Actual source code: pgmres.c

petsc-3.8.3 2017-12-09
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  2: /*
  3:     This file implements PGMRES (a Pipelined Generalized Minimal Residual method)
  4: */

  6:  #include <../src/ksp/ksp/impls/gmres/pgmres/pgmresimpl.h>
  7: #define PGMRES_DELTA_DIRECTIONS 10
  8: #define PGMRES_DEFAULT_MAXK     30

 10: static PetscErrorCode KSPPGMRESUpdateHessenberg(KSP,PetscInt,PetscBool*,PetscReal*);
 11: static PetscErrorCode KSPPGMRESBuildSoln(PetscScalar*,Vec,Vec,KSP,PetscInt);

 13: /*

 15:     KSPSetUp_PGMRES - Sets up the workspace needed by pgmres.

 17:     This is called once, usually automatically by KSPSolve() or KSPSetUp(),
 18:     but can be called directly by KSPSetUp().

 20: */
 21: static PetscErrorCode KSPSetUp_PGMRES(KSP ksp)
 22: {

 26:   KSPSetUp_GMRES(ksp);
 27:   return(0);
 28: }

 30: /*

 32:     KSPPGMRESCycle - Run pgmres, possibly with restart.  Return residual
 33:                   history if requested.

 35:     input parameters:
 36: .        pgmres  - structure containing parameters and work areas

 38:     output parameters:
 39: .        itcount - number of iterations used.  If null, ignored.
 40: .        converged - 0 if not converged

 42:     Notes:
 43:     On entry, the value in vector VEC_VV(0) should be
 44:     the initial residual.


 47:  */
 48: static PetscErrorCode KSPPGMRESCycle(PetscInt *itcount,KSP ksp)
 49: {
 50:   KSP_PGMRES     *pgmres = (KSP_PGMRES*)(ksp->data);
 51:   PetscReal      res_norm,res,newnorm;
 53:   PetscInt       it     = 0,j,k;
 54:   PetscBool      hapend = PETSC_FALSE;

 57:   if (itcount) *itcount = 0;
 58:   VecNormalize(VEC_VV(0),&res_norm);
 59:   KSPCheckNorm(ksp,res_norm);
 60:   res    = res_norm;
 61:   *RS(0) = res_norm;

 63:   /* check for the convergence */
 64:   PetscObjectSAWsTakeAccess((PetscObject)ksp);
 65:   ksp->rnorm = res;
 66:   PetscObjectSAWsGrantAccess((PetscObject)ksp);
 67:   pgmres->it = it-2;
 68:   KSPLogResidualHistory(ksp,res);
 69:   KSPMonitor(ksp,ksp->its,res);
 70:   if (!res) {
 71:     ksp->reason = KSP_CONVERGED_ATOL;
 72:     PetscInfo(ksp,"Converged due to zero residual norm on entry\n");
 73:     return(0);
 74:   }

 76:   (*ksp->converged)(ksp,ksp->its,res,&ksp->reason,ksp->cnvP);
 77:   for (; !ksp->reason; it++) {
 78:     Vec Zcur,Znext;
 79:     if (pgmres->vv_allocated <= it + VEC_OFFSET + 1) {
 80:       KSPGMRESGetNewVectors(ksp,it+1);
 81:     }
 82:     /* VEC_VV(it-1) is orthogonal, it will be normalized once the VecNorm arrives. */
 83:     Zcur  = VEC_VV(it);         /* Zcur is not yet orthogonal, but the VecMDot to orthogonalize it has been started. */
 84:     Znext = VEC_VV(it+1);       /* This iteration will compute Znext, update with a deferred correction once we know how
 85:                                  * Zcur relates to the previous vectors, and start the reduction to orthogonalize it. */

 87:     if (it < pgmres->max_k+1 && ksp->its+1 < PetscMax(2,ksp->max_it)) { /* We don't know whether what we have computed is enough, so apply the matrix. */
 88:       KSP_PCApplyBAorAB(ksp,Zcur,Znext,VEC_TEMP_MATOP);
 89:     }

 91:     if (it > 1) {               /* Complete the pending reduction */
 92:       VecNormEnd(VEC_VV(it-1),NORM_2,&newnorm);
 93:       *HH(it-1,it-2) = newnorm;
 94:     }
 95:     if (it > 0) {               /* Finish the reduction computing the latest column of H */
 96:       VecMDotEnd(Zcur,it,&(VEC_VV(0)),HH(0,it-1));
 97:     }

 99:     if (it > 1) {
100:       /* normalize the base vector from two iterations ago, basis is complete up to here */
101:       VecScale(VEC_VV(it-1),1./ *HH(it-1,it-2));

103:       KSPPGMRESUpdateHessenberg(ksp,it-2,&hapend,&res);
104:       pgmres->it = it-2;
105:       ksp->its++;
106:       ksp->rnorm = res;

108:       (*ksp->converged)(ksp,ksp->its,res,&ksp->reason,ksp->cnvP);
109:       if (it < pgmres->max_k+1 || ksp->reason || ksp->its == ksp->max_it) {  /* Monitor if we are done or still iterating, but not before a restart. */
110:         KSPLogResidualHistory(ksp,res);
111:         KSPMonitor(ksp,ksp->its,res);
112:       }
113:       if (ksp->reason) break;
114:       /* Catch error in happy breakdown and signal convergence and break from loop */
115:       if (hapend) {
116:         if (ksp->errorifnotconverged) SETERRQ1(PetscObjectComm((PetscObject)ksp),PETSC_ERR_NOT_CONVERGED,"You reached the happy break down, but convergence was not indicated. Residual norm = %g",(double)res);
117:         else {
118:           ksp->reason = KSP_DIVERGED_BREAKDOWN;
119:           break;
120:         }
121:       }

123:       if (!(it < pgmres->max_k+1 && ksp->its < ksp->max_it)) break;

125:       /* The it-2 column of H was not scaled when we computed Zcur, apply correction */
126:       VecScale(Zcur,1./ *HH(it-1,it-2));
127:       /* And Znext computed in this iteration was computed using the under-scaled Zcur */
128:       VecScale(Znext,1./ *HH(it-1,it-2));

130:       /* In the previous iteration, we projected an unnormalized Zcur against the Krylov basis, so we need to fix the column of H resulting from that projection. */
131:       for (k=0; k<it; k++) *HH(k,it-1) /= *HH(it-1,it-2);
132:       /* When Zcur was projected against the Krylov basis, VV(it-1) was still not normalized, so fix that too. This
133:        * column is complete except for HH(it,it-1) which we won't know until the next iteration. */
134:       *HH(it-1,it-1) /= *HH(it-1,it-2);
135:     }

137:     if (it > 0) {
138:       PetscScalar *work;
139:       if (!pgmres->orthogwork) {PetscMalloc1(pgmres->max_k + 2,&pgmres->orthogwork);}
140:       work = pgmres->orthogwork;
141:       /* Apply correction computed by the VecMDot in the last iteration to Znext. The original form is
142:        *
143:        *   Znext -= sum_{j=0}^{i-1} Z[j+1] * H[j,i-1]
144:        *
145:        * where
146:        *
147:        *   Z[j] = sum_{k=0}^j V[k] * H[k,j-1]
148:        *
149:        * substituting
150:        *
151:        *   Znext -= sum_{j=0}^{i-1} sum_{k=0}^{j+1} V[k] * H[k,j] * H[j,i-1]
152:        *
153:        * rearranging the iteration space from row-column to column-row
154:        *
155:        *   Znext -= sum_{k=0}^i sum_{j=k-1}^{i-1} V[k] * H[k,j] * H[j,i-1]
156:        *
157:        * Note that column it-1 of HH is correct. For all previous columns, we must look at HES because HH has already
158:        * been transformed to upper triangular form.
159:        */
160:       for (k=0; k<it+1; k++) {
161:         work[k] = 0;
162:         for (j=PetscMax(0,k-1); j<it-1; j++) work[k] -= *HES(k,j) * *HH(j,it-1);
163:       }
164:       VecMAXPY(Znext,it+1,work,&VEC_VV(0));
165:       VecAXPY(Znext,-*HH(it-1,it-1),Zcur);

167:       /* Orthogonalize Zcur against existing basis vectors. */
168:       for (k=0; k<it; k++) work[k] = -*HH(k,it-1);
169:       VecMAXPY(Zcur,it,work,&VEC_VV(0));
170:       /* Zcur is now orthogonal, and will be referred to as VEC_VV(it) again, though it is still not normalized. */
171:       /* Begin computing the norm of the new vector, will be normalized after the MatMult in the next iteration. */
172:       VecNormBegin(VEC_VV(it),NORM_2,&newnorm);
173:     }

175:     /* Compute column of H (to the diagonal, but not the subdiagonal) to be able to orthogonalize the newest vector. */
176:     VecMDotBegin(Znext,it+1,&VEC_VV(0),HH(0,it));

178:     /* Start an asynchronous split-mode reduction, the result of the MDot and Norm will be collected on the next iteration. */
179:     PetscCommSplitReductionBegin(PetscObjectComm((PetscObject)Znext));
180:   }

182:   if (itcount) *itcount = it-1; /* Number of iterations actually completed. */

184:   /*
185:     Down here we have to solve for the "best" coefficients of the Krylov
186:     columns, add the solution values together, and possibly unwind the
187:     preconditioning from the solution
188:    */
189:   /* Form the solution (or the solution so far) */
190:   KSPPGMRESBuildSoln(RS(0),ksp->vec_sol,ksp->vec_sol,ksp,it-2);
191:   return(0);
192: }

194: /*
195:     KSPSolve_PGMRES - This routine applies the PGMRES method.


198:    Input Parameter:
199: .     ksp - the Krylov space object that was set to use pgmres

201:    Output Parameter:
202: .     outits - number of iterations used

204: */
205: static PetscErrorCode KSPSolve_PGMRES(KSP ksp)
206: {
208:   PetscInt       its,itcount;
209:   KSP_PGMRES     *pgmres    = (KSP_PGMRES*)ksp->data;
210:   PetscBool      guess_zero = ksp->guess_zero;

213:   if (ksp->calc_sings && !pgmres->Rsvd) SETERRQ(PetscObjectComm((PetscObject)ksp),PETSC_ERR_ORDER,"Must call KSPSetComputeSingularValues() before KSPSetUp() is called");
214:   PetscObjectSAWsTakeAccess((PetscObject)ksp);
215:   ksp->its = 0;
216:   PetscObjectSAWsGrantAccess((PetscObject)ksp);

218:   itcount     = 0;
219:   ksp->reason = KSP_CONVERGED_ITERATING;
220:   while (!ksp->reason) {
221:     KSPInitialResidual(ksp,ksp->vec_sol,VEC_TEMP,VEC_TEMP_MATOP,VEC_VV(0),ksp->vec_rhs);
222:     KSPPGMRESCycle(&its,ksp);
223:     itcount += its;
224:     if (itcount >= ksp->max_it) {
225:       if (!ksp->reason) ksp->reason = KSP_DIVERGED_ITS;
226:       break;
227:     }
228:     ksp->guess_zero = PETSC_FALSE; /* every future call to KSPInitialResidual() will have nonzero guess */
229:   }
230:   ksp->guess_zero = guess_zero; /* restore if user provided nonzero initial guess */
231:   return(0);
232: }

234: static PetscErrorCode KSPDestroy_PGMRES(KSP ksp)
235: {

239:   KSPDestroy_GMRES(ksp);
240:   return(0);
241: }

243: /*
244:     KSPPGMRESBuildSoln - create the solution from the starting vector and the
245:                       current iterates.

247:     Input parameters:
248:         nrs - work area of size it + 1.
249:         vguess  - index of initial guess
250:         vdest - index of result.  Note that vguess may == vdest (replace
251:                 guess with the solution).
252:         it - HH upper triangular part is a block of size (it+1) x (it+1)

254:      This is an internal routine that knows about the PGMRES internals.
255:  */
256: static PetscErrorCode KSPPGMRESBuildSoln(PetscScalar *nrs,Vec vguess,Vec vdest,KSP ksp,PetscInt it)
257: {
258:   PetscScalar    tt;
260:   PetscInt       k,j;
261:   KSP_PGMRES     *pgmres = (KSP_PGMRES*)(ksp->data);

264:   /* Solve for solution vector that minimizes the residual */

266:   if (it < 0) {                                 /* no pgmres steps have been performed */
267:     VecCopy(vguess,vdest); /* VecCopy() is smart, exits immediately if vguess == vdest */
268:     return(0);
269:   }

271:   /* solve the upper triangular system - RS is the right side and HH is
272:      the upper triangular matrix  - put soln in nrs */
273:   if (*HH(it,it) != 0.0) nrs[it] = *RS(it) / *HH(it,it);
274:   else nrs[it] = 0.0;

276:   for (k=it-1; k>=0; k--) {
277:     tt = *RS(k);
278:     for (j=k+1; j<=it; j++) tt -= *HH(k,j) * nrs[j];
279:     nrs[k] = tt / *HH(k,k);
280:   }

282:   /* Accumulate the correction to the solution of the preconditioned problem in TEMP */
283:   VecZeroEntries(VEC_TEMP);
284:   VecMAXPY(VEC_TEMP,it+1,nrs,&VEC_VV(0));
285:   KSPUnwindPreconditioner(ksp,VEC_TEMP,VEC_TEMP_MATOP);
286:   /* add solution to previous solution */
287:   if (vdest == vguess) {
288:     VecAXPY(vdest,1.0,VEC_TEMP);
289:   } else {
290:     VecWAXPY(vdest,1.0,VEC_TEMP,vguess);
291:   }
292:   return(0);
293: }

295: /*

297:     KSPPGMRESUpdateHessenberg - Do the scalar work for the orthogonalization.
298:                             Return new residual.

300:     input parameters:

302: .        ksp -    Krylov space object
303: .        it  -    plane rotations are applied to the (it+1)th column of the
304:                   modified hessenberg (i.e. HH(:,it))
305: .        hapend - PETSC_FALSE not happy breakdown ending.

307:     output parameters:
308: .        res - the new residual

310:  */
311: /*
312: .  it - column of the Hessenberg that is complete, PGMRES is actually computing two columns ahead of this
313:  */
314: static PetscErrorCode KSPPGMRESUpdateHessenberg(KSP ksp,PetscInt it,PetscBool *hapend,PetscReal *res)
315: {
316:   PetscScalar    *hh,*cc,*ss,*rs;
317:   PetscInt       j;
318:   PetscReal      hapbnd;
319:   KSP_PGMRES     *pgmres = (KSP_PGMRES*)(ksp->data);

323:   hh = HH(0,it);   /* pointer to beginning of column to update */
324:   cc = CC(0);      /* beginning of cosine rotations */
325:   ss = SS(0);      /* beginning of sine rotations */
326:   rs = RS(0);      /* right hand side of least squares system */

328:   /* The Hessenberg matrix is now correct through column it, save that form for possible spectral analysis */
329:   for (j=0; j<=it+1; j++) *HES(j,it) = hh[j];

331:   /* check for the happy breakdown */
332:   hapbnd = PetscMin(PetscAbsScalar(hh[it+1] / rs[it]),pgmres->haptol);
333:   if (PetscAbsScalar(hh[it+1]) < hapbnd) {
334:     PetscInfo4(ksp,"Detected happy breakdown, current hapbnd = %14.12e H(%D,%D) = %14.12e\n",(double)hapbnd,it+1,it,(double)PetscAbsScalar(*HH(it+1,it)));
335:     *hapend = PETSC_TRUE;
336:   }

338:   /* Apply all the previously computed plane rotations to the new column
339:      of the Hessenberg matrix */
340:   /* Note: this uses the rotation [conj(c)  s ; -s   c], c= cos(theta), s= sin(theta),
341:      and some refs have [c   s ; -conj(s)  c] (don't be confused!) */

343:   for (j=0; j<it; j++) {
344:     PetscScalar hhj = hh[j];
345:     hh[j]   = PetscConj(cc[j])*hhj + ss[j]*hh[j+1];
346:     hh[j+1] =          -ss[j] *hhj + cc[j]*hh[j+1];
347:   }

349:   /*
350:     compute the new plane rotation, and apply it to:
351:      1) the right-hand-side of the Hessenberg system (RS)
352:         note: it affects RS(it) and RS(it+1)
353:      2) the new column of the Hessenberg matrix
354:         note: it affects HH(it,it) which is currently pointed to
355:         by hh and HH(it+1, it) (*(hh+1))
356:     thus obtaining the updated value of the residual...
357:   */

359:   /* compute new plane rotation */

361:   if (!*hapend) {
362:     PetscReal delta = PetscSqrtReal(PetscSqr(PetscAbsScalar(hh[it])) + PetscSqr(PetscAbsScalar(hh[it+1])));
363:     if (delta == 0.0) {
364:       ksp->reason = KSP_DIVERGED_NULL;
365:       return(0);
366:     }

368:     cc[it] = hh[it] / delta;    /* new cosine value */
369:     ss[it] = hh[it+1] / delta;  /* new sine value */

371:     hh[it]   = PetscConj(cc[it])*hh[it] + ss[it]*hh[it+1];
372:     rs[it+1] = -ss[it]*rs[it];
373:     rs[it]   = PetscConj(cc[it])*rs[it];
374:     *res     = PetscAbsScalar(rs[it+1]);
375:   } else { /* happy breakdown: HH(it+1, it) = 0, therefore we don't need to apply
376:             another rotation matrix (so RH doesn't change).  The new residual is
377:             always the new sine term times the residual from last time (RS(it)),
378:             but now the new sine rotation would be zero...so the residual should
379:             be zero...so we will multiply "zero" by the last residual.  This might
380:             not be exactly what we want to do here -could just return "zero". */

382:     *res = 0.0;
383:   }
384:   return(0);
385: }

387: /*
388:    KSPBuildSolution_PGMRES

390:      Input Parameter:
391: .     ksp - the Krylov space object
392: .     ptr-

394:    Output Parameter:
395: .     result - the solution

397:    Note: this calls KSPPGMRESBuildSoln - the same function that KSPPGMRESCycle
398:    calls directly.

400: */
401: PetscErrorCode KSPBuildSolution_PGMRES(KSP ksp,Vec ptr,Vec *result)
402: {
403:   KSP_PGMRES     *pgmres = (KSP_PGMRES*)ksp->data;

407:   if (!ptr) {
408:     if (!pgmres->sol_temp) {
409:       VecDuplicate(ksp->vec_sol,&pgmres->sol_temp);
410:       PetscLogObjectParent((PetscObject)ksp,(PetscObject)pgmres->sol_temp);
411:     }
412:     ptr = pgmres->sol_temp;
413:   }
414:   if (!pgmres->nrs) {
415:     /* allocate the work area */
416:     PetscMalloc1(pgmres->max_k,&pgmres->nrs);
417:     PetscLogObjectMemory((PetscObject)ksp,pgmres->max_k*sizeof(PetscScalar));
418:   }

420:   KSPPGMRESBuildSoln(pgmres->nrs,ksp->vec_sol,ptr,ksp,pgmres->it);
421:   if (result) *result = ptr;
422:   return(0);
423: }

425: PetscErrorCode KSPSetFromOptions_PGMRES(PetscOptionItems *PetscOptionsObject,KSP ksp)
426: {

430:   KSPSetFromOptions_GMRES(PetscOptionsObject,ksp);
431:   PetscOptionsHead(PetscOptionsObject,"KSP pipelined GMRES Options");
432:   PetscOptionsTail();
433:   return(0);
434: }

436: PetscErrorCode KSPReset_PGMRES(KSP ksp)
437: {

441:   KSPReset_GMRES(ksp);
442:   return(0);
443: }

445: /*MC
446:      KSPPGMRES - Implements the Pipelined Generalized Minimal Residual method.

448:    Options Database Keys:
449: +   -ksp_gmres_restart <restart> - the number of Krylov directions to orthogonalize against
450: .   -ksp_gmres_haptol <tol> - sets the tolerance for "happy ending" (exact convergence)
451: .   -ksp_gmres_preallocate - preallocate all the Krylov search directions initially (otherwise groups of
452:                              vectors are allocated as needed)
453: .   -ksp_gmres_classicalgramschmidt - use classical (unmodified) Gram-Schmidt to orthogonalize against the Krylov space (fast) (the default)
454: .   -ksp_gmres_modifiedgramschmidt - use modified Gram-Schmidt in the orthogonalization (more stable, but slower)
455: .   -ksp_gmres_cgs_refinement_type <never,ifneeded,always> - determine if iterative refinement is used to increase the
456:                                    stability of the classical Gram-Schmidt  orthogonalization.
457: -   -ksp_gmres_krylov_monitor - plot the Krylov space generated

459:    Level: beginner

461:    Notes:
462:    MPI configuration may be necessary for reductions to make asynchronous progress, which is important for performance of pipelined methods.
463:    See the FAQ on the PETSc website for details.

465:    Reference:
466:    Ghysels, Ashby, Meerbergen, Vanroose, Hiding global communication latencies in the GMRES algorithm on massively parallel machines, 2012.

468:    Developer Notes: This object is subclassed off of KSPGMRES

470: .seealso:  KSPCreate(), KSPSetType(), KSPType (for list of available types), KSP, KSPGMRES, KSPLGMRES, KSPPIPECG, KSPPIPECR,
471:            KSPGMRESSetRestart(), KSPGMRESSetHapTol(), KSPGMRESSetPreAllocateVectors(), KSPGMRESSetOrthogonalization(), KSPGMRESGetOrthogonalization(),
472:            KSPGMRESClassicalGramSchmidtOrthogonalization(), KSPGMRESModifiedGramSchmidtOrthogonalization(),
473:            KSPGMRESCGSRefinementType, KSPGMRESSetCGSRefinementType(),  KSPGMRESGetCGSRefinementType(), KSPGMRESMonitorKrylov()
474: M*/

476: PETSC_EXTERN PetscErrorCode KSPCreate_PGMRES(KSP ksp)
477: {
478:   KSP_PGMRES     *pgmres;

482:   PetscNewLog(ksp,&pgmres);

484:   ksp->data                              = (void*)pgmres;
485:   ksp->ops->buildsolution                = KSPBuildSolution_PGMRES;
486:   ksp->ops->setup                        = KSPSetUp_PGMRES;
487:   ksp->ops->solve                        = KSPSolve_PGMRES;
488:   ksp->ops->reset                        = KSPReset_PGMRES;
489:   ksp->ops->destroy                      = KSPDestroy_PGMRES;
490:   ksp->ops->view                         = KSPView_GMRES;
491:   ksp->ops->setfromoptions               = KSPSetFromOptions_PGMRES;
492:   ksp->ops->computeextremesingularvalues = KSPComputeExtremeSingularValues_GMRES;
493:   ksp->ops->computeeigenvalues           = KSPComputeEigenvalues_GMRES;

495:   KSPSetSupportedNorm(ksp,KSP_NORM_PRECONDITIONED,PC_LEFT,3);
496:   KSPSetSupportedNorm(ksp,KSP_NORM_UNPRECONDITIONED,PC_RIGHT,2);

498:   PetscObjectComposeFunction((PetscObject)ksp,"KSPGMRESSetPreAllocateVectors_C",KSPGMRESSetPreAllocateVectors_GMRES);
499:   PetscObjectComposeFunction((PetscObject)ksp,"KSPGMRESSetOrthogonalization_C",KSPGMRESSetOrthogonalization_GMRES);
500:   PetscObjectComposeFunction((PetscObject)ksp,"KSPGMRESGetOrthogonalization_C",KSPGMRESGetOrthogonalization_GMRES);
501:   PetscObjectComposeFunction((PetscObject)ksp,"KSPGMRESSetRestart_C",KSPGMRESSetRestart_GMRES);
502:   PetscObjectComposeFunction((PetscObject)ksp,"KSPGMRESGetRestart_C",KSPGMRESGetRestart_GMRES);
503:   PetscObjectComposeFunction((PetscObject)ksp,"KSPGMRESSetCGSRefinementType_C",KSPGMRESSetCGSRefinementType_GMRES);
504:   PetscObjectComposeFunction((PetscObject)ksp,"KSPGMRESGetCGSRefinementType_C",KSPGMRESGetCGSRefinementType_GMRES);

506:   pgmres->nextra_vecs    = 1;
507:   pgmres->haptol         = 1.0e-30;
508:   pgmres->q_preallocate  = 0;
509:   pgmres->delta_allocate = PGMRES_DELTA_DIRECTIONS;
510:   pgmres->orthog         = KSPGMRESClassicalGramSchmidtOrthogonalization;
511:   pgmres->nrs            = 0;
512:   pgmres->sol_temp       = 0;
513:   pgmres->max_k          = PGMRES_DEFAULT_MAXK;
514:   pgmres->Rsvd           = 0;
515:   pgmres->orthogwork     = 0;
516:   pgmres->cgstype        = KSP_GMRES_CGS_REFINE_NEVER;
517:   return(0);
518: }