xref: /petsc/src/mat/impls/aij/mpi/mpiptap.c (revision e497d3c8e8f8bb645f0a9e4d8aea8ec4981f22b9)

/*
  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>
#include <petsctime.h>

#define PTAP_PROFILE

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) {
    Mat_Merge_SeqsToMPI *merge=ptap->merge;
    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() */

    ierr = MatDestroy(&ptap->Rd);CHKERRQ(ierr);
    ierr = MatDestroy(&ptap->Ro);CHKERRQ(ierr);
    ierr = MatDestroy(&ptap->AP);CHKERRQ(ierr);
    ierr = MatDestroy(&ptap->AP_loc);CHKERRQ(ierr);

    if (ptap->api) {ierr = PetscFree(ptap->api);CHKERRQ(ierr);}
    if (ptap->apj) {ierr = PetscFree(ptap->apj);CHKERRQ(ierr);}
    if (ptap->apa) {ierr = PetscFree(ptap->apa);CHKERRQ(ierr);}
    if (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_MPIAIJ          *a     = (Mat_MPIAIJ*)A->data;
  Mat_PtAPMPI         *ptap  = a->ptap;
  Mat_Merge_SeqsToMPI *merge = ptap->merge;

  PetscFunctionBegin;
  ierr = (*merge->duplicate)(A,op,M);CHKERRQ(ierr);

  (*M)->ops->destroy   = merge->destroy;
  (*M)->ops->duplicate = merge->duplicate;
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "MatPtAP_MPIAIJ_MPIAIJ"
PetscErrorCode MatPtAP_MPIAIJ_MPIAIJ(Mat A,Mat P,MatReuse scall,PetscReal fill,Mat *C)
{
  PetscErrorCode ierr;
  PetscBool      newalg=PETSC_FALSE;

  PetscFunctionBegin;
  ierr = PetscOptionsGetBool(NULL,"-matptap_new",&newalg,NULL);CHKERRQ(ierr);
  if (scall == MAT_INITIAL_MATRIX) {
    ierr = PetscLogEventBegin(MAT_PtAPSymbolic,A,P,0,0);CHKERRQ(ierr);
    if (newalg) {
      ierr = MatPtAPSymbolic_MPIAIJ_MPIAIJ_new(A,P,fill,C);CHKERRQ(ierr);
    } else {
      ierr = MatPtAPSymbolic_MPIAIJ_MPIAIJ(A,P,fill,C);CHKERRQ(ierr);
    }
    ierr = PetscLogEventEnd(MAT_PtAPSymbolic,A,P,0,0);CHKERRQ(ierr);
  }
  ierr = PetscLogEventBegin(MAT_PtAPNumeric,A,P,0,0);CHKERRQ(ierr);
  if (newalg) {
    ierr = MatPtAPNumeric_MPIAIJ_MPIAIJ_new(A,P,*C);CHKERRQ(ierr);
  } else {
    ierr = MatPtAPNumeric_MPIAIJ_MPIAIJ(A,P,*C);CHKERRQ(ierr);
  }
  ierr = PetscLogEventEnd(MAT_PtAPNumeric,A,P,0,0);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "MatPtAPSymbolic_MPIAIJ_MPIAIJ_new"
PetscErrorCode MatPtAPSymbolic_MPIAIJ_MPIAIJ_new(Mat A,Mat P,PetscReal fill,Mat *C)
{
  PetscErrorCode ierr;
  Mat_PtAPMPI    *ptap;
  Mat_MPIAIJ     *c,*p=(Mat_MPIAIJ*)P->data;

  PetscFunctionBegin;
  ierr = MatPtAPSymbolic_MPIAIJ_MPIAIJ(A,P,fill,C);CHKERRQ(ierr); //rm later !!!
  c    = (Mat_MPIAIJ*)(*C)->data;
  ptap = c->ptap;
  ptap->reuse = MAT_INITIAL_MATRIX;
  ierr = MatTranspose_SeqAIJ(p->A,MAT_INITIAL_MATRIX,&ptap->Rd);CHKERRQ(ierr);
  ierr = MatTranspose_SeqAIJ(p->B,MAT_INITIAL_MATRIX,&ptap->Ro);CHKERRQ(ierr);

  ierr = MatMatMult(A,P,MAT_INITIAL_MATRIX,2.0,&ptap->AP);CHKERRQ(ierr);
  ierr = MatMPIAIJGetLocalMat(ptap->AP,MAT_INITIAL_MATRIX,&ptap->AP_loc);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "MatPtAPNumeric_MPIAIJ_MPIAIJ_new"
PetscErrorCode MatPtAPNumeric_MPIAIJ_MPIAIJ_new(Mat A,Mat P,Mat C)
{
  PetscErrorCode    ierr;
  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_PtAPMPI       *ptap = c->ptap;
  Mat               AP_loc,C_loc,Co;
  PetscInt          i,rstart,rend,cm,ncols,row;
  PetscMPIInt       rank;
  MPI_Comm          comm;
  const PetscInt    *cols;
  const PetscScalar *vals;
  PetscLogDouble    t0,t1,t2,t3,t4,eR,eAP,eCseq,eCmpi;

  PetscFunctionBegin;
  ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
  ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);

  ierr = MatZeroEntries(C);CHKERRQ(ierr);

  /* 1) get R = Pd^T,Ro = Po^T */
  ierr = PetscTime(&t0);CHKERRQ(ierr);
  ierr = MatTranspose_SeqAIJ(p->A,MAT_REUSE_MATRIX,&ptap->Rd);CHKERRQ(ierr);
  ierr = MatTranspose_SeqAIJ(p->B,MAT_REUSE_MATRIX,&ptap->Ro);CHKERRQ(ierr);
  ierr = PetscTime(&t1);CHKERRQ(ierr);
  eR = t1 - t0;

  /* 2) get AP_loc */
  AP_loc = ptap->AP_loc;
  Mat_SeqAIJ    *ap=(Mat_SeqAIJ*)AP_loc->data;

  /* 2-1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
  /*-----------------------------------------------------*/
  if (ptap->reuse == MAT_INITIAL_MATRIX) {
    /* P_oth and P_loc are obtained in MatPtASymbolic(), skip calling MatGetBrowsOfAoCols() and MatMPIAIJGetLocalMat() */
    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);
  }

  /* 2-2) compute numeric C_seq = P_loc^T*A_loc*P - dominating part */
  /*--------------------------------------------------------------*/
  /* get data from symbolic products */
  Mat_SeqAIJ  *p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
  Mat_SeqAIJ  *p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
  PetscInt    *api,*apj,am = A->rmap->n,j,col,apnz;
  PetscScalar *apa = ptap->apa;

  api = ptap->api; apj = ptap->apj;
  for (i=0; i<am; i++) {
    /* AP[i,:] = A[i,:]*P = Ad*P_loc Ao*P_oth */
    AProw_nonscalable(i,ad,ao,p_loc,p_oth,apa);
    apnz = api[i+1] - api[i];
    for (j=0; j<apnz; j++) {
      col = apj[j+api[i]];
      //if (ap->a[j+ap->i[i]] != apa[col]) SETERRQ4(NULL,PETSC_ERR_ARG_INCOMP,"%d %d ap %g != apa %g",i,col,ap->a[j+ap->i[i]],apa[col]);
      ap->a[j+ap->i[i]] = apa[col];
      apa[col] = 0.0;
    }
  }

  ierr = PetscTime(&t2);CHKERRQ(ierr);
  eAP = t2 - t1;

  /* 3) C_loc = R*AP_loc, Co = Ro*AP_loc */
  ierr = MatMatMult(ptap->Rd,AP_loc,MAT_INITIAL_MATRIX,2.0,&C_loc);CHKERRQ(ierr);
  ierr = MatMatMult(ptap->Ro,AP_loc,MAT_INITIAL_MATRIX,2.0,&Co);CHKERRQ(ierr);
  //printf("[%d] Co %d, %d\n", rank,Co->rmap->N,Co->cmap->N);
  ierr = PetscTime(&t3);CHKERRQ(ierr);
  eCseq = t3 - t2;

  /* add C_loc and Co to to C */
  ierr = MatGetOwnershipRange(C,&rstart,&rend);CHKERRQ(ierr);

  /* C_loc -> C */
  cm = C_loc->rmap->N;
  for (i=0; i<cm; i++) {
    ierr = MatGetRow(C_loc,i,&ncols,&cols,&vals);CHKERRQ(ierr);
    row = rstart + i;
    ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
    ierr = MatRestoreRow(C_loc,i,&ncols,&cols,&vals);CHKERRQ(ierr);
  }

  /* Co -> C, off-processor part */
  //printf("[%d] p->B %d, %d\n",rank,p->B->rmap->N,p->B->cmap->N);
  for (i=0; i<Co->rmap->N; i++) {
    ierr = MatGetRow(Co,i,&ncols,&cols,&vals);CHKERRQ(ierr);
    row = p->garray[i];
    //printf("[%d] row[%d] = %d\n",rank,i,row);
    ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
    ierr = MatRestoreRow(Co,i,&ncols,&cols,&vals);CHKERRQ(ierr);
  }
  ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = PetscTime(&t4);CHKERRQ(ierr);
  eCmpi = t4 - t3;

  ierr = MatDestroy(&C_loc);CHKERRQ(ierr);
  ierr = MatDestroy(&Co);CHKERRQ(ierr);

  if (rank==1) {
    ierr = PetscPrintf(MPI_COMM_SELF," R %g, AP %g, Cseq %g, Cmpi %g = %g\n", eR,eAP,eCseq,eCmpi,eR+eAP+eCseq+eCmpi);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=NULL,current_space=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            *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;
  PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri,icompleted=0;
  PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
  PetscInt            len,proc,*dnz,*onz,*owners;
  PetscInt            nzi,*pti,*ptj;
  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            rmax;

  PetscFunctionBegin;
  ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);

  /* 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(&ptap);CHKERRQ(ierr);
  ierr        = PetscNew(&merge);CHKERRQ(ierr);
  ptap->merge = merge;
  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);

  /* get P_loc by taking all local rows of P */
  ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);

  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;

  /* (1) compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth (api,apj) */
  /*--------------------------------------------------------------------------*/
  ierr   = PetscMalloc1(am+1,&api);CHKERRQ(ierr);
  api[0] = 0;

  /* create and initialize a linked list */
  ierr = PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt);CHKERRQ(ierr);

  /* Initial FreeSpace size is fill*(nnz(A) + nnz(P)) -OOM for ex56, np=8k on Intrepid! */
  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++) {
    /* 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 = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
    }
    /* 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 = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
    }
    apnz     = lnk[0];
    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 = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,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      = PetscMalloc1(api[am]+1,&apj);CHKERRQ(ierr);
  ierr      = PetscFreeSpaceContiguous(&free_space,apj);CHKERRQ(ierr);
  afill_tmp = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1);
  if (afill_tmp > afill) afill = afill_tmp;

  /* (2) determine symbolic Co=(p->B)^T*AP - send to others (coi,coj)*/
  /*-----------------------------------------------------------------*/
  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   = PetscMalloc1(pon+1,&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);CHKERRQ(ierr);
  current_space = free_space;

  for (i=0; i<pon; i++) {
    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 = PetscLLCondensedAddSorted(apnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
    }
    nnz = lnk[0];

    /* 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 = PetscLLCondensedClean(pN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);

    current_space->array           += nnz;
    current_space->local_used      += nnz;
    current_space->local_remaining -= nnz;

    coi[i+1] = coi[i] + nnz;
  }

  ierr      = PetscMalloc1(coi[pon],&coj);CHKERRQ(ierr);
  ierr      = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
  afill_tmp = (PetscReal)coi[pon]/(poti[pon] + api[am]+1);
  if (afill_tmp > afill) afill = afill_tmp;
  ierr = MatRestoreSymbolicTranspose_SeqAIJ(p->B,&poti,&potj);CHKERRQ(ierr);

  /* (3) send j-array (coj) of Co to other processors */
  /*--------------------------------------------------*/
  ierr = PetscCalloc1(size,&merge->len_s);CHKERRQ(ierr);
  len_s        = merge->len_s;
  merge->nsend = 0;


  /* determine row ownership */
  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 = PetscMalloc2(size,&len_si,size,&sstatus);CHKERRQ(ierr);
  ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
  ierr = PetscMalloc1(size+2,&owners_co);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(=Pt*AP) to be sent to [proc] */
    len_s[proc] += coi[i+1] - coi[i]; /* num of nonzeros in Co to be sent to [proc] */
  }

  len          = 0; /* max length of buf_si[], see (4) */
  owners_co[0] = 0;
  for (proc=0; proc<size; proc++) {
    owners_co[proc+1] = owners_co[proc] + len_si[proc];
    if (len_s[proc]) {
      merge->nsend++;
      len_si[proc] = 2*(len_si[proc] + 1); /* length of buf_si to be sent to [proc] */
      len         += len_si[proc];
    }
  }

  /* determine the number and length of messages to receive for coi and coj  */
  ierr = PetscGatherNumberOfMessages(comm,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 = PetscMalloc1(merge->nsend+1,&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 */
  for (i=0; i<merge->nrecv; i++) {
    ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
  }
  ierr = PetscFree(rwaits);CHKERRQ(ierr);
  if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}

  /* (4) 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   = PetscMalloc1(len+1,&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; /* num of rows in Co to be sent to [proc] */
    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,&icompleted,&rstatus);CHKERRQ(ierr);
  }
  ierr = PetscFree(rwaits);CHKERRQ(ierr);
  if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}

  ierr = PetscFree2(len_si,sstatus);CHKERRQ(ierr);
  ierr = PetscFree(len_ri);CHKERRQ(ierr);
  ierr = PetscFree(swaits);CHKERRQ(ierr);
  ierr = PetscFree(buf_s);CHKERRQ(ierr);

  /* (5) compute the local portion of C (mpi mat) */
  /*----------------------------------------------*/
  ierr = MatGetSymbolicTranspose_SeqAIJ(p->A,&pdti,&pdtj);CHKERRQ(ierr);

  /* allocate pti array and free space for accumulating nonzero column info */
  ierr   = PetscMalloc1(pn+1,&pti);CHKERRQ(ierr);
  pti[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);CHKERRQ(ierr);
  current_space = free_space;

  ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&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 */
    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 = PetscLLCondensedAddSorted(apnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
    }

    /* 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 = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
        nextrow[k]++; nextci[k]++;
      }
    }
    nnz = lnk[0];

    /* 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 = PetscLLCondensedClean(pN,nnz,current_space->array,lnk,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;

    pti[i+1] = pti[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      = PetscMalloc1(pti[pn]+1,&ptj);CHKERRQ(ierr);
  ierr      = PetscFreeSpaceContiguous(&free_space,ptj);CHKERRQ(ierr);
  afill_tmp = (PetscReal)pti[pn]/(pi_loc[pm] + api[am]+1);
  if (afill_tmp > afill) afill = afill_tmp;
  ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);

  /* (6) create symbolic parallel matrix Cmpi */
  /*------------------------------------------*/
  ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
  ierr = MatSetSizes(Cmpi,pn,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
  ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(P->cmap->bs));CHKERRQ(ierr);
  ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
  ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
  ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);

  merge->bi        = pti;      /* Cseq->i */
  merge->bj        = ptj;      /* Cseq->j */
  merge->coi       = coi;      /* Co->i   */
  merge->coj       = coj;      /* Co->j   */
  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->rmax  = ap_rmax;
  *C          = Cmpi;

  /* flag 'scalable' determines which implementations to be used:
       0: do dense axpy in MatPtAPNumeric() - fast, but requires storage of a nonscalable dense array apa;
       1: do sparse axpy in MatPtAPNumeric() - might slow, uses a sparse array apa */
  /* set default scalable */
  ptap->scalable = PETSC_FALSE; //PETSC_TRUE;

  ierr = PetscOptionsGetBool(((PetscObject)Cmpi)->prefix,"-matptap_scalable",&ptap->scalable,NULL);CHKERRQ(ierr);
  if (!ptap->scalable) {  /* Do dense axpy */
    ierr = PetscCalloc1(pN,&ptap->apa);CHKERRQ(ierr);
  } else {
    ierr = PetscCalloc1(ap_rmax+1,&ptap->apa);CHKERRQ(ierr);
  }

#if defined(PETSC_USE_INFO)
  if (pti[pn] != 0) {
    ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
    ierr = PetscInfo1(Cmpi,"Use MatPtAP(A,P,MatReuse,%g,&C) for best performance.\n",(double)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_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;
  Mat_Merge_SeqsToMPI *merge;
  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;
  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;
  PetscBool           scalable;
#if defined(PTAP_PROFILE)
  PetscLogDouble t0,t1,t2,eP,t3,t4,et2_AP=0.0,ePtAP=0.0,t2_0,t2_1,t2_2;
#endif

  PetscFunctionBegin;
  ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
#if defined(PTAP_PROFILE)
  ierr = PetscTime(&t0);CHKERRQ(ierr);
#endif
  ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
  ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);

  ptap = c->ptap;
  if (!ptap) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_INCOMP,"MatPtAP() has not been called to create matrix C yet, cannot use MAT_REUSE_MATRIX");
  merge    = ptap->merge;
  apa      = ptap->apa;
  scalable = ptap->scalable;

  /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
  /*-----------------------------------------------------*/
  if (ptap->reuse == MAT_INITIAL_MATRIX) {
    /* P_oth and P_loc are obtained in MatPtASymbolic(), skip calling MatGetBrowsOfAoCols() and MatMPIAIJGetLocalMat() */
    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);
  }
#if defined(PTAP_PROFILE)
  ierr = PetscTime(&t1);CHKERRQ(ierr);
  eP = t1-t0;
#endif
  /*
  printf("[%d] Ad: %d, %d; Ao: %d, %d; P_loc: %d, %d; P_oth %d, %d;\n",rank,
         a->A->rmap->N,a->A->cmap->N,a->B->rmap->N,a->B->cmap->N,
         ptap->P_loc->rmap->N,ptap->P_loc->cmap->N,
         ptap->P_oth->rmap->N,ptap->P_oth->cmap->N);
   */

  /* 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 = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);

  bi     = merge->bi; bj = merge->bj;
  owners = merge->rowmap->range;
  ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);  /* ba: Cseq->a */

  api = ptap->api; apj = ptap->apj;

  if (!scalable) { /* Do dense axpy on apa (length of pN, stores A[i,:]*P) - nonscalable, but faster (could take 1/3 scalable time) */
    ierr = PetscInfo(C,"Using non-scalable dense axpy\n");CHKERRQ(ierr);
#if 0
    /* ------ 10x slower -------------- */
    /*==================================*/
    Mat         R = ptap->R;
    Mat_SeqAIJ  *r = (Mat_SeqAIJ*)R->data;
    PetscInt    *ri=r->i,*rj=r->j,rnz,arow,l,prow,pcol,pN=P->cmap->N;
    PetscScalar *ra=r->a,tmp,cdense[pN];

    ierr = PetscMemzero(cdense,pN*sizeof(PetscScalar));CHKERRQ(ierr);
    for (i=0; i<cm; i++) { /* each row of C or R */
      rnz = ri[i+1] - ri[i];

      for (j=0; j<rnz; j++) { /* each nz of R */
        arow = rj[ri[i] + j];

        /* diagonal portion of A */
        anz  = ad->i[arow+1] - ad->i[arow];
        for (k=0; k<anz; k++) { /* each nz of Ad */
          tmp  = ra[ri[i] + j]*ad->a[ad->i[arow] + k];
          prow = ad->j[ad->i[arow] + k];
          pnz  = pi_loc[prow+1] - pi_loc[prow];

          for (l=0; l<pnz; l++) { /* each nz of P_loc */
            pcol = pj_loc[pi_loc[prow] + l];
            cdense[pcol] += tmp*pa_loc[pi_loc[prow] + l];
          }
        }

        /* off-diagonal portion of A */
        anz  = ao->i[arow+1] - ao->i[arow];
        for (k=0; k<anz; k++) { /* each nz of Ao */
          tmp  = ra[ri[i] + j]*ao->a[ao->i[arow] + k];
          prow = ao->j[ao->i[arow] + k];
          pnz  = pi_oth[prow+1] - pi_oth[prow];

          for (l=0; l<pnz; l++) { /* each nz of P_oth */
            pcol = pj_oth[pi_oth[prow] + l];
            cdense[pcol] += tmp*pa_oth[pi_oth[prow] + l];
          }
        }

      } //for (j=0; j<rnz; j++)

      /* copy cdense[] into ca; zero cdense[] */
      cnz = bi[i+1] - bi[i];
      cj  = bj + bi[i];
      ca  = ba + bi[i];
      for (j=0; j<cnz; j++) {
        ca[j] += cdense[cj[j]];
        cdense[cj[j]] = 0.0;
      }
#if 0
      if (rank == 0) {
        printf("[%d] row %d: ",rank,i);
        for (j=0; j<pN; j++) printf(" %g,",cdense[j]);
        printf("\n");
      }
      for (j=0; j<pN; j++) cdense[j]=0.0; // zero cdnese[]
#endif
    } //for (i=0; i<cm; i++) {
#endif

    //==========================================

    ierr = PetscTime(&t1);CHKERRQ(ierr);
    for (i=0; i<am; i++) {
#if defined(PTAP_PROFILE)
      ierr = PetscTime(&t2_0);CHKERRQ(ierr);
#endif
      /* 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);
      }
#if defined(PTAP_PROFILE)
      ierr    = PetscTime(&t2_1);CHKERRQ(ierr);
      et2_AP += t2_1 - t2_0;
#endif

      /* 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);
      }
#if 1
      /* 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);
      }
#endif
      /* zero the current row of A*P */
      for (k=0; k<apnz; k++) apa[apJ[k]] = 0.0;
#if defined(PTAP_PROFILE)
      ierr      = PetscTime(&t2_2);CHKERRQ(ierr);
      ePtAP += t2_2 - t2_1;
#endif
    }

    if (rank == 100) {
    for (row=0; row<cm; row++) {
      printf("[%d] row %d: ",rank,row);
      cnz = bi[row+1] - bi[row];
      for (j=0; j<cnz; j++) printf(" %g,",ba[bi[row]+j]);
      printf("\n");
    }
    }

  } else { /* Do sparse axpy on apa (length of ap_rmax, stores A[i,:]*P) - scalable, but slower */
    ierr = PetscInfo(C,"Using scalable sparse axpy\n");CHKERRQ(ierr);
    /*-----------------------------------------------------------------------------------------*/
    pA=pa_loc;
    for (i=0; i<am; i++) {
#if defined(PTAP_PROFILE)
      ierr = PetscTime(&t2_0);CHKERRQ(ierr);
#endif
      /* 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);
      }
#if defined(PTAP_PROFILE)
      ierr    = PetscTime(&t2_1);CHKERRQ(ierr);
      et2_AP += t2_1 - t2_0;
#endif

      /* 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);
#if defined(PTAP_PROFILE)
      ierr      = PetscTime(&t2_2);CHKERRQ(ierr);
      ePtAP += t2_2 - t2_1;
#endif
    }
  }
#if defined(PTAP_PROFILE)
  ierr = PetscTime(&t2);CHKERRQ(ierr);
#endif

  /* 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,&s_waits,size,&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);
#if defined(PTAP_PROFILE)
  ierr = PetscTime(&t3);CHKERRQ(ierr);
#endif

  /* 4) insert local Cseq and received values into Cmpi */
  /*------------------------------------------------------*/
  ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&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);
#if defined(PTAP_PROFILE)
  ierr = PetscTime(&t4);CHKERRQ(ierr);
  if (rank==1) {
    ierr = PetscPrintf(MPI_COMM_SELF,"  [%d] PtAPNum %g/P + %g/PtAP( %g/A*P + %g/Pt*AP ) + %g/comm + %g/Cloc = %g\n\n",rank,eP,t2-t1,et2_AP,ePtAP,t3-t2,t4-t3,t4-t0);CHKERRQ(ierr);
  }
#endif
  PetscFunctionReturn(0);
}