1186:   PetscCall(BVGetSignature(ctx->V,vomega));
    1187:   PetscCall(DSRestoreMatAndColumn(pep->ds,DS_MAT_D,0,&D,&vomega));

    1189:   PetscCall(PetscInfo(pep,"Start STOAR: sigma=%g in [%g,%g], for deflation: left=%" PetscInt_FMT " right=%" PetscInt_FMT ", searching: left=%" PetscInt_FMT " right=%" PetscInt_FMT "\n",(double)sr->sPres->value,(double)(sr->sPres->neighb[0]?sr->sPres->neighb[0]->value:sr->int0),(double)(sr->sPres->neighb[1]?sr->sPres->neighb[1]->value:sr->int1),sr->ndef0,sr->ndef1,sr->sPres->nsch[0],sr->sPres->nsch[1]));

    1191:   /* Restart loop */
    1192:   l = 0;
    1193:   pep->nconv = k;
    1194:   while (pep->reason == PEP_CONVERGED_ITERATING) {
    1195:     pep->its++;
    1196:     PetscCall(DSGetArrayReal(pep->ds,DS_MAT_T,&a));
    1197:     b = a+ldds;
    1198:     PetscCall(DSGetArrayReal(pep->ds,DS_MAT_D,&omega));

    1200:     /* Compute an nv-step Lanczos factorization */
    1201:     nv = PetscMin(pep->nconv+pep->mpd,pep->ncv);
    1202:     PetscCall(PEPSTOARrun_QSlice(pep,a,b,omega,pep->nconv+l,&nv,&breakdown,&symmlost,pep->work));
    1203:     beta = b[nv-1];
    1204:     if (symmlost && nv==pep->nconv+l) {
    1205:       pep->reason = PEP_DIVERGED_SYMMETRY_LOST;
    1206:       pep->nconv = nconv;
    1207:       PetscCall(PetscInfo(pep,"Symmetry lost in STOAR sigma=%g nconv=%" PetscInt_FMT "\n",(double)sr->sPres->value,nconv));
    1208:       if (falselock || !ctx->lock) {
    1209:         PetscCall(BVSetActiveColumns(ctx->V,0,pep->nconv));
    1210:         PetscCall(BVTensorCompress(ctx->V,0));
    1211:       }
    1212:       break;
    1213:     }
    1214:     PetscCall(DSRestoreArrayReal(pep->ds,DS_MAT_T,&a));
    1215:     PetscCall(DSRestoreArrayReal(pep->ds,DS_MAT_D,&omega));
    1216:     PetscCall(DSSetDimensions(pep->ds,nv,pep->nconv,pep->nconv+l));
    1217:     if (l==0) PetscCall(DSSetState(pep->ds,DS_STATE_INTERMEDIATE));
    1218:     else PetscCall(DSSetState(pep->ds,DS_STATE_RAW));

    1220:     /* Solve projected problem */
    1221:     PetscCall(DSSolve(pep->ds,pep->eigr,pep->eigi));
    1222:     PetscCall(DSSort(pep->ds,pep->eigr,pep->eigi,NULL,NULL,NULL));
    1223:     PetscCall(DSUpdateExtraRow(pep->ds));
    1224:     PetscCall(DSSynchronize(pep->ds,pep->eigr,pep->eigi));

    1226:     /* Check convergence */
    1227:     /* PetscCall(PEPSTOARpreKConvergence(pep,nv,&norm,pep->work));*/
    1228:     norm = 1.0;
    1229:     PetscCall(DSGetDimensions(pep->ds,NULL,NULL,NULL,&t));
    1230:     PetscCall(PEPKrylovConvergence(pep,PETSC_FALSE,pep->nconv,t-pep->nconv,PetscAbsReal(beta)*norm,&k));
    1231:     PetscCall((*pep->stopping)(pep,pep->its,pep->max_it,k,pep->nev,&pep->reason,pep->stoppingctx));
    1232:     for (j=0;j<k;j++) back[j] = pep->eigr[j];
    1233:     PetscCall(STBackTransform(pep->st,k,back,pep->eigi));
    1234:     count0=count1=0;
    1235:     for (j=0;j<k;j++) {
    1236:       lambda = PetscRealPart(back[j]);
    1237:       if (((sr->dir)*(sr->sPres->value - lambda) > 0) && ((sr->dir)*(lambda - sr->sPres->ext[0]) > 0)) count0++;
    1238:       if (((sr->dir)*(lambda - sr->sPres->value) > 0) && ((sr->dir)*(sr->sPres->ext[1] - lambda) > 0)) count1++;
    1239:     }
    1240:     if ((count0-sr->ndef0 >= sr->sPres->nsch[0]) && (count1-sr->ndef1 >= sr->sPres->nsch[1])) pep->reason = PEP_CONVERGED_TOL;
    1241:     /* Update l */
    1242:     if (pep->reason != PEP_CONVERGED_ITERATING || breakdown) l = 0;
    1243:     else {
    1244:       l = PetscMax(1,(PetscInt)((nv-k)/2));
    1245:       l = PetscMin(l,t);
    1246:       PetscCall(DSGetTruncateSize(pep->ds,k,t,&l));
    1247:       if (!breakdown) {
    1248:         /* Prepare the Rayleigh quotient for restart */
    1249:         PetscCall(DSTruncate(pep->ds,k+l,PETSC_FALSE));
    1250:       }
    1251:     }
    1252:     nconv = k;
    1253:     if (!ctx->lock && pep->reason == PEP_CONVERGED_ITERATING && !breakdown) { l += k; k = 0; } /* non-locking variant: reset no. of converged pairs */
    1254:     if (l) PetscCall(PetscInfo(pep,"Preparing to restart keeping l=%" PetscInt_FMT " vectors\n",l));

    1256:     /* Update S */
    1257:     PetscCall(DSGetMat(pep->ds,DS_MAT_Q,&MQ));
    1258:     PetscCall(BVMultInPlace(ctx->V,MQ,pep->nconv,k+l));
    1259:     PetscCall(DSRestoreMat(pep->ds,DS_MAT_Q,&MQ));

    1261:     /* Copy last column of S */
    1262:     PetscCall(BVCopyColumn(ctx->V,nv,k+l));
    1263:     PetscCall(BVSetActiveColumns(ctx->V,0,k+l));
    1264:     if (k+l) {
    1265:       PetscCall(DSSetDimensions(pep->ds,k+l,PETSC_DEFAULT,PETSC_DEFAULT));
    1266:       PetscCall(DSGetMatAndColumn(pep->ds,DS_MAT_D,0,&D,&vomega));
    1267:       PetscCall(BVSetSignature(ctx->V,vomega));
    1268:       PetscCall(DSRestoreMatAndColumn(pep->ds,DS_MAT_D,0,&D,&vomega));
    1269:     }

    1271:     if (breakdown && pep->reason == PEP_CONVERGED_ITERATING) {
    1272:       /* stop if breakdown */
    1273:       PetscCall(PetscInfo(pep,"Breakdown TOAR method (it=%" PetscInt_FMT " norm=%g)\n",pep->its,(double)beta));
    1274:       pep->reason = PEP_DIVERGED_BREAKDOWN;
    1275:     }
    1276:     if (pep->reason != PEP_CONVERGED_ITERATING) l--;
    1277:     PetscCall(BVGetActiveColumns(pep->V,NULL,&nq));
    1278:     if (k+l+deg<=nq) {
    1279:       PetscCall(BVSetActiveColumns(ctx->V,pep->nconv,k+l+1));
    1280:       if (!falselock && ctx->lock) PetscCall(BVTensorCompress(ctx->V,k-pep->nconv));
    1281:       else PetscCall(BVTensorCompress(ctx->V,0));
    1282:     }
    1283:     pep->nconv = k;
    1284:     PetscCall(PEPMonitor(pep,pep->its,nconv,pep->eigr,pep->eigi,pep->errest,nv));
    1285:   }
    1286:   sr->itsKs += pep->its;
    1287:   if (pep->nconv>0) {
    1288:     PetscCall(BVSetActiveColumns(ctx->V,0,pep->nconv));
    1289:     PetscCall(BVGetActiveColumns(pep->V,NULL,&nq));
    1290:     PetscCall(BVSetActiveColumns(pep->V,0,nq));
    1291:     if (nq>pep->nconv) {
    1292:       PetscCall(BVTensorCompress(ctx->V,pep->nconv));
    1293:       PetscCall(BVSetActiveColumns(pep->V,0,pep->nconv));
    1294:     }
    1295:     for (j=0;j<pep->nconv;j++) {
    1296:       pep->eigr[j] *= pep->sfactor;
    1297:       pep->eigi[j] *= pep->sfactor;
    1298:     }
    1299:   }
    1300:   PetscCall(PetscInfo(pep,"Finished STOAR: nconv=%" PetscInt_FMT " (deflated=%" PetscInt_FMT ", left=%" PetscInt_FMT ", right=%" PetscInt_FMT ")\n",pep->nconv,sr->ndef0+sr->ndef1,count0-sr->ndef0,count1-sr->ndef1));
    1301:   PetscCall(STScaleShift(pep->st,sinv?1.0/pep->sfactor:pep->sfactor));
    1302:   PetscCall(RGPopScale(pep->rg));

    1304:   PetscCheck(pep->reason!=PEP_DIVERGED_SYMMETRY_LOST || nconv>=sr->ndef0+sr->ndef1,PetscObjectComm((PetscObject)pep),PETSC_ERR_PLIB,"Symmetry lost at sigma=%g",(double)sr->sPres->value);
    1305:   if (pep->reason == PEP_DIVERGED_SYMMETRY_LOST && nconv==sr->ndef0+sr->ndef1) {
    1306:     PetscCheck(++sr->symmlost<=10,PetscObjectComm((PetscObject)pep),PETSC_ERR_PLIB,"Symmetry lost at sigma=%g",(double)sr->sPres->value);
    1307:   } else sr->symmlost = 0;

    1309:   PetscCall(DSTruncate(pep->ds,pep->nconv,PETSC_TRUE));
    1310:   PetscCall(PetscFree(back));
    1311:   PetscFunctionReturn(PETSC_SUCCESS);
    1312: }

    1314: #define SWAP(a,b,t) do {t=a;a=b;b=t;} while (0)

    1316: static PetscErrorCode PEPQSliceGetInertias(PEP pep,PetscInt *n,PetscReal **shifts,PetscInt **inertias)
    1317: {
    1318:   PEP_STOAR      *ctx=(PEP_STOAR*)pep->data;
    1319:   PEP_SR          sr=ctx->sr;
    1320:   PetscInt        i=0,j,tmpi;
    1321:   PetscReal       v,tmpr;
    1322:   PEP_shift       s;

    1324:   PetscFunctionBegin;
    1325:   PetscCheck(pep->state,PetscObjectComm((PetscObject)pep),PETSC_ERR_ARG_WRONGSTATE,"Must call PEPSetUp() first");
    1326:   PetscCheck(sr,PetscObjectComm((PetscObject)pep),PETSC_ERR_ARG_WRONGSTATE,"Only available in interval computations, see PEPSetInterval()");
    1327:   if (!sr->s0) {  /* PEPSolve not called yet */
    1328:     *n = 2;
    1329:   } else {
    1330:     *n = 1;
    1331:     s = sr->s0;
    1332:     while (s) {
    1333:       (*n)++;
    1334:       s = s->neighb[1];
    1335:     }
    1336:   }
    1337:   PetscCall(PetscMalloc1(*n,shifts));
    1338:   PetscCall(PetscMalloc1(*n,inertias));
    1339:   if (!sr->s0) {  /* PEPSolve not called yet */
    1340:     (*shifts)[0]   = sr->int0;
    1341:     (*shifts)[1]   = sr->int1;
    1342:     (*inertias)[0] = sr->inertia0;
    1343:     (*inertias)[1] = sr->inertia1;
    1344:   } else {
    1345:     s = sr->s0;
    1346:     while (s) {
    1347:       (*shifts)[i]     = s->value;
    1348:       (*inertias)[i++] = s->inertia;
    1349:       s = s->neighb[1];
    1350:     }
    1351:     (*shifts)[i]   = sr->int1;
    1352:     (*inertias)[i] = sr->inertia1;
    1353:   }
    1354:   /* remove possible duplicate in last position */
    1355:   if ((*shifts)[(*n)-1]==(*shifts)[(*n)-2]) (*n)--;
    1356:   /* sort result */
    1357:   for (i=0;i<*n;i++) {
    1358:     v = (*shifts)[i];
    1359:     for (j=i+1;j<*n;j++) {
    1360:       if (v > (*shifts)[j]) {
    1361:         SWAP((*shifts)[i],(*shifts)[j],tmpr);
    1362:         SWAP((*inertias)[i],(*inertias)[j],tmpi);
    1363:         v = (*shifts)[i];
    1364:       }
    1365:     }
    1366:   }
    1367:   PetscFunctionReturn(PETSC_SUCCESS);
    1368: }

    1370: PetscErrorCode PEPSolve_STOAR_QSlice(PEP pep)
    1371: {
    1372:   PetscInt       i,j,ti,deg=pep->nmat-1;
    1373:   PetscReal      newS;
    1374:   PEP_STOAR      *ctx=(PEP_STOAR*)pep->data;
    1375:   PEP_SR         sr=ctx->sr;
    1376:   Mat            S,B;
    1377:   PetscScalar    *pS;

    1379:   PetscFunctionBegin;
    1380:   PetscCall(PetscCitationsRegister(citation,&cited));

    1382:   /* Only with eigenvalues present in the interval ...*/
    1383:   if (sr->numEigs==0) {
    1384:     pep->reason = PEP_CONVERGED_TOL;
    1385:     PetscFunctionReturn(PETSC_SUCCESS);
    1386:   }

    1388:   /* Inner product matrix */
    1389:   PetscCall(PEPSTOARSetUpInnerMatrix(pep,&B));

    1391:   /* Array of pending shifts */
    1392:   sr->maxPend = 100; /* Initial size */
    1393:   sr->nPend = 0;
    1394:   PetscCall(PetscMalloc1(sr->maxPend,&sr->pending));
    1395:   PetscCall(PEPCreateShift(pep,sr->int0,NULL,NULL));
    1396:   /* extract first shift */
    1397:   sr->sPrev = NULL;
    1398:   sr->sPres = sr->pending[--sr->nPend];
    1399:   sr->sPres->inertia = sr->inertia0;
    1400:   pep->target = sr->sPres->value;
    1401:   sr->s0 = sr->sPres;
    1402:   sr->indexEig = 0;

    1404:   for (i=0;i<sr->numEigs;i++) {
    1405:     sr->eigr[i]   = 0.0;
    1406:     sr->eigi[i]   = 0.0;
    1407:     sr->errest[i] = 0.0;
    1408:     sr->perm[i]   = i;
    1409:   }
    1410:   /* Vectors for deflation */
    1411:   PetscCall(PetscMalloc2(sr->numEigs,&sr->idxDef0,sr->numEigs,&sr->idxDef1));
    1412:   sr->indexEig = 0;
    1413:   while (sr->sPres) {
    1414:     /* Search for deflation */
    1415:     PetscCall(PEPLookForDeflation(pep));
    1416:     /* KrylovSchur */
    1417:     PetscCall(PEPSTOAR_QSlice(pep,B));

    1419:     PetscCall(PEPStoreEigenpairs(pep));
    1420:     /* Select new shift */
    1421:     if (!sr->sPres->comp[1]) {
    1422:       PetscCall(PEPGetNewShiftValue(pep,1,&newS));
    1423:       PetscCall(PEPCreateShift(pep,newS,sr->sPres,sr->sPres->neighb[1]));
    1424:     }
    1425:     if (!sr->sPres->comp[0]) {
    1426:       /* Completing earlier interval */
    1427:       PetscCall(PEPGetNewShiftValue(pep,0,&newS));
    1428:       PetscCall(PEPCreateShift(pep,newS,sr->sPres->neighb[0],sr->sPres));
    1429:     }
    1430:     /* Preparing for a new search of values */
    1431:     PetscCall(PEPExtractShift(pep));
    1432:   }

    1434:   /* Updating pep values prior to exit */
    1435:   PetscCall(PetscFree2(sr->idxDef0,sr->idxDef1));
    1436:   PetscCall(PetscFree(sr->pending));
    1437:   pep->nconv  = sr->indexEig;
    1438:   pep->reason = PEP_CONVERGED_TOL;
    1439:   pep->its    = sr->itsKs;
    1440:   pep->nev    = sr->indexEig;
    1441:   PetscCall(MatCreateSeqDense(PETSC_COMM_SELF,pep->nconv,pep->nconv,NULL,&S));
    1442:   PetscCall(MatDenseGetArray(S,&pS));
    1443:   for (i=0;i<pep->nconv;i++) {
    1444:     for (j=0;j<sr->qinfo[i].nq;j++) pS[i*pep->nconv+sr->qinfo[i].q[j]] = *(sr->S+i*sr->ld*deg+j);
    1445:   }
    1446:   PetscCall(MatDenseRestoreArray(S,&pS));
    1447:   PetscCall(BVSetActiveColumns(sr->V,0,pep->nconv));
    1448:   PetscCall(BVMultInPlace(sr->V,S,0,pep->nconv));
    1449:   PetscCall(MatDestroy(&S));
    1450:   PetscCall(BVDestroy(&pep->V));
    1451:   pep->V = sr->V;
    1452:   PetscCall(PetscFree4(pep->eigr,pep->eigi,pep->errest,pep->perm));
    1453:   pep->eigr   = sr->eigr;
    1454:   pep->eigi   = sr->eigi;
    1455:   pep->perm   = sr->perm;
    1456:   pep->errest = sr->errest;
    1457:   if (sr->dir<0) {
    1458:     for (i=0;i<pep->nconv/2;i++) {
    1459:       ti = sr->perm[i]; sr->perm[i] = sr->perm[pep->nconv-1-i]; sr->perm[pep->nconv-1-i] = ti;
    1460:     }
    1461:   }
    1462:   PetscCall(PetscFree(ctx->inertias));
    1463:   PetscCall(PetscFree(ctx->shifts));
    1464:   PetscCall(MatDestroy(&B));
    1465:   PetscCall(PEPQSliceGetInertias(pep,&ctx->nshifts,&ctx->shifts,&ctx->inertias));
    1466:   PetscFunctionReturn(PETSC_SUCCESS);
    1467: }