Date: 2020-01-13T12:00:12+00:00

Actual source code: lanczos.c
slepc-3.12.2 2020-01-13
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      1: /*
      2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      4:    Copyright (c) 2002-2019, 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: "lanczos"

     13:    Method: Explicitly Restarted Symmetric/Hermitian Lanczos

     15:    Algorithm:

     17:        Lanczos method for symmetric (Hermitian) problems, with explicit
     18:        restart and deflation. Several reorthogonalization strategies can
     19:        be selected.

     21:    References:

     23:        [1] "Lanczos Methods in SLEPc", SLEPc Technical Report STR-5,
     24:            available at http://slepc.upv.es.
     25: */

     27: #include <slepc/private/epsimpl.h>                /*I "slepceps.h" I*/
     28: #include <slepcblaslapack.h>

     30: typedef struct {
     31:   EPSLanczosReorthogType reorthog;      /* user-provided reorthogonalization parameter */
     32:   PetscInt               allocsize;     /* number of columns of work BV's allocated at setup */
     33:   BV                     AV;            /* work BV used in selective reorthogonalization */
     34: } EPS_LANCZOS;

     36: PetscErrorCode EPSSetUp_Lanczos(EPS eps)
     37: {
     38:   EPS_LANCZOS        *lanczos = (EPS_LANCZOS*)eps->data;
     39:   BVOrthogRefineType refine;
     40:   BVOrthogBlockType  btype;
     41:   PetscReal          eta;
     42:   PetscErrorCode     ierr;

     45:   EPSSetDimensions_Default(eps,eps->nev,&eps->ncv,&eps->mpd);
     46:   if (eps->ncv>eps->nev+eps->mpd) SETERRQ(PetscObjectComm((PetscObject)eps),1,"The value of ncv must not be larger than nev+mpd");
     47:   if (!eps->max_it) eps->max_it = PetscMax(100,2*eps->n/eps->ncv);
     48:   if (!eps->which) { EPSSetWhichEigenpairs_Default(eps); }
     49:   switch (eps->which) {
     50:     case EPS_LARGEST_IMAGINARY:
     51:     case EPS_SMALLEST_IMAGINARY:
     52:     case EPS_TARGET_IMAGINARY:
     53:     case EPS_ALL:
     54:       SETERRQ(PetscObjectComm((PetscObject)eps),1,"Wrong value of eps->which");
     55:     default: ; /* default case to remove warning */
     56:   }
     57:   if (!eps->extraction) {
     58:     EPSSetExtraction(eps,EPS_RITZ);
     59:   } else if (eps->extraction!=EPS_RITZ) SETERRQ(PetscObjectComm((PetscObject)eps),PETSC_ERR_SUP,"Unsupported extraction type");
     60:   if (eps->arbitrary) SETERRQ(PetscObjectComm((PetscObject)eps),PETSC_ERR_SUP,"Arbitrary selection of eigenpairs not supported in this solver");

     62:   EPSAllocateSolution(eps,1);
     63:   EPS_SetInnerProduct(eps);
     64:   if (lanczos->reorthog != EPS_LANCZOS_REORTHOG_FULL) {
     65:     BVGetOrthogonalization(eps->V,NULL,&refine,&eta,&btype);
     66:     BVSetOrthogonalization(eps->V,BV_ORTHOG_MGS,refine,eta,btype);
     67:     PetscInfo(eps,"Switching to MGS orthogonalization\n");
     68:   }
     69:   if (lanczos->reorthog == EPS_LANCZOS_REORTHOG_SELECTIVE) {
     70:     if (!lanczos->allocsize) {
     71:       BVDuplicate(eps->V,&lanczos->AV);
     72:       BVGetSizes(lanczos->AV,NULL,NULL,&lanczos->allocsize);
     73:     } else { /* make sure V and AV have the same size */
     74:       BVGetSizes(eps->V,NULL,NULL,&lanczos->allocsize);
     75:       BVResize(lanczos->AV,lanczos->allocsize,PETSC_FALSE);
     76:     }
     77:   }

     79:   DSSetType(eps->ds,DSHEP);
     80:   DSSetCompact(eps->ds,PETSC_TRUE);
     81:   DSAllocate(eps->ds,eps->ncv+1);
     82:   if (lanczos->reorthog == EPS_LANCZOS_REORTHOG_LOCAL) {
     83:     EPSSetWorkVecs(eps,1);
     84:   }

     86:   if (!eps->ishermitian) SETERRQ(PetscObjectComm((PetscObject)eps),PETSC_ERR_SUP,"Requested method is only available for Hermitian problems");
     87:   if (eps->isgeneralized && eps->ishermitian && !eps->ispositive) SETERRQ(PetscObjectComm((PetscObject)eps),PETSC_ERR_SUP,"Requested method does not work for indefinite problems");
     88:   return(0);
     89: }

     91: /*
     92:    EPSLocalLanczos - Local reorthogonalization.

     94:    This is the simplest variant. At each Lanczos step, the corresponding Lanczos vector
     95:    is orthogonalized with respect to the two previous Lanczos vectors, according to
     96:    the three term Lanczos recurrence. WARNING: This variant does not track the loss of
     97:    orthogonality that occurs in finite-precision arithmetic and, therefore, the
     98:    generated vectors are not guaranteed to be (semi-)orthogonal.
     99: */
    100: static PetscErrorCode EPSLocalLanczos(EPS eps,PetscReal *alpha,PetscReal *beta,PetscInt k,PetscInt *M,PetscBool *breakdown)
    101: {
    103:   PetscInt       i,j,m = *M;
    104:   Vec            vj,vj1;
    105:   PetscBool      *which,lwhich[100];
    106:   PetscScalar    *hwork,lhwork[100];

    109:   if (m > 100) {
    110:     PetscMalloc2(m,&which,m,&hwork);
    111:   } else {
    112:     which = lwhich;
    113:     hwork = lhwork;
    114:   }
    115:   for (i=0;i<k;i++) which[i] = PETSC_TRUE;

    117:   BVSetActiveColumns(eps->V,0,m);
    118:   for (j=k;j<m;j++) {
    119:     BVGetColumn(eps->V,j,&vj);
    120:     BVGetColumn(eps->V,j+1,&vj1);
    121:     STApply(eps->st,vj,vj1);
    122:     BVRestoreColumn(eps->V,j,&vj);
    123:     BVRestoreColumn(eps->V,j+1,&vj1);
    124:     which[j] = PETSC_TRUE;
    125:     if (j-2>=k) which[j-2] = PETSC_FALSE;
    126:     BVOrthogonalizeSomeColumn(eps->V,j+1,which,hwork,beta+j,breakdown);
    127:     alpha[j] = PetscRealPart(hwork[j]);
    128:     if (*breakdown) {
    129:       *M = j+1;
    130:       break;
    131:     } else {
    132:       BVScaleColumn(eps->V,j+1,1/beta[j]);
    133:     }
    134:   }
    135:   if (m > 100) {
    136:     PetscFree2(which,hwork);
    137:   }
    138:   return(0);
    139: }

    141: /*
    142:    DenseTridiagonal - Solves a real tridiagonal Hermitian Eigenvalue Problem.

    144:    Input Parameters:
    145: +  n   - dimension of the eigenproblem
    146: .  D   - pointer to the array containing the diagonal elements
    147: -  E   - pointer to the array containing the off-diagonal elements

    149:    Output Parameters:
    150: +  w  - pointer to the array to store the computed eigenvalues
    151: -  V  - pointer to the array to store the eigenvectors

    153:    Notes:
    154:    If V is NULL then the eigenvectors are not computed.

    156:    This routine use LAPACK routines xSTEVR.
    157: */
    158: static PetscErrorCode DenseTridiagonal(PetscInt n_,PetscReal *D,PetscReal *E,PetscReal *w,PetscScalar *V)
    159: {
    160: #if defined(SLEPC_MISSING_LAPACK_STEVR)
    162:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"STEVR - Lapack routine is unavailable");
    163: #else
    165:   PetscReal      abstol = 0.0,vl,vu,*work;
    166:   PetscBLASInt   il,iu,m,*isuppz,n,lwork,*iwork,liwork,info;
    167:   const char     *jobz;
    168: #if defined(PETSC_USE_COMPLEX)
    169:   PetscInt       i,j;
    170:   PetscReal      *VV=NULL;
    171: #endif

    174:   PetscBLASIntCast(n_,&n);
    175:   PetscBLASIntCast(20*n_,&lwork);
    176:   PetscBLASIntCast(10*n_,&liwork);
    177:   if (V) {
    178:     jobz = "V";
    179: #if defined(PETSC_USE_COMPLEX)
    180:     PetscMalloc1(n*n,&VV);
    181: #endif
    182:   } else jobz = "N";
    183:   PetscMalloc3(2*n,&isuppz,lwork,&work,liwork,&iwork);
    184:   PetscFPTrapPush(PETSC_FP_TRAP_OFF);
    185: #if defined(PETSC_USE_COMPLEX)
    186:   PetscStackCallBLAS("LAPACKstevr",LAPACKstevr_(jobz,"A",&n,D,E,&vl,&vu,&il,&iu,&abstol,&m,w,VV,&n,isuppz,work,&lwork,iwork,&liwork,&info));
    187: #else
    188:   PetscStackCallBLAS("LAPACKstevr",LAPACKstevr_(jobz,"A",&n,D,E,&vl,&vu,&il,&iu,&abstol,&m,w,V,&n,isuppz,work,&lwork,iwork,&liwork,&info));
    189: #endif
    190:   PetscFPTrapPop();
    191:   SlepcCheckLapackInfo("stevr",info);
    192: #if defined(PETSC_USE_COMPLEX)
    193:   if (V) {
    194:     for (i=0;i<n;i++)
    195:       for (j=0;j<n;j++)
    196:         V[i*n+j] = VV[i*n+j];
    197:     PetscFree(VV);
    198:   }
    199: #endif
    200:   PetscFree3(isuppz,work,iwork);
    201:   return(0);
    202: #endif
    203: }

    205: /*
    206:    EPSSelectiveLanczos - Selective reorthogonalization.
    207: */
    208: static PetscErrorCode EPSSelectiveLanczos(EPS eps,PetscReal *alpha,PetscReal *beta,PetscInt k,PetscInt *M,PetscBool *breakdown,PetscReal anorm)
    209: {
    211:   EPS_LANCZOS    *lanczos = (EPS_LANCZOS*)eps->data;
    212:   PetscInt       i,j,m = *M,n,nritz=0,nritzo;
    213:   Vec            vj,vj1,av;
    214:   PetscReal      *d,*e,*ritz,norm;
    215:   PetscScalar    *Y,*hwork;
    216:   PetscBool      *which;

    219:   PetscCalloc6(m+1,&d,m,&e,m,&ritz,m*m,&Y,m,&which,m,&hwork);
    220:   for (i=0;i<k;i++) which[i] = PETSC_TRUE;

    222:   for (j=k;j<m;j++) {
    223:     BVSetActiveColumns(eps->V,0,m);

    225:     /* Lanczos step */
    226:     BVGetColumn(eps->V,j,&vj);
    227:     BVGetColumn(eps->V,j+1,&vj1);
    228:     STApply(eps->st,vj,vj1);
    229:     BVRestoreColumn(eps->V,j,&vj);
    230:     BVRestoreColumn(eps->V,j+1,&vj1);
    231:     which[j] = PETSC_TRUE;
    232:     if (j-2>=k) which[j-2] = PETSC_FALSE;
    233:     BVOrthogonalizeSomeColumn(eps->V,j+1,which,hwork,&norm,breakdown);
    234:     alpha[j] = PetscRealPart(hwork[j]);
    235:     beta[j] = norm;
    236:     if (*breakdown) {
    237:       *M = j+1;
    238:       break;
    239:     }

    241:     /* Compute eigenvalues and eigenvectors Y of the tridiagonal block */
    242:     n = j-k+1;
    243:     for (i=0;i<n;i++) {
    244:       d[i] = alpha[i+k];
    245:       e[i] = beta[i+k];
    246:     }
    247:     DenseTridiagonal(n,d,e,ritz,Y);

    249:     /* Estimate ||A|| */
    250:     for (i=0;i<n;i++)
    251:       if (PetscAbsReal(ritz[i]) > anorm) anorm = PetscAbsReal(ritz[i]);

    253:     /* Compute nearly converged Ritz vectors */
    254:     nritzo = 0;
    255:     for (i=0;i<n;i++) {
    256:       if (norm*PetscAbsScalar(Y[i*n+n-1]) < PETSC_SQRT_MACHINE_EPSILON*anorm) nritzo++;
    257:     }
    258:     if (nritzo>nritz) {
    259:       nritz = 0;
    260:       for (i=0;i<n;i++) {
    261:         if (norm*PetscAbsScalar(Y[i*n+n-1]) < PETSC_SQRT_MACHINE_EPSILON*anorm) {
    262:           BVSetActiveColumns(eps->V,k,k+n);
    263:           BVGetColumn(lanczos->AV,nritz,&av);
    264:           BVMultVec(eps->V,1.0,0.0,av,Y+i*n);
    265:           BVRestoreColumn(lanczos->AV,nritz,&av);
    266:           nritz++;
    267:         }
    268:       }
    269:     }
    270:     if (nritz > 0) {
    271:       BVGetColumn(eps->V,j+1,&vj1);
    272:       BVSetActiveColumns(lanczos->AV,0,nritz);
    273:       BVOrthogonalizeVec(lanczos->AV,vj1,hwork,&norm,breakdown);
    274:       BVRestoreColumn(eps->V,j+1,&vj1);
    275:       if (*breakdown) {
    276:         *M = j+1;
    277:         break;
    278:       }
    279:     }
    280:     BVScaleColumn(eps->V,j+1,1.0/norm);
    281:   }

    283:   PetscFree6(d,e,ritz,Y,which,hwork);
    284:   return(0);
    285: }

    287: static void update_omega(PetscReal *omega,PetscReal *omega_old,PetscInt j,PetscReal *alpha,PetscReal *beta,PetscReal eps1,PetscReal anorm)
    288: {
    289:   PetscInt  k;
    290:   PetscReal T,binv;

    293:   /* Estimate of contribution to roundoff errors from A*v
    294:        fl(A*v) = A*v + f,
    295:      where ||f|| \approx eps1*||A||.
    296:      For a full matrix A, a rule-of-thumb estimate is eps1 = sqrt(n)*eps */
    297:   T = eps1*anorm;
    298:   binv = 1.0/beta[j+1];

    300:   /* Update omega(1) using omega(0)==0 */
    301:   omega_old[0]= beta[1]*omega[1] + (alpha[0]-alpha[j])*omega[0] - beta[j]*omega_old[0];
    302:   if (omega_old[0] > 0) omega_old[0] = binv*(omega_old[0] + T);
    303:   else omega_old[0] = binv*(omega_old[0] - T);

    305:   /* Update remaining components */
    306:   for (k=1;k<j-1;k++) {
    307:     omega_old[k] = beta[k+1]*omega[k+1] + (alpha[k]-alpha[j])*omega[k] + beta[k]*omega[k-1] - beta[j]*omega_old[k];
    308:     if (omega_old[k] > 0) omega_old[k] = binv*(omega_old[k] + T);
    309:     else omega_old[k] = binv*(omega_old[k] - T);
    310:   }
    311:   omega_old[j-1] = binv*T;

    313:   /* Swap omega and omega_old */
    314:   for (k=0;k<j;k++) {
    315:     omega[k] = omega_old[k];
    316:     omega_old[k] = omega[k];
    317:   }
    318:   omega[j] = eps1;
    319:   PetscFunctionReturnVoid();
    320: }

    322: static void compute_int(PetscBool *which,PetscReal *mu,PetscInt j,PetscReal delta,PetscReal eta)
    323: {
    324:   PetscInt  i,k,maxpos;
    325:   PetscReal max;
    326:   PetscBool found;

    329:   /* initialize which */
    330:   found = PETSC_FALSE;
    331:   maxpos = 0;
    332:   max = 0.0;
    333:   for (i=0;i<j;i++) {
    334:     if (PetscAbsReal(mu[i]) >= delta) {
    335:       which[i] = PETSC_TRUE;
    336:       found = PETSC_TRUE;
    337:     } else which[i] = PETSC_FALSE;
    338:     if (PetscAbsReal(mu[i]) > max) {
    339:       maxpos = i;
    340:       max = PetscAbsReal(mu[i]);
    341:     }
    342:   }
    343:   if (!found) which[maxpos] = PETSC_TRUE;

    345:   for (i=0;i<j;i++) {
    346:     if (which[i]) {
    347:       /* find left interval */
    348:       for (k=i;k>=0;k--) {
    349:         if (PetscAbsReal(mu[k])<eta || which[k]) break;
    350:         else which[k] = PETSC_TRUE;
    351:       }
    352:       /* find right interval */
    353:       for (k=i+1;k<j;k++) {
    354:         if (PetscAbsReal(mu[k])<eta || which[k]) break;
    355:         else which[k] = PETSC_TRUE;
    356:       }
    357:     }
    358:   }
    359:   PetscFunctionReturnVoid();
    360: }

    362: /*
    363:    EPSPartialLanczos - Partial reorthogonalization.
    364: */
    365: static PetscErrorCode EPSPartialLanczos(EPS eps,PetscReal *alpha,PetscReal *beta,PetscInt k,PetscInt *M,PetscBool *breakdown,PetscReal anorm)
    366: {
    368:   EPS_LANCZOS    *lanczos = (EPS_LANCZOS*)eps->data;
    369:   PetscInt       i,j,m = *M;
    370:   Vec            vj,vj1;
    371:   PetscReal      norm,*omega,lomega[100],*omega_old,lomega_old[100],eps1,delta,eta;
    372:   PetscBool      *which,lwhich[100],*which2,lwhich2[100];
    373:   PetscBool      reorth = PETSC_FALSE,force_reorth = PETSC_FALSE;
    374:   PetscBool      fro = PETSC_FALSE,estimate_anorm = PETSC_FALSE;
    375:   PetscScalar    *hwork,lhwork[100];

    378:   if (m>100) {
    379:     PetscMalloc5(m,&omega,m,&omega_old,m,&which,m,&which2,m,&hwork);
    380:   } else {
    381:     omega     = lomega;
    382:     omega_old = lomega_old;
    383:     which     = lwhich;
    384:     which2    = lwhich2;
    385:     hwork     = lhwork;
    386:   }

    388:   eps1 = PetscSqrtReal((PetscReal)eps->n)*PETSC_MACHINE_EPSILON/2;
    389:   delta = PETSC_SQRT_MACHINE_EPSILON/PetscSqrtReal((PetscReal)eps->ncv);
    390:   eta = PetscPowReal(PETSC_MACHINE_EPSILON,3.0/4.0)/PetscSqrtReal((PetscReal)eps->ncv);
    391:   if (anorm < 0.0) {
    392:     anorm = 1.0;
    393:     estimate_anorm = PETSC_TRUE;
    394:   }
    395:   for (i=0;i<PetscMax(100,m);i++) omega[i] = omega_old[i] = 0.0;
    396:   for (i=0;i<k;i++) which[i] = PETSC_TRUE;

    398:   BVSetActiveColumns(eps->V,0,m);
    399:   for (j=k;j<m;j++) {
    400:     BVGetColumn(eps->V,j,&vj);
    401:     BVGetColumn(eps->V,j+1,&vj1);
    402:     STApply(eps->st,vj,vj1);
    403:     BVRestoreColumn(eps->V,j,&vj);
    404:     BVRestoreColumn(eps->V,j+1,&vj1);
    405:     if (fro) {
    406:       /* Lanczos step with full reorthogonalization */
    407:       BVOrthogonalizeColumn(eps->V,j+1,hwork,&norm,breakdown);
    408:       alpha[j] = PetscRealPart(hwork[j]);
    409:     } else {
    410:       /* Lanczos step */
    411:       which[j] = PETSC_TRUE;
    412:       if (j-2>=k) which[j-2] = PETSC_FALSE;
    413:       BVOrthogonalizeSomeColumn(eps->V,j+1,which,hwork,&norm,breakdown);
    414:       alpha[j] = PetscRealPart(hwork[j]);
    415:       beta[j] = norm;

    417:       /* Estimate ||A|| if needed */
    418:       if (estimate_anorm) {
    419:         if (j>k) anorm = PetscMax(anorm,PetscAbsReal(alpha[j])+norm+beta[j-1]);
    420:         else anorm = PetscMax(anorm,PetscAbsReal(alpha[j])+norm);
    421:       }

    423:       /* Check if reorthogonalization is needed */
    424:       reorth = PETSC_FALSE;
    425:       if (j>k) {
    426:         update_omega(omega,omega_old,j,alpha,beta-1,eps1,anorm);
    427:         for (i=0;i<j-k;i++) {
    428:           if (PetscAbsReal(omega[i]) > delta) reorth = PETSC_TRUE;
    429:         }
    430:       }
    431:       if (reorth || force_reorth) {
    432:         for (i=0;i<k;i++) which2[i] = PETSC_FALSE;
    433:         for (i=k;i<=j;i++) which2[i] = PETSC_TRUE;
    434:         if (lanczos->reorthog == EPS_LANCZOS_REORTHOG_PERIODIC) {
    435:           /* Periodic reorthogonalization */
    436:           if (force_reorth) force_reorth = PETSC_FALSE;
    437:           else force_reorth = PETSC_TRUE;
    438:           for (i=0;i<j-k;i++) omega[i] = eps1;
    439:         } else {
    440:           /* Partial reorthogonalization */
    441:           if (force_reorth) force_reorth = PETSC_FALSE;
    442:           else {
    443:             force_reorth = PETSC_TRUE;
    444:             compute_int(which2+k,omega,j-k,delta,eta);
    445:             for (i=0;i<j-k;i++) {
    446:               if (which2[i+k]) omega[i] = eps1;
    447:             }
    448:           }
    449:         }
    450:         BVOrthogonalizeSomeColumn(eps->V,j+1,which2,hwork,&norm,breakdown);
    451:       }
    452:     }

    454:     if (*breakdown || norm < eps->n*anorm*PETSC_MACHINE_EPSILON) {
    455:       *M = j+1;
    456:       break;
    457:     }
    458:     if (!fro && norm*delta < anorm*eps1) {
    459:       fro = PETSC_TRUE;
    460:       PetscInfo1(eps,"Switching to full reorthogonalization at iteration %D\n",eps->its);
    461:     }
    462:     beta[j] = norm;
    463:     BVScaleColumn(eps->V,j+1,1.0/norm);
    464:   }

    466:   if (m>100) {
    467:     PetscFree5(omega,omega_old,which,which2,hwork);
    468:   }
    469:   return(0);
    470: }

    472: /*
    473:    EPSBasicLanczos - Computes an m-step Lanczos factorization. The first k
    474:    columns are assumed to be locked and therefore they are not modified. On
    475:    exit, the following relation is satisfied:

    477:                     OP * V - V * T = f * e_m^T

    479:    where the columns of V are the Lanczos vectors, T is a tridiagonal matrix,
    480:    f is the residual vector and e_m is the m-th vector of the canonical basis.
    481:    The Lanczos vectors (together with vector f) are B-orthogonal (to working
    482:    accuracy) if full reorthogonalization is being used, otherwise they are
    483:    (B-)semi-orthogonal. On exit, beta contains the B-norm of f and the next
    484:    Lanczos vector can be computed as v_{m+1} = f / beta.

    486:    This function simply calls another function which depends on the selected
    487:    reorthogonalization strategy.
    488: */
    489: static PetscErrorCode EPSBasicLanczos(EPS eps,PetscReal *alpha,PetscReal *beta,PetscInt k,PetscInt *m,PetscBool *breakdown,PetscReal anorm)
    490: {
    491:   PetscErrorCode     ierr;
    492:   EPS_LANCZOS        *lanczos = (EPS_LANCZOS*)eps->data;
    493:   PetscScalar        *T;
    494:   PetscInt           i,n=*m;
    495:   PetscReal          betam;
    496:   BVOrthogRefineType orthog_ref;

    499:   switch (lanczos->reorthog) {
    500:     case EPS_LANCZOS_REORTHOG_LOCAL:
    501:       EPSLocalLanczos(eps,alpha,beta,k,m,breakdown);
    502:       break;
    503:     case EPS_LANCZOS_REORTHOG_FULL:
    504:       EPSFullLanczos(eps,alpha,beta,k,m,breakdown);
    505:       break;
    506:     case EPS_LANCZOS_REORTHOG_SELECTIVE:
    507:       EPSSelectiveLanczos(eps,alpha,beta,k,m,breakdown,anorm);
    508:       break;
    509:     case EPS_LANCZOS_REORTHOG_PERIODIC:
    510:     case EPS_LANCZOS_REORTHOG_PARTIAL:
    511:       EPSPartialLanczos(eps,alpha,beta,k,m,breakdown,anorm);
    512:       break;
    513:     case EPS_LANCZOS_REORTHOG_DELAYED:
    514:       PetscMalloc1(n*n,&T);
    515:       BVGetOrthogonalization(eps->V,NULL,&orthog_ref,NULL,NULL);
    516:       if (orthog_ref == BV_ORTHOG_REFINE_NEVER) {
    517:         EPSDelayedArnoldi1(eps,T,n,k,m,&betam,breakdown);
    518:       } else {
    519:         EPSDelayedArnoldi(eps,T,n,k,m,&betam,breakdown);
    520:       }
    521:       for (i=k;i<n-1;i++) {
    522:         alpha[i] = PetscRealPart(T[n*i+i]);
    523:         beta[i] = PetscRealPart(T[n*i+i+1]);
    524:       }
    525:       alpha[n-1] = PetscRealPart(T[n*(n-1)+n-1]);
    526:       beta[n-1] = betam;
    527:       PetscFree(T);
    528:       break;
    529:   }
    530:   return(0);
    531: }

    533: PetscErrorCode EPSSolve_Lanczos(EPS eps)
    534: {
    535:   EPS_LANCZOS    *lanczos = (EPS_LANCZOS*)eps->data;
    537:   PetscInt       nconv,i,j,k,l,x,n,*perm,restart,ncv=eps->ncv,r,ld;
    538:   Vec            vi,vj,w;
    539:   Mat            U;
    540:   PetscScalar    *Y,*ritz,stmp;
    541:   PetscReal      *d,*e,*bnd,anorm,beta,norm,rtmp,resnorm;
    542:   PetscBool      breakdown;
    543:   char           *conv,ctmp;

    546:   DSGetLeadingDimension(eps->ds,&ld);
    547:   PetscMalloc4(ncv,&ritz,ncv,&bnd,ncv,&perm,ncv,&conv);

    549:   /* The first Lanczos vector is the normalized initial vector */
    550:   EPSGetStartVector(eps,0,NULL);

    552:   anorm = -1.0;
    553:   nconv = 0;

    555:   /* Restart loop */
    556:   while (eps->reason == EPS_CONVERGED_ITERATING) {
    557:     eps->its++;

    559:     /* Compute an ncv-step Lanczos factorization */
    560:     n = PetscMin(nconv+eps->mpd,ncv);
    561:     DSGetArrayReal(eps->ds,DS_MAT_T,&d);
    562:     e = d + ld;
    563:     EPSBasicLanczos(eps,d,e,nconv,&n,&breakdown,anorm);
    564:     beta = e[n-1];
    565:     DSRestoreArrayReal(eps->ds,DS_MAT_T,&d);
    566:     DSSetDimensions(eps->ds,n,0,nconv,0);
    567:     DSSetState(eps->ds,DS_STATE_INTERMEDIATE);
    568:     BVSetActiveColumns(eps->V,nconv,n);

    570:     /* Solve projected problem */
    571:     DSSolve(eps->ds,ritz,NULL);
    572:     DSSort(eps->ds,ritz,NULL,NULL,NULL,NULL);
    573:     DSSynchronize(eps->ds,ritz,NULL);

    575:     /* Estimate ||A|| */
    576:     for (i=nconv;i<n;i++)
    577:       anorm = PetscMax(anorm,PetscAbsReal(PetscRealPart(ritz[i])));

    579:     /* Compute residual norm estimates as beta*abs(Y(m,:)) + eps*||A|| */
    580:     DSGetArray(eps->ds,DS_MAT_Q,&Y);
    581:     for (i=nconv;i<n;i++) {
    582:       resnorm = beta*PetscAbsScalar(Y[n-1+i*ld]) + PETSC_MACHINE_EPSILON*anorm;
    583:       (*eps->converged)(eps,ritz[i],eps->eigi[i],resnorm,&bnd[i],eps->convergedctx);
    584:       if (bnd[i]<eps->tol) conv[i] = 'C';
    585:       else conv[i] = 'N';
    586:     }
    587:     DSRestoreArray(eps->ds,DS_MAT_Q,&Y);

    589:     /* purge repeated ritz values */
    590:     if (lanczos->reorthog == EPS_LANCZOS_REORTHOG_LOCAL) {
    591:       for (i=nconv+1;i<n;i++) {
    592:         if (conv[i] == 'C' && PetscAbsScalar((ritz[i]-ritz[i-1])/ritz[i]) < eps->tol) conv[i] = 'R';
    593:       }
    594:     }

    596:     /* Compute restart vector */
    597:     if (breakdown) {
    598:       PetscInfo2(eps,"Breakdown in Lanczos method (it=%D norm=%g)\n",eps->its,(double)beta);
    599:     } else {
    600:       restart = nconv;
    601:       while (restart<n && conv[restart] != 'N') restart++;
    602:       if (restart >= n) {
    603:         breakdown = PETSC_TRUE;
    604:       } else {
    605:         for (i=restart+1;i<n;i++) {
    606:           if (conv[i] == 'N') {
    607:             SlepcSCCompare(eps->sc,ritz[restart],0.0,ritz[i],0.0,&r);
    608:             if (r>0) restart = i;
    609:           }
    610:         }
    611:         DSGetArray(eps->ds,DS_MAT_Q,&Y);
    612:         BVMultColumn(eps->V,1.0,0.0,n,Y+restart*ld+nconv);
    613:         DSRestoreArray(eps->ds,DS_MAT_Q,&Y);
    614:       }
    615:     }

    617:     /* Count and put converged eigenvalues first */
    618:     for (i=nconv;i<n;i++) perm[i] = i;
    619:     for (k=nconv;k<n;k++) {
    620:       if (conv[perm[k]] != 'C') {
    621:         j = k + 1;
    622:         while (j<n && conv[perm[j]] != 'C') j++;
    623:         if (j>=n) break;
    624:         l = perm[k]; perm[k] = perm[j]; perm[j] = l;
    625:       }
    626:     }

    628:     /* Sort eigenvectors according to permutation */
    629:     DSGetArray(eps->ds,DS_MAT_Q,&Y);
    630:     for (i=nconv;i<k;i++) {
    631:       x = perm[i];
    632:       if (x != i) {
    633:         j = i + 1;
    634:         while (perm[j] != i) j++;
    635:         /* swap eigenvalues i and j */
    636:         stmp = ritz[x]; ritz[x] = ritz[i]; ritz[i] = stmp;
    637:         rtmp = bnd[x]; bnd[x] = bnd[i]; bnd[i] = rtmp;
    638:         ctmp = conv[x]; conv[x] = conv[i]; conv[i] = ctmp;
    639:         perm[j] = x; perm[i] = i;
    640:         /* swap eigenvectors i and j */
    641:         for (l=0;l<n;l++) {
    642:           stmp = Y[l+x*ld]; Y[l+x*ld] = Y[l+i*ld]; Y[l+i*ld] = stmp;
    643:         }
    644:       }
    645:     }
    646:     DSRestoreArray(eps->ds,DS_MAT_Q,&Y);

    648:     /* compute converged eigenvectors */
    649:     DSGetMat(eps->ds,DS_MAT_Q,&U);
    650:     BVMultInPlace(eps->V,U,nconv,k);
    651:     MatDestroy(&U);

    653:     /* purge spurious ritz values */
    654:     if (lanczos->reorthog == EPS_LANCZOS_REORTHOG_LOCAL) {
    655:       for (i=nconv;i<k;i++) {
    656:         BVGetColumn(eps->V,i,&vi);
    657:         VecNorm(vi,NORM_2,&norm);
    658:         VecScale(vi,1.0/norm);
    659:         w = eps->work[0];
    660:         STApply(eps->st,vi,w);
    661:         VecAXPY(w,-ritz[i],vi);
    662:         BVRestoreColumn(eps->V,i,&vi);
    663:         VecNorm(w,NORM_2,&norm);
    664:         (*eps->converged)(eps,ritz[i],eps->eigi[i],norm,&bnd[i],eps->convergedctx);
    665:         if (bnd[i]>=eps->tol) conv[i] = 'S';
    666:       }
    667:       for (i=nconv;i<k;i++) {
    668:         if (conv[i] != 'C') {
    669:           j = i + 1;
    670:           while (j<k && conv[j] != 'C') j++;
    671:           if (j>=k) break;
    672:           /* swap eigenvalues i and j */
    673:           stmp = ritz[j]; ritz[j] = ritz[i]; ritz[i] = stmp;
    674:           rtmp = bnd[j]; bnd[j] = bnd[i]; bnd[i] = rtmp;
    675:           ctmp = conv[j]; conv[j] = conv[i]; conv[i] = ctmp;
    676:           /* swap eigenvectors i and j */
    677:           BVGetColumn(eps->V,i,&vi);
    678:           BVGetColumn(eps->V,j,&vj);
    679:           VecSwap(vi,vj);
    680:           BVRestoreColumn(eps->V,i,&vi);
    681:           BVRestoreColumn(eps->V,j,&vj);
    682:         }
    683:       }
    684:       k = i;
    685:     }

    687:     /* store ritz values and estimated errors */
    688:     for (i=nconv;i<n;i++) {
    689:       eps->eigr[i] = ritz[i];
    690:       eps->errest[i] = bnd[i];
    691:     }
    692:     nconv = k;
    693:     EPSMonitor(eps,eps->its,nconv,eps->eigr,eps->eigi,eps->errest,n);
    694:     (*eps->stopping)(eps,eps->its,eps->max_it,nconv,eps->nev,&eps->reason,eps->stoppingctx);

    696:     if (eps->reason == EPS_CONVERGED_ITERATING) { /* copy restart vector */
    697:       BVCopyColumn(eps->V,n,nconv);
    698:       if (lanczos->reorthog == EPS_LANCZOS_REORTHOG_LOCAL && !breakdown) {
    699:         /* Reorthonormalize restart vector */
    700:         BVOrthonormalizeColumn(eps->V,nconv,PETSC_FALSE,NULL,&breakdown);
    701:       }
    702:       if (breakdown) {
    703:         /* Use random vector for restarting */
    704:         PetscInfo(eps,"Using random vector for restart\n");
    705:         EPSGetStartVector(eps,nconv,&breakdown);
    706:       }
    707:       if (breakdown) { /* give up */
    708:         eps->reason = EPS_DIVERGED_BREAKDOWN;
    709:         PetscInfo(eps,"Unable to generate more start vectors\n");
    710:       }
    711:     }
    712:   }
    713:   eps->nconv = nconv;

    715:   PetscFree4(ritz,bnd,perm,conv);
    716:   return(0);
    717: }

    719: PetscErrorCode EPSSetFromOptions_Lanczos(PetscOptionItems *PetscOptionsObject,EPS eps)
    720: {
    721:   PetscErrorCode         ierr;
    722:   EPS_LANCZOS            *lanczos = (EPS_LANCZOS*)eps->data;
    723:   PetscBool              flg;
    724:   EPSLanczosReorthogType reorthog;

    727:   PetscOptionsHead(PetscOptionsObject,"EPS Lanczos Options");

    729:     PetscOptionsEnum("-eps_lanczos_reorthog","Lanczos reorthogonalization","EPSLanczosSetReorthog",EPSLanczosReorthogTypes,(PetscEnum)lanczos->reorthog,(PetscEnum*)&reorthog,&flg);
    730:     if (flg) { EPSLanczosSetReorthog(eps,reorthog); }

    732:   PetscOptionsTail();
    733:   return(0);
    734: }

    736: static PetscErrorCode EPSLanczosSetReorthog_Lanczos(EPS eps,EPSLanczosReorthogType reorthog)
    737: {
    738:   EPS_LANCZOS *lanczos = (EPS_LANCZOS*)eps->data;

    741:   switch (reorthog) {
    742:     case EPS_LANCZOS_REORTHOG_LOCAL:
    743:     case EPS_LANCZOS_REORTHOG_FULL:
    744:     case EPS_LANCZOS_REORTHOG_DELAYED:
    745:     case EPS_LANCZOS_REORTHOG_SELECTIVE:
    746:     case EPS_LANCZOS_REORTHOG_PERIODIC:
    747:     case EPS_LANCZOS_REORTHOG_PARTIAL:
    748:       if (lanczos->reorthog != reorthog) {
    749:         lanczos->reorthog = reorthog;
    750:         eps->state = EPS_STATE_INITIAL;
    751:       }
    752:       break;
    753:     default:
    754:       SETERRQ(PetscObjectComm((PetscObject)eps),PETSC_ERR_ARG_OUTOFRANGE,"Invalid reorthogonalization type");
    755:   }
    756:   return(0);
    757: }

    759: /*@
    760:    EPSLanczosSetReorthog - Sets the type of reorthogonalization used during the Lanczos
    761:    iteration.

    763:    Logically Collective on eps

    765:    Input Parameters:
    766: +  eps - the eigenproblem solver context
    767: -  reorthog - the type of reorthogonalization

    769:    Options Database Key:
    770: .  -eps_lanczos_reorthog - Sets the reorthogonalization type (either 'local', 'selective',
    771:                          'periodic', 'partial', 'full' or 'delayed')

    773:    Level: advanced

    775: .seealso: EPSLanczosGetReorthog(), EPSLanczosReorthogType
    776: @*/
    777: PetscErrorCode EPSLanczosSetReorthog(EPS eps,EPSLanczosReorthogType reorthog)
    778: {

    784:   PetscTryMethod(eps,"EPSLanczosSetReorthog_C",(EPS,EPSLanczosReorthogType),(eps,reorthog));
    785:   return(0);
    786: }

    788: static PetscErrorCode EPSLanczosGetReorthog_Lanczos(EPS eps,EPSLanczosReorthogType *reorthog)
    789: {
    790:   EPS_LANCZOS *lanczos = (EPS_LANCZOS*)eps->data;

    793:   *reorthog = lanczos->reorthog;
    794:   return(0);
    795: }

    797: /*@
    798:    EPSLanczosGetReorthog - Gets the type of reorthogonalization used during
    799:    the Lanczos iteration.

    801:    Not Collective

    803:    Input Parameter:
    804: .  eps - the eigenproblem solver context

    806:    Output Parameter:
    807: .  reorthog - the type of reorthogonalization

    809:    Level: advanced

    811: .seealso: EPSLanczosSetReorthog(), EPSLanczosReorthogType
    812: @*/
    813: PetscErrorCode EPSLanczosGetReorthog(EPS eps,EPSLanczosReorthogType *reorthog)
    814: {

    820:   PetscUseMethod(eps,"EPSLanczosGetReorthog_C",(EPS,EPSLanczosReorthogType*),(eps,reorthog));
    821:   return(0);
    822: }

    824: PetscErrorCode EPSReset_Lanczos(EPS eps)
    825: {
    827:   EPS_LANCZOS    *lanczos = (EPS_LANCZOS*)eps->data;

    830:   BVDestroy(&lanczos->AV);
    831:   lanczos->allocsize = 0;
    832:   return(0);
    833: }

    835: PetscErrorCode EPSDestroy_Lanczos(EPS eps)
    836: {

    840:   PetscFree(eps->data);
    841:   PetscObjectComposeFunction((PetscObject)eps,"EPSLanczosSetReorthog_C",NULL);
    842:   PetscObjectComposeFunction((PetscObject)eps,"EPSLanczosGetReorthog_C",NULL);
    843:   return(0);
    844: }

    846: PetscErrorCode EPSView_Lanczos(EPS eps,PetscViewer viewer)
    847: {
    849:   EPS_LANCZOS    *lanczos = (EPS_LANCZOS*)eps->data;
    850:   PetscBool      isascii;

    853:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii);
    854:   if (isascii) {
    855:     PetscViewerASCIIPrintf(viewer,"  %s reorthogonalization\n",EPSLanczosReorthogTypes[lanczos->reorthog]);
    856:   }
    857:   return(0);
    858: }

    860: SLEPC_EXTERN PetscErrorCode EPSCreate_Lanczos(EPS eps)
    861: {
    862:   EPS_LANCZOS    *ctx;

    866:   PetscNewLog(eps,&ctx);
    867:   eps->data = (void*)ctx;

    869:   eps->useds = PETSC_TRUE;

    871:   eps->ops->solve          = EPSSolve_Lanczos;
    872:   eps->ops->setup          = EPSSetUp_Lanczos;
    873:   eps->ops->setfromoptions = EPSSetFromOptions_Lanczos;
    874:   eps->ops->destroy        = EPSDestroy_Lanczos;
    875:   eps->ops->reset          = EPSReset_Lanczos;
    876:   eps->ops->view           = EPSView_Lanczos;
    877:   eps->ops->backtransform  = EPSBackTransform_Default;
    878:   eps->ops->computevectors = EPSComputeVectors_Hermitian;

    880:   PetscObjectComposeFunction((PetscObject)eps,"EPSLanczosSetReorthog_C",EPSLanczosSetReorthog_Lanczos);
    881:   PetscObjectComposeFunction((PetscObject)eps,"EPSLanczosGetReorthog_C",EPSLanczosGetReorthog_Lanczos);
    882:   return(0);
    883: }