Actual source code: lsqr.c

      2: /* lourens.vanzanen@shell.com contributed the standard error estimates of the solution, Jul 25, 2006 */
      3: /* Bas van't Hof contributed the preconditioned aspects Feb 10, 2010 */

      5: #define SWAP(a,b,c) { c = a; a = b; b = c; }

      7: #include <petsc/private/kspimpl.h>
      8: #include <petscdraw.h>

     10: typedef struct {
     11:   PetscInt  nwork_n,nwork_m;
     12:   Vec       *vwork_m;   /* work vectors of length m, where the system is size m x n */
     13:   Vec       *vwork_n;   /* work vectors of length n */
     14:   Vec       se;         /* Optional standard error vector */
     15:   PetscBool se_flg;     /* flag for -ksp_lsqr_set_standard_error */
     16:   PetscBool exact_norm; /* flag for -ksp_lsqr_exact_mat_norm */
     17:   PetscReal arnorm;     /* Good estimate of norm((A*inv(Pmat))'*r), where r = A*x - b, used in specific stopping criterion */
     18:   PetscReal anorm;      /* Poor estimate of norm(A*inv(Pmat),'fro') used in specific stopping criterion */
     19:   /* Backup previous convergence test */
     20:   PetscErrorCode        (*converged)(KSP,PetscInt,PetscReal,KSPConvergedReason*,void*);
     21:   PetscErrorCode        (*convergeddestroy)(void*);
     22:   void                  *cnvP;
     23: } KSP_LSQR;

     25: static PetscErrorCode  VecSquare(Vec v)
     26: {
     28:   PetscScalar    *x;
     29:   PetscInt       i, n;

     32:   VecGetLocalSize(v, &n);
     33:   VecGetArray(v, &x);
     34:   for (i = 0; i < n; i++) x[i] *= PetscConj(x[i]);
     35:   VecRestoreArray(v, &x);
     36:   return(0);
     37: }

     39: static PetscErrorCode KSPSetUp_LSQR(KSP ksp)
     40: {
     42:   KSP_LSQR       *lsqr = (KSP_LSQR*)ksp->data;
     43:   PetscBool      nopreconditioner;

     46:   PetscObjectTypeCompare((PetscObject)ksp->pc,PCNONE,&nopreconditioner);

     48:   if (lsqr->vwork_m) {
     49:     VecDestroyVecs(lsqr->nwork_m,&lsqr->vwork_m);
     50:   }

     52:   if (lsqr->vwork_n) {
     53:     VecDestroyVecs(lsqr->nwork_n,&lsqr->vwork_n);
     54:   }

     56:   lsqr->nwork_m = 2;
     57:   if (nopreconditioner) lsqr->nwork_n = 4;
     58:   else lsqr->nwork_n = 5;
     59:   KSPCreateVecs(ksp,lsqr->nwork_n,&lsqr->vwork_n,lsqr->nwork_m,&lsqr->vwork_m);

     61:   if (lsqr->se_flg && !lsqr->se) {
     62:     VecDuplicate(lsqr->vwork_n[0],&lsqr->se);
     63:     VecSet(lsqr->se,PETSC_INFINITY);
     64:   } else if (!lsqr->se_flg) {
     65:     VecDestroy(&lsqr->se);
     66:   }
     67:   return(0);
     68: }

     70: static PetscErrorCode KSPSolve_LSQR(KSP ksp)
     71: {
     73:   PetscInt       i,size1,size2;
     74:   PetscScalar    rho,rhobar,phi,phibar,theta,c,s,tmp,tau;
     75:   PetscReal      beta,alpha,rnorm;
     76:   Vec            X,B,V,V1,U,U1,TMP,W,W2,Z = NULL;
     77:   Mat            Amat,Pmat;
     78:   KSP_LSQR       *lsqr = (KSP_LSQR*)ksp->data;
     79:   PetscBool      diagonalscale,nopreconditioner;

     82:   PCGetDiagonalScale(ksp->pc,&diagonalscale);
     83:   if (diagonalscale) SETERRQ1(PetscObjectComm((PetscObject)ksp),PETSC_ERR_SUP,"Krylov method %s does not support diagonal scaling",((PetscObject)ksp)->type_name);

     85:   PCGetOperators(ksp->pc,&Amat,&Pmat);
     86:   PetscObjectTypeCompare((PetscObject)ksp->pc,PCNONE,&nopreconditioner);

     88:   /* vectors of length m, where system size is mxn */
     89:   B  = ksp->vec_rhs;
     90:   U  = lsqr->vwork_m[0];
     91:   U1 = lsqr->vwork_m[1];

     93:   /* vectors of length n */
     94:   X  = ksp->vec_sol;
     95:   W  = lsqr->vwork_n[0];
     96:   V  = lsqr->vwork_n[1];
     97:   V1 = lsqr->vwork_n[2];
     98:   W2 = lsqr->vwork_n[3];
     99:   if (!nopreconditioner) Z = lsqr->vwork_n[4];

    101:   /* standard error vector */
    102:   if (lsqr->se) {
    103:     VecSet(lsqr->se,0.0);
    104:   }

    106:   /* Compute initial residual, temporarily use work vector u */
    107:   if (!ksp->guess_zero) {
    108:     KSP_MatMult(ksp,Amat,X,U);       /*   u <- b - Ax     */
    109:     VecAYPX(U,-1.0,B);
    110:   } else {
    111:     VecCopy(B,U);            /*   u <- b (x is 0) */
    112:   }

    114:   /* Test for nothing to do */
    115:   VecNorm(U,NORM_2,&rnorm);
    116:   KSPCheckNorm(ksp,rnorm);
    117:   PetscObjectSAWsTakeAccess((PetscObject)ksp);
    118:   ksp->its   = 0;
    119:   ksp->rnorm = rnorm;
    120:   PetscObjectSAWsGrantAccess((PetscObject)ksp);
    121:   KSPLogResidualHistory(ksp,rnorm);
    122:   KSPMonitor(ksp,0,rnorm);
    123:   (*ksp->converged)(ksp,0,rnorm,&ksp->reason,ksp->cnvP);
    124:   if (ksp->reason) return(0);

    126:   beta = rnorm;
    127:   VecScale(U,1.0/beta);
    128:   KSP_MatMultTranspose(ksp,Amat,U,V);
    129:   if (nopreconditioner) {
    130:     VecNorm(V,NORM_2,&alpha);
    131:     KSPCheckNorm(ksp,rnorm);
    132:   } else {
    133:     /* this is an application of the preconditioner for the normal equations; not the operator, see the manual page */
    134:     PCApply(ksp->pc,V,Z);
    135:     VecDotRealPart(V,Z,&alpha);
    136:     if (alpha <= 0.0) {
    137:       ksp->reason = KSP_DIVERGED_BREAKDOWN;
    138:       return(0);
    139:     }
    140:     alpha = PetscSqrtReal(alpha);
    141:     VecScale(Z,1.0/alpha);
    142:   }
    143:   VecScale(V,1.0/alpha);

    145:   if (nopreconditioner) {
    146:     VecCopy(V,W);
    147:   } else {
    148:     VecCopy(Z,W);
    149:   }

    151:   if (lsqr->exact_norm) {
    152:     MatNorm(Amat,NORM_FROBENIUS,&lsqr->anorm);
    153:   } else lsqr->anorm = 0.0;

    155:   lsqr->arnorm = alpha * beta;
    156:   phibar       = beta;
    157:   rhobar       = alpha;
    158:   i            = 0;
    159:   do {
    160:     if (nopreconditioner) {
    161:       KSP_MatMult(ksp,Amat,V,U1);
    162:     } else {
    163:       KSP_MatMult(ksp,Amat,Z,U1);
    164:     }
    165:     VecAXPY(U1,-alpha,U);
    166:     VecNorm(U1,NORM_2,&beta);
    167:     KSPCheckNorm(ksp,beta);
    168:     if (beta > 0.0) {
    169:       VecScale(U1,1.0/beta); /* beta*U1 = Amat*V - alpha*U */
    170:       if (!lsqr->exact_norm) {
    171:         lsqr->anorm = PetscSqrtScalar(PetscSqr(lsqr->anorm) + PetscSqr(alpha) + PetscSqr(beta));
    172:       }
    173:     }

    175:     KSP_MatMultTranspose(ksp,Amat,U1,V1);
    176:     VecAXPY(V1,-beta,V);
    177:     if (nopreconditioner) {
    178:       VecNorm(V1,NORM_2,&alpha);
    179:       KSPCheckNorm(ksp,alpha);
    180:     } else {
    181:       PCApply(ksp->pc,V1,Z);
    182:       VecDotRealPart(V1,Z,&alpha);
    183:       if (alpha <= 0.0) {
    184:         ksp->reason = KSP_DIVERGED_BREAKDOWN;
    185:         break;
    186:       }
    187:       alpha = PetscSqrtReal(alpha);
    188:       VecScale(Z,1.0/alpha);
    189:     }
    190:     VecScale(V1,1.0/alpha); /* alpha*V1 = Amat^T*U1 - beta*V */
    191:     rho    = PetscSqrtScalar(rhobar*rhobar + beta*beta);
    192:     c      = rhobar / rho;
    193:     s      = beta / rho;
    194:     theta  = s * alpha;
    195:     rhobar = -c * alpha;
    196:     phi    = c * phibar;
    197:     phibar = s * phibar;
    198:     tau    = s * phi;

    200:     VecAXPY(X,phi/rho,W);  /*    x <- x + (phi/rho) w   */

    202:     if (lsqr->se) {
    203:       VecCopy(W,W2);
    204:       VecSquare(W2);
    205:       VecScale(W2,1.0/(rho*rho));
    206:       VecAXPY(lsqr->se, 1.0, W2); /* lsqr->se <- lsqr->se + (w^2/rho^2) */
    207:     }
    208:     if (nopreconditioner) {
    209:       VecAYPX(W,-theta/rho,V1);  /* w <- v - (theta/rho) w */
    210:     } else {
    211:       VecAYPX(W,-theta/rho,Z);   /* w <- z - (theta/rho) w */
    212:     }

    214:     lsqr->arnorm = alpha*PetscAbsScalar(tau);
    215:     rnorm        = PetscRealPart(phibar);

    217:     PetscObjectSAWsTakeAccess((PetscObject)ksp);
    218:     ksp->its++;
    219:     ksp->rnorm = rnorm;
    220:     PetscObjectSAWsGrantAccess((PetscObject)ksp);
    221:     KSPLogResidualHistory(ksp,rnorm);
    222:     KSPMonitor(ksp,i+1,rnorm);
    223:     (*ksp->converged)(ksp,i+1,rnorm,&ksp->reason,ksp->cnvP);
    224:     if (ksp->reason) break;
    225:     SWAP(U1,U,TMP);
    226:     SWAP(V1,V,TMP);

    228:     i++;
    229:   } while (i<ksp->max_it);
    230:   if (i >= ksp->max_it && !ksp->reason) ksp->reason = KSP_DIVERGED_ITS;

    232:   /* Finish off the standard error estimates */
    233:   if (lsqr->se) {
    234:     tmp  = 1.0;
    235:     MatGetSize(Amat,&size1,&size2);
    236:     if (size1 > size2) tmp = size1 - size2;
    237:     tmp  = rnorm / PetscSqrtScalar(tmp);
    238:     VecSqrtAbs(lsqr->se);
    239:     VecScale(lsqr->se,tmp);
    240:   }
    241:   return(0);
    242: }

    245: PetscErrorCode KSPDestroy_LSQR(KSP ksp)
    246: {
    247:   KSP_LSQR       *lsqr = (KSP_LSQR*)ksp->data;

    251:   /* Free work vectors */
    252:   if (lsqr->vwork_n) {
    253:     VecDestroyVecs(lsqr->nwork_n,&lsqr->vwork_n);
    254:   }
    255:   if (lsqr->vwork_m) {
    256:     VecDestroyVecs(lsqr->nwork_m,&lsqr->vwork_m);
    257:   }
    258:   VecDestroy(&lsqr->se);
    259:   /* Revert convergence test */
    260:   KSPSetConvergenceTest(ksp,lsqr->converged,lsqr->cnvP,lsqr->convergeddestroy);
    261:   /* Free the KSP_LSQR context */
    262:   PetscFree(ksp->data);
    263:   PetscObjectComposeFunction((PetscObject)ksp,"KSPLSQRMonitorResidual_C",NULL);
    264:   PetscObjectComposeFunction((PetscObject)ksp,"KSPLSQRMonitorResidualDrawLG_C",NULL);
    265:   return(0);
    266: }

    268: /*@
    269:    KSPLSQRSetComputeStandardErrorVec - Compute vector of standard error estimates during KSPSolve_LSQR().

    271:    Not Collective

    273:    Input Parameters:
    274: +  ksp   - iterative context
    275: -  flg   - compute the vector of standard estimates or not

    277:    Developer notes:
    278:    Vaclav: I'm not sure whether this vector is useful for anything.

    280:    Level: intermediate

    282: .seealso: KSPSolve(), KSPLSQR, KSPLSQRGetStandardErrorVec()
    283: @*/
    284: PetscErrorCode  KSPLSQRSetComputeStandardErrorVec(KSP ksp, PetscBool flg)
    285: {
    286:   KSP_LSQR       *lsqr = (KSP_LSQR*)ksp->data;

    289:   lsqr->se_flg = flg;
    290:   return(0);
    291: }

    293: /*@
    294:    KSPLSQRSetExactMatNorm - Compute exact matrix norm instead of iteratively refined estimate.

    296:    Not Collective

    298:    Input Parameters:
    299: +  ksp   - iterative context
    300: -  flg   - compute exact matrix norm or not

    302:    Notes:
    303:    By default, flg=PETSC_FALSE. This is usually preferred to avoid possibly expensive computation of the norm.
    304:    For flg=PETSC_TRUE, we call MatNorm(Amat,NORM_FROBENIUS,&lsqr->anorm) which will work only for some types of explicitly assembled matrices.
    305:    This can affect convergence rate as KSPLSQRConvergedDefault() assumes different value of ||A|| used in normal equation stopping criterion.

    307:    Level: intermediate

    309: .seealso: KSPSolve(), KSPLSQR, KSPLSQRGetNorms(), KSPLSQRConvergedDefault()
    310: @*/
    311: PetscErrorCode  KSPLSQRSetExactMatNorm(KSP ksp, PetscBool flg)
    312: {
    313:   KSP_LSQR       *lsqr = (KSP_LSQR*)ksp->data;

    316:   lsqr->exact_norm = flg;
    317:   return(0);
    318: }

    320: /*@
    321:    KSPLSQRGetStandardErrorVec - Get vector of standard error estimates.
    322:    Only available if -ksp_lsqr_set_standard_error was set to true
    323:    or KSPLSQRSetComputeStandardErrorVec(ksp, PETSC_TRUE) was called.
    324:    Otherwise returns NULL.

    326:    Not Collective

    328:    Input Parameters:
    329: .  ksp   - iterative context

    331:    Output Parameters:
    332: .  se - vector of standard estimates

    334:    Options Database Keys:
    335: .   -ksp_lsqr_set_standard_error  - set standard error estimates of solution

    337:    Developer notes:
    338:    Vaclav: I'm not sure whether this vector is useful for anything.

    340:    Level: intermediate

    342: .seealso: KSPSolve(), KSPLSQR, KSPLSQRSetComputeStandardErrorVec()
    343: @*/
    344: PetscErrorCode  KSPLSQRGetStandardErrorVec(KSP ksp,Vec *se)
    345: {
    346:   KSP_LSQR *lsqr = (KSP_LSQR*)ksp->data;

    349:   *se = lsqr->se;
    350:   return(0);
    351: }

    353: /*@
    354:    KSPLSQRGetNorms - Get norm estimates that LSQR computes internally during KSPSolve().

    356:    Not Collective

    358:    Input Parameters:
    359: .  ksp   - iterative context

    361:    Output Parameters:
    362: +  arnorm - good estimate of norm((A*inv(Pmat))'*r), where r = A*x - b, used in specific stopping criterion
    363: -  anorm - poor estimate of norm(A*inv(Pmat),'fro') used in specific stopping criterion

    365:    Notes:
    366:    Output parameters are meaningful only after KSPSolve().
    367:    These are the same quantities as normar and norma in MATLAB's lsqr(), whose output lsvec is a vector of normar / norma for all iterations.
    368:    If -ksp_lsqr_exact_mat_norm is set or KSPLSQRSetExactMatNorm(ksp, PETSC_TRUE) called, then anorm is exact Frobenius norm.

    370:    Level: intermediate

    372: .seealso: KSPSolve(), KSPLSQR, KSPLSQRSetExactMatNorm()
    373: @*/
    374: PetscErrorCode  KSPLSQRGetNorms(KSP ksp,PetscReal *arnorm, PetscReal *anorm)
    375: {
    376:   KSP_LSQR       *lsqr = (KSP_LSQR*)ksp->data;

    379:   if (arnorm)   *arnorm = lsqr->arnorm;
    380:   if (anorm)    *anorm = lsqr->anorm;
    381:   return(0);
    382: }

    384: PetscErrorCode KSPLSQRMonitorResidual_LSQR(KSP ksp, PetscInt n, PetscReal rnorm, PetscViewerAndFormat *vf)
    385: {
    386:   KSP_LSQR          *lsqr  = (KSP_LSQR*)ksp->data;
    387:   PetscViewer       viewer = vf->viewer;
    388:   PetscViewerFormat format = vf->format;
    389:   char              normtype[256];
    390:   PetscInt          tablevel;
    391:   const char        *prefix;
    392:   PetscErrorCode    ierr;

    395:   PetscObjectGetTabLevel((PetscObject) ksp, &tablevel);
    396:   PetscObjectGetOptionsPrefix((PetscObject) ksp, &prefix);
    397:   PetscStrncpy(normtype, KSPNormTypes[ksp->normtype], sizeof(normtype));
    398:   PetscStrtolower(normtype);
    399:   PetscViewerPushFormat(viewer, format);
    400:   PetscViewerASCIIAddTab(viewer, tablevel);
    401:   if (n == 0 && prefix) {PetscViewerASCIIPrintf(viewer, "  Residual norm, norm of normal equations, and matrix norm for %s solve.\n", prefix);}
    402:   if (!n) {
    403:     PetscViewerASCIIPrintf(viewer,"%3D KSP resid norm %14.12e\n",n,(double)rnorm);
    404:   } else {
    405:     PetscViewerASCIIPrintf(viewer,"%3D KSP resid norm %14.12e normal eq resid norm %14.12e matrix norm %14.12e\n",n,(double)rnorm,(double)lsqr->arnorm,(double)lsqr->anorm);
    406:   }
    407:   PetscViewerASCIISubtractTab(viewer, tablevel);
    408:   PetscViewerPopFormat(viewer);
    409:   return(0);
    410: }

    412: /*@C
    413:   KSPLSQRMonitorResidual - Prints the residual norm, as well as the normal equation residual norm, at each iteration of an iterative solver.

    415:   Collective on ksp

    417:   Input Parameters:
    418: + ksp   - iterative context
    419: . n     - iteration number
    420: . rnorm - 2-norm (preconditioned) residual value (may be estimated).
    421: - vf    - The viewer context

    423:   Options Database Key:
    424: . -ksp_lsqr_monitor - Activates KSPLSQRMonitorResidual()

    426:   Level: intermediate

    428: .seealso: KSPMonitorSet(), KSPMonitorResidual(),KSPMonitorTrueResidualMaxNorm()
    429: @*/
    430: PetscErrorCode KSPLSQRMonitorResidual(KSP ksp, PetscInt n, PetscReal rnorm, PetscViewerAndFormat *vf)
    431: {

    438:   PetscTryMethod(ksp, "KSPLSQRMonitorResidual_C", (KSP,PetscInt,PetscReal,PetscViewerAndFormat*), (ksp,n,rnorm,vf));
    439:   return(0);
    440: }

    442: PetscErrorCode KSPLSQRMonitorResidualDrawLG_LSQR(KSP ksp, PetscInt n, PetscReal rnorm, PetscViewerAndFormat *vf)
    443: {
    444:   KSP_LSQR           *lsqr  = (KSP_LSQR*)ksp->data;
    445:   PetscViewer        viewer = vf->viewer;
    446:   PetscViewerFormat  format = vf->format;
    447:   PetscDrawLG        lg     = vf->lg;
    448:   KSPConvergedReason reason;
    449:   PetscReal          x[2], y[2];
    450:   PetscErrorCode     ierr;

    453:   PetscViewerPushFormat(viewer, format);
    454:   if (!n) {PetscDrawLGReset(lg);}
    455:   x[0] = (PetscReal) n;
    456:   if (rnorm > 0.0) y[0] = PetscLog10Real(rnorm);
    457:   else y[0] = -15.0;
    458:   x[1] = (PetscReal) n;
    459:   if (lsqr->arnorm > 0.0) y[1] = PetscLog10Real(lsqr->arnorm);
    460:   else y[1] = -15.0;
    461:   PetscDrawLGAddPoint(lg, x, y);
    462:   KSPGetConvergedReason(ksp, &reason);
    463:   if (n <= 20 || !(n % 5) || reason) {
    464:     PetscDrawLGDraw(lg);
    465:     PetscDrawLGSave(lg);
    466:   }
    467:   PetscViewerPopFormat(viewer);
    468:   return(0);
    469: }

    471: /*@C
    472:   KSPLSQRMonitorResidualDrawLG - Plots the true residual norm at each iteration of an iterative solver.

    474:   Collective on ksp

    476:   Input Parameters:
    477: + ksp   - iterative context
    478: . n     - iteration number
    479: . rnorm - 2-norm (preconditioned) residual value (may be estimated).
    480: - vf    - The viewer context

    482:   Options Database Key:
    483: . -ksp_lsqr_monitor draw::draw_lg - Activates KSPMonitorTrueResidualDrawLG()

    485:   Level: intermediate

    487: .seealso: KSPMonitorSet(), KSPMonitorTrueResidual()
    488: @*/
    489: PetscErrorCode KSPLSQRMonitorResidualDrawLG(KSP ksp, PetscInt n, PetscReal rnorm, PetscViewerAndFormat *vf)
    490: {

    498:   PetscTryMethod(ksp, "KSPLSQRMonitorResidualDrawLG_C", (KSP,PetscInt,PetscReal,PetscViewerAndFormat*), (ksp,n,rnorm,vf));
    499:   return(0);
    500: }

    502: /*@C
    503:   KSPLSQRMonitorResidualDrawLGCreate - Creates the plotter for the LSQR residual and normal eqn residual.

    505:   Collective on ksp

    507:   Input Parameters:
    508: + viewer - The PetscViewer
    509: . format - The viewer format
    510: - ctx    - An optional user context

    512:   Output Parameter:
    513: . vf    - The viewer context

    515:   Level: intermediate

    517: .seealso: KSPMonitorSet(), KSPLSQRMonitorResidual()
    518: @*/
    519: PetscErrorCode KSPLSQRMonitorResidualDrawLGCreate(PetscViewer viewer, PetscViewerFormat format, void *ctx, PetscViewerAndFormat **vf)
    520: {
    521:   const char    *names[] = {"residual", "normal eqn residual"};

    525:   PetscViewerAndFormatCreate(viewer, format, vf);
    526:   (*vf)->data = ctx;
    527:   KSPMonitorLGCreate(PetscObjectComm((PetscObject) viewer), NULL, NULL, "Log Residual Norm", 2, names, PETSC_DECIDE, PETSC_DECIDE, 400, 300, &(*vf)->lg);
    528:   return(0);
    529: }

    531: PetscErrorCode KSPSetFromOptions_LSQR(PetscOptionItems *PetscOptionsObject,KSP ksp)
    532: {
    534:   KSP_LSQR       *lsqr = (KSP_LSQR*)ksp->data;

    537:   PetscOptionsHead(PetscOptionsObject,"KSP LSQR Options");
    538:   PetscOptionsBool("-ksp_lsqr_compute_standard_error","Set Standard Error Estimates of Solution","KSPLSQRSetComputeStandardErrorVec",lsqr->se_flg,&lsqr->se_flg,NULL);
    539:   PetscOptionsBool("-ksp_lsqr_exact_mat_norm","Compute exact matrix norm instead of iteratively refined estimate","KSPLSQRSetExactMatNorm",lsqr->exact_norm,&lsqr->exact_norm,NULL);
    540:   KSPMonitorSetFromOptions(ksp, "-ksp_lsqr_monitor", "lsqr_residual", NULL);
    541:   PetscOptionsTail();
    542:   return(0);
    543: }

    545: PetscErrorCode KSPView_LSQR(KSP ksp,PetscViewer viewer)
    546: {
    547:   KSP_LSQR       *lsqr = (KSP_LSQR*)ksp->data;
    549:   PetscBool      iascii;

    552:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);
    553:   if (iascii) {
    554:     if (lsqr->se) {
    555:       PetscReal rnorm;
    556:       VecNorm(lsqr->se,NORM_2,&rnorm);
    557:       PetscViewerASCIIPrintf(viewer,"  norm of standard error %g, iterations %d\n",(double)rnorm,ksp->its);
    558:     } else {
    559:       PetscViewerASCIIPrintf(viewer,"  standard error not computed\n");
    560:     }
    561:     if (lsqr->exact_norm) {
    562:       PetscViewerASCIIPrintf(viewer,"  using exact matrix norm\n");
    563:     } else {
    564:       PetscViewerASCIIPrintf(viewer,"  using inexact matrix norm\n");
    565:     }
    566:   }
    567:   return(0);
    568: }

    570: /*@C
    571:    KSPLSQRConvergedDefault - Determines convergence of the LSQR Krylov method.

    573:    Collective on ksp

    575:    Input Parameters:
    576: +  ksp   - iterative context
    577: .  n     - iteration number
    578: .  rnorm - 2-norm residual value (may be estimated)
    579: -  ctx - convergence context which must be created by KSPConvergedDefaultCreate()

    581:    reason is set to:
    582: +   positive - if the iteration has converged;
    583: .   negative - if residual norm exceeds divergence threshold;
    584: -   0 - otherwise.

    586:    Notes:
    587:    KSPConvergedDefault() is called first to check for convergence in A*x=b.
    588:    If that does not determine convergence then checks convergence for the least squares problem, i.e. in min{|b-A*x|}.
    589:    Possible convergence for the least squares problem (which is based on the residual of the normal equations) are KSP_CONVERGED_RTOL_NORMAL norm and KSP_CONVERGED_ATOL_NORMAL.
    590:    KSP_CONVERGED_RTOL_NORMAL is returned if ||A'*r|| < rtol * ||A|| * ||r||.
    591:    Matrix norm ||A|| is iteratively refined estimate, see KSPLSQRGetNorms().
    592:    This criterion is now largely compatible with that in MATLAB lsqr().

    594:    Level: intermediate

    596: .seealso: KSPLSQR, KSPSetConvergenceTest(), KSPSetTolerances(), KSPConvergedSkip(), KSPConvergedReason, KSPGetConvergedReason(),
    597:           KSPConvergedDefaultSetUIRNorm(), KSPConvergedDefaultSetUMIRNorm(), KSPConvergedDefaultCreate(), KSPConvergedDefaultDestroy(), KSPConvergedDefault(), KSPLSQRGetNorms(), KSPLSQRSetExactMatNorm()
    598: @*/
    599: PetscErrorCode  KSPLSQRConvergedDefault(KSP ksp,PetscInt n,PetscReal rnorm,KSPConvergedReason *reason,void *ctx)
    600: {
    602:   KSP_LSQR       *lsqr = (KSP_LSQR*)ksp->data;

    605:   /* check for convergence in A*x=b */
    606:   KSPConvergedDefault(ksp,n,rnorm,reason,ctx);
    607:   if (!n || *reason) return(0);

    609:   /* check for convergence in min{|b-A*x|} */
    610:   if (lsqr->arnorm < ksp->abstol) {
    611:     PetscInfo3(ksp,"LSQR solver has converged. Normal equation residual %14.12e is less than absolute tolerance %14.12e at iteration %D\n",(double)lsqr->arnorm,(double)ksp->abstol,n);
    612:     *reason = KSP_CONVERGED_ATOL_NORMAL;
    613:   } else if (lsqr->arnorm < ksp->rtol * lsqr->anorm * rnorm) {
    614:     PetscInfo6(ksp,"LSQR solver has converged. Normal equation residual %14.12e is less than rel. tol. %14.12e times %s Frobenius norm of matrix %14.12e times residual %14.12e at iteration %D\n",(double)lsqr->arnorm,(double)ksp->rtol,lsqr->exact_norm?"exact":"approx.",(double)lsqr->anorm,(double)rnorm,n);
    615:     *reason = KSP_CONVERGED_RTOL_NORMAL;
    616:   }
    617:   return(0);
    618: }

    620: /*MC
    621:      KSPLSQR - This implements LSQR

    623:    Options Database Keys:
    624: +   -ksp_lsqr_set_standard_error  - set standard error estimates of solution, see KSPLSQRSetComputeStandardErrorVec() and KSPLSQRGetStandardErrorVec()
    625: .   -ksp_lsqr_exact_mat_norm - compute exact matrix norm instead of iteratively refined estimate, see KSPLSQRSetExactMatNorm()
    626: -   -ksp_lsqr_monitor - monitor residual norm, norm of residual of normal equations A'*A x = A' b, and estimate of matrix norm ||A||

    628:    Level: beginner

    630:    Notes:
    631:      Supports non-square (rectangular) matrices.

    633:      This varient, when applied with no preconditioning is identical to the original algorithm in exact arithematic; however, in practice, with no preconditioning
    634:      due to inexact arithematic, it can converge differently. Hence when no preconditioner is used (PCType PCNONE) it automatically reverts to the original algorithm.

    636:      With the PETSc built-in preconditioners, such as ICC, one should call KSPSetOperators(ksp,A,A'*A)) since the preconditioner needs to work
    637:      for the normal equations A'*A.

    639:      Supports only left preconditioning.

    641:      For least squares problems wit nonzero residual A*x - b, there are additional convergence tests for the residual of the normal equations, A'*(b - Ax), see KSPLSQRConvergedDefault().

    643:    References:
    644: .  1. - The original unpreconditioned algorithm can be found in Paige and Saunders, ACM Transactions on Mathematical Software, Vol 8, 1982.

    646:      In exact arithmetic the LSQR method (with no preconditioning) is identical to the KSPCG algorithm applied to the normal equations.
    647:      The preconditioned variant was implemented by Bas van't Hof and is essentially a left preconditioning for the Normal Equations. It appears the implementation with preconditioner
    648:      track the true norm of the residual and uses that in the convergence test.

    650:    Developer Notes:
    651:     How is this related to the KSPCGNE implementation? One difference is that KSPCGNE applies
    652:             the preconditioner transpose times the preconditioner,  so one does not need to pass A'*A as the third argument to KSPSetOperators().

    656: .seealso:  KSPCreate(), KSPSetType(), KSPType (for list of available types), KSP, KSPSolve(), KSPLSQRConvergedDefault(), KSPLSQRSetComputeStandardErrorVec(), KSPLSQRGetStandardErrorVec(), KSPLSQRSetExactMatNorm()

    658: M*/
    659: PETSC_EXTERN PetscErrorCode KSPCreate_LSQR(KSP ksp)
    660: {
    661:   KSP_LSQR       *lsqr;
    662:   void           *ctx;

    666:   PetscNewLog(ksp,&lsqr);
    667:   lsqr->se     = NULL;
    668:   lsqr->se_flg = PETSC_FALSE;
    669:   lsqr->exact_norm = PETSC_FALSE;
    670:   lsqr->anorm  = -1.0;
    671:   lsqr->arnorm = -1.0;
    672:   ksp->data    = (void*)lsqr;
    673:   KSPSetSupportedNorm(ksp,KSP_NORM_UNPRECONDITIONED,PC_LEFT,3);

    675:   ksp->ops->setup          = KSPSetUp_LSQR;
    676:   ksp->ops->solve          = KSPSolve_LSQR;
    677:   ksp->ops->destroy        = KSPDestroy_LSQR;
    678:   ksp->ops->setfromoptions = KSPSetFromOptions_LSQR;
    679:   ksp->ops->view           = KSPView_LSQR;

    681:   /* Backup current convergence test; remove destroy routine from KSP to prevent destroying the convergence context in KSPSetConvergenceTest() */
    682:   KSPGetAndClearConvergenceTest(ksp,&lsqr->converged,&lsqr->cnvP,&lsqr->convergeddestroy);
    683:   /* Override current convergence test */
    684:   KSPConvergedDefaultCreate(&ctx);
    685:   KSPSetConvergenceTest(ksp,KSPLSQRConvergedDefault,ctx,KSPConvergedDefaultDestroy);
    686:   PetscObjectComposeFunction((PetscObject)ksp,"KSPLSQRMonitorResidual_C",KSPLSQRMonitorResidual_LSQR);
    687:   PetscObjectComposeFunction((PetscObject)ksp,"KSPLSQRMonitorResidualDrawLG_C",KSPLSQRMonitorResidualDrawLG_LSQR);
    688:   return(0);
    689: }