Actual source code: svdsolve.c

  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 - the singular value solver context

 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: [](ch:svd), `SVDCreate()`, `SVDSetUp()`, `SVDDestroy()`
115: @*/
116: PetscErrorCode SVDSolve(SVD svd)
117: {
118:   PetscInt       i,m,n,*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));

128:   /* safeguard for matrices with zero rows or columns */
129:   PetscCall(MatGetSize(svd->OP,&m,&n));
130:   if (m == 0 || n == 0) {
131:     svd->nconv  = 0;
132:     svd->reason = SVD_CONVERGED_TOL;
133:     svd->state  = SVD_STATE_SOLVED;
134:     PetscFunctionReturn(PETSC_SUCCESS);
135:   }

137:   svd->its = 0;
138:   svd->nconv = 0;
139:   for (i=0;i<svd->ncv;i++) {
140:     svd->sigma[i]  = 0.0;
141:     svd->errest[i] = 0.0;
142:     svd->perm[i]   = i;
143:   }
144:   PetscCall(SVDViewFromOptions(svd,NULL,"-svd_view_pre"));

146:   switch (svd->problem_type) {
147:     case SVD_STANDARD:
148:       PetscUseTypeMethod(svd,solve);
149:       break;
150:     case SVD_GENERALIZED:
151:       PetscUseTypeMethod(svd,solveg);
152:       break;
153:     case SVD_HYPERBOLIC:
154:       PetscUseTypeMethod(svd,solveh);
155:       break;
156:   }
157:   svd->state = SVD_STATE_SOLVED;

159:   /* sort singular triplets */
160:   if (svd->which == SVD_SMALLEST) PetscCall(PetscSortRealWithPermutation(svd->nconv,svd->sigma,svd->perm));
161:   else {
162:     PetscCall(PetscMalloc1(svd->nconv,&workperm));
163:     for (i=0;i<svd->nconv;i++) workperm[i] = i;
164:     PetscCall(PetscSortRealWithPermutation(svd->nconv,svd->sigma,workperm));
165:     for (i=0;i<svd->nconv;i++) svd->perm[i] = workperm[svd->nconv-i-1];
166:     PetscCall(PetscFree(workperm));
167:   }
168:   PetscCall(PetscLogEventEnd(SVD_Solve,svd,0,0,0));

170:   /* various viewers */
171:   PetscCall(SVDViewFromOptions(svd,NULL,"-svd_view"));
172:   PetscCall(SVDConvergedReasonViewFromOptions(svd));
173:   PetscCall(SVDErrorViewFromOptions(svd));
174:   PetscCall(SVDValuesViewFromOptions(svd));
175:   PetscCall(SVDVectorsViewFromOptions(svd));
176:   PetscCall(MatViewFromOptions(svd->OP,(PetscObject)svd,"-svd_view_mat0"));
177:   if (svd->isgeneralized) PetscCall(MatViewFromOptions(svd->OPb,(PetscObject)svd,"-svd_view_mat1"));
178:   if (svd->ishyperbolic) PetscCall(VecViewFromOptions(svd->omega,(PetscObject)svd,"-svd_view_signature"));

180:   /* Remove the initial subspaces */
181:   svd->nini = 0;
182:   svd->ninil = 0;
183:   PetscFunctionReturn(PETSC_SUCCESS);
184: }

186: /*@
187:    SVDGetIterationNumber - Gets the current iteration number. If the
188:    call to SVDSolve() is complete, then it returns the number of iterations
189:    carried out by the solution method.

191:    Not Collective

193:    Input Parameter:
194: .  svd - the singular value solver context

196:    Output Parameter:
197: .  its - number of iterations

199:    Note:
200:    During the i-th iteration this call returns i-1. If SVDSolve() is
201:    complete, then parameter "its" contains either the iteration number at
202:    which convergence was successfully reached, or failure was detected.
203:    Call SVDGetConvergedReason() to determine if the solver converged or
204:    failed and why.

206:    Level: intermediate

208: .seealso: [](ch:svd), `SVDGetConvergedReason()`, `SVDSetTolerances()`
209: @*/
210: PetscErrorCode SVDGetIterationNumber(SVD svd,PetscInt *its)
211: {
212:   PetscFunctionBegin;
214:   PetscAssertPointer(its,2);
215:   *its = svd->its;
216:   PetscFunctionReturn(PETSC_SUCCESS);
217: }

219: /*@
220:    SVDGetConvergedReason - Gets the reason why the `SVDSolve()` iteration was
221:    stopped.

223:    Not Collective

225:    Input Parameter:
226: .  svd - the singular value solver context

228:    Output Parameter:
229: .  reason - negative value indicates diverged, positive value converged, see
230:    `SVDConvergedReason` for the possible values

232:    Options Database Key:
233: .  -svd_converged_reason - print reason for convergence/divergence, and number of iterations

235:    Note:
236:    If this routine is called before or doing the `SVDSolve()` the value of
237:    `SVD_CONVERGED_ITERATING` is returned.

239:    Level: intermediate

241: .seealso: [](ch:svd), `SVDSetTolerances()`, `SVDSolve()`, `SVDConvergedReason`
242: @*/
243: PetscErrorCode SVDGetConvergedReason(SVD svd,SVDConvergedReason *reason)
244: {
245:   PetscFunctionBegin;
247:   PetscAssertPointer(reason,2);
248:   SVDCheckSolved(svd,1);
249:   *reason = svd->reason;
250:   PetscFunctionReturn(PETSC_SUCCESS);
251: }

253: /*@
254:    SVDGetConverged - Gets the number of converged singular values.

256:    Not Collective

258:    Input Parameter:
259: .  svd - the singular value solver context

261:    Output Parameter:
262: .  nconv - number of converged singular values

264:    Note:
265:    This function should be called after SVDSolve() has finished.

267:    Level: beginner

269: .seealso: [](ch:svd), `SVDSetDimensions()`, `SVDSolve()`, `SVDGetSingularTriplet()`
270: @*/
271: PetscErrorCode SVDGetConverged(SVD svd,PetscInt *nconv)
272: {
273:   PetscFunctionBegin;
275:   PetscAssertPointer(nconv,2);
276:   SVDCheckSolved(svd,1);
277:   *nconv = svd->nconv;
278:   PetscFunctionReturn(PETSC_SUCCESS);
279: }

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

286:    Collective

288:    Input Parameters:
289: +  svd - the singular value solver context
290: -  i   - index of the solution

292:    Output Parameters:
293: +  sigma - singular value
294: .  u     - left singular vector
295: -  v     - right singular vector

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

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

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

313:    Level: beginner

315: .seealso: [](ch:svd), `SVDSolve()`, `SVDGetConverged()`, `SVDSetWhichSingularTriplets()`
316: @*/
317: PetscErrorCode SVDGetSingularTriplet(SVD svd,PetscInt i,PetscReal *sigma,Vec u,Vec v)
318: {
319:   PetscInt       M,N;
320:   Vec            w;

322:   PetscFunctionBegin;
325:   SVDCheckSolved(svd,1);
328:   PetscCheck(i>=0,PetscObjectComm((PetscObject)svd),PETSC_ERR_ARG_OUTOFRANGE,"The index cannot be negative");
329:   PetscCheck(i<svd->nconv,PetscObjectComm((PetscObject)svd),PETSC_ERR_ARG_OUTOFRANGE,"The index can be nconv-1 at most, see SVDGetConverged()");
330:   if (sigma) *sigma = svd->sigma[svd->perm[i]];
331:   if (u || v) {
332:     if (!svd->isgeneralized) {
333:       PetscCall(MatGetSize(svd->OP,&M,&N));
334:       if (M<N) { w = u; u = v; v = w; }
335:     }
336:     PetscCall(SVDComputeVectors(svd));
337:     if (u) PetscCall(BVCopyVec(svd->U,svd->perm[i],u));
338:     if (v) PetscCall(BVCopyVec(svd->V,svd->perm[i],v));
339:   }
340:   PetscFunctionReturn(PETSC_SUCCESS);
341: }

343: /*
344:    SVDComputeResidualNorms_Standard - Computes the norms of the left and
345:    right residuals associated with the i-th computed singular triplet.

347:    Input Parameters:
348:      sigma - singular value
349:      u,v   - singular vectors
350:      x,y   - two work vectors with the same dimensions as u,v
351: */
352: static PetscErrorCode SVDComputeResidualNorms_Standard(SVD svd,PetscReal sigma,Vec u,Vec v,Vec x,Vec y,PetscReal *norm1,PetscReal *norm2)
353: {
354:   PetscInt       M,N;

356:   PetscFunctionBegin;
357:   /* norm1 = ||A*v-sigma*u||_2 */
358:   if (norm1) {
359:     PetscCall(MatMult(svd->OP,v,x));
360:     PetscCall(VecAXPY(x,-sigma,u));
361:     PetscCall(VecNorm(x,NORM_2,norm1));
362:   }
363:   /* norm2 = ||A^T*u-sigma*v||_2 */
364:   if (norm2) {
365:     PetscCall(MatGetSize(svd->OP,&M,&N));
366:     if (M<N) PetscCall(MatMult(svd->A,u,y));
367:     else PetscCall(MatMult(svd->AT,u,y));
368:     PetscCall(VecAXPY(y,-sigma,v));
369:     PetscCall(VecNorm(y,NORM_2,norm2));
370:   }
371:   PetscFunctionReturn(PETSC_SUCCESS);
372: }

374: /*
375:    SVDComputeResidualNorms_Generalized - In GSVD, compute the residual norms
376:    norm1 = ||s^2*A'*u-c*B'*B*x||_2 and norm2 = ||c^2*B'*v-s*A'*A*x||_2.

378:    Input Parameters:
379:      sigma - singular value
380:      uv    - left singular vectors [u;v]
381:      x     - right singular vector
382:      y,z   - two work vectors with the same dimension as x
383: */
384: static PetscErrorCode SVDComputeResidualNorms_Generalized(SVD svd,PetscReal sigma,Vec uv,Vec x,Vec y,Vec z,PetscReal *norm1,PetscReal *norm2)
385: {
386:   Vec            u,v,ax,bx,nest,aux[2];
387:   PetscReal      c,s;

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

397:   s = 1.0/PetscSqrtReal(1.0+sigma*sigma);
398:   c = sigma*s;

400:   /* norm1 = ||s^2*A'*u-c*B'*B*x||_2 */
401:   if (norm1) {
402:     PetscCall(VecDuplicate(v,&bx));
403:     PetscCall(MatMultHermitianTranspose(svd->OP,u,z));
404:     PetscCall(MatMult(svd->OPb,x,bx));
405:     PetscCall(MatMultHermitianTranspose(svd->OPb,bx,y));
406:     PetscCall(VecAXPBY(y,s*s,-c,z));
407:     PetscCall(VecNorm(y,NORM_2,norm1));
408:     PetscCall(VecDestroy(&bx));
409:   }
410:   /* norm2 = ||c^2*B'*v-s*A'*A*x||_2 */
411:   if (norm2) {
412:     PetscCall(VecDuplicate(u,&ax));
413:     PetscCall(MatMultHermitianTranspose(svd->OPb,v,z));
414:     PetscCall(MatMult(svd->OP,x,ax));
415:     PetscCall(MatMultHermitianTranspose(svd->OP,ax,y));
416:     PetscCall(VecAXPBY(y,c*c,-s,z));
417:     PetscCall(VecNorm(y,NORM_2,norm2));
418:     PetscCall(VecDestroy(&ax));
419:   }

421:   PetscCall(VecDestroy(&v));
422:   PetscCall(VecDestroy(&u));
423:   PetscCall(VecDestroy(&nest));
424:   PetscFunctionReturn(PETSC_SUCCESS);
425: }

427: /*
428:    SVDComputeResidualNorms_Hyperbolic - Computes the norms of the left and
429:    right residuals associated with the i-th computed singular triplet.

431:    Input Parameters:
432:      sigma - singular value
433:      sign  - corresponding element of the signature Omega2
434:      u,v   - singular vectors
435:      x,y,z - three work vectors with the same dimensions as u,v,u
436: */
437: static PetscErrorCode SVDComputeResidualNorms_Hyperbolic(SVD svd,PetscReal sigma,PetscReal sign,Vec u,Vec v,Vec x,Vec y,Vec z,PetscReal *norm1,PetscReal *norm2)
438: {
439:   PetscInt       M,N;

441:   PetscFunctionBegin;
442:   /* norm1 = ||A*v-sigma*u||_2 */
443:   if (norm1) {
444:     PetscCall(MatMult(svd->OP,v,x));
445:     PetscCall(VecAXPY(x,-sigma,u));
446:     PetscCall(VecNorm(x,NORM_2,norm1));
447:   }
448:   /* norm2 = ||A^T*Omega*u-sigma*sign*v||_2 */
449:   if (norm2) {
450:     PetscCall(MatGetSize(svd->OP,&M,&N));
451:     PetscCall(VecPointwiseMult(z,u,svd->omega));
452:     if (M<N) PetscCall(MatMult(svd->A,z,y));
453:     else PetscCall(MatMult(svd->AT,z,y));
454:     PetscCall(VecAXPY(y,-sigma*sign,v));
455:     PetscCall(VecNorm(y,NORM_2,norm2));
456:   }
457:   PetscFunctionReturn(PETSC_SUCCESS);
458: }

460: /*@
461:    SVDComputeError - Computes the error (based on the residual norm) associated
462:    with the i-th singular triplet.

464:    Collective

466:    Input Parameters:
467: +  svd  - the singular value solver context
468: .  i    - the solution index
469: -  type - the type of error to compute

471:    Output Parameter:
472: .  error - the error

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

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

485:    Level: beginner

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

495:   PetscFunctionBegin;
499:   PetscAssertPointer(error,4);
500:   SVDCheckSolved(svd,1);

502:   /* allocate work vectors */
503:   switch (svd->problem_type) {
504:     case SVD_STANDARD:
505:       PetscCall(SVDSetWorkVecs(svd,2,2));
506:       u = svd->workl[0];
507:       v = svd->workr[0];
508:       x = svd->workl[1];
509:       y = svd->workr[1];
510:       break;
511:     case SVD_GENERALIZED:
512:       PetscCall(SVDSetWorkVecs(svd,1,3));
513:       u = svd->workl[0];
514:       v = svd->workr[0];
515:       x = svd->workr[1];
516:       y = svd->workr[2];
517:       break;
518:     case SVD_HYPERBOLIC:
519:       PetscCall(SVDSetWorkVecs(svd,3,2));
520:       u = svd->workl[0];
521:       v = svd->workr[0];
522:       x = svd->workl[1];
523:       y = svd->workr[1];
524:       z = svd->workl[2];
525:       break;
526:   }

528:   /* compute residual norm */
529:   PetscCall(SVDGetSingularTriplet(svd,i,&sigma,u,v));
530:   switch (svd->problem_type) {
531:     case SVD_STANDARD:
532:       PetscCall(SVDComputeResidualNorms_Standard(svd,sigma,u,v,x,y,&norm1,&norm2));
533:       break;
534:     case SVD_GENERALIZED:
535:       PetscCall(SVDComputeResidualNorms_Generalized(svd,sigma,u,v,x,y,&norm1,&norm2));
536:       break;
537:     case SVD_HYPERBOLIC:
538:       PetscCall(SVDComputeResidualNorms_Hyperbolic(svd,sigma,svd->sign[svd->perm[i]],u,v,x,y,z,&norm1,&norm2));
539:       break;
540:   }
541:   *error = SlepcAbs(norm1,norm2);

543:   /* compute 2-norm of eigenvector of the cyclic form */
544:   if (type!=SVD_ERROR_ABSOLUTE) {
545:     switch (svd->problem_type) {
546:       case SVD_STANDARD:
547:         vecnorm = PETSC_SQRT2;
548:         break;
549:       case SVD_GENERALIZED:
550:         PetscCall(VecNorm(v,NORM_2,&vecnorm));
551:         vecnorm = PetscSqrtReal(1.0+vecnorm*vecnorm);
552:         break;
553:       case SVD_HYPERBOLIC:
554:         PetscCall(VecNorm(u,NORM_2,&vecnorm));
555:         vecnorm = PetscSqrtReal(1.0+vecnorm*vecnorm);
556:         break;
557:     }
558:   }

560:   /* compute error */
561:   switch (type) {
562:     case SVD_ERROR_ABSOLUTE:
563:       break;
564:     case SVD_ERROR_RELATIVE:
565:       if (svd->isgeneralized) {
566:         s = 1.0/PetscSqrtReal(1.0+sigma*sigma);
567:         c = sigma*s;
568:         norm1 /= c*vecnorm;
569:         norm2 /= s*vecnorm;
570:         *error = PetscMax(norm1,norm2);
571:       } else *error /= sigma*vecnorm;
572:       break;
573:     case SVD_ERROR_NORM:
574:       if (!svd->nrma) PetscCall(MatNorm(svd->OP,NORM_INFINITY,&svd->nrma));
575:       if (svd->isgeneralized && !svd->nrmb) PetscCall(MatNorm(svd->OPb,NORM_INFINITY,&svd->nrmb));
576:       *error /= PetscMax(svd->nrma,svd->nrmb)*vecnorm;
577:       break;
578:     default:
579:       SETERRQ(PetscObjectComm((PetscObject)svd),PETSC_ERR_ARG_OUTOFRANGE,"Invalid error type");
580:   }
581:   PetscFunctionReturn(PETSC_SUCCESS);
582: }