Actual source code: ptoar.c

slepc-3.16.1 2021-11-17
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  1: /*
  2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
  4:    Copyright (c) 2002-2021, Universitat Politecnica de Valencia, Spain

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

 13:    Method: TOAR

 15:    Algorithm:

 17:        Two-Level Orthogonal Arnoldi.

 19:    References:

 21:        [1] Y. Su, J. Zhang and Z. Bai, "A compact Arnoldi algorithm for
 22:            polynomial eigenvalue problems", talk presented at RANMEP 2008.

 24:        [2] C. Campos and J.E. Roman, "Parallel Krylov solvers for the
 25:            polynomial eigenvalue problem in SLEPc", SIAM J. Sci. Comput.
 26:            38(5):S385-S411, 2016.

 28:        [3] D. Lu, Y. Su and Z. Bai, "Stability analysis of the two-level
 29:            orthogonal Arnoldi procedure", SIAM J. Matrix Anal. App.
 30:            37(1):195-214, 2016.
 31: */

 33: #include <slepc/private/pepimpl.h>
 34: #include "../src/pep/impls/krylov/pepkrylov.h"
 35: #include <slepcblaslapack.h>

 37: static PetscBool  cited = PETSC_FALSE;
 38: static const char citation[] =
 39:   "@Article{slepc-pep,\n"
 40:   "   author = \"C. Campos and J. E. Roman\",\n"
 41:   "   title = \"Parallel {Krylov} solvers for the polynomial eigenvalue problem in {SLEPc}\",\n"
 42:   "   journal = \"{SIAM} J. Sci. Comput.\",\n"
 43:   "   volume = \"38\",\n"
 44:   "   number = \"5\",\n"
 45:   "   pages = \"S385--S411\",\n"
 46:   "   year = \"2016,\"\n"
 47:   "   doi = \"https://doi.org/10.1137/15M1022458\"\n"
 48:   "}\n";

 50: PetscErrorCode PEPSetUp_TOAR(PEP pep)
 51: {
 53:   PEP_TOAR       *ctx = (PEP_TOAR*)pep->data;
 54:   PetscBool      sinv,flg;
 55:   PetscInt       i;

 58:   PEPCheckShiftSinvert(pep);
 59:   PEPSetDimensions_Default(pep,pep->nev,&pep->ncv,&pep->mpd);
 60:   if (!ctx->lock && pep->mpd<pep->ncv) SETERRQ(PetscObjectComm((PetscObject)pep),PETSC_ERR_SUP,"Should not use mpd parameter in non-locking variant");
 61:   if (pep->max_it==PETSC_DEFAULT) pep->max_it = PetscMax(100,2*(pep->nmat-1)*pep->n/pep->ncv);
 62:   if (!pep->which) { PEPSetWhichEigenpairs_Default(pep); }
 63:   if (pep->which==PEP_ALL) SETERRQ(PetscObjectComm((PetscObject)pep),PETSC_ERR_SUP,"This solver does not support computing all eigenvalues");
 64:   if (pep->problem_type!=PEP_GENERAL) {
 65:     PetscInfo(pep,"Problem type ignored, performing a non-symmetric linearization\n");
 66:   }

 68:   if (!ctx->keep) ctx->keep = 0.5;

 70:   PEPAllocateSolution(pep,pep->nmat-1);
 71:   PEPSetWorkVecs(pep,3);
 72:   DSSetType(pep->ds,DSNHEP);
 73:   DSSetExtraRow(pep->ds,PETSC_TRUE);
 74:   DSAllocate(pep->ds,pep->ncv+1);

 76:   PEPBasisCoefficients(pep,pep->pbc);
 77:   STGetTransform(pep->st,&flg);
 78:   if (!flg) {
 79:     PetscMalloc1(pep->nmat,&pep->solvematcoeffs);
 80:     PetscLogObjectMemory((PetscObject)pep,pep->nmat*sizeof(PetscScalar));
 81:     PetscObjectTypeCompare((PetscObject)pep->st,STSINVERT,&sinv);
 82:     if (sinv) {
 83:       PEPEvaluateBasis(pep,pep->target,0,pep->solvematcoeffs,NULL);
 84:     } else {
 85:       for (i=0;i<pep->nmat-1;i++) pep->solvematcoeffs[i] = 0.0;
 86:       pep->solvematcoeffs[pep->nmat-1] = 1.0;
 87:     }
 88:   }
 89:   BVDestroy(&ctx->V);
 90:   BVCreateTensor(pep->V,pep->nmat-1,&ctx->V);
 91:   return(0);
 92: }

 94: /*
 95:   Extend the TOAR basis by applying the the matrix operator
 96:   over a vector which is decomposed in the TOAR way
 97:   Input:
 98:     - pbc: array containing the polynomial basis coefficients
 99:     - S,V: define the latest Arnoldi vector (nv vectors in V)
100:   Output:
101:     - t: new vector extending the TOAR basis
102:     - r: temporary coefficients to compute the TOAR coefficients
103:          for the new Arnoldi vector
104:   Workspace: t_ (two vectors)
105: */
106: static PetscErrorCode PEPTOARExtendBasis(PEP pep,PetscBool sinvert,PetscScalar sigma,PetscScalar *S,PetscInt ls,PetscInt nv,BV V,Vec t,PetscScalar *r,PetscInt lr,Vec *t_)
107: {
109:   PetscInt       nmat=pep->nmat,deg=nmat-1,k,j,off=0,lss;
110:   Vec            v=t_[0],ve=t_[1],q=t_[2];
111:   PetscScalar    alpha=1.0,*ss,a;
112:   PetscReal      *ca=pep->pbc,*cb=pep->pbc+nmat,*cg=pep->pbc+2*nmat;
113:   PetscBool      flg;

116:   BVSetActiveColumns(pep->V,0,nv);
117:   STGetTransform(pep->st,&flg);
118:   if (sinvert) {
119:     for (j=0;j<nv;j++) {
120:       if (deg>1) r[lr+j] = S[j]/ca[0];
121:       if (deg>2) r[2*lr+j] = (S[ls+j]+(sigma-cb[1])*r[lr+j])/ca[1];
122:     }
123:     for (k=2;k<deg-1;k++) {
124:       for (j=0;j<nv;j++) r[(k+1)*lr+j] = (S[k*ls+j]+(sigma-cb[k])*r[k*lr+j]-cg[k]*r[(k-1)*lr+j])/ca[k];
125:     }
126:     k = deg-1;
127:     for (j=0;j<nv;j++) r[j] = (S[k*ls+j]+(sigma-cb[k])*r[k*lr+j]-cg[k]*r[(k-1)*lr+j])/ca[k];
128:     ss = r; lss = lr; off = 1; alpha = -1.0; a = pep->sfactor;
129:   } else {
130:     ss = S; lss = ls; off = 0; alpha = -ca[deg-1]; a = 1.0;
131:   }
132:   BVMultVec(V,1.0,0.0,v,ss+off*lss);
133:   if (pep->Dr) { /* balancing */
134:     VecPointwiseMult(v,v,pep->Dr);
135:   }
136:   STMatMult(pep->st,off,v,q);
137:   VecScale(q,a);
138:   for (j=1+off;j<deg+off-1;j++) {
139:     BVMultVec(V,1.0,0.0,v,ss+j*lss);
140:     if (pep->Dr) {
141:       VecPointwiseMult(v,v,pep->Dr);
142:     }
143:     STMatMult(pep->st,j,v,t);
144:     a *= pep->sfactor;
145:     VecAXPY(q,a,t);
146:   }
147:   if (sinvert) {
148:     BVMultVec(V,1.0,0.0,v,ss);
149:     if (pep->Dr) {
150:       VecPointwiseMult(v,v,pep->Dr);
151:     }
152:     STMatMult(pep->st,deg,v,t);
153:     a *= pep->sfactor;
154:     VecAXPY(q,a,t);
155:   } else {
156:     BVMultVec(V,1.0,0.0,ve,ss+(deg-1)*lss);
157:     if (pep->Dr) {
158:       VecPointwiseMult(ve,ve,pep->Dr);
159:     }
160:     a *= pep->sfactor;
161:     STMatMult(pep->st,deg-1,ve,t);
162:     VecAXPY(q,a,t);
163:     a *= pep->sfactor;
164:   }
165:   if (flg || !sinvert) alpha /= a;
166:   STMatSolve(pep->st,q,t);
167:   VecScale(t,alpha);
168:   if (!sinvert) {
169:     if (cg[deg-1]!=0) { VecAXPY(t,cg[deg-1],v); }
170:     if (cb[deg-1]!=0) { VecAXPY(t,cb[deg-1],ve); }
171:   }
172:   if (pep->Dr) {
173:     VecPointwiseDivide(t,t,pep->Dr);
174:   }
175:   return(0);
176: }

178: /*
179:   Compute TOAR coefficients of the blocks of the new Arnoldi vector computed
180: */
181: static PetscErrorCode PEPTOARCoefficients(PEP pep,PetscBool sinvert,PetscScalar sigma,PetscInt nv,PetscScalar *S,PetscInt ls,PetscScalar *r,PetscInt lr,PetscScalar *x)
182: {
183:   PetscInt    k,j,nmat=pep->nmat,d=nmat-1;
184:   PetscReal   *ca=pep->pbc,*cb=pep->pbc+nmat,*cg=pep->pbc+2*nmat;
185:   PetscScalar t=1.0,tp=0.0,tt;

188:   if (sinvert) {
189:     for (k=1;k<d;k++) {
190:       tt = t;
191:       t = ((sigma-cb[k-1])*t-cg[k-1]*tp)/ca[k-1]; /* k-th basis polynomial */
192:       tp = tt;
193:       for (j=0;j<=nv;j++) r[k*lr+j] += t*x[j];
194:     }
195:   } else {
196:     for (j=0;j<=nv;j++) r[j] = (cb[0]-sigma)*S[j]+ca[0]*S[ls+j];
197:     for (k=1;k<d-1;k++) {
198:       for (j=0;j<=nv;j++) r[k*lr+j] = (cb[k]-sigma)*S[k*ls+j]+ca[k]*S[(k+1)*ls+j]+cg[k]*S[(k-1)*ls+j];
199:     }
200:     if (sigma!=0.0) for (j=0;j<=nv;j++) r[(d-1)*lr+j] -= sigma*S[(d-1)*ls+j];
201:   }
202:   return(0);
203: }

205: /*
206:   Compute a run of Arnoldi iterations dim(work)=ld
207: */
208: static PetscErrorCode PEPTOARrun(PEP pep,PetscScalar sigma,PetscScalar *H,PetscInt ldh,PetscInt k,PetscInt *M,PetscBool *breakdown,Vec *t_)
209: {
211:   PEP_TOAR       *ctx = (PEP_TOAR*)pep->data;
212:   PetscInt       j,m=*M,deg=pep->nmat-1,ld;
213:   PetscInt       lds,nqt,l;
214:   Vec            t;
215:   PetscReal      norm;
216:   PetscBool      flg,sinvert=PETSC_FALSE,lindep;
217:   PetscScalar    *x,*S;
218:   Mat            MS;

221:   BVTensorGetFactors(ctx->V,NULL,&MS);
222:   MatDenseGetArray(MS,&S);
223:   BVGetSizes(pep->V,NULL,NULL,&ld);
224:   lds = ld*deg;
225:   BVGetActiveColumns(pep->V,&l,&nqt);
226:   STGetTransform(pep->st,&flg);
227:   if (!flg) {
228:     /* spectral transformation handled by the solver */
229:     PetscObjectTypeCompareAny((PetscObject)pep->st,&flg,STSINVERT,STSHIFT,"");
230:     if (!flg) SETERRQ(PetscObjectComm((PetscObject)pep),PETSC_ERR_SUP,"ST type not supported for TOAR without transforming matrices");
231:     PetscObjectTypeCompare((PetscObject)pep->st,STSINVERT,&sinvert);
232:   }
233:   BVSetActiveColumns(ctx->V,0,m);
234:   for (j=k;j<m;j++) {
235:     /* apply operator */
236:     BVGetColumn(pep->V,nqt,&t);
237:     PEPTOARExtendBasis(pep,sinvert,sigma,S+j*lds,ld,nqt,pep->V,t,S+(j+1)*lds,ld,t_);
238:     BVRestoreColumn(pep->V,nqt,&t);

240:     /* orthogonalize */
241:     if (sinvert) x = S+(j+1)*lds;
242:     else x = S+(deg-1)*ld+(j+1)*lds;
243:     BVOrthogonalizeColumn(pep->V,nqt,x,&norm,&lindep);
244:     if (!lindep) {
245:       x[nqt] = norm;
246:       BVScaleColumn(pep->V,nqt,1.0/norm);
247:       nqt++;
248:     }

250:     PEPTOARCoefficients(pep,sinvert,sigma,nqt-1,S+j*lds,ld,S+(j+1)*lds,ld,x);

252:     /* level-2 orthogonalization */
253:     BVOrthogonalizeColumn(ctx->V,j+1,H+j*ldh,&norm,breakdown);
254:     H[j+1+ldh*j] = norm;
255:     if (*breakdown) {
256:       *M = j+1;
257:       break;
258:     }
259:     BVScaleColumn(ctx->V,j+1,1.0/norm);
260:     BVSetActiveColumns(pep->V,l,nqt);
261:   }
262:   BVSetActiveColumns(ctx->V,0,*M);
263:   MatDenseRestoreArray(MS,&S);
264:   BVTensorRestoreFactors(ctx->V,NULL,&MS);
265:   return(0);
266: }

268: /*
269:   Computes T_j = phi_idx(T). In T_j and T_p are phi_{idx-1}(T)
270:    and phi_{idx-2}(T) respectively or null if idx=0,1.
271:    Tp and Tj are input/output arguments
272: */
273: static PetscErrorCode PEPEvaluateBasisM(PEP pep,PetscInt k,PetscScalar *T,PetscInt ldt,PetscInt idx,PetscScalar **Tp,PetscScalar **Tj)
274: {
276:   PetscInt       i;
277:   PetscReal      *ca,*cb,*cg;
278:   PetscScalar    *pt,g,a;
279:   PetscBLASInt   k_,ldt_;

282:   if (idx==0) {
283:     PetscArrayzero(*Tj,k*k);
284:     PetscArrayzero(*Tp,k*k);
285:     for (i=0;i<k;i++) (*Tj)[i+i*k] = 1.0;
286:   } else {
287:     PetscBLASIntCast(ldt,&ldt_);
288:     PetscBLASIntCast(k,&k_);
289:     ca = pep->pbc; cb = pep->pbc+pep->nmat; cg = pep->pbc+2*pep->nmat;
290:     for (i=0;i<k;i++) T[i*ldt+i] -= cb[idx-1];
291:     a = 1/ca[idx-1];
292:     g = (idx==1)?0.0:-cg[idx-1]/ca[idx-1];
293:     PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&k_,&k_,&k_,&a,T,&ldt_,*Tj,&k_,&g,*Tp,&k_));
294:     pt = *Tj; *Tj = *Tp; *Tp = pt;
295:     for (i=0;i<k;i++) T[i*ldt+i] += cb[idx-1];
296:   }
297:   return(0);
298: }

300: static PetscErrorCode PEPExtractInvariantPair(PEP pep,PetscScalar sigma,PetscInt sr,PetscInt k,PetscScalar *S,PetscInt ld,PetscInt deg,PetscScalar *H,PetscInt ldh)
301: {
303:   PetscInt       i,j,jj,lds,ldt,d=pep->nmat-1,idxcpy=0;
304:   PetscScalar    *At,*Bt,*Hj,*Hp,*T,sone=1.0,g,a,*pM,*work;
305:   PetscBLASInt   k_,sr_,lds_,ldh_,info,*p,lwork,ldt_;
306:   PetscBool      transf=PETSC_FALSE,flg;
307:   PetscReal      norm,maxnrm,*rwork;
308:   BV             *R,Y;
309:   Mat            M,*A;

312:   if (k==0) return(0);
313:   lds = deg*ld;
314:   PetscCalloc6(k,&p,sr*k,&At,k*k,&Bt,k*k,&Hj,k*k,&Hp,sr*k,&work);
315:   PetscBLASIntCast(sr,&sr_);
316:   PetscBLASIntCast(k,&k_);
317:   PetscBLASIntCast(lds,&lds_);
318:   PetscBLASIntCast(ldh,&ldh_);
319:   STGetTransform(pep->st,&flg);
320:   if (!flg) {
321:      PetscObjectTypeCompare((PetscObject)pep->st,STSINVERT,&flg);
322:     if (flg || sigma!=0.0) transf=PETSC_TRUE;
323:   }
324:   if (transf) {
325:     PetscMalloc1(k*k,&T);
326:     ldt = k;
327:     for (i=0;i<k;i++) {
328:       PetscArraycpy(T+k*i,H+i*ldh,k);
329:     }
330:     if (flg) {
331:       PetscFPTrapPush(PETSC_FP_TRAP_OFF);
332:       PetscStackCallBLAS("LAPACKgetrf",LAPACKgetrf_(&k_,&k_,T,&k_,p,&info));
333:       SlepcCheckLapackInfo("getrf",info);
334:       PetscBLASIntCast(sr*k,&lwork);
335:       PetscStackCallBLAS("LAPACKgetri",LAPACKgetri_(&k_,T,&k_,p,work,&lwork,&info));
336:       SlepcCheckLapackInfo("getri",info);
337:       PetscFPTrapPop();
338:     }
339:     if (sigma!=0.0) for (i=0;i<k;i++) T[i+k*i] += sigma;
340:   } else {
341:     T = H; ldt = ldh;
342:   }
343:   PetscBLASIntCast(ldt,&ldt_);
344:   switch (pep->extract) {
345:   case PEP_EXTRACT_NONE:
346:     break;
347:   case PEP_EXTRACT_NORM:
348:     if (pep->basis == PEP_BASIS_MONOMIAL) {
349:       PetscBLASIntCast(ldt,&ldt_);
350:       PetscMalloc1(k,&rwork);
351:       norm = LAPACKlange_("F",&k_,&k_,T,&ldt_,rwork);
352:       PetscFree(rwork);
353:       if (norm>1.0) idxcpy = d-1;
354:     } else {
355:       PetscBLASIntCast(ldt,&ldt_);
356:       PetscMalloc1(k,&rwork);
357:       maxnrm = 0.0;
358:       for (i=0;i<pep->nmat-1;i++) {
359:         PEPEvaluateBasisM(pep,k,T,ldt,i,&Hp,&Hj);
360:         norm = LAPACKlange_("F",&k_,&k_,Hj,&k_,rwork);
361:         if (norm > maxnrm) {
362:           idxcpy = i;
363:           maxnrm = norm;
364:         }
365:       }
366:       PetscFree(rwork);
367:     }
368:     if (idxcpy>0) {
369:       /* copy block idxcpy of S to the first one */
370:       for (j=0;j<k;j++) {
371:         PetscArraycpy(S+j*lds,S+idxcpy*ld+j*lds,sr);
372:       }
373:     }
374:     break;
375:   case PEP_EXTRACT_RESIDUAL:
376:     STGetTransform(pep->st,&flg);
377:     if (flg) {
378:       PetscMalloc1(pep->nmat,&A);
379:       for (i=0;i<pep->nmat;i++) {
380:         STGetMatrixTransformed(pep->st,i,A+i);
381:       }
382:     } else A = pep->A;
383:     PetscMalloc1(pep->nmat-1,&R);
384:     for (i=0;i<pep->nmat-1;i++) {
385:       BVDuplicateResize(pep->V,k,R+i);
386:     }
387:     BVDuplicateResize(pep->V,sr,&Y);
388:     MatCreateSeqDense(PETSC_COMM_SELF,sr,k,NULL,&M);
389:     g = 0.0; a = 1.0;
390:     BVSetActiveColumns(pep->V,0,sr);
391:     for (j=0;j<pep->nmat;j++) {
392:       BVMatMult(pep->V,A[j],Y);
393:       PEPEvaluateBasisM(pep,k,T,ldt,i,&Hp,&Hj);
394:       for (i=0;i<pep->nmat-1;i++) {
395:         PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&sr_,&k_,&k_,&a,S+i*ld,&lds_,Hj,&k_,&g,At,&sr_));
396:         MatDenseGetArray(M,&pM);
397:         for (jj=0;jj<k;jj++) {
398:           PetscArraycpy(pM+jj*sr,At+jj*sr,sr);
399:         }
400:         MatDenseRestoreArray(M,&pM);
401:         BVMult(R[i],1.0,(i==0)?0.0:1.0,Y,M);
402:       }
403:     }

405:     /* frobenius norm */
406:     maxnrm = 0.0;
407:     for (i=0;i<pep->nmat-1;i++) {
408:       BVNorm(R[i],NORM_FROBENIUS,&norm);
409:       if (maxnrm > norm) {
410:         maxnrm = norm;
411:         idxcpy = i;
412:       }
413:     }
414:     if (idxcpy>0) {
415:       /* copy block idxcpy of S to the first one */
416:       for (j=0;j<k;j++) {
417:         PetscArraycpy(S+j*lds,S+idxcpy*ld+j*lds,sr);
418:       }
419:     }
420:     if (flg) { PetscFree(A); }
421:     for (i=0;i<pep->nmat-1;i++) {
422:       BVDestroy(&R[i]);
423:     }
424:     PetscFree(R);
425:     BVDestroy(&Y);
426:     MatDestroy(&M);
427:     break;
428:   case PEP_EXTRACT_STRUCTURED:
429:     for (j=0;j<k;j++) Bt[j+j*k] = 1.0;
430:     for (j=0;j<sr;j++) {
431:       for (i=0;i<k;i++) At[j*k+i] = PetscConj(S[i*lds+j]);
432:     }
433:     PEPEvaluateBasisM(pep,k,T,ldt,0,&Hp,&Hj);
434:     for (i=1;i<deg;i++) {
435:       PEPEvaluateBasisM(pep,k,T,ldt,i,&Hp,&Hj);
436:       PetscStackCallBLAS("BLASgemm",BLASgemm_("N","C",&k_,&sr_,&k_,&sone,Hj,&k_,S+i*ld,&lds_,&sone,At,&k_));
437:       PetscStackCallBLAS("BLASgemm",BLASgemm_("N","C",&k_,&k_,&k_,&sone,Hj,&k_,Hj,&k_,&sone,Bt,&k_));
438:     }
439:     PetscFPTrapPush(PETSC_FP_TRAP_OFF);
440:     PetscStackCallBLAS("LAPACKgesv",LAPACKgesv_(&k_,&sr_,Bt,&k_,p,At,&k_,&info));
441:     PetscFPTrapPop();
442:     SlepcCheckLapackInfo("gesv",info);
443:     for (j=0;j<sr;j++) {
444:       for (i=0;i<k;i++) S[i*lds+j] = PetscConj(At[j*k+i]);
445:     }
446:     break;
447:   }
448:   if (transf) { PetscFree(T); }
449:   PetscFree6(p,At,Bt,Hj,Hp,work);
450:   return(0);
451: }

453: PetscErrorCode PEPSolve_TOAR(PEP pep)
454: {
456:   PEP_TOAR       *ctx = (PEP_TOAR*)pep->data;
457:   PetscInt       i,j,k,l,nv=0,ld,lds,ldds,nq=0,nconv=0;
458:   PetscInt       nmat=pep->nmat,deg=nmat-1;
459:   PetscScalar    *S,*H,sigma;
460:   PetscReal      beta;
461:   PetscBool      breakdown=PETSC_FALSE,flg,falselock=PETSC_FALSE,sinv=PETSC_FALSE;
462:   Mat            MS,MQ;

465:   PetscCitationsRegister(citation,&cited);
466:   if (ctx->lock) {
467:     /* undocumented option to use a cheaper locking instead of the true locking */
468:     PetscOptionsGetBool(NULL,NULL,"-pep_toar_falselocking",&falselock,NULL);
469:   }
470:   DSGetLeadingDimension(pep->ds,&ldds);
471:   STGetShift(pep->st,&sigma);

473:   /* update polynomial basis coefficients */
474:   STGetTransform(pep->st,&flg);
475:   if (pep->sfactor!=1.0) {
476:     for (i=0;i<nmat;i++) {
477:       pep->pbc[nmat+i] /= pep->sfactor;
478:       pep->pbc[2*nmat+i] /= pep->sfactor*pep->sfactor;
479:     }
480:     if (!flg) {
481:       pep->target /= pep->sfactor;
482:       RGPushScale(pep->rg,1.0/pep->sfactor);
483:       STScaleShift(pep->st,1.0/pep->sfactor);
484:       sigma /= pep->sfactor;
485:     } else {
486:       PetscObjectTypeCompare((PetscObject)pep->st,STSINVERT,&sinv);
487:       pep->target = sinv?pep->target*pep->sfactor:pep->target/pep->sfactor;
488:       RGPushScale(pep->rg,sinv?pep->sfactor:1.0/pep->sfactor);
489:       STScaleShift(pep->st,sinv?pep->sfactor:1.0/pep->sfactor);
490:     }
491:   }

493:   if (flg) sigma = 0.0;

495:   /* clean projected matrix (including the extra-arrow) */
496:   DSGetArray(pep->ds,DS_MAT_A,&H);
497:   PetscArrayzero(H,ldds*ldds);
498:   DSRestoreArray(pep->ds,DS_MAT_A,&H);

500:   /* Get the starting Arnoldi vector */
501:   BVTensorBuildFirstColumn(ctx->V,pep->nini);

503:   /* restart loop */
504:   l = 0;
505:   while (pep->reason == PEP_CONVERGED_ITERATING) {
506:     pep->its++;

508:     /* compute an nv-step Lanczos factorization */
509:     nv = PetscMax(PetscMin(nconv+pep->mpd,pep->ncv),nv);
510:     DSGetArray(pep->ds,DS_MAT_A,&H);
511:     PEPTOARrun(pep,sigma,H,ldds,pep->nconv+l,&nv,&breakdown,pep->work);
512:     beta = PetscAbsScalar(H[(nv-1)*ldds+nv]);
513:     DSRestoreArray(pep->ds,DS_MAT_A,&H);
514:     DSSetDimensions(pep->ds,nv,pep->nconv,pep->nconv+l);
515:     if (l==0) {
516:       DSSetState(pep->ds,DS_STATE_INTERMEDIATE);
517:     } else {
518:       DSSetState(pep->ds,DS_STATE_RAW);
519:     }
520:     BVSetActiveColumns(ctx->V,pep->nconv,nv);

522:     /* solve projected problem */
523:     DSSolve(pep->ds,pep->eigr,pep->eigi);
524:     DSSort(pep->ds,pep->eigr,pep->eigi,NULL,NULL,NULL);
525:     DSUpdateExtraRow(pep->ds);
526:     DSSynchronize(pep->ds,pep->eigr,pep->eigi);

528:     /* check convergence */
529:     PEPKrylovConvergence(pep,PETSC_FALSE,pep->nconv,nv-pep->nconv,beta,&k);
530:     (*pep->stopping)(pep,pep->its,pep->max_it,k,pep->nev,&pep->reason,pep->stoppingctx);

532:     /* update l */
533:     if (pep->reason != PEP_CONVERGED_ITERATING || breakdown) l = 0;
534:     else {
535:       l = (nv==k)?0:PetscMax(1,(PetscInt)((nv-k)*ctx->keep));
536:       DSGetTruncateSize(pep->ds,k,nv,&l);
537:       if (!breakdown) {
538:         /* prepare the Rayleigh quotient for restart */
539:         DSTruncate(pep->ds,k+l,PETSC_FALSE);
540:       }
541:     }
542:     nconv = k;
543:     if (!ctx->lock && pep->reason == PEP_CONVERGED_ITERATING && !breakdown) { l += k; k = 0; } /* non-locking variant: reset no. of converged pairs */
544:     if (l) { PetscInfo1(pep,"Preparing to restart keeping l=%D vectors\n",l); }

546:     /* update S */
547:     DSGetMat(pep->ds,DS_MAT_Q,&MQ);
548:     BVMultInPlace(ctx->V,MQ,pep->nconv,k+l);
549:     MatDestroy(&MQ);

551:     /* copy last column of S */
552:     BVCopyColumn(ctx->V,nv,k+l);

554:     if (breakdown && pep->reason == PEP_CONVERGED_ITERATING) {
555:       /* stop if breakdown */
556:       PetscInfo2(pep,"Breakdown TOAR method (it=%D norm=%g)\n",pep->its,(double)beta);
557:       pep->reason = PEP_DIVERGED_BREAKDOWN;
558:     }
559:     if (pep->reason != PEP_CONVERGED_ITERATING) l--;
560:     /* truncate S */
561:     BVGetActiveColumns(pep->V,NULL,&nq);
562:     if (k+l+deg<=nq) {
563:       BVSetActiveColumns(ctx->V,pep->nconv,k+l+1);
564:       if (!falselock && ctx->lock) {
565:         BVTensorCompress(ctx->V,k-pep->nconv);
566:       } else {
567:         BVTensorCompress(ctx->V,0);
568:       }
569:     }
570:     pep->nconv = k;
571:     PEPMonitor(pep,pep->its,nconv,pep->eigr,pep->eigi,pep->errest,nv);
572:   }
573:   if (pep->nconv>0) {
574:     /* {V*S_nconv^i}_{i=0}^{d-1} has rank nconv instead of nconv+d-1. Force zeros in each S_nconv^i block */
575:     BVSetActiveColumns(ctx->V,0,pep->nconv);
576:     BVGetActiveColumns(pep->V,NULL,&nq);
577:     BVSetActiveColumns(pep->V,0,nq);
578:     if (nq>pep->nconv) {
579:       BVTensorCompress(ctx->V,pep->nconv);
580:       BVSetActiveColumns(pep->V,0,pep->nconv);
581:       nq = pep->nconv;
582:     }

584:     /* perform Newton refinement if required */
585:     if (pep->refine==PEP_REFINE_MULTIPLE && pep->rits>0) {
586:       /* extract invariant pair */
587:       BVTensorGetFactors(ctx->V,NULL,&MS);
588:       MatDenseGetArray(MS,&S);
589:       DSGetArray(pep->ds,DS_MAT_A,&H);
590:       BVGetSizes(pep->V,NULL,NULL,&ld);
591:       lds = deg*ld;
592:       PEPExtractInvariantPair(pep,sigma,nq,pep->nconv,S,ld,deg,H,ldds);
593:       DSRestoreArray(pep->ds,DS_MAT_A,&H);
594:       DSSetDimensions(pep->ds,pep->nconv,0,0);
595:       DSSetState(pep->ds,DS_STATE_RAW);
596:       PEPNewtonRefinement_TOAR(pep,sigma,&pep->rits,NULL,pep->nconv,S,lds);
597:       DSSolve(pep->ds,pep->eigr,pep->eigi);
598:       DSSort(pep->ds,pep->eigr,pep->eigi,NULL,NULL,NULL);
599:       DSSynchronize(pep->ds,pep->eigr,pep->eigi);
600:       DSGetMat(pep->ds,DS_MAT_Q,&MQ);
601:       BVMultInPlace(ctx->V,MQ,0,pep->nconv);
602:       MatDestroy(&MQ);
603:       MatDenseRestoreArray(MS,&S);
604:       BVTensorRestoreFactors(ctx->V,NULL,&MS);
605:     }
606:   }
607:   STGetTransform(pep->st,&flg);
608:   if (pep->refine!=PEP_REFINE_MULTIPLE || pep->rits==0) {
609:     if (!flg && pep->ops->backtransform) {
610:         (*pep->ops->backtransform)(pep);
611:     }
612:     if (pep->sfactor!=1.0) {
613:       for (j=0;j<pep->nconv;j++) {
614:         pep->eigr[j] *= pep->sfactor;
615:         pep->eigi[j] *= pep->sfactor;
616:       }
617:       /* restore original values */
618:       for (i=0;i<pep->nmat;i++) {
619:         pep->pbc[pep->nmat+i] *= pep->sfactor;
620:         pep->pbc[2*pep->nmat+i] *= pep->sfactor*pep->sfactor;
621:       }
622:     }
623:   }
624:   /* restore original values */
625:   if (!flg) {
626:     pep->target *= pep->sfactor;
627:     STScaleShift(pep->st,pep->sfactor);
628:   } else {
629:     STScaleShift(pep->st,sinv?1.0/pep->sfactor:pep->sfactor);
630:     pep->target = (sinv)?pep->target/pep->sfactor:pep->target*pep->sfactor;
631:   }
632:   if (pep->sfactor!=1.0) { RGPopScale(pep->rg); }

634:   /* change the state to raw so that DSVectors() computes eigenvectors from scratch */
635:   DSSetDimensions(pep->ds,pep->nconv,0,0);
636:   DSSetState(pep->ds,DS_STATE_RAW);
637:   return(0);
638: }

640: static PetscErrorCode PEPTOARSetRestart_TOAR(PEP pep,PetscReal keep)
641: {
642:   PEP_TOAR *ctx = (PEP_TOAR*)pep->data;

645:   if (keep==PETSC_DEFAULT) ctx->keep = 0.5;
646:   else {
647:     if (keep<0.1 || keep>0.9) SETERRQ(PetscObjectComm((PetscObject)pep),PETSC_ERR_ARG_OUTOFRANGE,"The keep argument must be in the range [0.1,0.9]");
648:     ctx->keep = keep;
649:   }
650:   return(0);
651: }

653: /*@
654:    PEPTOARSetRestart - Sets the restart parameter for the TOAR
655:    method, in particular the proportion of basis vectors that must be kept
656:    after restart.

658:    Logically Collective on pep

660:    Input Parameters:
661: +  pep  - the eigenproblem solver context
662: -  keep - the number of vectors to be kept at restart

664:    Options Database Key:
665: .  -pep_toar_restart - Sets the restart parameter

667:    Notes:
668:    Allowed values are in the range [0.1,0.9]. The default is 0.5.

670:    Level: advanced

672: .seealso: PEPTOARGetRestart()
673: @*/
674: PetscErrorCode PEPTOARSetRestart(PEP pep,PetscReal keep)
675: {

681:   PetscTryMethod(pep,"PEPTOARSetRestart_C",(PEP,PetscReal),(pep,keep));
682:   return(0);
683: }

685: static PetscErrorCode PEPTOARGetRestart_TOAR(PEP pep,PetscReal *keep)
686: {
687:   PEP_TOAR *ctx = (PEP_TOAR*)pep->data;

690:   *keep = ctx->keep;
691:   return(0);
692: }

694: /*@
695:    PEPTOARGetRestart - Gets the restart parameter used in the TOAR method.

697:    Not Collective

699:    Input Parameter:
700: .  pep - the eigenproblem solver context

702:    Output Parameter:
703: .  keep - the restart parameter

705:    Level: advanced

707: .seealso: PEPTOARSetRestart()
708: @*/
709: PetscErrorCode PEPTOARGetRestart(PEP pep,PetscReal *keep)
710: {

716:   PetscUseMethod(pep,"PEPTOARGetRestart_C",(PEP,PetscReal*),(pep,keep));
717:   return(0);
718: }

720: static PetscErrorCode PEPTOARSetLocking_TOAR(PEP pep,PetscBool lock)
721: {
722:   PEP_TOAR *ctx = (PEP_TOAR*)pep->data;

725:   ctx->lock = lock;
726:   return(0);
727: }

729: /*@
730:    PEPTOARSetLocking - Choose between locking and non-locking variants of
731:    the TOAR method.

733:    Logically Collective on pep

735:    Input Parameters:
736: +  pep  - the eigenproblem solver context
737: -  lock - true if the locking variant must be selected

739:    Options Database Key:
740: .  -pep_toar_locking - Sets the locking flag

742:    Notes:
743:    The default is to lock converged eigenpairs when the method restarts.
744:    This behaviour can be changed so that all directions are kept in the
745:    working subspace even if already converged to working accuracy (the
746:    non-locking variant).

748:    Level: advanced

750: .seealso: PEPTOARGetLocking()
751: @*/
752: PetscErrorCode PEPTOARSetLocking(PEP pep,PetscBool lock)
753: {

759:   PetscTryMethod(pep,"PEPTOARSetLocking_C",(PEP,PetscBool),(pep,lock));
760:   return(0);
761: }

763: static PetscErrorCode PEPTOARGetLocking_TOAR(PEP pep,PetscBool *lock)
764: {
765:   PEP_TOAR *ctx = (PEP_TOAR*)pep->data;

768:   *lock = ctx->lock;
769:   return(0);
770: }

772: /*@
773:    PEPTOARGetLocking - Gets the locking flag used in the TOAR method.

775:    Not Collective

777:    Input Parameter:
778: .  pep - the eigenproblem solver context

780:    Output Parameter:
781: .  lock - the locking flag

783:    Level: advanced

785: .seealso: PEPTOARSetLocking()
786: @*/
787: PetscErrorCode PEPTOARGetLocking(PEP pep,PetscBool *lock)
788: {

794:   PetscUseMethod(pep,"PEPTOARGetLocking_C",(PEP,PetscBool*),(pep,lock));
795:   return(0);
796: }

798: PetscErrorCode PEPSetFromOptions_TOAR(PetscOptionItems *PetscOptionsObject,PEP pep)
799: {
801:   PetscBool      flg,lock;
802:   PetscReal      keep;

805:   PetscOptionsHead(PetscOptionsObject,"PEP TOAR Options");

807:     PetscOptionsReal("-pep_toar_restart","Proportion of vectors kept after restart","PEPTOARSetRestart",0.5,&keep,&flg);
808:     if (flg) { PEPTOARSetRestart(pep,keep); }

810:     PetscOptionsBool("-pep_toar_locking","Choose between locking and non-locking variants","PEPTOARSetLocking",PETSC_FALSE,&lock,&flg);
811:     if (flg) { PEPTOARSetLocking(pep,lock); }

813:   PetscOptionsTail();
814:   return(0);
815: }

817: PetscErrorCode PEPView_TOAR(PEP pep,PetscViewer viewer)
818: {
820:   PEP_TOAR       *ctx = (PEP_TOAR*)pep->data;
821:   PetscBool      isascii;

824:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii);
825:   if (isascii) {
826:     PetscViewerASCIIPrintf(viewer,"  %d%% of basis vectors kept after restart\n",(int)(100*ctx->keep));
827:     PetscViewerASCIIPrintf(viewer,"  using the %slocking variant\n",ctx->lock?"":"non-");
828:   }
829:   return(0);
830: }

832: PetscErrorCode PEPDestroy_TOAR(PEP pep)
833: {
835:   PEP_TOAR       *ctx = (PEP_TOAR*)pep->data;

838:   BVDestroy(&ctx->V);
839:   PetscFree(pep->data);
840:   PetscObjectComposeFunction((PetscObject)pep,"PEPTOARSetRestart_C",NULL);
841:   PetscObjectComposeFunction((PetscObject)pep,"PEPTOARGetRestart_C",NULL);
842:   PetscObjectComposeFunction((PetscObject)pep,"PEPTOARSetLocking_C",NULL);
843:   PetscObjectComposeFunction((PetscObject)pep,"PEPTOARGetLocking_C",NULL);
844:   return(0);
845: }

847: SLEPC_EXTERN PetscErrorCode PEPCreate_TOAR(PEP pep)
848: {
849:   PEP_TOAR       *ctx;

853:   PetscNewLog(pep,&ctx);
854:   pep->data = (void*)ctx;

856:   pep->lineariz = PETSC_TRUE;
857:   ctx->lock     = PETSC_TRUE;

859:   pep->ops->solve          = PEPSolve_TOAR;
860:   pep->ops->setup          = PEPSetUp_TOAR;
861:   pep->ops->setfromoptions = PEPSetFromOptions_TOAR;
862:   pep->ops->destroy        = PEPDestroy_TOAR;
863:   pep->ops->view           = PEPView_TOAR;
864:   pep->ops->backtransform  = PEPBackTransform_Default;
865:   pep->ops->computevectors = PEPComputeVectors_Default;
866:   pep->ops->extractvectors = PEPExtractVectors_TOAR;

868:   PetscObjectComposeFunction((PetscObject)pep,"PEPTOARSetRestart_C",PEPTOARSetRestart_TOAR);
869:   PetscObjectComposeFunction((PetscObject)pep,"PEPTOARGetRestart_C",PEPTOARGetRestart_TOAR);
870:   PetscObjectComposeFunction((PetscObject)pep,"PEPTOARSetLocking_C",PEPTOARSetLocking_TOAR);
871:   PetscObjectComposeFunction((PetscObject)pep,"PEPTOARGetLocking_C",PEPTOARGetLocking_TOAR);
872:   return(0);
873: }