xref: /petsc/src/mat/impls/aij/mpi/mpiptap.c (revision b7f45c76db6feae4085b9eefce882ba3e3973571)

/*
  Defines projective product routines where A is a MPIAIJ matrix
          C = P^T * A * P
*/

#include <../src/mat/impls/aij/seq/aij.h>   /*I "petscmat.h" I*/
#include <../src/mat/utils/freespace.h>
#include <../src/mat/impls/aij/mpi/mpiaij.h>
#include <petscbt.h>

/*
#define DEBUG_MATPTAP
 */

extern PetscErrorCode MatDestroy_MPIAIJ(Mat);
#undef __FUNCT__
#define __FUNCT__ "MatDestroy_MPIAIJ_PtAP"
PetscErrorCode MatDestroy_MPIAIJ_PtAP(Mat A)
{
  PetscErrorCode       ierr;
  Mat_MPIAIJ           *a=(Mat_MPIAIJ*)A->data;
  Mat_PtAPMPI          *ptap=a->ptap;

  PetscFunctionBegin;
  if (ptap){
    ierr = PetscFree2(ptap->startsj_s,ptap->startsj_r);CHKERRQ(ierr);
    ierr = PetscFree(ptap->bufa);CHKERRQ(ierr);
    ierr = MatDestroy(&ptap->P_loc);CHKERRQ(ierr);
    ierr = MatDestroy(&ptap->P_oth);CHKERRQ(ierr);
    ierr = MatDestroy(&ptap->A_loc);CHKERRQ(ierr); /* used by MatTransposeMatMult() */
    if (ptap->api){ierr = PetscFree(ptap->api);CHKERRQ(ierr);}
    if (ptap->apj){ierr = PetscFree(ptap->apj);CHKERRQ(ierr);}
  }
  if (ptap->merge) {
    Mat_Merge_SeqsToMPI  *merge=ptap->merge;
    ierr = PetscFree(merge->id_r);CHKERRQ(ierr);
    ierr = PetscFree(merge->len_s);CHKERRQ(ierr);
    ierr = PetscFree(merge->len_r);CHKERRQ(ierr);
    ierr = PetscFree(merge->bi);CHKERRQ(ierr);
    ierr = PetscFree(merge->bj);CHKERRQ(ierr);
    ierr = PetscFree(merge->buf_ri[0]);CHKERRQ(ierr);
    ierr = PetscFree(merge->buf_ri);CHKERRQ(ierr);
    ierr = PetscFree(merge->buf_rj[0]);CHKERRQ(ierr);
    ierr = PetscFree(merge->buf_rj);CHKERRQ(ierr);
    ierr = PetscFree(merge->coi);CHKERRQ(ierr);
    ierr = PetscFree(merge->coj);CHKERRQ(ierr);
    ierr = PetscFree(merge->owners_co);CHKERRQ(ierr);
    ierr = PetscLayoutDestroy(&merge->rowmap);CHKERRQ(ierr);
    ierr = merge->destroy(A);CHKERRQ(ierr);
    ierr = PetscFree(ptap->merge);CHKERRQ(ierr);
  }
  ierr = PetscFree(ptap);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "MatDuplicate_MPIAIJ_MatPtAP"
PetscErrorCode MatDuplicate_MPIAIJ_MatPtAP(Mat A, MatDuplicateOption op, Mat *M)
{
  PetscErrorCode       ierr;
  Mat_Merge_SeqsToMPI  *merge;
  PetscContainer       container;

  PetscFunctionBegin;
  ierr = PetscObjectQuery((PetscObject)A,"MatMergeSeqsToMPI",(PetscObject *)&container);CHKERRQ(ierr);
  if (container) {
    ierr  = PetscContainerGetPointer(container,(void **)&merge);CHKERRQ(ierr);
  } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Container does not exit");
  ierr = (*merge->duplicate)(A,op,M);CHKERRQ(ierr);
  (*M)->ops->destroy   = merge->destroy;   /* =MatDestroy_MPIAIJ, *M doesn't duplicate A's container! */
  (*M)->ops->duplicate = merge->duplicate; /* =MatDuplicate_ MPIAIJ */
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "MatPtAPSymbolic_MPIAIJ"
PetscErrorCode MatPtAPSymbolic_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat *C)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (!P->ops->ptapsymbolic_mpiaij) SETERRQ2(((PetscObject)A)->comm,PETSC_ERR_SUP,"Not implemented for A=%s and P=%s",((PetscObject)A)->type_name,((PetscObject)P)->type_name);
  ierr = (*P->ops->ptapsymbolic_mpiaij)(A,P,fill,C);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "MatPtAPNumeric_MPIAIJ"
PetscErrorCode MatPtAPNumeric_MPIAIJ(Mat A,Mat P,Mat C)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (!P->ops->ptapnumeric_mpiaij) SETERRQ2(((PetscObject)A)->comm,PETSC_ERR_SUP,"Not implemented for A=%s and P=%s",((PetscObject)A)->type_name,((PetscObject)P)->type_name);
  ierr = (*P->ops->ptapnumeric_mpiaij)(A,P,C);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "MatPtAPSymbolic_MPIAIJ_MPIAIJ"
PetscErrorCode MatPtAPSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat *C)
{
  PetscErrorCode       ierr;
  Mat                  Cmpi;
  Mat_PtAPMPI          *ptap;
  PetscFreeSpaceList   free_space=PETSC_NULL,current_space=PETSC_NULL;
  Mat_MPIAIJ           *a=(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data,*c;
  Mat_SeqAIJ           *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
  Mat_SeqAIJ           *p_loc,*p_oth;
  PetscInt             *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pdti,*pdtj,*poti,*potj,*ptJ;
  PetscInt             *adi=ad->i,*aj,*aoi=ao->i,nnz;
  PetscInt             nlnk,*lnk,*owners_co,*coi,*coj,i,k,pnz,row;
  PetscInt             am=A->rmap->n,pN=P->cmap->N,pm=P->rmap->n,pn=P->cmap->n;
  PetscBT              lnkbt;
  MPI_Comm             comm=((PetscObject)A)->comm;
  PetscMPIInt          size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
  PetscInt             **buf_rj,**buf_ri,**buf_ri_k;
  PetscInt             len,proc,*dnz,*onz,*owners;
  PetscInt             nzi,*bi,*bj;
  PetscInt             nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
  MPI_Request          *swaits,*rwaits;
  MPI_Status           *sstatus,rstatus;
  Mat_Merge_SeqsToMPI  *merge;
  PetscInt             *api,*apj,*Jptr,apnz,*prmap=p->garray,pon,nspacedouble=0,j,ap_rmax=0;
  PetscReal            afill=1.0,afill_tmp;
  PetscInt             rstart = P->cmap->rstart,rmax;
  PetscScalar          *vals;

  PetscFunctionBegin;
  /* check if matrix local sizes are compatible */
  if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend){
    SETERRQ4(comm,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, Arow (%D, %D) != Prow (%D,%D)",A->rmap->rstart,A->rmap->rend,P->rmap->rstart,P->rmap->rend);
  }
  if (A->cmap->rstart != P->rmap->rstart || A->cmap->rend != P->rmap->rend){
    SETERRQ4(comm,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, Acol (%D, %D) != Prow (%D,%D)",A->cmap->rstart,A->cmap->rend,P->rmap->rstart,P->rmap->rend);
  }

  ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
  ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);

  /* create struct Mat_PtAPMPI and attached it to C later */
  ierr = PetscNew(Mat_PtAPMPI,&ptap);CHKERRQ(ierr);
  ptap->reuse    = MAT_INITIAL_MATRIX;


  /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
  ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
#if defined(DEBUG_MATPTAP)
   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] P_oth is done...\n",rank);
#endif

  /* get P_loc by taking all local rows of P */
  ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
#if defined(DEBUG_MATPTAP)
  ierr = MPI_Barrier(comm);CHKERRQ(ierr);
  if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] P_loc is done...\n",rank);
#endif

  p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
  p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
  pi_loc = p_loc->i; pj_loc = p_loc->j;
  pi_oth = p_oth->i; pj_oth = p_oth->j;

  /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
  /*-------------------------------------------------------------------*/
  ierr = PetscMalloc((am+1)*sizeof(PetscInt),&api);CHKERRQ(ierr);
  api[0] = 0;

  /* create and initialize a linked list */
  nlnk = pN+1;
#if defined(DEBUG_MATPTAP)
  ierr = MPI_Barrier(comm);CHKERRQ(ierr);
  if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] call PetscLLCreate(), pN %D, P_rmax %D %D; Annz %D\n",rank,pN,p_loc->rmax,p_oth->rmax,adi[am]+aoi[am]);
#endif
  ierr = PetscLLCreate(pN,pN,nlnk,lnk,lnkbt);CHKERRQ(ierr);
#if defined(DEBUG_MATPTAP)
  ierr = MPI_Barrier(comm);CHKERRQ(ierr);
  if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] call PetscLLCreate() is done\n",rank);
#endif

  /* Initial FreeSpace size is fill*(nnz(A) + nnz(P)) */
  ierr = PetscFreeSpaceGet((PetscInt)(fill*(adi[am]+aoi[am]+pi_loc[pm])),&free_space);CHKERRQ(ierr);
  current_space = free_space;

  for (i=0; i<am; i++) {
    apnz = 0;
    /* diagonal portion of A */
    nzi = adi[i+1] - adi[i];
    aj  = ad->j + adi[i];
    for (j=0; j<nzi; j++){
      row  = aj[j];
      pnz  = pi_loc[row+1] - pi_loc[row];
      Jptr = pj_loc + pi_loc[row];
      /* add non-zero cols of P into the sorted linked list lnk */
      ierr = PetscLLAddSorted(pnz,Jptr,pN,nlnk,lnk,lnkbt);CHKERRQ(ierr);
      apnz += nlnk;
    }
    /* off-diagonal portion of A */
    nzi = aoi[i+1] - aoi[i];
    aj  = ao->j + aoi[i];
    for (j=0; j<nzi; j++){
      row = aj[j];
      pnz = pi_oth[row+1] - pi_oth[row];
      Jptr  = pj_oth + pi_oth[row];
      ierr = PetscLLAddSorted(pnz,Jptr,pN,nlnk,lnk,lnkbt);CHKERRQ(ierr);
      apnz += nlnk;
    }

    api[i+1] = api[i] + apnz;
    if (ap_rmax < apnz) ap_rmax = apnz;

    /* if free space is not available, double the total space in the list */
    if (current_space->local_remaining<apnz) {
      ierr = PetscFreeSpaceGet(apnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
      nspacedouble++;
    }

    /* Copy data into free space, then initialize lnk */
    ierr = PetscLLClean(pN,pN,apnz,lnk,current_space->array,lnkbt);CHKERRQ(ierr);
    current_space->array           += apnz;
    current_space->local_used      += apnz;
    current_space->local_remaining -= apnz;
  }
  /* Allocate space for apj, initialize apj, and */
  /* destroy list of free space and other temporary array(s) */
  ierr = PetscMalloc((api[am]+1)*sizeof(PetscInt),&apj);CHKERRQ(ierr);
  ierr = PetscFreeSpaceContiguous(&free_space,apj);CHKERRQ(ierr);
  afill_tmp = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]);
  if (afill_tmp > afill) afill = afill_tmp;

  /* determine symbolic Co=(p->B)^T*AP - send to others */
  /*----------------------------------------------------*/
  ierr = MatGetSymbolicTranspose_SeqAIJ(p->B,&poti,&potj);CHKERRQ(ierr);

  /* then, compute symbolic Co = (p->B)^T*AP */
  pon = (p->B)->cmap->n; /* total num of rows to be sent to other processors
                         >= (num of nonzero rows of C_seq) - pn */
  ierr = PetscMalloc((pon+1)*sizeof(PetscInt),&coi);CHKERRQ(ierr);
  coi[0] = 0;

  /* set initial free space to be fill*(nnz(p->B) + nnz(AP)) */
  nnz           = fill*(poti[pon] + api[am]);
  ierr          = PetscFreeSpaceGet(nnz,&free_space);
  current_space = free_space;

  for (i=0; i<pon; i++) {
    nnz = 0;
    pnz = poti[i+1] - poti[i];
    ptJ = potj + poti[i];
    for (j=0; j<pnz; j++){
      row  = ptJ[j]; /* row of AP == col of Pot */
      apnz = api[row+1] - api[row];
      Jptr = apj + api[row];
      /* add non-zero cols of AP into the sorted linked list lnk */
      ierr = PetscLLAddSorted(apnz,Jptr,pN,nlnk,lnk,lnkbt);CHKERRQ(ierr);
      nnz += nlnk;
    }

    /* If free space is not available, double the total space in the list */
    if (current_space->local_remaining<nnz) {
      ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
      nspacedouble++;
    }

    /* Copy data into free space, and zero out denserows */
    ierr = PetscLLClean(pN,pN,nnz,lnk,current_space->array,lnkbt);CHKERRQ(ierr);
    current_space->array           += nnz;
    current_space->local_used      += nnz;
    current_space->local_remaining -= nnz;
    coi[i+1] = coi[i] + nnz;
  }
  ierr = PetscMalloc((coi[pon]+1)*sizeof(PetscInt),&coj);CHKERRQ(ierr);
  ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
  afill_tmp = (PetscReal)coi[pon]/(poti[pon] + api[am]);
  if (afill_tmp > afill) afill = afill_tmp;
  ierr = MatRestoreSymbolicTranspose_SeqAIJ(p->B,&poti,&potj);CHKERRQ(ierr);

#if defined(DEBUG_MATPTAP)
  ierr = MPI_Barrier(comm);CHKERRQ(ierr);
  if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] local Co is done\n",rank);
#endif

  /* send j-array (coj) of Co to other processors */
  /*----------------------------------------------*/
  /* determine row ownership */
  ierr = PetscNew(Mat_Merge_SeqsToMPI,&merge);CHKERRQ(ierr);
  ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
  merge->rowmap->n = pn;
  merge->rowmap->bs = 1;
  ierr = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
  owners = merge->rowmap->range;

  /* determine the number of messages to send, their lengths */
  ierr = PetscMalloc(size*sizeof(PetscMPIInt),&len_si);CHKERRQ(ierr);
  ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
  ierr = PetscMalloc(size*sizeof(PetscMPIInt),&merge->len_s);CHKERRQ(ierr);
  len_s = merge->len_s;
  merge->nsend = 0;

  ierr = PetscMalloc((size+2)*sizeof(PetscInt),&owners_co);CHKERRQ(ierr);
  ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);

  proc = 0;
  for (i=0; i<pon; i++){
    while (prmap[i] >= owners[proc+1]) proc++;
    len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
    len_s[proc] += coi[i+1] - coi[i];
  }

  len   = 0;  /* max length of buf_si[] */
  owners_co[0] = 0;
  for (proc=0; proc<size; proc++){
    owners_co[proc+1] = owners_co[proc] + len_si[proc];
    if (len_si[proc]){
      merge->nsend++;
      len_si[proc] = 2*(len_si[proc] + 1);
      len += len_si[proc];
    }
  }

  /* determine the number and length of messages to receive for coi and coj  */
  ierr = PetscGatherNumberOfMessages(comm,PETSC_NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
  ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);

  /* post the Irecv and Isend of coj */
  ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
  ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
  ierr = PetscMalloc((merge->nsend+1)*sizeof(MPI_Request),&swaits);CHKERRQ(ierr);
  for (proc=0, k=0; proc<size; proc++){
    if (!len_s[proc]) continue;
    i = owners_co[proc];
    ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
    k++;
  }

  /* receives and sends of coj are complete */
  ierr = PetscMalloc(size*sizeof(MPI_Status),&sstatus);CHKERRQ(ierr);
  for (i=0; i<merge->nrecv; i++){
    ierr = MPI_Waitany(merge->nrecv,rwaits,&j,&rstatus);CHKERRQ(ierr);
  }
  ierr = PetscFree(rwaits);CHKERRQ(ierr);
  if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}

  /* send and recv coi */
  /*-------------------*/
  ierr = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
  ierr = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
  ierr = PetscMalloc((len+1)*sizeof(PetscInt),&buf_s);CHKERRQ(ierr);
  buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
  for (proc=0,k=0; proc<size; proc++){
    if (!len_s[proc]) continue;
    /* form outgoing message for i-structure:
         buf_si[0]:                 nrows to be sent
               [1:nrows]:           row index (global)
               [nrows+1:2*nrows+1]: i-structure index
    */
    /*-------------------------------------------*/
    nrows = len_si[proc]/2 - 1;
    buf_si_i    = buf_si + nrows+1;
    buf_si[0]   = nrows;
    buf_si_i[0] = 0;
    nrows = 0;
    for (i=owners_co[proc]; i<owners_co[proc+1]; i++){
      nzi = coi[i+1] - coi[i];
      buf_si_i[nrows+1] = buf_si_i[nrows] + nzi; /* i-structure */
      buf_si[nrows+1] =prmap[i] -owners[proc]; /* local row index */
      nrows++;
    }
    ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
    k++;
    buf_si += len_si[proc];
  }
  i = merge->nrecv;
  while (i--) {
    ierr = MPI_Waitany(merge->nrecv,rwaits,&j,&rstatus);CHKERRQ(ierr);
  }
  ierr = PetscFree(rwaits);CHKERRQ(ierr);
  if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
  /*
  ierr = PetscInfo2(A,"nsend: %d, nrecv: %d\n",merge->nsend,merge->nrecv);CHKERRQ(ierr);
  for (i=0; i<merge->nrecv; i++){
    ierr = PetscInfo3(A,"recv len_ri=%d, len_rj=%d from [%d]\n",len_ri[i],merge->len_r[i],merge->id_r[i]);CHKERRQ(ierr);
  }
  */
  ierr = PetscFree(len_si);CHKERRQ(ierr);
  ierr = PetscFree(len_ri);CHKERRQ(ierr);
  ierr = PetscFree(swaits);CHKERRQ(ierr);
  ierr = PetscFree(sstatus);CHKERRQ(ierr);
  ierr = PetscFree(buf_s);CHKERRQ(ierr);

#if defined(DEBUG_MATPTAP)
  ierr = MPI_Barrier(comm);CHKERRQ(ierr);
  if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] Co is done\n",rank);
#endif

  /* compute the local portion of C (mpi mat) */
  /*------------------------------------------*/
  ierr = MatGetSymbolicTranspose_SeqAIJ(p->A,&pdti,&pdtj);CHKERRQ(ierr);

  /* allocate bi array and free space for accumulating nonzero column info */
  ierr = PetscMalloc((pn+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
  bi[0] = 0;

  /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
  nnz           = fill*(pi_loc[pm] + api[am]);
  ierr          = PetscFreeSpaceGet(nnz,&free_space);
  current_space = free_space;

  ierr = PetscMalloc3(merge->nrecv,PetscInt**,&buf_ri_k,merge->nrecv,PetscInt*,&nextrow,merge->nrecv,PetscInt*,&nextci);CHKERRQ(ierr);
  for (k=0; k<merge->nrecv; k++){
    buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
    nrows       = *buf_ri_k[k];
    nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
    nextci[k]   = buf_ri_k[k] + (nrows + 1);/* poins to the next i-structure of k-th recved i-structure  */
  }
  ierr = MatPreallocateInitialize(comm,pn,pn,dnz,onz);CHKERRQ(ierr);
  rmax = 0;
  for (i=0; i<pn; i++) {
    /* add pdt[i,:]*AP into lnk */
    nnz = 0;
    pnz = pdti[i+1] - pdti[i];
    ptJ = pdtj + pdti[i];
    for (j=0; j<pnz; j++){
      row  = ptJ[j];  /* row of AP == col of Pt */
      apnz = api[row+1] - api[row];
      Jptr = apj + api[row];
      /* add non-zero cols of AP into the sorted linked list lnk */
      ierr = PetscLLAddSorted(apnz,Jptr,pN,nlnk,lnk,lnkbt);CHKERRQ(ierr);
      nnz += nlnk;
    }
    /* add received col data into lnk */
    for (k=0; k<merge->nrecv; k++){ /* k-th received message */
      if (i == *nextrow[k]) { /* i-th row */
        nzi = *(nextci[k]+1) - *nextci[k];
        Jptr  = buf_rj[k] + *nextci[k];
        ierr = PetscLLAddSorted(nzi,Jptr,pN,nlnk,lnk,lnkbt);CHKERRQ(ierr);
        nnz += nlnk;
        nextrow[k]++; nextci[k]++;
      }
    }

    /* if free space is not available, make more free space */
    if (current_space->local_remaining<nnz) {
      ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
      nspacedouble++;
    }
    /* copy data into free space, then initialize lnk */
    ierr = PetscLLClean(pN,pN,nnz,lnk,current_space->array,lnkbt);CHKERRQ(ierr);
    ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
    current_space->array           += nnz;
    current_space->local_used      += nnz;
    current_space->local_remaining -= nnz;
    bi[i+1] = bi[i] + nnz;
    if (nnz > rmax) rmax = nnz;
  }
  ierr = MatRestoreSymbolicTranspose_SeqAIJ(p->A,&pdti,&pdtj);CHKERRQ(ierr);
  ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);

  ierr = PetscMalloc((bi[pn]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
  ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
  afill_tmp = (PetscReal)bi[pn]/(pi_loc[pm] + api[am]);
  if (afill_tmp > afill) afill = afill_tmp;
  ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);

  /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part - check runex55_SA ?  */
  /*------------------------------------------------------------------------------------------------------*/
  ierr = PetscMalloc((rmax+1)*sizeof(PetscScalar),&vals);CHKERRQ(ierr);
  ierr = PetscMemzero(vals,rmax*sizeof(PetscScalar));CHKERRQ(ierr);

  ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
  ierr = MatSetSizes(Cmpi,pn,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
  ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
  ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
  ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
  ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
  for (i=0; i<pn; i++){
    row = i + rstart;
    nnz = bi[i+1] - bi[i];
    Jptr = bj + bi[i];
    ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
  }
  ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = PetscFree(vals);CHKERRQ(ierr);

  merge->bi            = bi;
  merge->bj            = bj;
  merge->coi           = coi;
  merge->coj           = coj;
  merge->buf_ri        = buf_ri;
  merge->buf_rj        = buf_rj;
  merge->owners_co     = owners_co;
  merge->destroy       = Cmpi->ops->destroy;
  merge->duplicate     = Cmpi->ops->duplicate;

  /* Cmpi is not ready for use - assembly will be done by MatPtAPNumeric() */
  /* Cmpi->assembled      = PETSC_FALSE; */
  Cmpi->ops->destroy   = MatDestroy_MPIAIJ_PtAP;
  Cmpi->ops->duplicate = MatDuplicate_MPIAIJ_MatPtAP;

  /* attach the supporting struct to Cmpi for reuse */
  c = (Mat_MPIAIJ*)Cmpi->data;
  c->ptap        = ptap;
  ptap->api      = api;
  ptap->apj      = apj;
  ptap->merge    = merge;
  ptap->rmax     = ap_rmax;

  *C = Cmpi;
#if defined(PETSC_USE_INFO)
  if (bi[pn] != 0) {
    ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %G needed %G.\n",nspacedouble,fill,afill);CHKERRQ(ierr);
    ierr = PetscInfo1(Cmpi,"Use MatPtAP(A,P,MatReuse,%G,&C) for best performance.\n",afill);CHKERRQ(ierr);
  } else {
    ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
  }
#endif
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "MatPtAPNumeric_MPIAIJ_MPIAIJ"
PetscErrorCode MatPtAPNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
{
  PetscErrorCode       ierr;
  Mat_Merge_SeqsToMPI  *merge;
  Mat_MPIAIJ           *a=(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
  Mat_SeqAIJ           *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
  Mat_SeqAIJ           *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
  Mat_SeqAIJ           *p_loc,*p_oth;
  Mat_PtAPMPI          *ptap;
  PetscInt             *adi=ad->i,*aoi=ao->i,*adj,*aoj,*apJ,nextp;
  PetscInt             *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pJ,*pj;
  PetscInt             i,j,k,anz,pnz,apnz,nextap,row,*cj;
  MatScalar            *ada,*aoa,*apa,*pa,*ca,*pa_loc,*pa_oth,valtmp;
  PetscInt             am=A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
  MPI_Comm             comm=((PetscObject)C)->comm;
  PetscMPIInt          size,rank,taga,*len_s;
  PetscInt             *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
  PetscInt             **buf_ri,**buf_rj;
  PetscInt             cnz=0,*bj_i,*bi,*bj,bnz,nextcj; /* bi,bj,ba: local array of C(mpi mat) */
  MPI_Request          *s_waits,*r_waits;
  MPI_Status           *status;
  MatScalar            **abuf_r,*ba_i,*pA,*coa,*ba;
  PetscInt             *api,*apj,*coi,*coj;
  PetscInt             *poJ=po->j,*pdJ=pd->j,pcstart=P->cmap->rstart,pcend=P->cmap->rend;
  PetscInt             sparse_axpy;

  PetscFunctionBegin;
  ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
  ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);

  ptap  = c->ptap;
  merge = ptap->merge;

  /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
  /*--------------------------------------------------*/
  if (ptap->reuse == MAT_INITIAL_MATRIX){
    ptap->reuse = MAT_REUSE_MATRIX;
  } else { /* update numerical values of P_oth and P_loc */
    ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
    ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
  }

  /* 2) compute numeric C_seq = P_loc^T*A_loc*P - dominating part */
  /*--------------------------------------------------------------*/
  /* get data from symbolic products */
  p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
  p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
  pi_loc=p_loc->i; pj_loc=p_loc->j; pJ=pj_loc; pa_loc=p_loc->a;
  pi_oth=p_oth->i; pj_oth=p_oth->j; pa_oth=p_oth->a;

  coi = merge->coi; coj = merge->coj;
  ierr = PetscMalloc((coi[pon]+1)*sizeof(MatScalar),&coa);CHKERRQ(ierr);
  ierr = PetscMemzero(coa,coi[pon]*sizeof(MatScalar));CHKERRQ(ierr);

  bi     = merge->bi; bj = merge->bj;
  owners = merge->rowmap->range;
  ierr   = PetscMalloc((bi[cm]+1)*sizeof(MatScalar),&ba);CHKERRQ(ierr);
  ierr   = PetscMemzero(ba,bi[cm]*sizeof(MatScalar));CHKERRQ(ierr);

  api = ptap->api; apj = ptap->apj;
  /* flag 'sparse_axpy' determines which implementations to be used:
       0: do dense axpy in MatPtAPNumeric() - fastest, but requires storage of a dense array apa; (default)
       1: do one sparse axpy - uses same memory as sparse_axpy=0 and might execute less flops
          (apnz vs. cnz in the outerproduct), slower than case '0' when cnz is not too large than apnz;
       2: do two sparse axpy in MatPtAPNumeric() - slowest, uses a sparse array apa */
  /* set default sparse_axpy */
  sparse_axpy = 0;
  ierr = PetscOptionsGetInt(PETSC_NULL,"-matptap_sparseaxpy",&sparse_axpy,PETSC_NULL);CHKERRQ(ierr);
  if (sparse_axpy == 0){  /* Do not perform sparse axpy */
    /*--------------------------------------------------*/
    /* malloc apa to store dense row A[i,:]*P */
    ierr = PetscMalloc((P->cmap->N)*sizeof(PetscScalar),&apa);CHKERRQ(ierr);
    ierr = PetscMemzero(apa,P->cmap->N*sizeof(PetscScalar));CHKERRQ(ierr);

    for (i=0; i<am; i++) {
      /* 2-a) form i-th sparse row of A_loc*P = Ad*P_loc + Ao*P_oth */
      /*------------------------------------------------------------*/
      apJ = apj + api[i];

      /* diagonal portion of A */
      anz = adi[i+1] - adi[i];
      adj = ad->j + adi[i];
      ada = ad->a + adi[i];
      for (j=0; j<anz; j++) {
        row = adj[j];
        pnz = pi_loc[row+1] - pi_loc[row];
        pj  = pj_loc + pi_loc[row];
        pa  = pa_loc + pi_loc[row];

        /* perform dense axpy */
        valtmp = ada[j];
        for (k=0; k<pnz; k++){
          apa[pj[k]] += valtmp*pa[k];
        }
        ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
      }

      /* off-diagonal portion of A */
      anz = aoi[i+1] - aoi[i];
      aoj = ao->j + aoi[i];
      aoa = ao->a + aoi[i];
      for (j=0; j<anz; j++) {
        row = aoj[j];
        pnz = pi_oth[row+1] - pi_oth[row];
        pj  = pj_oth + pi_oth[row];
        pa  = pa_oth + pi_oth[row];

        /* perform dense axpy */
        valtmp = aoa[j];
        for (k=0; k<pnz; k++){
          apa[pj[k]] += valtmp*pa[k];
        }
        ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
      }

      /* 2-b) Compute Cseq = P_loc[i,:]^T*AP[i,:] using outer product */
      /*--------------------------------------------------------------*/
      apnz = api[i+1] - api[i];
      /* put the value into Co=(p->B)^T*AP (off-diagonal part, send to others) */
      pnz = po->i[i+1] - po->i[i];
      poJ = po->j + po->i[i];
      pA  = po->a + po->i[i];
      for (j=0; j<pnz; j++){
        row = poJ[j];
        cnz = coi[row+1] - coi[row];
        cj  = coj + coi[row];
        ca  = coa + coi[row];
        /* perform dense axpy */
        valtmp = pA[j];
        for (k=0; k<cnz; k++) {
          ca[k] += valtmp*apa[cj[k]];
        }
        ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
      }

      /* put the value into Cd (diagonal part) */
      pnz = pd->i[i+1] - pd->i[i];
      pdJ = pd->j + pd->i[i];
      pA  = pd->a + pd->i[i];
      for (j=0; j<pnz; j++){
        row = pdJ[j];
        cnz = bi[row+1] - bi[row];
        cj  = bj + bi[row];
        ca  = ba + bi[row];
        /* perform dense axpy */
        valtmp = pA[j];
        for (k=0; k<cnz; k++) {
          ca[k] += valtmp*apa[cj[k]];
        }
        ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
      }

      /* zero the current row of A*P */
      for (k=0; k<apnz; k++) apa[apJ[k]] = 0.0;
    }
  } else if (sparse_axpy == 1){  /* Perform one sparse axpy */
    /*------------------------------------------------------*/
    /* malloc apa to store dense row A[i,:]*P */
    ierr = PetscMalloc((P->cmap->N)*sizeof(PetscScalar),&apa);CHKERRQ(ierr);
    ierr = PetscMemzero(apa,P->cmap->N*sizeof(PetscScalar));CHKERRQ(ierr);

    for (i=0; i<am; i++) {
      /* 2-a) form i-th sparse row of A_loc*P = Ad*P_loc + Ao*P_oth */
      /*------------------------------------------------------------*/
      apJ = apj + api[i];

      /* diagonal portion of A */
      anz = adi[i+1] - adi[i];
      adj = ad->j + adi[i];
      ada = ad->a + adi[i];
      for (j=0; j<anz; j++) {
        row = adj[j];
        pnz = pi_loc[row+1] - pi_loc[row];
        pj  = pj_loc + pi_loc[row];
        pa  = pa_loc + pi_loc[row];

        /* perform dense axpy */
        valtmp = ada[j];
        for (k=0; k<pnz; k++){
          apa[pj[k]] += valtmp*pa[k];
        }
        ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
      }

      /* off-diagonal portion of A */
      anz = aoi[i+1] - aoi[i];
      aoj = ao->j + aoi[i];
      aoa = ao->a + aoi[i];
      for (j=0; j<anz; j++) {
        row = aoj[j];
        pnz = pi_oth[row+1] - pi_oth[row];
        pj  = pj_oth + pi_oth[row];
        pa  = pa_oth + pi_oth[row];

        /* perform dense axpy */
        valtmp = aoa[j];
        for (k=0; k<pnz; k++){
          apa[pj[k]] += valtmp*pa[k];
        }
        ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
      }

      /* 2-b) Compute Cseq = P_loc[i,:]^T*AP[i,:] using outer product */
      /*--------------------------------------------------------------*/
      apnz = api[i+1] - api[i];
      /* put the value into Co=(p->B)^T*AP (off-diagonal part, send to others) */
      pnz = po->i[i+1] - po->i[i];
      poJ = po->j + po->i[i];
      pA  = po->a + po->i[i];
      for (j=0; j<pnz; j++){
        row    = poJ[j];
        cj     = coj + coi[row];
        ca     = coa + coi[row];
        valtmp = pA[j];
        /* perform sparse axpy */
        nextap = 0;
        for (k=0; nextap<apnz; k++) {
          if (cj[k]==apJ[nextap]) { /* global column index */
            ca[k] += valtmp*apa[cj[k]]; nextap++;
          }
        }
        ierr = PetscLogFlops(2.0*apnz);CHKERRQ(ierr);
      }

      /* put the value into Cd (diagonal part) */
      pnz = pd->i[i+1] - pd->i[i];
      pdJ = pd->j + pd->i[i];
      pA  = pd->a + pd->i[i];
      for (j=0; j<pnz; j++){
        row    = pdJ[j];
        cj     = bj + bi[row];
        ca     = ba + bi[row];
        valtmp = pA[j];
        /* perform sparse axpy */
        nextap = 0;
        for (k=0; nextap<apnz; k++) {
          if (cj[k]==apJ[nextap]) { /* global column index */
            ca[k] += valtmp*apa[cj[k]];
            nextap++;
          }
        }
        ierr = PetscLogFlops(2.0*apnz);CHKERRQ(ierr);
      }

      /* zero the current row of A*P */
      for (k=0; k<apnz; k++) apa[apJ[k]] = 0.0;
    }
  } else if (sparse_axpy == 2){/* Perform two sparse axpy */
    /*----------------------------------------------------*/
    /* malloc apa to store sparse row A[i,:]*P */
    ierr = PetscMalloc((ptap->rmax+1)*sizeof(MatScalar),&apa);CHKERRQ(ierr);
    ierr = PetscMemzero(apa,ptap->rmax*sizeof(MatScalar));CHKERRQ(ierr);

    pA=pa_loc;
    for (i=0; i<am; i++) {
      /* form i-th sparse row of A*P */
      apnz = api[i+1] - api[i];
      apJ  = apj + api[i];
      /* diagonal portion of A */
      anz = adi[i+1] - adi[i];
      adj = ad->j + adi[i];
      ada = ad->a + adi[i];
      for (j=0; j<anz; j++) {
        row = adj[j];
        pnz = pi_loc[row+1] - pi_loc[row];
        pj  = pj_loc + pi_loc[row];
        pa  = pa_loc + pi_loc[row];
        valtmp = ada[j];
        nextp  = 0;
        for (k=0; nextp<pnz; k++) {
          if (apJ[k] == pj[nextp]) { /* col of AP == col of P */
            apa[k] += valtmp*pa[nextp++];
          }
        }
        ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
      }
      /* off-diagonal portion of A */
      anz = aoi[i+1] - aoi[i];
      aoj = ao->j + aoi[i];
      aoa = ao->a + aoi[i];
      for (j=0; j<anz; j++) {
        row = aoj[j];
        pnz = pi_oth[row+1] - pi_oth[row];
        pj  = pj_oth + pi_oth[row];
        pa  = pa_oth + pi_oth[row];
        valtmp = aoa[j];
        nextp  = 0;
        for (k=0; nextp<pnz; k++) {
          if (apJ[k] == pj[nextp]) { /* col of AP == col of P */
            apa[k] += valtmp*pa[nextp++];
          }
        }
        ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
      }

      /* 2-b) Compute Cseq = P_loc[i,:]^T*AP[i,:] using outer product */
      /*--------------------------------------------------------------*/
      pnz = pi_loc[i+1] - pi_loc[i];
      pJ  = pj_loc + pi_loc[i];
      for (j=0; j<pnz; j++) {
        nextap = 0;
        row    = pJ[j]; /* global index */
        if (row < pcstart || row >=pcend) { /* put the value into Co */
          row = *poJ;
          cj  = coj + coi[row];
          ca  = coa + coi[row]; poJ++;
        } else {                            /* put the value into Cd */
          row  = *pdJ;
          cj   = bj + bi[row];
          ca   = ba + bi[row]; pdJ++;
        }
        valtmp = pA[j];
        for (k=0; nextap<apnz; k++) {
          if (cj[k]==apJ[nextap]) ca[k] += valtmp*apa[nextap++];
        }
        ierr = PetscLogFlops(2.0*apnz);CHKERRQ(ierr);
      }
      pA += pnz;
      /* zero the current row info for A*P */
      ierr = PetscMemzero(apa,apnz*sizeof(MatScalar));CHKERRQ(ierr);
    }
  } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"sparse_axpy only takes values 0, 1 and 2");
  ierr = PetscFree(apa);CHKERRQ(ierr);

  /* 3) send and recv matrix values coa */
  /*------------------------------------*/
  buf_ri = merge->buf_ri;
  buf_rj = merge->buf_rj;
  len_s  = merge->len_s;
  ierr = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
  ierr = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);

  ierr = PetscMalloc2(merge->nsend+1,MPI_Request,&s_waits,size,MPI_Status,&status);CHKERRQ(ierr);
  for (proc=0,k=0; proc<size; proc++){
    if (!len_s[proc]) continue;
    i = merge->owners_co[proc];
    ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
    k++;
  }
  if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
  if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}

  ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
  ierr = PetscFree(r_waits);CHKERRQ(ierr);
  ierr = PetscFree(coa);CHKERRQ(ierr);

  /* 4) insert local Cseq and received values into Cmpi */
  /*------------------------------------------------------*/
  ierr = PetscMalloc3(merge->nrecv,PetscInt**,&buf_ri_k,merge->nrecv,PetscInt*,&nextrow,merge->nrecv,PetscInt*,&nextci);CHKERRQ(ierr);
  for (k=0; k<merge->nrecv; k++){
    buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
    nrows       = *(buf_ri_k[k]);
    nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
    nextci[k]   = buf_ri_k[k] + (nrows + 1);/* poins to the next i-structure of k-th recved i-structure  */
  }

  for (i=0; i<cm; i++) {
    row = owners[rank] + i; /* global row index of C_seq */
    bj_i = bj + bi[i];  /* col indices of the i-th row of C */
    ba_i = ba + bi[i];
    bnz  = bi[i+1] - bi[i];
    /* add received vals into ba */
    for (k=0; k<merge->nrecv; k++){ /* k-th received message */
      /* i-th row */
      if (i == *nextrow[k]) {
        cnz = *(nextci[k]+1) - *nextci[k];
        cj  = buf_rj[k] + *(nextci[k]);
        ca  = abuf_r[k] + *(nextci[k]);
        nextcj = 0;
        for (j=0; nextcj<cnz; j++){
          if (bj_i[j] == cj[nextcj]){ /* bcol == ccol */
            ba_i[j] += ca[nextcj++];
          }
        }
        nextrow[k]++; nextci[k]++;
        ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
      }
    }
    ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
  }
  ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);

  ierr = PetscFree(ba);CHKERRQ(ierr);
  ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
  ierr = PetscFree(abuf_r);CHKERRQ(ierr);
  ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}