xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision d0609ced746bc51b019815ca91d747429db24893)

/*
  Defines matrix-matrix product routines for pairs of MPIAIJ matrices
          C = A * B
*/
#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 <../src/mat/impls/dense/mpi/mpidense.h>
#include <petsc/private/vecimpl.h>
#include <petsc/private/sfimpl.h>

#if defined(PETSC_HAVE_HYPRE)
PETSC_INTERN PetscErrorCode MatMatMultSymbolic_AIJ_AIJ_wHYPRE(Mat,Mat,PetscReal,Mat);
#endif

PETSC_INTERN PetscErrorCode MatProductSymbolic_AB_MPIAIJ_MPIAIJ(Mat C)
{
  Mat_Product         *product = C->product;
  Mat                 A=product->A,B=product->B;
  MatProductAlgorithm alg=product->alg;
  PetscReal           fill=product->fill;
  PetscBool           flg;

  PetscFunctionBegin;
  /* scalable */
  PetscCall(PetscStrcmp(alg,"scalable",&flg));
  if (flg) {
    PetscCall(MatMatMultSymbolic_MPIAIJ_MPIAIJ(A,B,fill,C));
    PetscFunctionReturn(0);
  }

  /* nonscalable */
  PetscCall(PetscStrcmp(alg,"nonscalable",&flg));
  if (flg) {
    PetscCall(MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(A,B,fill,C));
    PetscFunctionReturn(0);
  }

  /* seqmpi */
  PetscCall(PetscStrcmp(alg,"seqmpi",&flg));
  if (flg) {
    PetscCall(MatMatMultSymbolic_MPIAIJ_MPIAIJ_seqMPI(A,B,fill,C));
    PetscFunctionReturn(0);
  }

  /* backend general code */
  PetscCall(PetscStrcmp(alg,"backend",&flg));
  if (flg) {
    PetscCall(MatProductSymbolic_MPIAIJBACKEND(C));
    PetscFunctionReturn(0);
  }

#if defined(PETSC_HAVE_HYPRE)
  PetscCall(PetscStrcmp(alg,"hypre",&flg));
  if (flg) {
    PetscCall(MatMatMultSymbolic_AIJ_AIJ_wHYPRE(A,B,fill,C));
    PetscFunctionReturn(0);
  }
#endif
  SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_SUP,"Mat Product Algorithm is not supported");
}

PetscErrorCode MatDestroy_MPIAIJ_MatMatMult(void *data)
{
  Mat_APMPI      *ptap = (Mat_APMPI*)data;

  PetscFunctionBegin;
  PetscCall(PetscFree2(ptap->startsj_s,ptap->startsj_r));
  PetscCall(PetscFree(ptap->bufa));
  PetscCall(MatDestroy(&ptap->P_loc));
  PetscCall(MatDestroy(&ptap->P_oth));
  PetscCall(MatDestroy(&ptap->Pt));
  PetscCall(PetscFree(ptap->api));
  PetscCall(PetscFree(ptap->apj));
  PetscCall(PetscFree(ptap->apa));
  PetscCall(PetscFree(ptap));
  PetscFunctionReturn(0);
}

PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,Mat C)
{
  Mat_MPIAIJ        *a  =(Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
  Mat_SeqAIJ        *ad =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
  Mat_SeqAIJ        *cd =(Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
  PetscScalar       *cda=cd->a,*coa=co->a;
  Mat_SeqAIJ        *p_loc,*p_oth;
  PetscScalar       *apa,*ca;
  PetscInt          cm =C->rmap->n;
  Mat_APMPI         *ptap;
  PetscInt          *api,*apj,*apJ,i,k;
  PetscInt          cstart=C->cmap->rstart;
  PetscInt          cdnz,conz,k0,k1;
  const PetscScalar *dummy;
  MPI_Comm          comm;
  PetscMPIInt       size;

  PetscFunctionBegin;
  MatCheckProduct(C,3);
  ptap = (Mat_APMPI*)C->product->data;
  PetscCheck(ptap,PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be computed. Missing data");
  PetscCall(PetscObjectGetComm((PetscObject)A,&comm));
  PetscCallMPI(MPI_Comm_size(comm,&size));

  PetscCheckFalse(!ptap->P_oth && size>1,PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"AP cannot be reused. Do not call MatProductClear()");

  /* flag CPU mask for C */
#if defined(PETSC_HAVE_DEVICE)
  if (C->offloadmask != PETSC_OFFLOAD_UNALLOCATED) C->offloadmask = PETSC_OFFLOAD_CPU;
  if (c->A->offloadmask != PETSC_OFFLOAD_UNALLOCATED) c->A->offloadmask = PETSC_OFFLOAD_CPU;
  if (c->B->offloadmask != PETSC_OFFLOAD_UNALLOCATED) c->B->offloadmask = PETSC_OFFLOAD_CPU;
#endif

  /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
  /*-----------------------------------------------------*/
  /* update numerical values of P_oth and P_loc */
  PetscCall(MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth));
  PetscCall(MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc));

  /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
  /*----------------------------------------------------------*/
  /* get data from symbolic products */
  p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
  p_oth = NULL;
  if (size >1) {
    p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
  }

  /* get apa for storing dense row A[i,:]*P */
  apa = ptap->apa;

  api = ptap->api;
  apj = ptap->apj;
  /* trigger copy to CPU */
  PetscCall(MatSeqAIJGetArrayRead(a->A,&dummy));
  PetscCall(MatSeqAIJRestoreArrayRead(a->A,&dummy));
  PetscCall(MatSeqAIJGetArrayRead(a->B,&dummy));
  PetscCall(MatSeqAIJRestoreArrayRead(a->B,&dummy));
  for (i=0; i<cm; i++) {
    /* compute apa = A[i,:]*P */
    AProw_nonscalable(i,ad,ao,p_loc,p_oth,apa);

    /* set values in C */
    apJ  = apj + api[i];
    cdnz = cd->i[i+1] - cd->i[i];
    conz = co->i[i+1] - co->i[i];

    /* 1st off-diagonal part of C */
    ca = coa + co->i[i];
    k  = 0;
    for (k0=0; k0<conz; k0++) {
      if (apJ[k] >= cstart) break;
      ca[k0]      = apa[apJ[k]];
      apa[apJ[k++]] = 0.0;
    }

    /* diagonal part of C */
    ca = cda + cd->i[i];
    for (k1=0; k1<cdnz; k1++) {
      ca[k1]      = apa[apJ[k]];
      apa[apJ[k++]] = 0.0;
    }

    /* 2nd off-diagonal part of C */
    ca = coa + co->i[i];
    for (; k0<conz; k0++) {
      ca[k0]      = apa[apJ[k]];
      apa[apJ[k++]] = 0.0;
    }
  }
  PetscCall(MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY));
  PetscFunctionReturn(0);
}

PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,PetscReal fill,Mat C)
{
  MPI_Comm           comm;
  PetscMPIInt        size;
  Mat_APMPI          *ptap;
  PetscFreeSpaceList free_space=NULL,current_space=NULL;
  Mat_MPIAIJ         *a=(Mat_MPIAIJ*)A->data;
  Mat_SeqAIJ         *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
  PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
  PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
  PetscInt           *lnk,i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi;
  PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
  PetscBT            lnkbt;
  PetscReal          afill;
  MatType            mtype;

  PetscFunctionBegin;
  MatCheckProduct(C,4);
  PetscCheck(!C->product->data,PETSC_COMM_SELF,PETSC_ERR_PLIB,"Extra product struct not empty");
  PetscCall(PetscObjectGetComm((PetscObject)A,&comm));
  PetscCallMPI(MPI_Comm_size(comm,&size));

  /* create struct Mat_APMPI and attached it to C later */
  PetscCall(PetscNew(&ptap));

  /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
  PetscCall(MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth));

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

  p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
  pi_loc = p_loc->i; pj_loc = p_loc->j;
  if (size > 1) {
    p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
    pi_oth = p_oth->i; pj_oth = p_oth->j;
  } else {
    p_oth = NULL;
    pi_oth = NULL; pj_oth = NULL;
  }

  /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
  /*-------------------------------------------------------------------*/
  PetscCall(PetscMalloc1(am+2,&api));
  ptap->api = api;
  api[0]    = 0;

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

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

  MatPreallocateBegin(comm,am,pn,dnz,onz);
  for (i=0; i<am; i++) {
    /* diagonal portion of A */
    nzi = adi[i+1] - adi[i];
    for (j=0; j<nzi; j++) {
      row  = *adj++;
      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 */
      PetscCall(PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt));
    }
    /* off-diagonal portion of A */
    nzi = aoi[i+1] - aoi[i];
    for (j=0; j<nzi; j++) {
      row  = *aoj++;
      pnz  = pi_oth[row+1] - pi_oth[row];
      Jptr = pj_oth + pi_oth[row];
      PetscCall(PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt));
    }
    /* add possible missing diagonal entry */
    if (C->force_diagonals) {
      j = i + rstart; /* column index */
      PetscCall(PetscLLCondensedAddSorted(1,&j,lnk,lnkbt));
    }

    apnz     = lnk[0];
    api[i+1] = api[i] + apnz;

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

    /* Copy data into free space, then initialize lnk */
    PetscCall(PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt));
    PetscCall(MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz));

    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) */
  PetscCall(PetscMalloc1(api[am]+1,&ptap->apj));
  apj  = ptap->apj;
  PetscCall(PetscFreeSpaceContiguous(&free_space,ptap->apj));
  PetscCall(PetscLLDestroy(lnk,lnkbt));

  /* malloc apa to store dense row A[i,:]*P */
  PetscCall(PetscCalloc1(pN,&ptap->apa));

  /* set and assemble symbolic parallel matrix C */
  /*---------------------------------------------*/
  PetscCall(MatSetSizes(C,am,pn,PETSC_DETERMINE,PETSC_DETERMINE));
  PetscCall(MatSetBlockSizesFromMats(C,A,P));

  PetscCall(MatGetType(A,&mtype));
  PetscCall(MatSetType(C,mtype));
  PetscCall(MatMPIAIJSetPreallocation(C,0,dnz,0,onz));
  MatPreallocateEnd(dnz,onz);

  PetscCall(MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(C, apj, api));
  PetscCall(MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY));
  PetscCall(MatSetOption(C,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE));

  C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
  C->ops->productnumeric = MatProductNumeric_AB;

  /* attach the supporting struct to C for reuse */
  C->product->data    = ptap;
  C->product->destroy = MatDestroy_MPIAIJ_MatMatMult;

  /* set MatInfo */
  afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
  if (afill < 1.0) afill = 1.0;
  C->info.mallocs           = nspacedouble;
  C->info.fill_ratio_given  = fill;
  C->info.fill_ratio_needed = afill;

#if defined(PETSC_USE_INFO)
  if (api[am]) {
    PetscCall(PetscInfo(C,"Reallocs %" PetscInt_FMT "; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill));
    PetscCall(PetscInfo(C,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill));
  } else {
    PetscCall(PetscInfo(C,"Empty matrix product\n"));
  }
#endif
  PetscFunctionReturn(0);
}

/* ------------------------------------------------------- */
static PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat,Mat,PetscReal,Mat);
static PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat,Mat,Mat);

static PetscErrorCode MatProductSetFromOptions_MPIAIJ_MPIDense_AB(Mat C)
{
  Mat_Product *product = C->product;
  Mat         A = product->A,B=product->B;

  PetscFunctionBegin;
  if (A->cmap->rstart != B->rmap->rstart || A->cmap->rend != B->rmap->rend)
    SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, (%" PetscInt_FMT ", %" PetscInt_FMT ") != (%" PetscInt_FMT ",%" PetscInt_FMT ")",A->cmap->rstart,A->cmap->rend,B->rmap->rstart,B->rmap->rend);

  C->ops->matmultsymbolic = MatMatMultSymbolic_MPIAIJ_MPIDense;
  C->ops->productsymbolic = MatProductSymbolic_AB;
  PetscFunctionReturn(0);
}
/* -------------------------------------------------------------------- */
static PetscErrorCode MatProductSetFromOptions_MPIAIJ_MPIDense_AtB(Mat C)
{
  Mat_Product *product = C->product;
  Mat         A = product->A,B=product->B;

  PetscFunctionBegin;
  if (A->rmap->rstart != B->rmap->rstart || A->rmap->rend != B->rmap->rend)
    SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, (%" PetscInt_FMT ", %" PetscInt_FMT ") != (%" PetscInt_FMT ",%" PetscInt_FMT ")",A->rmap->rstart,A->rmap->rend,B->rmap->rstart,B->rmap->rend);

  C->ops->transposematmultsymbolic = MatTransposeMatMultSymbolic_MPIAIJ_MPIDense;
  C->ops->productsymbolic          = MatProductSymbolic_AtB;
  PetscFunctionReturn(0);
}

/* --------------------------------------------------------------------- */
PETSC_INTERN PetscErrorCode MatProductSetFromOptions_MPIAIJ_MPIDense(Mat C)
{
  Mat_Product    *product = C->product;

  PetscFunctionBegin;
  switch (product->type) {
  case MATPRODUCT_AB:
    PetscCall(MatProductSetFromOptions_MPIAIJ_MPIDense_AB(C));
    break;
  case MATPRODUCT_AtB:
    PetscCall(MatProductSetFromOptions_MPIAIJ_MPIDense_AtB(C));
    break;
  default:
    break;
  }
  PetscFunctionReturn(0);
}
/* ------------------------------------------------------- */

typedef struct {
  Mat          workB,workB1;
  MPI_Request  *rwaits,*swaits;
  PetscInt     nsends,nrecvs;
  MPI_Datatype *stype,*rtype;
  PetscInt     blda;
} MPIAIJ_MPIDense;

PetscErrorCode MatMPIAIJ_MPIDenseDestroy(void *ctx)
{
  MPIAIJ_MPIDense *contents = (MPIAIJ_MPIDense*)ctx;
  PetscInt        i;

  PetscFunctionBegin;
  PetscCall(MatDestroy(&contents->workB));
  PetscCall(MatDestroy(&contents->workB1));
  for (i=0; i<contents->nsends; i++) {
    PetscCallMPI(MPI_Type_free(&contents->stype[i]));
  }
  for (i=0; i<contents->nrecvs; i++) {
    PetscCallMPI(MPI_Type_free(&contents->rtype[i]));
  }
  PetscCall(PetscFree4(contents->stype,contents->rtype,contents->rwaits,contents->swaits));
  PetscCall(PetscFree(contents));
  PetscFunctionReturn(0);
}

static PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat A,Mat B,PetscReal fill,Mat C)
{
  Mat_MPIAIJ      *aij=(Mat_MPIAIJ*)A->data;
  PetscInt        nz=aij->B->cmap->n,nsends,nrecvs,i,nrows_to,j,blda,m,M,n,N;
  MPIAIJ_MPIDense *contents;
  VecScatter      ctx=aij->Mvctx;
  PetscInt        Am=A->rmap->n,Bm=B->rmap->n,BN=B->cmap->N,Bbn,Bbn1,bs,nrows_from,numBb;
  MPI_Comm        comm;
  MPI_Datatype    type1,*stype,*rtype;
  const PetscInt  *sindices,*sstarts,*rstarts;
  PetscMPIInt     *disp;
  PetscBool       cisdense;

  PetscFunctionBegin;
  MatCheckProduct(C,4);
  PetscCheck(!C->product->data,PetscObjectComm((PetscObject)C),PETSC_ERR_PLIB,"Product data not empty");
  PetscCall(PetscObjectGetComm((PetscObject)A,&comm));
  PetscCall(PetscObjectBaseTypeCompare((PetscObject)C,MATMPIDENSE,&cisdense));
  if (!cisdense) {
    PetscCall(MatSetType(C,((PetscObject)B)->type_name));
  }
  PetscCall(MatGetLocalSize(C,&m,&n));
  PetscCall(MatGetSize(C,&M,&N));
  if (m == PETSC_DECIDE || n == PETSC_DECIDE || M == PETSC_DECIDE || N == PETSC_DECIDE) {
    PetscCall(MatSetSizes(C,Am,B->cmap->n,A->rmap->N,BN));
  }
  PetscCall(MatSetBlockSizesFromMats(C,A,B));
  PetscCall(MatSetUp(C));
  PetscCall(MatDenseGetLDA(B,&blda));
  PetscCall(PetscNew(&contents));

  PetscCall(VecScatterGetRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,NULL,NULL));
  PetscCall(VecScatterGetRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,NULL,NULL));

  /* Create column block of B and C for memory scalability when BN is too large */
  /* Estimate Bbn, column size of Bb */
  if (nz) {
    Bbn1 = 2*Am*BN/nz;
    if (!Bbn1) Bbn1 = 1;
  } else Bbn1 = BN;

  bs = PetscAbs(B->cmap->bs);
  Bbn1 = Bbn1/bs *bs; /* Bbn1 is a multiple of bs */
  if (Bbn1 > BN) Bbn1 = BN;
  PetscCallMPI(MPI_Allreduce(&Bbn1,&Bbn,1,MPIU_INT,MPI_MAX,comm));

  /* Enable runtime option for Bbn */
  PetscOptionsBegin(comm,((PetscObject)C)->prefix,"MatMatMult","Mat");
  PetscCall(PetscOptionsInt("-matmatmult_Bbn","Number of columns in Bb","MatMatMult",Bbn,&Bbn,NULL));
  PetscOptionsEnd();
  Bbn  = PetscMin(Bbn,BN);

  if (Bbn > 0 && Bbn < BN) {
    numBb = BN/Bbn;
    Bbn1 = BN - numBb*Bbn;
  } else numBb = 0;

  if (numBb) {
    PetscCall(PetscInfo(C,"use Bb, BN=%" PetscInt_FMT ", Bbn=%" PetscInt_FMT "; numBb=%" PetscInt_FMT "\n",BN,Bbn,numBb));
    if (Bbn1) { /* Create workB1 for the remaining columns */
      PetscCall(PetscInfo(C,"use Bb1, BN=%" PetscInt_FMT ", Bbn1=%" PetscInt_FMT "\n",BN,Bbn1));
      /* Create work matrix used to store off processor rows of B needed for local product */
      PetscCall(MatCreateSeqDense(PETSC_COMM_SELF,nz,Bbn1,NULL,&contents->workB1));
    } else contents->workB1 = NULL;
  }

  /* Create work matrix used to store off processor rows of B needed for local product */
  PetscCall(MatCreateSeqDense(PETSC_COMM_SELF,nz,Bbn,NULL,&contents->workB));

  /* Use MPI derived data type to reduce memory required by the send/recv buffers */
  PetscCall(PetscMalloc4(nsends,&stype,nrecvs,&rtype,nrecvs,&contents->rwaits,nsends,&contents->swaits));
  contents->stype  = stype;
  contents->nsends = nsends;

  contents->rtype  = rtype;
  contents->nrecvs = nrecvs;
  contents->blda   = blda;

  PetscCall(PetscMalloc1(Bm+1,&disp));
  for (i=0; i<nsends; i++) {
    nrows_to = sstarts[i+1]-sstarts[i];
    for (j=0; j<nrows_to; j++) disp[j] = sindices[sstarts[i]+j]; /* rowB to be sent */
    PetscCallMPI(MPI_Type_create_indexed_block(nrows_to,1,disp,MPIU_SCALAR,&type1));
    PetscCallMPI(MPI_Type_create_resized(type1,0,blda*sizeof(PetscScalar),&stype[i]));
    PetscCallMPI(MPI_Type_commit(&stype[i]));
    PetscCallMPI(MPI_Type_free(&type1));
  }

  for (i=0; i<nrecvs; i++) {
    /* received values from a process form a (nrows_from x Bbn) row block in workB (column-wise) */
    nrows_from = rstarts[i+1]-rstarts[i];
    disp[0] = 0;
    PetscCallMPI(MPI_Type_create_indexed_block(1,nrows_from,disp,MPIU_SCALAR,&type1));
    PetscCallMPI(MPI_Type_create_resized(type1,0,nz*sizeof(PetscScalar),&rtype[i]));
    PetscCallMPI(MPI_Type_commit(&rtype[i]));
    PetscCallMPI(MPI_Type_free(&type1));
  }

  PetscCall(PetscFree(disp));
  PetscCall(VecScatterRestoreRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,NULL,NULL));
  PetscCall(VecScatterRestoreRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,NULL,NULL));
  PetscCall(MatSetOption(C,MAT_NO_OFF_PROC_ENTRIES,PETSC_TRUE));
  PetscCall(MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY));
  PetscCall(MatSetOption(C,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE));

  C->product->data = contents;
  C->product->destroy = MatMPIAIJ_MPIDenseDestroy;
  C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
  PetscFunctionReturn(0);
}

PETSC_INTERN PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat,Mat,Mat,const PetscBool);
/*
    Performs an efficient scatter on the rows of B needed by this process; this is
    a modification of the VecScatterBegin_() routines.

    Input: Bbidx = 0: B = Bb
                 = 1: B = Bb1, see MatMatMultSymbolic_MPIAIJ_MPIDense()
*/
PetscErrorCode MatMPIDenseScatter(Mat A,Mat B,PetscInt Bbidx,Mat C,Mat *outworkB)
{
  Mat_MPIAIJ        *aij = (Mat_MPIAIJ*)A->data;
  const PetscScalar *b;
  PetscScalar       *rvalues;
  VecScatter        ctx = aij->Mvctx;
  const PetscInt    *sindices,*sstarts,*rstarts;
  const PetscMPIInt *sprocs,*rprocs;
  PetscInt          i,nsends,nrecvs;
  MPI_Request       *swaits,*rwaits;
  MPI_Comm          comm;
  PetscMPIInt       tag=((PetscObject)ctx)->tag,ncols=B->cmap->N,nrows=aij->B->cmap->n,nsends_mpi,nrecvs_mpi;
  MPIAIJ_MPIDense   *contents;
  Mat               workB;
  MPI_Datatype      *stype,*rtype;
  PetscInt          blda;

  PetscFunctionBegin;
  MatCheckProduct(C,4);
  PetscCheck(C->product->data,PetscObjectComm((PetscObject)C),PETSC_ERR_PLIB,"Product data empty");
  contents = (MPIAIJ_MPIDense*)C->product->data;
  PetscCall(VecScatterGetRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,&sprocs,NULL/*bs*/));
  PetscCall(VecScatterGetRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,&rprocs,NULL/*bs*/));
  PetscCall(PetscMPIIntCast(nsends,&nsends_mpi));
  PetscCall(PetscMPIIntCast(nrecvs,&nrecvs_mpi));
  if (Bbidx == 0) workB = *outworkB = contents->workB;
  else workB = *outworkB = contents->workB1;
  PetscCheck(nrows == workB->rmap->n,PETSC_COMM_SELF,PETSC_ERR_PLIB,"Number of rows of workB %" PetscInt_FMT " not equal to columns of aij->B %d",workB->cmap->n,nrows);
  swaits = contents->swaits;
  rwaits = contents->rwaits;

  PetscCall(MatDenseGetArrayRead(B,&b));
  PetscCall(MatDenseGetLDA(B,&blda));
  PetscCheck(blda == contents->blda,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Cannot reuse an input matrix with lda %" PetscInt_FMT " != %" PetscInt_FMT,blda,contents->blda);
  PetscCall(MatDenseGetArray(workB,&rvalues));

  /* Post recv, use MPI derived data type to save memory */
  PetscCall(PetscObjectGetComm((PetscObject)C,&comm));
  rtype = contents->rtype;
  for (i=0; i<nrecvs; i++) {
    PetscCallMPI(MPI_Irecv(rvalues+(rstarts[i]-rstarts[0]),ncols,rtype[i],rprocs[i],tag,comm,rwaits+i));
  }

  stype = contents->stype;
  for (i=0; i<nsends; i++) {
    PetscCallMPI(MPI_Isend(b,ncols,stype[i],sprocs[i],tag,comm,swaits+i));
  }

  if (nrecvs) PetscCallMPI(MPI_Waitall(nrecvs_mpi,rwaits,MPI_STATUSES_IGNORE));
  if (nsends) PetscCallMPI(MPI_Waitall(nsends_mpi,swaits,MPI_STATUSES_IGNORE));

  PetscCall(VecScatterRestoreRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,&sprocs,NULL));
  PetscCall(VecScatterRestoreRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,&rprocs,NULL));
  PetscCall(MatDenseRestoreArrayRead(B,&b));
  PetscCall(MatDenseRestoreArray(workB,&rvalues));
  PetscFunctionReturn(0);
}

static PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat A,Mat B,Mat C)
{
  Mat_MPIAIJ      *aij    = (Mat_MPIAIJ*)A->data;
  Mat_MPIDense    *bdense = (Mat_MPIDense*)B->data;
  Mat_MPIDense    *cdense = (Mat_MPIDense*)C->data;
  Mat             workB;
  MPIAIJ_MPIDense *contents;

  PetscFunctionBegin;
  MatCheckProduct(C,3);
  PetscCheck(C->product->data,PetscObjectComm((PetscObject)C),PETSC_ERR_PLIB,"Product data empty");
  contents = (MPIAIJ_MPIDense*)C->product->data;
  /* diagonal block of A times all local rows of B */
  /* TODO: this calls a symbolic multiplication every time, which could be avoided */
  PetscCall(MatMatMult(aij->A,bdense->A,MAT_REUSE_MATRIX,PETSC_DEFAULT,&cdense->A));
  if (contents->workB->cmap->n == B->cmap->N) {
    /* get off processor parts of B needed to complete C=A*B */
    PetscCall(MatMPIDenseScatter(A,B,0,C,&workB));

    /* off-diagonal block of A times nonlocal rows of B */
    PetscCall(MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A,PETSC_TRUE));
  } else {
    Mat       Bb,Cb;
    PetscInt  BN=B->cmap->N,n=contents->workB->cmap->n,i;
    PetscBool ccpu;

    PetscCheck(n > 0,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Column block size %" PetscInt_FMT " must be positive",n);
    /* Prevent from unneeded copies back and forth from the GPU
       when getting and restoring the submatrix
       We need a proper GPU code for AIJ * dense in parallel */
    PetscCall(MatBoundToCPU(C,&ccpu));
    PetscCall(MatBindToCPU(C,PETSC_TRUE));
    for (i=0; i<BN; i+=n) {
      PetscCall(MatDenseGetSubMatrix(B,i,PetscMin(i+n,BN),&Bb));
      PetscCall(MatDenseGetSubMatrix(C,i,PetscMin(i+n,BN),&Cb));

      /* get off processor parts of B needed to complete C=A*B */
      PetscCall(MatMPIDenseScatter(A,Bb,(i+n)>BN,C,&workB));

      /* off-diagonal block of A times nonlocal rows of B */
      cdense = (Mat_MPIDense*)Cb->data;
      PetscCall(MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A,PETSC_TRUE));
      PetscCall(MatDenseRestoreSubMatrix(B,&Bb));
      PetscCall(MatDenseRestoreSubMatrix(C,&Cb));
    }
    PetscCall(MatBindToCPU(C,ccpu));
  }
  PetscFunctionReturn(0);
}

PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
{
  Mat_MPIAIJ        *a   = (Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
  Mat_SeqAIJ        *ad  = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
  Mat_SeqAIJ        *cd  = (Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
  PetscInt          *adi = ad->i,*adj,*aoi=ao->i,*aoj;
  PetscScalar       *ada,*aoa,*cda=cd->a,*coa=co->a;
  Mat_SeqAIJ        *p_loc,*p_oth;
  PetscInt          *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
  PetscScalar       *pa_loc,*pa_oth,*pa,valtmp,*ca;
  PetscInt          cm    = C->rmap->n,anz,pnz;
  Mat_APMPI         *ptap;
  PetscScalar       *apa_sparse;
  const PetscScalar *dummy;
  PetscInt          *api,*apj,*apJ,i,j,k,row;
  PetscInt          cstart = C->cmap->rstart;
  PetscInt          cdnz,conz,k0,k1,nextp;
  MPI_Comm          comm;
  PetscMPIInt       size;

  PetscFunctionBegin;
  MatCheckProduct(C,3);
  ptap = (Mat_APMPI*)C->product->data;
  PetscCheck(ptap,PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be computed. Missing data");
  PetscCall(PetscObjectGetComm((PetscObject)C,&comm));
  PetscCallMPI(MPI_Comm_size(comm,&size));
  PetscCheckFalse(!ptap->P_oth && size>1,PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"AP cannot be reused. Do not call MatProductClear()");

  /* flag CPU mask for C */
#if defined(PETSC_HAVE_DEVICE)
  if (C->offloadmask != PETSC_OFFLOAD_UNALLOCATED) C->offloadmask = PETSC_OFFLOAD_CPU;
  if (c->A->offloadmask != PETSC_OFFLOAD_UNALLOCATED) c->A->offloadmask = PETSC_OFFLOAD_CPU;
  if (c->B->offloadmask != PETSC_OFFLOAD_UNALLOCATED) c->B->offloadmask = PETSC_OFFLOAD_CPU;
#endif
  apa_sparse = ptap->apa;

  /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
  /*-----------------------------------------------------*/
  /* update numerical values of P_oth and P_loc */
  PetscCall(MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth));
  PetscCall(MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc));

  /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
  /*----------------------------------------------------------*/
  /* get data from symbolic products */
  p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
  pi_loc = p_loc->i; pj_loc = p_loc->j; pa_loc = p_loc->a;
  if (size >1) {
    p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
    pi_oth = p_oth->i; pj_oth = p_oth->j; pa_oth = p_oth->a;
  } else {
    p_oth = NULL; pi_oth = NULL; pj_oth = NULL; pa_oth = NULL;
  }

  /* trigger copy to CPU */
  PetscCall(MatSeqAIJGetArrayRead(a->A,&dummy));
  PetscCall(MatSeqAIJRestoreArrayRead(a->A,&dummy));
  PetscCall(MatSeqAIJGetArrayRead(a->B,&dummy));
  PetscCall(MatSeqAIJRestoreArrayRead(a->B,&dummy));
  api = ptap->api;
  apj = ptap->apj;
  for (i=0; i<cm; 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];
      /* perform sparse axpy */
      valtmp = ada[j];
      nextp  = 0;
      for (k=0; nextp<pnz; k++) {
        if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
          apa_sparse[k] += valtmp*pa[nextp++];
        }
      }
      PetscCall(PetscLogFlops(2.0*pnz));
    }

    /* 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 sparse axpy */
      valtmp = aoa[j];
      nextp  = 0;
      for (k=0; nextp<pnz; k++) {
        if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
          apa_sparse[k] += valtmp*pa[nextp++];
        }
      }
      PetscCall(PetscLogFlops(2.0*pnz));
    }

    /* set values in C */
    cdnz = cd->i[i+1] - cd->i[i];
    conz = co->i[i+1] - co->i[i];

    /* 1st off-diagonal part of C */
    ca = coa + co->i[i];
    k  = 0;
    for (k0=0; k0<conz; k0++) {
      if (apJ[k] >= cstart) break;
      ca[k0]        = apa_sparse[k];
      apa_sparse[k] = 0.0;
      k++;
    }

    /* diagonal part of C */
    ca = cda + cd->i[i];
    for (k1=0; k1<cdnz; k1++) {
      ca[k1]        = apa_sparse[k];
      apa_sparse[k] = 0.0;
      k++;
    }

    /* 2nd off-diagonal part of C */
    ca = coa + co->i[i];
    for (; k0<conz; k0++) {
      ca[k0]        = apa_sparse[k];
      apa_sparse[k] = 0.0;
      k++;
    }
  }
  PetscCall(MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY));
  PetscFunctionReturn(0);
}

/* same as MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(), except using LLCondensed to avoid O(BN) memory requirement */
PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat C)
{
  MPI_Comm           comm;
  PetscMPIInt        size;
  Mat_APMPI          *ptap;
  PetscFreeSpaceList free_space = NULL,current_space=NULL;
  Mat_MPIAIJ         *a  = (Mat_MPIAIJ*)A->data;
  Mat_SeqAIJ         *ad = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
  PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
  PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
  PetscInt           i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi,*lnk,apnz_max=1;
  PetscInt           am=A->rmap->n,pn=P->cmap->n,pm=P->rmap->n,lsize=pn+20;
  PetscReal          afill;
  MatType            mtype;

  PetscFunctionBegin;
  MatCheckProduct(C,4);
  PetscCheck(!C->product->data,PETSC_COMM_SELF,PETSC_ERR_PLIB,"Extra product struct not empty");
  PetscCall(PetscObjectGetComm((PetscObject)A,&comm));
  PetscCallMPI(MPI_Comm_size(comm,&size));

  /* create struct Mat_APMPI and attached it to C later */
  PetscCall(PetscNew(&ptap));

  /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
  PetscCall(MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth));

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

  p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
  pi_loc = p_loc->i; pj_loc = p_loc->j;
  if (size > 1) {
    p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
    pi_oth = p_oth->i; pj_oth = p_oth->j;
  } else {
    p_oth  = NULL;
    pi_oth = NULL; pj_oth = NULL;
  }

  /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
  /*-------------------------------------------------------------------*/
  PetscCall(PetscMalloc1(am+2,&api));
  ptap->api = api;
  api[0]    = 0;

  PetscCall(PetscLLCondensedCreate_Scalable(lsize,&lnk));

  /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
  PetscCall(PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space));
  current_space = free_space;
  MatPreallocateBegin(comm,am,pn,dnz,onz);
  for (i=0; i<am; i++) {
    /* diagonal portion of A */
    nzi = adi[i+1] - adi[i];
    for (j=0; j<nzi; j++) {
      row  = *adj++;
      pnz  = pi_loc[row+1] - pi_loc[row];
      Jptr = pj_loc + pi_loc[row];
      /* Expand list if it is not long enough */
      if (pnz+apnz_max > lsize) {
        lsize = pnz+apnz_max;
        PetscCall(PetscLLCondensedExpand_Scalable(lsize, &lnk));
      }
      /* add non-zero cols of P into the sorted linked list lnk */
      PetscCall(PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk));
      apnz     = *lnk; /* The first element in the list is the number of items in the list */
      api[i+1] = api[i] + apnz;
      if (apnz > apnz_max) apnz_max = apnz + 1; /* '1' for diagonal entry */
    }
    /* off-diagonal portion of A */
    nzi = aoi[i+1] - aoi[i];
    for (j=0; j<nzi; j++) {
      row  = *aoj++;
      pnz  = pi_oth[row+1] - pi_oth[row];
      Jptr = pj_oth + pi_oth[row];
      /* Expand list if it is not long enough */
      if (pnz+apnz_max > lsize) {
        lsize = pnz + apnz_max;
        PetscCall(PetscLLCondensedExpand_Scalable(lsize, &lnk));
      }
      /* add non-zero cols of P into the sorted linked list lnk */
      PetscCall(PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk));
      apnz     = *lnk;  /* The first element in the list is the number of items in the list */
      api[i+1] = api[i] + apnz;
      if (apnz > apnz_max) apnz_max = apnz + 1; /* '1' for diagonal entry */
    }

    /* add missing diagonal entry */
    if (C->force_diagonals) {
      j = i + rstart; /* column index */
      PetscCall(PetscLLCondensedAddSorted_Scalable(1,&j,lnk));
    }

    apnz     = *lnk;
    api[i+1] = api[i] + apnz;
    if (apnz > apnz_max) apnz_max = apnz;

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

    /* Copy data into free space, then initialize lnk */
    PetscCall(PetscLLCondensedClean_Scalable(apnz,current_space->array,lnk));
    PetscCall(MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz));

    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) */
  PetscCall(PetscMalloc1(api[am]+1,&ptap->apj));
  apj  = ptap->apj;
  PetscCall(PetscFreeSpaceContiguous(&free_space,ptap->apj));
  PetscCall(PetscLLCondensedDestroy_Scalable(lnk));

  /* create and assemble symbolic parallel matrix C */
  /*----------------------------------------------------*/
  PetscCall(MatSetSizes(C,am,pn,PETSC_DETERMINE,PETSC_DETERMINE));
  PetscCall(MatSetBlockSizesFromMats(C,A,P));
  PetscCall(MatGetType(A,&mtype));
  PetscCall(MatSetType(C,mtype));
  PetscCall(MatMPIAIJSetPreallocation(C,0,dnz,0,onz));
  MatPreallocateEnd(dnz,onz);

  /* malloc apa for assembly C */
  PetscCall(PetscCalloc1(apnz_max,&ptap->apa));

  PetscCall(MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(C, apj, api));
  PetscCall(MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY));
  PetscCall(MatSetOption(C,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE));

  C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ;
  C->ops->productnumeric = MatProductNumeric_AB;

  /* attach the supporting struct to C for reuse */
  C->product->data    = ptap;
  C->product->destroy = MatDestroy_MPIAIJ_MatMatMult;

  /* set MatInfo */
  afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
  if (afill < 1.0) afill = 1.0;
  C->info.mallocs           = nspacedouble;
  C->info.fill_ratio_given  = fill;
  C->info.fill_ratio_needed = afill;

#if defined(PETSC_USE_INFO)
  if (api[am]) {
    PetscCall(PetscInfo(C,"Reallocs %" PetscInt_FMT "; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill));
    PetscCall(PetscInfo(C,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill));
  } else {
    PetscCall(PetscInfo(C,"Empty matrix product\n"));
  }
#endif
  PetscFunctionReturn(0);
}

/* This function is needed for the seqMPI matrix-matrix multiplication.  */
/* Three input arrays are merged to one output array. The size of the    */
/* output array is also output. Duplicate entries only show up once.     */
static void Merge3SortedArrays(PetscInt  size1, PetscInt *in1,
                               PetscInt  size2, PetscInt *in2,
                               PetscInt  size3, PetscInt *in3,
                               PetscInt *size4, PetscInt *out)
{
  int i = 0, j = 0, k = 0, l = 0;

  /* Traverse all three arrays */
  while (i<size1 && j<size2 && k<size3) {
    if (in1[i] < in2[j] && in1[i] < in3[k]) {
      out[l++] = in1[i++];
    }
    else if (in2[j] < in1[i] && in2[j] < in3[k]) {
      out[l++] = in2[j++];
    }
    else if (in3[k] < in1[i] && in3[k] < in2[j]) {
      out[l++] = in3[k++];
    }
    else if (in1[i] == in2[j] && in1[i] < in3[k]) {
      out[l++] = in1[i];
      i++, j++;
    }
    else if (in1[i] == in3[k] && in1[i] < in2[j]) {
      out[l++] = in1[i];
      i++, k++;
    }
    else if (in3[k] == in2[j] && in2[j] < in1[i])  {
      out[l++] = in2[j];
      k++, j++;
    }
    else if (in1[i] == in2[j] && in1[i] == in3[k]) {
      out[l++] = in1[i];
      i++, j++, k++;
    }
  }

  /* Traverse two remaining arrays */
  while (i<size1 && j<size2) {
    if (in1[i] < in2[j]) {
      out[l++] = in1[i++];
    }
    else if (in1[i] > in2[j]) {
      out[l++] = in2[j++];
    }
    else {
      out[l++] = in1[i];
      i++, j++;
    }
  }

  while (i<size1 && k<size3) {
    if (in1[i] < in3[k]) {
      out[l++] = in1[i++];
    }
    else if (in1[i] > in3[k]) {
      out[l++] = in3[k++];
    }
    else {
      out[l++] = in1[i];
      i++, k++;
    }
  }

  while (k<size3 && j<size2)  {
    if (in3[k] < in2[j]) {
      out[l++] = in3[k++];
    }
    else if (in3[k] > in2[j]) {
      out[l++] = in2[j++];
    }
    else {
      out[l++] = in3[k];
      k++, j++;
    }
  }

  /* Traverse one remaining array */
  while (i<size1) out[l++] = in1[i++];
  while (j<size2) out[l++] = in2[j++];
  while (k<size3) out[l++] = in3[k++];

  *size4 = l;
}

/* This matrix-matrix multiplication algorithm divides the multiplication into three multiplications and  */
/* adds up the products. Two of these three multiplications are performed with existing (sequential)      */
/* matrix-matrix multiplications.  */
PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_seqMPI(Mat A, Mat P, PetscReal fill, Mat C)
{
  MPI_Comm           comm;
  PetscMPIInt        size;
  Mat_APMPI          *ptap;
  PetscFreeSpaceList free_space_diag=NULL, current_space=NULL;
  Mat_MPIAIJ         *a  =(Mat_MPIAIJ*)A->data;
  Mat_SeqAIJ         *ad =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc;
  Mat_MPIAIJ         *p  =(Mat_MPIAIJ*)P->data;
  Mat_SeqAIJ         *adpd_seq, *p_off, *aopoth_seq;
  PetscInt           adponz, adpdnz;
  PetscInt           *pi_loc,*dnz,*onz;
  PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,rstart=A->rmap->rstart;
  PetscInt           *lnk,i, i1=0,pnz,row,*adpoi,*adpoj, *api, *adpoJ, *aopJ, *apJ,*Jptr, aopnz, nspacedouble=0,j,nzi,
                     *apj,apnz, *adpdi, *adpdj, *adpdJ, *poff_i, *poff_j, *j_temp, *aopothi, *aopothj;
  PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n, p_colstart, p_colend;
  PetscBT            lnkbt;
  PetscReal          afill;
  PetscMPIInt        rank;
  Mat                adpd, aopoth;
  MatType            mtype;
  const char         *prefix;

  PetscFunctionBegin;
  MatCheckProduct(C,4);
  PetscCheck(!C->product->data,PETSC_COMM_SELF,PETSC_ERR_PLIB,"Extra product struct not empty");
  PetscCall(PetscObjectGetComm((PetscObject)A,&comm));
  PetscCallMPI(MPI_Comm_size(comm,&size));
  PetscCallMPI(MPI_Comm_rank(comm, &rank));
  PetscCall(MatGetOwnershipRangeColumn(P, &p_colstart, &p_colend));

  /* create struct Mat_APMPI and attached it to C later */
  PetscCall(PetscNew(&ptap));

  /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
  PetscCall(MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth));

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

  p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
  pi_loc = p_loc->i;

  /* Allocate memory for the i arrays of the matrices A*P, A_diag*P_off and A_offd * P */
  PetscCall(PetscMalloc1(am+2,&api));
  PetscCall(PetscMalloc1(am+2,&adpoi));

  adpoi[0]    = 0;
  ptap->api = api;
  api[0] = 0;

  /* create and initialize a linked list, will be used for both A_diag * P_loc_off and A_offd * P_oth */
  PetscCall(PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt));
  MatPreallocateBegin(comm,am,pn,dnz,onz);

  /* Symbolic calc of A_loc_diag * P_loc_diag */
  PetscCall(MatGetOptionsPrefix(A,&prefix));
  PetscCall(MatProductCreate(a->A,p->A,NULL,&adpd));
  PetscCall(MatGetOptionsPrefix(A,&prefix));
  PetscCall(MatSetOptionsPrefix(adpd,prefix));
  PetscCall(MatAppendOptionsPrefix(adpd,"inner_diag_"));

  PetscCall(MatProductSetType(adpd,MATPRODUCT_AB));
  PetscCall(MatProductSetAlgorithm(adpd,"sorted"));
  PetscCall(MatProductSetFill(adpd,fill));
  PetscCall(MatProductSetFromOptions(adpd));

  adpd->force_diagonals = C->force_diagonals;
  PetscCall(MatProductSymbolic(adpd));

  adpd_seq = (Mat_SeqAIJ*)((adpd)->data);
  adpdi = adpd_seq->i; adpdj = adpd_seq->j;
  p_off = (Mat_SeqAIJ*)((p->B)->data);
  poff_i = p_off->i; poff_j = p_off->j;

  /* j_temp stores indices of a result row before they are added to the linked list */
  PetscCall(PetscMalloc1(pN+2,&j_temp));

  /* Symbolic calc of the A_diag * p_loc_off */
  /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
  PetscCall(PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space_diag));
  current_space = free_space_diag;

  for (i=0; i<am; i++) {
    /* A_diag * P_loc_off */
    nzi = adi[i+1] - adi[i];
    for (j=0; j<nzi; j++) {
      row  = *adj++;
      pnz  = poff_i[row+1] - poff_i[row];
      Jptr = poff_j + poff_i[row];
      for (i1 = 0; i1 < pnz; i1++) {
        j_temp[i1] = p->garray[Jptr[i1]];
      }
      /* add non-zero cols of P into the sorted linked list lnk */
      PetscCall(PetscLLCondensedAddSorted(pnz,j_temp,lnk,lnkbt));
    }

    adponz     = lnk[0];
    adpoi[i+1] = adpoi[i] + adponz;

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

    /* Copy data into free space, then initialize lnk */
    PetscCall(PetscLLCondensedClean(pN,adponz,current_space->array,lnk,lnkbt));

    current_space->array           += adponz;
    current_space->local_used      += adponz;
    current_space->local_remaining -= adponz;
  }

  /* Symbolic calc of A_off * P_oth */
  PetscCall(MatSetOptionsPrefix(a->B,prefix));
  PetscCall(MatAppendOptionsPrefix(a->B,"inner_offdiag_"));
  PetscCall(MatCreate(PETSC_COMM_SELF,&aopoth));
  PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ(a->B, ptap->P_oth, fill, aopoth));
  aopoth_seq = (Mat_SeqAIJ*)((aopoth)->data);
  aopothi = aopoth_seq->i; aopothj = aopoth_seq->j;

  /* Allocate space for apj, adpj, aopj, ... */
  /* destroy lists of free space and other temporary array(s) */

  PetscCall(PetscMalloc1(aopothi[am] + adpoi[am] + adpdi[am]+2, &ptap->apj));
  PetscCall(PetscMalloc1(adpoi[am]+2, &adpoj));

  /* Copy from linked list to j-array */
  PetscCall(PetscFreeSpaceContiguous(&free_space_diag,adpoj));
  PetscCall(PetscLLDestroy(lnk,lnkbt));

  adpoJ = adpoj;
  adpdJ = adpdj;
  aopJ = aopothj;
  apj  = ptap->apj;
  apJ = apj; /* still empty */

  /* Merge j-arrays of A_off * P, A_diag * P_loc_off, and */
  /* A_diag * P_loc_diag to get A*P */
  for (i = 0; i < am; i++) {
    aopnz  =  aopothi[i+1] -  aopothi[i];
    adponz = adpoi[i+1] - adpoi[i];
    adpdnz = adpdi[i+1] - adpdi[i];

    /* Correct indices from A_diag*P_diag */
    for (i1 = 0; i1 < adpdnz; i1++) {
      adpdJ[i1] += p_colstart;
    }
    /* Merge j-arrays of A_diag * P_loc_off and A_diag * P_loc_diag and A_off * P_oth */
    Merge3SortedArrays(adponz, adpoJ, adpdnz, adpdJ, aopnz, aopJ, &apnz, apJ);
    PetscCall(MatPreallocateSet(i+rstart, apnz, apJ, dnz, onz));

    aopJ += aopnz;
    adpoJ += adponz;
    adpdJ += adpdnz;
    apJ += apnz;
    api[i+1] = api[i] + apnz;
  }

  /* malloc apa to store dense row A[i,:]*P */
  PetscCall(PetscCalloc1(pN+2,&ptap->apa));

  /* create and assemble symbolic parallel matrix C */
  PetscCall(MatSetSizes(C,am,pn,PETSC_DETERMINE,PETSC_DETERMINE));
  PetscCall(MatSetBlockSizesFromMats(C,A,P));
  PetscCall(MatGetType(A,&mtype));
  PetscCall(MatSetType(C,mtype));
  PetscCall(MatMPIAIJSetPreallocation(C,0,dnz,0,onz));
  MatPreallocateEnd(dnz,onz);

  PetscCall(MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(C, apj, api));
  PetscCall(MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY));
  PetscCall(MatSetOption(C,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE));

  C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
  C->ops->productnumeric = MatProductNumeric_AB;

  /* attach the supporting struct to C for reuse */
  C->product->data    = ptap;
  C->product->destroy = MatDestroy_MPIAIJ_MatMatMult;

  /* set MatInfo */
  afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
  if (afill < 1.0) afill = 1.0;
  C->info.mallocs           = nspacedouble;
  C->info.fill_ratio_given  = fill;
  C->info.fill_ratio_needed = afill;

#if defined(PETSC_USE_INFO)
  if (api[am]) {
    PetscCall(PetscInfo(C,"Reallocs %" PetscInt_FMT "; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill));
    PetscCall(PetscInfo(C,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill));
  } else {
    PetscCall(PetscInfo(C,"Empty matrix product\n"));
  }
#endif

  PetscCall(MatDestroy(&aopoth));
  PetscCall(MatDestroy(&adpd));
  PetscCall(PetscFree(j_temp));
  PetscCall(PetscFree(adpoj));
  PetscCall(PetscFree(adpoi));
  PetscFunctionReturn(0);
}

/*-------------------------------------------------------------------------*/
/* This routine only works when scall=MAT_REUSE_MATRIX! */
PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult(Mat P,Mat A,Mat C)
{
  Mat_APMPI      *ptap;
  Mat            Pt;

  PetscFunctionBegin;
  MatCheckProduct(C,3);
  ptap = (Mat_APMPI*)C->product->data;
  PetscCheck(ptap,PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be computed. Missing data");
  PetscCheck(ptap->Pt,PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtA cannot be reused. Do not call MatProductClear()");

  Pt   = ptap->Pt;
  PetscCall(MatTranspose(P,MAT_REUSE_MATRIX,&Pt));
  PetscCall(MatMatMultNumeric_MPIAIJ_MPIAIJ(Pt,A,C));
  PetscFunctionReturn(0);
}

/* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,PetscReal fill,Mat C)
{
  Mat_APMPI           *ptap;
  Mat_MPIAIJ          *p=(Mat_MPIAIJ*)P->data;
  MPI_Comm            comm;
  PetscMPIInt         size,rank;
  PetscFreeSpaceList  free_space=NULL,current_space=NULL;
  PetscInt            pn=P->cmap->n,aN=A->cmap->N,an=A->cmap->n;
  PetscInt            *lnk,i,k,nsend,rstart;
  PetscBT             lnkbt;
  PetscMPIInt         tagi,tagj,*len_si,*len_s,*len_ri,nrecv;
  PETSC_UNUSED PetscMPIInt icompleted=0;
  PetscInt            **buf_rj,**buf_ri,**buf_ri_k,row,ncols,*cols;
  PetscInt            len,proc,*dnz,*onz,*owners,nzi;
  PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
  MPI_Request         *swaits,*rwaits;
  MPI_Status          *sstatus,rstatus;
  PetscLayout         rowmap;
  PetscInt            *owners_co,*coi,*coj;    /* i and j array of (p->B)^T*A*P - used in the communication */
  PetscMPIInt         *len_r,*id_r;    /* array of length of comm->size, store send/recv matrix values */
  PetscInt            *Jptr,*prmap=p->garray,con,j,Crmax;
  Mat_SeqAIJ          *a_loc,*c_loc,*c_oth;
  PetscTable          ta;
  MatType             mtype;
  const char          *prefix;

  PetscFunctionBegin;
  PetscCall(PetscObjectGetComm((PetscObject)A,&comm));
  PetscCallMPI(MPI_Comm_size(comm,&size));
  PetscCallMPI(MPI_Comm_rank(comm,&rank));

  /* create symbolic parallel matrix C */
  PetscCall(MatGetType(A,&mtype));
  PetscCall(MatSetType(C,mtype));

  C->ops->transposematmultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;

  /* create struct Mat_APMPI and attached it to C later */
  PetscCall(PetscNew(&ptap));
  ptap->reuse = MAT_INITIAL_MATRIX;

  /* (0) compute Rd = Pd^T, Ro = Po^T  */
  /* --------------------------------- */
  PetscCall(MatTranspose_SeqAIJ(p->A,MAT_INITIAL_MATRIX,&ptap->Rd));
  PetscCall(MatTranspose_SeqAIJ(p->B,MAT_INITIAL_MATRIX,&ptap->Ro));

  /* (1) compute symbolic A_loc */
  /* ---------------------------*/
  PetscCall(MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&ptap->A_loc));

  /* (2-1) compute symbolic C_oth = Ro*A_loc  */
  /* ------------------------------------ */
  PetscCall(MatGetOptionsPrefix(A,&prefix));
  PetscCall(MatSetOptionsPrefix(ptap->Ro,prefix));
  PetscCall(MatAppendOptionsPrefix(ptap->Ro,"inner_offdiag_"));
  PetscCall(MatCreate(PETSC_COMM_SELF,&ptap->C_oth));
  PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Ro,ptap->A_loc,fill,ptap->C_oth));

  /* (3) send coj of C_oth to other processors  */
  /* ------------------------------------------ */
  /* determine row ownership */
  PetscCall(PetscLayoutCreate(comm,&rowmap));
  rowmap->n  = pn;
  rowmap->bs = 1;
  PetscCall(PetscLayoutSetUp(rowmap));
  owners = rowmap->range;

  /* determine the number of messages to send, their lengths */
  PetscCall(PetscMalloc4(size,&len_s,size,&len_si,size,&sstatus,size+2,&owners_co));
  PetscCall(PetscArrayzero(len_s,size));
  PetscCall(PetscArrayzero(len_si,size));

  c_oth = (Mat_SeqAIJ*)ptap->C_oth->data;
  coi   = c_oth->i; coj = c_oth->j;
  con   = ptap->C_oth->rmap->n;
  proc  = 0;
  for (i=0; i<con; i++) {
    while (prmap[i] >= owners[proc+1]) proc++;
    len_si[proc]++;               /* num of rows in Co(=Pt*A) 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;
  nsend        = 0;
  for (proc=0; proc<size; proc++) {
    owners_co[proc+1] = owners_co[proc] + len_si[proc];
    if (len_s[proc]) {
      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  */
  PetscCall(PetscGatherNumberOfMessages(comm,NULL,len_s,&nrecv));
  PetscCall(PetscGatherMessageLengths2(comm,nsend,nrecv,len_s,len_si,&id_r,&len_r,&len_ri));

  /* post the Irecv and Isend of coj */
  PetscCall(PetscCommGetNewTag(comm,&tagj));
  PetscCall(PetscPostIrecvInt(comm,tagj,nrecv,id_r,len_r,&buf_rj,&rwaits));
  PetscCall(PetscMalloc1(nsend+1,&swaits));
  for (proc=0, k=0; proc<size; proc++) {
    if (!len_s[proc]) continue;
    i    = owners_co[proc];
    PetscCallMPI(MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k));
    k++;
  }

  /* (2-2) compute symbolic C_loc = Rd*A_loc */
  /* ---------------------------------------- */
  PetscCall(MatSetOptionsPrefix(ptap->Rd,prefix));
  PetscCall(MatAppendOptionsPrefix(ptap->Rd,"inner_diag_"));
  PetscCall(MatCreate(PETSC_COMM_SELF,&ptap->C_loc));
  PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Rd,ptap->A_loc,fill,ptap->C_loc));
  c_loc = (Mat_SeqAIJ*)ptap->C_loc->data;

  /* receives coj are complete */
  for (i=0; i<nrecv; i++) {
    PetscCallMPI(MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus));
  }
  PetscCall(PetscFree(rwaits));
  if (nsend) PetscCallMPI(MPI_Waitall(nsend,swaits,sstatus));

  /* add received column indices into ta to update Crmax */
  a_loc = (Mat_SeqAIJ*)(ptap->A_loc)->data;

  /* create and initialize a linked list */
  PetscCall(PetscTableCreate(an,aN,&ta)); /* for compute Crmax */
  MatRowMergeMax_SeqAIJ(a_loc,ptap->A_loc->rmap->N,ta);

  for (k=0; k<nrecv; k++) {/* k-th received message */
    Jptr = buf_rj[k];
    for (j=0; j<len_r[k]; j++) {
      PetscCall(PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES));
    }
  }
  PetscCall(PetscTableGetCount(ta,&Crmax));
  PetscCall(PetscTableDestroy(&ta));

  /* (4) send and recv coi */
  /*-----------------------*/
  PetscCall(PetscCommGetNewTag(comm,&tagi));
  PetscCall(PetscPostIrecvInt(comm,tagi,nrecv,id_r,len_ri,&buf_ri,&rwaits));
  PetscCall(PetscMalloc1(len+1,&buf_s));
  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++;
    }
    PetscCallMPI(MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k));
    k++;
    buf_si += len_si[proc];
  }
  for (i=0; i<nrecv; i++) {
    PetscCallMPI(MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus));
  }
  PetscCall(PetscFree(rwaits));
  if (nsend) PetscCallMPI(MPI_Waitall(nsend,swaits,sstatus));

  PetscCall(PetscFree4(len_s,len_si,sstatus,owners_co));
  PetscCall(PetscFree(len_ri));
  PetscCall(PetscFree(swaits));
  PetscCall(PetscFree(buf_s));

  /* (5) compute the local portion of C      */
  /* ------------------------------------------ */
  /* set initial free space to be Crmax, sufficient for holding nozeros in each row of C */
  PetscCall(PetscFreeSpaceGet(Crmax,&free_space));
  current_space = free_space;

  PetscCall(PetscMalloc3(nrecv,&buf_ri_k,nrecv,&nextrow,nrecv,&nextci));
  for (k=0; k<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); /* points to the next i-structure of k-th recved i-structure  */
  }

  MatPreallocateBegin(comm,pn,an,dnz,onz);
  PetscCall(PetscLLCondensedCreate(Crmax,aN,&lnk,&lnkbt));
  for (i=0; i<pn; i++) { /* for each local row of C */
    /* add C_loc into C */
    nzi  = c_loc->i[i+1] - c_loc->i[i];
    Jptr = c_loc->j + c_loc->i[i];
    PetscCall(PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt));

    /* add received col data into lnk */
    for (k=0; k<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];
        PetscCall(PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt));
        nextrow[k]++; nextci[k]++;
      }
    }

    /* add missing diagonal entry */
    if (C->force_diagonals) {
      k = i + owners[rank]; /* column index */
      PetscCall(PetscLLCondensedAddSorted(1,&k,lnk,lnkbt));
    }

    nzi = lnk[0];

    /* copy data into free space, then initialize lnk */
    PetscCall(PetscLLCondensedClean(aN,nzi,current_space->array,lnk,lnkbt));
    PetscCall(MatPreallocateSet(i+owners[rank],nzi,current_space->array,dnz,onz));
  }
  PetscCall(PetscFree3(buf_ri_k,nextrow,nextci));
  PetscCall(PetscLLDestroy(lnk,lnkbt));
  PetscCall(PetscFreeSpaceDestroy(free_space));

  /* local sizes and preallocation */
  PetscCall(MatSetSizes(C,pn,an,PETSC_DETERMINE,PETSC_DETERMINE));
  if (P->cmap->bs > 0) PetscCall(PetscLayoutSetBlockSize(C->rmap,P->cmap->bs));
  if (A->cmap->bs > 0) PetscCall(PetscLayoutSetBlockSize(C->cmap,A->cmap->bs));
  PetscCall(MatMPIAIJSetPreallocation(C,0,dnz,0,onz));
  MatPreallocateEnd(dnz,onz);

  /* add C_loc and C_oth to C */
  PetscCall(MatGetOwnershipRange(C,&rstart,NULL));
  for (i=0; i<pn; i++) {
    ncols = c_loc->i[i+1] - c_loc->i[i];
    cols  = c_loc->j + c_loc->i[i];
    row   = rstart + i;
    PetscCall(MatSetValues(C,1,(const PetscInt*)&row,ncols,(const PetscInt*)cols,NULL,INSERT_VALUES));

    if (C->force_diagonals) {
      PetscCall(MatSetValues(C,1,(const PetscInt*)&row,1,(const PetscInt*)&row,NULL,INSERT_VALUES));
    }
  }
  for (i=0; i<con; i++) {
    ncols = c_oth->i[i+1] - c_oth->i[i];
    cols  = c_oth->j + c_oth->i[i];
    row   = prmap[i];
    PetscCall(MatSetValues(C,1,(const PetscInt*)&row,ncols,(const PetscInt*)cols,NULL,INSERT_VALUES));
  }
  PetscCall(MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY));
  PetscCall(MatSetOption(C,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE));

  /* members in merge */
  PetscCall(PetscFree(id_r));
  PetscCall(PetscFree(len_r));
  PetscCall(PetscFree(buf_ri[0]));
  PetscCall(PetscFree(buf_ri));
  PetscCall(PetscFree(buf_rj[0]));
  PetscCall(PetscFree(buf_rj));
  PetscCall(PetscLayoutDestroy(&rowmap));

  /* attach the supporting struct to C for reuse */
  C->product->data    = ptap;
  C->product->destroy = MatDestroy_MPIAIJ_PtAP;
  PetscFunctionReturn(0);
}

PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,Mat C)
{
  Mat_MPIAIJ        *p=(Mat_MPIAIJ*)P->data;
  Mat_SeqAIJ        *c_seq;
  Mat_APMPI         *ptap;
  Mat               A_loc,C_loc,C_oth;
  PetscInt          i,rstart,rend,cm,ncols,row;
  const PetscInt    *cols;
  const PetscScalar *vals;

  PetscFunctionBegin;
  MatCheckProduct(C,3);
  ptap = (Mat_APMPI*)C->product->data;
  PetscCheck(ptap,PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be computed. Missing data");
  PetscCheck(ptap->A_loc,PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtA cannot be reused. Do not call MatProductClear()");
  PetscCall(MatZeroEntries(C));

  if (ptap->reuse == MAT_REUSE_MATRIX) {
    /* These matrices are obtained in MatTransposeMatMultSymbolic() */
    /* 1) get R = Pd^T, Ro = Po^T */
    /*----------------------------*/
    PetscCall(MatTranspose_SeqAIJ(p->A,MAT_REUSE_MATRIX,&ptap->Rd));
    PetscCall(MatTranspose_SeqAIJ(p->B,MAT_REUSE_MATRIX,&ptap->Ro));

    /* 2) compute numeric A_loc */
    /*--------------------------*/
    PetscCall(MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&ptap->A_loc));
  }

  /* 3) C_loc = Rd*A_loc, C_oth = Ro*A_loc */
  A_loc = ptap->A_loc;
  PetscCall(((ptap->C_loc)->ops->matmultnumeric)(ptap->Rd,A_loc,ptap->C_loc));
  PetscCall(((ptap->C_oth)->ops->matmultnumeric)(ptap->Ro,A_loc,ptap->C_oth));
  C_loc = ptap->C_loc;
  C_oth = ptap->C_oth;

  /* add C_loc and C_oth to C */
  PetscCall(MatGetOwnershipRange(C,&rstart,&rend));

  /* C_loc -> C */
  cm    = C_loc->rmap->N;
  c_seq = (Mat_SeqAIJ*)C_loc->data;
  cols = c_seq->j;
  vals = c_seq->a;
  for (i=0; i<cm; i++) {
    ncols = c_seq->i[i+1] - c_seq->i[i];
    row = rstart + i;
    PetscCall(MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES));
    cols += ncols; vals += ncols;
  }

  /* Co -> C, off-processor part */
  cm    = C_oth->rmap->N;
  c_seq = (Mat_SeqAIJ*)C_oth->data;
  cols  = c_seq->j;
  vals  = c_seq->a;
  for (i=0; i<cm; i++) {
    ncols = c_seq->i[i+1] - c_seq->i[i];
    row = p->garray[i];
    PetscCall(MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES));
    cols += ncols; vals += ncols;
  }
  PetscCall(MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY));
  PetscCall(MatSetOption(C,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE));

  ptap->reuse = MAT_REUSE_MATRIX;
  PetscFunctionReturn(0);
}

PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ(Mat P,Mat A,Mat C)
{
  Mat_Merge_SeqsToMPI *merge;
  Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data;
  Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
  Mat_APMPI           *ptap;
  PetscInt            *adj;
  PetscInt            i,j,k,anz,pnz,row,*cj,nexta;
  MatScalar           *ada,*ca,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;
  const PetscScalar   *dummy;
  PetscInt            *ai,*aj,*coi,*coj,*poJ,*pdJ;
  Mat                 A_loc;
  Mat_SeqAIJ          *a_loc;

  PetscFunctionBegin;
  MatCheckProduct(C,3);
  ptap = (Mat_APMPI*)C->product->data;
  PetscCheck(ptap,PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be computed. Missing data");
  PetscCheck(ptap->A_loc,PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtA cannot be reused. Do not call MatProductClear()");
  PetscCall(PetscObjectGetComm((PetscObject)C,&comm));
  PetscCallMPI(MPI_Comm_size(comm,&size));
  PetscCallMPI(MPI_Comm_rank(comm,&rank));

  merge = ptap->merge;

  /* 2) compute numeric C_seq = P_loc^T*A_loc */
  /*------------------------------------------*/
  /* get data from symbolic products */
  coi    = merge->coi; coj = merge->coj;
  PetscCall(PetscCalloc1(coi[pon]+1,&coa));
  bi     = merge->bi; bj = merge->bj;
  owners = merge->rowmap->range;
  PetscCall(PetscCalloc1(bi[cm]+1,&ba));

  /* get A_loc by taking all local rows of A */
  A_loc = ptap->A_loc;
  PetscCall(MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc));
  a_loc = (Mat_SeqAIJ*)(A_loc)->data;
  ai    = a_loc->i;
  aj    = a_loc->j;

  /* trigger copy to CPU */
  PetscCall(MatSeqAIJGetArrayRead(p->A,&dummy));
  PetscCall(MatSeqAIJRestoreArrayRead(p->A,&dummy));
  PetscCall(MatSeqAIJGetArrayRead(p->B,&dummy));
  PetscCall(MatSeqAIJRestoreArrayRead(p->B,&dummy));
  for (i=0; i<am; i++) {
    anz = ai[i+1] - ai[i];
    adj = aj + ai[i];
    ada = a_loc->a + ai[i];

    /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
    /*-------------------------------------------------------------*/
    /* put the value into Co=(p->B)^T*A (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];
      /* perform sparse axpy */
      nexta  = 0;
      valtmp = pA[j];
      for (k=0; nexta<anz; k++) {
        if (cj[k] == adj[nexta]) {
          ca[k] += valtmp*ada[nexta];
          nexta++;
        }
      }
      PetscCall(PetscLogFlops(2.0*anz));
    }

    /* 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];
      /* perform sparse axpy */
      nexta  = 0;
      valtmp = pA[j];
      for (k=0; nexta<anz; k++) {
        if (cj[k] == adj[nexta]) {
          ca[k] += valtmp*ada[nexta];
          nexta++;
        }
      }
      PetscCall(PetscLogFlops(2.0*anz));
    }
  }

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

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

  PetscCall(PetscFree2(s_waits,status));
  PetscCall(PetscFree(r_waits));
  PetscCall(PetscFree(coa));

  /* 4) insert local Cseq and received values into Cmpi */
  /*----------------------------------------------------*/
  PetscCall(PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci));
  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); /* points 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]++;
        PetscCall(PetscLogFlops(2.0*cnz));
      }
    }
    PetscCall(MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES));
  }
  PetscCall(MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY));

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

PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P,Mat A,PetscReal fill,Mat C)
{
  Mat                 A_loc;
  Mat_APMPI           *ptap;
  PetscFreeSpaceList  free_space=NULL,current_space=NULL;
  Mat_MPIAIJ          *p=(Mat_MPIAIJ*)P->data,*a=(Mat_MPIAIJ*)A->data;
  PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
  PetscInt            nnz;
  PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
  PetscInt            am  =A->rmap->n,pn=P->cmap->n;
  MPI_Comm            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            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
  PetscReal           afill  =1.0,afill_tmp;
  PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Armax;
  Mat_SeqAIJ          *a_loc;
  PetscTable          ta;
  MatType             mtype;

  PetscFunctionBegin;
  PetscCall(PetscObjectGetComm((PetscObject)A,&comm));
  /* check if matrix local sizes are compatible */
  PetscCheckFalse(A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend,comm,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, A (%" PetscInt_FMT ", %" PetscInt_FMT ") != P (%" PetscInt_FMT ",%" PetscInt_FMT ")",A->rmap->rstart,A->rmap->rend,P->rmap->rstart,P->rmap->rend);

  PetscCallMPI(MPI_Comm_size(comm,&size));
  PetscCallMPI(MPI_Comm_rank(comm,&rank));

  /* create struct Mat_APMPI and attached it to C later */
  PetscCall(PetscNew(&ptap));

  /* get A_loc by taking all local rows of A */
  PetscCall(MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc));

  ptap->A_loc = A_loc;
  a_loc       = (Mat_SeqAIJ*)(A_loc)->data;
  ai          = a_loc->i;
  aj          = a_loc->j;

  /* determine symbolic Co=(p->B)^T*A - send to others */
  /*----------------------------------------------------*/
  PetscCall(MatGetSymbolicTranspose_SeqAIJ(p->A,&pdti,&pdtj));
  PetscCall(MatGetSymbolicTranspose_SeqAIJ(p->B,&poti,&potj));
  pon = (p->B)->cmap->n; /* total num of rows to be sent to other processors
                         >= (num of nonzero rows of C_seq) - pn */
  PetscCall(PetscMalloc1(pon+1,&coi));
  coi[0] = 0;

  /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
  nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(poti[pon],ai[am]));
  PetscCall(PetscFreeSpaceGet(nnz,&free_space));
  current_space = free_space;

  /* create and initialize a linked list */
  PetscCall(PetscTableCreate(A->cmap->n + a->B->cmap->N,aN,&ta));
  MatRowMergeMax_SeqAIJ(a_loc,am,ta);
  PetscCall(PetscTableGetCount(ta,&Armax));

  PetscCall(PetscLLCondensedCreate_Scalable(Armax,&lnk));

  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 A_loc == col of Pot */
      anz  = ai[row+1] - ai[row];
      Jptr = aj + ai[row];
      /* add non-zero cols of AP into the sorted linked list lnk */
      PetscCall(PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk));
    }
    nnz = lnk[0];

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

    /* Copy data into free space, and zero out denserows */
    PetscCall(PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk));

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

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

  PetscCall(PetscMalloc1(coi[pon]+1,&coj));
  PetscCall(PetscFreeSpaceContiguous(&free_space,coj));
  PetscCall(PetscLLCondensedDestroy_Scalable(lnk)); /* must destroy to get a new one for C */

  afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
  if (afill_tmp > afill) afill = afill_tmp;

  /* send j-array (coj) of Co to other processors */
  /*----------------------------------------------*/
  /* determine row ownership */
  PetscCall(PetscNew(&merge));
  PetscCall(PetscLayoutCreate(comm,&merge->rowmap));

  merge->rowmap->n  = pn;
  merge->rowmap->bs = 1;

  PetscCall(PetscLayoutSetUp(merge->rowmap));
  owners = merge->rowmap->range;

  /* determine the number of messages to send, their lengths */
  PetscCall(PetscCalloc1(size,&len_si));
  PetscCall(PetscCalloc1(size,&merge->len_s));

  len_s        = merge->len_s;
  merge->nsend = 0;

  PetscCall(PetscMalloc1(size+2,&owners_co));

  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  */
  PetscCall(PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv));
  PetscCall(PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri));

  /* post the Irecv and Isend of coj */
  PetscCall(PetscCommGetNewTag(comm,&tagj));
  PetscCall(PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits));
  PetscCall(PetscMalloc1(merge->nsend+1,&swaits));
  for (proc=0, k=0; proc<size; proc++) {
    if (!len_s[proc]) continue;
    i    = owners_co[proc];
    PetscCallMPI(MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k));
    k++;
  }

  /* receives and sends of coj are complete */
  PetscCall(PetscMalloc1(size,&sstatus));
  for (i=0; i<merge->nrecv; i++) {
    PETSC_UNUSED PetscMPIInt icompleted;
    PetscCallMPI(MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus));
  }
  PetscCall(PetscFree(rwaits));
  if (merge->nsend) PetscCallMPI(MPI_Waitall(merge->nsend,swaits,sstatus));

  /* add received column indices into table to update Armax */
  /* Armax can be as large as aN if a P[row,:] is dense, see src/ksp/ksp/tutorials/ex56.c! */
  for (k=0; k<merge->nrecv; k++) {/* k-th received message */
    Jptr = buf_rj[k];
    for (j=0; j<merge->len_r[k]; j++) {
      PetscCall(PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES));
    }
  }
  PetscCall(PetscTableGetCount(ta,&Armax));

  /* send and recv coi */
  /*-------------------*/
  PetscCall(PetscCommGetNewTag(comm,&tagi));
  PetscCall(PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits));
  PetscCall(PetscMalloc1(len+1,&buf_s));
  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++;
    }
    PetscCallMPI(MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k));
    k++;
    buf_si += len_si[proc];
  }
  i = merge->nrecv;
  while (i--) {
    PETSC_UNUSED PetscMPIInt icompleted;
    PetscCallMPI(MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus));
  }
  PetscCall(PetscFree(rwaits));
  if (merge->nsend) PetscCallMPI(MPI_Waitall(merge->nsend,swaits,sstatus));
  PetscCall(PetscFree(len_si));
  PetscCall(PetscFree(len_ri));
  PetscCall(PetscFree(swaits));
  PetscCall(PetscFree(sstatus));
  PetscCall(PetscFree(buf_s));

  /* compute the local portion of C (mpi mat) */
  /*------------------------------------------*/
  /* allocate bi array and free space for accumulating nonzero column info */
  PetscCall(PetscMalloc1(pn+1,&bi));
  bi[0] = 0;

  /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
  nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(pdti[pn],PetscIntSumTruncate(poti[pon],ai[am])));
  PetscCall(PetscFreeSpaceGet(nnz,&free_space));
  current_space = free_space;

  PetscCall(PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci));
  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); /* points to the next i-structure of k-th received i-structure  */
  }

  PetscCall(PetscLLCondensedCreate_Scalable(Armax,&lnk));
  MatPreallocateBegin(comm,pn,A->cmap->n,dnz,onz);
  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 */
      anz  = ai[row+1] - ai[row];
      Jptr = aj + ai[row];
      /* add non-zero cols of AP into the sorted linked list lnk */
      PetscCall(PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk));
    }

    /* 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];
        PetscCall(PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk));
        nextrow[k]++; nextci[k]++;
      }
    }

    /* add missing diagonal entry */
    if (C->force_diagonals) {
      k = i + owners[rank]; /* column index */
      PetscCall(PetscLLCondensedAddSorted_Scalable(1,&k,lnk));
    }

    nnz = lnk[0];

    /* if free space is not available, make more free space */
    if (current_space->local_remaining<nnz) {
      PetscCall(PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space));
      nspacedouble++;
    }
    /* copy data into free space, then initialize lnk */
    PetscCall(PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk));
    PetscCall(MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz));

    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;
  }
  PetscCall(PetscFree3(buf_ri_k,nextrow,nextci));

  PetscCall(PetscMalloc1(bi[pn]+1,&bj));
  PetscCall(PetscFreeSpaceContiguous(&free_space,bj));
  afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
  if (afill_tmp > afill) afill = afill_tmp;
  PetscCall(PetscLLCondensedDestroy_Scalable(lnk));
  PetscCall(PetscTableDestroy(&ta));
  PetscCall(MatRestoreSymbolicTranspose_SeqAIJ(p->A,&pdti,&pdtj));
  PetscCall(MatRestoreSymbolicTranspose_SeqAIJ(p->B,&poti,&potj));

  /* create symbolic parallel matrix C - why cannot be assembled in Numeric part   */
  /*-------------------------------------------------------------------------------*/
  PetscCall(MatSetSizes(C,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE));
  PetscCall(MatSetBlockSizes(C,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs)));
  PetscCall(MatGetType(A,&mtype));
  PetscCall(MatSetType(C,mtype));
  PetscCall(MatMPIAIJSetPreallocation(C,0,dnz,0,onz));
  MatPreallocateEnd(dnz,onz);
  PetscCall(MatSetBlockSize(C,1));
  PetscCall(MatSetOption(C,MAT_NO_OFF_PROC_ENTRIES,PETSC_TRUE));
  for (i=0; i<pn; i++) {
    row  = i + rstart;
    nnz  = bi[i+1] - bi[i];
    Jptr = bj + bi[i];
    PetscCall(MatSetValues(C,1,&row,nnz,Jptr,NULL,INSERT_VALUES));
  }
  PetscCall(MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY));
  PetscCall(MatSetOption(C,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE));
  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;

  /* attach the supporting struct to C for reuse */
  C->product->data    = ptap;
  C->product->destroy = MatDestroy_MPIAIJ_PtAP;
  ptap->merge         = merge;

  C->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ;

#if defined(PETSC_USE_INFO)
  if (bi[pn] != 0) {
    PetscCall(PetscInfo(C,"Reallocs %" PetscInt_FMT "; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill));
    PetscCall(PetscInfo(C,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill));
  } else {
    PetscCall(PetscInfo(C,"Empty matrix product\n"));
  }
#endif
  PetscFunctionReturn(0);
}

/* ---------------------------------------------------------------- */
static PetscErrorCode MatProductSymbolic_AtB_MPIAIJ_MPIAIJ(Mat C)
{
  Mat_Product    *product = C->product;
  Mat            A=product->A,B=product->B;
  PetscReal      fill=product->fill;
  PetscBool      flg;

  PetscFunctionBegin;
  /* scalable */
  PetscCall(PetscStrcmp(product->alg,"scalable",&flg));
  if (flg) {
    PetscCall(MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(A,B,fill,C));
    goto next;
  }

  /* nonscalable */
  PetscCall(PetscStrcmp(product->alg,"nonscalable",&flg));
  if (flg) {
    PetscCall(MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(A,B,fill,C));
    goto next;
  }

  /* matmatmult */
  PetscCall(PetscStrcmp(product->alg,"at*b",&flg));
  if (flg) {
    Mat       At;
    Mat_APMPI *ptap;

    PetscCall(MatTranspose(A,MAT_INITIAL_MATRIX,&At));
    PetscCall(MatMatMultSymbolic_MPIAIJ_MPIAIJ(At,B,fill,C));
    ptap = (Mat_APMPI*)C->product->data;
    if (ptap) {
      ptap->Pt = At;
      C->product->destroy = MatDestroy_MPIAIJ_PtAP;
    }
    C->ops->transposematmultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult;
    goto next;
  }

  /* backend general code */
  PetscCall(PetscStrcmp(product->alg,"backend",&flg));
  if (flg) {
    PetscCall(MatProductSymbolic_MPIAIJBACKEND(C));
    PetscFunctionReturn(0);
  }

  SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"MatProduct type is not supported");

next:
  C->ops->productnumeric = MatProductNumeric_AtB;
  PetscFunctionReturn(0);
}

/* ---------------------------------------------------------------- */
/* Set options for MatMatMultxxx_MPIAIJ_MPIAIJ */
static PetscErrorCode MatProductSetFromOptions_MPIAIJ_AB(Mat C)
{
  Mat_Product    *product = C->product;
  Mat            A=product->A,B=product->B;
#if defined(PETSC_HAVE_HYPRE)
  const char     *algTypes[5] = {"scalable","nonscalable","seqmpi","backend","hypre"};
  PetscInt       nalg = 5;
#else
  const char     *algTypes[4] = {"scalable","nonscalable","seqmpi","backend",};
  PetscInt       nalg = 4;
#endif
  PetscInt       alg = 1; /* set nonscalable algorithm as default */
  PetscBool      flg;
  MPI_Comm       comm;

  PetscFunctionBegin;
  /* Check matrix local sizes */
  PetscCall(PetscObjectGetComm((PetscObject)C,&comm));
  PetscCheckFalse(A->cmap->rstart != B->rmap->rstart || A->cmap->rend != B->rmap->rend,PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, (%" PetscInt_FMT ", %" PetscInt_FMT ") != (%" PetscInt_FMT ",%" PetscInt_FMT ")",A->cmap->rstart,A->cmap->rend,B->rmap->rstart,B->rmap->rend);

  /* Set "nonscalable" as default algorithm */
  PetscCall(PetscStrcmp(C->product->alg,"default",&flg));
  if (flg) {
    PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));

    /* Set "scalable" as default if BN and local nonzeros of A and B are large */
    if (B->cmap->N > 100000) { /* may switch to scalable algorithm as default */
      MatInfo     Ainfo,Binfo;
      PetscInt    nz_local;
      PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;

      PetscCall(MatGetInfo(A,MAT_LOCAL,&Ainfo));
      PetscCall(MatGetInfo(B,MAT_LOCAL,&Binfo));
      nz_local = (PetscInt)(Ainfo.nz_allocated + Binfo.nz_allocated);

      if (B->cmap->N > product->fill*nz_local) alg_scalable_loc = PETSC_TRUE;
      PetscCall(MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm));

      if (alg_scalable) {
        alg  = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
        PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));
        PetscCall(PetscInfo(B,"Use scalable algorithm, BN %" PetscInt_FMT ", fill*nz_allocated %g\n",B->cmap->N,(double)(product->fill*nz_local)));
      }
    }
  }

  /* Get runtime option */
  if (product->api_user) {
    PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatMatMult","Mat");
    PetscCall(PetscOptionsEList("-matmatmult_via","Algorithmic approach","MatMatMult",algTypes,nalg,algTypes[alg],&alg,&flg));
    PetscOptionsEnd();
  } else {
    PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_AB","Mat");
    PetscCall(PetscOptionsEList("-mat_product_algorithm","Algorithmic approach","MatMatMult",algTypes,nalg,algTypes[alg],&alg,&flg));
    PetscOptionsEnd();
  }
  if (flg) {
    PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));
  }

  C->ops->productsymbolic = MatProductSymbolic_AB_MPIAIJ_MPIAIJ;
  PetscFunctionReturn(0);
}

/* Set options for MatTransposeMatMultXXX_MPIAIJ_MPIAIJ */
static PetscErrorCode MatProductSetFromOptions_MPIAIJ_AtB(Mat C)
{
  Mat_Product    *product = C->product;
  Mat            A=product->A,B=product->B;
  const char     *algTypes[4] = {"scalable","nonscalable","at*b","backend"};
  PetscInt       nalg = 4;
  PetscInt       alg = 1; /* set default algorithm  */
  PetscBool      flg;
  MPI_Comm       comm;

  PetscFunctionBegin;
  /* Check matrix local sizes */
  PetscCall(PetscObjectGetComm((PetscObject)C,&comm));
  PetscCheckFalse(A->rmap->rstart != B->rmap->rstart || A->rmap->rend != B->rmap->rend,PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, A (%" PetscInt_FMT ", %" PetscInt_FMT ") != B (%" PetscInt_FMT ",%" PetscInt_FMT ")",A->rmap->rstart,A->rmap->rend,B->rmap->rstart,B->rmap->rend);

  /* Set default algorithm */
  PetscCall(PetscStrcmp(C->product->alg,"default",&flg));
  if (flg) {
    PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));
  }

  /* Set "scalable" as default if BN and local nonzeros of A and B are large */
  if (alg && B->cmap->N > 100000) { /* may switch to scalable algorithm as default */
    MatInfo     Ainfo,Binfo;
    PetscInt    nz_local;
    PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;

    PetscCall(MatGetInfo(A,MAT_LOCAL,&Ainfo));
    PetscCall(MatGetInfo(B,MAT_LOCAL,&Binfo));
    nz_local = (PetscInt)(Ainfo.nz_allocated + Binfo.nz_allocated);

    if (B->cmap->N > product->fill*nz_local) alg_scalable_loc = PETSC_TRUE;
    PetscCall(MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm));

    if (alg_scalable) {
      alg  = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
      PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));
      PetscCall(PetscInfo(B,"Use scalable algorithm, BN %" PetscInt_FMT ", fill*nz_allocated %g\n",B->cmap->N,(double)(product->fill*nz_local)));
    }
  }

  /* Get runtime option */
  if (product->api_user) {
    PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatTransposeMatMult","Mat");
    PetscCall(PetscOptionsEList("-mattransposematmult_via","Algorithmic approach","MatTransposeMatMult",algTypes,nalg,algTypes[alg],&alg,&flg));
    PetscOptionsEnd();
  } else {
    PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_AtB","Mat");
    PetscCall(PetscOptionsEList("-mat_product_algorithm","Algorithmic approach","MatTransposeMatMult",algTypes,nalg,algTypes[alg],&alg,&flg));
    PetscOptionsEnd();
  }
  if (flg) {
    PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));
  }

  C->ops->productsymbolic = MatProductSymbolic_AtB_MPIAIJ_MPIAIJ;
  PetscFunctionReturn(0);
}

static PetscErrorCode MatProductSetFromOptions_MPIAIJ_PtAP(Mat C)
{
  Mat_Product    *product = C->product;
  Mat            A=product->A,P=product->B;
  MPI_Comm       comm;
  PetscBool      flg;
  PetscInt       alg=1; /* set default algorithm */
#if !defined(PETSC_HAVE_HYPRE)
  const char     *algTypes[5] = {"scalable","nonscalable","allatonce","allatonce_merged","backend"};
  PetscInt       nalg=5;
#else
  const char     *algTypes[6] = {"scalable","nonscalable","allatonce","allatonce_merged","backend","hypre"};
  PetscInt       nalg=6;
#endif
  PetscInt       pN=P->cmap->N;

  PetscFunctionBegin;
  /* Check matrix local sizes */
  PetscCall(PetscObjectGetComm((PetscObject)C,&comm));
  PetscCheckFalse(A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend,PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, Arow (%" PetscInt_FMT ", %" PetscInt_FMT ") != Prow (%" PetscInt_FMT ",%" PetscInt_FMT ")",A->rmap->rstart,A->rmap->rend,P->rmap->rstart,P->rmap->rend);
  PetscCheckFalse(A->cmap->rstart != P->rmap->rstart || A->cmap->rend != P->rmap->rend,PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, Acol (%" PetscInt_FMT ", %" PetscInt_FMT ") != Prow (%" PetscInt_FMT ",%" PetscInt_FMT ")",A->cmap->rstart,A->cmap->rend,P->rmap->rstart,P->rmap->rend);

  /* Set "nonscalable" as default algorithm */
  PetscCall(PetscStrcmp(C->product->alg,"default",&flg));
  if (flg) {
    PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));

    /* Set "scalable" as default if BN and local nonzeros of A and B are large */
    if (pN > 100000) {
      MatInfo     Ainfo,Pinfo;
      PetscInt    nz_local;
      PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;

      PetscCall(MatGetInfo(A,MAT_LOCAL,&Ainfo));
      PetscCall(MatGetInfo(P,MAT_LOCAL,&Pinfo));
      nz_local = (PetscInt)(Ainfo.nz_allocated + Pinfo.nz_allocated);

      if (pN > product->fill*nz_local) alg_scalable_loc = PETSC_TRUE;
      PetscCall(MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm));

      if (alg_scalable) {
        alg = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
        PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));
      }
    }
  }

  /* Get runtime option */
  if (product->api_user) {
    PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatPtAP","Mat");
    PetscCall(PetscOptionsEList("-matptap_via","Algorithmic approach","MatPtAP",algTypes,nalg,algTypes[alg],&alg,&flg));
    PetscOptionsEnd();
  } else {
    PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_PtAP","Mat");
    PetscCall(PetscOptionsEList("-mat_product_algorithm","Algorithmic approach","MatPtAP",algTypes,nalg,algTypes[alg],&alg,&flg));
    PetscOptionsEnd();
  }
  if (flg) {
    PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));
  }

  C->ops->productsymbolic = MatProductSymbolic_PtAP_MPIAIJ_MPIAIJ;
  PetscFunctionReturn(0);
}

static PetscErrorCode MatProductSetFromOptions_MPIAIJ_RARt(Mat C)
{
  Mat_Product *product = C->product;
  Mat         A = product->A,R=product->B;

  PetscFunctionBegin;
  /* Check matrix local sizes */
  PetscCheckFalse(A->cmap->n != R->cmap->n || A->rmap->n != R->cmap->n,PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, A local (%" PetscInt_FMT ", %" PetscInt_FMT "), R local (%" PetscInt_FMT ",%" PetscInt_FMT ")",A->rmap->n,A->rmap->n,R->rmap->n,R->cmap->n);

  C->ops->productsymbolic = MatProductSymbolic_RARt_MPIAIJ_MPIAIJ;
  PetscFunctionReturn(0);
}

/*
 Set options for ABC = A*B*C = A*(B*C); ABC's algorithm must be chosen from AB's algorithm
*/
static PetscErrorCode MatProductSetFromOptions_MPIAIJ_ABC(Mat C)
{
  Mat_Product    *product = C->product;
  PetscBool      flg = PETSC_FALSE;
  PetscInt       alg = 1; /* default algorithm */
  const char     *algTypes[3] = {"scalable","nonscalable","seqmpi"};
  PetscInt       nalg = 3;

  PetscFunctionBegin;
  /* Set default algorithm */
  PetscCall(PetscStrcmp(C->product->alg,"default",&flg));
  if (flg) {
    PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));
  }

  /* Get runtime option */
  if (product->api_user) {
    PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatMatMatMult","Mat");
    PetscCall(PetscOptionsEList("-matmatmatmult_via","Algorithmic approach","MatMatMatMult",algTypes,nalg,algTypes[alg],&alg,&flg));
    PetscOptionsEnd();
  } else {
    PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_ABC","Mat");
    PetscCall(PetscOptionsEList("-mat_product_algorithm","Algorithmic approach","MatProduct_ABC",algTypes,nalg,algTypes[alg],&alg,&flg));
    PetscOptionsEnd();
  }
  if (flg) {
    PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));
  }

  C->ops->matmatmultsymbolic = MatMatMatMultSymbolic_MPIAIJ_MPIAIJ_MPIAIJ;
  C->ops->productsymbolic    = MatProductSymbolic_ABC;
  PetscFunctionReturn(0);
}

PETSC_INTERN PetscErrorCode MatProductSetFromOptions_MPIAIJ(Mat C)
{
  Mat_Product    *product = C->product;

  PetscFunctionBegin;
  switch (product->type) {
  case MATPRODUCT_AB:
    PetscCall(MatProductSetFromOptions_MPIAIJ_AB(C));
    break;
  case MATPRODUCT_AtB:
    PetscCall(MatProductSetFromOptions_MPIAIJ_AtB(C));
    break;
  case MATPRODUCT_PtAP:
    PetscCall(MatProductSetFromOptions_MPIAIJ_PtAP(C));
    break;
  case MATPRODUCT_RARt:
    PetscCall(MatProductSetFromOptions_MPIAIJ_RARt(C));
    break;
  case MATPRODUCT_ABC:
    PetscCall(MatProductSetFromOptions_MPIAIJ_ABC(C));
    break;
  default:
    break;
  }
  PetscFunctionReturn(0);
}