Actual source code: bvorthog.c

slepc-3.18.2 2023-01-26
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  1: /*
  2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
  4:    Copyright (c) 2002-, Universitat Politecnica de Valencia, Spain

  6:    This file is part of SLEPc.
  7:    SLEPc is distributed under a 2-clause BSD license (see LICENSE).
  8:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  9: */
 10: /*
 11:    BV orthogonalization routines
 12: */

 14: #include <slepc/private/bvimpl.h>

 16: /*
 17:    BV_NormVecOrColumn - Compute the 2-norm of the working vector, irrespective of
 18:    whether it is in a column or not
 19: */
 20: static inline PetscErrorCode BV_NormVecOrColumn(BV bv,PetscInt j,Vec v,PetscReal *nrm)
 21: {
 22:   if (v) BVNormVec(bv,v,NORM_2,nrm);
 23:   else BVNormColumn(bv,j,NORM_2,nrm);
 24:   return 0;
 25: }

 27: /*
 28:    BVDotColumnInc - Same as BVDotColumn() but also including column j, which
 29:    is multiplied by itself
 30: */
 31: static inline PetscErrorCode BVDotColumnInc(BV X,PetscInt j,PetscScalar *q)
 32: {
 33:   PetscInt       ksave;
 34:   Vec            y;

 36:   PetscLogEventBegin(BV_DotVec,X,0,0,0);
 37:   ksave = X->k;
 38:   X->k = j+1;
 39:   BVGetColumn(X,j,&y);
 40:   PetscUseTypeMethod(X,dotvec,y,q);
 41:   BVRestoreColumn(X,j,&y);
 42:   X->k = ksave;
 43:   PetscLogEventEnd(BV_DotVec,X,0,0,0);
 44:   return 0;
 45: }

 47: /*
 48:    BVOrthogonalizeMGS1 - Compute one step of Modified Gram-Schmidt
 49: */
 50: static PetscErrorCode BVOrthogonalizeMGS1(BV bv,PetscInt j,Vec v,PetscBool *which,PetscScalar *h,PetscScalar *c,PetscReal *onrm,PetscReal *nrm)
 51: {
 52:   PetscInt          i;
 53:   PetscScalar       dot;
 54:   PetscBool         indef=bv->indef;
 55:   Vec               vi,z,w=v;
 56:   const PetscScalar *omega;

 58:   if (!v) BVGetColumn(bv,j,&w);
 59:   if (onrm) BVNormVec(bv,w,NORM_2,onrm);
 60:   z = w;
 61:   if (indef) VecGetArrayRead(bv->omega,&omega);
 62:   for (i=-bv->nc;i<j;i++) {
 63:     if (which && i>=0 && !which[i]) continue;
 64:     BVGetColumn(bv,i,&vi);
 65:     /* h_i = (v, v_i) */
 66:     if (bv->matrix) {
 67:       BV_IPMatMult(bv,w);
 68:       z = bv->Bx;
 69:     }
 70:     VecDot(z,vi,&dot);
 71:     /* v <- v - h_i v_i */
 72:     BV_SetValue(bv,i,0,c,dot);
 73:     if (indef) dot /= PetscRealPart(omega[bv->nc+i]);
 74:     VecAXPY(w,-dot,vi);
 75:     BVRestoreColumn(bv,i,&vi);
 76:   }
 77:   if (nrm) BVNormVec(bv,w,NORM_2,nrm);
 78:   if (!v) BVRestoreColumn(bv,j,&w);
 79:   BV_AddCoefficients(bv,j,h,c);
 80:   if (indef) VecRestoreArrayRead(bv->omega,&omega);
 81:   return 0;
 82: }

 84: /*
 85:    BVOrthogonalizeCGS1 - Compute |v'| (estimated), |v| and one step of CGS with
 86:    only one global synchronization
 87: */
 88: static PetscErrorCode BVOrthogonalizeCGS1(BV bv,PetscInt j,Vec v,PetscBool *which,PetscScalar *h,PetscScalar *c,PetscReal *onorm,PetscReal *norm)
 89: {
 90:   PetscReal      sum,beta;

 92:   /* h = W^* v ; alpha = (v, v) */
 93:   bv->k = j;
 94:   if (onorm || norm) {
 95:     if (!v) {
 96:       BVDotColumnInc(bv,j,c);
 97:       BV_SquareRoot(bv,j,c,&beta);
 98:     } else {
 99:       BVDotVec(bv,v,c);
100:       BVNormVec(bv,v,NORM_2,&beta);
101:     }
102:   } else {
103:     if (!v) BVDotColumn(bv,j,c);
104:     else BVDotVec(bv,v,c);
105:   }

107:   /* q = v - V h */
108:   if (PetscUnlikely(bv->indef)) BV_ApplySignature(bv,j,c,PETSC_TRUE);
109:   if (!v) BVMultColumn(bv,-1.0,1.0,j,c);
110:   else BVMultVec(bv,-1.0,1.0,v,c);
111:   if (PetscUnlikely(bv->indef)) BV_ApplySignature(bv,j,c,PETSC_FALSE);

113:   /* compute |v| */
114:   if (onorm) *onorm = beta;

116:   if (norm) {
117:     if (PetscUnlikely(bv->indef)) BV_NormVecOrColumn(bv,j,v,norm);
118:     else {
119:       /* estimate |v'| from |v| */
120:       BV_SquareSum(bv,j,c,&sum);
121:       *norm = beta*beta-sum;
122:       if (PetscUnlikely(*norm <= 0.0)) BV_NormVecOrColumn(bv,j,v,norm);
123:       else *norm = PetscSqrtReal(*norm);
124:     }
125:   }
126:   BV_AddCoefficients(bv,j,h,c);
127:   return 0;
128: }

130: #define BVOrthogonalizeGS1(a,b,c,d,e,f,g,h) ((bv->ops->gramschmidt)?(*bv->ops->gramschmidt):(mgs?BVOrthogonalizeMGS1:BVOrthogonalizeCGS1))(a,b,c,d,e,f,g,h)

132: /*
133:    BVOrthogonalizeGS - Orthogonalize with (classical or modified) Gram-Schmidt

135:    j      - the index of the column to orthogonalize (cannot use both j and v)
136:    v      - the vector to orthogonalize (cannot use both j and v)
137:    which  - logical array indicating selected columns (only used in MGS)
138:    norm   - (optional) norm of the vector after being orthogonalized
139:    lindep - (optional) flag indicating possible linear dependence
140: */
141: static PetscErrorCode BVOrthogonalizeGS(BV bv,PetscInt j,Vec v,PetscBool *which,PetscReal *norm,PetscBool *lindep)
142: {
143:   PetscScalar    *h,*c,*omega;
144:   PetscReal      onrm,nrm;
145:   PetscInt       k,l;
146:   PetscBool      mgs,dolindep,signature;

148:   if (v) {
149:     k = bv->k;
150:     h = bv->h;
151:     c = bv->c;
152:   } else {
153:     k = j;
154:     h = NULL;
155:     c = NULL;
156:   }

158:   mgs = (bv->orthog_type==BV_ORTHOG_MGS)? PETSC_TRUE: PETSC_FALSE;

160:   /* if indefinite inner product, skip the computation of lindep */
161:   if (bv->indef && lindep) *lindep = PETSC_FALSE;
162:   dolindep = (!bv->indef && lindep)? PETSC_TRUE: PETSC_FALSE;

164:   /* if indefinite and we are orthogonalizing a column, the norm must always be computed */
165:   signature = (bv->indef && !v)? PETSC_TRUE: PETSC_FALSE;

167:   BV_CleanCoefficients(bv,k,h);

169:   switch (bv->orthog_ref) {

171:   case BV_ORTHOG_REFINE_IFNEEDED:
172:     BVOrthogonalizeGS1(bv,k,v,which,h,c,&onrm,&nrm);
173:     /* repeat if ||q|| < eta ||h|| */
174:     l = 1;
175:     while (l<3 && nrm && PetscAbsReal(nrm) < bv->orthog_eta*PetscAbsReal(onrm)) {
176:       l++;
177:       if (mgs||bv->indef) onrm = nrm;
178:       BVOrthogonalizeGS1(bv,k,v,which,h,c,(mgs||bv->indef)?NULL:&onrm,&nrm);
179:     }
180:     /* linear dependence check: criterion not satisfied in the last iteration */
181:     if (dolindep) *lindep = PetscNot(nrm && PetscAbsReal(nrm) >= bv->orthog_eta*PetscAbsReal(onrm));
182:     break;

184:   case BV_ORTHOG_REFINE_NEVER:
185:     BVOrthogonalizeGS1(bv,k,v,which,h,c,NULL,NULL);
186:     /* compute ||v|| */
187:     if (norm || dolindep || signature) BV_NormVecOrColumn(bv,k,v,&nrm);
188:     /* linear dependence check: just test for exactly zero norm */
189:     if (dolindep) *lindep = PetscNot(nrm);
190:     break;

192:   case BV_ORTHOG_REFINE_ALWAYS:
193:     BVOrthogonalizeGS1(bv,k,v,which,h,c,NULL,NULL);
194:     BVOrthogonalizeGS1(bv,k,v,which,h,c,dolindep?&onrm:NULL,(norm||dolindep||signature)?&nrm:NULL);
195:     /* linear dependence check: criterion not satisfied in the second iteration */
196:     if (dolindep) *lindep = PetscNot(nrm && PetscAbsReal(nrm) >= bv->orthog_eta*PetscAbsReal(onrm));
197:     break;
198:   }
199:   if (signature) {
200:     VecGetArray(bv->omega,&omega);
201:     omega[bv->nc+k] = (nrm<0.0)? -1.0: 1.0;
202:     VecRestoreArray(bv->omega,&omega);
203:   }
204:   if (norm) {
205:     *norm = nrm;
206:     if (!v) { /* store norm value next to the orthogonalization coefficients */
207:       if (dolindep && *lindep) BV_SetValue(bv,k,k,h,0.0);
208:       else BV_SetValue(bv,k,k,h,nrm);
209:     }
210:   }
211:   return 0;
212: }

214: /*@
215:    BVOrthogonalizeVec - Orthogonalize a given vector with respect to all
216:    active columns.

218:    Collective on bv

220:    Input Parameters:
221: +  bv     - the basis vectors context
222: -  v      - the vector

224:    Output Parameters:
225: +  H      - (optional) coefficients computed during orthogonalization
226: .  norm   - (optional) norm of the vector after being orthogonalized
227: -  lindep - (optional) flag indicating that refinement did not improve the quality
228:             of orthogonalization

230:    Notes:
231:    This function is equivalent to BVOrthogonalizeColumn() but orthogonalizes
232:    a vector as an argument rather than taking one of the BV columns. The
233:    vector is orthogonalized against all active columns (k) and the constraints.
234:    If H is given, it must have enough space to store k-l coefficients, where l
235:    is the number of leading columns.

237:    In the case of an indefinite inner product, the lindep parameter is not
238:    computed (set to false).

240:    Level: advanced

242: .seealso: BVOrthogonalizeColumn(), BVSetOrthogonalization(), BVSetActiveColumns(), BVGetNumConstraints()
243: @*/
244: PetscErrorCode BVOrthogonalizeVec(BV bv,Vec v,PetscScalar *H,PetscReal *norm,PetscBool *lindep)
245: {
246:   PetscInt       ksave,lsave;

251:   BVCheckSizes(bv,1);

255:   PetscLogEventBegin(BV_OrthogonalizeVec,bv,0,0,0);
256:   ksave = bv->k;
257:   lsave = bv->l;
258:   bv->l = -bv->nc;  /* must also orthogonalize against constraints and leading columns */
259:   BV_AllocateCoeffs(bv);
260:   BV_AllocateSignature(bv);
261:   BVOrthogonalizeGS(bv,0,v,NULL,norm,lindep);
262:   bv->k = ksave;
263:   bv->l = lsave;
264:   if (H) BV_StoreCoefficients(bv,bv->k,bv->h,H);
265:   PetscLogEventEnd(BV_OrthogonalizeVec,bv,0,0,0);
266:   return 0;
267: }

269: /*@
270:    BVOrthogonalizeColumn - Orthogonalize one of the column vectors with respect to
271:    the previous ones.

273:    Collective on bv

275:    Input Parameters:
276: +  bv     - the basis vectors context
277: -  j      - index of column to be orthogonalized

279:    Output Parameters:
280: +  H      - (optional) coefficients computed during orthogonalization
281: .  norm   - (optional) norm of the vector after being orthogonalized
282: -  lindep - (optional) flag indicating that refinement did not improve the quality
283:             of orthogonalization

285:    Notes:
286:    This function applies an orthogonal projector to project vector V[j] onto
287:    the orthogonal complement of the span of the columns V[0..j-1],
288:    where V[.] are the vectors of BV. The columns V[0..j-1] are assumed to be
289:    mutually orthonormal.

291:    Leading columns V[0..l-1] also participate in the orthogonalization, as well
292:    as the constraints. If H is given, it must have enough space to store
293:    j-l+1 coefficients (the last coefficient will contain the value norm, unless
294:    the norm argument is NULL).

296:    If a non-standard inner product has been specified with BVSetMatrix(),
297:    then the vector is B-orthogonalized, using the non-standard inner product
298:    defined by matrix B. The output vector satisfies V[j]'*B*V[0..j-1] = 0.

300:    This routine does not normalize the resulting vector, see BVOrthonormalizeColumn().

302:    In the case of an indefinite inner product, the lindep parameter is not
303:    computed (set to false).

305:    Level: advanced

307: .seealso: BVSetOrthogonalization(), BVSetMatrix(), BVSetActiveColumns(), BVOrthogonalize(), BVOrthogonalizeVec(), BVGetNumConstraints(), BVOrthonormalizeColumn()
308: @*/
309: PetscErrorCode BVOrthogonalizeColumn(BV bv,PetscInt j,PetscScalar *H,PetscReal *norm,PetscBool *lindep)
310: {
311:   PetscInt       ksave,lsave;

316:   BVCheckSizes(bv,1);

320:   PetscLogEventBegin(BV_OrthogonalizeVec,bv,0,0,0);
321:   ksave = bv->k;
322:   lsave = bv->l;
323:   bv->l = -bv->nc;  /* must also orthogonalize against constraints and leading columns */
324:   if (!bv->buffer) BVGetBufferVec(bv,&bv->buffer);
325:   BV_AllocateSignature(bv);
326:   BVOrthogonalizeGS(bv,j,NULL,NULL,norm,lindep);
327:   bv->k = ksave;
328:   bv->l = lsave;
329:   if (H) BV_StoreCoefficients(bv,j,NULL,H);
330:   PetscLogEventEnd(BV_OrthogonalizeVec,bv,0,0,0);
331:   PetscObjectStateIncrease((PetscObject)bv);
332:   return 0;
333: }

335: /*@
336:    BVOrthonormalizeColumn - Orthonormalize one of the column vectors with respect to
337:    the previous ones.

339:    Collective on bv

341:    Input Parameters:
342: +  bv      - the basis vectors context
343: .  j       - index of column to be orthonormalized
344: -  replace - whether it is allowed to set the vector randomly

346:    Output Parameters:
347: +  norm    - (optional) norm of the vector after orthogonalization and before normalization
348: -  lindep  - (optional) flag indicating that linear dependence was determined during
349:              orthogonalization

351:    Notes:
352:    This is equivalent to a call to BVOrthogonalizeColumn() followed by a
353:    call to BVScaleColumn() with the reciprocal of the norm.

355:    This function first orthogonalizes vector V[j] with respect to V[0..j-1],
356:    where V[.] are the vectors of BV. A byproduct of this computation is norm,
357:    the norm of the vector after orthogonalization. Secondly, it scales the
358:    vector with 1/norm, so that the resulting vector has unit norm.

360:    If after orthogonalization the vector V[j] is exactly zero, it cannot be normalized
361:    because norm=0. In that case, it could be left as zero or replaced by a random
362:    vector that is then orthonormalized. The latter is achieved by setting the
363:    argument replace to TRUE. The vector will be replaced by a random vector also
364:    if lindep was set to TRUE, even if the norm is not exactly zero.

366:    If the vector has been replaced by a random vector, the output arguments norm and
367:    lindep will be set according to the orthogonalization of this new vector.

369:    Level: advanced

371: .seealso: BVOrthogonalizeColumn(), BVScaleColumn()
372: @*/
373: PetscErrorCode BVOrthonormalizeColumn(BV bv,PetscInt j,PetscBool replace,PetscReal *norm,PetscBool *lindep)
374: {
375:   PetscScalar    alpha;
376:   PetscReal      nrm;
377:   PetscInt       ksave,lsave;
378:   PetscBool      lndep;

383:   BVCheckSizes(bv,1);

387:   /* orthogonalize */
388:   PetscLogEventBegin(BV_OrthogonalizeVec,bv,0,0,0);
389:   ksave = bv->k;
390:   lsave = bv->l;
391:   bv->l = -bv->nc;  /* must also orthogonalize against constraints and leading columns */
392:   if (!bv->buffer) BVGetBufferVec(bv,&bv->buffer);
393:   BV_AllocateSignature(bv);
394:   BVOrthogonalizeGS(bv,j,NULL,NULL,&nrm,&lndep);
395:   if (replace && (nrm==0.0 || lndep)) {
396:     PetscInfo(bv,"Vector was linearly dependent, generating a new random vector\n");
397:     BVSetRandomColumn(bv,j);
398:     BVOrthogonalizeGS(bv,j,NULL,NULL,&nrm,&lndep);
399:     if (nrm==0.0 || lndep) {  /* yet another attempt */
400:       BVSetRandomColumn(bv,j);
401:       BVOrthogonalizeGS(bv,j,NULL,NULL,&nrm,&lndep);
402:     }
403:   }
404:   bv->k = ksave;
405:   bv->l = lsave;
406:   PetscLogEventEnd(BV_OrthogonalizeVec,bv,0,0,0);

408:   /* scale */
409:   if (nrm!=1.0 && nrm!=0.0) {
410:     alpha = 1.0/nrm;
411:     PetscLogEventBegin(BV_Scale,bv,0,0,0);
412:     if (bv->n) PetscUseTypeMethod(bv,scale,j,alpha);
413:     PetscLogEventEnd(BV_Scale,bv,0,0,0);
414:   }
415:   if (norm) *norm = nrm;
416:   if (lindep) *lindep = lndep;
417:   PetscObjectStateIncrease((PetscObject)bv);
418:   return 0;
419: }

421: /*@
422:    BVOrthogonalizeSomeColumn - Orthogonalize one of the column vectors with
423:    respect to some of the previous ones.

425:    Collective on bv

427:    Input Parameters:
428: +  bv     - the basis vectors context
429: .  j      - index of column to be orthogonalized
430: -  which  - logical array indicating selected columns

432:    Output Parameters:
433: +  H      - (optional) coefficients computed during orthogonalization
434: .  norm   - (optional) norm of the vector after being orthogonalized
435: -  lindep - (optional) flag indicating that refinement did not improve the quality
436:             of orthogonalization

438:    Notes:
439:    This function is similar to BVOrthogonalizeColumn(), but V[j] is
440:    orthogonalized only against columns V[i] having which[i]=PETSC_TRUE.
441:    The length of array which must be j at least.

443:    The use of this operation is restricted to MGS orthogonalization type.

445:    In the case of an indefinite inner product, the lindep parameter is not
446:    computed (set to false).

448:    Level: advanced

450: .seealso: BVOrthogonalizeColumn(), BVSetOrthogonalization()
451: @*/
452: PetscErrorCode BVOrthogonalizeSomeColumn(BV bv,PetscInt j,PetscBool *which,PetscScalar *H,PetscReal *norm,PetscBool *lindep)
453: {
454:   PetscInt       ksave,lsave;

460:   BVCheckSizes(bv,1);

465:   PetscLogEventBegin(BV_OrthogonalizeVec,bv,0,0,0);
466:   ksave = bv->k;
467:   lsave = bv->l;
468:   bv->l = -bv->nc;  /* must also orthogonalize against constraints and leading columns */
469:   if (!bv->buffer) BVGetBufferVec(bv,&bv->buffer);
470:   BV_AllocateSignature(bv);
471:   BVOrthogonalizeGS(bv,j,NULL,which,norm,lindep);
472:   bv->k = ksave;
473:   bv->l = lsave;
474:   if (H) BV_StoreCoefficients(bv,j,NULL,H);
475:   PetscLogEventEnd(BV_OrthogonalizeVec,bv,0,0,0);
476:   PetscObjectStateIncrease((PetscObject)bv);
477:   return 0;
478: }

480: /*
481:    Block Gram-Schmidt: V2 = V2 - V1*R12, where R12 = V1'*V2
482:  */
483: static PetscErrorCode BVOrthogonalize_BlockGS(BV V,Mat R)
484: {
485:   BV             V1;

487:   BVGetSplit(V,&V1,NULL);
488:   BVDot(V,V1,R);
489:   BVMult(V,-1.0,1.0,V1,R);
490:   BVRestoreSplit(V,&V1,NULL);
491:   return 0;
492: }

494: /*
495:    Orthogonalize a set of vectors with Gram-Schmidt, column by column.
496:  */
497: static PetscErrorCode BVOrthogonalize_GS(BV V,Mat R)
498: {
499:   PetscScalar    *r=NULL;
500:   PetscReal      norm;
501:   PetscInt       j,ldr,lsave;
502:   Vec            v,w;

504:   if (R) {
505:     MatDenseGetLDA(R,&ldr);
506:     MatDenseGetArray(R,&r);
507:   }
508:   if (V->matrix) {
509:     BVGetCachedBV(V,&V->cached);
510:     BVSetActiveColumns(V->cached,V->l,V->k);
511:   }
512:   for (j=V->l;j<V->k;j++) {
513:     if (V->matrix && V->orthog_type==BV_ORTHOG_MGS) {  /* fill cached BV */
514:       BVGetColumn(V->cached,j,&v);
515:       BVGetColumn(V,j,&w);
516:       MatMult(V->matrix,w,v);
517:       BVRestoreColumn(V,j,&w);
518:       BVRestoreColumn(V->cached,j,&v);
519:     }
520:     if (R) {
521:       BVOrthogonalizeColumn(V,j,NULL,&norm,NULL);
522:       lsave = V->l;
523:       V->l = -V->nc;
524:       BV_StoreCoefficients(V,j,NULL,r+j*ldr);
525:       V->l = lsave;
526:       r[j+j*ldr] = norm;
527:     } else BVOrthogonalizeColumn(V,j,NULL,&norm,NULL);
529:     if (V->matrix && V->orthog_type==BV_ORTHOG_CGS) {  /* fill cached BV */
530:       BVGetColumn(V->cached,j,&v);
531:       VecCopy(V->Bx,v);
532:       BVRestoreColumn(V->cached,j,&v);
533:     }
534:     BVScaleColumn(V,j,1.0/norm);
535:   }
536:   if (R) MatDenseRestoreArray(R,&r);
537:   return 0;
538: }

540: /*
541:   BV_GetBufferMat - Create auxiliary seqdense matrix that wraps the bv->buffer.
542: */
543: static inline PetscErrorCode BV_GetBufferMat(BV bv)
544: {
545:   PetscInt       ld;
546:   PetscScalar    *array;

548:   if (!bv->Abuffer) {
549:     if (!bv->buffer) BVGetBufferVec(bv,&bv->buffer);
550:     ld = bv->m+bv->nc;
551:     VecGetArray(bv->buffer,&array);
552:     MatCreateSeqDense(PETSC_COMM_SELF,ld,bv->m,array,&bv->Abuffer);
553:     VecRestoreArray(bv->buffer,&array);
554:   }
555:   return 0;
556: }

558: /*
559:    BV_StoreCoeffsBlock_Default - Copy the contents of the BV buffer to a dense Mat
560:    provided by the caller. Only columns l:k-1 are copied, restricting to the upper
561:    triangular part if tri=PETSC_TRUE.
562: */
563: static inline PetscErrorCode BV_StoreCoeffsBlock_Default(BV bv,Mat R,PetscBool tri)
564: {
565:   const PetscScalar *bb;
566:   PetscScalar       *rr;
567:   PetscInt          j,ldr,ldb;

569:   MatDenseGetLDA(R,&ldr);
570:   MatDenseGetArray(R,&rr);
571:   ldb  = bv->m+bv->nc;
572:   VecGetArrayRead(bv->buffer,&bb);
573:   for (j=bv->l;j<bv->k;j++) PetscArraycpy(rr+j*ldr,bb+j*ldb,(tri?(j+1):bv->k)+bv->nc);
574:   VecRestoreArrayRead(bv->buffer,&bb);
575:   MatDenseRestoreArray(R,&rr);
576:   return 0;
577: }

579: /*
580:    Orthogonalize a set of vectors with Cholesky: R=chol(V'*V), Q=V*inv(R)
581:  */
582: static PetscErrorCode BVOrthogonalize_Chol(BV V,Mat Rin)
583: {
584:   Mat            R,S;

586:   BV_GetBufferMat(V);
587:   R = V->Abuffer;
588:   if (Rin) S = Rin;   /* use Rin as a workspace for S */
589:   else S = R;
590:   if (V->l) BVOrthogonalize_BlockGS(V,R);
591:   BVDot(V,V,R);
592:   BVMatCholInv_LAPACK_Private(V,R,S);
593:   BVMultInPlace(V,S,V->l,V->k);
594:   if (Rin) BV_StoreCoeffsBlock_Default(V,Rin,PETSC_TRUE);
595:   return 0;
596: }

598: /*
599:    Orthogonalize a set of vectors with the Tall-Skinny QR method
600:  */
601: static PetscErrorCode BVOrthogonalize_TSQR(BV V,Mat Rin)
602: {
603:   PetscScalar    *pv,*r=NULL;
604:   PetscInt       ldr;
605:   Mat            R;

607:   BV_GetBufferMat(V);
608:   R = V->Abuffer;
609:   if (V->l) BVOrthogonalize_BlockGS(V,R);
610:   MatDenseGetLDA(R,&ldr);
611:   MatDenseGetArray(R,&r);
612:   BVGetArray(V,&pv);
613:   BVOrthogonalize_LAPACK_TSQR(V,V->n,V->k-V->l,pv+(V->nc+V->l)*V->n,r+V->l*ldr+V->l,ldr);
614:   BVRestoreArray(V,&pv);
615:   MatDenseRestoreArray(R,&r);
616:   if (Rin) BV_StoreCoeffsBlock_Default(V,Rin,PETSC_TRUE);
617:   return 0;
618: }

620: /*
621:    Orthogonalize a set of vectors with TSQR, but computing R only, then doing Q=V*inv(R)
622:  */
623: static PetscErrorCode BVOrthogonalize_TSQRCHOL(BV V,Mat Rin)
624: {
625:   PetscScalar    *pv,*r=NULL;
626:   PetscInt       ldr;
627:   Mat            R,S;

629:   BV_GetBufferMat(V);
630:   R = V->Abuffer;
631:   if (Rin) S = Rin;   /* use Rin as a workspace for S */
632:   else S = R;
633:   if (V->l) BVOrthogonalize_BlockGS(V,R);
634:   MatDenseGetLDA(R,&ldr);
635:   MatDenseGetArray(R,&r);
636:   BVGetArray(V,&pv);
637:   BVOrthogonalize_LAPACK_TSQR_OnlyR(V,V->n,V->k-V->l,pv+(V->nc+V->l)*V->n,r+V->l*ldr+V->l,ldr);
638:   BVRestoreArray(V,&pv);
639:   MatDenseRestoreArray(R,&r);
640:   BVMatTriInv_LAPACK_Private(V,R,S);
641:   BVMultInPlace(V,S,V->l,V->k);
642:   if (Rin) BV_StoreCoeffsBlock_Default(V,Rin,PETSC_TRUE);
643:   return 0;
644: }

646: /*
647:    Orthogonalize a set of vectors with SVQB
648:  */
649: static PetscErrorCode BVOrthogonalize_SVQB(BV V,Mat Rin)
650: {
651:   Mat            R,S;

653:   BV_GetBufferMat(V);
654:   R = V->Abuffer;
655:   if (Rin) S = Rin;   /* use Rin as a workspace for S */
656:   else S = R;
657:   if (V->l) BVOrthogonalize_BlockGS(V,R);
658:   BVDot(V,V,R);
659:   BVMatSVQB_LAPACK_Private(V,R,S);
660:   BVMultInPlace(V,S,V->l,V->k);
661:   if (Rin) BV_StoreCoeffsBlock_Default(V,Rin,PETSC_FALSE);
662:   return 0;
663: }

665: /*@
666:    BVOrthogonalize - Orthogonalize all columns (starting from the leading ones),
667:    that is, compute the QR decomposition.

669:    Collective on V

671:    Input Parameters:
672: +  V - basis vectors to be orthogonalized (or B-orthogonalized), modified on output
673: -  R - a sequential dense matrix (or NULL), on output the triangular factor of
674:        the QR decomposition

676:    Notes:
677:    On input, matrix R must be a square sequential dense Mat, with at least as many
678:    rows and columns as the number of active columns of V. The output satisfies
679:    V0 = V*R (where V0 represent the input V) and V'*V = I (or V'*B*V = I if an
680:    inner product matrix B has been specified with BVSetMatrix()).

682:    If V has leading columns, then they are not modified (are assumed to be already
683:    orthonormal) and the leading columns of R are not referenced. Let the
684:    decomposition be
685: .vb
686:    [ V01 V02 ] = [ V1 V2 ] [ R11 R12 ]
687:                            [  0  R22 ]
688: .ve
689:    then V1 is left unchanged (equal to V01) as well as R11 (it should satisfy
690:    V01 = V1*R11).

692:    Can pass NULL if R is not required.

694:    The method to be used for block orthogonalization can be set with
695:    BVSetOrthogonalization(). If set to GS, the computation is done column by
696:    column with successive calls to BVOrthogonalizeColumn(). Note that in the
697:    SVQB method the R factor is not upper triangular.

699:    If V is rank-deficient or very ill-conditioned, that is, one or more columns are
700:    (almost) linearly dependent with respect to the rest, then the algorithm may
701:    break down or result in larger numerical error. Linearly dependent columns are
702:    essentially replaced by random directions, and the corresponding diagonal entry
703:    in R is set to (nearly) zero.

705:    Level: intermediate

707: .seealso: BVOrthogonalizeColumn(), BVOrthogonalizeVec(), BVSetMatrix(), BVSetActiveColumns(), BVSetOrthogonalization(), BVOrthogBlockType
708: @*/
709: PetscErrorCode BVOrthogonalize(BV V,Mat R)
710: {
711:   PetscInt       m,n;

715:   BVCheckSizes(V,1);
716:   if (R) {
720:     MatGetSize(R,&m,&n);
723:   }

726:   PetscLogEventBegin(BV_Orthogonalize,V,R,0,0);
727:   switch (V->orthog_block) {
728:   case BV_ORTHOG_BLOCK_GS: /* proceed column by column with Gram-Schmidt */
729:     BVOrthogonalize_GS(V,R);
730:     break;
731:   case BV_ORTHOG_BLOCK_CHOL:
732:     BVOrthogonalize_Chol(V,R);
733:     break;
734:   case BV_ORTHOG_BLOCK_TSQR:
736:     BVOrthogonalize_TSQR(V,R);
737:     break;
738:   case BV_ORTHOG_BLOCK_TSQRCHOL:
740:     BVOrthogonalize_TSQRCHOL(V,R);
741:     break;
742:   case BV_ORTHOG_BLOCK_SVQB:
743:     BVOrthogonalize_SVQB(V,R);
744:     break;
745:   }
746:   PetscLogEventEnd(BV_Orthogonalize,V,R,0,0);
747:   PetscObjectStateIncrease((PetscObject)V);
748:   return 0;
749: }