Actual source code: epssolve.c

slepc-3.12.1 2019-11-08
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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:    EPS routines related to the solution process
 12: */

 14: #include <slepc/private/epsimpl.h>   /*I "slepceps.h" I*/
 15: #include <slepc/private/bvimpl.h>    /*I "slepcbv.h" I*/
 16: #include <petscdraw.h>

 18: PetscErrorCode EPSComputeVectors(EPS eps)
 19: {

 23:   EPSCheckSolved(eps,1);
 24:   if (eps->state==EPS_STATE_SOLVED && eps->ops->computevectors) {
 25:     (*eps->ops->computevectors)(eps);
 26:   }
 27:   eps->state = EPS_STATE_EIGENVECTORS;
 28:   return(0);
 29: }

 31: #define SWAP(a,b,t) {t=a;a=b;b=t;}

 33: static PetscErrorCode EPSComputeValues(EPS eps)
 34: {
 36:   PetscBool      injective,iscomp,isfilter;
 37:   PetscInt       i,n,aux,nconv0;
 38:   Mat            A,B=NULL,G,Z;

 41:   switch (eps->categ) {
 42:     case EPS_CATEGORY_KRYLOV:
 43:     case EPS_CATEGORY_OTHER:
 44:       STIsInjective(eps->st,&injective);
 45:       if (injective) {
 46:         /* one-to-one mapping: backtransform eigenvalues */
 47:         if (eps->ops->backtransform) {
 48:           (*eps->ops->backtransform)(eps);
 49:         } else SETERRQ(PetscObjectComm((PetscObject)eps),PETSC_ERR_PLIB,"Internal error, spectral transform should have a backtransform operation");
 50:       } else {
 51:         /* compute eigenvalues from Rayleigh quotient */
 52:         DSGetDimensions(eps->ds,&n,NULL,NULL,NULL,NULL);
 53:         if (!n) break;
 54:         EPSGetOperators(eps,&A,&B);
 55:         BVSetActiveColumns(eps->V,0,n);
 56:         DSGetCompact(eps->ds,&iscomp);
 57:         DSSetCompact(eps->ds,PETSC_FALSE);
 58:         DSGetMat(eps->ds,DS_MAT_A,&G);
 59:         BVMatProject(eps->V,A,eps->V,G);
 60:         DSRestoreMat(eps->ds,DS_MAT_A,&G);
 61:         if (B) {
 62:           DSGetMat(eps->ds,DS_MAT_B,&G);
 63:           BVMatProject(eps->V,B,eps->V,G);
 64:           DSRestoreMat(eps->ds,DS_MAT_A,&G);
 65:         }
 66:         DSSolve(eps->ds,eps->eigr,eps->eigi);
 67:         DSSort(eps->ds,eps->eigr,eps->eigi,NULL,NULL,NULL);
 68:         DSSynchronize(eps->ds,eps->eigr,eps->eigi);
 69:         DSSetCompact(eps->ds,iscomp);
 70:         if (eps->ishermitian && (!eps->isgeneralized || eps->ispositive)) { /* V = V * Z */
 71:           DSVectors(eps->ds,DS_MAT_X,NULL,NULL);
 72:           DSGetMat(eps->ds,DS_MAT_X,&Z);
 73:           BVMultInPlace(eps->V,Z,0,n);
 74:           MatDestroy(&Z);
 75:         }
 76:         /* in case of STFILTER discard computed eigenvalues that lie outside the wanted interval */
 77:         PetscObjectTypeCompare((PetscObject)eps->st,STFILTER,&isfilter);
 78:         if (isfilter) {
 79:           nconv0 = eps->nconv;
 80:           for (i=0;i<eps->nconv;i++) {
 81:             if (PetscRealPart(eps->eigr[eps->perm[i]])<eps->inta || PetscRealPart(eps->eigr[eps->perm[i]])>eps->intb) {
 82:               eps->nconv--;
 83:               if (i<eps->nconv) { SWAP(eps->perm[i],eps->perm[eps->nconv],aux); i--; }
 84:             }
 85:           }
 86:           if (nconv0>eps->nconv) {
 87:             PetscInfo1(eps,"Discarded %D computed eigenvalues lying outside the interval\n",nconv0-eps->nconv);
 88:           }
 89:         }
 90:       }
 91:       break;
 92:     case EPS_CATEGORY_PRECOND:
 93:     case EPS_CATEGORY_CONTOUR:
 94:       /* eigenvalues already available as an output of the solver */
 95:       break;
 96:   }
 97:   return(0);
 98: }

100: /*@
101:    EPSSolve - Solves the eigensystem.

103:    Collective on eps

105:    Input Parameter:
106: .  eps - eigensolver context obtained from EPSCreate()

108:    Options Database Keys:
109: +  -eps_view - print information about the solver used
110: .  -eps_view_mat0 binary - save the first matrix (A) to the default binary viewer
111: .  -eps_view_mat1 binary - save the second matrix (B) to the default binary viewer
112: .  -eps_view_vectors binary - save the computed eigenvectors to the default binary viewer
113: .  -eps_view_values - print computed eigenvalues
114: .  -eps_converged_reason - print reason for convergence, and number of iterations
115: .  -eps_error_absolute - print absolute errors of each eigenpair
116: .  -eps_error_relative - print relative errors of each eigenpair
117: -  -eps_error_backward - print backward errors of each eigenpair

119:    Level: beginner

121: .seealso: EPSCreate(), EPSSetUp(), EPSDestroy(), EPSSetTolerances()
122: @*/
123: PetscErrorCode EPSSolve(EPS eps)
124: {
126:   PetscInt       i;
127:   STMatMode      matmode;
128:   Mat            A,B;

132:   if (eps->state>=EPS_STATE_SOLVED) return(0);
133:   PetscLogEventBegin(EPS_Solve,eps,0,0,0);

135:   /* call setup */
136:   EPSSetUp(eps);
137:   eps->nconv = 0;
138:   eps->its   = 0;
139:   for (i=0;i<eps->ncv;i++) {
140:     eps->eigr[i]   = 0.0;
141:     eps->eigi[i]   = 0.0;
142:     eps->errest[i] = 0.0;
143:     eps->perm[i]   = i;
144:   }
145:   EPSViewFromOptions(eps,NULL,"-eps_view_pre");
146:   RGViewFromOptions(eps->rg,NULL,"-rg_view");

148:   /* call solver */
149:   (*eps->ops->solve)(eps);
150:   eps->state = EPS_STATE_SOLVED;

152:   STGetMatMode(eps->st,&matmode);
153:   if (matmode == ST_MATMODE_INPLACE && eps->ispositive) {
154:     /* Purify eigenvectors before reverting operator */
155:     EPSComputeVectors(eps);
156:   }
157:   STPostSolve(eps->st);

159:   if (!eps->reason) SETERRQ(PetscObjectComm((PetscObject)eps),PETSC_ERR_PLIB,"Internal error, solver returned without setting converged reason");

161:   /* Map eigenvalues back to the original problem if appropriate */
162:   EPSComputeValues(eps);

164: #if !defined(PETSC_USE_COMPLEX)
165:   /* reorder conjugate eigenvalues (positive imaginary first) */
166:   for (i=0;i<eps->nconv-1;i++) {
167:     if (eps->eigi[i] != 0) {
168:       if (eps->eigi[i] < 0) {
169:         eps->eigi[i] = -eps->eigi[i];
170:         eps->eigi[i+1] = -eps->eigi[i+1];
171:         /* the next correction only works with eigenvectors */
172:         EPSComputeVectors(eps);
173:         BVScaleColumn(eps->V,i+1,-1.0);
174:       }
175:       i++;
176:     }
177:   }
178: #endif

180:   /* sort eigenvalues according to eps->which parameter */
181:   SlepcSortEigenvalues(eps->sc,eps->nconv,eps->eigr,eps->eigi,eps->perm);
182:   PetscLogEventEnd(EPS_Solve,eps,0,0,0);

184:   /* various viewers */
185:   EPSViewFromOptions(eps,NULL,"-eps_view");
186:   EPSReasonViewFromOptions(eps);
187:   EPSErrorViewFromOptions(eps);
188:   EPSValuesViewFromOptions(eps);
189:   EPSVectorsViewFromOptions(eps);
190:   EPSGetOperators(eps,&A,&B);
191:   MatViewFromOptions(A,(PetscObject)eps,"-eps_view_mat0");
192:   if (eps->isgeneralized) {
193:     MatViewFromOptions(B,(PetscObject)eps,"-eps_view_mat1");
194:   }

196:   /* Remove deflation and initial subspaces */
197:   if (eps->nds) {
198:     BVSetNumConstraints(eps->V,0);
199:     eps->nds = 0;
200:   }
201:   eps->nini = 0;
202:   return(0);
203: }

205: /*@
206:    EPSGetIterationNumber - Gets the current iteration number. If the
207:    call to EPSSolve() is complete, then it returns the number of iterations
208:    carried out by the solution method.

210:    Not Collective

212:    Input Parameter:
213: .  eps - the eigensolver context

215:    Output Parameter:
216: .  its - number of iterations

218:    Note:
219:    During the i-th iteration this call returns i-1. If EPSSolve() is
220:    complete, then parameter "its" contains either the iteration number at
221:    which convergence was successfully reached, or failure was detected.
222:    Call EPSGetConvergedReason() to determine if the solver converged or
223:    failed and why.

225:    Level: intermediate

227: .seealso: EPSGetConvergedReason(), EPSSetTolerances()
228: @*/
229: PetscErrorCode EPSGetIterationNumber(EPS eps,PetscInt *its)
230: {
234:   *its = eps->its;
235:   return(0);
236: }

238: /*@
239:    EPSGetConverged - Gets the number of converged eigenpairs.

241:    Not Collective

243:    Input Parameter:
244: .  eps - the eigensolver context

246:    Output Parameter:
247: .  nconv - number of converged eigenpairs

249:    Note:
250:    This function should be called after EPSSolve() has finished.

252:    Level: beginner

254: .seealso: EPSSetDimensions(), EPSSolve()
255: @*/
256: PetscErrorCode EPSGetConverged(EPS eps,PetscInt *nconv)
257: {
261:   EPSCheckSolved(eps,1);
262:   *nconv = eps->nconv;
263:   return(0);
264: }

266: /*@
267:    EPSGetConvergedReason - Gets the reason why the EPSSolve() iteration was
268:    stopped.

270:    Not Collective

272:    Input Parameter:
273: .  eps - the eigensolver context

275:    Output Parameter:
276: .  reason - negative value indicates diverged, positive value converged

278:    Notes:
279:    Possible values for reason are
280: +  EPS_CONVERGED_TOL - converged up to tolerance
281: .  EPS_CONVERGED_USER - converged due to a user-defined condition
282: .  EPS_DIVERGED_ITS - required more than max_it iterations to reach convergence
283: .  EPS_DIVERGED_BREAKDOWN - generic breakdown in method
284: -  EPS_DIVERGED_SYMMETRY_LOST - pseudo-Lanczos was not able to keep symmetry

286:    Can only be called after the call to EPSSolve() is complete.

288:    Level: intermediate

290: .seealso: EPSSetTolerances(), EPSSolve(), EPSConvergedReason
291: @*/
292: PetscErrorCode EPSGetConvergedReason(EPS eps,EPSConvergedReason *reason)
293: {
297:   EPSCheckSolved(eps,1);
298:   *reason = eps->reason;
299:   return(0);
300: }

302: /*@
303:    EPSGetInvariantSubspace - Gets an orthonormal basis of the computed invariant
304:    subspace.

306:    Not Collective, but vectors are shared by all processors that share the EPS

308:    Input Parameter:
309: .  eps - the eigensolver context

311:    Output Parameter:
312: .  v - an array of vectors

314:    Notes:
315:    This function should be called after EPSSolve() has finished.

317:    The user should provide in v an array of nconv vectors, where nconv is
318:    the value returned by EPSGetConverged().

320:    The first k vectors returned in v span an invariant subspace associated
321:    with the first k computed eigenvalues (note that this is not true if the
322:    k-th eigenvalue is complex and matrix A is real; in this case the first
323:    k+1 vectors should be used). An invariant subspace X of A satisfies Ax
324:    in X for all x in X (a similar definition applies for generalized
325:    eigenproblems).

327:    Level: intermediate

329: .seealso: EPSGetEigenpair(), EPSGetConverged(), EPSSolve()
330: @*/
331: PetscErrorCode EPSGetInvariantSubspace(EPS eps,Vec *v)
332: {
334:   PetscInt       i;
335:   BV             V=eps->V;
336:   Vec            w;

342:   EPSCheckSolved(eps,1);
343:   if (!eps->ishermitian && eps->state==EPS_STATE_EIGENVECTORS) SETERRQ(PetscObjectComm((PetscObject)eps),PETSC_ERR_ARG_WRONGSTATE,"EPSGetInvariantSubspace must be called before EPSGetEigenpair,EPSGetEigenvector or EPSComputeError");
344:   if (eps->balance!=EPS_BALANCE_NONE && eps->D) {
345:     BVDuplicateResize(eps->V,eps->nconv,&V);
346:     BVSetActiveColumns(eps->V,0,eps->nconv);
347:     BVCopy(eps->V,V);
348:     for (i=0;i<eps->nconv;i++) {
349:       BVGetColumn(V,i,&w);
350:       VecPointwiseDivide(w,w,eps->D);
351:       BVRestoreColumn(V,i,&w);
352:     }
353:     BVOrthogonalize(V,NULL);
354:   }
355:   for (i=0;i<eps->nconv;i++) {
356:     BVCopyVec(V,i,v[i]);
357:   }
358:   if (eps->balance!=EPS_BALANCE_NONE && eps->D) {
359:     BVDestroy(&V);
360:   }
361:   return(0);
362: }

364: /*@C
365:    EPSGetEigenpair - Gets the i-th solution of the eigenproblem as computed by
366:    EPSSolve(). The solution consists in both the eigenvalue and the eigenvector.

368:    Logically Collective on eps

370:    Input Parameters:
371: +  eps - eigensolver context
372: -  i   - index of the solution

374:    Output Parameters:
375: +  eigr - real part of eigenvalue
376: .  eigi - imaginary part of eigenvalue
377: .  Vr   - real part of eigenvector
378: -  Vi   - imaginary part of eigenvector

380:    Notes:
381:    It is allowed to pass NULL for Vr and Vi, if the eigenvector is not
382:    required. Otherwise, the caller must provide valid Vec objects, i.e.,
383:    they must be created by the calling program with e.g. MatCreateVecs().

385:    If the eigenvalue is real, then eigi and Vi are set to zero. If PETSc is
386:    configured with complex scalars the eigenvalue is stored
387:    directly in eigr (eigi is set to zero) and the eigenvector in Vr (Vi is
388:    set to zero). In both cases, the user can pass NULL in eigi and Vi.

390:    The index i should be a value between 0 and nconv-1 (see EPSGetConverged()).
391:    Eigenpairs are indexed according to the ordering criterion established
392:    with EPSSetWhichEigenpairs().

394:    The 2-norm of the eigenvector is one unless the problem is generalized
395:    Hermitian. In this case the eigenvector is normalized with respect to the
396:    norm defined by the B matrix.

398:    Level: beginner

400: .seealso: EPSGetEigenvalue(), EPSGetEigenvector(), EPSGetLeftEigenvector(), EPSSolve(),
401:           EPSGetConverged(), EPSSetWhichEigenpairs(), EPSGetInvariantSubspace()
402: @*/
403: PetscErrorCode EPSGetEigenpair(EPS eps,PetscInt i,PetscScalar *eigr,PetscScalar *eigi,Vec Vr,Vec Vi)
404: {

410:   EPSCheckSolved(eps,1);
411:   if (i<0 || i>=eps->nconv) SETERRQ(PetscObjectComm((PetscObject)eps),PETSC_ERR_ARG_OUTOFRANGE,"Argument 2 out of range");
412:   EPSGetEigenvalue(eps,i,eigr,eigi);
413:   if (Vr || Vi) { EPSGetEigenvector(eps,i,Vr,Vi); }
414:   return(0);
415: }

417: /*@C
418:    EPSGetEigenvalue - Gets the i-th eigenvalue as computed by EPSSolve().

420:    Not Collective

422:    Input Parameters:
423: +  eps - eigensolver context
424: -  i   - index of the solution

426:    Output Parameters:
427: +  eigr - real part of eigenvalue
428: -  eigi - imaginary part of eigenvalue

430:    Notes:
431:    If the eigenvalue is real, then eigi is set to zero. If PETSc is
432:    configured with complex scalars the eigenvalue is stored
433:    directly in eigr (eigi is set to zero).

435:    The index i should be a value between 0 and nconv-1 (see EPSGetConverged()).
436:    Eigenpairs are indexed according to the ordering criterion established
437:    with EPSSetWhichEigenpairs().

439:    Level: beginner

441: .seealso: EPSSolve(), EPSGetConverged(), EPSSetWhichEigenpairs(), EPSGetEigenpair()
442: @*/
443: PetscErrorCode EPSGetEigenvalue(EPS eps,PetscInt i,PetscScalar *eigr,PetscScalar *eigi)
444: {
445:   PetscInt k;

449:   EPSCheckSolved(eps,1);
450:   if (i<0 || i>=eps->nconv) SETERRQ(PetscObjectComm((PetscObject)eps),PETSC_ERR_ARG_OUTOFRANGE,"Argument 2 out of range");
451:   k = eps->perm[i];
452: #if defined(PETSC_USE_COMPLEX)
453:   if (eigr) *eigr = eps->eigr[k];
454:   if (eigi) *eigi = 0;
455: #else
456:   if (eigr) *eigr = eps->eigr[k];
457:   if (eigi) *eigi = eps->eigi[k];
458: #endif
459:   return(0);
460: }

462: /*@
463:    EPSGetEigenvector - Gets the i-th right eigenvector as computed by EPSSolve().

465:    Logically Collective on eps

467:    Input Parameters:
468: +  eps - eigensolver context
469: -  i   - index of the solution

471:    Output Parameters:
472: +  Vr   - real part of eigenvector
473: -  Vi   - imaginary part of eigenvector

475:    Notes:
476:    The caller must provide valid Vec objects, i.e., they must be created
477:    by the calling program with e.g. MatCreateVecs().

479:    If the corresponding eigenvalue is real, then Vi is set to zero. If PETSc is
480:    configured with complex scalars the eigenvector is stored
481:    directly in Vr (Vi is set to zero). In any case, the user can pass NULL in Vr
482:    or Vi if one of them is not required.

484:    The index i should be a value between 0 and nconv-1 (see EPSGetConverged()).
485:    Eigenpairs are indexed according to the ordering criterion established
486:    with EPSSetWhichEigenpairs().

488:    The 2-norm of the eigenvector is one unless the problem is generalized
489:    Hermitian. In this case the eigenvector is normalized with respect to the
490:    norm defined by the B matrix.

492:    Level: beginner

494: .seealso: EPSSolve(), EPSGetConverged(), EPSSetWhichEigenpairs(), EPSGetEigenpair(), EPSGetLeftEigenvector()
495: @*/
496: PetscErrorCode EPSGetEigenvector(EPS eps,PetscInt i,Vec Vr,Vec Vi)
497: {
499:   PetscInt       k;

506:   EPSCheckSolved(eps,1);
507:   if (i<0 || i>=eps->nconv) SETERRQ(PetscObjectComm((PetscObject)eps),PETSC_ERR_ARG_OUTOFRANGE,"Argument 2 out of range");
508:   EPSComputeVectors(eps);
509:   k = eps->perm[i];
510:   BV_GetEigenvector(eps->V,k,eps->eigi[k],Vr,Vi);
511:   return(0);
512: }

514: /*@
515:    EPSGetLeftEigenvector - Gets the i-th left eigenvector as computed by EPSSolve().

517:    Logically Collective on eps

519:    Input Parameters:
520: +  eps - eigensolver context
521: -  i   - index of the solution

523:    Output Parameters:
524: +  Wr   - real part of left eigenvector
525: -  Wi   - imaginary part of left eigenvector

527:    Notes:
528:    The caller must provide valid Vec objects, i.e., they must be created
529:    by the calling program with e.g. MatCreateVecs().

531:    If the corresponding eigenvalue is real, then Wi is set to zero. If PETSc is
532:    configured with complex scalars the eigenvector is stored directly in Wr
533:    (Wi is set to zero). In any case, the user can pass NULL in Wr or Wi if
534:    one of them is not required.

536:    The index i should be a value between 0 and nconv-1 (see EPSGetConverged()).
537:    Eigensolutions are indexed according to the ordering criterion established
538:    with EPSSetWhichEigenpairs().

540:    Left eigenvectors are available only if the twosided flag was set, see
541:    EPSSetTwoSided().

543:    Level: intermediate

545: .seealso: EPSGetEigenvector(), EPSGetConverged(), EPSSetWhichEigenpairs(), EPSSetTwoSided()
546: @*/
547: PetscErrorCode EPSGetLeftEigenvector(EPS eps,PetscInt i,Vec Wr,Vec Wi)
548: {
550:   PetscInt       k;

557:   EPSCheckSolved(eps,1);
558:   if (!eps->twosided) SETERRQ(PetscObjectComm((PetscObject)eps),PETSC_ERR_ARG_WRONGSTATE,"Must request left vectors with EPSSetTwoSided");
559:   if (i<0 || i>=eps->nconv) SETERRQ(PetscObjectComm((PetscObject)eps),PETSC_ERR_ARG_OUTOFRANGE,"Argument 2 out of range");
560:   EPSComputeVectors(eps);
561:   k = eps->perm[i];
562:   BV_GetEigenvector(eps->W,k,eps->eigi[k],Wr,Wi);
563:   return(0);
564: }

566: /*@
567:    EPSGetErrorEstimate - Returns the error estimate associated to the i-th
568:    computed eigenpair.

570:    Not Collective

572:    Input Parameter:
573: +  eps - eigensolver context
574: -  i   - index of eigenpair

576:    Output Parameter:
577: .  errest - the error estimate

579:    Notes:
580:    This is the error estimate used internally by the eigensolver. The actual
581:    error bound can be computed with EPSComputeError(). See also the users
582:    manual for details.

584:    Level: advanced

586: .seealso: EPSComputeError()
587: @*/
588: PetscErrorCode EPSGetErrorEstimate(EPS eps,PetscInt i,PetscReal *errest)
589: {
593:   EPSCheckSolved(eps,1);
594:   if (i<0 || i>=eps->nconv) SETERRQ(PetscObjectComm((PetscObject)eps),PETSC_ERR_ARG_OUTOFRANGE,"Argument 2 out of range");
595:   *errest = eps->errest[eps->perm[i]];
596:   return(0);
597: }

599: /*
600:    EPSComputeResidualNorm_Private - Computes the norm of the residual vector
601:    associated with an eigenpair.

603:    Input Parameters:
604:      trans - whether A' must be used instead of A
605:      kr,ki - eigenvalue
606:      xr,xi - eigenvector
607:      z     - three work vectors (the second one not referenced in complex scalars)
608: */
609: PetscErrorCode EPSComputeResidualNorm_Private(EPS eps,PetscBool trans,PetscScalar kr,PetscScalar ki,Vec xr,Vec xi,Vec *z,PetscReal *norm)
610: {
612:   PetscInt       nmat;
613:   Mat            A,B;
614:   Vec            u,w;
615:   PetscScalar    alpha;
616: #if !defined(PETSC_USE_COMPLEX)
617:   Vec            v;
618:   PetscReal      ni,nr;
619: #endif
620:   PetscErrorCode (*matmult)(Mat,Vec,Vec) = trans? MatMultHermitianTranspose: MatMult;

623:   u = z[0]; w = z[2];
624:   STGetNumMatrices(eps->st,&nmat);
625:   STGetMatrix(eps->st,0,&A);
626:   if (nmat>1) { STGetMatrix(eps->st,1,&B); }

628: #if !defined(PETSC_USE_COMPLEX)
629:   v = z[1];
630:   if (ki == 0 || PetscAbsScalar(ki) < PetscAbsScalar(kr*PETSC_MACHINE_EPSILON)) {
631: #endif
632:     (*matmult)(A,xr,u);                          /* u=A*x */
633:     if (PetscAbsScalar(kr) > PETSC_MACHINE_EPSILON) {
634:       if (nmat>1) { (*matmult)(B,xr,w); }
635:       else { VecCopy(xr,w); }                    /* w=B*x */
636:       alpha = trans? -PetscConj(kr): -kr;
637:       VecAXPY(u,alpha,w);                        /* u=A*x-k*B*x */
638:     }
639:     VecNorm(u,NORM_2,norm);
640: #if !defined(PETSC_USE_COMPLEX)
641:   } else {
642:     (*matmult)(A,xr,u);                          /* u=A*xr */
643:     if (SlepcAbsEigenvalue(kr,ki) > PETSC_MACHINE_EPSILON) {
644:       if (nmat>1) { (*matmult)(B,xr,v); }
645:       else { VecCopy(xr,v); }                    /* v=B*xr */
646:       VecAXPY(u,-kr,v);                          /* u=A*xr-kr*B*xr */
647:       if (nmat>1) { (*matmult)(B,xi,w); }
648:       else { VecCopy(xi,w); }                    /* w=B*xi */
649:       VecAXPY(u,trans?-ki:ki,w);                 /* u=A*xr-kr*B*xr+ki*B*xi */
650:     }
651:     VecNorm(u,NORM_2,&nr);
652:     (*matmult)(A,xi,u);                          /* u=A*xi */
653:     if (SlepcAbsEigenvalue(kr,ki) > PETSC_MACHINE_EPSILON) {
654:       VecAXPY(u,-kr,w);                          /* u=A*xi-kr*B*xi */
655:       VecAXPY(u,trans?ki:-ki,v);                 /* u=A*xi-kr*B*xi-ki*B*xr */
656:     }
657:     VecNorm(u,NORM_2,&ni);
658:     *norm = SlepcAbsEigenvalue(nr,ni);
659:   }
660: #endif
661:   return(0);
662: }

664: /*@
665:    EPSComputeError - Computes the error (based on the residual norm) associated
666:    with the i-th computed eigenpair.

668:    Collective on eps

670:    Input Parameter:
671: +  eps  - the eigensolver context
672: .  i    - the solution index
673: -  type - the type of error to compute

675:    Output Parameter:
676: .  error - the error

678:    Notes:
679:    The error can be computed in various ways, all of them based on the residual
680:    norm ||Ax-kBx||_2 where k is the eigenvalue and x is the eigenvector.

682:    Level: beginner

684: .seealso: EPSErrorType, EPSSolve(), EPSGetErrorEstimate()
685: @*/
686: PetscErrorCode EPSComputeError(EPS eps,PetscInt i,EPSErrorType type,PetscReal *error)
687: {
689:   Mat            A,B;
690:   Vec            xr,xi,w[3];
691:   PetscReal      t,vecnorm=1.0,errorl;
692:   PetscScalar    kr,ki;
693:   PetscBool      flg;

700:   EPSCheckSolved(eps,1);

702:   /* allocate work vectors */
703: #if defined(PETSC_USE_COMPLEX)
704:   EPSSetWorkVecs(eps,3);
705:   xi   = NULL;
706:   w[1] = NULL;
707: #else
708:   EPSSetWorkVecs(eps,5);
709:   xi   = eps->work[3];
710:   w[1] = eps->work[4];
711: #endif
712:   xr   = eps->work[0];
713:   w[0] = eps->work[1];
714:   w[2] = eps->work[2];

716:   /* compute residual norm */
717:   EPSGetEigenpair(eps,i,&kr,&ki,xr,xi);
718:   EPSComputeResidualNorm_Private(eps,PETSC_FALSE,kr,ki,xr,xi,w,error);

720:   /* compute 2-norm of eigenvector */
721:   if (eps->problem_type==EPS_GHEP) {
722:     VecNorm(xr,NORM_2,&vecnorm);
723:   }

725:   /* if two-sided, compute left residual norm and take the maximum */
726:   if (eps->twosided) {
727:     EPSGetLeftEigenvector(eps,i,xr,xi);
728:     EPSComputeResidualNorm_Private(eps,PETSC_TRUE,kr,ki,xr,xi,w,&errorl);
729:     *error = PetscMax(*error,errorl);
730:   }

732:   /* compute error */
733:   switch (type) {
734:     case EPS_ERROR_ABSOLUTE:
735:       break;
736:     case EPS_ERROR_RELATIVE:
737:       *error /= SlepcAbsEigenvalue(kr,ki)*vecnorm;
738:       break;
739:     case EPS_ERROR_BACKWARD:
740:       /* initialization of matrix norms */
741:       if (!eps->nrma) {
742:         STGetMatrix(eps->st,0,&A);
743:         MatHasOperation(A,MATOP_NORM,&flg);
744:         if (!flg) SETERRQ(PetscObjectComm((PetscObject)eps),PETSC_ERR_ARG_WRONG,"The computation of backward errors requires a matrix norm operation");
745:         MatNorm(A,NORM_INFINITY,&eps->nrma);
746:       }
747:       if (eps->isgeneralized) {
748:         if (!eps->nrmb) {
749:           STGetMatrix(eps->st,1,&B);
750:           MatHasOperation(B,MATOP_NORM,&flg);
751:           if (!flg) SETERRQ(PetscObjectComm((PetscObject)eps),PETSC_ERR_ARG_WRONG,"The computation of backward errors requires a matrix norm operation");
752:           MatNorm(B,NORM_INFINITY,&eps->nrmb);
753:         }
754:       } else eps->nrmb = 1.0;
755:       t = SlepcAbsEigenvalue(kr,ki);
756:       *error /= (eps->nrma+t*eps->nrmb)*vecnorm;
757:       break;
758:     default:
759:       SETERRQ(PetscObjectComm((PetscObject)eps),PETSC_ERR_ARG_OUTOFRANGE,"Invalid error type");
760:   }
761:   return(0);
762: }

764: /*
765:    EPSGetStartVector - Generate a suitable vector to be used as the starting vector
766:    for the recurrence that builds the right subspace.

768:    Collective on eps

770:    Input Parameters:
771: +  eps - the eigensolver context
772: -  i   - iteration number

774:    Output Parameters:
775: .  breakdown - flag indicating that a breakdown has occurred

777:    Notes:
778:    The start vector is computed from another vector: for the first step (i=0),
779:    the first initial vector is used (see EPSSetInitialSpace()); otherwise a random
780:    vector is created. Then this vector is forced to be in the range of OP (only
781:    for generalized definite problems) and orthonormalized with respect to all
782:    V-vectors up to i-1. The resulting vector is placed in V[i].

784:    The flag breakdown is set to true if either i=0 and the vector belongs to the
785:    deflation space, or i>0 and the vector is linearly dependent with respect
786:    to the V-vectors.
787: */
788: PetscErrorCode EPSGetStartVector(EPS eps,PetscInt i,PetscBool *breakdown)
789: {
791:   PetscReal      norm;
792:   PetscBool      lindep;
793:   Vec            w,z;


799:   /* For the first step, use the first initial vector, otherwise a random one */
800:   if (i>0 || eps->nini==0) {
801:     BVSetRandomColumn(eps->V,i);
802:   }

804:   /* Force the vector to be in the range of OP for definite generalized problems */
805:   if (eps->ispositive || (eps->isgeneralized && eps->ishermitian)) {
806:     BVCreateVec(eps->V,&w);
807:     BVCopyVec(eps->V,i,w);
808:     BVGetColumn(eps->V,i,&z);
809:     STApply(eps->st,w,z);
810:     BVRestoreColumn(eps->V,i,&z);
811:     VecDestroy(&w);
812:   }

814:   /* Orthonormalize the vector with respect to previous vectors */
815:   BVOrthogonalizeColumn(eps->V,i,NULL,&norm,&lindep);
816:   if (breakdown) *breakdown = lindep;
817:   else if (lindep || norm == 0.0) {
818:     if (i==0) SETERRQ(PetscObjectComm((PetscObject)eps),1,"Initial vector is zero or belongs to the deflation space");
819:     else SETERRQ(PetscObjectComm((PetscObject)eps),1,"Unable to generate more start vectors");
820:   }
821:   BVScaleColumn(eps->V,i,1.0/norm);
822:   return(0);
823: }