Actual source code: svdsolve.c

slepc-3.20.2 2024-03-15
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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:    SVD routines related to the solution process
 12: */

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

 17: /*
 18:   SVDComputeVectors_Left - Compute left singular vectors as U=A*V.
 19:   Only done if the leftbasis flag is false. Assumes V is available.
 20:  */
 21: PetscErrorCode SVDComputeVectors_Left(SVD svd)
 22: {
 23:   Vec                tl,omega2,u,v,w;
 24:   PetscInt           i,oldsize;
 25:   VecType            vtype;
 26:   const PetscScalar* varray;

 28:   PetscFunctionBegin;
 29:   if (!svd->leftbasis) {
 30:     /* generate left singular vectors on U */
 31:     if (!svd->U) PetscCall(SVDGetBV(svd,NULL,&svd->U));
 32:     PetscCall(BVGetSizes(svd->U,NULL,NULL,&oldsize));
 33:     if (!oldsize) {
 34:       if (!((PetscObject)(svd->U))->type_name) PetscCall(BVSetType(svd->U,((PetscObject)(svd->V))->type_name));
 35:       PetscCall(MatCreateVecsEmpty(svd->A,NULL,&tl));
 36:       PetscCall(BVSetSizesFromVec(svd->U,tl,svd->ncv));
 37:       PetscCall(VecDestroy(&tl));
 38:     }
 39:     PetscCall(BVSetActiveColumns(svd->V,0,svd->nconv));
 40:     PetscCall(BVSetActiveColumns(svd->U,0,svd->nconv));
 41:     if (!svd->ishyperbolic) PetscCall(BVMatMult(svd->V,svd->A,svd->U));
 42:     else if (svd->swapped) {  /* compute right singular vectors as V=A'*Omega*U */
 43:       PetscCall(MatCreateVecs(svd->A,&w,NULL));
 44:       for (i=0;i<svd->nconv;i++) {
 45:         PetscCall(BVGetColumn(svd->V,i,&v));
 46:         PetscCall(BVGetColumn(svd->U,i,&u));
 47:         PetscCall(VecPointwiseMult(w,v,svd->omega));
 48:         PetscCall(MatMult(svd->A,w,u));
 49:         PetscCall(BVRestoreColumn(svd->V,i,&v));
 50:         PetscCall(BVRestoreColumn(svd->U,i,&u));
 51:       }
 52:       PetscCall(VecDestroy(&w));
 53:     } else {  /* compute left singular vectors as usual U=A*V, and set-up Omega-orthogonalization of U */
 54:       PetscCall(BV_SetMatrixDiagonal(svd->U,svd->omega,svd->A));
 55:       PetscCall(BVMatMult(svd->V,svd->A,svd->U));
 56:     }
 57:     PetscCall(BVOrthogonalize(svd->U,NULL));
 58:     if (svd->ishyperbolic && !svd->swapped) {  /* store signature after Omega-orthogonalization */
 59:       PetscCall(MatGetVecType(svd->A,&vtype));
 60:       PetscCall(VecCreate(PETSC_COMM_SELF,&omega2));
 61:       PetscCall(VecSetSizes(omega2,svd->nconv,svd->nconv));
 62:       PetscCall(VecSetType(omega2,vtype));
 63:       PetscCall(BVGetSignature(svd->U,omega2));
 64:       PetscCall(VecGetArrayRead(omega2,&varray));
 65:       for (i=0;i<svd->nconv;i++) {
 66:         svd->sign[i] = PetscRealPart(varray[i]);
 67:         if (PetscRealPart(varray[i])<0.0) PetscCall(BVScaleColumn(svd->U,i,-1.0));
 68:       }
 69:       PetscCall(VecRestoreArrayRead(omega2,&varray));
 70:       PetscCall(VecDestroy(&omega2));
 71:     }
 72:   }
 73:   PetscFunctionReturn(PETSC_SUCCESS);
 74: }

 76: PetscErrorCode SVDComputeVectors(SVD svd)
 77: {
 78:   PetscFunctionBegin;
 79:   SVDCheckSolved(svd,1);
 80:   if (svd->state==SVD_STATE_SOLVED) PetscTryTypeMethod(svd,computevectors);
 81:   svd->state = SVD_STATE_VECTORS;
 82:   PetscFunctionReturn(PETSC_SUCCESS);
 83: }

 85: /*@
 86:    SVDSolve - Solves the singular value problem.

 88:    Collective

 90:    Input Parameter:
 91: .  svd - singular value solver context obtained from SVDCreate()

 93:    Options Database Keys:
 94: +  -svd_view - print information about the solver used
 95: .  -svd_view_mat0 - view the first matrix (A)
 96: .  -svd_view_mat1 - view the second matrix (B)
 97: .  -svd_view_signature - view the signature matrix (omega)
 98: .  -svd_view_vectors - view the computed singular vectors
 99: .  -svd_view_values - view the computed singular values
100: .  -svd_converged_reason - print reason for convergence, and number of iterations
101: .  -svd_error_absolute - print absolute errors of each singular triplet
102: .  -svd_error_relative - print relative errors of each singular triplet
103: -  -svd_error_norm     - print errors relative to the matrix norms of each singular triplet

105:    Notes:
106:    All the command-line options listed above admit an optional argument specifying
107:    the viewer type and options. For instance, use '-svd_view_mat0 binary:amatrix.bin'
108:    to save the A matrix to a binary file, '-svd_view_values draw' to draw the computed
109:    singular values graphically, or '-svd_error_relative :myerr.m:ascii_matlab' to save
110:    the errors in a file that can be executed in Matlab.

112:    Level: beginner

114: .seealso: SVDCreate(), SVDSetUp(), SVDDestroy()
115: @*/
116: PetscErrorCode SVDSolve(SVD svd)
117: {
118:   PetscInt       i,*workperm;

120:   PetscFunctionBegin;
122:   if (svd->state>=SVD_STATE_SOLVED) PetscFunctionReturn(PETSC_SUCCESS);
123:   PetscCall(PetscLogEventBegin(SVD_Solve,svd,0,0,0));

125:   /* call setup */
126:   PetscCall(SVDSetUp(svd));
127:   svd->its = 0;
128:   svd->nconv = 0;
129:   for (i=0;i<svd->ncv;i++) {
130:     svd->sigma[i]  = 0.0;
131:     svd->errest[i] = 0.0;
132:     svd->perm[i]   = i;
133:   }
134:   PetscCall(SVDViewFromOptions(svd,NULL,"-svd_view_pre"));

136:   switch (svd->problem_type) {
137:     case SVD_STANDARD:
138:       PetscUseTypeMethod(svd,solve);
139:       break;
140:     case SVD_GENERALIZED:
141:       PetscUseTypeMethod(svd,solveg);
142:       break;
143:     case SVD_HYPERBOLIC:
144:       PetscUseTypeMethod(svd,solveh);
145:       break;
146:   }
147:   svd->state = SVD_STATE_SOLVED;

149:   /* sort singular triplets */
150:   if (svd->which == SVD_SMALLEST) PetscCall(PetscSortRealWithPermutation(svd->nconv,svd->sigma,svd->perm));
151:   else {
152:     PetscCall(PetscMalloc1(svd->nconv,&workperm));
153:     for (i=0;i<svd->nconv;i++) workperm[i] = i;
154:     PetscCall(PetscSortRealWithPermutation(svd->nconv,svd->sigma,workperm));
155:     for (i=0;i<svd->nconv;i++) svd->perm[i] = workperm[svd->nconv-i-1];
156:     PetscCall(PetscFree(workperm));
157:   }
158:   PetscCall(PetscLogEventEnd(SVD_Solve,svd,0,0,0));

160:   /* various viewers */
161:   PetscCall(SVDViewFromOptions(svd,NULL,"-svd_view"));
162:   PetscCall(SVDConvergedReasonViewFromOptions(svd));
163:   PetscCall(SVDErrorViewFromOptions(svd));
164:   PetscCall(SVDValuesViewFromOptions(svd));
165:   PetscCall(SVDVectorsViewFromOptions(svd));
166:   PetscCall(MatViewFromOptions(svd->OP,(PetscObject)svd,"-svd_view_mat0"));
167:   if (svd->isgeneralized) PetscCall(MatViewFromOptions(svd->OPb,(PetscObject)svd,"-svd_view_mat1"));
168:   if (svd->ishyperbolic) PetscCall(VecViewFromOptions(svd->omega,(PetscObject)svd,"-svd_view_signature"));

170:   /* Remove the initial subspaces */
171:   svd->nini = 0;
172:   svd->ninil = 0;
173:   PetscFunctionReturn(PETSC_SUCCESS);
174: }

176: /*@
177:    SVDGetIterationNumber - Gets the current iteration number. If the
178:    call to SVDSolve() is complete, then it returns the number of iterations
179:    carried out by the solution method.

181:    Not Collective

183:    Input Parameter:
184: .  svd - the singular value solver context

186:    Output Parameter:
187: .  its - number of iterations

189:    Note:
190:    During the i-th iteration this call returns i-1. If SVDSolve() is
191:    complete, then parameter "its" contains either the iteration number at
192:    which convergence was successfully reached, or failure was detected.
193:    Call SVDGetConvergedReason() to determine if the solver converged or
194:    failed and why.

196:    Level: intermediate

198: .seealso: SVDGetConvergedReason(), SVDSetTolerances()
199: @*/
200: PetscErrorCode SVDGetIterationNumber(SVD svd,PetscInt *its)
201: {
202:   PetscFunctionBegin;
204:   PetscAssertPointer(its,2);
205:   *its = svd->its;
206:   PetscFunctionReturn(PETSC_SUCCESS);
207: }

209: /*@
210:    SVDGetConvergedReason - Gets the reason why the SVDSolve() iteration was
211:    stopped.

213:    Not Collective

215:    Input Parameter:
216: .  svd - the singular value solver context

218:    Output Parameter:
219: .  reason - negative value indicates diverged, positive value converged
220:    (see SVDConvergedReason)

222:    Options Database Key:
223: .  -svd_converged_reason - print the reason to a viewer

225:    Notes:
226:    Possible values for reason are
227: +  SVD_CONVERGED_TOL - converged up to tolerance
228: .  SVD_CONVERGED_USER - converged due to a user-defined condition
229: .  SVD_CONVERGED_MAXIT - reached the maximum number of iterations with SVD_CONV_MAXIT criterion
230: .  SVD_DIVERGED_ITS - required more than max_it iterations to reach convergence
231: .  SVD_DIVERGED_BREAKDOWN - generic breakdown in method
232: -  SVD_DIVERGED_SYMMETRY_LOST - underlying indefinite eigensolver was not able to keep symmetry

234:    Can only be called after the call to SVDSolve() is complete.

236:    Level: intermediate

238: .seealso: SVDSetTolerances(), SVDSolve(), SVDConvergedReason
239: @*/
240: PetscErrorCode SVDGetConvergedReason(SVD svd,SVDConvergedReason *reason)
241: {
242:   PetscFunctionBegin;
244:   PetscAssertPointer(reason,2);
245:   SVDCheckSolved(svd,1);
246:   *reason = svd->reason;
247:   PetscFunctionReturn(PETSC_SUCCESS);
248: }

250: /*@
251:    SVDGetConverged - Gets the number of converged singular values.

253:    Not Collective

255:    Input Parameter:
256: .  svd - the singular value solver context

258:    Output Parameter:
259: .  nconv - number of converged singular values

261:    Note:
262:    This function should be called after SVDSolve() has finished.

264:    Level: beginner

266: .seealso: SVDSetDimensions(), SVDSolve(), SVDGetSingularTriplet()
267: @*/
268: PetscErrorCode SVDGetConverged(SVD svd,PetscInt *nconv)
269: {
270:   PetscFunctionBegin;
272:   PetscAssertPointer(nconv,2);
273:   SVDCheckSolved(svd,1);
274:   *nconv = svd->nconv;
275:   PetscFunctionReturn(PETSC_SUCCESS);
276: }

278: /*@C
279:    SVDGetSingularTriplet - Gets the i-th triplet of the singular value decomposition
280:    as computed by SVDSolve(). The solution consists in the singular value and its left
281:    and right singular vectors.

283:    Collective

285:    Input Parameters:
286: +  svd - singular value solver context
287: -  i   - index of the solution

289:    Output Parameters:
290: +  sigma - singular value
291: .  u     - left singular vector
292: -  v     - right singular vector

294:    Note:
295:    Both u or v can be NULL if singular vectors are not required.
296:    Otherwise, the caller must provide valid Vec objects, i.e.,
297:    they must be created by the calling program with e.g. MatCreateVecs().

299:    The index i should be a value between 0 and nconv-1 (see SVDGetConverged()).
300:    Singular triplets are indexed according to the ordering criterion established
301:    with SVDSetWhichSingularTriplets().

303:    In the case of GSVD, the solution consists in three vectors u,v,x that are
304:    returned as follows. Vector x is returned in the right singular vector
305:    (argument v) and has length equal to the number of columns of A and B.
306:    The other two vectors are returned stacked on top of each other [u;v] in
307:    the left singular vector argument, with length equal to m+n (number of rows
308:    of A plus number of rows of B).

310:    Level: beginner

312: .seealso: SVDSolve(), SVDGetConverged(), SVDSetWhichSingularTriplets()
313: @*/
314: PetscErrorCode SVDGetSingularTriplet(SVD svd,PetscInt i,PetscReal *sigma,Vec u,Vec v)
315: {
316:   PetscInt       M,N;
317:   Vec            w;

319:   PetscFunctionBegin;
322:   SVDCheckSolved(svd,1);
325:   PetscCheck(i>=0,PetscObjectComm((PetscObject)svd),PETSC_ERR_ARG_OUTOFRANGE,"The index cannot be negative");
326:   PetscCheck(i<svd->nconv,PetscObjectComm((PetscObject)svd),PETSC_ERR_ARG_OUTOFRANGE,"The index can be nconv-1 at most, see SVDGetConverged()");
327:   if (sigma) *sigma = svd->sigma[svd->perm[i]];
328:   if (u || v) {
329:     if (!svd->isgeneralized) {
330:       PetscCall(MatGetSize(svd->OP,&M,&N));
331:       if (M<N) { w = u; u = v; v = w; }
332:     }
333:     PetscCall(SVDComputeVectors(svd));
334:     if (u) PetscCall(BVCopyVec(svd->U,svd->perm[i],u));
335:     if (v) PetscCall(BVCopyVec(svd->V,svd->perm[i],v));
336:   }
337:   PetscFunctionReturn(PETSC_SUCCESS);
338: }

340: /*
341:    SVDComputeResidualNorms_Standard - Computes the norms of the left and
342:    right residuals associated with the i-th computed singular triplet.

344:    Input Parameters:
345:      sigma - singular value
346:      u,v   - singular vectors
347:      x,y   - two work vectors with the same dimensions as u,v
348: */
349: static PetscErrorCode SVDComputeResidualNorms_Standard(SVD svd,PetscReal sigma,Vec u,Vec v,Vec x,Vec y,PetscReal *norm1,PetscReal *norm2)
350: {
351:   PetscInt       M,N;

353:   PetscFunctionBegin;
354:   /* norm1 = ||A*v-sigma*u||_2 */
355:   if (norm1) {
356:     PetscCall(MatMult(svd->OP,v,x));
357:     PetscCall(VecAXPY(x,-sigma,u));
358:     PetscCall(VecNorm(x,NORM_2,norm1));
359:   }
360:   /* norm2 = ||A^T*u-sigma*v||_2 */
361:   if (norm2) {
362:     PetscCall(MatGetSize(svd->OP,&M,&N));
363:     if (M<N) PetscCall(MatMult(svd->A,u,y));
364:     else PetscCall(MatMult(svd->AT,u,y));
365:     PetscCall(VecAXPY(y,-sigma,v));
366:     PetscCall(VecNorm(y,NORM_2,norm2));
367:   }
368:   PetscFunctionReturn(PETSC_SUCCESS);
369: }

371: /*
372:    SVDComputeResidualNorms_Generalized - In GSVD, compute the residual norms
373:    norm1 = ||s^2*A'*u-c*B'*B*x||_2 and norm2 = ||c^2*B'*v-s*A'*A*x||_2.

375:    Input Parameters:
376:      sigma - singular value
377:      uv    - left singular vectors [u;v]
378:      x     - right singular vector
379:      y,z   - two work vectors with the same dimension as x
380: */
381: static PetscErrorCode SVDComputeResidualNorms_Generalized(SVD svd,PetscReal sigma,Vec uv,Vec x,Vec y,Vec z,PetscReal *norm1,PetscReal *norm2)
382: {
383:   Vec            u,v,ax,bx,nest,aux[2];
384:   PetscReal      c,s;

386:   PetscFunctionBegin;
387:   PetscCall(MatCreateVecs(svd->OP,NULL,&u));
388:   PetscCall(MatCreateVecs(svd->OPb,NULL,&v));
389:   aux[0] = u;
390:   aux[1] = v;
391:   PetscCall(VecCreateNest(PetscObjectComm((PetscObject)svd),2,NULL,aux,&nest));
392:   PetscCall(VecCopy(uv,nest));

394:   s = 1.0/PetscSqrtReal(1.0+sigma*sigma);
395:   c = sigma*s;

397:   /* norm1 = ||s^2*A'*u-c*B'*B*x||_2 */
398:   if (norm1) {
399:     PetscCall(VecDuplicate(v,&bx));
400:     PetscCall(MatMultHermitianTranspose(svd->OP,u,z));
401:     PetscCall(MatMult(svd->OPb,x,bx));
402:     PetscCall(MatMultHermitianTranspose(svd->OPb,bx,y));
403:     PetscCall(VecAXPBY(y,s*s,-c,z));
404:     PetscCall(VecNorm(y,NORM_2,norm1));
405:     PetscCall(VecDestroy(&bx));
406:   }
407:   /* norm2 = ||c^2*B'*v-s*A'*A*x||_2 */
408:   if (norm2) {
409:     PetscCall(VecDuplicate(u,&ax));
410:     PetscCall(MatMultHermitianTranspose(svd->OPb,v,z));
411:     PetscCall(MatMult(svd->OP,x,ax));
412:     PetscCall(MatMultHermitianTranspose(svd->OP,ax,y));
413:     PetscCall(VecAXPBY(y,c*c,-s,z));
414:     PetscCall(VecNorm(y,NORM_2,norm2));
415:     PetscCall(VecDestroy(&ax));
416:   }

418:   PetscCall(VecDestroy(&v));
419:   PetscCall(VecDestroy(&u));
420:   PetscCall(VecDestroy(&nest));
421:   PetscFunctionReturn(PETSC_SUCCESS);
422: }

424: /*
425:    SVDComputeResidualNorms_Hyperbolic - Computes the norms of the left and
426:    right residuals associated with the i-th computed singular triplet.

428:    Input Parameters:
429:      sigma - singular value
430:      sign  - corresponding element of the signature Omega2
431:      u,v   - singular vectors
432:      x,y,z - three work vectors with the same dimensions as u,v,u
433: */
434: static PetscErrorCode SVDComputeResidualNorms_Hyperbolic(SVD svd,PetscReal sigma,PetscReal sign,Vec u,Vec v,Vec x,Vec y,Vec z,PetscReal *norm1,PetscReal *norm2)
435: {
436:   PetscInt       M,N;

438:   PetscFunctionBegin;
439:   /* norm1 = ||A*v-sigma*u||_2 */
440:   if (norm1) {
441:     PetscCall(MatMult(svd->OP,v,x));
442:     PetscCall(VecAXPY(x,-sigma,u));
443:     PetscCall(VecNorm(x,NORM_2,norm1));
444:   }
445:   /* norm2 = ||A^T*Omega*u-sigma*sign*v||_2 */
446:   if (norm2) {
447:     PetscCall(MatGetSize(svd->OP,&M,&N));
448:     PetscCall(VecPointwiseMult(z,u,svd->omega));
449:     if (M<N) PetscCall(MatMult(svd->A,z,y));
450:     else PetscCall(MatMult(svd->AT,z,y));
451:     PetscCall(VecAXPY(y,-sigma*sign,v));
452:     PetscCall(VecNorm(y,NORM_2,norm2));
453:   }
454:   PetscFunctionReturn(PETSC_SUCCESS);
455: }

457: /*@
458:    SVDComputeError - Computes the error (based on the residual norm) associated
459:    with the i-th singular triplet.

461:    Collective

463:    Input Parameters:
464: +  svd  - the singular value solver context
465: .  i    - the solution index
466: -  type - the type of error to compute

468:    Output Parameter:
469: .  error - the error

471:    Notes:
472:    The error can be computed in various ways, all of them based on the residual
473:    norm obtained as sqrt(n1^2+n2^2) with n1 = ||A*v-sigma*u||_2 and
474:    n2 = ||A^T*u-sigma*v||_2, where sigma is the singular value, u is the left
475:    singular vector and v is the right singular vector.

477:    In the case of the GSVD, the two components of the residual norm are
478:    n1 = ||s^2*A'*u-c*B'*B*x||_2 and n2 = ||c^2*B'*v-s*A'*A*x||_2, where [u;v]
479:    are the left singular vectors and x is the right singular vector, with
480:    sigma=c/s.

482:    Level: beginner

484: .seealso: SVDErrorType, SVDSolve()
485: @*/
486: PetscErrorCode SVDComputeError(SVD svd,PetscInt i,SVDErrorType type,PetscReal *error)
487: {
488:   PetscReal      sigma,norm1,norm2,c,s;
489:   Vec            u=NULL,v=NULL,x=NULL,y=NULL,z=NULL;
490:   PetscReal      vecnorm=1.0;

492:   PetscFunctionBegin;
496:   PetscAssertPointer(error,4);
497:   SVDCheckSolved(svd,1);

499:   /* allocate work vectors */
500:   switch (svd->problem_type) {
501:     case SVD_STANDARD:
502:       PetscCall(SVDSetWorkVecs(svd,2,2));
503:       u = svd->workl[0];
504:       v = svd->workr[0];
505:       x = svd->workl[1];
506:       y = svd->workr[1];
507:       break;
508:     case SVD_GENERALIZED:
509:       PetscCall(SVDSetWorkVecs(svd,1,3));
510:       u = svd->workl[0];
511:       v = svd->workr[0];
512:       x = svd->workr[1];
513:       y = svd->workr[2];
514:       break;
515:     case SVD_HYPERBOLIC:
516:       PetscCall(SVDSetWorkVecs(svd,3,2));
517:       u = svd->workl[0];
518:       v = svd->workr[0];
519:       x = svd->workl[1];
520:       y = svd->workr[1];
521:       z = svd->workl[2];
522:       break;
523:   }

525:   /* compute residual norm */
526:   PetscCall(SVDGetSingularTriplet(svd,i,&sigma,u,v));
527:   switch (svd->problem_type) {
528:     case SVD_STANDARD:
529:       PetscCall(SVDComputeResidualNorms_Standard(svd,sigma,u,v,x,y,&norm1,&norm2));
530:       break;
531:     case SVD_GENERALIZED:
532:       PetscCall(SVDComputeResidualNorms_Generalized(svd,sigma,u,v,x,y,&norm1,&norm2));
533:       break;
534:     case SVD_HYPERBOLIC:
535:       PetscCall(SVDComputeResidualNorms_Hyperbolic(svd,sigma,svd->sign[svd->perm[i]],u,v,x,y,z,&norm1,&norm2));
536:       break;
537:   }
538:   *error = SlepcAbs(norm1,norm2);

540:   /* compute 2-norm of eigenvector of the cyclic form */
541:   if (type!=SVD_ERROR_ABSOLUTE) {
542:     switch (svd->problem_type) {
543:       case SVD_STANDARD:
544:         vecnorm = PETSC_SQRT2;
545:         break;
546:       case SVD_GENERALIZED:
547:         PetscCall(VecNorm(v,NORM_2,&vecnorm));
548:         vecnorm = PetscSqrtReal(1.0+vecnorm*vecnorm);
549:         break;
550:       case SVD_HYPERBOLIC:
551:         PetscCall(VecNorm(u,NORM_2,&vecnorm));
552:         vecnorm = PetscSqrtReal(1.0+vecnorm*vecnorm);
553:         break;
554:     }
555:   }

557:   /* compute error */
558:   switch (type) {
559:     case SVD_ERROR_ABSOLUTE:
560:       break;
561:     case SVD_ERROR_RELATIVE:
562:       if (svd->isgeneralized) {
563:         s = 1.0/PetscSqrtReal(1.0+sigma*sigma);
564:         c = sigma*s;
565:         norm1 /= c*vecnorm;
566:         norm2 /= s*vecnorm;
567:         *error = PetscMax(norm1,norm2);
568:       } else *error /= sigma*vecnorm;
569:       break;
570:     case SVD_ERROR_NORM:
571:       if (!svd->nrma) PetscCall(MatNorm(svd->OP,NORM_INFINITY,&svd->nrma));
572:       if (svd->isgeneralized && !svd->nrmb) PetscCall(MatNorm(svd->OPb,NORM_INFINITY,&svd->nrmb));
573:       *error /= PetscMax(svd->nrma,svd->nrmb)*vecnorm;
574:       break;
575:     default:
576:       SETERRQ(PetscObjectComm((PetscObject)svd),PETSC_ERR_ARG_OUTOFRANGE,"Invalid error type");
577:   }
578:   PetscFunctionReturn(PETSC_SUCCESS);
579: }