Date: 2026-05-01T10:02:33+00:00

Actual source code: ks-slice.c

      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 eigensolver: "krylovschur"

     13:    Method: Krylov-Schur with spectrum slicing for symmetric eigenproblems

     15:    References:

     17:        [1] R.G. Grimes et al., "A shifted block Lanczos algorithm for
     18:            solving sparse symmetric generalized eigenproblems", SIAM J.
     19:            Matrix Anal. Appl. 15(1):228-272, 1994.

     21:        [2] C. Campos and J.E. Roman, "Spectrum slicing strategies based
     22:            on restarted Lanczos methods", Numer. Algor. 60(2):279-295,
     23:            2012.
     24: */

     26: #include <slepc/private/epsimpl.h>
     27: #include "krylovschur.h"

     29: static PetscBool  cited = PETSC_FALSE;
     30: static const char citation[] =
     31:   "@Article{slepc-slice,\n"
     32:   "   author = \"C. Campos and J. E. Roman\",\n"
     33:   "   title = \"Strategies for spectrum slicing based on restarted {Lanczos} methods\",\n"
     34:   "   journal = \"Numer. Algorithms\",\n"
     35:   "   volume = \"60\",\n"
     36:   "   number = \"2\",\n"
     37:   "   pages = \"279--295\",\n"
     38:   "   year = \"2012,\"\n"
     39:   "   doi = \"https://doi.org/10.1007/s11075-012-9564-z\"\n"
     40:   "}\n";

     42: #define SLICE_PTOL PETSC_SQRT_MACHINE_EPSILON

     44: static PetscErrorCode EPSSliceResetSR(EPS eps)
     45: {
     46:   EPS_KRYLOVSCHUR *ctx=(EPS_KRYLOVSCHUR*)eps->data;
     47:   EPS_SR          sr=ctx->sr;
     48:   EPS_shift       s;

     50:   PetscFunctionBegin;
     51:   if (sr) {
     52:     if (ctx->npart>1) {
     53:       PetscCall(BVDestroy(&sr->V));
     54:       PetscCall(PetscFree4(sr->eigr,sr->eigi,sr->errest,sr->perm));
     55:     }
     56:     /* Reviewing list of shifts to free memory */
     57:     s = sr->s0;
     58:     if (s) {
     59:       while (s->neighb[1]) {
     60:         s = s->neighb[1];
     61:         PetscCall(PetscFree(s->neighb[0]));
     62:       }
     63:       PetscCall(PetscFree(s));
     64:     }
     65:     PetscCall(PetscFree(sr));
     66:   }
     67:   ctx->sr = NULL;
     68:   PetscFunctionReturn(PETSC_SUCCESS);
     69: }

     71: PetscErrorCode EPSReset_KrylovSchur_Slice(EPS eps)
     72: {
     73:   EPS_KRYLOVSCHUR *ctx=(EPS_KRYLOVSCHUR*)eps->data;

     75:   PetscFunctionBegin;
     76:   if (!ctx->global) PetscFunctionReturn(PETSC_SUCCESS);
     77:   /* Reset auxiliary EPS */
     78:   PetscCall(EPSSliceResetSR(ctx->eps));
     79:   PetscCall(EPSReset(ctx->eps));
     80:   PetscCall(EPSSliceResetSR(eps));
     81:   PetscCall(PetscFree(ctx->inertias));
     82:   PetscCall(PetscFree(ctx->shifts));
     83:   PetscFunctionReturn(PETSC_SUCCESS);
     84: }

     86: PetscErrorCode EPSDestroy_KrylovSchur_Slice(EPS eps)
     87: {
     88:   EPS_KRYLOVSCHUR *ctx=(EPS_KRYLOVSCHUR*)eps->data;

     90:   PetscFunctionBegin;
     91:   if (!ctx->global) PetscFunctionReturn(PETSC_SUCCESS);
     92:   /* Destroy auxiliary EPS */
     93:   PetscCall(EPSReset_KrylovSchur_Slice(eps));
     94:   PetscCall(EPSDestroy(&ctx->eps));
     95:   if (ctx->npart>1) {
     96:     PetscCall(PetscSubcommDestroy(&ctx->subc));
     97:     if (ctx->commset) {
     98:       PetscCallMPI(MPI_Comm_free(&ctx->commrank));
     99:       ctx->commset = PETSC_FALSE;
    100:     }
    101:     PetscCall(ISDestroy(&ctx->isrow));
    102:     PetscCall(ISDestroy(&ctx->iscol));
    103:     PetscCall(MatDestroyMatrices(1,&ctx->submata));
    104:     PetscCall(MatDestroyMatrices(1,&ctx->submatb));
    105:   }
    106:   PetscCall(PetscFree(ctx->subintervals));
    107:   PetscCall(PetscFree(ctx->nconv_loc));
    108:   PetscFunctionReturn(PETSC_SUCCESS);
    109: }

    111: /*
    112:   EPSSliceAllocateSolution - Allocate memory storage for common variables such
    113:   as eigenvalues and eigenvectors. The argument extra is used for methods
    114:   that require a working basis slightly larger than ncv.
    115: */
    116: static PetscErrorCode EPSSliceAllocateSolution(EPS eps,PetscInt extra)
    117: {
    118:   EPS_KRYLOVSCHUR    *ctx=(EPS_KRYLOVSCHUR*)eps->data;
    119:   PetscReal          eta;
    120:   PetscInt           k;
    121:   BVType             type;
    122:   BVOrthogType       orthog_type;
    123:   BVOrthogRefineType orthog_ref;
    124:   BVOrthogBlockType  ob_type;
    125:   Mat                matrix;
    126:   Vec                t;
    127:   EPS_SR             sr = ctx->sr;

    129:   PetscFunctionBegin;
    130:   /* allocate space for eigenvalues and friends */
    131:   k = PetscMax(1,sr->numEigs);
    132:   PetscCall(PetscFree4(sr->eigr,sr->eigi,sr->errest,sr->perm));
    133:   PetscCall(PetscMalloc4(k,&sr->eigr,k,&sr->eigi,k,&sr->errest,k,&sr->perm));

    135:   /* allocate sr->V and transfer options from eps->V */
    136:   PetscCall(BVDestroy(&sr->V));
    137:   PetscCall(BVCreate(PetscObjectComm((PetscObject)eps),&sr->V));
    138:   if (!eps->V) PetscCall(EPSGetBV(eps,&eps->V));
    139:   if (!((PetscObject)eps->V)->type_name) PetscCall(BVSetType(sr->V,BVMAT));
    140:   else {
    141:     PetscCall(BVGetType(eps->V,&type));
    142:     PetscCall(BVSetType(sr->V,type));
    143:   }
    144:   PetscCall(STMatCreateVecsEmpty(eps->st,&t,NULL));
    145:   PetscCall(BVSetSizesFromVec(sr->V,t,k));
    146:   PetscCall(VecDestroy(&t));
    147:   PetscCall(EPS_SetInnerProduct(eps));
    148:   PetscCall(BVGetMatrix(eps->V,&matrix,NULL));
    149:   PetscCall(BVSetMatrix(sr->V,matrix,PETSC_FALSE));
    150:   PetscCall(BVGetOrthogonalization(eps->V,&orthog_type,&orthog_ref,&eta,&ob_type));
    151:   PetscCall(BVSetOrthogonalization(sr->V,orthog_type,orthog_ref,eta,ob_type));
    152:   PetscFunctionReturn(PETSC_SUCCESS);
    153: }

    155: static PetscErrorCode EPSSliceGetEPS(EPS eps)
    156: {
    157:   EPS_KRYLOVSCHUR    *ctx=(EPS_KRYLOVSCHUR*)eps->data,*ctx_local;
    158:   BV                 V;
    159:   BVType             type;
    160:   PetscReal          eta;
    161:   BVOrthogType       orthog_type;
    162:   BVOrthogRefineType orthog_ref;
    163:   BVOrthogBlockType  ob_type;
    164:   PetscInt           i;
    165:   PetscReal          h,a,b;
    166:   PetscRandom        rand;
    167:   EPS_SR             sr=ctx->sr;

    169:   PetscFunctionBegin;
    170:   if (!ctx->eps) PetscCall(EPSKrylovSchurGetChildEPS(eps,&ctx->eps));

    172:   /* Determine subintervals */
    173:   if (ctx->npart==1) {
    174:     a = eps->inta; b = eps->intb;
    175:   } else {
    176:     if (!ctx->subintset) { /* uniform distribution if no set by user */
    177:       PetscCheck(sr->hasEnd,PetscObjectComm((PetscObject)eps),PETSC_ERR_ARG_WRONG,"Global interval must be bounded for splitting it in uniform subintervals");
    178:       h = (eps->intb-eps->inta)/ctx->npart;
    179:       a = eps->inta+ctx->subc->color*h;
    180:       b = (ctx->subc->color==ctx->npart-1)?eps->intb:eps->inta+(ctx->subc->color+1)*h;
    181:       PetscCall(PetscFree(ctx->subintervals));
    182:       PetscCall(PetscMalloc1(ctx->npart+1,&ctx->subintervals));
    183:       for (i=0;i<ctx->npart;i++) ctx->subintervals[i] = eps->inta+h*i;
    184:       ctx->subintervals[ctx->npart] = eps->intb;
    185:     } else {
    186:       a = ctx->subintervals[ctx->subc->color];
    187:       b = ctx->subintervals[ctx->subc->color+1];
    188:     }
    189:   }
    190:   PetscCall(EPSSetInterval(ctx->eps,a,b));
    191:   PetscCall(EPSSetConvergenceTest(ctx->eps,eps->conv));
    192:   PetscCall(EPSSetDimensions(ctx->eps,ctx->nev,ctx->ncv,ctx->mpd));
    193:   PetscCall(EPSKrylovSchurSetLocking(ctx->eps,ctx->lock));

    195:   ctx_local = (EPS_KRYLOVSCHUR*)ctx->eps->data;
    196:   ctx_local->detect = ctx->detect;

    198:   /* transfer options from eps->V */
    199:   PetscCall(EPSGetBV(ctx->eps,&V));
    200:   PetscCall(BVGetRandomContext(V,&rand));  /* make sure the random context is available when duplicating */
    201:   if (!eps->V) PetscCall(EPSGetBV(eps,&eps->V));
    202:   if (!((PetscObject)eps->V)->type_name) PetscCall(BVSetType(V,BVMAT));
    203:   else {
    204:     PetscCall(BVGetType(eps->V,&type));
    205:     PetscCall(BVSetType(V,type));
    206:   }
    207:   PetscCall(BVGetOrthogonalization(eps->V,&orthog_type,&orthog_ref,&eta,&ob_type));
    208:   PetscCall(BVSetOrthogonalization(V,orthog_type,orthog_ref,eta,ob_type));

    210:   ctx->eps->which = eps->which;
    211:   ctx->eps->max_it = eps->max_it;
    212:   ctx->eps->tol = eps->tol;
    213:   ctx->eps->purify = eps->purify;
    214:   if (eps->tol==(PetscReal)PETSC_DETERMINE) eps->tol = SLEPC_DEFAULT_TOL;
    215:   PetscCall(EPSSetProblemType(ctx->eps,eps->problem_type));
    216:   PetscCall(EPSSetUp(ctx->eps));
    217:   ctx->eps->nconv = 0;
    218:   ctx->eps->its   = 0;
    219:   for (i=0;i<ctx->eps->ncv;i++) {
    220:     ctx->eps->eigr[i]   = 0.0;
    221:     ctx->eps->eigi[i]   = 0.0;
    222:     ctx->eps->errest[i] = 0.0;
    223:   }
    224:   PetscFunctionReturn(PETSC_SUCCESS);
    225: }

    227: static PetscErrorCode EPSSliceGetInertia(EPS eps,PetscReal shift,PetscInt *inertia,PetscInt *zeros)
    228: {
    229:   KSP            ksp,kspr;
    230:   PC             pc;
    231:   Mat            F;
    232:   PetscReal      nzshift=shift;
    233:   PetscBool      flg;

    235:   PetscFunctionBegin;
    236:   if (shift >= PETSC_MAX_REAL) { /* Right-open interval */
    237:     if (inertia) *inertia = eps->n;
    238:   } else if (shift <= PETSC_MIN_REAL) {
    239:     if (inertia) *inertia = 0;
    240:     if (zeros) *zeros = 0;
    241:   } else {
    242:     /* If the shift is zero, perturb it to a very small positive value.
    243:        The goal is that the nonzero pattern is the same in all cases and reuse
    244:        the symbolic factorizations */
    245:     nzshift = (shift==0.0)? 10.0/PETSC_MAX_REAL: shift;
    246:     PetscCall(STSetShift(eps->st,nzshift));
    247:     PetscCall(STGetKSP(eps->st,&ksp));
    248:     PetscCall(KSPGetPC(ksp,&pc));
    249:     PetscCall(PetscObjectTypeCompare((PetscObject)pc,PCREDUNDANT,&flg));
    250:     if (flg) {
    251:       PetscCall(PCRedundantGetKSP(pc,&kspr));
    252:       PetscCall(KSPGetPC(kspr,&pc));
    253:     }
    254:     PetscCall(PetscObjectTypeCompare((PetscObject)pc,PCTELESCOPE,&flg));
    255:     if (flg) {
    256:       PetscCall(PCTelescopeGetKSP(pc,&kspr));
    257:       PetscCall(KSPGetPC(kspr,&pc));
    258:     }
    259:     PetscCall(PCFactorGetMatrix(pc,&F));
    260:     PetscCall(PCSetUp(pc));
    261:     PetscCall(MatGetInertia(F,inertia,zeros,NULL));
    262:   }
    263:   if (inertia) PetscCall(PetscInfo(eps,"Computed inertia at shift %g: %" PetscInt_FMT "\n",(double)nzshift,*inertia));
    264:   PetscFunctionReturn(PETSC_SUCCESS);
    265: }

    267: /*
    268:    Dummy backtransform operation
    269:  */
    270: static PetscErrorCode EPSBackTransform_Skip(EPS eps)
    271: {
    272:   PetscFunctionBegin;
    273:   PetscFunctionReturn(PETSC_SUCCESS);
    274: }

    276: PetscErrorCode EPSSetUp_KrylovSchur_Slice(EPS eps)
    277: {
    278:   EPS_KRYLOVSCHUR *ctx = (EPS_KRYLOVSCHUR*)eps->data,*ctx_glob;
    279:   EPS_SR          sr,sr_loc,sr_glob;
    280:   PetscInt        nEigs,dssz=1,i,zeros=0,off=0,method,hiteig=0;
    281:   PetscMPIInt     nproc,rank=0,aux;
    282:   PetscReal       r;
    283:   MPI_Request     req;
    284:   Mat             A,B=NULL;
    285:   DSParallelType  ptype;
    286:   MPI_Comm        child;

    288:   PetscFunctionBegin;
    289:   if (eps->nev==0) eps->nev = 1;
    290:   if (ctx->global) {
    291:     EPSCheckHermitianDefiniteCondition(eps,PETSC_TRUE," with spectrum slicing");
    292:     EPSCheckSinvertCayleyCondition(eps,PETSC_TRUE," with spectrum slicing");
    293:     PetscCheck(eps->inta!=eps->intb,PetscObjectComm((PetscObject)eps),PETSC_ERR_SUP,"This solver does not support computing all eigenvalues unless you provide a computational interval with EPSSetInterval()");
    294:     PetscCheck(eps->intb<PETSC_MAX_REAL || eps->inta>PETSC_MIN_REAL,PetscObjectComm((PetscObject)eps),PETSC_ERR_ARG_WRONG,"The defined computational interval should have at least one of their sides bounded");
    295:     PetscCheck(eps->nds==0,PetscObjectComm((PetscObject)eps),PETSC_ERR_SUP,"Spectrum slicing not supported in combination with deflation space");
    296:     EPSCheckUnsupportedCondition(eps,EPS_FEATURE_ARBITRARY | EPS_FEATURE_REGION | EPS_FEATURE_STOPPING,PETSC_TRUE," with spectrum slicing");
    297:     EPSCheckIgnoredCondition(eps,EPS_FEATURE_BALANCE,PETSC_TRUE," with spectrum slicing");
    298:     if (eps->tol==(PetscReal)PETSC_DETERMINE) {
    299: #if defined(PETSC_USE_REAL_SINGLE)
    300:       eps->tol = SLEPC_DEFAULT_TOL;
    301: #else
    302:       /* use tighter tolerance */
    303:       eps->tol = SLEPC_DEFAULT_TOL*1e-2;
    304: #endif
    305:     }
    306:     if (eps->max_it==PETSC_DETERMINE) eps->max_it = 100;
    307:     if (ctx->nev==1) ctx->nev = PetscMin(40,eps->n);  /* nev not set, use default value */
    308:     PetscCheck(eps->n<=10 || ctx->nev>=10,PetscObjectComm((PetscObject)eps),PETSC_ERR_ARG_WRONG,"nev cannot be less than 10 in spectrum slicing runs");
    309:   }
    310:   eps->ops->backtransform = EPSBackTransform_Skip;

    312:   /* create spectrum slicing context and initialize it */
    313:   PetscCall(EPSSliceResetSR(eps));
    314:   PetscCall(PetscNew(&sr));
    315:   ctx->sr = sr;
    316:   sr->itsKs = 0;
    317:   sr->nleap = 0;
    318:   sr->nMAXCompl = eps->nev/4;
    319:   sr->iterCompl = eps->max_it/4;
    320:   sr->sPres = NULL;
    321:   sr->nS = 0;

    323:   if (ctx->npart==1 || ctx->global) {
    324:     /* check presence of ends and finding direction */
    325:     if ((eps->inta > PETSC_MIN_REAL && !(ctx->subintervals && ctx->subintervals[0]==ctx->subintervals[1])) || eps->intb >= PETSC_MAX_REAL) {
    326:       sr->int0 = eps->inta;
    327:       sr->int1 = eps->intb;
    328:       sr->dir = 1;
    329:       if (eps->intb >= PETSC_MAX_REAL) { /* Right-open interval */
    330:         sr->hasEnd = PETSC_FALSE;
    331:       } else sr->hasEnd = PETSC_TRUE;
    332:     } else {
    333:       sr->int0 = eps->intb;
    334:       sr->int1 = eps->inta;
    335:       sr->dir = -1;
    336:       sr->hasEnd = PetscNot(eps->inta <= PETSC_MIN_REAL);
    337:     }
    338:   }
    339:   if (ctx->global) {
    340:     PetscCall(EPSSetDimensions_Default(eps,&ctx->nev,&ctx->ncv,&ctx->mpd));
    341:     /* create subintervals and initialize auxiliary eps for slicing runs */
    342:     PetscCall(EPSKrylovSchurGetChildEPS(eps,&ctx->eps));
    343:     /* prevent computation of factorization in global eps */
    344:     PetscCall(STSetTransform(eps->st,PETSC_FALSE));
    345:     PetscCall(EPSSliceGetEPS(eps));
    346:     sr_loc = ((EPS_KRYLOVSCHUR*)ctx->eps->data)->sr;
    347:     if (ctx->npart>1) {
    348:       PetscCall(PetscSubcommGetChild(ctx->subc,&child));
    349:       if ((sr->dir>0&&ctx->subc->color==0)||(sr->dir<0&&ctx->subc->color==ctx->npart-1)) sr->inertia0 = sr_loc->inertia0;
    350:       PetscCallMPI(MPI_Comm_rank(child,&rank));
    351:       if (!rank) {
    352:         PetscCall(PetscMPIIntCast((sr->dir>0)?0:ctx->npart-1,&aux));
    353:         PetscCallMPI(MPI_Bcast(&sr->inertia0,1,MPIU_INT,aux,ctx->commrank));
    354:       }
    355:       PetscCallMPI(MPI_Bcast(&sr->inertia0,1,MPIU_INT,0,child));
    356:       PetscCall(PetscFree(ctx->nconv_loc));
    357:       PetscCall(PetscMalloc1(ctx->npart,&ctx->nconv_loc));
    358:       PetscCallMPI(MPI_Comm_size(((PetscObject)eps)->comm,&nproc));
    359:       if (sr->dir<0) off = 1;
    360:       if (nproc%ctx->npart==0) { /* subcommunicators with the same size */
    361:         PetscCall(PetscMPIIntCast(sr_loc->numEigs,&aux));
    362:         PetscCallMPI(MPI_Allgather(&aux,1,MPI_INT,ctx->nconv_loc,1,MPI_INT,ctx->commrank));
    363:         PetscCallMPI(MPI_Allgather(sr_loc->dir==sr->dir?&sr_loc->int0:&sr_loc->int1,1,MPIU_REAL,ctx->subintervals+off,1,MPIU_REAL,ctx->commrank));
    364:       } else {
    365:         PetscCallMPI(MPI_Comm_rank(child,&rank));
    366:         if (!rank) {
    367:           PetscCall(PetscMPIIntCast(sr_loc->numEigs,&aux));
    368:           PetscCallMPI(MPI_Allgather(&aux,1,MPI_INT,ctx->nconv_loc,1,MPI_INT,ctx->commrank));
    369:           PetscCallMPI(MPI_Allgather(sr_loc->dir==sr->dir?&sr_loc->int0:&sr_loc->int1,1,MPIU_REAL,ctx->subintervals+off,1,MPIU_REAL,ctx->commrank));
    370:         }
    371:         PetscCall(PetscMPIIntCast(ctx->npart,&aux));
    372:         PetscCallMPI(MPI_Bcast(ctx->nconv_loc,aux,MPI_INT,0,child));
    373:         PetscCallMPI(MPI_Bcast(ctx->subintervals+off,aux,MPIU_REAL,0,child));
    374:       }
    375:       nEigs = 0;
    376:       for (i=0;i<ctx->npart;i++) nEigs += ctx->nconv_loc[i];
    377:     } else {
    378:       nEigs = sr_loc->numEigs;
    379:       sr->inertia0 = sr_loc->inertia0;
    380:       sr->dir = sr_loc->dir;
    381:     }
    382:     sr->inertia1 = sr->inertia0+sr->dir*nEigs;
    383:     sr->numEigs = nEigs;
    384:     eps->nev = nEigs;
    385:     eps->ncv = nEigs;
    386:     eps->mpd = nEigs;
    387:   } else {
    388:     ctx_glob = (EPS_KRYLOVSCHUR*)ctx->eps->data;
    389:     sr_glob = ctx_glob->sr;
    390:     if (ctx->npart>1) {
    391:       sr->dir = sr_glob->dir;
    392:       sr->int0 = (sr->dir==1)?eps->inta:eps->intb;
    393:       sr->int1 = (sr->dir==1)?eps->intb:eps->inta;
    394:       if ((sr->dir>0&&ctx->subc->color==ctx->npart-1)||(sr->dir<0&&ctx->subc->color==0)) sr->hasEnd = sr_glob->hasEnd;
    395:       else sr->hasEnd = PETSC_TRUE;
    396:     }
    397:     /* sets first shift */
    398:     PetscCall(STSetShift(eps->st,(sr->int0==0.0)?10.0/PETSC_MAX_REAL:sr->int0));
    399:     PetscCall(STSetUp(eps->st));

    401:     /* compute inertia0 */
    402:     PetscCall(EPSSliceGetInertia(eps,sr->int0,&sr->inertia0,ctx->detect?&zeros:NULL));
    403:     /* undocumented option to control what to do when an eigenvalue is found:
    404:        - error out if it's the endpoint of the user-provided interval (or sub-interval)
    405:        - if it's an endpoint computed internally:
    406:           + if hiteig=0 error out
    407:           + else if hiteig=1 the subgroup that hit the eigenvalue does nothing
    408:           + otherwise the subgroup that hit the eigenvalue perturbs the shift and recomputes inertia
    409:     */
    410:     PetscCall(PetscOptionsGetInt(NULL,NULL,"-eps_krylovschur_hiteigenvalue",&hiteig,NULL));
    411:     if (zeros) { /* error in factorization */
    412:       PetscCheck(sr->int0!=ctx->eps->inta && sr->int0!=ctx->eps->intb,((PetscObject)eps)->comm,PETSC_ERR_USER,"Found singular matrix for the transformed problem in the interval endpoint");
    413:       PetscCheck(!ctx_glob->subintset || hiteig,((PetscObject)eps)->comm,PETSC_ERR_USER,"Found singular matrix for the transformed problem in an interval endpoint defined by user");
    414:       if (hiteig==1) { /* idle subgroup */
    415:         sr->inertia0 = -1;
    416:       } else { /* perturb shift */
    417:         sr->int0 *= (1.0+SLICE_PTOL);
    418:         PetscCall(EPSSliceGetInertia(eps,sr->int0,&sr->inertia0,&zeros));
    419:         PetscCheck(zeros==0,((PetscObject)eps)->comm,PETSC_ERR_CONV_FAILED,"Inertia computation fails in %g",(double)sr->int1);
    420:       }
    421:     }
    422:     if (ctx->npart>1) {
    423:       PetscCall(PetscSubcommGetChild(ctx->subc,&child));
    424:       /* inertia1 is received from neighbour */
    425:       PetscCallMPI(MPI_Comm_rank(child,&rank));
    426:       if (!rank) {
    427:         if (sr->inertia0!=-1 && ((sr->dir>0 && ctx->subc->color>0) || (sr->dir<0 && ctx->subc->color<ctx->npart-1))) { /* send inertia0 to neighbour0 */
    428:           PetscCall(PetscMPIIntCast(ctx->subc->color-sr->dir,&aux));
    429:           PetscCallMPI(MPIU_Isend(&sr->inertia0,1,MPIU_INT,aux,0,ctx->commrank,&req));
    430:           PetscCallMPI(MPIU_Isend(&sr->int0,1,MPIU_REAL,aux,0,ctx->commrank,&req));
    431:         }
    432:         if ((sr->dir>0 && ctx->subc->color<ctx->npart-1)|| (sr->dir<0 && ctx->subc->color>0)) { /* receive inertia1 from neighbour1 */
    433:           PetscCall(PetscMPIIntCast(ctx->subc->color+sr->dir,&aux));
    434:           PetscCallMPI(MPI_Recv(&sr->inertia1,1,MPIU_INT,aux,0,ctx->commrank,MPI_STATUS_IGNORE));
    435:           PetscCallMPI(MPI_Recv(&sr->int1,1,MPIU_REAL,aux,0,ctx->commrank,MPI_STATUS_IGNORE));
    436:         }
    437:         if (sr->inertia0==-1 && !(sr->dir>0 && ctx->subc->color==ctx->npart-1) && !(sr->dir<0 && ctx->subc->color==0)) {
    438:           sr->inertia0 = sr->inertia1; sr->int0 = sr->int1;
    439:           PetscCall(PetscMPIIntCast(ctx->subc->color-sr->dir,&aux));
    440:           PetscCallMPI(MPIU_Isend(&sr->inertia0,1,MPIU_INT,aux,0,ctx->commrank,&req));
    441:           PetscCallMPI(MPIU_Isend(&sr->int0,1,MPIU_REAL,aux,0,ctx->commrank,&req));
    442:         }
    443:       }
    444:       if ((sr->dir>0 && ctx->subc->color<ctx->npart-1)||(sr->dir<0 && ctx->subc->color>0)) {
    445:         PetscCallMPI(MPI_Bcast(&sr->inertia1,1,MPIU_INT,0,child));
    446:         PetscCallMPI(MPI_Bcast(&sr->int1,1,MPIU_REAL,0,child));
    447:       } else sr_glob->inertia1 = sr->inertia1;
    448:     }

    450:     /* last process in eps comm computes inertia1 */
    451:     if (ctx->npart==1 || ((sr->dir>0 && ctx->subc->color==ctx->npart-1) || (sr->dir<0 && ctx->subc->color==0))) {
    452:       PetscCall(EPSSliceGetInertia(eps,sr->int1,&sr->inertia1,ctx->detect?&zeros:NULL));
    453:       PetscCheck(zeros==0,((PetscObject)eps)->comm,PETSC_ERR_USER,"Found singular matrix for the transformed problem in an interval endpoint defined by user");
    454:       if (!rank && sr->inertia0==-1) {
    455:         sr->inertia0 = sr->inertia1; sr->int0 = sr->int1;
    456:         PetscCall(PetscMPIIntCast(ctx->subc->color-sr->dir,&aux));
    457:         PetscCallMPI(MPIU_Isend(&sr->inertia0,1,MPIU_INT,aux,0,ctx->commrank,&req));
    458:         PetscCallMPI(MPIU_Isend(&sr->int0,1,MPIU_REAL,aux,0,ctx->commrank,&req));
    459:       }
    460:       if (sr->hasEnd) {
    461:         sr->dir = -sr->dir; r = sr->int0; sr->int0 = sr->int1; sr->int1 = r;
    462:         i = sr->inertia0; sr->inertia0 = sr->inertia1; sr->inertia1 = i;
    463:       }
    464:     }

    466:     /* number of eigenvalues in interval */
    467:     sr->numEigs = (sr->dir)*(sr->inertia1 - sr->inertia0);
    468:     if (ctx->npart>1) {
    469:       /* memory allocate for subinterval eigenpairs */
    470:       PetscCall(EPSSliceAllocateSolution(eps,1));
    471:     }
    472:     dssz = eps->ncv+1;
    473:     PetscCall(DSGetParallel(ctx->eps->ds,&ptype));
    474:     PetscCall(DSSetParallel(eps->ds,ptype));
    475:     PetscCall(DSGetMethod(ctx->eps->ds,&method));
    476:     PetscCall(DSSetMethod(eps->ds,method));
    477:   }
    478:   PetscCall(DSSetType(eps->ds,DSHEP));
    479:   PetscCall(DSSetCompact(eps->ds,PETSC_TRUE));
    480:   PetscCall(DSAllocate(eps->ds,dssz));
    481:   /* keep state of subcomm matrices to check that the user does not modify them */
    482:   PetscCall(EPSGetOperators(eps,&A,&B));
    483:   PetscCall(MatGetState(A,&ctx->Astate));
    484:   PetscCall(PetscObjectGetId((PetscObject)A,&ctx->Aid));
    485:   if (B) {
    486:     PetscCall(MatGetState(B,&ctx->Bstate));
    487:     PetscCall(PetscObjectGetId((PetscObject)B,&ctx->Bid));
    488:   } else {
    489:     ctx->Bstate=0;
    490:     ctx->Bid=0;
    491:   }
    492:   PetscFunctionReturn(PETSC_SUCCESS);
    493: }

    495: static PetscErrorCode EPSSliceGatherEigenVectors(EPS eps)
    496: {
    497:   Vec             v,vg,v_loc;
    498:   IS              is1,is2;
    499:   VecScatter      vec_sc;
    500:   EPS_KRYLOVSCHUR *ctx=(EPS_KRYLOVSCHUR*)eps->data;
    501:   PetscInt        nloc,m0,n0,i,si,idx,*idx1,*idx2,j;
    502:   PetscScalar     *array;
    503:   EPS_SR          sr_loc;
    504:   BV              V_loc;

    506:   PetscFunctionBegin;
    507:   sr_loc = ((EPS_KRYLOVSCHUR*)ctx->eps->data)->sr;
    508:   V_loc = sr_loc->V;

    510:   /* Gather parallel eigenvectors */
    511:   PetscCall(BVGetColumn(eps->V,0,&v));
    512:   PetscCall(VecGetOwnershipRange(v,&n0,&m0));
    513:   PetscCall(BVRestoreColumn(eps->V,0,&v));
    514:   PetscCall(BVGetColumn(ctx->eps->V,0,&v));
    515:   PetscCall(VecGetLocalSize(v,&nloc));
    516:   PetscCall(BVRestoreColumn(ctx->eps->V,0,&v));
    517:   PetscCall(PetscMalloc2(m0-n0,&idx1,m0-n0,&idx2));
    518:   PetscCall(VecCreateMPI(PetscObjectComm((PetscObject)eps),nloc,PETSC_DECIDE,&vg));
    519:   idx = -1;
    520:   for (si=0;si<ctx->npart;si++) {
    521:     j = 0;
    522:     for (i=n0;i<m0;i++) {
    523:       idx1[j]   = i;
    524:       idx2[j++] = i+eps->n*si;
    525:     }
    526:     PetscCall(ISCreateGeneral(PetscObjectComm((PetscObject)eps),(m0-n0),idx1,PETSC_COPY_VALUES,&is1));
    527:     PetscCall(ISCreateGeneral(PetscObjectComm((PetscObject)eps),(m0-n0),idx2,PETSC_COPY_VALUES,&is2));
    528:     PetscCall(BVGetColumn(eps->V,0,&v));
    529:     PetscCall(VecScatterCreate(v,is1,vg,is2,&vec_sc));
    530:     PetscCall(BVRestoreColumn(eps->V,0,&v));
    531:     PetscCall(ISDestroy(&is1));
    532:     PetscCall(ISDestroy(&is2));
    533:     for (i=0;i<ctx->nconv_loc[si];i++) {
    534:       PetscCall(BVGetColumn(eps->V,++idx,&v));
    535:       if (ctx->subc->color==si) {
    536:         PetscCall(BVGetColumn(V_loc,i,&v_loc));
    537:         PetscCall(VecGetArray(v_loc,&array));
    538:         PetscCall(VecPlaceArray(vg,array));
    539:       }
    540:       PetscCall(VecScatterBegin(vec_sc,vg,v,INSERT_VALUES,SCATTER_REVERSE));
    541:       PetscCall(VecScatterEnd(vec_sc,vg,v,INSERT_VALUES,SCATTER_REVERSE));
    542:       if (ctx->subc->color==si) {
    543:         PetscCall(VecResetArray(vg));
    544:         PetscCall(VecRestoreArray(v_loc,&array));
    545:         PetscCall(BVRestoreColumn(V_loc,i,&v_loc));
    546:       }
    547:       PetscCall(BVRestoreColumn(eps->V,idx,&v));
    548:     }
    549:     PetscCall(VecScatterDestroy(&vec_sc));
    550:   }
    551:   PetscCall(PetscFree2(idx1,idx2));
    552:   PetscCall(VecDestroy(&vg));
    553:   PetscFunctionReturn(PETSC_SUCCESS);
    554: }

    556: /*
    557:   EPSComputeVectors_Slice - Recover Eigenvectors from subcomunicators
    558:  */
    559: PetscErrorCode EPSComputeVectors_Slice(EPS eps)
    560: {
    561:   EPS_KRYLOVSCHUR *ctx=(EPS_KRYLOVSCHUR*)eps->data;

    563:   PetscFunctionBegin;
    564:   if (ctx->global && ctx->npart>1) {
    565:     PetscCall(EPSComputeVectors(ctx->eps));
    566:     PetscCall(EPSSliceGatherEigenVectors(eps));
    567:   }
    568:   PetscFunctionReturn(PETSC_SUCCESS);
    569: }

    571: static PetscErrorCode EPSSliceGetInertias(EPS eps,PetscInt *n,PetscReal **shifts,PetscInt **inertias)
    572: {
    573:   EPS_KRYLOVSCHUR *ctx=(EPS_KRYLOVSCHUR*)eps->data;
    574:   PetscInt        i=0,j,tmpi;
    575:   PetscReal       v,tmpr;
    576:   EPS_shift       s;

    578:   PetscFunctionBegin;
    579:   PetscCheck(eps->state,PetscObjectComm((PetscObject)eps),PETSC_ERR_ARG_WRONGSTATE,"Must call EPSSetUp() first");
    580:   PetscCheck(ctx->sr,PetscObjectComm((PetscObject)eps),PETSC_ERR_ARG_WRONGSTATE,"Only available in interval computations, see EPSSetInterval()");
    581:   if (!ctx->sr->s0) {  /* EPSSolve not called yet */
    582:     *n = 2;
    583:   } else {
    584:     *n = 1;
    585:     s = ctx->sr->s0;
    586:     while (s) {
    587:       (*n)++;
    588:       s = s->neighb[1];
    589:     }
    590:   }
    591:   PetscCall(PetscMalloc1(*n,shifts));
    592:   PetscCall(PetscMalloc1(*n,inertias));
    593:   if (!ctx->sr->s0) {  /* EPSSolve not called yet */
    594:     (*shifts)[0]   = ctx->sr->int0;
    595:     (*shifts)[1]   = ctx->sr->int1;
    596:     (*inertias)[0] = ctx->sr->inertia0;
    597:     (*inertias)[1] = ctx->sr->inertia1;
    598:   } else {
    599:     s = ctx->sr->s0;
    600:     while (s) {
    601:       (*shifts)[i]     = s->value;
    602:       (*inertias)[i++] = s->inertia;
    603:       s = s->neighb[1];
    604:     }
    605:     (*shifts)[i]   = ctx->sr->int1;
    606:     (*inertias)[i] = ctx->sr->inertia1;
    607:   }
    608:   /* remove possible duplicate in last position */
    609:   if ((*shifts)[(*n)-1]==(*shifts)[(*n)-2]) (*n)--;
    610:   /* sort result */
    611:   for (i=0;i<*n;i++) {
    612:     v = (*shifts)[i];
    613:     for (j=i+1;j<*n;j++) {
    614:       if (v > (*shifts)[j]) {
    615:         SlepcSwap((*shifts)[i],(*shifts)[j],tmpr);
    616:         SlepcSwap((*inertias)[i],(*inertias)[j],tmpi);
    617:         v = (*shifts)[i];
    618:       }
    619:     }
    620:   }
    621:   PetscFunctionReturn(PETSC_SUCCESS);
    622: }

    624: static PetscErrorCode EPSSliceGatherSolution(EPS eps)
    625: {
    626:   PetscMPIInt     rank,nproc,*disp,*ns_loc,aux;
    627:   EPS_KRYLOVSCHUR *ctx=(EPS_KRYLOVSCHUR*)eps->data;
    628:   PetscInt        i,idx,j,*perm_loc,off=0,*inertias_loc,ns;
    629:   PetscScalar     *eigr_loc;
    630:   EPS_SR          sr_loc;
    631:   PetscReal       *shifts_loc;
    632:   MPI_Comm        child;

    634:   PetscFunctionBegin;
    635:   eps->nconv = 0;
    636:   for (i=0;i<ctx->npart;i++) eps->nconv += ctx->nconv_loc[i];
    637:   sr_loc = ((EPS_KRYLOVSCHUR*)ctx->eps->data)->sr;

    639:   /* Gather the shifts used and the inertias computed */
    640:   PetscCall(EPSSliceGetInertias(ctx->eps,&ns,&shifts_loc,&inertias_loc));
    641:   if (ctx->sr->dir>0 && shifts_loc[ns-1]==sr_loc->int1 && ctx->subc->color<ctx->npart-1) ns--;
    642:   if (ctx->sr->dir<0 && shifts_loc[ns-1]==sr_loc->int0 && ctx->subc->color>0) {
    643:     ns--;
    644:     for (i=0;i<ns;i++) {
    645:       inertias_loc[i] = inertias_loc[i+1];
    646:       shifts_loc[i] = shifts_loc[i+1];
    647:     }
    648:   }
    649:   PetscCall(PetscMalloc1(ctx->npart,&ns_loc));
    650:   PetscCall(PetscSubcommGetChild(ctx->subc,&child));
    651:   PetscCallMPI(MPI_Comm_rank(child,&rank));
    652:   PetscCall(PetscMPIIntCast(ns,&aux));
    653:   if (!rank) PetscCallMPI(MPI_Allgather(&aux,1,MPI_INT,ns_loc,1,MPI_INT,ctx->commrank));
    654:   PetscCall(PetscMPIIntCast(ctx->npart,&aux));
    655:   PetscCallMPI(MPI_Bcast(ns_loc,aux,MPI_INT,0,child));
    656:   ctx->nshifts = 0;
    657:   for (i=0;i<ctx->npart;i++) ctx->nshifts += ns_loc[i];
    658:   PetscCall(PetscFree(ctx->inertias));
    659:   PetscCall(PetscFree(ctx->shifts));
    660:   PetscCall(PetscMalloc1(ctx->nshifts,&ctx->inertias));
    661:   PetscCall(PetscMalloc1(ctx->nshifts,&ctx->shifts));

    663:   /* Gather eigenvalues (same ranks have fully set of eigenvalues)*/
    664:   eigr_loc = sr_loc->eigr;
    665:   perm_loc = sr_loc->perm;
    666:   PetscCallMPI(MPI_Comm_size(((PetscObject)eps)->comm,&nproc));
    667:   PetscCall(PetscMalloc1(ctx->npart,&disp));
    668:   disp[0] = 0;
    669:   for (i=1;i<ctx->npart;i++) disp[i] = disp[i-1]+ctx->nconv_loc[i-1];
    670:   if (nproc%ctx->npart==0) { /* subcommunicators with the same size */
    671:     PetscCall(PetscMPIIntCast(sr_loc->numEigs,&aux));
    672:     PetscCallMPI(MPI_Allgatherv(eigr_loc,aux,MPIU_SCALAR,eps->eigr,ctx->nconv_loc,disp,MPIU_SCALAR,ctx->commrank)); /* eigenvalues */
    673:     PetscCallMPI(MPI_Allgatherv(perm_loc,aux,MPIU_INT,eps->perm,ctx->nconv_loc,disp,MPIU_INT,ctx->commrank)); /* perm */
    674:     for (i=1;i<ctx->npart;i++) disp[i] = disp[i-1]+ns_loc[i-1];
    675:     PetscCall(PetscMPIIntCast(ns,&aux));
    676:     PetscCallMPI(MPI_Allgatherv(shifts_loc,aux,MPIU_REAL,ctx->shifts,ns_loc,disp,MPIU_REAL,ctx->commrank)); /* shifts */
    677:     PetscCallMPI(MPI_Allgatherv(inertias_loc,aux,MPIU_INT,ctx->inertias,ns_loc,disp,MPIU_INT,ctx->commrank)); /* inertias */
    678:     PetscCallMPI(MPIU_Allreduce(&sr_loc->itsKs,&eps->its,1,MPIU_INT,MPI_SUM,ctx->commrank));
    679:   } else { /* subcommunicators with different size */
    680:     if (!rank) {
    681:       PetscCall(PetscMPIIntCast(sr_loc->numEigs,&aux));
    682:       PetscCallMPI(MPI_Allgatherv(eigr_loc,aux,MPIU_SCALAR,eps->eigr,ctx->nconv_loc,disp,MPIU_SCALAR,ctx->commrank)); /* eigenvalues */
    683:       PetscCallMPI(MPI_Allgatherv(perm_loc,aux,MPIU_INT,eps->perm,ctx->nconv_loc,disp,MPIU_INT,ctx->commrank)); /* perm */
    684:       for (i=1;i<ctx->npart;i++) disp[i] = disp[i-1]+ns_loc[i-1];
    685:       PetscCall(PetscMPIIntCast(ns,&aux));
    686:       PetscCallMPI(MPI_Allgatherv(shifts_loc,aux,MPIU_REAL,ctx->shifts,ns_loc,disp,MPIU_REAL,ctx->commrank)); /* shifts */
    687:       PetscCallMPI(MPI_Allgatherv(inertias_loc,aux,MPIU_INT,ctx->inertias,ns_loc,disp,MPIU_INT,ctx->commrank)); /* inertias */
    688:       PetscCallMPI(MPIU_Allreduce(&sr_loc->itsKs,&eps->its,1,MPIU_INT,MPI_SUM,ctx->commrank));
    689:     }
    690:     PetscCall(PetscMPIIntCast(eps->nconv,&aux));
    691:     PetscCallMPI(MPI_Bcast(eps->eigr,aux,MPIU_SCALAR,0,child));
    692:     PetscCallMPI(MPI_Bcast(eps->perm,aux,MPIU_INT,0,child));
    693:     PetscCall(PetscMPIIntCast(ctx->nshifts,&aux));
    694:     PetscCallMPI(MPI_Bcast(ctx->shifts,aux,MPIU_REAL,0,child));
    695:     PetscCallMPI(MPI_Bcast(ctx->inertias,aux,MPIU_INT,0,child));
    696:     PetscCallMPI(MPI_Bcast(&eps->its,1,MPIU_INT,0,child));
    697:   }
    698:   /* Update global array eps->perm */
    699:   idx = ctx->nconv_loc[0];
    700:   for (i=1;i<ctx->npart;i++) {
    701:     off += ctx->nconv_loc[i-1];
    702:     for (j=0;j<ctx->nconv_loc[i];j++) eps->perm[idx++] += off;
    703:   }

    705:   /* Gather parallel eigenvectors */
    706:   PetscCall(PetscFree(ns_loc));
    707:   PetscCall(PetscFree(disp));
    708:   PetscCall(PetscFree(shifts_loc));
    709:   PetscCall(PetscFree(inertias_loc));
    710:   PetscFunctionReturn(PETSC_SUCCESS);
    711: }

    713: /*
    714:    Fills the fields of a shift structure
    715: */
    716: static PetscErrorCode EPSCreateShift(EPS eps,PetscReal val,EPS_shift neighb0,EPS_shift neighb1)
    717: {
    718:   EPS_shift       s,*pending2;
    719:   PetscInt        i;
    720:   EPS_SR          sr;
    721:   EPS_KRYLOVSCHUR *ctx=(EPS_KRYLOVSCHUR*)eps->data;

    723:   PetscFunctionBegin;
    724:   sr = ctx->sr;
    725:   if ((neighb0 && val==neighb0->value) || (neighb1 && val==neighb1->value)) {
    726:     sr->nPend++;
    727:     PetscFunctionReturn(PETSC_SUCCESS);
    728:   }
    729:   PetscCall(PetscNew(&s));
    730:   s->value = val;
    731:   s->neighb[0] = neighb0;
    732:   if (neighb0) neighb0->neighb[1] = s;
    733:   s->neighb[1] = neighb1;
    734:   if (neighb1) neighb1->neighb[0] = s;
    735:   s->comp[0] = PETSC_FALSE;
    736:   s->comp[1] = PETSC_FALSE;
    737:   s->index = -1;
    738:   s->neigs = 0;
    739:   s->nconv[0] = s->nconv[1] = 0;
    740:   s->nsch[0] = s->nsch[1]=0;
    741:   /* Inserts in the stack of pending shifts */
    742:   /* If needed, the array is resized */
    743:   if (sr->nPend >= sr->maxPend) {
    744:     sr->maxPend *= 2;
    745:     PetscCall(PetscMalloc1(sr->maxPend,&pending2));
    746:     for (i=0;i<sr->nPend;i++) pending2[i] = sr->pending[i];
    747:     PetscCall(PetscFree(sr->pending));
    748:     sr->pending = pending2;
    749:   }
    750:   sr->pending[sr->nPend++]=s;
    751:   PetscFunctionReturn(PETSC_SUCCESS);
    752: }

    754: /* Prepare for Rational Krylov update */
    755: static PetscErrorCode EPSPrepareRational(EPS eps)
    756: {
    757:   EPS_KRYLOVSCHUR *ctx=(EPS_KRYLOVSCHUR*)eps->data;
    758:   PetscInt        dir,i,k,ld,nv;
    759:   PetscScalar     *A;
    760:   EPS_SR          sr = ctx->sr;
    761:   Vec             v;

    763:   PetscFunctionBegin;
    764:   PetscCall(DSGetLeadingDimension(eps->ds,&ld));
    765:   dir = (sr->sPres->neighb[0] == sr->sPrev)?1:-1;
    766:   dir*=sr->dir;
    767:   k = 0;
    768:   for (i=0;i<sr->nS;i++) {
    769:     if (dir*PetscRealPart(sr->S[i])>0.0) {
    770:       sr->S[k] = sr->S[i];
    771:       sr->S[sr->nS+k] = sr->S[sr->nS+i];
    772:       PetscCall(BVGetColumn(sr->Vnext,k,&v));
    773:       PetscCall(BVCopyVec(eps->V,eps->nconv+i,v));
    774:       PetscCall(BVRestoreColumn(sr->Vnext,k,&v));
    775:       k++;
    776:       if (k>=sr->nS/2) break;
    777:     }
    778:   }
    779:   /* Copy to DS */
    780:   PetscCall(DSSetCompact(eps->ds,PETSC_FALSE));  /* make sure DS_MAT_A is allocated */
    781:   PetscCall(DSGetArray(eps->ds,DS_MAT_A,&A));
    782:   PetscCall(PetscArrayzero(A,ld*ld));
    783:   for (i=0;i<k;i++) {
    784:     A[i*(1+ld)] = sr->S[i];
    785:     A[k+i*ld] = sr->S[sr->nS+i];
    786:   }
    787:   sr->nS = k;
    788:   PetscCall(DSRestoreArray(eps->ds,DS_MAT_A,&A));
    789:   PetscCall(DSGetDimensions(eps->ds,&nv,NULL,NULL,NULL));
    790:   PetscCall(DSSetDimensions(eps->ds,nv,0,k));
    791:   /* Append u to V */
    792:   PetscCall(BVGetColumn(sr->Vnext,sr->nS,&v));
    793:   PetscCall(BVCopyVec(eps->V,sr->nv,v));
    794:   PetscCall(BVRestoreColumn(sr->Vnext,sr->nS,&v));
    795:   PetscFunctionReturn(PETSC_SUCCESS);
    796: }

    798: /* Provides next shift to be computed */
    799: static PetscErrorCode EPSExtractShift(EPS eps)
    800: {
    801:   PetscInt        iner,zeros=0;
    802:   EPS_KRYLOVSCHUR *ctx=(EPS_KRYLOVSCHUR*)eps->data;
    803:   EPS_SR          sr;
    804:   PetscReal       newShift,diam,ptol;
    805:   EPS_shift       sPres;

    807:   PetscFunctionBegin;
    808:   sr = ctx->sr;
    809:   if (sr->nPend > 0) {
    810:     if (sr->sPres==sr->pending[sr->nPend-1]) {
    811:       eps->reason = EPS_CONVERGED_ITERATING;
    812:       eps->its = 0;
    813:       sr->nPend--;
    814:       sr->sPres->rep = PETSC_TRUE;
    815:       PetscFunctionReturn(PETSC_SUCCESS);
    816:     }
    817:     sr->sPrev = sr->sPres;
    818:     sr->sPres = sr->pending[--sr->nPend];
    819:     sPres = sr->sPres;
    820:     PetscCall(EPSSliceGetInertia(eps,sPres->value,&iner,ctx->detect?&zeros:NULL));
    821:     if (zeros) {
    822:       diam = PetscMin(PetscAbsReal(sPres->neighb[0]->value-sPres->value),PetscAbsReal(sPres->neighb[1]->value-sPres->value));
    823:       ptol = PetscMin(SLICE_PTOL,diam/2);
    824:       newShift = sPres->value*(1.0+ptol);
    825:       if (sr->dir*(sPres->neighb[0] && newShift-sPres->neighb[0]->value) < 0) newShift = (sPres->value+sPres->neighb[0]->value)/2;
    826:       else if (sPres->neighb[1] && sr->dir*(sPres->neighb[1]->value-newShift) < 0) newShift = (sPres->value+sPres->neighb[1]->value)/2;
    827:       PetscCall(EPSSliceGetInertia(eps,newShift,&iner,&zeros));
    828:       PetscCheck(zeros==0,((PetscObject)eps)->comm,PETSC_ERR_CONV_FAILED,"Inertia computation fails in %g",(double)newShift);
    829:       sPres->value = newShift;
    830:     }
    831:     sr->sPres->inertia = iner;
    832:     eps->target = sr->sPres->value;
    833:     eps->reason = EPS_CONVERGED_ITERATING;
    834:     eps->its = 0;
    835:   } else sr->sPres = NULL;
    836:   PetscFunctionReturn(PETSC_SUCCESS);
    837: }

    839: /*
    840:    Symmetric KrylovSchur adapted to spectrum slicing:
    841:    Allows searching an specific amount of eigenvalues in the subintervals left and right.
    842:    Returns whether the search has succeeded
    843: */
    844: static PetscErrorCode EPSKrylovSchur_Slice(EPS eps)
    845: {
    846:   EPS_KRYLOVSCHUR *ctx=(EPS_KRYLOVSCHUR*)eps->data;
    847:   PetscInt        i,k,l,ld,nv,*iwork,j,count0,count1,iterCompl=0,n0,n1;
    848:   Mat             U,Op,T;
    849:   PetscScalar     *Q,*A;
    850:   PetscReal       *a,*b,beta,lambda;
    851:   EPS_shift       sPres;
    852:   PetscBool       breakdown,complIterating,sch0,sch1;
    853:   EPS_SR          sr = ctx->sr;

    855:   PetscFunctionBegin;
    856:   /* Spectrum slicing data */
    857:   sPres = sr->sPres;
    858:   complIterating =PETSC_FALSE;
    859:   sch1 = sch0 = PETSC_TRUE;
    860:   PetscCall(DSGetLeadingDimension(eps->ds,&ld));
    861:   PetscCall(PetscMalloc1(2*ld,&iwork));
    862:   count0=0;count1=0; /* Found on both sides */
    863:   if (!sPres->rep && sr->nS > 0 && (sPres->neighb[0] == sr->sPrev || sPres->neighb[1] == sr->sPrev)) {
    864:     /* Rational Krylov */
    865:     PetscCall(DSTranslateRKS(eps->ds,sr->sPrev->value-sPres->value));
    866:     PetscCall(DSGetDimensions(eps->ds,NULL,NULL,&l,NULL));
    867:     PetscCall(DSSetDimensions(eps->ds,l+1,0,0));
    868:     PetscCall(BVSetActiveColumns(eps->V,0,l+1));
    869:     PetscCall(DSGetMat(eps->ds,DS_MAT_Q,&U));
    870:     PetscCall(BVMultInPlace(eps->V,U,0,l+1));
    871:     PetscCall(DSRestoreMat(eps->ds,DS_MAT_Q,&U));
    872:   } else {
    873:     /* Get the starting Lanczos vector */
    874:     PetscCall(EPSGetStartVector(eps,0,NULL));
    875:     l = 0;
    876:   }
    877:   /* Restart loop */
    878:   while (eps->reason == EPS_CONVERGED_ITERATING) {
    879:     eps->its++; sr->itsKs++;
    880:     /* Compute an nv-step Lanczos factorization */
    881:     nv = PetscMin(eps->nconv+eps->mpd,eps->ncv);
    882:     PetscCall(DSSetDimensions(eps->ds,nv,eps->nconv,eps->nconv+l));
    883:     PetscCall(DSGetMat(eps->ds,DS_MAT_T,&T));
    884:     PetscCall(STGetOperator(eps->st,&Op));
    885:     PetscCall(BVMatLanczos(eps->V,Op,T,eps->nconv+l,&nv,&beta,&breakdown));
    886:     PetscCall(STRestoreOperator(eps->st,&Op));
    887:     sr->nv = nv;
    888:     PetscCall(DSRestoreMat(eps->ds,DS_MAT_T,&T));
    889:     PetscCall(DSSetDimensions(eps->ds,nv,eps->nconv,eps->nconv+l));
    890:     if (l==0) PetscCall(DSSetState(eps->ds,DS_STATE_INTERMEDIATE));
    891:     else PetscCall(DSSetState(eps->ds,DS_STATE_RAW));
    892:     PetscCall(BVSetActiveColumns(eps->V,eps->nconv,nv));

    894:     /* Solve projected problem and compute residual norm estimates */
    895:     if (eps->its == 1 && l > 0) { /* After rational update, DS_MAT_A is available */
    896:       PetscCall(DSGetArray(eps->ds,DS_MAT_A,&A));
    897:       PetscCall(DSGetArrayReal(eps->ds,DS_MAT_T,&a));
    898:       b = a + ld;
    899:       k = eps->nconv+l;
    900:       A[k*ld+k-1] = A[(k-1)*ld+k];
    901:       A[k*ld+k] = a[k];
    902:       for (j=k+1; j< nv; j++) {
    903:         A[j*ld+j] = a[j];
    904:         A[j*ld+j-1] = b[j-1] ;
    905:         A[(j-1)*ld+j] = b[j-1];
    906:       }
    907:       PetscCall(DSRestoreArray(eps->ds,DS_MAT_A,&A));
    908:       PetscCall(DSRestoreArrayReal(eps->ds,DS_MAT_T,&a));
    909:       PetscCall(DSSolve(eps->ds,eps->eigr,NULL));
    910:       PetscCall(DSSort(eps->ds,eps->eigr,NULL,NULL,NULL,NULL));
    911:       PetscCall(DSSetCompact(eps->ds,PETSC_TRUE));
    912:     } else { /* Restart */
    913:       PetscCall(DSSolve(eps->ds,eps->eigr,NULL));
    914:       PetscCall(DSSort(eps->ds,eps->eigr,NULL,NULL,NULL,NULL));
    915:     }
    916:     PetscCall(DSSynchronize(eps->ds,eps->eigr,NULL));

    918:     /* Residual */
    919:     PetscCall(EPSKrylovConvergence(eps,PETSC_TRUE,eps->nconv,nv-eps->nconv,beta,0.0,1.0,&k));
    920:     /* Checking values obtained for completing */
    921:     for (i=0;i<k;i++) {
    922:       sr->back[i]=eps->eigr[i];
    923:     }
    924:     PetscCall(STBackTransform(eps->st,k,sr->back,eps->eigi));
    925:     count0=count1=0;
    926:     for (i=0;i<k;i++) {
    927:       lambda = PetscRealPart(sr->back[i]);
    928:       if ((sr->dir*(sPres->value - lambda) > 0) && (sr->dir*(lambda - sPres->ext[0]) > 0)) count0++;
    929:       if ((sr->dir*(lambda - sPres->value) > 0) && (sr->dir*(sPres->ext[1] - lambda) > 0)) count1++;
    930:     }
    931:     if (k>eps->nev && eps->ncv-k<5) eps->reason = EPS_CONVERGED_TOL;
    932:     else {
    933:       /* Checks completion */
    934:       if ((!sch0||count0 >= sPres->nsch[0]) && (!sch1 ||count1 >= sPres->nsch[1])) {
    935:         eps->reason = EPS_CONVERGED_TOL;
    936:       } else {
    937:         if (!complIterating && eps->its >= eps->max_it) eps->reason = EPS_DIVERGED_ITS;
    938:         if (complIterating) {
    939:           if (--iterCompl <= 0) eps->reason = EPS_DIVERGED_ITS;
    940:         } else if (k >= eps->nev) {
    941:           n0 = sPres->nsch[0]-count0;
    942:           n1 = sPres->nsch[1]-count1;
    943:           if (sr->iterCompl>0 && ((n0>0 && n0<= sr->nMAXCompl)||(n1>0&&n1<=sr->nMAXCompl))) {
    944:             /* Iterating for completion*/
    945:             complIterating = PETSC_TRUE;
    946:             if (n0 >sr->nMAXCompl)sch0 = PETSC_FALSE;
    947:             if (n1 >sr->nMAXCompl)sch1 = PETSC_FALSE;
    948:             iterCompl = sr->iterCompl;
    949:           } else eps->reason = EPS_CONVERGED_TOL;
    950:         }
    951:       }
    952:     }
    953:     /* Update l */
    954:     if (eps->reason == EPS_CONVERGED_ITERATING) l = PetscMax(1,(PetscInt)((nv-k)*ctx->keep));
    955:     else l = nv-k;
    956:     if (breakdown) l=0;
    957:     if (!ctx->lock && l>0 && eps->reason == EPS_CONVERGED_ITERATING) { l += k; k = 0; } /* non-locking variant: reset no. of converged pairs */

    959:     if (eps->reason == EPS_CONVERGED_ITERATING) {
    960:       if (breakdown) {
    961:         /* Start a new Lanczos factorization */
    962:         PetscCall(PetscInfo(eps,"Breakdown in Krylov-Schur method (it=%" PetscInt_FMT " norm=%g)\n",eps->its,(double)beta));
    963:         PetscCall(EPSGetStartVector(eps,k,&breakdown));
    964:         if (breakdown) {
    965:           eps->reason = EPS_DIVERGED_BREAKDOWN;
    966:           PetscCall(PetscInfo(eps,"Unable to generate more start vectors\n"));
    967:         }
    968:       } else {
    969:         /* Prepare the Rayleigh quotient for restart */
    970:         PetscCall(DSGetArrayReal(eps->ds,DS_MAT_T,&a));
    971:         PetscCall(DSGetArray(eps->ds,DS_MAT_Q,&Q));
    972:         b = a + ld;
    973:         for (i=k;i<k+l;i++) {
    974:           a[i] = PetscRealPart(eps->eigr[i]);
    975:           b[i] = PetscRealPart(Q[nv-1+i*ld]*beta);
    976:         }
    977:         PetscCall(DSRestoreArrayReal(eps->ds,DS_MAT_T,&a));
    978:         PetscCall(DSRestoreArray(eps->ds,DS_MAT_Q,&Q));
    979:       }
    980:     }
    981:     /* Update the corresponding vectors V(:,idx) = V*Q(:,idx) */
    982:     PetscCall(DSGetMat(eps->ds,DS_MAT_Q,&U));
    983:     PetscCall(BVMultInPlace(eps->V,U,eps->nconv,k+l));
    984:     PetscCall(DSRestoreMat(eps->ds,DS_MAT_Q,&U));

    986:     /* Normalize u and append it to V */
    987:     if (eps->reason == EPS_CONVERGED_ITERATING && !breakdown) PetscCall(BVCopyColumn(eps->V,nv,k+l));
    988:     eps->nconv = k;
    989:     if (eps->reason != EPS_CONVERGED_ITERATING) {
    990:       /* Store approximated values for next shift */
    991:       PetscCall(DSGetArray(eps->ds,DS_MAT_Q,&Q));
    992:       sr->nS = l;
    993:       for (i=0;i<l;i++) {
    994:         sr->S[i] = eps->eigr[i+k];/* Diagonal elements */
    995:         sr->S[i+l] = Q[nv-1+(i+k)*ld]*beta; /* Out of diagonal elements */
    996:       }
    997:       PetscCall(DSRestoreArray(eps->ds,DS_MAT_Q,&Q));
    998:     }
    999:   }
    1000:   /* Check for completion */
    1001:   for (i=0;i< eps->nconv; i++) {
    1002:     if (sr->dir*PetscRealPart(eps->eigr[i])>0) sPres->nconv[1]++;
    1003:     else sPres->nconv[0]++;
    1004:   }
    1005:   sPres->comp[0] = PetscNot(count0 < sPres->nsch[0]);
    1006:   sPres->comp[1] = PetscNot(count1 < sPres->nsch[1]);
    1007:   PetscCall(PetscInfo(eps,"Lanczos: %" PetscInt_FMT " evals in [%g,%g]%s and %" PetscInt_FMT " evals in [%g,%g]%s\n",count0,(double)(sr->dir==1?sPres->ext[0]:sPres->value),(double)(sr->dir==1?sPres->value:sPres->ext[0]),sPres->comp[0]?"*":"",count1,(double)(sr->dir==1?sPres->value:sPres->ext[1]),(double)(sr->dir==1?sPres->ext[1]:sPres->value),sPres->comp[1]?"*":""));
    1008:   PetscCheck(count0<=sPres->nsch[0] && count1<=sPres->nsch[1],PetscObjectComm((PetscObject)eps),PETSC_ERR_PLIB,"Mismatch between number of values found and information from inertia%s",ctx->detect?"":", consider using EPSKrylovSchurSetDetectZeros()");
    1009:   PetscCall(PetscFree(iwork));
    1010:   PetscFunctionReturn(PETSC_SUCCESS);
    1011: }

    1013: /*
    1014:   Obtains value of subsequent shift
    1015: */
    1016: static PetscErrorCode EPSGetNewShiftValue(EPS eps,PetscInt side,PetscReal *newS)
    1017: {
    1018:   PetscReal       lambda,d_prev;
    1019:   PetscInt        i,idxP;
    1020:   EPS_SR          sr;
    1021:   EPS_shift       sPres,s;
    1022:   EPS_KRYLOVSCHUR *ctx=(EPS_KRYLOVSCHUR*)eps->data;

    1024:   PetscFunctionBegin;
    1025:   sr = ctx->sr;
    1026:   sPres = sr->sPres;
    1027:   if (sPres->neighb[side]) {
    1028:     /* Completing a previous interval */
    1029:     *newS = (sPres->value + sPres->neighb[side]->value)/2;
    1030:     if (PetscAbsReal(sPres->value - *newS)/PetscAbsReal(sPres->value)<=100*PETSC_SQRT_MACHINE_EPSILON) *newS = sPres->value;
    1031:   } else { /* (Only for side=1). Creating a new interval. */
    1032:     if (sPres->neigs==0) {/* No value has been accepted*/
    1033:       if (sPres->neighb[0]) {
    1034:         /* Multiplying by 10 the previous distance */
    1035:         *newS = sPres->value + 10*sr->dir*PetscAbsReal(sPres->value - sPres->neighb[0]->value);
    1036:         sr->nleap++;
    1037:         /* Stops when the interval is open and no values are found in the last 5 shifts (there might be infinite eigenvalues) */
    1038:         PetscCheck(sr->hasEnd || sr->nleap<=5,PetscObjectComm((PetscObject)eps),PETSC_ERR_PLIB,"Unable to compute the wanted eigenvalues with open interval");
    1039:       } else { /* First shift */
    1040:         PetscCheck(eps->nconv!=0,PetscObjectComm((PetscObject)eps),PETSC_ERR_PLIB,"First shift renders no information");
    1041:         /* Unaccepted values give information for next shift */
    1042:         idxP=0;/* Number of values left from shift */
    1043:         for (i=0;i<eps->nconv;i++) {
    1044:           lambda = PetscRealPart(eps->eigr[i]);
    1045:           if (sr->dir*(lambda - sPres->value) <0) idxP++;
    1046:           else break;
    1047:         }
    1048:         /* Avoiding subtraction of eigenvalues (might be the same).*/
    1049:         if (idxP>0) {
    1050:           d_prev = PetscAbsReal(sPres->value - PetscRealPart(eps->eigr[0]))/(idxP+0.3);
    1051:         } else {
    1052:           d_prev = PetscAbsReal(sPres->value - PetscRealPart(eps->eigr[eps->nconv-1]))/(eps->nconv+0.3);
    1053:         }
    1054:         *newS = sPres->value + (sr->dir*d_prev*eps->nev)/2;
    1055:       }
    1056:     } else { /* Accepted values found */
    1057:       sr->nleap = 0;
    1058:       /* Average distance of values in previous subinterval */
    1059:       s = sPres->neighb[0];
    1060:       while (s && PetscAbs(s->inertia - sPres->inertia)==0) {
    1061:         s = s->neighb[0];/* Looking for previous shifts with eigenvalues within */
    1062:       }
    1063:       if (s) {
    1064:         d_prev = PetscAbsReal((sPres->value - s->value)/(sPres->inertia - s->inertia));
    1065:       } else { /* First shift. Average distance obtained with values in this shift */
    1066:         /* first shift might be too far from first wanted eigenvalue (no values found outside the interval)*/
    1067:         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(eps->tol)) {
    1068:           d_prev =  PetscAbsReal((PetscRealPart(sr->eigr[sr->indexEig-1]) - PetscRealPart(sr->eigr[0])))/(sPres->neigs+0.3);
    1069:         } else {
    1070:           d_prev = PetscAbsReal(PetscRealPart(sr->eigr[sr->indexEig-1]) - sPres->value)/(sPres->neigs+0.3);
    1071:         }
    1072:       }
    1073:       /* Average distance is used for next shift by adding it to value on the right or to shift */
    1074:       if (sr->dir*(PetscRealPart(sr->eigr[sPres->index + sPres->neigs -1]) - sPres->value)>0) {
    1075:         *newS = PetscRealPart(sr->eigr[sPres->index + sPres->neigs -1])+ (sr->dir*d_prev*eps->nev)/2;
    1076:       } else { /* Last accepted value is on the left of shift. Adding to shift */
    1077:         *newS = sPres->value + (sr->dir*d_prev*eps->nev)/2;
    1078:       }
    1079:     }
    1080:     /* End of interval can not be surpassed */
    1081:     if (sr->dir*(sr->int1 - *newS) < 0) *newS = sr->int1;
    1082:   }/* of neighb[side]==null */
    1083:   PetscFunctionReturn(PETSC_SUCCESS);
    1084: }

    1086: /*
    1087:   Function for sorting an array of real values
    1088: */
    1089: static PetscErrorCode sortRealEigenvalues(PetscScalar *r,PetscInt *perm,PetscInt nr,PetscBool prev,PetscInt dir)
    1090: {
    1091:   PetscReal re;
    1092:   PetscInt  i,j,tmp;

    1094:   PetscFunctionBegin;
    1095:   if (!prev) for (i=0;i<nr;i++) perm[i] = i;
    1096:   /* Insertion sort */
    1097:   for (i=1;i<nr;i++) {
    1098:     re = PetscRealPart(r[perm[i]]);
    1099:     j = i-1;
    1100:     while (j>=0 && dir*(re - PetscRealPart(r[perm[j]])) <= 0) {
    1101:       tmp = perm[j]; perm[j] = perm[j+1]; perm[j+1] = tmp; j--;
    1102:     }
    1103:   }
    1104:   PetscFunctionReturn(PETSC_SUCCESS);
    1105: }

    1107: /* Stores the pairs obtained since the last shift in the global arrays */
    1108: static PetscErrorCode EPSStoreEigenpairs(EPS eps)
    1109: {
    1110:   EPS_KRYLOVSCHUR *ctx=(EPS_KRYLOVSCHUR*)eps->data;
    1111:   PetscReal       lambda,err,norm;
    1112:   PetscInt        i,count;
    1113:   PetscBool       iscayley;
    1114:   EPS_SR          sr = ctx->sr;
    1115:   EPS_shift       sPres;
    1116:   Vec             v,w;

    1118:   PetscFunctionBegin;
    1119:   sPres = sr->sPres;
    1120:   sPres->index = sr->indexEig;
    1121:   count = sr->indexEig;
    1122:   /* Back-transform */
    1123:   PetscCall(STBackTransform(eps->st,eps->nconv,eps->eigr,eps->eigi));
    1124:   PetscCall(PetscObjectTypeCompare((PetscObject)eps->st,STCAYLEY,&iscayley));
    1125:   /* Sort eigenvalues */
    1126:   PetscCall(sortRealEigenvalues(eps->eigr,eps->perm,eps->nconv,PETSC_FALSE,sr->dir));
    1127:   /* Values stored in global array */
    1128:   for (i=0;i<eps->nconv;i++) {
    1129:     lambda = PetscRealPart(eps->eigr[eps->perm[i]]);
    1130:     err = eps->errest[eps->perm[i]];

    1132:     if (sr->dir*(lambda - sPres->ext[0]) > 0 && (sr->dir)*(sPres->ext[1] - lambda) > 0) {/* Valid value */
    1133:       PetscCheck(count<sr->numEigs,PetscObjectComm((PetscObject)eps),PETSC_ERR_PLIB,"Unexpected error in Spectrum Slicing");
    1134:       sr->eigr[count] = lambda;
    1135:       sr->errest[count] = err;
    1136:       /* Explicit purification */
    1137:       PetscCall(BVGetColumn(eps->V,eps->perm[i],&w));
    1138:       if (eps->purify) {
    1139:         PetscCall(BVGetColumn(sr->V,count,&v));
    1140:         PetscCall(STApply(eps->st,w,v));
    1141:         PetscCall(BVRestoreColumn(sr->V,count,&v));
    1142:       } else PetscCall(BVInsertVec(sr->V,count,w));
    1143:       PetscCall(BVRestoreColumn(eps->V,eps->perm[i],&w));
    1144:       PetscCall(BVNormColumn(sr->V,count,NORM_2,&norm));
    1145:       PetscCall(BVScaleColumn(sr->V,count,1.0/norm));
    1146:       count++;
    1147:     }
    1148:   }
    1149:   sPres->neigs = count - sr->indexEig;
    1150:   sr->indexEig = count;
    1151:   /* Global ordering array updating */
    1152:   PetscCall(sortRealEigenvalues(sr->eigr,sr->perm,count,PETSC_TRUE,sr->dir));
    1153:   PetscFunctionReturn(PETSC_SUCCESS);
    1154: }

    1156: static PetscErrorCode EPSLookForDeflation(EPS eps)
    1157: {
    1158:   PetscReal       val;
    1159:   PetscInt        i,count0=0,count1=0;
    1160:   EPS_shift       sPres;
    1161:   PetscInt        ini,fin,k,idx0,idx1;
    1162:   EPS_SR          sr;
    1163:   Vec             v;
    1164:   EPS_KRYLOVSCHUR *ctx=(EPS_KRYLOVSCHUR*)eps->data;

    1166:   PetscFunctionBegin;
    1167:   sr = ctx->sr;
    1168:   sPres = sr->sPres;

    1170:   if (sPres->neighb[0]) ini = (sr->dir)*(sPres->neighb[0]->inertia - sr->inertia0);
    1171:   else ini = 0;
    1172:   fin = sr->indexEig;
    1173:   /* Selection of ends for searching new values */
    1174:   if (!sPres->neighb[0]) sPres->ext[0] = sr->int0;/* First shift */
    1175:   else sPres->ext[0] = sPres->neighb[0]->value;
    1176:   if (!sPres->neighb[1]) {
    1177:     if (sr->hasEnd) sPres->ext[1] = sr->int1;
    1178:     else sPres->ext[1] = (sr->dir > 0)?PETSC_MAX_REAL:PETSC_MIN_REAL;
    1179:   } else sPres->ext[1] = sPres->neighb[1]->value;
    1180:   /* Selection of values between right and left ends */
    1181:   for (i=ini;i<fin;i++) {
    1182:     val=PetscRealPart(sr->eigr[sr->perm[i]]);
    1183:     /* Values to the right of left shift */
    1184:     if (sr->dir*(val - sPres->ext[1]) < 0) {
    1185:       if (sr->dir*(val - sPres->value) < 0) count0++;
    1186:       else count1++;
    1187:     } else break;
    1188:   }
    1189:   /* The number of values on each side are found */
    1190:   if (sPres->neighb[0]) {
    1191:     sPres->nsch[0] = (sr->dir)*(sPres->inertia - sPres->neighb[0]->inertia)-count0;
    1192:     PetscCheck(sPres->nsch[0]>=0,PetscObjectComm((PetscObject)eps),PETSC_ERR_PLIB,"Mismatch between number of values found and information from inertia%s",ctx->detect?"":", consider using EPSKrylovSchurSetDetectZeros()");
    1193:   } else sPres->nsch[0] = 0;

    1195:   if (sPres->neighb[1]) {
    1196:     sPres->nsch[1] = (sr->dir)*(sPres->neighb[1]->inertia - sPres->inertia) - count1;
    1197:     PetscCheck(sPres->nsch[1]>=0,PetscObjectComm((PetscObject)eps),PETSC_ERR_PLIB,"Mismatch between number of values found and information from inertia%s",ctx->detect?"":", consider using EPSKrylovSchurSetDetectZeros()");
    1198:   } else sPres->nsch[1] = (sr->dir)*(sr->inertia1 - sPres->inertia);

    1200:   /* Completing vector of indexes for deflation */
    1201:   idx0 = ini;
    1202:   idx1 = ini+count0+count1;
    1203:   k=0;
    1204:   for (i=idx0;i<idx1;i++) sr->idxDef[k++]=sr->perm[i];
    1205:   PetscCall(BVDuplicateResize(eps->V,k+eps->ncv+1,&sr->Vnext));
    1206:   PetscCall(BVSetNumConstraints(sr->Vnext,k));
    1207:   for (i=0;i<k;i++) {
    1208:     PetscCall(BVGetColumn(sr->Vnext,-i-1,&v));
    1209:     PetscCall(BVCopyVec(sr->V,sr->idxDef[i],v));
    1210:     PetscCall(BVRestoreColumn(sr->Vnext,-i-1,&v));
    1211:   }

    1213:   /* For rational Krylov */
    1214:   if (!sr->sPres->rep && sr->nS>0 && (sr->sPrev == sr->sPres->neighb[0] || sr->sPrev == sr->sPres->neighb[1])) PetscCall(EPSPrepareRational(eps));
    1215:   eps->nconv = 0;
    1216:   /* Get rid of temporary Vnext */
    1217:   PetscCall(BVDestroy(&eps->V));
    1218:   eps->V = sr->Vnext;
    1219:   sr->Vnext = NULL;
    1220:   PetscFunctionReturn(PETSC_SUCCESS);
    1221: }

    1223: PetscErrorCode EPSSolve_KrylovSchur_Slice(EPS eps)
    1224: {
    1225:   PetscInt         i,lds,ti;
    1226:   PetscReal        newS;
    1227:   EPS_KRYLOVSCHUR  *ctx=(EPS_KRYLOVSCHUR*)eps->data;
    1228:   EPS_SR           sr=ctx->sr;
    1229:   Mat              A,B=NULL;
    1230:   PetscObjectState Astate,Bstate=0;
    1231:   PetscObjectId    Aid,Bid=0;

    1233:   PetscFunctionBegin;
    1234:   PetscCall(PetscCitationsRegister(citation,&cited));
    1235:   if (ctx->global) {
    1236:     PetscCall(EPSSolve_KrylovSchur_Slice(ctx->eps));
    1237:     ctx->eps->state = EPS_STATE_SOLVED;
    1238:     eps->reason = EPS_CONVERGED_TOL;
    1239:     if (ctx->npart>1) {
    1240:       /* Gather solution from subsolvers */
    1241:       PetscCall(EPSSliceGatherSolution(eps));
    1242:     } else {
    1243:       eps->nconv = sr->numEigs;
    1244:       eps->its   = ctx->eps->its;
    1245:       PetscCall(PetscFree(ctx->inertias));
    1246:       PetscCall(PetscFree(ctx->shifts));
    1247:       PetscCall(EPSSliceGetInertias(ctx->eps,&ctx->nshifts,&ctx->shifts,&ctx->inertias));
    1248:     }
    1249:   } else {
    1250:     if (ctx->npart==1) {
    1251:       sr->eigr   = ctx->eps->eigr;
    1252:       sr->eigi   = ctx->eps->eigi;
    1253:       sr->perm   = ctx->eps->perm;
    1254:       sr->errest = ctx->eps->errest;
    1255:       sr->V      = ctx->eps->V;
    1256:     }
    1257:     /* Check that the user did not modify subcomm matrices */
    1258:     PetscCall(EPSGetOperators(eps,&A,&B));
    1259:     PetscCall(MatGetState(A,&Astate));
    1260:     PetscCall(PetscObjectGetId((PetscObject)A,&Aid));
    1261:     if (B) {
    1262:       PetscCall(MatGetState(B,&Bstate));
    1263:       PetscCall(PetscObjectGetId((PetscObject)B,&Bid));
    1264:     }
    1265:     PetscCheck(Astate==ctx->Astate && (!B || Bstate==ctx->Bstate) && Aid==ctx->Aid && (!B || Bid==ctx->Bid),PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"Subcomm matrices have been modified by user");
    1266:     /* Only with eigenvalues present in the interval ...*/
    1267:     if (sr->numEigs==0) {
    1268:       eps->reason = EPS_CONVERGED_TOL;
    1269:       PetscFunctionReturn(PETSC_SUCCESS);
    1270:     }
    1271:     /* Array of pending shifts */
    1272:     sr->maxPend = 100; /* Initial size */
    1273:     sr->nPend = 0;
    1274:     PetscCall(PetscMalloc1(sr->maxPend,&sr->pending));
    1275:     PetscCall(EPSCreateShift(eps,sr->int0,NULL,NULL));
    1276:     /* extract first shift */
    1277:     sr->sPrev = NULL;
    1278:     sr->sPres = sr->pending[--sr->nPend];
    1279:     sr->sPres->inertia = sr->inertia0;
    1280:     eps->target = sr->sPres->value;
    1281:     sr->s0 = sr->sPres;
    1282:     sr->indexEig = 0;
    1283:     /* Memory reservation for auxiliary variables */
    1284:     lds = PetscMin(eps->mpd,eps->ncv);
    1285:     PetscCall(PetscCalloc1(lds*lds,&sr->S));
    1286:     PetscCall(PetscMalloc1(eps->ncv,&sr->back));
    1287:     for (i=0;i<sr->numEigs;i++) {
    1288:       sr->eigr[i]   = 0.0;
    1289:       sr->eigi[i]   = 0.0;
    1290:       sr->errest[i] = 0.0;
    1291:       sr->perm[i]   = i;
    1292:     }
    1293:     /* Vectors for deflation */
    1294:     PetscCall(PetscMalloc1(sr->numEigs,&sr->idxDef));
    1295:     sr->indexEig = 0;
    1296:     /* Main loop */
    1297:     while (sr->sPres) {
    1298:       /* Search for deflation */
    1299:       PetscCall(EPSLookForDeflation(eps));
    1300:       /* KrylovSchur */
    1301:       PetscCall(EPSKrylovSchur_Slice(eps));

    1303:       PetscCall(EPSStoreEigenpairs(eps));
    1304:       /* Select new shift */
    1305:       if (!sr->sPres->comp[1]) {
    1306:         PetscCall(EPSGetNewShiftValue(eps,1,&newS));
    1307:         PetscCall(EPSCreateShift(eps,newS,sr->sPres,sr->sPres->neighb[1]));
    1308:       }
    1309:       if (!sr->sPres->comp[0]) {
    1310:         /* Completing earlier interval */
    1311:         PetscCall(EPSGetNewShiftValue(eps,0,&newS));
    1312:         PetscCall(EPSCreateShift(eps,newS,sr->sPres->neighb[0],sr->sPres));
    1313:       }
    1314:       /* Preparing for a new search of values */
    1315:       PetscCall(EPSExtractShift(eps));
    1316:     }

    1318:     /* Updating eps values prior to exit */
    1319:     PetscCall(PetscFree(sr->S));
    1320:     PetscCall(PetscFree(sr->idxDef));
    1321:     PetscCall(PetscFree(sr->pending));
    1322:     PetscCall(PetscFree(sr->back));
    1323:     PetscCall(BVDuplicateResize(eps->V,eps->ncv+1,&sr->Vnext));
    1324:     PetscCall(BVSetNumConstraints(sr->Vnext,0));
    1325:     PetscCall(BVDestroy(&eps->V));
    1326:     eps->V      = sr->Vnext;
    1327:     eps->nconv  = sr->indexEig;
    1328:     eps->reason = EPS_CONVERGED_TOL;
    1329:     eps->its    = sr->itsKs;
    1330:     eps->nds    = 0;
    1331:     if (sr->dir<0) {
    1332:       for (i=0;i<eps->nconv/2;i++) {
    1333:         ti = sr->perm[i]; sr->perm[i] = sr->perm[eps->nconv-1-i]; sr->perm[eps->nconv-1-i] = ti;
    1334:       }
    1335:     }
    1336:   }
    1337:   PetscFunctionReturn(PETSC_SUCCESS);
    1338: }