xref: /petsc/src/mat/impls/aij/seq/matmatmult.c (revision 48a46eb9bd028bec07ec0f396b1a3abb43f14558)

/*
  Defines matrix-matrix product routines for pairs of SeqAIJ matrices
          C = A * B
*/

#include <../src/mat/impls/aij/seq/aij.h> /*I "petscmat.h" I*/
#include <../src/mat/utils/freespace.h>
#include <petscbt.h>
#include <petsc/private/isimpl.h>
#include <../src/mat/impls/dense/seq/dense.h>

PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ(Mat A, Mat B, Mat C) {
  PetscFunctionBegin;
  if (C->ops->matmultnumeric) PetscCall((*C->ops->matmultnumeric)(A, B, C));
  else PetscCall(MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted(A, B, C));
  PetscFunctionReturn(0);
}

/* Modified from MatCreateSeqAIJWithArrays() */
PETSC_INTERN PetscErrorCode MatSetSeqAIJWithArrays_private(MPI_Comm comm, PetscInt m, PetscInt n, PetscInt i[], PetscInt j[], PetscScalar a[], MatType mtype, Mat mat) {
  PetscInt    ii;
  Mat_SeqAIJ *aij;
  PetscBool   isseqaij, osingle, ofree_a, ofree_ij;

  PetscFunctionBegin;
  PetscCheck(m <= 0 || !i[0], PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "i (row indices) must start with 0");
  PetscCall(MatSetSizes(mat, m, n, m, n));

  if (!mtype) {
    PetscCall(PetscObjectBaseTypeCompare((PetscObject)mat, MATSEQAIJ, &isseqaij));
    if (!isseqaij) PetscCall(MatSetType(mat, MATSEQAIJ));
  } else {
    PetscCall(MatSetType(mat, mtype));
  }

  aij      = (Mat_SeqAIJ *)(mat)->data;
  osingle  = aij->singlemalloc;
  ofree_a  = aij->free_a;
  ofree_ij = aij->free_ij;
  /* changes the free flags */
  PetscCall(MatSeqAIJSetPreallocation_SeqAIJ(mat, MAT_SKIP_ALLOCATION, NULL));

  PetscCall(PetscFree(aij->ilen));
  PetscCall(PetscFree(aij->imax));
  PetscCall(PetscMalloc1(m, &aij->imax));
  PetscCall(PetscMalloc1(m, &aij->ilen));
  for (ii = 0, aij->nonzerorowcnt = 0, aij->rmax = 0; ii < m; ii++) {
    const PetscInt rnz = i[ii + 1] - i[ii];
    aij->nonzerorowcnt += !!rnz;
    aij->rmax     = PetscMax(aij->rmax, rnz);
    aij->ilen[ii] = aij->imax[ii] = i[ii + 1] - i[ii];
  }
  aij->maxnz = i[m];
  aij->nz    = i[m];

  if (osingle) {
    PetscCall(PetscFree3(aij->a, aij->j, aij->i));
  } else {
    if (ofree_a) PetscCall(PetscFree(aij->a));
    if (ofree_ij) PetscCall(PetscFree(aij->j));
    if (ofree_ij) PetscCall(PetscFree(aij->i));
  }
  aij->i            = i;
  aij->j            = j;
  aij->a            = a;
  aij->nonew        = -1; /* this indicates that inserting a new value in the matrix that generates a new nonzero is an error */
  /* default to not retain ownership */
  aij->singlemalloc = PETSC_FALSE;
  aij->free_a       = PETSC_FALSE;
  aij->free_ij      = PETSC_FALSE;
  PetscCall(MatCheckCompressedRow(mat, aij->nonzerorowcnt, &aij->compressedrow, aij->i, m, 0.6));
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  if (product) {
    alg = product->alg;
  } else {
    alg = "sorted";
  }
  /* sorted */
  PetscCall(PetscStrcmp(alg, "sorted", &flg));
  if (flg) {
    PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ_Sorted(A, B, fill, C));
    PetscFunctionReturn(0);
  }

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

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

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

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

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

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

#if defined(PETSC_HAVE_HYPRE)
  PetscCall(PetscStrcmp(alg, "hypre", &flg));
  if (flg) {
    PetscCall(MatMatMultSymbolic_AIJ_AIJ_wHYPRE(A, B, fill, C));
    PetscFunctionReturn(0);
  }
#endif

  SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Mat Product Algorithm is not supported");
}

PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_LLCondensed(Mat A, Mat B, PetscReal fill, Mat C) {
  Mat_SeqAIJ        *a = (Mat_SeqAIJ *)A->data, *b = (Mat_SeqAIJ *)B->data, *c;
  PetscInt          *ai = a->i, *bi = b->i, *ci, *cj;
  PetscInt           am = A->rmap->N, bn = B->cmap->N, bm = B->rmap->N;
  PetscReal          afill;
  PetscInt           i, j, anzi, brow, bnzj, cnzi, *bj, *aj, *lnk, ndouble = 0, Crmax;
  PetscTable         ta;
  PetscBT            lnkbt;
  PetscFreeSpaceList free_space = NULL, current_space = NULL;

  PetscFunctionBegin;
  /* Get ci and cj */
  /*---------------*/
  /* Allocate ci array, arrays for fill computation and */
  /* free space for accumulating nonzero column info */
  PetscCall(PetscMalloc1(am + 2, &ci));
  ci[0] = 0;

  /* create and initialize a linked list */
  PetscCall(PetscTableCreate(bn, bn, &ta));
  MatRowMergeMax_SeqAIJ(b, bm, ta);
  PetscCall(PetscTableGetCount(ta, &Crmax));
  PetscCall(PetscTableDestroy(&ta));

  PetscCall(PetscLLCondensedCreate(Crmax, bn, &lnk, &lnkbt));

  /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
  PetscCall(PetscFreeSpaceGet(PetscRealIntMultTruncate(fill, PetscIntSumTruncate(ai[am], bi[bm])), &free_space));

  current_space = free_space;

  /* Determine ci and cj */
  for (i = 0; i < am; i++) {
    anzi = ai[i + 1] - ai[i];
    aj   = a->j + ai[i];
    for (j = 0; j < anzi; j++) {
      brow = aj[j];
      bnzj = bi[brow + 1] - bi[brow];
      bj   = b->j + bi[brow];
      /* add non-zero cols of B into the sorted linked list lnk */
      PetscCall(PetscLLCondensedAddSorted(bnzj, bj, lnk, lnkbt));
    }
    /* add possible missing diagonal entry */
    if (C->force_diagonals) PetscCall(PetscLLCondensedAddSorted(1, &i, lnk, lnkbt));
    cnzi = lnk[0];

    /* If free space is not available, make more free space */
    /* Double the amount of total space in the list */
    if (current_space->local_remaining < cnzi) {
      PetscCall(PetscFreeSpaceGet(PetscIntSumTruncate(cnzi, current_space->total_array_size), &current_space));
      ndouble++;
    }

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

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

    ci[i + 1] = ci[i] + cnzi;
  }

  /* Column indices are in the list of free space */
  /* Allocate space for cj, initialize cj, and */
  /* destroy list of free space and other temporary array(s) */
  PetscCall(PetscMalloc1(ci[am] + 1, &cj));
  PetscCall(PetscFreeSpaceContiguous(&free_space, cj));
  PetscCall(PetscLLCondensedDestroy(lnk, lnkbt));

  /* put together the new symbolic matrix */
  PetscCall(MatSetSeqAIJWithArrays_private(PetscObjectComm((PetscObject)A), am, bn, ci, cj, NULL, ((PetscObject)A)->type_name, C));
  PetscCall(MatSetBlockSizesFromMats(C, A, B));

  /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
  /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
  c          = (Mat_SeqAIJ *)(C->data);
  c->free_a  = PETSC_FALSE;
  c->free_ij = PETSC_TRUE;
  c->nonew   = 0;

  /* fast, needs non-scalable O(bn) array 'abdense' */
  C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;

  /* set MatInfo */
  afill = (PetscReal)ci[am] / (ai[am] + bi[bm]) + 1.e-5;
  if (afill < 1.0) afill = 1.0;
  C->info.mallocs           = ndouble;
  C->info.fill_ratio_given  = fill;
  C->info.fill_ratio_needed = afill;

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

PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted(Mat A, Mat B, Mat C) {
  PetscLogDouble     flops = 0.0;
  Mat_SeqAIJ        *a     = (Mat_SeqAIJ *)A->data;
  Mat_SeqAIJ        *b     = (Mat_SeqAIJ *)B->data;
  Mat_SeqAIJ        *c     = (Mat_SeqAIJ *)C->data;
  PetscInt          *ai = a->i, *aj = a->j, *bi = b->i, *bj = b->j, *bjj, *ci = c->i, *cj = c->j;
  PetscInt           am = A->rmap->n, cm = C->rmap->n;
  PetscInt           i, j, k, anzi, bnzi, cnzi, brow;
  PetscScalar       *ca, valtmp;
  PetscScalar       *ab_dense;
  PetscContainer     cab_dense;
  const PetscScalar *aa, *ba, *baj;

  PetscFunctionBegin;
  PetscCall(MatSeqAIJGetArrayRead(A, &aa));
  PetscCall(MatSeqAIJGetArrayRead(B, &ba));
  if (!c->a) { /* first call of MatMatMultNumeric_SeqAIJ_SeqAIJ, allocate ca and matmult_abdense */
    PetscCall(PetscMalloc1(ci[cm] + 1, &ca));
    c->a      = ca;
    c->free_a = PETSC_TRUE;
  } else ca = c->a;

  /* TODO this should be done in the symbolic phase */
  /* However, this function is so heavily used (sometimes in an hidden way through multnumeric function pointers
     that is hard to eradicate) */
  PetscCall(PetscObjectQuery((PetscObject)C, "__PETSc__ab_dense", (PetscObject *)&cab_dense));
  if (!cab_dense) {
    PetscCall(PetscMalloc1(B->cmap->N, &ab_dense));
    PetscCall(PetscContainerCreate(PETSC_COMM_SELF, &cab_dense));
    PetscCall(PetscContainerSetPointer(cab_dense, ab_dense));
    PetscCall(PetscContainerSetUserDestroy(cab_dense, PetscContainerUserDestroyDefault));
    PetscCall(PetscObjectCompose((PetscObject)C, "__PETSc__ab_dense", (PetscObject)cab_dense));
    PetscCall(PetscObjectDereference((PetscObject)cab_dense));
  }
  PetscCall(PetscContainerGetPointer(cab_dense, (void **)&ab_dense));
  PetscCall(PetscArrayzero(ab_dense, B->cmap->N));

  /* clean old values in C */
  PetscCall(PetscArrayzero(ca, ci[cm]));
  /* Traverse A row-wise. */
  /* Build the ith row in C by summing over nonzero columns in A, */
  /* the rows of B corresponding to nonzeros of A. */
  for (i = 0; i < am; i++) {
    anzi = ai[i + 1] - ai[i];
    for (j = 0; j < anzi; j++) {
      brow   = aj[j];
      bnzi   = bi[brow + 1] - bi[brow];
      bjj    = bj + bi[brow];
      baj    = ba + bi[brow];
      /* perform dense axpy */
      valtmp = aa[j];
      for (k = 0; k < bnzi; k++) { ab_dense[bjj[k]] += valtmp * baj[k]; }
      flops += 2 * bnzi;
    }
    aj += anzi;
    aa += anzi;

    cnzi = ci[i + 1] - ci[i];
    for (k = 0; k < cnzi; k++) {
      ca[k] += ab_dense[cj[k]];
      ab_dense[cj[k]] = 0.0; /* zero ab_dense */
    }
    flops += cnzi;
    cj += cnzi;
    ca += cnzi;
  }
#if defined(PETSC_HAVE_DEVICE)
  if (C->offloadmask != PETSC_OFFLOAD_UNALLOCATED) C->offloadmask = PETSC_OFFLOAD_CPU;
#endif
  PetscCall(MatAssemblyBegin(C, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(C, MAT_FINAL_ASSEMBLY));
  PetscCall(PetscLogFlops(flops));
  PetscCall(MatSeqAIJRestoreArrayRead(A, &aa));
  PetscCall(MatSeqAIJRestoreArrayRead(B, &ba));
  PetscFunctionReturn(0);
}

PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable(Mat A, Mat B, Mat C) {
  PetscLogDouble     flops = 0.0;
  Mat_SeqAIJ        *a     = (Mat_SeqAIJ *)A->data;
  Mat_SeqAIJ        *b     = (Mat_SeqAIJ *)B->data;
  Mat_SeqAIJ        *c     = (Mat_SeqAIJ *)C->data;
  PetscInt          *ai = a->i, *aj = a->j, *bi = b->i, *bj = b->j, *bjj, *ci = c->i, *cj = c->j;
  PetscInt           am = A->rmap->N, cm = C->rmap->N;
  PetscInt           i, j, k, anzi, bnzi, cnzi, brow;
  PetscScalar       *ca = c->a, valtmp;
  const PetscScalar *aa, *ba, *baj;
  PetscInt           nextb;

  PetscFunctionBegin;
  PetscCall(MatSeqAIJGetArrayRead(A, &aa));
  PetscCall(MatSeqAIJGetArrayRead(B, &ba));
  if (!ca) { /* first call of MatMatMultNumeric_SeqAIJ_SeqAIJ, allocate ca and matmult_abdense */
    PetscCall(PetscMalloc1(ci[cm] + 1, &ca));
    c->a      = ca;
    c->free_a = PETSC_TRUE;
  }

  /* clean old values in C */
  PetscCall(PetscArrayzero(ca, ci[cm]));
  /* Traverse A row-wise. */
  /* Build the ith row in C by summing over nonzero columns in A, */
  /* the rows of B corresponding to nonzeros of A. */
  for (i = 0; i < am; i++) {
    anzi = ai[i + 1] - ai[i];
    cnzi = ci[i + 1] - ci[i];
    for (j = 0; j < anzi; j++) {
      brow   = aj[j];
      bnzi   = bi[brow + 1] - bi[brow];
      bjj    = bj + bi[brow];
      baj    = ba + bi[brow];
      /* perform sparse axpy */
      valtmp = aa[j];
      nextb  = 0;
      for (k = 0; nextb < bnzi; k++) {
        if (cj[k] == bjj[nextb]) { /* ccol == bcol */
          ca[k] += valtmp * baj[nextb++];
        }
      }
      flops += 2 * bnzi;
    }
    aj += anzi;
    aa += anzi;
    cj += cnzi;
    ca += cnzi;
  }
#if defined(PETSC_HAVE_DEVICE)
  if (C->offloadmask != PETSC_OFFLOAD_UNALLOCATED) C->offloadmask = PETSC_OFFLOAD_CPU;
#endif
  PetscCall(MatAssemblyBegin(C, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(C, MAT_FINAL_ASSEMBLY));
  PetscCall(PetscLogFlops(flops));
  PetscCall(MatSeqAIJRestoreArrayRead(A, &aa));
  PetscCall(MatSeqAIJRestoreArrayRead(B, &ba));
  PetscFunctionReturn(0);
}

PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable_fast(Mat A, Mat B, PetscReal fill, Mat C) {
  Mat_SeqAIJ        *a = (Mat_SeqAIJ *)A->data, *b = (Mat_SeqAIJ *)B->data, *c;
  PetscInt          *ai = a->i, *bi = b->i, *ci, *cj;
  PetscInt           am = A->rmap->N, bn = B->cmap->N, bm = B->rmap->N;
  MatScalar         *ca;
  PetscReal          afill;
  PetscInt           i, j, anzi, brow, bnzj, cnzi, *bj, *aj, *lnk, ndouble = 0, Crmax;
  PetscTable         ta;
  PetscFreeSpaceList free_space = NULL, current_space = NULL;

  PetscFunctionBegin;
  /* Get ci and cj - same as MatMatMultSymbolic_SeqAIJ_SeqAIJ except using PetscLLxxx_fast() */
  /*-----------------------------------------------------------------------------------------*/
  /* Allocate arrays for fill computation and free space for accumulating nonzero column */
  PetscCall(PetscMalloc1(am + 2, &ci));
  ci[0] = 0;

  /* create and initialize a linked list */
  PetscCall(PetscTableCreate(bn, bn, &ta));
  MatRowMergeMax_SeqAIJ(b, bm, ta);
  PetscCall(PetscTableGetCount(ta, &Crmax));
  PetscCall(PetscTableDestroy(&ta));

  PetscCall(PetscLLCondensedCreate_fast(Crmax, &lnk));

  /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
  PetscCall(PetscFreeSpaceGet(PetscRealIntMultTruncate(fill, PetscIntSumTruncate(ai[am], bi[bm])), &free_space));
  current_space = free_space;

  /* Determine ci and cj */
  for (i = 0; i < am; i++) {
    anzi = ai[i + 1] - ai[i];
    aj   = a->j + ai[i];
    for (j = 0; j < anzi; j++) {
      brow = aj[j];
      bnzj = bi[brow + 1] - bi[brow];
      bj   = b->j + bi[brow];
      /* add non-zero cols of B into the sorted linked list lnk */
      PetscCall(PetscLLCondensedAddSorted_fast(bnzj, bj, lnk));
    }
    /* add possible missing diagonal entry */
    if (C->force_diagonals) PetscCall(PetscLLCondensedAddSorted_fast(1, &i, lnk));
    cnzi = lnk[1];

    /* If free space is not available, make more free space */
    /* Double the amount of total space in the list */
    if (current_space->local_remaining < cnzi) {
      PetscCall(PetscFreeSpaceGet(PetscIntSumTruncate(cnzi, current_space->total_array_size), &current_space));
      ndouble++;
    }

    /* Copy data into free space, then initialize lnk */
    PetscCall(PetscLLCondensedClean_fast(cnzi, current_space->array, lnk));

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

    ci[i + 1] = ci[i] + cnzi;
  }

  /* Column indices are in the list of free space */
  /* Allocate space for cj, initialize cj, and */
  /* destroy list of free space and other temporary array(s) */
  PetscCall(PetscMalloc1(ci[am] + 1, &cj));
  PetscCall(PetscFreeSpaceContiguous(&free_space, cj));
  PetscCall(PetscLLCondensedDestroy_fast(lnk));

  /* Allocate space for ca */
  PetscCall(PetscCalloc1(ci[am] + 1, &ca));

  /* put together the new symbolic matrix */
  PetscCall(MatSetSeqAIJWithArrays_private(PetscObjectComm((PetscObject)A), am, bn, ci, cj, ca, ((PetscObject)A)->type_name, C));
  PetscCall(MatSetBlockSizesFromMats(C, A, B));

  /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
  /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
  c          = (Mat_SeqAIJ *)(C->data);
  c->free_a  = PETSC_TRUE;
  c->free_ij = PETSC_TRUE;
  c->nonew   = 0;

  /* slower, less memory */
  C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable;

  /* set MatInfo */
  afill = (PetscReal)ci[am] / (ai[am] + bi[bm]) + 1.e-5;
  if (afill < 1.0) afill = 1.0;
  C->info.mallocs           = ndouble;
  C->info.fill_ratio_given  = fill;
  C->info.fill_ratio_needed = afill;

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

PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable(Mat A, Mat B, PetscReal fill, Mat C) {
  Mat_SeqAIJ        *a = (Mat_SeqAIJ *)A->data, *b = (Mat_SeqAIJ *)B->data, *c;
  PetscInt          *ai = a->i, *bi = b->i, *ci, *cj;
  PetscInt           am = A->rmap->N, bn = B->cmap->N, bm = B->rmap->N;
  MatScalar         *ca;
  PetscReal          afill;
  PetscInt           i, j, anzi, brow, bnzj, cnzi, *bj, *aj, *lnk, ndouble = 0, Crmax;
  PetscTable         ta;
  PetscFreeSpaceList free_space = NULL, current_space = NULL;

  PetscFunctionBegin;
  /* Get ci and cj - same as MatMatMultSymbolic_SeqAIJ_SeqAIJ except using PetscLLxxx_Scalalbe() */
  /*---------------------------------------------------------------------------------------------*/
  /* Allocate arrays for fill computation and free space for accumulating nonzero column */
  PetscCall(PetscMalloc1(am + 2, &ci));
  ci[0] = 0;

  /* create and initialize a linked list */
  PetscCall(PetscTableCreate(bn, bn, &ta));
  MatRowMergeMax_SeqAIJ(b, bm, ta);
  PetscCall(PetscTableGetCount(ta, &Crmax));
  PetscCall(PetscTableDestroy(&ta));
  PetscCall(PetscLLCondensedCreate_Scalable(Crmax, &lnk));

  /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
  PetscCall(PetscFreeSpaceGet(PetscRealIntMultTruncate(fill, PetscIntSumTruncate(ai[am], bi[bm])), &free_space));
  current_space = free_space;

  /* Determine ci and cj */
  for (i = 0; i < am; i++) {
    anzi = ai[i + 1] - ai[i];
    aj   = a->j + ai[i];
    for (j = 0; j < anzi; j++) {
      brow = aj[j];
      bnzj = bi[brow + 1] - bi[brow];
      bj   = b->j + bi[brow];
      /* add non-zero cols of B into the sorted linked list lnk */
      PetscCall(PetscLLCondensedAddSorted_Scalable(bnzj, bj, lnk));
    }
    /* add possible missing diagonal entry */
    if (C->force_diagonals) PetscCall(PetscLLCondensedAddSorted_Scalable(1, &i, lnk));

    cnzi = lnk[0];

    /* If free space is not available, make more free space */
    /* Double the amount of total space in the list */
    if (current_space->local_remaining < cnzi) {
      PetscCall(PetscFreeSpaceGet(PetscIntSumTruncate(cnzi, current_space->total_array_size), &current_space));
      ndouble++;
    }

    /* Copy data into free space, then initialize lnk */
    PetscCall(PetscLLCondensedClean_Scalable(cnzi, current_space->array, lnk));

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

    ci[i + 1] = ci[i] + cnzi;
  }

  /* Column indices are in the list of free space */
  /* Allocate space for cj, initialize cj, and */
  /* destroy list of free space and other temporary array(s) */
  PetscCall(PetscMalloc1(ci[am] + 1, &cj));
  PetscCall(PetscFreeSpaceContiguous(&free_space, cj));
  PetscCall(PetscLLCondensedDestroy_Scalable(lnk));

  /* Allocate space for ca */
  /*-----------------------*/
  PetscCall(PetscCalloc1(ci[am] + 1, &ca));

  /* put together the new symbolic matrix */
  PetscCall(MatSetSeqAIJWithArrays_private(PetscObjectComm((PetscObject)A), am, bn, ci, cj, ca, ((PetscObject)A)->type_name, C));
  PetscCall(MatSetBlockSizesFromMats(C, A, B));

  /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
  /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
  c          = (Mat_SeqAIJ *)(C->data);
  c->free_a  = PETSC_TRUE;
  c->free_ij = PETSC_TRUE;
  c->nonew   = 0;

  /* slower, less memory */
  C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable;

  /* set MatInfo */
  afill = (PetscReal)ci[am] / (ai[am] + bi[bm]) + 1.e-5;
  if (afill < 1.0) afill = 1.0;
  C->info.mallocs           = ndouble;
  C->info.fill_ratio_given  = fill;
  C->info.fill_ratio_needed = afill;

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

PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Heap(Mat A, Mat B, PetscReal fill, Mat C) {
  Mat_SeqAIJ        *a = (Mat_SeqAIJ *)A->data, *b = (Mat_SeqAIJ *)B->data, *c;
  const PetscInt    *ai = a->i, *bi = b->i, *aj = a->j, *bj = b->j;
  PetscInt          *ci, *cj, *bb;
  PetscInt           am = A->rmap->N, bn = B->cmap->N, bm = B->rmap->N;
  PetscReal          afill;
  PetscInt           i, j, col, ndouble = 0;
  PetscFreeSpaceList free_space = NULL, current_space = NULL;
  PetscHeap          h;

  PetscFunctionBegin;
  /* Get ci and cj - by merging sorted rows using a heap */
  /*---------------------------------------------------------------------------------------------*/
  /* Allocate arrays for fill computation and free space for accumulating nonzero column */
  PetscCall(PetscMalloc1(am + 2, &ci));
  ci[0] = 0;

  /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
  PetscCall(PetscFreeSpaceGet(PetscRealIntMultTruncate(fill, PetscIntSumTruncate(ai[am], bi[bm])), &free_space));
  current_space = free_space;

  PetscCall(PetscHeapCreate(a->rmax, &h));
  PetscCall(PetscMalloc1(a->rmax, &bb));

  /* Determine ci and cj */
  for (i = 0; i < am; i++) {
    const PetscInt  anzi = ai[i + 1] - ai[i]; /* number of nonzeros in this row of A, this is the number of rows of B that we merge */
    const PetscInt *acol = aj + ai[i];        /* column indices of nonzero entries in this row */
    ci[i + 1]            = ci[i];
    /* Populate the min heap */
    for (j = 0; j < anzi; j++) {
      bb[j] = bi[acol[j]];           /* bb points at the start of the row */
      if (bb[j] < bi[acol[j] + 1]) { /* Add if row is nonempty */
        PetscCall(PetscHeapAdd(h, j, bj[bb[j]++]));
      }
    }
    /* Pick off the min element, adding it to free space */
    PetscCall(PetscHeapPop(h, &j, &col));
    while (j >= 0) {
      if (current_space->local_remaining < 1) { /* double the size, but don't exceed 16 MiB */
        PetscCall(PetscFreeSpaceGet(PetscMin(PetscIntMultTruncate(2, current_space->total_array_size), 16 << 20), &current_space));
        ndouble++;
      }
      *(current_space->array++) = col;
      current_space->local_used++;
      current_space->local_remaining--;
      ci[i + 1]++;

      /* stash if anything else remains in this row of B */
      if (bb[j] < bi[acol[j] + 1]) PetscCall(PetscHeapStash(h, j, bj[bb[j]++]));
      while (1) { /* pop and stash any other rows of B that also had an entry in this column */
        PetscInt j2, col2;
        PetscCall(PetscHeapPeek(h, &j2, &col2));
        if (col2 != col) break;
        PetscCall(PetscHeapPop(h, &j2, &col2));
        if (bb[j2] < bi[acol[j2] + 1]) PetscCall(PetscHeapStash(h, j2, bj[bb[j2]++]));
      }
      /* Put any stashed elements back into the min heap */
      PetscCall(PetscHeapUnstash(h));
      PetscCall(PetscHeapPop(h, &j, &col));
    }
  }
  PetscCall(PetscFree(bb));
  PetscCall(PetscHeapDestroy(&h));

  /* Column indices are in the list of free space */
  /* Allocate space for cj, initialize cj, and */
  /* destroy list of free space and other temporary array(s) */
  PetscCall(PetscMalloc1(ci[am], &cj));
  PetscCall(PetscFreeSpaceContiguous(&free_space, cj));

  /* put together the new symbolic matrix */
  PetscCall(MatSetSeqAIJWithArrays_private(PetscObjectComm((PetscObject)A), am, bn, ci, cj, NULL, ((PetscObject)A)->type_name, C));
  PetscCall(MatSetBlockSizesFromMats(C, A, B));

  /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
  /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
  c          = (Mat_SeqAIJ *)(C->data);
  c->free_a  = PETSC_TRUE;
  c->free_ij = PETSC_TRUE;
  c->nonew   = 0;

  C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;

  /* set MatInfo */
  afill = (PetscReal)ci[am] / (ai[am] + bi[bm]) + 1.e-5;
  if (afill < 1.0) afill = 1.0;
  C->info.mallocs           = ndouble;
  C->info.fill_ratio_given  = fill;
  C->info.fill_ratio_needed = afill;

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

PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_BTHeap(Mat A, Mat B, PetscReal fill, Mat C) {
  Mat_SeqAIJ        *a = (Mat_SeqAIJ *)A->data, *b = (Mat_SeqAIJ *)B->data, *c;
  const PetscInt    *ai = a->i, *bi = b->i, *aj = a->j, *bj = b->j;
  PetscInt          *ci, *cj, *bb;
  PetscInt           am = A->rmap->N, bn = B->cmap->N, bm = B->rmap->N;
  PetscReal          afill;
  PetscInt           i, j, col, ndouble = 0;
  PetscFreeSpaceList free_space = NULL, current_space = NULL;
  PetscHeap          h;
  PetscBT            bt;

  PetscFunctionBegin;
  /* Get ci and cj - using a heap for the sorted rows, but use BT so that each index is only added once */
  /*---------------------------------------------------------------------------------------------*/
  /* Allocate arrays for fill computation and free space for accumulating nonzero column */
  PetscCall(PetscMalloc1(am + 2, &ci));
  ci[0] = 0;

  /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
  PetscCall(PetscFreeSpaceGet(PetscRealIntMultTruncate(fill, PetscIntSumTruncate(ai[am], bi[bm])), &free_space));

  current_space = free_space;

  PetscCall(PetscHeapCreate(a->rmax, &h));
  PetscCall(PetscMalloc1(a->rmax, &bb));
  PetscCall(PetscBTCreate(bn, &bt));

  /* Determine ci and cj */
  for (i = 0; i < am; i++) {
    const PetscInt  anzi = ai[i + 1] - ai[i];    /* number of nonzeros in this row of A, this is the number of rows of B that we merge */
    const PetscInt *acol = aj + ai[i];           /* column indices of nonzero entries in this row */
    const PetscInt *fptr = current_space->array; /* Save beginning of the row so we can clear the BT later */
    ci[i + 1]            = ci[i];
    /* Populate the min heap */
    for (j = 0; j < anzi; j++) {
      PetscInt brow = acol[j];
      for (bb[j] = bi[brow]; bb[j] < bi[brow + 1]; bb[j]++) {
        PetscInt bcol = bj[bb[j]];
        if (!PetscBTLookupSet(bt, bcol)) { /* new entry */
          PetscCall(PetscHeapAdd(h, j, bcol));
          bb[j]++;
          break;
        }
      }
    }
    /* Pick off the min element, adding it to free space */
    PetscCall(PetscHeapPop(h, &j, &col));
    while (j >= 0) {
      if (current_space->local_remaining < 1) { /* double the size, but don't exceed 16 MiB */
        fptr = NULL;                            /* need PetscBTMemzero */
        PetscCall(PetscFreeSpaceGet(PetscMin(PetscIntMultTruncate(2, current_space->total_array_size), 16 << 20), &current_space));
        ndouble++;
      }
      *(current_space->array++) = col;
      current_space->local_used++;
      current_space->local_remaining--;
      ci[i + 1]++;

      /* stash if anything else remains in this row of B */
      for (; bb[j] < bi[acol[j] + 1]; bb[j]++) {
        PetscInt bcol = bj[bb[j]];
        if (!PetscBTLookupSet(bt, bcol)) { /* new entry */
          PetscCall(PetscHeapAdd(h, j, bcol));
          bb[j]++;
          break;
        }
      }
      PetscCall(PetscHeapPop(h, &j, &col));
    }
    if (fptr) { /* Clear the bits for this row */
      for (; fptr < current_space->array; fptr++) PetscCall(PetscBTClear(bt, *fptr));
    } else { /* We reallocated so we don't remember (easily) how to clear only the bits we changed */
      PetscCall(PetscBTMemzero(bn, bt));
    }
  }
  PetscCall(PetscFree(bb));
  PetscCall(PetscHeapDestroy(&h));
  PetscCall(PetscBTDestroy(&bt));

  /* Column indices are in the list of free space */
  /* Allocate space for cj, initialize cj, and */
  /* destroy list of free space and other temporary array(s) */
  PetscCall(PetscMalloc1(ci[am], &cj));
  PetscCall(PetscFreeSpaceContiguous(&free_space, cj));

  /* put together the new symbolic matrix */
  PetscCall(MatSetSeqAIJWithArrays_private(PetscObjectComm((PetscObject)A), am, bn, ci, cj, NULL, ((PetscObject)A)->type_name, C));
  PetscCall(MatSetBlockSizesFromMats(C, A, B));

  /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
  /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
  c          = (Mat_SeqAIJ *)(C->data);
  c->free_a  = PETSC_TRUE;
  c->free_ij = PETSC_TRUE;
  c->nonew   = 0;

  C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;

  /* set MatInfo */
  afill = (PetscReal)ci[am] / (ai[am] + bi[bm]) + 1.e-5;
  if (afill < 1.0) afill = 1.0;
  C->info.mallocs           = ndouble;
  C->info.fill_ratio_given  = fill;
  C->info.fill_ratio_needed = afill;

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

PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_RowMerge(Mat A, Mat B, PetscReal fill, Mat C) {
  Mat_SeqAIJ     *a = (Mat_SeqAIJ *)A->data, *b = (Mat_SeqAIJ *)B->data, *c;
  const PetscInt *ai = a->i, *bi = b->i, *aj = a->j, *bj = b->j, *inputi, *inputj, *inputcol, *inputcol_L1;
  PetscInt       *ci, *cj, *outputj, worki_L1[9], worki_L2[9];
  PetscInt        c_maxmem, a_maxrownnz = 0, a_rownnz;
  const PetscInt  workcol[8] = {0, 1, 2, 3, 4, 5, 6, 7};
  const PetscInt  am = A->rmap->N, bn = B->cmap->N, bm = B->rmap->N;
  const PetscInt *brow_ptr[8], *brow_end[8];
  PetscInt        window[8];
  PetscInt        window_min, old_window_min, ci_nnz, outputi_nnz = 0, L1_nrows, L2_nrows;
  PetscInt        i, k, ndouble = 0, L1_rowsleft, rowsleft;
  PetscReal       afill;
  PetscInt       *workj_L1, *workj_L2, *workj_L3;
  PetscInt        L1_nnz, L2_nnz;

  /* Step 1: Get upper bound on memory required for allocation.
             Because of the way virtual memory works,
             only the memory pages that are actually needed will be physically allocated. */
  PetscFunctionBegin;
  PetscCall(PetscMalloc1(am + 1, &ci));
  for (i = 0; i < am; i++) {
    const PetscInt  anzi = ai[i + 1] - ai[i]; /* number of nonzeros in this row of A, this is the number of rows of B that we merge */
    const PetscInt *acol = aj + ai[i];        /* column indices of nonzero entries in this row */
    a_rownnz             = 0;
    for (k = 0; k < anzi; ++k) {
      a_rownnz += bi[acol[k] + 1] - bi[acol[k]];
      if (a_rownnz > bn) {
        a_rownnz = bn;
        break;
      }
    }
    a_maxrownnz = PetscMax(a_maxrownnz, a_rownnz);
  }
  /* temporary work areas for merging rows */
  PetscCall(PetscMalloc1(a_maxrownnz * 8, &workj_L1));
  PetscCall(PetscMalloc1(a_maxrownnz * 8, &workj_L2));
  PetscCall(PetscMalloc1(a_maxrownnz, &workj_L3));

  /* This should be enough for almost all matrices. If not, memory is reallocated later. */
  c_maxmem = 8 * (ai[am] + bi[bm]);
  /* Step 2: Populate pattern for C */
  PetscCall(PetscMalloc1(c_maxmem, &cj));

  ci_nnz      = 0;
  ci[0]       = 0;
  worki_L1[0] = 0;
  worki_L2[0] = 0;
  for (i = 0; i < am; i++) {
    const PetscInt  anzi = ai[i + 1] - ai[i]; /* number of nonzeros in this row of A, this is the number of rows of B that we merge */
    const PetscInt *acol = aj + ai[i];        /* column indices of nonzero entries in this row */
    rowsleft             = anzi;
    inputcol_L1          = acol;
    L2_nnz               = 0;
    L2_nrows             = 1; /* Number of rows to be merged on Level 3. output of L3 already exists -> initial value 1   */
    worki_L2[1]          = 0;
    outputi_nnz          = 0;

    /* If the number of indices in C so far + the max number of columns in the next row > c_maxmem  -> allocate more memory */
    while (ci_nnz + a_maxrownnz > c_maxmem) {
      c_maxmem *= 2;
      ndouble++;
      PetscCall(PetscRealloc(sizeof(PetscInt) * c_maxmem, &cj));
    }

    while (rowsleft) {
      L1_rowsleft = PetscMin(64, rowsleft); /* In the inner loop max 64 rows of B can be merged */
      L1_nrows    = 0;
      L1_nnz      = 0;
      inputcol    = inputcol_L1;
      inputi      = bi;
      inputj      = bj;

      /* The following macro is used to specialize for small rows in A.
         This helps with compiler unrolling, improving performance substantially.
          Input:  inputj   inputi  inputcol  bn
          Output: outputj  outputi_nnz                       */
#define MatMatMultSymbolic_RowMergeMacro(ANNZ) \
  window_min  = bn; \
  outputi_nnz = 0; \
  for (k = 0; k < ANNZ; ++k) { \
    brow_ptr[k] = inputj + inputi[inputcol[k]]; \
    brow_end[k] = inputj + inputi[inputcol[k] + 1]; \
    window[k]   = (brow_ptr[k] != brow_end[k]) ? *brow_ptr[k] : bn; \
    window_min  = PetscMin(window[k], window_min); \
  } \
  while (window_min < bn) { \
    outputj[outputi_nnz++] = window_min; \
    /* advance front and compute new minimum */ \
    old_window_min         = window_min; \
    window_min             = bn; \
    for (k = 0; k < ANNZ; ++k) { \
      if (window[k] == old_window_min) { \
        brow_ptr[k]++; \
        window[k] = (brow_ptr[k] != brow_end[k]) ? *brow_ptr[k] : bn; \
      } \
      window_min = PetscMin(window[k], window_min); \
    } \
  }

      /************** L E V E L  1 ***************/
      /* Merge up to 8 rows of B to L1 work array*/
      while (L1_rowsleft) {
        outputi_nnz = 0;
        if (anzi > 8) outputj = workj_L1 + L1_nnz; /* Level 1 rowmerge*/
        else outputj = cj + ci_nnz;                /* Merge directly to C */

        switch (L1_rowsleft) {
        case 1:
          brow_ptr[0] = inputj + inputi[inputcol[0]];
          brow_end[0] = inputj + inputi[inputcol[0] + 1];
          for (; brow_ptr[0] != brow_end[0]; ++brow_ptr[0]) outputj[outputi_nnz++] = *brow_ptr[0]; /* copy row in b over */
          inputcol += L1_rowsleft;
          rowsleft -= L1_rowsleft;
          L1_rowsleft = 0;
          break;
        case 2:
          MatMatMultSymbolic_RowMergeMacro(2);
          inputcol += L1_rowsleft;
          rowsleft -= L1_rowsleft;
          L1_rowsleft = 0;
          break;
        case 3:
          MatMatMultSymbolic_RowMergeMacro(3);
          inputcol += L1_rowsleft;
          rowsleft -= L1_rowsleft;
          L1_rowsleft = 0;
          break;
        case 4:
          MatMatMultSymbolic_RowMergeMacro(4);
          inputcol += L1_rowsleft;
          rowsleft -= L1_rowsleft;
          L1_rowsleft = 0;
          break;
        case 5:
          MatMatMultSymbolic_RowMergeMacro(5);
          inputcol += L1_rowsleft;
          rowsleft -= L1_rowsleft;
          L1_rowsleft = 0;
          break;
        case 6:
          MatMatMultSymbolic_RowMergeMacro(6);
          inputcol += L1_rowsleft;
          rowsleft -= L1_rowsleft;
          L1_rowsleft = 0;
          break;
        case 7:
          MatMatMultSymbolic_RowMergeMacro(7);
          inputcol += L1_rowsleft;
          rowsleft -= L1_rowsleft;
          L1_rowsleft = 0;
          break;
        default:
          MatMatMultSymbolic_RowMergeMacro(8);
          inputcol += 8;
          rowsleft -= 8;
          L1_rowsleft -= 8;
          break;
        }
        inputcol_L1 = inputcol;
        L1_nnz += outputi_nnz;
        worki_L1[++L1_nrows] = L1_nnz;
      }

      /********************** L E V E L  2 ************************/
      /* Merge from L1 work array to either C or to L2 work array */
      if (anzi > 8) {
        inputi      = worki_L1;
        inputj      = workj_L1;
        inputcol    = workcol;
        outputi_nnz = 0;

        if (anzi <= 64) outputj = cj + ci_nnz; /* Merge from L1 work array to C */
        else outputj = workj_L2 + L2_nnz;      /* Merge from L1 work array to L2 work array */

        switch (L1_nrows) {
        case 1:
          brow_ptr[0] = inputj + inputi[inputcol[0]];
          brow_end[0] = inputj + inputi[inputcol[0] + 1];
          for (; brow_ptr[0] != brow_end[0]; ++brow_ptr[0]) outputj[outputi_nnz++] = *brow_ptr[0]; /* copy row in b over */
          break;
        case 2: MatMatMultSymbolic_RowMergeMacro(2); break;
        case 3: MatMatMultSymbolic_RowMergeMacro(3); break;
        case 4: MatMatMultSymbolic_RowMergeMacro(4); break;
        case 5: MatMatMultSymbolic_RowMergeMacro(5); break;
        case 6: MatMatMultSymbolic_RowMergeMacro(6); break;
        case 7: MatMatMultSymbolic_RowMergeMacro(7); break;
        case 8: MatMatMultSymbolic_RowMergeMacro(8); break;
        default: SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "MatMatMult logic error: Not merging 1-8 rows from L1 work array!");
        }
        L2_nnz += outputi_nnz;
        worki_L2[++L2_nrows] = L2_nnz;

        /************************ L E V E L  3 **********************/
        /* Merge from L2 work array to either C or to L2 work array */
        if (anzi > 64 && (L2_nrows == 8 || rowsleft == 0)) {
          inputi      = worki_L2;
          inputj      = workj_L2;
          inputcol    = workcol;
          outputi_nnz = 0;
          if (rowsleft) outputj = workj_L3;
          else outputj = cj + ci_nnz;
          switch (L2_nrows) {
          case 1:
            brow_ptr[0] = inputj + inputi[inputcol[0]];
            brow_end[0] = inputj + inputi[inputcol[0] + 1];
            for (; brow_ptr[0] != brow_end[0]; ++brow_ptr[0]) outputj[outputi_nnz++] = *brow_ptr[0]; /* copy row in b over */
            break;
          case 2: MatMatMultSymbolic_RowMergeMacro(2); break;
          case 3: MatMatMultSymbolic_RowMergeMacro(3); break;
          case 4: MatMatMultSymbolic_RowMergeMacro(4); break;
          case 5: MatMatMultSymbolic_RowMergeMacro(5); break;
          case 6: MatMatMultSymbolic_RowMergeMacro(6); break;
          case 7: MatMatMultSymbolic_RowMergeMacro(7); break;
          case 8: MatMatMultSymbolic_RowMergeMacro(8); break;
          default: SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "MatMatMult logic error: Not merging 1-8 rows from L2 work array!");
          }
          L2_nrows    = 1;
          L2_nnz      = outputi_nnz;
          worki_L2[1] = outputi_nnz;
          /* Copy to workj_L2 */
          if (rowsleft) {
            for (k = 0; k < outputi_nnz; ++k) workj_L2[k] = outputj[k];
          }
        }
      }
    } /* while (rowsleft) */
#undef MatMatMultSymbolic_RowMergeMacro

    /* terminate current row */
    ci_nnz += outputi_nnz;
    ci[i + 1] = ci_nnz;
  }

  /* Step 3: Create the new symbolic matrix */
  PetscCall(MatSetSeqAIJWithArrays_private(PetscObjectComm((PetscObject)A), am, bn, ci, cj, NULL, ((PetscObject)A)->type_name, C));
  PetscCall(MatSetBlockSizesFromMats(C, A, B));

  /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
  /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
  c          = (Mat_SeqAIJ *)(C->data);
  c->free_a  = PETSC_TRUE;
  c->free_ij = PETSC_TRUE;
  c->nonew   = 0;

  C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;

  /* set MatInfo */
  afill = (PetscReal)ci[am] / (ai[am] + bi[bm]) + 1.e-5;
  if (afill < 1.0) afill = 1.0;
  C->info.mallocs           = ndouble;
  C->info.fill_ratio_given  = fill;
  C->info.fill_ratio_needed = afill;

#if defined(PETSC_USE_INFO)
  if (ci[am]) {
    PetscCall(PetscInfo(C, "Reallocs %" PetscInt_FMT "; Fill ratio: given %g needed %g.\n", ndouble, (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

  /* Step 4: Free temporary work areas */
  PetscCall(PetscFree(workj_L1));
  PetscCall(PetscFree(workj_L2));
  PetscCall(PetscFree(workj_L3));
  PetscFunctionReturn(0);
}

/* concatenate unique entries and then sort */
PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Sorted(Mat A, Mat B, PetscReal fill, Mat C) {
  Mat_SeqAIJ     *a = (Mat_SeqAIJ *)A->data, *b = (Mat_SeqAIJ *)B->data, *c;
  const PetscInt *ai = a->i, *bi = b->i, *aj = a->j, *bj = b->j;
  PetscInt       *ci, *cj, bcol;
  PetscInt        am = A->rmap->N, bn = B->cmap->N, bm = B->rmap->N;
  PetscReal       afill;
  PetscInt        i, j, ndouble = 0;
  PetscSegBuffer  seg, segrow;
  char           *seen;

  PetscFunctionBegin;
  PetscCall(PetscMalloc1(am + 1, &ci));
  ci[0] = 0;

  /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
  PetscCall(PetscSegBufferCreate(sizeof(PetscInt), (PetscInt)(fill * (ai[am] + bi[bm])), &seg));
  PetscCall(PetscSegBufferCreate(sizeof(PetscInt), 100, &segrow));
  PetscCall(PetscCalloc1(bn, &seen));

  /* Determine ci and cj */
  for (i = 0; i < am; i++) {
    const PetscInt  anzi = ai[i + 1] - ai[i]; /* number of nonzeros in this row of A, this is the number of rows of B that we merge */
    const PetscInt *acol = aj + ai[i];        /* column indices of nonzero entries in this row */
    PetscInt packlen     = 0, *PETSC_RESTRICT crow;

    /* Pack segrow */
    for (j = 0; j < anzi; j++) {
      PetscInt brow = acol[j], bjstart = bi[brow], bjend = bi[brow + 1], k;
      for (k = bjstart; k < bjend; k++) {
        bcol = bj[k];
        if (!seen[bcol]) { /* new entry */
          PetscInt *PETSC_RESTRICT slot;
          PetscCall(PetscSegBufferGetInts(segrow, 1, &slot));
          *slot      = bcol;
          seen[bcol] = 1;
          packlen++;
        }
      }
    }

    /* Check i-th diagonal entry */
    if (C->force_diagonals && !seen[i]) {
      PetscInt *PETSC_RESTRICT slot;
      PetscCall(PetscSegBufferGetInts(segrow, 1, &slot));
      *slot   = i;
      seen[i] = 1;
      packlen++;
    }

    PetscCall(PetscSegBufferGetInts(seg, packlen, &crow));
    PetscCall(PetscSegBufferExtractTo(segrow, crow));
    PetscCall(PetscSortInt(packlen, crow));
    ci[i + 1] = ci[i] + packlen;
    for (j = 0; j < packlen; j++) seen[crow[j]] = 0;
  }
  PetscCall(PetscSegBufferDestroy(&segrow));
  PetscCall(PetscFree(seen));

  /* Column indices are in the segmented buffer */
  PetscCall(PetscSegBufferExtractAlloc(seg, &cj));
  PetscCall(PetscSegBufferDestroy(&seg));

  /* put together the new symbolic matrix */
  PetscCall(MatSetSeqAIJWithArrays_private(PetscObjectComm((PetscObject)A), am, bn, ci, cj, NULL, ((PetscObject)A)->type_name, C));
  PetscCall(MatSetBlockSizesFromMats(C, A, B));

  /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
  /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
  c          = (Mat_SeqAIJ *)(C->data);
  c->free_a  = PETSC_TRUE;
  c->free_ij = PETSC_TRUE;
  c->nonew   = 0;

  C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;

  /* set MatInfo */
  afill = (PetscReal)ci[am] / PetscMax(ai[am] + bi[bm], 1) + 1.e-5;
  if (afill < 1.0) afill = 1.0;
  C->info.mallocs           = ndouble;
  C->info.fill_ratio_given  = fill;
  C->info.fill_ratio_needed = afill;

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

PetscErrorCode MatDestroy_SeqAIJ_MatMatMultTrans(void *data) {
  Mat_MatMatTransMult *abt = (Mat_MatMatTransMult *)data;

  PetscFunctionBegin;
  PetscCall(MatTransposeColoringDestroy(&abt->matcoloring));
  PetscCall(MatDestroy(&abt->Bt_den));
  PetscCall(MatDestroy(&abt->ABt_den));
  PetscCall(PetscFree(abt));
  PetscFunctionReturn(0);
}

PetscErrorCode MatMatTransposeMultSymbolic_SeqAIJ_SeqAIJ(Mat A, Mat B, PetscReal fill, Mat C) {
  Mat                  Bt;
  Mat_MatMatTransMult *abt;
  Mat_Product         *product = C->product;
  char                *alg;

  PetscFunctionBegin;
  PetscCheck(product, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Missing product struct");
  PetscCheck(!product->data, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Extra product struct not empty");

  /* create symbolic Bt */
  PetscCall(MatTransposeSymbolic(B, &Bt));
  PetscCall(MatSetBlockSizes(Bt, PetscAbs(A->cmap->bs), PetscAbs(B->cmap->bs)));
  PetscCall(MatSetType(Bt, ((PetscObject)A)->type_name));

  /* get symbolic C=A*Bt */
  PetscCall(PetscStrallocpy(product->alg, &alg));
  PetscCall(MatProductSetAlgorithm(C, "sorted")); /* set algorithm for C = A*Bt */
  PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ(A, Bt, fill, C));
  PetscCall(MatProductSetAlgorithm(C, alg)); /* resume original algorithm for ABt product */
  PetscCall(PetscFree(alg));

  /* create a supporting struct for reuse intermediate dense matrices with matcoloring */
  PetscCall(PetscNew(&abt));

  product->data    = abt;
  product->destroy = MatDestroy_SeqAIJ_MatMatMultTrans;

  C->ops->mattransposemultnumeric = MatMatTransposeMultNumeric_SeqAIJ_SeqAIJ;

  abt->usecoloring = PETSC_FALSE;
  PetscCall(PetscStrcmp(product->alg, "color", &abt->usecoloring));
  if (abt->usecoloring) {
    /* Create MatTransposeColoring from symbolic C=A*B^T */
    MatTransposeColoring matcoloring;
    MatColoring          coloring;
    ISColoring           iscoloring;
    Mat                  Bt_dense, C_dense;

    /* inode causes memory problem */
    PetscCall(MatSetOption(C, MAT_USE_INODES, PETSC_FALSE));

    PetscCall(MatColoringCreate(C, &coloring));
    PetscCall(MatColoringSetDistance(coloring, 2));
    PetscCall(MatColoringSetType(coloring, MATCOLORINGSL));
    PetscCall(MatColoringSetFromOptions(coloring));
    PetscCall(MatColoringApply(coloring, &iscoloring));
    PetscCall(MatColoringDestroy(&coloring));
    PetscCall(MatTransposeColoringCreate(C, iscoloring, &matcoloring));

    abt->matcoloring = matcoloring;

    PetscCall(ISColoringDestroy(&iscoloring));

    /* Create Bt_dense and C_dense = A*Bt_dense */
    PetscCall(MatCreate(PETSC_COMM_SELF, &Bt_dense));
    PetscCall(MatSetSizes(Bt_dense, A->cmap->n, matcoloring->ncolors, A->cmap->n, matcoloring->ncolors));
    PetscCall(MatSetType(Bt_dense, MATSEQDENSE));
    PetscCall(MatSeqDenseSetPreallocation(Bt_dense, NULL));

    Bt_dense->assembled = PETSC_TRUE;
    abt->Bt_den         = Bt_dense;

    PetscCall(MatCreate(PETSC_COMM_SELF, &C_dense));
    PetscCall(MatSetSizes(C_dense, A->rmap->n, matcoloring->ncolors, A->rmap->n, matcoloring->ncolors));
    PetscCall(MatSetType(C_dense, MATSEQDENSE));
    PetscCall(MatSeqDenseSetPreallocation(C_dense, NULL));

    Bt_dense->assembled = PETSC_TRUE;
    abt->ABt_den        = C_dense;

#if defined(PETSC_USE_INFO)
    {
      Mat_SeqAIJ *c = (Mat_SeqAIJ *)C->data;
      PetscCall(PetscInfo(C, "Use coloring of C=A*B^T; B^T: %" PetscInt_FMT " %" PetscInt_FMT ", Bt_dense: %" PetscInt_FMT ",%" PetscInt_FMT "; Cnz %" PetscInt_FMT " / (cm*ncolors %" PetscInt_FMT ") = %g\n", B->cmap->n, B->rmap->n, Bt_dense->rmap->n,
                          Bt_dense->cmap->n, c->nz, A->rmap->n * matcoloring->ncolors, (double)(((PetscReal)(c->nz)) / ((PetscReal)(A->rmap->n * matcoloring->ncolors)))));
    }
#endif
  }
  /* clean up */
  PetscCall(MatDestroy(&Bt));
  PetscFunctionReturn(0);
}

PetscErrorCode MatMatTransposeMultNumeric_SeqAIJ_SeqAIJ(Mat A, Mat B, Mat C) {
  Mat_SeqAIJ          *a = (Mat_SeqAIJ *)A->data, *b = (Mat_SeqAIJ *)B->data, *c = (Mat_SeqAIJ *)C->data;
  PetscInt            *ai = a->i, *aj = a->j, *bi = b->i, *bj = b->j, anzi, bnzj, nexta, nextb, *acol, *bcol, brow;
  PetscInt             cm = C->rmap->n, *ci = c->i, *cj = c->j, i, j, cnzi, *ccol;
  PetscLogDouble       flops = 0.0;
  MatScalar           *aa = a->a, *aval, *ba = b->a, *bval, *ca, *cval;
  Mat_MatMatTransMult *abt;
  Mat_Product         *product = C->product;

  PetscFunctionBegin;
  PetscCheck(product, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Missing product struct");
  abt = (Mat_MatMatTransMult *)product->data;
  PetscCheck(abt, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Missing product struct");
  /* clear old values in C */
  if (!c->a) {
    PetscCall(PetscCalloc1(ci[cm] + 1, &ca));
    c->a      = ca;
    c->free_a = PETSC_TRUE;
  } else {
    ca = c->a;
    PetscCall(PetscArrayzero(ca, ci[cm] + 1));
  }

  if (abt->usecoloring) {
    MatTransposeColoring matcoloring = abt->matcoloring;
    Mat                  Bt_dense, C_dense = abt->ABt_den;

    /* Get Bt_dense by Apply MatTransposeColoring to B */
    Bt_dense = abt->Bt_den;
    PetscCall(MatTransColoringApplySpToDen(matcoloring, B, Bt_dense));

    /* C_dense = A*Bt_dense */
    PetscCall(MatMatMultNumeric_SeqAIJ_SeqDense(A, Bt_dense, C_dense));

    /* Recover C from C_dense */
    PetscCall(MatTransColoringApplyDenToSp(matcoloring, C_dense, C));
    PetscFunctionReturn(0);
  }

  for (i = 0; i < cm; i++) {
    anzi = ai[i + 1] - ai[i];
    acol = aj + ai[i];
    aval = aa + ai[i];
    cnzi = ci[i + 1] - ci[i];
    ccol = cj + ci[i];
    cval = ca + ci[i];
    for (j = 0; j < cnzi; j++) {
      brow = ccol[j];
      bnzj = bi[brow + 1] - bi[brow];
      bcol = bj + bi[brow];
      bval = ba + bi[brow];

      /* perform sparse inner-product c(i,j)=A[i,:]*B[j,:]^T */
      nexta = 0;
      nextb = 0;
      while (nexta < anzi && nextb < bnzj) {
        while (nexta < anzi && acol[nexta] < bcol[nextb]) nexta++;
        if (nexta == anzi) break;
        while (nextb < bnzj && acol[nexta] > bcol[nextb]) nextb++;
        if (nextb == bnzj) break;
        if (acol[nexta] == bcol[nextb]) {
          cval[j] += aval[nexta] * bval[nextb];
          nexta++;
          nextb++;
          flops += 2;
        }
      }
    }
  }
  PetscCall(MatAssemblyBegin(C, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(C, MAT_FINAL_ASSEMBLY));
  PetscCall(PetscLogFlops(flops));
  PetscFunctionReturn(0);
}

PetscErrorCode MatDestroy_SeqAIJ_MatTransMatMult(void *data) {
  Mat_MatTransMatMult *atb = (Mat_MatTransMatMult *)data;

  PetscFunctionBegin;
  PetscCall(MatDestroy(&atb->At));
  if (atb->destroy) PetscCall((*atb->destroy)(atb->data));
  PetscCall(PetscFree(atb));
  PetscFunctionReturn(0);
}

PetscErrorCode MatTransposeMatMultSymbolic_SeqAIJ_SeqAIJ(Mat A, Mat B, PetscReal fill, Mat C) {
  Mat          At      = NULL;
  Mat_Product *product = C->product;
  PetscBool    flg, def, square;

  PetscFunctionBegin;
  MatCheckProduct(C, 4);
  square = (PetscBool)(A == B && A->symmetric == PETSC_BOOL3_TRUE);
  /* outerproduct */
  PetscCall(PetscStrcmp(product->alg, "outerproduct", &flg));
  if (flg) {
    /* create symbolic At */
    if (!square) {
      PetscCall(MatTransposeSymbolic(A, &At));
      PetscCall(MatSetBlockSizes(At, PetscAbs(A->cmap->bs), PetscAbs(B->cmap->bs)));
      PetscCall(MatSetType(At, ((PetscObject)A)->type_name));
    }
    /* get symbolic C=At*B */
    PetscCall(MatProductSetAlgorithm(C, "sorted"));
    PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ(square ? A : At, B, fill, C));

    /* clean up */
    if (!square) PetscCall(MatDestroy(&At));

    C->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_SeqAIJ_SeqAIJ; /* outerproduct */
    PetscCall(MatProductSetAlgorithm(C, "outerproduct"));
    PetscFunctionReturn(0);
  }

  /* matmatmult */
  PetscCall(PetscStrcmp(product->alg, "default", &def));
  PetscCall(PetscStrcmp(product->alg, "at*b", &flg));
  if (flg || def) {
    Mat_MatTransMatMult *atb;

    PetscCheck(!product->data, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Extra product struct not empty");
    PetscCall(PetscNew(&atb));
    if (!square) PetscCall(MatTranspose(A, MAT_INITIAL_MATRIX, &At));
    PetscCall(MatProductSetAlgorithm(C, "sorted"));
    PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ(square ? A : At, B, fill, C));
    PetscCall(MatProductSetAlgorithm(C, "at*b"));
    product->data    = atb;
    product->destroy = MatDestroy_SeqAIJ_MatTransMatMult;
    atb->At          = At;

    C->ops->mattransposemultnumeric = NULL; /* see MatProductNumeric_AtB_SeqAIJ_SeqAIJ */
    PetscFunctionReturn(0);
  }

  SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Mat Product Algorithm is not supported");
}

PetscErrorCode MatTransposeMatMultNumeric_SeqAIJ_SeqAIJ(Mat A, Mat B, Mat C) {
  Mat_SeqAIJ    *a = (Mat_SeqAIJ *)A->data, *b = (Mat_SeqAIJ *)B->data, *c = (Mat_SeqAIJ *)C->data;
  PetscInt       am = A->rmap->n, anzi, *ai = a->i, *aj = a->j, *bi = b->i, *bj, bnzi, nextb;
  PetscInt       cm = C->rmap->n, *ci = c->i, *cj = c->j, crow, *cjj, i, j, k;
  PetscLogDouble flops = 0.0;
  MatScalar     *aa    = a->a, *ba, *ca, *caj;

  PetscFunctionBegin;
  if (!c->a) {
    PetscCall(PetscCalloc1(ci[cm] + 1, &ca));

    c->a      = ca;
    c->free_a = PETSC_TRUE;
  } else {
    ca = c->a;
    PetscCall(PetscArrayzero(ca, ci[cm]));
  }

  /* compute A^T*B using outer product (A^T)[:,i]*B[i,:] */
  for (i = 0; i < am; i++) {
    bj   = b->j + bi[i];
    ba   = b->a + bi[i];
    bnzi = bi[i + 1] - bi[i];
    anzi = ai[i + 1] - ai[i];
    for (j = 0; j < anzi; j++) {
      nextb = 0;
      crow  = *aj++;
      cjj   = cj + ci[crow];
      caj   = ca + ci[crow];
      /* perform sparse axpy operation.  Note cjj includes bj. */
      for (k = 0; nextb < bnzi; k++) {
        if (cjj[k] == *(bj + nextb)) { /* ccol == bcol */
          caj[k] += (*aa) * (*(ba + nextb));
          nextb++;
        }
      }
      flops += 2 * bnzi;
      aa++;
    }
  }

  /* Assemble the final matrix and clean up */
  PetscCall(MatAssemblyBegin(C, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(C, MAT_FINAL_ASSEMBLY));
  PetscCall(PetscLogFlops(flops));
  PetscFunctionReturn(0);
}

PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqDense(Mat A, Mat B, PetscReal fill, Mat C) {
  PetscFunctionBegin;
  PetscCall(MatMatMultSymbolic_SeqDense_SeqDense(A, B, 0.0, C));
  C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqDense;
  PetscFunctionReturn(0);
}

PETSC_INTERN PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat A, Mat B, Mat C, const PetscBool add) {
  Mat_SeqAIJ        *a = (Mat_SeqAIJ *)A->data;
  PetscScalar       *c, r1, r2, r3, r4, *c1, *c2, *c3, *c4;
  const PetscScalar *aa, *b, *b1, *b2, *b3, *b4, *av;
  const PetscInt    *aj;
  PetscInt           cm = C->rmap->n, cn = B->cmap->n, bm, am = A->rmap->n;
  PetscInt           clda;
  PetscInt           am4, bm4, col, i, j, n;

  PetscFunctionBegin;
  if (!cm || !cn) PetscFunctionReturn(0);
  PetscCall(MatSeqAIJGetArrayRead(A, &av));
  if (add) {
    PetscCall(MatDenseGetArray(C, &c));
  } else {
    PetscCall(MatDenseGetArrayWrite(C, &c));
  }
  PetscCall(MatDenseGetArrayRead(B, &b));
  PetscCall(MatDenseGetLDA(B, &bm));
  PetscCall(MatDenseGetLDA(C, &clda));
  am4 = 4 * clda;
  bm4 = 4 * bm;
  b1  = b;
  b2  = b1 + bm;
  b3  = b2 + bm;
  b4  = b3 + bm;
  c1  = c;
  c2  = c1 + clda;
  c3  = c2 + clda;
  c4  = c3 + clda;
  for (col = 0; col < (cn / 4) * 4; col += 4) { /* over columns of C */
    for (i = 0; i < am; i++) {                  /* over rows of A in those columns */
      r1 = r2 = r3 = r4 = 0.0;
      n                 = a->i[i + 1] - a->i[i];
      aj                = a->j + a->i[i];
      aa                = av + a->i[i];
      for (j = 0; j < n; j++) {
        const PetscScalar aatmp = aa[j];
        const PetscInt    ajtmp = aj[j];
        r1 += aatmp * b1[ajtmp];
        r2 += aatmp * b2[ajtmp];
        r3 += aatmp * b3[ajtmp];
        r4 += aatmp * b4[ajtmp];
      }
      if (add) {
        c1[i] += r1;
        c2[i] += r2;
        c3[i] += r3;
        c4[i] += r4;
      } else {
        c1[i] = r1;
        c2[i] = r2;
        c3[i] = r3;
        c4[i] = r4;
      }
    }
    b1 += bm4;
    b2 += bm4;
    b3 += bm4;
    b4 += bm4;
    c1 += am4;
    c2 += am4;
    c3 += am4;
    c4 += am4;
  }
  /* process remaining columns */
  if (col != cn) {
    PetscInt rc = cn - col;

    if (rc == 1) {
      for (i = 0; i < am; i++) {
        r1 = 0.0;
        n  = a->i[i + 1] - a->i[i];
        aj = a->j + a->i[i];
        aa = av + a->i[i];
        for (j = 0; j < n; j++) r1 += aa[j] * b1[aj[j]];
        if (add) c1[i] += r1;
        else c1[i] = r1;
      }
    } else if (rc == 2) {
      for (i = 0; i < am; i++) {
        r1 = r2 = 0.0;
        n       = a->i[i + 1] - a->i[i];
        aj      = a->j + a->i[i];
        aa      = av + a->i[i];
        for (j = 0; j < n; j++) {
          const PetscScalar aatmp = aa[j];
          const PetscInt    ajtmp = aj[j];
          r1 += aatmp * b1[ajtmp];
          r2 += aatmp * b2[ajtmp];
        }
        if (add) {
          c1[i] += r1;
          c2[i] += r2;
        } else {
          c1[i] = r1;
          c2[i] = r2;
        }
      }
    } else {
      for (i = 0; i < am; i++) {
        r1 = r2 = r3 = 0.0;
        n            = a->i[i + 1] - a->i[i];
        aj           = a->j + a->i[i];
        aa           = av + a->i[i];
        for (j = 0; j < n; j++) {
          const PetscScalar aatmp = aa[j];
          const PetscInt    ajtmp = aj[j];
          r1 += aatmp * b1[ajtmp];
          r2 += aatmp * b2[ajtmp];
          r3 += aatmp * b3[ajtmp];
        }
        if (add) {
          c1[i] += r1;
          c2[i] += r2;
          c3[i] += r3;
        } else {
          c1[i] = r1;
          c2[i] = r2;
          c3[i] = r3;
        }
      }
    }
  }
  PetscCall(PetscLogFlops(cn * (2.0 * a->nz)));
  if (add) {
    PetscCall(MatDenseRestoreArray(C, &c));
  } else {
    PetscCall(MatDenseRestoreArrayWrite(C, &c));
  }
  PetscCall(MatDenseRestoreArrayRead(B, &b));
  PetscCall(MatSeqAIJRestoreArrayRead(A, &av));
  PetscFunctionReturn(0);
}

PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqDense(Mat A, Mat B, Mat C) {
  PetscFunctionBegin;
  PetscCheck(B->rmap->n == A->cmap->n, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Number columns in A %" PetscInt_FMT " not equal rows in B %" PetscInt_FMT, A->cmap->n, B->rmap->n);
  PetscCheck(A->rmap->n == C->rmap->n, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Number rows in C %" PetscInt_FMT " not equal rows in A %" PetscInt_FMT, C->rmap->n, A->rmap->n);
  PetscCheck(B->cmap->n == C->cmap->n, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Number columns in B %" PetscInt_FMT " not equal columns in C %" PetscInt_FMT, B->cmap->n, C->cmap->n);

  PetscCall(MatMatMultNumericAdd_SeqAIJ_SeqDense(A, B, C, PETSC_FALSE));
  PetscFunctionReturn(0);
}

/* ------------------------------------------------------- */
static PetscErrorCode MatProductSetFromOptions_SeqAIJ_SeqDense_AB(Mat C) {
  PetscFunctionBegin;
  C->ops->matmultsymbolic = MatMatMultSymbolic_SeqAIJ_SeqDense;
  C->ops->productsymbolic = MatProductSymbolic_AB;
  PetscFunctionReturn(0);
}

PETSC_INTERN PetscErrorCode MatTMatTMultSymbolic_SeqAIJ_SeqDense(Mat, Mat, PetscReal, Mat);

static PetscErrorCode MatProductSetFromOptions_SeqAIJ_SeqDense_AtB(Mat C) {
  PetscFunctionBegin;
  C->ops->transposematmultsymbolic = MatTMatTMultSymbolic_SeqAIJ_SeqDense;
  C->ops->productsymbolic          = MatProductSymbolic_AtB;
  PetscFunctionReturn(0);
}

static PetscErrorCode MatProductSetFromOptions_SeqAIJ_SeqDense_ABt(Mat C) {
  PetscFunctionBegin;
  C->ops->mattransposemultsymbolic = MatTMatTMultSymbolic_SeqAIJ_SeqDense;
  C->ops->productsymbolic          = MatProductSymbolic_ABt;
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  switch (product->type) {
  case MATPRODUCT_AB: PetscCall(MatProductSetFromOptions_SeqAIJ_SeqDense_AB(C)); break;
  case MATPRODUCT_AtB: PetscCall(MatProductSetFromOptions_SeqAIJ_SeqDense_AtB(C)); break;
  case MATPRODUCT_ABt: PetscCall(MatProductSetFromOptions_SeqAIJ_SeqDense_ABt(C)); break;
  default: break;
  }
  PetscFunctionReturn(0);
}
/* ------------------------------------------------------- */
static PetscErrorCode MatProductSetFromOptions_SeqXBAIJ_SeqDense_AB(Mat C) {
  Mat_Product *product = C->product;
  Mat          A       = product->A;
  PetscBool    baij;

  PetscFunctionBegin;
  PetscCall(PetscObjectTypeCompare((PetscObject)A, MATSEQBAIJ, &baij));
  if (!baij) { /* A is seqsbaij */
    PetscBool sbaij;
    PetscCall(PetscObjectTypeCompare((PetscObject)A, MATSEQSBAIJ, &sbaij));
    PetscCheck(sbaij, PetscObjectComm((PetscObject)C), PETSC_ERR_ARG_WRONGSTATE, "Mat must be either seqbaij or seqsbaij format");

    C->ops->matmultsymbolic = MatMatMultSymbolic_SeqSBAIJ_SeqDense;
  } else { /* A is seqbaij */
    C->ops->matmultsymbolic = MatMatMultSymbolic_SeqBAIJ_SeqDense;
  }

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

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

  PetscFunctionBegin;
  MatCheckProduct(C, 1);
  PetscCheck(product->A, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Missing A");
  if (product->type == MATPRODUCT_AB || (product->type == MATPRODUCT_AtB && product->A->symmetric == PETSC_BOOL3_TRUE)) PetscCall(MatProductSetFromOptions_SeqXBAIJ_SeqDense_AB(C));
  PetscFunctionReturn(0);
}

/* ------------------------------------------------------- */
static PetscErrorCode MatProductSetFromOptions_SeqDense_SeqAIJ_AB(Mat C) {
  PetscFunctionBegin;
  C->ops->matmultsymbolic = MatMatMultSymbolic_SeqDense_SeqAIJ;
  C->ops->productsymbolic = MatProductSymbolic_AB;
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  if (product->type == MATPRODUCT_AB) PetscCall(MatProductSetFromOptions_SeqDense_SeqAIJ_AB(C));
  PetscFunctionReturn(0);
}
/* ------------------------------------------------------- */

PetscErrorCode MatTransColoringApplySpToDen_SeqAIJ(MatTransposeColoring coloring, Mat B, Mat Btdense) {
  Mat_SeqAIJ   *b       = (Mat_SeqAIJ *)B->data;
  Mat_SeqDense *btdense = (Mat_SeqDense *)Btdense->data;
  PetscInt     *bi = b->i, *bj = b->j;
  PetscInt      m = Btdense->rmap->n, n = Btdense->cmap->n, j, k, l, col, anz, *btcol, brow, ncolumns;
  MatScalar    *btval, *btval_den, *ba = b->a;
  PetscInt     *columns = coloring->columns, *colorforcol = coloring->colorforcol, ncolors = coloring->ncolors;

  PetscFunctionBegin;
  btval_den = btdense->v;
  PetscCall(PetscArrayzero(btval_den, m * n));
  for (k = 0; k < ncolors; k++) {
    ncolumns = coloring->ncolumns[k];
    for (l = 0; l < ncolumns; l++) { /* insert a row of B to a column of Btdense */
      col   = *(columns + colorforcol[k] + l);
      btcol = bj + bi[col];
      btval = ba + bi[col];
      anz   = bi[col + 1] - bi[col];
      for (j = 0; j < anz; j++) {
        brow            = btcol[j];
        btval_den[brow] = btval[j];
      }
    }
    btval_den += m;
  }
  PetscFunctionReturn(0);
}

PetscErrorCode MatTransColoringApplyDenToSp_SeqAIJ(MatTransposeColoring matcoloring, Mat Cden, Mat Csp) {
  Mat_SeqAIJ        *csp = (Mat_SeqAIJ *)Csp->data;
  const PetscScalar *ca_den, *ca_den_ptr;
  PetscScalar       *ca = csp->a;
  PetscInt           k, l, m = Cden->rmap->n, ncolors = matcoloring->ncolors;
  PetscInt           brows = matcoloring->brows, *den2sp = matcoloring->den2sp;
  PetscInt           nrows, *row, *idx;
  PetscInt          *rows = matcoloring->rows, *colorforrow = matcoloring->colorforrow;

  PetscFunctionBegin;
  PetscCall(MatDenseGetArrayRead(Cden, &ca_den));

  if (brows > 0) {
    PetscInt *lstart, row_end, row_start;
    lstart = matcoloring->lstart;
    PetscCall(PetscArrayzero(lstart, ncolors));

    row_end = brows;
    if (row_end > m) row_end = m;
    for (row_start = 0; row_start < m; row_start += brows) { /* loop over row blocks of Csp */
      ca_den_ptr = ca_den;
      for (k = 0; k < ncolors; k++) { /* loop over colors (columns of Cden) */
        nrows = matcoloring->nrows[k];
        row   = rows + colorforrow[k];
        idx   = den2sp + colorforrow[k];
        for (l = lstart[k]; l < nrows; l++) {
          if (row[l] >= row_end) {
            lstart[k] = l;
            break;
          } else {
            ca[idx[l]] = ca_den_ptr[row[l]];
          }
        }
        ca_den_ptr += m;
      }
      row_end += brows;
      if (row_end > m) row_end = m;
    }
  } else { /* non-blocked impl: loop over columns of Csp - slow if Csp is large */
    ca_den_ptr = ca_den;
    for (k = 0; k < ncolors; k++) {
      nrows = matcoloring->nrows[k];
      row   = rows + colorforrow[k];
      idx   = den2sp + colorforrow[k];
      for (l = 0; l < nrows; l++) { ca[idx[l]] = ca_den_ptr[row[l]]; }
      ca_den_ptr += m;
    }
  }

  PetscCall(MatDenseRestoreArrayRead(Cden, &ca_den));
#if defined(PETSC_USE_INFO)
  if (matcoloring->brows > 0) {
    PetscCall(PetscInfo(Csp, "Loop over %" PetscInt_FMT " row blocks for den2sp\n", brows));
  } else {
    PetscCall(PetscInfo(Csp, "Loop over colors/columns of Cden, inefficient for large sparse matrix product \n"));
  }
#endif
  PetscFunctionReturn(0);
}

PetscErrorCode MatTransposeColoringCreate_SeqAIJ(Mat mat, ISColoring iscoloring, MatTransposeColoring c) {
  PetscInt        i, n, nrows, Nbs, j, k, m, ncols, col, cm;
  const PetscInt *is, *ci, *cj, *row_idx;
  PetscInt        nis = iscoloring->n, *rowhit, bs = 1;
  IS             *isa;
  Mat_SeqAIJ     *csp = (Mat_SeqAIJ *)mat->data;
  PetscInt       *colorforrow, *rows, *rows_i, *idxhit, *spidx, *den2sp, *den2sp_i;
  PetscInt       *colorforcol, *columns, *columns_i, brows;
  PetscBool       flg;

  PetscFunctionBegin;
  PetscCall(ISColoringGetIS(iscoloring, PETSC_USE_POINTER, PETSC_IGNORE, &isa));

  /* bs >1 is not being tested yet! */
  Nbs       = mat->cmap->N / bs;
  c->M      = mat->rmap->N / bs; /* set total rows, columns and local rows */
  c->N      = Nbs;
  c->m      = c->M;
  c->rstart = 0;
  c->brows  = 100;

  c->ncolors = nis;
  PetscCall(PetscMalloc3(nis, &c->ncolumns, nis, &c->nrows, nis + 1, &colorforrow));
  PetscCall(PetscMalloc1(csp->nz + 1, &rows));
  PetscCall(PetscMalloc1(csp->nz + 1, &den2sp));

  brows = c->brows;
  PetscCall(PetscOptionsGetInt(NULL, NULL, "-matden2sp_brows", &brows, &flg));
  if (flg) c->brows = brows;
  if (brows > 0) PetscCall(PetscMalloc1(nis + 1, &c->lstart));

  colorforrow[0] = 0;
  rows_i         = rows;
  den2sp_i       = den2sp;

  PetscCall(PetscMalloc1(nis + 1, &colorforcol));
  PetscCall(PetscMalloc1(Nbs + 1, &columns));

  colorforcol[0] = 0;
  columns_i      = columns;

  /* get column-wise storage of mat */
  PetscCall(MatGetColumnIJ_SeqAIJ_Color(mat, 0, PETSC_FALSE, PETSC_FALSE, &ncols, &ci, &cj, &spidx, NULL));

  cm = c->m;
  PetscCall(PetscMalloc1(cm + 1, &rowhit));
  PetscCall(PetscMalloc1(cm + 1, &idxhit));
  for (i = 0; i < nis; i++) { /* loop over color */
    PetscCall(ISGetLocalSize(isa[i], &n));
    PetscCall(ISGetIndices(isa[i], &is));

    c->ncolumns[i] = n;
    if (n) PetscCall(PetscArraycpy(columns_i, is, n));
    colorforcol[i + 1] = colorforcol[i] + n;
    columns_i += n;

    /* fast, crude version requires O(N*N) work */
    PetscCall(PetscArrayzero(rowhit, cm));

    for (j = 0; j < n; j++) { /* loop over columns*/
      col     = is[j];
      row_idx = cj + ci[col];
      m       = ci[col + 1] - ci[col];
      for (k = 0; k < m; k++) { /* loop over columns marking them in rowhit */
        idxhit[*row_idx]   = spidx[ci[col] + k];
        rowhit[*row_idx++] = col + 1;
      }
    }
    /* count the number of hits */
    nrows = 0;
    for (j = 0; j < cm; j++) {
      if (rowhit[j]) nrows++;
    }
    c->nrows[i]        = nrows;
    colorforrow[i + 1] = colorforrow[i] + nrows;

    nrows = 0;
    for (j = 0; j < cm; j++) { /* loop over rows */
      if (rowhit[j]) {
        rows_i[nrows]   = j;
        den2sp_i[nrows] = idxhit[j];
        nrows++;
      }
    }
    den2sp_i += nrows;

    PetscCall(ISRestoreIndices(isa[i], &is));
    rows_i += nrows;
  }
  PetscCall(MatRestoreColumnIJ_SeqAIJ_Color(mat, 0, PETSC_FALSE, PETSC_FALSE, &ncols, &ci, &cj, &spidx, NULL));
  PetscCall(PetscFree(rowhit));
  PetscCall(ISColoringRestoreIS(iscoloring, PETSC_USE_POINTER, &isa));
  PetscCheck(csp->nz == colorforrow[nis], PETSC_COMM_SELF, PETSC_ERR_PLIB, "csp->nz %" PetscInt_FMT " != colorforrow[nis] %" PetscInt_FMT, csp->nz, colorforrow[nis]);

  c->colorforrow = colorforrow;
  c->rows        = rows;
  c->den2sp      = den2sp;
  c->colorforcol = colorforcol;
  c->columns     = columns;

  PetscCall(PetscFree(idxhit));
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  if (C->ops->mattransposemultnumeric) {
    /* Alg: "outerproduct" */
    PetscCall((*C->ops->mattransposemultnumeric)(A, B, C));
  } else {
    /* Alg: "matmatmult" -- C = At*B */
    Mat_MatTransMatMult *atb = (Mat_MatTransMatMult *)product->data;

    PetscCheck(atb, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Missing product struct");
    if (atb->At) {
      /* At is computed in MatTransposeMatMultSymbolic_SeqAIJ_SeqAIJ();
         user may have called MatProductReplaceMats() to get this A=product->A */
      PetscCall(MatTransposeSetPrecursor(A, atb->At));
      PetscCall(MatTranspose(A, MAT_REUSE_MATRIX, &atb->At));
    }
    PetscCall(MatMatMultNumeric_SeqAIJ_SeqAIJ(atb->At ? atb->At : A, B, C));
  }
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  PetscCall(MatTransposeMatMultSymbolic_SeqAIJ_SeqAIJ(A, B, fill, C));

  C->ops->productnumeric = MatProductNumeric_AtB_SeqAIJ_SeqAIJ;
  PetscFunctionReturn(0);
}

/* --------------------------------------------------------------- */
static PetscErrorCode MatProductSetFromOptions_SeqAIJ_AB(Mat C) {
  Mat_Product *product = C->product;
  PetscInt     alg     = 0; /* default algorithm */
  PetscBool    flg     = PETSC_FALSE;
#if !defined(PETSC_HAVE_HYPRE)
  const char *algTypes[7] = {"sorted", "scalable", "scalable_fast", "heap", "btheap", "llcondensed", "rowmerge"};
  PetscInt    nalg        = 7;
#else
  const char *algTypes[8] = {"sorted", "scalable", "scalable_fast", "heap", "btheap", "llcondensed", "rowmerge", "hypre"};
  PetscInt    nalg        = 8;
#endif

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

  /* Get runtime option */
  if (product->api_user) {
    PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatMatMult", "Mat");
    PetscCall(PetscOptionsEList("-matmatmult_via", "Algorithmic approach", "MatMatMult", algTypes, nalg, algTypes[0], &alg, &flg));
    PetscOptionsEnd();
  } else {
    PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatProduct_AB", "Mat");
    PetscCall(PetscOptionsEList("-mat_product_algorithm", "Algorithmic approach", "MatProduct_AB", algTypes, nalg, algTypes[0], &alg, &flg));
    PetscOptionsEnd();
  }
  if (flg) PetscCall(MatProductSetAlgorithm(C, (MatProductAlgorithm)algTypes[alg]));

  C->ops->productsymbolic = MatProductSymbolic_AB;
  C->ops->matmultsymbolic = MatMatMultSymbolic_SeqAIJ_SeqAIJ;
  PetscFunctionReturn(0);
}

static PetscErrorCode MatProductSetFromOptions_SeqAIJ_AtB(Mat C) {
  Mat_Product *product     = C->product;
  PetscInt     alg         = 0; /* default algorithm */
  PetscBool    flg         = PETSC_FALSE;
  const char  *algTypes[3] = {"default", "at*b", "outerproduct"};
  PetscInt     nalg        = 3;

  PetscFunctionBegin;
  /* 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", "MatProduct_AtB", algTypes, nalg, algTypes[alg], &alg, &flg));
    PetscOptionsEnd();
  }
  if (flg) PetscCall(MatProductSetAlgorithm(C, (MatProductAlgorithm)algTypes[alg]));

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

static PetscErrorCode MatProductSetFromOptions_SeqAIJ_ABt(Mat C) {
  Mat_Product *product     = C->product;
  PetscInt     alg         = 0; /* default algorithm */
  PetscBool    flg         = PETSC_FALSE;
  const char  *algTypes[2] = {"default", "color"};
  PetscInt     nalg        = 2;

  PetscFunctionBegin;
  /* Set default algorithm */
  PetscCall(PetscStrcmp(C->product->alg, "default", &flg));
  if (!flg) {
    alg = 1;
    PetscCall(MatProductSetAlgorithm(C, (MatProductAlgorithm)algTypes[alg]));
  }

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

  C->ops->mattransposemultsymbolic = MatMatTransposeMultSymbolic_SeqAIJ_SeqAIJ;
  C->ops->productsymbolic          = MatProductSymbolic_ABt;
  PetscFunctionReturn(0);
}

static PetscErrorCode MatProductSetFromOptions_SeqAIJ_PtAP(Mat C) {
  Mat_Product *product = C->product;
  PetscBool    flg     = PETSC_FALSE;
  PetscInt     alg     = 0; /* default algorithm -- alg=1 should be default!!! */
#if !defined(PETSC_HAVE_HYPRE)
  const char *algTypes[2] = {"scalable", "rap"};
  PetscInt    nalg        = 2;
#else
  const char *algTypes[3] = {"scalable", "rap", "hypre"};
  PetscInt    nalg        = 3;
#endif

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

  /* Get runtime option */
  if (product->api_user) {
    PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatPtAP", "Mat");
    PetscCall(PetscOptionsEList("-matptap_via", "Algorithmic approach", "MatPtAP", algTypes, nalg, algTypes[0], &alg, &flg));
    PetscOptionsEnd();
  } else {
    PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatProduct_PtAP", "Mat");
    PetscCall(PetscOptionsEList("-mat_product_algorithm", "Algorithmic approach", "MatProduct_PtAP", algTypes, nalg, algTypes[0], &alg, &flg));
    PetscOptionsEnd();
  }
  if (flg) PetscCall(MatProductSetAlgorithm(C, (MatProductAlgorithm)algTypes[alg]));

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

static PetscErrorCode MatProductSetFromOptions_SeqAIJ_RARt(Mat C) {
  Mat_Product *product     = C->product;
  PetscBool    flg         = PETSC_FALSE;
  PetscInt     alg         = 0; /* default algorithm */
  const char  *algTypes[3] = {"r*a*rt", "r*art", "coloring_rart"};
  PetscInt     nalg        = 3;

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

  /* Get runtime option */
  if (product->api_user) {
    PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatRARt", "Mat");
    PetscCall(PetscOptionsEList("-matrart_via", "Algorithmic approach", "MatRARt", algTypes, nalg, algTypes[0], &alg, &flg));
    PetscOptionsEnd();
  } else {
    PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatProduct_RARt", "Mat");
    PetscCall(PetscOptionsEList("-mat_product_algorithm", "Algorithmic approach", "MatProduct_RARt", algTypes, nalg, algTypes[0], &alg, &flg));
    PetscOptionsEnd();
  }
  if (flg) PetscCall(MatProductSetAlgorithm(C, (MatProductAlgorithm)algTypes[alg]));

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

/* ABC = A*B*C = A*(B*C); ABC's algorithm must be chosen from AB's algorithm */
static PetscErrorCode MatProductSetFromOptions_SeqAIJ_ABC(Mat C) {
  Mat_Product *product     = C->product;
  PetscInt     alg         = 0; /* default algorithm */
  PetscBool    flg         = PETSC_FALSE;
  const char  *algTypes[7] = {"sorted", "scalable", "scalable_fast", "heap", "btheap", "llcondensed", "rowmerge"};
  PetscInt     nalg        = 7;

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

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

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

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

  PetscFunctionBegin;
  switch (product->type) {
  case MATPRODUCT_AB: PetscCall(MatProductSetFromOptions_SeqAIJ_AB(C)); break;
  case MATPRODUCT_AtB: PetscCall(MatProductSetFromOptions_SeqAIJ_AtB(C)); break;
  case MATPRODUCT_ABt: PetscCall(MatProductSetFromOptions_SeqAIJ_ABt(C)); break;
  case MATPRODUCT_PtAP: PetscCall(MatProductSetFromOptions_SeqAIJ_PtAP(C)); break;
  case MATPRODUCT_RARt: PetscCall(MatProductSetFromOptions_SeqAIJ_RARt(C)); break;
  case MATPRODUCT_ABC: PetscCall(MatProductSetFromOptions_SeqAIJ_ABC(C)); break;
  default: break;
  }
  PetscFunctionReturn(0);
}