Date: 2015-09-03T15:04:35+00:00

Actual source code: epskrylov.c
slepc-3.6.1 2015-09-03
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      1: /*
      2:    Common subroutines for all Krylov-type solvers.

      4:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      5:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      6:    Copyright (c) 2002-2015, Universitat Politecnica de Valencia, Spain

      8:    This file is part of SLEPc.

     10:    SLEPc is free software: you can redistribute it and/or modify it under  the
     11:    terms of version 3 of the GNU Lesser General Public License as published by
     12:    the Free Software Foundation.

     14:    SLEPc  is  distributed in the hope that it will be useful, but WITHOUT  ANY
     15:    WARRANTY;  without even the implied warranty of MERCHANTABILITY or  FITNESS
     16:    FOR  A  PARTICULAR PURPOSE. See the GNU Lesser General Public  License  for
     17:    more details.

     19:    You  should have received a copy of the GNU Lesser General  Public  License
     20:    along with SLEPc. If not, see <http://www.gnu.org/licenses/>.
     21:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     22: */

     24: #include <slepc/private/epsimpl.h>
     25: #include <slepc/private/slepcimpl.h>
     26: #include <slepcblaslapack.h>

     30: /*
     31:    EPSBasicArnoldi - Computes an m-step Arnoldi factorization. The first k
     32:    columns are assumed to be locked and therefore they are not modified. On
     33:    exit, the following relation is satisfied:

     35:                     OP * V - V * H = beta*v_m * e_m^T

     37:    where the columns of V are the Arnoldi vectors (which are B-orthonormal),
     38:    H is an upper Hessenberg matrix, e_m is the m-th vector of the canonical basis.
     39:    On exit, beta contains the B-norm of V[m] before normalization.
     40: */
     41: PetscErrorCode EPSBasicArnoldi(EPS eps,PetscBool trans,PetscScalar *H,PetscInt ldh,PetscInt k,PetscInt *M,PetscReal *beta,PetscBool *breakdown)
     42: {
     44:   PetscInt       j,m = *M;
     45:   Vec            vj,vj1;

     48:   BVSetActiveColumns(eps->V,0,m);
     49:   for (j=k;j<m;j++) {
     50:     BVGetColumn(eps->V,j,&vj);
     51:     BVGetColumn(eps->V,j+1,&vj1);
     52:     if (trans) {
     53:       STApplyTranspose(eps->st,vj,vj1);
     54:     } else {
     55:       STApply(eps->st,vj,vj1);
     56:     }
     57:     BVRestoreColumn(eps->V,j,&vj);
     58:     BVRestoreColumn(eps->V,j+1,&vj1);
     59:     BVOrthogonalizeColumn(eps->V,j+1,H+ldh*j,beta,breakdown);
     60:     H[j+1+ldh*j] = *beta;
     61:     if (*breakdown) {
     62:       *M = j+1;
     63:       break;
     64:     } else {
     65:       BVScaleColumn(eps->V,j+1,1.0/(*beta));
     66:     }
     67:   }
     68:   return(0);
     69: }

     73: /*
     74:    EPSDelayedArnoldi - This function is equivalent to EPSBasicArnoldi but
     75:    performs the computation in a different way. The main idea is that
     76:    reorthogonalization is delayed to the next Arnoldi step. This version is
     77:    more scalable but in some cases convergence may stagnate.
     78: */
     79: PetscErrorCode EPSDelayedArnoldi(EPS eps,PetscScalar *H,PetscInt ldh,PetscInt k,PetscInt *M,PetscReal *beta,PetscBool *breakdown)
     80: {
     82:   PetscInt       i,j,m=*M;
     83:   Vec            u,t;
     84:   PetscScalar    shh[100],*lhh,dot,dot2;
     85:   PetscReal      norm1=0.0,norm2;
     86:   Vec            vj,vj1,vj2;

     89:   if (m<=100) lhh = shh;
     90:   else {
     91:     PetscMalloc1(m,&lhh);
     92:   }
     93:   BVCreateVec(eps->V,&u);
     94:   BVCreateVec(eps->V,&t);

     96:   BVSetActiveColumns(eps->V,0,m);
     97:   for (j=k;j<m;j++) {
     98:     BVGetColumn(eps->V,j,&vj);
     99:     BVGetColumn(eps->V,j+1,&vj1);
    100:     STApply(eps->st,vj,vj1);
    101:     BVRestoreColumn(eps->V,j,&vj);
    102:     BVRestoreColumn(eps->V,j+1,&vj1);

    104:     BVDotColumnBegin(eps->V,j+1,H+ldh*j);
    105:     if (j>k) {
    106:       BVDotColumnBegin(eps->V,j,lhh);
    107:       BVGetColumn(eps->V,j,&vj);
    108:       VecDotBegin(vj,vj,&dot);
    109:     }
    110:     if (j>k+1) {
    111:       BVNormVecBegin(eps->V,u,NORM_2,&norm2);
    112:       BVGetColumn(eps->V,j-2,&vj2);
    113:       VecDotBegin(u,vj2,&dot2);
    114:     }

    116:     BVDotColumnEnd(eps->V,j+1,H+ldh*j);
    117:     if (j>k) {
    118:       BVDotColumnEnd(eps->V,j,lhh);
    119:       VecDotEnd(vj,vj,&dot);
    120:       BVRestoreColumn(eps->V,j,&vj);
    121:     }
    122:     if (j>k+1) {
    123:       BVNormVecEnd(eps->V,u,NORM_2,&norm2);
    124:       VecDotEnd(u,vj2,&dot2);
    125:       BVRestoreColumn(eps->V,j-2,&vj2);
    126:     }

    128:     if (j>k) {
    129:       norm1 = PetscSqrtReal(PetscRealPart(dot));
    130:       for (i=0;i<j;i++)
    131:         H[ldh*j+i] = H[ldh*j+i]/norm1;
    132:       H[ldh*j+j] = H[ldh*j+j]/dot;

    134:       BVCopyVec(eps->V,j,t);
    135:       BVScaleColumn(eps->V,j,1.0/norm1);
    136:       BVScaleColumn(eps->V,j+1,1.0/norm1);
    137:     }

    139:     BVMultColumn(eps->V,-1.0,1.0,j+1,H+ldh*j);

    141:     if (j>k) {
    142:       BVSetActiveColumns(eps->V,0,j);
    143:       BVMultVec(eps->V,-1.0,1.0,t,lhh);
    144:       BVSetActiveColumns(eps->V,0,m);
    145:       for (i=0;i<j;i++)
    146:         H[ldh*(j-1)+i] += lhh[i];
    147:     }

    149:     if (j>k+1) {
    150:       BVGetColumn(eps->V,j-1,&vj1);
    151:       VecCopy(u,vj1);
    152:       BVRestoreColumn(eps->V,j-1,&vj1);
    153:       BVScaleColumn(eps->V,j-1,1.0/norm2);
    154:       H[ldh*(j-2)+j-1] = norm2;
    155:     }

    157:     if (j<m-1) {
    158:       VecCopy(t,u);
    159:     }
    160:   }

    162:   BVNormVec(eps->V,t,NORM_2,&norm2);
    163:   VecScale(t,1.0/norm2);
    164:   BVGetColumn(eps->V,m-1,&vj1);
    165:   VecCopy(t,vj1);
    166:   BVRestoreColumn(eps->V,m-1,&vj1);
    167:   H[ldh*(m-2)+m-1] = norm2;

    169:   BVDotColumn(eps->V,m,lhh);

    171:   BVMultColumn(eps->V,-1.0,1.0,m,lhh);
    172:   for (i=0;i<m;i++)
    173:     H[ldh*(m-1)+i] += lhh[i];

    175:   BVNormColumn(eps->V,m,NORM_2,beta);
    176:   BVScaleColumn(eps->V,m,1.0 / *beta);
    177:   *breakdown = PETSC_FALSE;

    179:   if (m>100) { PetscFree(lhh); }
    180:   VecDestroy(&u);
    181:   VecDestroy(&t);
    182:   return(0);
    183: }

    187: /*
    188:    EPSDelayedArnoldi1 - This function is similar to EPSDelayedArnoldi,
    189:    but without reorthogonalization (only delayed normalization).
    190: */
    191: PetscErrorCode EPSDelayedArnoldi1(EPS eps,PetscScalar *H,PetscInt ldh,PetscInt k,PetscInt *M,PetscReal *beta,PetscBool *breakdown)
    192: {
    194:   PetscInt       i,j,m=*M;
    195:   PetscScalar    dot;
    196:   PetscReal      norm=0.0;
    197:   Vec            vj,vj1;

    200:   BVSetActiveColumns(eps->V,0,m);
    201:   for (j=k;j<m;j++) {
    202:     BVGetColumn(eps->V,j,&vj);
    203:     BVGetColumn(eps->V,j+1,&vj1);
    204:     STApply(eps->st,vj,vj1);
    205:     BVRestoreColumn(eps->V,j+1,&vj1);
    206:     BVDotColumnBegin(eps->V,j+1,H+ldh*j);
    207:     if (j>k) {
    208:       VecDotBegin(vj,vj,&dot);
    209:     }
    210:     BVDotColumnEnd(eps->V,j+1,H+ldh*j);
    211:     if (j>k) {
    212:       VecDotEnd(vj,vj,&dot);
    213:     }
    214:     BVRestoreColumn(eps->V,j,&vj);

    216:     if (j>k) {
    217:       norm = PetscSqrtReal(PetscRealPart(dot));
    218:       BVScaleColumn(eps->V,j,1.0/norm);
    219:       H[ldh*(j-1)+j] = norm;

    221:       for (i=0;i<j;i++)
    222:         H[ldh*j+i] = H[ldh*j+i]/norm;
    223:       H[ldh*j+j] = H[ldh*j+j]/dot;
    224:       BVScaleColumn(eps->V,j+1,1.0/norm);
    225:       *beta = norm;
    226:     }
    227:     BVMultColumn(eps->V,-1.0,1.0,j+1,H+ldh*j);
    228:   }

    230:   *breakdown = PETSC_FALSE;
    231:   return(0);
    232: }

    236: /*
    237:    EPSKrylovConvergence - Implements the loop that checks for convergence
    238:    in Krylov methods.

    240:    Input Parameters:
    241:      eps   - the eigensolver; some error estimates are updated in eps->errest
    242:      getall - whether all residuals must be computed
    243:      kini  - initial value of k (the loop variable)
    244:      nits  - number of iterations of the loop
    245:      V     - set of basis vectors (used only if trueresidual is activated)
    246:      nv    - number of vectors to process (dimension of Q, columns of V)
    247:      beta  - norm of f (the residual vector of the Arnoldi/Lanczos factorization)
    248:      corrf - correction factor for residual estimates (only in harmonic KS)

    250:    Output Parameters:
    251:      kout  - the first index where the convergence test failed
    252: */
    253: PetscErrorCode EPSKrylovConvergence(EPS eps,PetscBool getall,PetscInt kini,PetscInt nits,PetscReal beta,PetscReal corrf,PetscInt *kout)
    254: {
    256:   PetscInt       k,newk,marker,ld,inside;
    257:   PetscScalar    re,im,*Zr,*Zi,*X;
    258:   PetscReal      resnorm;
    259:   PetscBool      isshift,refined,istrivial;
    260:   Vec            x,y,w[3];

    263:   RGIsTrivial(eps->rg,&istrivial);
    264:   if (eps->trueres) {
    265:     BVCreateVec(eps->V,&x);
    266:     BVCreateVec(eps->V,&y);
    267:     BVCreateVec(eps->V,&w[0]);
    268:     BVCreateVec(eps->V,&w[2]);
    269: #if !defined(PETSC_USE_COMPLEX)
    270:     BVCreateVec(eps->V,&w[1]);
    271: #else
    272:     w[1] = NULL;
    273: #endif
    274:   }
    275:   DSGetLeadingDimension(eps->ds,&ld);
    276:   DSGetRefined(eps->ds,&refined);
    277:   PetscObjectTypeCompare((PetscObject)eps->st,STSHIFT,&isshift);
    278:   marker = -1;
    279:   if (eps->trackall) getall = PETSC_TRUE;
    280:   for (k=kini;k<kini+nits;k++) {
    281:     /* eigenvalue */
    282:     re = eps->eigr[k];
    283:     im = eps->eigi[k];
    284:     if (!istrivial || eps->trueres || isshift || eps->conv==EPS_CONV_NORM) {
    285:       STBackTransform(eps->st,1,&re,&im);
    286:     }
    287:     if (!istrivial) {
    288:       RGCheckInside(eps->rg,1,&re,&im,&inside);
    289:       if (marker==-1 && inside<=0) marker = k;
    290:       if (!(eps->trueres || isshift || eps->conv==EPS_CONV_NORM)) {  /* make sure eps->converged below uses the right value */
    291:         re = eps->eigr[k];
    292:         im = eps->eigi[k];
    293:       }
    294:     }
    295:     newk = k;
    296:     DSVectors(eps->ds,DS_MAT_X,&newk,&resnorm);
    297:     if (eps->trueres) {
    298:       DSGetArray(eps->ds,DS_MAT_X,&X);
    299:       Zr = X+k*ld;
    300:       if (newk==k+1) Zi = X+newk*ld;
    301:       else Zi = NULL;
    302:       EPSComputeRitzVector(eps,Zr,Zi,eps->V,x,y);
    303:       DSRestoreArray(eps->ds,DS_MAT_X,&X);
    304:       EPSComputeResidualNorm_Private(eps,re,im,x,y,w,&resnorm);
    305:     }
    306:     else if (!refined) resnorm *= beta*corrf;
    307:     /* error estimate */
    308:     (*eps->converged)(eps,re,im,resnorm,&eps->errest[k],eps->convergedctx);
    309:     if (marker==-1 && eps->errest[k] >= eps->tol) marker = k;
    310:     if (newk==k+1) {
    311:       eps->errest[k+1] = eps->errest[k];
    312:       k++;
    313:     }
    314:     if (marker!=-1 && !getall) break;
    315:   }
    316:   if (marker!=-1) k = marker;
    317:   *kout = k;
    318:   if (eps->trueres) {
    319:     VecDestroy(&x);
    320:     VecDestroy(&y);
    321:     VecDestroy(&w[0]);
    322:     VecDestroy(&w[2]);
    323: #if !defined(PETSC_USE_COMPLEX)
    324:     VecDestroy(&w[1]);
    325: #endif
    326:   }
    327:   return(0);
    328: }

    332: /*
    333:    EPSFullLanczos - Computes an m-step Lanczos factorization with full
    334:    reorthogonalization.  At each Lanczos step, the corresponding Lanczos
    335:    vector is orthogonalized with respect to all previous Lanczos vectors.
    336:    This is equivalent to computing an m-step Arnoldi factorization and
    337:    exploting symmetry of the operator.

    339:    The first k columns are assumed to be locked and therefore they are
    340:    not modified. On exit, the following relation is satisfied:

    342:                     OP * V - V * T = beta_m*v_m * e_m^T

    344:    where the columns of V are the Lanczos vectors (which are B-orthonormal),
    345:    T is a real symmetric tridiagonal matrix, and e_m is the m-th vector of
    346:    the canonical basis. The tridiagonal is stored as two arrays: alpha
    347:    contains the diagonal elements, beta the off-diagonal. On exit, the last
    348:    element of beta contains the B-norm of V[m] before normalization.
    349: */
    350: PetscErrorCode EPSFullLanczos(EPS eps,PetscReal *alpha,PetscReal *beta,PetscInt k,PetscInt *M,PetscBool *breakdown)
    351: {
    353:   PetscInt       j,m = *M;
    354:   Vec            vj,vj1;
    355:   PetscScalar    *hwork,lhwork[100];

    358:   if (m > 100) {
    359:     PetscMalloc1(m,&hwork);
    360:   } else hwork = lhwork;

    362:   BVSetActiveColumns(eps->V,0,m);
    363:   for (j=k;j<m;j++) {
    364:     BVGetColumn(eps->V,j,&vj);
    365:     BVGetColumn(eps->V,j+1,&vj1);
    366:     STApply(eps->st,vj,vj1);
    367:     BVRestoreColumn(eps->V,j,&vj);
    368:     BVRestoreColumn(eps->V,j+1,&vj1);
    369:     BVOrthogonalizeColumn(eps->V,j+1,hwork,beta+j,breakdown);
    370:     alpha[j] = PetscRealPart(hwork[j]);
    371:     if (*breakdown) {
    372:       *M = j+1;
    373:       break;
    374:     } else {
    375:       BVScaleColumn(eps->V,j+1,1.0/beta[j]);
    376:     }
    377:   }
    378:   if (m > 100) {
    379:     PetscFree(hwork);
    380:   }
    381:   return(0);
    382: }

    386: PetscErrorCode EPSPseudoLanczos(EPS eps,PetscReal *alpha,PetscReal *beta,PetscReal *omega,PetscInt k,PetscInt *M,PetscBool *breakdown,PetscBool *symmlost,PetscReal *cos,Vec w)
    387: {
    389:   PetscInt       j,m = *M,i,ld,l;
    390:   Vec            vj,vj1;
    391:   PetscScalar    *hwork,lhwork[100];
    392:   PetscReal      norm,norm1,norm2,t,*f,sym=0.0,fro=0.0;
    393:   PetscBLASInt   j_,one=1;

    396:   DSGetLeadingDimension(eps->ds,&ld);
    397:   DSGetDimensions(eps->ds,NULL,NULL,&l,NULL,NULL);
    398:   if (cos) *cos = 1.0;
    399:   if (m > 100) {
    400:     PetscMalloc1(m,&hwork);
    401:   } else hwork = lhwork;

    403:   BVSetActiveColumns(eps->V,0,m);
    404:   for (j=k;j<m;j++) {
    405:     BVGetColumn(eps->V,j,&vj);
    406:     BVGetColumn(eps->V,j+1,&vj1);
    407:     STApply(eps->st,vj,vj1);
    408:     BVRestoreColumn(eps->V,j,&vj);
    409:     BVRestoreColumn(eps->V,j+1,&vj1);
    410:     BVOrthogonalizeColumn(eps->V,j+1,hwork,&norm,breakdown);
    411:     alpha[j] = PetscRealPart(hwork[j]);
    412:     beta[j] = PetscAbsReal(norm);
    413:     DSGetArrayReal(eps->ds,DS_MAT_T,&f);
    414:     if (j==k) {
    415:       for (i=l;i<j-1;i++) hwork[i]-= f[2*ld+i];
    416:       for (i=0;i<l;i++) hwork[i] = 0.0;
    417:     }
    418:     DSRestoreArrayReal(eps->ds,DS_MAT_T,&f);
    419:     hwork[j-1] -= beta[j-1];
    420:     PetscBLASIntCast(j,&j_);
    421:     sym = SlepcAbs(BLASnrm2_(&j_,hwork,&one),sym);
    422:     fro = SlepcAbs(fro,SlepcAbs(alpha[j],beta[j]));
    423:     if (j>0) fro = SlepcAbs(fro,beta[j-1]);
    424:     if (sym/fro>PetscMax(PETSC_SQRT_MACHINE_EPSILON,10*eps->tol)) { *symmlost = PETSC_TRUE; *M=j+1; break; }
    425:     omega[j+1] = (norm<0.0)? -1.0: 1.0;
    426:     BVScaleColumn(eps->V,j+1,1.0/norm);
    427:     /* */
    428:     if (cos) {
    429:       BVGetColumn(eps->V,j+1,&vj1);
    430:       VecNorm(vj1,NORM_2,&norm1);
    431:       BVApplyMatrix(eps->V,vj1,w);
    432:       BVRestoreColumn(eps->V,j+1,&vj1);
    433:       VecNorm(w,NORM_2,&norm2);
    434:       t = 1.0/(norm1*norm2);
    435:       if (*cos>t) *cos = t;
    436:     }
    437:   }
    438:   if (m > 100) {
    439:     PetscFree(hwork);
    440:   }
    441:   return(0);
    442: }