Actual source code: iporthog.c

  1: /*
  2:      Routines related to orthogonalization.
  3:      See the SLEPc Technical Report STR-1 for a detailed explanation.

  5:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  6:    SLEPc - Scalable Library for Eigenvalue Problem Computations
  7:    Copyright (c) 2002-2010, Universidad Politecnica de Valencia, Spain

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

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

 20:    You  should have received a copy of the GNU Lesser General  Public  License
 21:    along with SLEPc. If not, see <http://www.gnu.org/licenses/>.
 22:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 23: */

 25:  #include private/ipimpl.h
 26:  #include slepcblaslapack.h

 28: /* 
 29:    IPOrthogonalizeMGS1 - Compute one step of Modified Gram-Schmidt 
 30: */
 33: static PetscErrorCode IPOrthogonalizeMGS1(IP ip,PetscInt n,PetscTruth *which,Vec *V,Vec v,PetscScalar *H)
 34: {
 36:   PetscInt       j;
 37:   PetscScalar    dot;
 38: 
 40:   for (j=0; j<n; j++)
 41:     if (!which || which[j]) {
 42:       /* h_j = ( v, v_j ) */
 43:       IPInnerProduct(ip,v,V[j],&dot);
 44:       /* v <- v - h_j v_j */
 45:       VecAXPY(v,-dot,V[j]);
 46:       if (H) H[j] += dot;
 47:     }
 48:   return(0);
 49: }

 51: /* 
 52:    IPOrthogonalizeCGS1 - Compute |v'| (estimated), |v| and one step of CGS with only one global synchronization
 53: */
 56: PetscErrorCode IPOrthogonalizeCGS1(IP ip,PetscInt nds,Vec *DS,PetscInt n,PetscTruth *which,Vec *V,Vec v,PetscScalar *H,PetscReal *onorm,PetscReal *norm)
 57: {
 59:   PetscInt       j;
 60:   PetscScalar    alpha;
 61:   PetscReal      sum;

 64:   /* h = W^* v ; alpha = (v , v) */
 65:   if (nds==0 && !which && !onorm && !norm) {
 66:     /* use simpler function */
 67:     IPMInnerProduct(ip,v,n,V,H);
 68:   } else {
 69:     /* merge comunications */
 70:     IPMInnerProductBegin(ip,v,nds,DS,H);
 71:     if (which) { /* use select array */
 72:       for (j=0; j<n; j++)
 73:         if (which[j]) { IPInnerProductBegin(ip,v,V[j],H+nds+j); }
 74:     } else {
 75:       IPMInnerProductBegin(ip,v,n,V,H+nds);
 76:     }
 77:     if (onorm || (norm && !ip->matrix)) {
 78:       IPInnerProductBegin(ip,v,v,&alpha);
 79:     }

 81:     IPMInnerProductEnd(ip,v,nds,DS,H);
 82:     if (which) { /* use select array */
 83:       for (j=0; j<n; j++)
 84:         if (which[j]) { IPInnerProductEnd(ip,v,V[j],H+nds+j); }
 85:     } else {
 86:       IPMInnerProductEnd(ip,v,n,V,H+nds);
 87:     }
 88:     if (onorm || (norm && !ip->matrix)) {
 89:       IPInnerProductEnd(ip,v,v,&alpha);
 90:     }
 91:   }

 93:   /* q = v - V h */
 94:   SlepcVecMAXPBY(v,1.0,-1.0,nds,H,DS);
 95:   if (which) {
 96:     for (j=0; j<n; j++)
 97:       if (which[j]) { VecAXPBY(v,-H[nds+j],1.0,V[j]); }
 98:   } else {
 99:     SlepcVecMAXPBY(v,1.0,-1.0,n,H+nds,V);
100:   }
101: 
102:   /* compute |v| */
103:   if (onorm) *onorm = sqrt(PetscRealPart(alpha));

105:   if (norm) {
106:     if (!ip->matrix) {
107:       /* estimate |v'| from |v| */
108:       sum = 0.0;
109:       for (j=0; j<nds; j++)
110:         sum += PetscRealPart(H[j] * PetscConj(H[j]));
111:       for (j=0; j<n; j++)
112:         if (!which || which[j])
113:           sum += PetscRealPart(H[nds+j] * PetscConj(H[nds+j]));
114:       *norm = PetscRealPart(alpha)-sum;
115:       if (*norm <= 0.0) {
116:         IPNorm(ip,v,norm);
117:       } else *norm = sqrt(*norm);
118:     } else {
119:       /* compute |v'| */
120:       IPNorm(ip,v,norm);
121:     }
122:   }
123:   return(0);
124: }

126: /* 
127:   IPOrthogonalizeMGS - Orthogonalize with modified Gram-Schmidt
128: */
131: static PetscErrorCode IPOrthogonalizeMGS(IP ip,PetscInt nds,Vec *DS,PetscInt n,PetscTruth *which,Vec *V,Vec v,PetscScalar *H,PetscReal *norm,PetscTruth *lindep)
132: {
134:   PetscInt       i,k;
135:   PetscReal      onrm,nrm;

138: 
139:   if (H) {
140:     for (i=0;i<n;i++)
141:       H[i] = 0;
142:   }
143: 
144:   switch (ip->orthog_ref) {
145: 
146:   case IP_ORTH_REFINE_NEVER:
147:     IPOrthogonalizeMGS1(ip,nds,PETSC_NULL,DS,v,PETSC_NULL);
148:     IPOrthogonalizeMGS1(ip,n,which,V,v,H);
149:     /* compute |v| */
150:     if (norm) { IPNorm(ip,v,norm); }
151:     /* linear dependence check does not work without refinement */
152:     if (lindep) *lindep = PETSC_FALSE;
153:     break;
154: 
155:   case IP_ORTH_REFINE_ALWAYS:
156:     /* first step */
157:     IPOrthogonalizeMGS1(ip,nds,PETSC_NULL,DS,v,PETSC_NULL);
158:     IPOrthogonalizeMGS1(ip,n,which,V,v,H);
159:     if (lindep) { IPNorm(ip,v,&onrm); }
160:     /* second step */
161:     IPOrthogonalizeMGS1(ip,nds,PETSC_NULL,DS,v,PETSC_NULL);
162:     IPOrthogonalizeMGS1(ip,n,which,V,v,H);
163:     if (lindep || norm) { IPNorm(ip,v,&nrm); }
164:     if (lindep) {
165:       if (nrm < ip->orthog_eta * onrm) *lindep = PETSC_TRUE;
166:       else *lindep = PETSC_FALSE;
167:     }
168:     if (norm) *norm = nrm;
169:     break;
170: 
171:   case IP_ORTH_REFINE_IFNEEDED:
172:     /* first step */
173:     IPNorm(ip,v,&onrm);
174:     IPOrthogonalizeMGS1(ip,nds,PETSC_NULL,DS,v,PETSC_NULL);
175:     IPOrthogonalizeMGS1(ip,n,which,V,v,H);
176:     IPNorm(ip,v,&nrm);
177:     /* ||q|| < eta ||h|| */
178:     k = 1;
179:     while (k<3 && nrm < ip->orthog_eta * onrm) {
180:       k++;
181:       onrm = nrm;
182:       IPOrthogonalizeMGS1(ip,nds,PETSC_NULL,DS,v,PETSC_NULL);
183:       IPOrthogonalizeMGS1(ip,n,which,V,v,H);
184:       IPNorm(ip,v,&nrm);
185:     }
186:     if (lindep) {
187:       if (nrm < ip->orthog_eta * onrm) *lindep = PETSC_TRUE;
188:       else *lindep = PETSC_FALSE;
189:     }
190:     if (norm) *norm = nrm;
191:     break;
192: 
193:   default:
194:     SETERRQ(PETSC_ERR_ARG_WRONG,"Unknown orthogonalization refinement");
195:   }

197:   return(0);
198: }

200: /*
201:   IPOrthogonalizeCGS - Orthogonalize with classical Gram-Schmidt
202: */
205: static PetscErrorCode IPOrthogonalizeCGS(IP ip,PetscInt nds,Vec *DS,PetscInt n,PetscTruth *which,Vec *V,Vec v,PetscScalar *H,PetscReal *norm,PetscTruth *lindep)
206: {
208:   PetscScalar    lh[100],*h,lc[100],*c;
209:   PetscTruth     allocatedh = PETSC_FALSE,allocatedc = PETSC_FALSE;
210:   PetscReal      onrm,nrm;
211:   PetscInt       j,k;

214:   /* allocate h and c if needed */
215:   if (!H || nds>0) {
216:     if (nds+n<=100) h = lh;
217:     else {
218:       PetscMalloc((nds+n)*sizeof(PetscScalar),&h);
219:       allocatedh = PETSC_TRUE;
220:     }
221:   } else h = H;
222:   if (ip->orthog_ref != IP_ORTH_REFINE_NEVER) {
223:     if (nds+n<=100) c = lc;
224:     else {
225:       PetscMalloc((nds+n)*sizeof(PetscScalar),&c);
226:       allocatedc = PETSC_TRUE;
227:     }
228:   }

230:   /* orthogonalize and compute onorm */
231:   switch (ip->orthog_ref) {
232: 
233:   case IP_ORTH_REFINE_NEVER:
234:     IPOrthogonalizeCGS1(ip,nds,DS,n,which,V,v,h,PETSC_NULL,PETSC_NULL);
235:     /* compute |v| */
236:     if (norm) { IPNorm(ip,v,norm); }
237:     /* linear dependence check does not work without refinement */
238:     if (lindep) *lindep = PETSC_FALSE;
239:     break;
240: 
241:   case IP_ORTH_REFINE_ALWAYS:
242:     IPOrthogonalizeCGS1(ip,nds,DS,n,which,V,v,h,PETSC_NULL,PETSC_NULL);
243:     if (lindep) {
244:       IPOrthogonalizeCGS1(ip,nds,DS,n,which,V,v,c,&onrm,&nrm);
245:       if (norm) *norm = nrm;
246:       if (nrm < ip->orthog_eta * onrm) *lindep = PETSC_TRUE;
247:       else *lindep = PETSC_FALSE;
248:     } else {
249:       IPOrthogonalizeCGS1(ip,nds,DS,n,which,V,v,c,PETSC_NULL,norm);
250:     }
251:     for (j=0;j<n;j++)
252:       if (!which || which[j]) h[nds+j] += c[nds+j];
253:     break;
254: 
255:   case IP_ORTH_REFINE_IFNEEDED:
256:     IPOrthogonalizeCGS1(ip,nds,DS,n,which,V,v,h,&onrm,&nrm);
257:     /* ||q|| < eta ||h|| */
258:     k = 1;
259:     while (k<3 && nrm < ip->orthog_eta * onrm) {
260:       k++;
261:       if (!ip->matrix) {
262:         IPOrthogonalizeCGS1(ip,nds,DS,n,which,V,v,c,&onrm,&nrm);
263:       } else {
264:         onrm = nrm;
265:         IPOrthogonalizeCGS1(ip,nds,DS,n,which,V,v,c,PETSC_NULL,&nrm);
266:       }
267:       for (j=0;j<n;j++)
268:         if (!which || which[j]) h[nds+j] += c[nds+j];
269:     }
270:     if (norm) *norm = nrm;
271:     if (lindep) {
272:       if (nrm < ip->orthog_eta * onrm) *lindep = PETSC_TRUE;
273:       else *lindep = PETSC_FALSE;
274:     }
275:     break;

277:   default:
278:     SETERRQ(PETSC_ERR_ARG_WRONG,"Unknown orthogonalization refinement");
279:   }

281:   /* recover H from workspace */
282:   if (H && nds>0) {
283:     for (j=0;j<n;j++)
284:       if (!which || which[j]) H[j] = h[nds+j];
285:   }

287:   /* free work space */
288:   if (allocatedc) { PetscFree(c); }
289:   if (allocatedh) { PetscFree(h); }
290:   return(0);
291: }

295: /*@
296:    IPOrthogonalize - Orthogonalize a vector with respect to two set of vectors.

298:    Collective on IP

300:    Input Parameters:
301: +  ip     - the inner product (IP) context
302: .  nds    - number of columns of DS
303: .  DS     - first set of vectors
304: .  n      - number of columns of V
305: .  which  - logical array indicating columns of V to be used
306: -  V      - second set of vectors

308:    Input/Output Parameter:
309: .  v      - (input) vector to be orthogonalized and (output) result of 
310:             orthogonalization

312:    Output Parameter:
313: +  H      - coefficients computed during orthogonalization with V
314: .  norm   - norm of the vector after being orthogonalized
315: -  lindep - flag indicating that refinement did not improve the quality
316:             of orthogonalization

318:    Notes:
319:    This function applies an orthogonal projector to project vector v onto the
320:    orthogonal complement of the span of the columns of DS and V.

322:    On exit, v0 = [V v]*H, where v0 is the original vector v.

324:    This routine does not normalize the resulting vector.

326:    Level: developer

328: .seealso: IPSetOrthogonalization(), IPBiOrthogonalize()
329: @*/
330: PetscErrorCode IPOrthogonalize(IP ip,PetscInt nds,Vec *DS,PetscInt n,PetscTruth *which,Vec *V,Vec v,PetscScalar *H,PetscReal *norm,PetscTruth *lindep)
331: {

335:   PetscLogEventBegin(IP_Orthogonalize,ip,0,0,0);
336: 
337:   if (nds==0 && n==0) {
338:     if (norm) { IPNorm(ip,v,norm); }
339:     if (lindep) *lindep = PETSC_FALSE;
340:   } else {
341:     switch (ip->orthog_type) {
342:     case IP_ORTH_CGS:
343:       IPOrthogonalizeCGS(ip,nds,DS,n,which,V,v,H,norm,lindep);
344:       break;
345:     case IP_ORTH_MGS:
346:       IPOrthogonalizeMGS(ip,nds,DS,n,which,V,v,H,norm,lindep);
347:       break;
348:     default:
349:       SETERRQ(PETSC_ERR_ARG_WRONG,"Unknown orthogonalization type");
350:     }
351:   }
352: 
353:   PetscLogEventEnd(IP_Orthogonalize,ip,0,0,0);
354:   return(0);
355: }

359: /*@
360:    IPQRDecomposition - Compute the QR factorization of a set of vectors.

362:    Collective on IP

364:    Input Parameters:
365: +  ip - the eigenproblem solver context
366: .  V - set of vectors
367: .  m - starting column
368: .  n - ending column
369: -  ldr - leading dimension of R

371:    Output Parameter:
372: .  R  - triangular matrix of QR factorization

374:    Notes:
375:    This routine orthonormalizes the columns of V so that V'*V=I up to 
376:    working precision. It assumes that the first m columns (from 0 to m-1) 
377:    are already orthonormal. The coefficients of the orthogonalization are
378:    stored in R so that V*R is equal to the original V.

380:    In some cases, this routine makes V B-orthonormal, that is, V'*B*V=I.

382:    If one of the vectors is linearly dependent wrt the rest (at working
383:    precision) then it is replaced by a random vector.

385:    Level: developer

387: .seealso: IPOrthogonalize(), IPNorm(), IPInnerProduct().
388: @*/
389: PetscErrorCode IPQRDecomposition(IP ip,Vec *V,PetscInt m,PetscInt n,PetscScalar *R,PetscInt ldr)
390: {
392:   PetscInt       k;
393:   PetscReal      norm;
394:   PetscTruth     lindep;
395:   PetscRandom    rctx=PETSC_NULL;
396: 

399:   for (k=m; k<n; k++) {

401:     /* orthogonalize v_k with respect to v_0, ..., v_{k-1} */
402:     if (R) { IPOrthogonalize(ip,0,PETSC_NULL,k,PETSC_NULL,V,V[k],&R[0+ldr*k],&norm,&lindep); }
403:     else   { IPOrthogonalize(ip,0,PETSC_NULL,k,PETSC_NULL,V,V[k],PETSC_NULL,&norm,&lindep); }

405:     /* normalize v_k: r_{k,k} = ||v_k||_2; v_k = v_k/r_{k,k} */
406:     if (norm==0.0 || lindep) {
407:       PetscInfo(ip,"Linearly dependent vector found, generating a new random vector\n");
408:       if (!rctx) {
409:         PetscRandomCreate(((PetscObject)ip)->comm,&rctx);
410:         PetscRandomSetFromOptions(rctx);
411:       }
412:       SlepcVecSetRandom(V[k],rctx);
413:       IPNorm(ip,V[k],&norm);
414:     }
415:     VecScale(V[k],1.0/norm);
416:     if (R) R[k+ldr*k] = norm;

418:   }
419:   if (rctx) { PetscRandomDestroy(rctx); }

421:   return(0);
422: }

424: /*
425:     Biorthogonalization routine using classical Gram-Schmidt with refinement.
426:  */
429: static PetscErrorCode IPCGSBiOrthogonalization(IP ip,PetscInt n_,Vec *V,Vec *W,Vec v,PetscScalar *H,PetscReal *hnorm,PetscReal *norm)
430: {
431: #if defined(SLEPC_MISSING_LAPACK_GELQF) || defined(SLEPC_MISSING_LAPACK_ORMLQ)
433:   SETERRQ(PETSC_ERR_SUP,"xGELQF - Lapack routine is unavailable.");
434: #else
436:   PetscBLASInt   j,ione=1,lwork,info,n=n_;
437:   PetscScalar    shh[100],*lhh,*vw,*tau,one=1.0,*work;


441:   /* Don't allocate small arrays */
442:   if (n<=100) lhh = shh;
443:   else { PetscMalloc(n*sizeof(PetscScalar),&lhh); }
444:   PetscMalloc(n*n*sizeof(PetscScalar),&vw);
445: 
446:   for (j=0;j<n;j++) {
447:     IPMInnerProduct(ip,V[j],n,W,vw+j*n);
448:   }
449:   lwork = n;
450:   PetscMalloc(n*sizeof(PetscScalar),&tau);
451:   PetscMalloc(lwork*sizeof(PetscScalar),&work);
452:   LAPACKgelqf_(&n,&n,vw,&n,tau,work,&lwork,&info);
453:   if (info) SETERRQ1(PETSC_ERR_LIB,"Error in Lapack xGELQF %i",info);
454: 
455:   /*** First orthogonalization ***/

457:   /* h = W^* v */
458:   /* q = v - V h */
459:   IPMInnerProduct(ip,v,n,W,H);
460:   BLAStrsm_("L","L","N","N",&n,&ione,&one,vw,&n,H,&n);
461:   LAPACKormlq_("L","N",&n,&ione,&n,vw,&n,tau,H,&n,work,&lwork,&info);
462:   if (info) SETERRQ1(PETSC_ERR_LIB,"Error in Lapack xORMLQ %i",info);
463:   SlepcVecMAXPBY(v,1.0,-1.0,n,H,V);

465:   /* compute norm of v */
466:   if (norm) { IPNorm(ip,v,norm); }
467: 
468:   if (n>100) { PetscFree(lhh); }
469:   PetscFree(vw);
470:   PetscFree(tau);
471:   PetscFree(work);
472:   return(0);
473: #endif
474: }

478: /*@
479:    IPBiOrthogonalize - Bi-orthogonalize a vector with respect to a set of vectors.

481:    Collective on IP

483:    Input Parameters:
484: +  ip - the inner product context
485: .  n - number of columns of V
486: .  V - set of vectors
487: -  W - set of vectors

489:    Input/Output Parameter:
490: .  v - vector to be orthogonalized

492:    Output Parameter:
493: +  H  - coefficients computed during orthogonalization
494: -  norm - norm of the vector after being orthogonalized

496:    Notes:
497:    This function applies an oblique projector to project vector v onto the
498:    span of the columns of V along the orthogonal complement of the column
499:    space of W. 

501:    On exit, v0 = [V v]*H, where v0 is the original vector v.

503:    This routine does not normalize the resulting vector.

505:    Level: developer

507: .seealso: IPSetOrthogonalization(), IPOrthogonalize()
508: @*/
509: PetscErrorCode IPBiOrthogonalize(IP ip,PetscInt n,Vec *V,Vec *W,Vec v,PetscScalar *H,PetscReal *norm)
510: {
512:   PetscScalar    lh[100],*h;
513:   PetscTruth     allocated = PETSC_FALSE;
514:   PetscReal      lhnrm,*hnrm,lnrm,*nrm;
516:   if (n==0) {
517:     if (norm) { IPNorm(ip,v,norm); }
518:   } else {
519:     PetscLogEventBegin(IP_Orthogonalize,ip,0,0,0);
520: 
521:     /* allocate H if needed */
522:     if (!H) {
523:       if (n<=100) h = lh;
524:       else {
525:         PetscMalloc(n*sizeof(PetscScalar),&h);
526:         allocated = PETSC_TRUE;
527:       }
528:     } else h = H;
529: 
530:     /* retrieve hnrm and nrm for linear dependence check or conditional refinement */
531:     if (ip->orthog_ref == IP_ORTH_REFINE_IFNEEDED) {
532:       hnrm = &lhnrm;
533:       if (norm) nrm = norm;
534:       else nrm = &lnrm;
535:     } else {
536:       hnrm = PETSC_NULL;
537:       nrm = norm;
538:     }
539: 
540:     switch (ip->orthog_type) {
541:       case IP_ORTH_CGS:
542:         IPCGSBiOrthogonalization(ip,n,V,W,v,h,hnrm,nrm);
543:         break;
544:       default:
545:         SETERRQ(PETSC_ERR_ARG_WRONG,"Unknown orthogonalization type");
546:     }
547: 
548:     if (allocated) { PetscFree(h); }
549: 
550:     PetscLogEventEnd(IP_Orthogonalize,ip,0,0,0);
551:   }
552:   return(0);
553: }