xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision 99aa8e072276db95456e154c852d5b07033be0d8)

/*
  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_ABt_MPIAIJ_MPIAIJ(Mat C)
{
  Mat_Product *product = C->product;
  Mat          B       = product->B;

  PetscFunctionBegin;
  PetscCall(MatTranspose(B, MAT_INITIAL_MATRIX, &product->B));
  PetscCall(MatDestroy(&B));
  PetscCall(MatProductSymbolic_AB_MPIAIJ_MPIAIJ(C));
  PetscFunctionReturn(PETSC_SUCCESS);
}

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(PETSC_SUCCESS);
  }

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

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

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

#if defined(PETSC_HAVE_HYPRE)
  PetscCall(PetscStrcmp(alg, "hypre", &flg));
  if (flg) {
    PetscCall(MatMatMultSymbolic_AIJ_AIJ_wHYPRE(A, B, fill, C));
    PetscFunctionReturn(PETSC_SUCCESS);
  }
#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(PETSC_SUCCESS);
}

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, *coa;
  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 *dummy1, *dummy2, *dummy3, *dummy4;
  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));
  PetscCheck(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, &dummy1));
  PetscCall(MatSeqAIJGetArrayRead(a->B, &dummy2));
  PetscCall(MatSeqAIJGetArrayRead(ptap->P_loc, &dummy3));
  if (ptap->P_oth) PetscCall(MatSeqAIJGetArrayRead(ptap->P_oth, &dummy4));
  PetscCall(MatSeqAIJGetArrayWrite(c->A, &cda));
  PetscCall(MatSeqAIJGetArrayWrite(c->B, &coa));
  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  = PetscSafePointerPlusOffset(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 = PetscSafePointerPlusOffset(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 = PetscSafePointerPlusOffset(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 = PetscSafePointerPlusOffset(coa, co->i[i]);
    for (; k0 < conz; k0++) {
      ca[k0]        = apa[apJ[k]];
      apa[apJ[k++]] = 0.0;
    }
  }
  PetscCall(MatSeqAIJRestoreArrayRead(a->A, &dummy1));
  PetscCall(MatSeqAIJRestoreArrayRead(a->B, &dummy2));
  PetscCall(MatSeqAIJRestoreArrayRead(ptap->P_loc, &dummy3));
  if (ptap->P_oth) PetscCall(MatSeqAIJRestoreArrayRead(ptap->P_oth, &dummy4));
  PetscCall(MatSeqAIJRestoreArrayWrite(c->A, &cda));
  PetscCall(MatSeqAIJRestoreArrayWrite(c->B, &coa));

  PetscCall(MatAssemblyBegin(C, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(C, MAT_FINAL_ASSEMBLY));
  PetscFunctionReturn(PETSC_SUCCESS);
}

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 + 1, &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], &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(PETSC_SUCCESS);
}

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(PETSC_SUCCESS);
}

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(PETSC_SUCCESS);
}

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(PETSC_SUCCESS);
}

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

static 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(PETSC_SUCCESS);
}

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, 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, numBb;
  MPI_Comm         comm;
  MPI_Datatype     type1, *stype, *rtype;
  const PetscInt  *sindices, *sstarts, *rstarts;
  PetscMPIInt     *disp, nsends, nrecvs, nrows_to, nrows_from;
  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   = B->cmap->bs;
  Bbn1 = Bbn1 / bs * bs; /* Bbn1 is a multiple of bs */
  if (Bbn1 > BN) Bbn1 = BN;
  PetscCallMPI(MPIU_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 (PetscMPIInt i = 0; i < nsends; i++) {
    PetscCall(PetscMPIIntCast(sstarts[i + 1] - sstarts[i], &nrows_to));
    for (PetscInt j = 0; j < nrows_to; j++) PetscCall(PetscMPIIntCast(sindices[sstarts[i] + j], &disp[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 (PetscMPIInt i = 0; i < nrecvs; i++) {
    /* received values from a process form a (nrows_from x Bbn) row block in workB (column-wise) */
    PetscCall(PetscMPIIntCast(rstarts[i + 1] - rstarts[i], &nrows_from));
    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));

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

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: If Bbidx = 0, uses B = Bb, else B = Bb1, see MatMatMultSymbolic_MPIAIJ_MPIDense()
*/

static 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;
  PetscMPIInt        nsends, nrecvs;
  MPI_Request       *swaits, *rwaits;
  MPI_Comm           comm;
  PetscMPIInt        tag = ((PetscObject)ctx)->tag, ncols, nrows, 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");
  PetscCall(PetscMPIIntCast(B->cmap->N, &ncols));
  PetscCall(PetscMPIIntCast(aij->B->cmap->n, &nrows));
  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 (PetscMPIInt i = 0; i < nrecvs; i++) PetscCallMPI(MPIU_Irecv(rvalues + (rstarts[i] - rstarts[0]), ncols, rtype[i], rprocs[i], tag, comm, rwaits + i));

  stype = contents->stype;
  for (PetscMPIInt i = 0; i < nsends; i++) PetscCallMPI(MPIU_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(PETSC_SUCCESS);
}

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_CURRENT, &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;
    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 (PetscInt i = 0; i < BN; i += n) {
      PetscCall(MatDenseGetSubMatrix(B, PETSC_DECIDE, PETSC_DECIDE, i, PetscMin(i + n, BN), &Bb));
      PetscCall(MatDenseGetSubMatrix(C, PETSC_DECIDE, PETSC_DECIDE, 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(PETSC_SUCCESS);
}

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));
  PetscCheck(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 = PetscSafePointerPlusOffset(ao->j, aoi[i]);
    aoa = PetscSafePointerPlusOffset(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 = PetscSafePointerPlusOffset(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 = PetscSafePointerPlusOffset(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(PETSC_SUCCESS);
}

/* 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 + 1, &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], &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(PETSC_SUCCESS);
}

/* 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 + 1, &api));
  PetscCall(PetscMalloc1(am + 1, &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, &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], &ptap->apj));
  PetscCall(PetscMalloc1(adpoi[am], &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, &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(PETSC_SUCCESS);
}

/* 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(MatTransposeSetPrecursor(P, Pt));
  PetscCall(MatTranspose(P, MAT_REUSE_MATRIX, &Pt));
  PetscCall(MatMatMultNumeric_MPIAIJ_MPIAIJ(Pt, A, C));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* 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, rstart;
  PetscBT                  lnkbt;
  PetscMPIInt              tagi, tagj, *len_si, *len_s, *len_ri, nrecv, proc, nsend;
  PETSC_UNUSED PetscMPIInt icompleted = 0;
  PetscInt               **buf_rj, **buf_ri, **buf_ri_k, row, ncols, *cols;
  PetscInt                 len, *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;
  PetscHMapI               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));

  /* (0) compute Rd = Pd^T, Ro = Po^T  */
  PetscCall(MatTranspose(p->A, MAT_INITIAL_MATRIX, &ptap->Rd));
  PetscCall(MatTranspose(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 + 1, &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, &swaits));
  for (proc = 0, k = 0; proc < size; proc++) {
    if (!len_s[proc]) continue;
    i = owners_co[proc];
    PetscCallMPI(MPIU_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(PetscHMapICreateWithSize(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(PetscHMapISet(ta, *(Jptr + j) + 1, 1));
  }
  PetscCall(PetscHMapIGetSize(ta, &Crmax));
  PetscCall(PetscHMapIDestroy(&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, &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(MPIU_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 nonzeros 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));
  PetscCall(PetscLayoutSetBlockSize(C->rmap, P->cmap->bs));
  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(PETSC_SUCCESS);
}

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));

  /* These matrices are obtained in MatTransposeMatMultSymbolic() */
  /* 1) get R = Pd^T, Ro = Po^T */
  PetscCall(MatTransposeSetPrecursor(p->A, ptap->Rd));
  PetscCall(MatTranspose(p->A, MAT_REUSE_MATRIX, &ptap->Rd));
  PetscCall(MatTransposeSetPrecursor(p->B, ptap->Ro));
  PetscCall(MatTranspose(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));
  PetscFunctionReturn(PETSC_SUCCESS);
}

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           *ap;
  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, proc;
  PetscInt            *owners, 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);
  ap = (Mat_APMPI *)C->product->data;
  PetscCheck(ap, PetscObjectComm((PetscObject)C), PETSC_ERR_ARG_WRONGSTATE, "PtA cannot be computed. Missing data");
  PetscCheck(ap->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 = ap->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], &coa));
  bi     = merge->bi;
  bj     = merge->bj;
  owners = merge->rowmap->range;
  PetscCall(PetscCalloc1(bi[cm], &ba));

  /* get A_loc by taking all local rows of A */
  A_loc = ap->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, &s_waits, size, &status));
  for (proc = 0, k = 0; proc < size; proc++) {
    if (!len_s[proc]) continue;
    i = merge->owners_co[proc];
    PetscCallMPI(MPIU_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(PETSC_SUCCESS);
}

PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P, Mat A, PetscReal fill, Mat C)
{
  Mat                  A_loc;
  Mat_APMPI           *ap;
  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, proc;
  PetscInt           **buf_rj, **buf_ri, **buf_ri_k;
  PetscInt             len, *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, Armax;
  Mat_SeqAIJ          *a_loc;
  PetscHMapI           ta;
  MatType              mtype;

  PetscFunctionBegin;
  PetscCall(PetscObjectGetComm((PetscObject)A, &comm));
  /* check if matrix local sizes are compatible */
  PetscCheck(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(&ap));

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

  ap->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(PetscHMapICreateWithSize(A->cmap->n + a->B->cmap->N, &ta));
  MatRowMergeMax_SeqAIJ(a_loc, am, ta);
  PetscCall(PetscHMapIGetSize(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], &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 + 1, &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_s[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, &swaits));
  for (proc = 0, k = 0; proc < size; proc++) {
    if (!len_s[proc]) continue;
    i = owners_co[proc];
    PetscCallMPI(MPIU_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(PetscHMapISet(ta, *(Jptr + j) + 1, 1));
  }
  PetscCall(PetscHMapIGetSize(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, &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(MPIU_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], &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(PetscHMapIDestroy(&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, P->cmap->bs, 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    = ap;
  C->product->destroy = MatDestroy_MPIAIJ_PtAP;
  ap->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(PETSC_SUCCESS);
}

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(PETSC_SUCCESS);
  }

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

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

/* 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;
  PetscCall(PetscObjectGetComm((PetscObject)C, &comm));

  /* Set "nonscalable" as default algorithm */
  PetscCall(PetscStrcmp(C->product->alg, "default", &flg));
  if (flg) {
    PetscCall(MatProductSetAlgorithm(C, 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;
      PetscCallMPI(MPIU_Allreduce(&alg_scalable_loc, &alg_scalable, 1, MPI_C_BOOL, MPI_LOR, comm));

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

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

static PetscErrorCode MatProductSetFromOptions_MPIAIJ_ABt(Mat C)
{
  PetscFunctionBegin;
  PetscCall(MatProductSetFromOptions_MPIAIJ_AB(C));
  C->ops->productsymbolic = MatProductSymbolic_ABt_MPIAIJ_MPIAIJ;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* 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));
  PetscCheck(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, 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;
    PetscCallMPI(MPIU_Allreduce(&alg_scalable_loc, &alg_scalable, 1, MPI_C_BOOL, MPI_LOR, comm));

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

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

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));
  PetscCheck(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);
  PetscCheck(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, 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;
      PetscCallMPI(MPIU_Allreduce(&alg_scalable_loc, &alg_scalable, 1, MPI_C_BOOL, MPI_LOR, comm));

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

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

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

  PetscFunctionBegin;
  /* Check matrix local sizes */
  PetscCheck(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(PETSC_SUCCESS);
}

/*
 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, 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, algTypes[alg]));

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

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_ABt:
    PetscCall(MatProductSetFromOptions_MPIAIJ_ABt(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(PETSC_SUCCESS);
}