impls/ncg/snesncg.c

#include <../src/snes/impls/ncg/snesncgimpl.h> /*I "petscsnes.h" I*/
const char *const SNESNCGTypes[] = {"FR","PRP","HS","DY","CD","SNESNCGType","SNES_NCG_",0};

#undef __FUNCT__
#define __FUNCT__ "SNESReset_NCG"
PetscErrorCode SNESReset_NCG(SNES snes)
{
  PetscFunctionBegin;
  PetscFunctionReturn(0);
}

#define SNESLINESEARCHNCGLINEAR "linear"

/*
  SNESDestroy_NCG - Destroys the private SNES_NCG context that was created with SNESCreate_NCG().

  Input Parameter:
. snes - the SNES context

  Application Interface Routine: SNESDestroy()
*/
#undef __FUNCT__
#define __FUNCT__ "SNESDestroy_NCG"
PetscErrorCode SNESDestroy_NCG(SNES snes)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscFree(snes->data);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*
   SNESSetUp_NCG - Sets up the internal data structures for the later use
   of the SNESNCG nonlinear solver.

   Input Parameters:
+  snes - the SNES context
-  x - the solution vector

   Application Interface Routine: SNESSetUp()
 */

PETSC_EXTERN PetscErrorCode SNESLineSearchCreate_NCGLinear(SNESLineSearch);

#undef __FUNCT__
#define __FUNCT__ "SNESSetUp_NCG"
PetscErrorCode SNESSetUp_NCG(SNES snes)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = SNESSetWorkVecs(snes,2);CHKERRQ(ierr);
  ierr = SNESSetUpMatrices(snes);CHKERRQ(ierr);
  if (snes->pcside == PC_RIGHT) SETERRQ(PetscObjectComm((PetscObject)snes),PETSC_ERR_ARG_WRONGSTATE, "SNESNCG only supports left preconditioning");
  if (snes->functype == SNES_FUNCTION_DEFAULT) snes->functype = SNES_FUNCTION_UNPRECONDITIONED;
  ierr = SNESLineSearchRegister(SNESLINESEARCHNCGLINEAR, SNESLineSearchCreate_NCGLinear);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}
/*
  SNESSetFromOptions_NCG - Sets various parameters for the SNESNCG method.

  Input Parameter:
. snes - the SNES context

  Application Interface Routine: SNESSetFromOptions()
*/
#undef __FUNCT__
#define __FUNCT__ "SNESSetFromOptions_NCG"
static PetscErrorCode SNESSetFromOptions_NCG(SNES snes)
{
  SNES_NCG       *ncg = (SNES_NCG*)snes->data;
  PetscErrorCode ierr;
  PetscBool      debug;
  SNESLineSearch linesearch;
  SNESNCGType    ncgtype=ncg->type;

  PetscFunctionBegin;
  ierr = PetscOptionsHead("SNES NCG options");CHKERRQ(ierr);
  ierr = PetscOptionsBool("-snes_ncg_monitor","Monitor NCG iterations","SNES",ncg->monitor ? PETSC_TRUE : PETSC_FALSE, &debug, NULL);CHKERRQ(ierr);
  ierr = PetscOptionsEnum("-snes_ncg_type","NCG Beta type used","SNESNCGSetType",SNESNCGTypes,(PetscEnum)ncg->type,(PetscEnum*)&ncgtype,NULL);CHKERRQ(ierr);
  ierr = SNESNCGSetType(snes, ncgtype);CHKERRQ(ierr);
  if (debug) {
    ncg->monitor = PETSC_VIEWER_STDOUT_(PetscObjectComm((PetscObject)snes));CHKERRQ(ierr);
  }
  ierr = PetscOptionsTail();CHKERRQ(ierr);
  if (!snes->linesearch) {
    ierr = SNESGetLineSearch(snes, &linesearch);CHKERRQ(ierr);
    if (!snes->pc) {
      ierr = SNESLineSearchSetType(linesearch, SNESLINESEARCHCP);CHKERRQ(ierr);
    } else {
      ierr = SNESLineSearchSetType(linesearch, SNESLINESEARCHL2);CHKERRQ(ierr);
    }
  }
  PetscFunctionReturn(0);
}

/*
  SNESView_NCG - Prints info from the SNESNCG data structure.

  Input Parameters:
+ SNES - the SNES context
- viewer - visualization context

  Application Interface Routine: SNESView()
*/
#undef __FUNCT__
#define __FUNCT__ "SNESView_NCG"
static PetscErrorCode SNESView_NCG(SNES snes, PetscViewer viewer)
{
  PetscBool      iascii;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscObjectTypeCompare((PetscObject) viewer, PETSCVIEWERASCII, &iascii);CHKERRQ(ierr);
  if (iascii) {
  }
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "SNESLineSearchApply_NCGLinear"
PetscErrorCode SNESLineSearchApply_NCGLinear(SNESLineSearch linesearch)
{
  PetscScalar    alpha, ptAp;
  Vec            X, Y, F, W;
  SNES           snes;
  PetscErrorCode ierr;
  PetscReal      *fnorm, *xnorm, *ynorm;
  MatStructure   flg = DIFFERENT_NONZERO_PATTERN;

  PetscFunctionBegin;
  ierr  = SNESLineSearchGetSNES(linesearch, &snes);CHKERRQ(ierr);
  X     = linesearch->vec_sol;
  W     = linesearch->vec_sol_new;
  F     = linesearch->vec_func;
  Y     = linesearch->vec_update;
  fnorm = &linesearch->fnorm;
  xnorm = &linesearch->xnorm;
  ynorm = &linesearch->ynorm;

  /*

   The exact step size for unpreconditioned linear CG is just:
   alpha = (r, r) / (p, Ap) = (f, f) / (y, Jy)
   */
  ierr  = SNESComputeJacobian(snes, X, &snes->jacobian, &snes->jacobian_pre, &flg);CHKERRQ(ierr);
  ierr  = VecDot(F, F, &alpha);CHKERRQ(ierr);
  ierr  = MatMult(snes->jacobian, Y, W);CHKERRQ(ierr);
  ierr  = VecDot(Y, W, &ptAp);CHKERRQ(ierr);
  alpha = alpha / ptAp;
  ierr  = PetscPrintf(PetscObjectComm((PetscObject)snes), "alpha: %G\n", PetscRealPart(alpha));CHKERRQ(ierr);
  ierr  = VecAXPY(X, alpha, Y);CHKERRQ(ierr);
  ierr  = SNESComputeFunction(snes, X, F);CHKERRQ(ierr);

  ierr = VecNorm(F, NORM_2, fnorm);CHKERRQ(ierr);
  ierr = VecNorm(X, NORM_2, xnorm);CHKERRQ(ierr);
  ierr = VecNorm(Y, NORM_2, ynorm);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "SNESLineSearchCreate_NCGLinear"
PETSC_EXTERN PetscErrorCode SNESLineSearchCreate_NCGLinear(SNESLineSearch linesearch)
{
  PetscFunctionBegin;
  linesearch->ops->apply          = SNESLineSearchApply_NCGLinear;
  linesearch->ops->destroy        = NULL;
  linesearch->ops->setfromoptions = NULL;
  linesearch->ops->reset          = NULL;
  linesearch->ops->view           = NULL;
  linesearch->ops->setup          = NULL;
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "SNESNCGComputeYtJtF_Private"
/*

 Assuming F = SNESComputeFunction(X) compute Y^tJ^tF using a simple secant approximation of the jacobian.

 */
PetscErrorCode SNESNCGComputeYtJtF_Private(SNES snes, Vec X, Vec F, Vec Y, Vec W, Vec G, PetscScalar * ytJtf)
{
  PetscErrorCode ierr;
  PetscScalar    ftf, ftg, fty, h;

  PetscFunctionBegin;
  ierr   = VecDot(F, F, &ftf);CHKERRQ(ierr);
  ierr   = VecDot(F, Y, &fty);CHKERRQ(ierr);
  h      = 1e-5*fty / fty;
  ierr   = VecCopy(X, W);CHKERRQ(ierr);
  ierr   = VecAXPY(W, -h, Y);CHKERRQ(ierr);          /* this is arbitrary */
  ierr   = SNESComputeFunction(snes, W, G);CHKERRQ(ierr);
  ierr   = VecDot(G, F, &ftg);CHKERRQ(ierr);
  *ytJtf = (ftg - ftf) / h;
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "SNESNCGSetType"
/*@
    SNESNCGSetType - Sets the conjugate update type for SNESNCG.

    Logically Collective on SNES

    Input Parameters:
+   snes - the iterative context
-   btype - update type

    Options Database:
.   -snes_ncg_type<prp,fr,hs,dy,cd>

    Level: intermediate

    SNESNCGSelectTypes:
+   SNES_NCG_FR - Fletcher-Reeves update
.   SNES_NCG_PRP - Polak-Ribiere-Polyak update
.   SNES_NCG_HS - Hestenes-Steifel update
.   SNES_NCG_DY - Dai-Yuan update
-   SNES_NCG_CD - Conjugate Descent update

   Notes:
   PRP is the default, and the only one that tolerates generalized search directions.

.keywords: SNES, SNESNCG, selection, type, set
@*/
PetscErrorCode SNESNCGSetType(SNES snes, SNESNCGType btype)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(snes,SNES_CLASSID,1);
  ierr = PetscTryMethod(snes,"SNESNCGSetType_C",(SNES,SNESNCGType),(snes,btype));CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "SNESNCGSetType_NCG"
PetscErrorCode SNESNCGSetType_NCG(SNES snes, SNESNCGType btype)
{
  SNES_NCG *ncg = (SNES_NCG*)snes->data;

  PetscFunctionBegin;
  ncg->type = btype;
  PetscFunctionReturn(0);
}

/*
  SNESSolve_NCG - Solves a nonlinear system with the Nonlinear Conjugate Gradient method.

  Input Parameters:
. snes - the SNES context

  Output Parameter:
. outits - number of iterations until termination

  Application Interface Routine: SNESSolve()
*/
#undef __FUNCT__
#define __FUNCT__ "SNESSolve_NCG"
PetscErrorCode SNESSolve_NCG(SNES snes)
{
  SNES_NCG            *ncg = (SNES_NCG*)snes->data;
  Vec                 X,dX,lX,F,dXold;
  PetscReal           fnorm, ynorm, xnorm, beta = 0.0;
  PetscScalar         dXdotdX, dXolddotdXold, dXdotdXold, lXdotdX, lXdotdXold;
  PetscInt            maxits, i;
  PetscErrorCode      ierr;
  PetscBool           lsSuccess = PETSC_TRUE;
  SNESLineSearch      linesearch;
  SNESConvergedReason reason;

  PetscFunctionBegin;
  snes->reason = SNES_CONVERGED_ITERATING;

  maxits = snes->max_its;            /* maximum number of iterations */
  X      = snes->vec_sol;            /* X^n */
  dXold  = snes->work[0];            /* The previous iterate of X */
  dX     = snes->work[1];            /* the preconditioned direction */
  lX     = snes->vec_sol_update;     /* the conjugate direction */
  F      = snes->vec_func;           /* residual vector */

  ierr = SNESGetLineSearch(snes, &linesearch);CHKERRQ(ierr);

  ierr       = PetscObjectAMSTakeAccess((PetscObject)snes);CHKERRQ(ierr);
  snes->iter = 0;
  snes->norm = 0.;
  ierr       = PetscObjectAMSGrantAccess((PetscObject)snes);CHKERRQ(ierr);

  /* compute the initial function and preconditioned update dX */

  if (snes->pc && snes->functype == SNES_FUNCTION_PRECONDITIONED) {
    ierr = SNESApplyPC(snes,X,NULL,NULL,dX);CHKERRQ(ierr);
    ierr = SNESGetConvergedReason(snes->pc,&reason);CHKERRQ(ierr);
    if (reason < 0  && reason != SNES_DIVERGED_MAX_IT) {
      snes->reason = SNES_DIVERGED_INNER;
      PetscFunctionReturn(0);
    }
    ierr = VecCopy(dX,F);CHKERRQ(ierr);
    ierr = VecNorm(F,NORM_2,&fnorm);CHKERRQ(ierr);
  } else {
    if (!snes->vec_func_init_set) {
      ierr = SNESComputeFunction(snes,X,F);CHKERRQ(ierr);
      if (snes->domainerror) {
        snes->reason = SNES_DIVERGED_FUNCTION_DOMAIN;
        PetscFunctionReturn(0);
      }
    } else snes->vec_func_init_set = PETSC_FALSE;
    if (!snes->norm_init_set) {
      /* convergence test */
      ierr = VecNorm(F,NORM_2,&fnorm);CHKERRQ(ierr);
      if (PetscIsInfOrNanReal(fnorm)) {
        snes->reason = SNES_DIVERGED_FNORM_NAN;
        PetscFunctionReturn(0);
      }
    } else {
      fnorm               = snes->norm_init;
      snes->norm_init_set = PETSC_FALSE;
    }
    ierr = VecCopy(F,dX);CHKERRQ(ierr);
  }
  if (snes->pc) {
    if (snes->functype == SNES_FUNCTION_UNPRECONDITIONED) {
      ierr = SNESApplyPC(snes,X,F,&fnorm,dX);CHKERRQ(ierr);
      ierr = SNESGetConvergedReason(snes->pc,&reason);CHKERRQ(ierr);
      if (reason < 0  && reason != SNES_DIVERGED_MAX_IT) {
        snes->reason = SNES_DIVERGED_INNER;
        PetscFunctionReturn(0);
      }
    }
  }
  ierr = VecCopy(dX,lX);CHKERRQ(ierr);
  ierr = VecDot(dX, dX, &dXdotdX);CHKERRQ(ierr);


  ierr       = PetscObjectAMSTakeAccess((PetscObject)snes);CHKERRQ(ierr);
  snes->norm = fnorm;
  ierr       = PetscObjectAMSGrantAccess((PetscObject)snes);CHKERRQ(ierr);
  ierr       = SNESLogConvergenceHistory(snes,fnorm,0);CHKERRQ(ierr);
  ierr       = SNESMonitor(snes,0,fnorm);CHKERRQ(ierr);

  /* set parameter for default relative tolerance convergence test */
  snes->ttol = fnorm*snes->rtol;
  /* test convergence */
  ierr = (*snes->ops->converged)(snes,0,0.0,0.0,fnorm,&snes->reason,snes->cnvP);CHKERRQ(ierr);
  if (snes->reason) PetscFunctionReturn(0);

  /* Call general purpose update function */
  if (snes->ops->update) {
    ierr = (*snes->ops->update)(snes, snes->iter);CHKERRQ(ierr);
  }

  /* first update -- just use the (preconditioned) residual direction for the initial conjugate direction */

  for (i = 1; i < maxits + 1; i++) {
    lsSuccess = PETSC_TRUE;
    /* some update types require the old update direction or conjugate direction */
    if (ncg->type != SNES_NCG_FR) {
      ierr = VecCopy(dX, dXold);CHKERRQ(ierr);
    }
    ierr = SNESLineSearchApply(linesearch,X,F,&fnorm,lX);CHKERRQ(ierr);
    ierr = SNESLineSearchGetSuccess(linesearch, &lsSuccess);CHKERRQ(ierr);
    if (!lsSuccess) {
      if (++snes->numFailures >= snes->maxFailures) {
        snes->reason = SNES_DIVERGED_LINE_SEARCH;
        PetscFunctionReturn(0);
      }
    }
    if (snes->nfuncs >= snes->max_funcs) {
      snes->reason = SNES_DIVERGED_FUNCTION_COUNT;
      PetscFunctionReturn(0);
    }
    if (snes->domainerror) {
      snes->reason = SNES_DIVERGED_FUNCTION_DOMAIN;
      PetscFunctionReturn(0);
    }
    ierr = SNESLineSearchGetNorms(linesearch, &xnorm, &fnorm, &ynorm);CHKERRQ(ierr);
    /* Monitor convergence */
    ierr       = PetscObjectAMSTakeAccess((PetscObject)snes);CHKERRQ(ierr);
    snes->iter = i;
    snes->norm = fnorm;
    ierr       = PetscObjectAMSGrantAccess((PetscObject)snes);CHKERRQ(ierr);
    ierr       = SNESLogConvergenceHistory(snes,snes->norm,0);CHKERRQ(ierr);
    ierr       = SNESMonitor(snes,snes->iter,snes->norm);CHKERRQ(ierr);

    /* Test for convergence */
    ierr = (*snes->ops->converged)(snes,snes->iter,xnorm,ynorm,fnorm,&snes->reason,snes->cnvP);CHKERRQ(ierr);
    if (snes->reason) PetscFunctionReturn(0);

    /* Call general purpose update function */
    if (snes->ops->update) {
      ierr = (*snes->ops->update)(snes, snes->iter);CHKERRQ(ierr);
    }
    if (snes->pc) {
      if (snes->functype == SNES_FUNCTION_PRECONDITIONED) {
        ierr = SNESApplyPC(snes,X,NULL,NULL,dX);CHKERRQ(ierr);
        ierr = SNESGetConvergedReason(snes->pc,&reason);CHKERRQ(ierr);
        if (reason < 0  && reason != SNES_DIVERGED_MAX_IT) {
          snes->reason = SNES_DIVERGED_INNER;
          PetscFunctionReturn(0);
        }
        ierr = VecCopy(dX,F);CHKERRQ(ierr);
      } else {
        ierr = SNESApplyPC(snes,X,F,&fnorm,dX);CHKERRQ(ierr);
        ierr = SNESGetConvergedReason(snes->pc,&reason);CHKERRQ(ierr);
        if (reason < 0  && reason != SNES_DIVERGED_MAX_IT) {
          snes->reason = SNES_DIVERGED_INNER;
          PetscFunctionReturn(0);
        }
      }
    } else {
      ierr = VecCopy(F,dX);CHKERRQ(ierr);
    }

    /* compute the conjugate direction lX = dX + beta*lX with beta = ((dX, dX) / (dX_old, dX_old) (Fletcher-Reeves update)*/
    switch (ncg->type) {
    case SNES_NCG_FR: /* Fletcher-Reeves */
      dXolddotdXold= dXdotdX;
      ierr         = VecDot(dX, dX, &dXdotdX);CHKERRQ(ierr);
      beta         = PetscRealPart(dXdotdX / dXolddotdXold);
      break;
    case SNES_NCG_PRP: /* Polak-Ribiere-Poylak */
      dXolddotdXold= dXdotdX;
      ierr         = VecDotBegin(dX, dX, &dXdotdXold);CHKERRQ(ierr);
      ierr         = VecDotBegin(dXold, dX, &dXdotdXold);CHKERRQ(ierr);
      ierr         = VecDotEnd(dX, dX, &dXdotdX);CHKERRQ(ierr);
      ierr         = VecDotEnd(dXold, dX, &dXdotdXold);CHKERRQ(ierr);
      beta         = PetscRealPart(((dXdotdX - dXdotdXold) / dXolddotdXold));
      if (beta < 0.0) beta = 0.0; /* restart */
      break;
    case SNES_NCG_HS: /* Hestenes-Stiefel */
      ierr = VecDotBegin(dX, dX, &dXdotdX);CHKERRQ(ierr);
      ierr = VecDotBegin(dX, dXold, &dXdotdXold);CHKERRQ(ierr);
      ierr = VecDotBegin(lX, dX, &lXdotdX);CHKERRQ(ierr);
      ierr = VecDotBegin(lX, dXold, &lXdotdXold);CHKERRQ(ierr);
      ierr = VecDotEnd(dX, dX, &dXdotdX);CHKERRQ(ierr);
      ierr = VecDotEnd(dX, dXold, &dXdotdXold);CHKERRQ(ierr);
      ierr = VecDotEnd(lX, dX, &lXdotdX);CHKERRQ(ierr);
      ierr = VecDotEnd(lX, dXold, &lXdotdXold);CHKERRQ(ierr);
      beta = PetscRealPart((dXdotdX - dXdotdXold) / (lXdotdX - lXdotdXold));
      break;
    case SNES_NCG_DY: /* Dai-Yuan */
      ierr = VecDotBegin(dX, dX, &dXdotdX);CHKERRQ(ierr);
      ierr = VecDotBegin(lX, dX, &lXdotdX);CHKERRQ(ierr);
      ierr = VecDotBegin(lX, dXold, &lXdotdXold);CHKERRQ(ierr);
      ierr = VecDotEnd(dX, dX, &dXdotdX);CHKERRQ(ierr);
      ierr = VecDotEnd(lX, dX, &lXdotdX);CHKERRQ(ierr);
      ierr = VecDotEnd(lX, dXold, &lXdotdXold);CHKERRQ(ierr);
      beta = PetscRealPart(dXdotdX / (lXdotdXold - lXdotdX));CHKERRQ(ierr);
      break;
    case SNES_NCG_CD: /* Conjugate Descent */
      ierr = VecDotBegin(dX, dX, &dXdotdX);CHKERRQ(ierr);
      ierr = VecDotBegin(lX, dXold, &lXdotdXold);CHKERRQ(ierr);
      ierr = VecDotEnd(dX, dX, &dXdotdX);CHKERRQ(ierr);
      ierr = VecDotEnd(lX, dXold, &lXdotdXold);CHKERRQ(ierr);
      beta = PetscRealPart(dXdotdX / lXdotdXold);CHKERRQ(ierr);
      break;
    }
    if (ncg->monitor) {
      ierr = PetscViewerASCIIPrintf(ncg->monitor, "beta = %e\n", beta);CHKERRQ(ierr);
    }
    ierr = VecAYPX(lX, beta, dX);CHKERRQ(ierr);
  }
  ierr = PetscInfo1(snes, "Maximum number of iterations has been reached: %D\n", maxits);CHKERRQ(ierr);
  if (!snes->reason) snes->reason = SNES_DIVERGED_MAX_IT;
  PetscFunctionReturn(0);
}


/*MC
  SNESNCG - Nonlinear Conjugate-Gradient method for the solution of general nonlinear systems.

  Level: beginner

  Options Database:
+   -snes_ncg_type <fr, prp, dy, hs, cd> - Choice of conjugate-gradient update parameter.
.   -snes_linesearch_type <cp,l2,basic> - Line search type.
-   -snes_ncg_monitor - Print relevant information about the ncg iteration.

Notes: This solves the nonlinear system of equations F(x) = 0 using the nonlinear generalization of the conjugate
gradient method.  This may be used with a nonlinear preconditioner used to pick the new search directions, but otherwise
chooses the initial search direction as F(x) for the initial guess x.


.seealso:  SNESCreate(), SNES, SNESSetType(), SNESNEWTONLS, SNESNEWTONTR, SNESNGMRES, SNESNQN
M*/
#undef __FUNCT__
#define __FUNCT__ "SNESCreate_NCG"
PETSC_EXTERN PetscErrorCode SNESCreate_NCG(SNES snes)
{
  PetscErrorCode ierr;
  SNES_NCG       * neP;

  PetscFunctionBegin;
  snes->ops->destroy        = SNESDestroy_NCG;
  snes->ops->setup          = SNESSetUp_NCG;
  snes->ops->setfromoptions = SNESSetFromOptions_NCG;
  snes->ops->view           = SNESView_NCG;
  snes->ops->solve          = SNESSolve_NCG;
  snes->ops->reset          = SNESReset_NCG;

  snes->usesksp = PETSC_FALSE;
  snes->usespc  = PETSC_TRUE;
  snes->pcside  = PC_LEFT;

  if (!snes->tolerancesset) {
    snes->max_funcs = 30000;
    snes->max_its   = 10000;
    snes->stol      = 1e-20;
  }

  ierr         = PetscNewLog(snes, SNES_NCG, &neP);CHKERRQ(ierr);
  snes->data   = (void*) neP;
  neP->monitor = NULL;
  neP->type    = SNES_NCG_PRP;
  ierr         = PetscObjectComposeFunction((PetscObject)snes,"SNESNCGSetType_C", SNESNCGSetType_NCG);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}