xref: /petsc/src/mat/impls/aij/mpi/mpiaij.c (revision f1f2ae845fd5815e1ad61e968c2c51b97d3840d2)

#include <../src/mat/impls/aij/mpi/mpiaij.h>   /*I "petscmat.h" I*/
#include <petsc/private/vecimpl.h>
#include <petsc/private/sfimpl.h>
#include <petsc/private/isimpl.h>
#include <petscblaslapack.h>
#include <petscsf.h>
#include <petsc/private/hashmapi.h>

PetscErrorCode MatGetRowIJ_MPIAIJ(Mat A,PetscInt oshift,PetscBool symmetric,PetscBool inodecompressed,PetscInt *m,const PetscInt *ia[],const PetscInt *ja[],PetscBool  *done)
{
  Mat            B;

  PetscFunctionBegin;
  PetscCall(MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&B));
  PetscCall(PetscObjectCompose((PetscObject)A,"MatGetRowIJ_MPIAIJ",(PetscObject)B));
  PetscCall(MatGetRowIJ(B,oshift,symmetric,inodecompressed,m,ia,ja,done));
  PetscFunctionReturn(0);
}

PetscErrorCode MatRestoreRowIJ_MPIAIJ(Mat A,PetscInt oshift,PetscBool symmetric,PetscBool inodecompressed,PetscInt *m,const PetscInt *ia[],const PetscInt *ja[],PetscBool  *done)
{
  Mat            B;

  PetscFunctionBegin;
  PetscCall(PetscObjectQuery((PetscObject)A,"MatGetRowIJ_MPIAIJ",(PetscObject*)&B));
  PetscCall(MatRestoreRowIJ(B,oshift,symmetric,inodecompressed,m,ia,ja,done));
  PetscCall(MatDestroy(&B));
  PetscFunctionReturn(0);
}

/*MC
   MATAIJ - MATAIJ = "aij" - A matrix type to be used for sparse matrices.

   This matrix type is identical to MATSEQAIJ when constructed with a single process communicator,
   and MATMPIAIJ otherwise.  As a result, for single process communicators,
  MatSeqAIJSetPreallocation is supported, and similarly MatMPIAIJSetPreallocation() is supported
  for communicators controlling multiple processes.  It is recommended that you call both of
  the above preallocation routines for simplicity.

   Options Database Keys:
. -mat_type aij - sets the matrix type to "aij" during a call to MatSetFromOptions()

  Developer Notes:
    Subclasses include MATAIJCUSPARSE, MATAIJPERM, MATAIJSELL, MATAIJMKL, MATAIJCRL, and also automatically switches over to use inodes when
   enough exist.

  Level: beginner

.seealso: `MatCreateAIJ()`, `MatCreateSeqAIJ()`, `MATSEQAIJ`, `MATMPIAIJ`
M*/

/*MC
   MATAIJCRL - MATAIJCRL = "aijcrl" - A matrix type to be used for sparse matrices.

   This matrix type is identical to MATSEQAIJCRL when constructed with a single process communicator,
   and MATMPIAIJCRL otherwise.  As a result, for single process communicators,
   MatSeqAIJSetPreallocation() is supported, and similarly MatMPIAIJSetPreallocation() is supported
  for communicators controlling multiple processes.  It is recommended that you call both of
  the above preallocation routines for simplicity.

   Options Database Keys:
. -mat_type aijcrl - sets the matrix type to "aijcrl" during a call to MatSetFromOptions()

  Level: beginner

.seealso: `MatCreateMPIAIJCRL`, `MATSEQAIJCRL`, `MATMPIAIJCRL`, `MATSEQAIJCRL`, `MATMPIAIJCRL`
M*/

static PetscErrorCode MatBindToCPU_MPIAIJ(Mat A,PetscBool flg)
{
  Mat_MPIAIJ     *a = (Mat_MPIAIJ*)A->data;

  PetscFunctionBegin;
#if defined(PETSC_HAVE_CUDA) || defined(PETSC_HAVE_VIENNACL)
  A->boundtocpu = flg;
#endif
  if (a->A) PetscCall(MatBindToCPU(a->A,flg));
  if (a->B) PetscCall(MatBindToCPU(a->B,flg));

  /* In addition to binding the diagonal and off-diagonal matrices, bind the local vectors used for matrix-vector products.
   * This maybe seems a little odd for a MatBindToCPU() call to do, but it makes no sense for the binding of these vectors
   * to differ from the parent matrix. */
  if (a->lvec) PetscCall(VecBindToCPU(a->lvec,flg));
  if (a->diag) PetscCall(VecBindToCPU(a->diag,flg));

  PetscFunctionReturn(0);
}

PetscErrorCode MatSetBlockSizes_MPIAIJ(Mat M, PetscInt rbs, PetscInt cbs)
{
  Mat_MPIAIJ     *mat = (Mat_MPIAIJ*)M->data;

  PetscFunctionBegin;
  if (mat->A) {
    PetscCall(MatSetBlockSizes(mat->A,rbs,cbs));
    PetscCall(MatSetBlockSizes(mat->B,rbs,1));
  }
  PetscFunctionReturn(0);
}

PetscErrorCode MatFindNonzeroRows_MPIAIJ(Mat M,IS *keptrows)
{
  Mat_MPIAIJ      *mat = (Mat_MPIAIJ*)M->data;
  Mat_SeqAIJ      *a   = (Mat_SeqAIJ*)mat->A->data;
  Mat_SeqAIJ      *b   = (Mat_SeqAIJ*)mat->B->data;
  const PetscInt  *ia,*ib;
  const MatScalar *aa,*bb,*aav,*bav;
  PetscInt        na,nb,i,j,*rows,cnt=0,n0rows;
  PetscInt        m = M->rmap->n,rstart = M->rmap->rstart;

  PetscFunctionBegin;
  *keptrows = NULL;

  ia   = a->i;
  ib   = b->i;
  PetscCall(MatSeqAIJGetArrayRead(mat->A,&aav));
  PetscCall(MatSeqAIJGetArrayRead(mat->B,&bav));
  for (i=0; i<m; i++) {
    na = ia[i+1] - ia[i];
    nb = ib[i+1] - ib[i];
    if (!na && !nb) {
      cnt++;
      goto ok1;
    }
    aa = aav + ia[i];
    for (j=0; j<na; j++) {
      if (aa[j] != 0.0) goto ok1;
    }
    bb = bav + ib[i];
    for (j=0; j <nb; j++) {
      if (bb[j] != 0.0) goto ok1;
    }
    cnt++;
ok1:;
  }
  PetscCall(MPIU_Allreduce(&cnt,&n0rows,1,MPIU_INT,MPI_SUM,PetscObjectComm((PetscObject)M)));
  if (!n0rows) {
    PetscCall(MatSeqAIJRestoreArrayRead(mat->A,&aav));
    PetscCall(MatSeqAIJRestoreArrayRead(mat->B,&bav));
    PetscFunctionReturn(0);
  }
  PetscCall(PetscMalloc1(M->rmap->n-cnt,&rows));
  cnt  = 0;
  for (i=0; i<m; i++) {
    na = ia[i+1] - ia[i];
    nb = ib[i+1] - ib[i];
    if (!na && !nb) continue;
    aa = aav + ia[i];
    for (j=0; j<na;j++) {
      if (aa[j] != 0.0) {
        rows[cnt++] = rstart + i;
        goto ok2;
      }
    }
    bb = bav + ib[i];
    for (j=0; j<nb; j++) {
      if (bb[j] != 0.0) {
        rows[cnt++] = rstart + i;
        goto ok2;
      }
    }
ok2:;
  }
  PetscCall(ISCreateGeneral(PetscObjectComm((PetscObject)M),cnt,rows,PETSC_OWN_POINTER,keptrows));
  PetscCall(MatSeqAIJRestoreArrayRead(mat->A,&aav));
  PetscCall(MatSeqAIJRestoreArrayRead(mat->B,&bav));
  PetscFunctionReturn(0);
}

PetscErrorCode  MatDiagonalSet_MPIAIJ(Mat Y,Vec D,InsertMode is)
{
  Mat_MPIAIJ        *aij = (Mat_MPIAIJ*) Y->data;
  PetscBool         cong;

  PetscFunctionBegin;
  PetscCall(MatHasCongruentLayouts(Y,&cong));
  if (Y->assembled && cong) {
    PetscCall(MatDiagonalSet(aij->A,D,is));
  } else {
    PetscCall(MatDiagonalSet_Default(Y,D,is));
  }
  PetscFunctionReturn(0);
}

PetscErrorCode MatFindZeroDiagonals_MPIAIJ(Mat M,IS *zrows)
{
  Mat_MPIAIJ     *aij = (Mat_MPIAIJ*)M->data;
  PetscInt       i,rstart,nrows,*rows;

  PetscFunctionBegin;
  *zrows = NULL;
  PetscCall(MatFindZeroDiagonals_SeqAIJ_Private(aij->A,&nrows,&rows));
  PetscCall(MatGetOwnershipRange(M,&rstart,NULL));
  for (i=0; i<nrows; i++) rows[i] += rstart;
  PetscCall(ISCreateGeneral(PetscObjectComm((PetscObject)M),nrows,rows,PETSC_OWN_POINTER,zrows));
  PetscFunctionReturn(0);
}

PetscErrorCode MatGetColumnReductions_MPIAIJ(Mat A,PetscInt type,PetscReal *reductions)
{
  Mat_MPIAIJ        *aij = (Mat_MPIAIJ*)A->data;
  PetscInt          i,m,n,*garray = aij->garray;
  Mat_SeqAIJ        *a_aij = (Mat_SeqAIJ*) aij->A->data;
  Mat_SeqAIJ        *b_aij = (Mat_SeqAIJ*) aij->B->data;
  PetscReal         *work;
  const PetscScalar *dummy;

  PetscFunctionBegin;
  PetscCall(MatGetSize(A,&m,&n));
  PetscCall(PetscCalloc1(n,&work));
  PetscCall(MatSeqAIJGetArrayRead(aij->A,&dummy));
  PetscCall(MatSeqAIJRestoreArrayRead(aij->A,&dummy));
  PetscCall(MatSeqAIJGetArrayRead(aij->B,&dummy));
  PetscCall(MatSeqAIJRestoreArrayRead(aij->B,&dummy));
  if (type == NORM_2) {
    for (i=0; i<a_aij->i[aij->A->rmap->n]; i++) {
      work[A->cmap->rstart + a_aij->j[i]] += PetscAbsScalar(a_aij->a[i]*a_aij->a[i]);
    }
    for (i=0; i<b_aij->i[aij->B->rmap->n]; i++) {
      work[garray[b_aij->j[i]]] += PetscAbsScalar(b_aij->a[i]*b_aij->a[i]);
    }
  } else if (type == NORM_1) {
    for (i=0; i<a_aij->i[aij->A->rmap->n]; i++) {
      work[A->cmap->rstart + a_aij->j[i]] += PetscAbsScalar(a_aij->a[i]);
    }
    for (i=0; i<b_aij->i[aij->B->rmap->n]; i++) {
      work[garray[b_aij->j[i]]] += PetscAbsScalar(b_aij->a[i]);
    }
  } else if (type == NORM_INFINITY) {
    for (i=0; i<a_aij->i[aij->A->rmap->n]; i++) {
      work[A->cmap->rstart + a_aij->j[i]] = PetscMax(PetscAbsScalar(a_aij->a[i]), work[A->cmap->rstart + a_aij->j[i]]);
    }
    for (i=0; i<b_aij->i[aij->B->rmap->n]; i++) {
      work[garray[b_aij->j[i]]] = PetscMax(PetscAbsScalar(b_aij->a[i]),work[garray[b_aij->j[i]]]);
    }
  } else if (type == REDUCTION_SUM_REALPART || type == REDUCTION_MEAN_REALPART) {
    for (i=0; i<a_aij->i[aij->A->rmap->n]; i++) {
      work[A->cmap->rstart + a_aij->j[i]] += PetscRealPart(a_aij->a[i]);
    }
    for (i=0; i<b_aij->i[aij->B->rmap->n]; i++) {
      work[garray[b_aij->j[i]]] += PetscRealPart(b_aij->a[i]);
    }
  } else if (type == REDUCTION_SUM_IMAGINARYPART || type == REDUCTION_MEAN_IMAGINARYPART) {
    for (i=0; i<a_aij->i[aij->A->rmap->n]; i++) {
      work[A->cmap->rstart + a_aij->j[i]] += PetscImaginaryPart(a_aij->a[i]);
    }
    for (i=0; i<b_aij->i[aij->B->rmap->n]; i++) {
      work[garray[b_aij->j[i]]] += PetscImaginaryPart(b_aij->a[i]);
    }
  } else SETERRQ(PetscObjectComm((PetscObject)A),PETSC_ERR_ARG_WRONG,"Unknown reduction type");
  if (type == NORM_INFINITY) {
    PetscCall(MPIU_Allreduce(work,reductions,n,MPIU_REAL,MPIU_MAX,PetscObjectComm((PetscObject)A)));
  } else {
    PetscCall(MPIU_Allreduce(work,reductions,n,MPIU_REAL,MPIU_SUM,PetscObjectComm((PetscObject)A)));
  }
  PetscCall(PetscFree(work));
  if (type == NORM_2) {
    for (i=0; i<n; i++) reductions[i] = PetscSqrtReal(reductions[i]);
  } else if (type == REDUCTION_MEAN_REALPART || type == REDUCTION_MEAN_IMAGINARYPART) {
    for (i=0; i<n; i++) reductions[i] /= m;
  }
  PetscFunctionReturn(0);
}

PetscErrorCode MatFindOffBlockDiagonalEntries_MPIAIJ(Mat A,IS *is)
{
  Mat_MPIAIJ      *a  = (Mat_MPIAIJ*)A->data;
  IS              sis,gis;
  const PetscInt  *isis,*igis;
  PetscInt        n,*iis,nsis,ngis,rstart,i;

  PetscFunctionBegin;
  PetscCall(MatFindOffBlockDiagonalEntries(a->A,&sis));
  PetscCall(MatFindNonzeroRows(a->B,&gis));
  PetscCall(ISGetSize(gis,&ngis));
  PetscCall(ISGetSize(sis,&nsis));
  PetscCall(ISGetIndices(sis,&isis));
  PetscCall(ISGetIndices(gis,&igis));

  PetscCall(PetscMalloc1(ngis+nsis,&iis));
  PetscCall(PetscArraycpy(iis,igis,ngis));
  PetscCall(PetscArraycpy(iis+ngis,isis,nsis));
  n    = ngis + nsis;
  PetscCall(PetscSortRemoveDupsInt(&n,iis));
  PetscCall(MatGetOwnershipRange(A,&rstart,NULL));
  for (i=0; i<n; i++) iis[i] += rstart;
  PetscCall(ISCreateGeneral(PetscObjectComm((PetscObject)A),n,iis,PETSC_OWN_POINTER,is));

  PetscCall(ISRestoreIndices(sis,&isis));
  PetscCall(ISRestoreIndices(gis,&igis));
  PetscCall(ISDestroy(&sis));
  PetscCall(ISDestroy(&gis));
  PetscFunctionReturn(0);
}

/*
  Local utility routine that creates a mapping from the global column
number to the local number in the off-diagonal part of the local
storage of the matrix.  When PETSC_USE_CTABLE is used this is scalable at
a slightly higher hash table cost; without it it is not scalable (each processor
has an order N integer array but is fast to access.
*/
PetscErrorCode MatCreateColmap_MPIAIJ_Private(Mat mat)
{
  Mat_MPIAIJ     *aij = (Mat_MPIAIJ*)mat->data;
  PetscInt       n = aij->B->cmap->n,i;

  PetscFunctionBegin;
  PetscCheck(!n || aij->garray,PETSC_COMM_SELF,PETSC_ERR_PLIB,"MPIAIJ Matrix was assembled but is missing garray");
#if defined(PETSC_USE_CTABLE)
  PetscCall(PetscTableCreate(n,mat->cmap->N+1,&aij->colmap));
  for (i=0; i<n; i++) {
    PetscCall(PetscTableAdd(aij->colmap,aij->garray[i]+1,i+1,INSERT_VALUES));
  }
#else
  PetscCall(PetscCalloc1(mat->cmap->N+1,&aij->colmap));
  PetscCall(PetscLogObjectMemory((PetscObject)mat,(mat->cmap->N+1)*sizeof(PetscInt)));
  for (i=0; i<n; i++) aij->colmap[aij->garray[i]] = i+1;
#endif
  PetscFunctionReturn(0);
}

#define MatSetValues_SeqAIJ_A_Private(row,col,value,addv,orow,ocol)     \
{ \
    if (col <= lastcol1)  low1 = 0;     \
    else                 high1 = nrow1; \
    lastcol1 = col;\
    while (high1-low1 > 5) { \
      t = (low1+high1)/2; \
      if (rp1[t] > col) high1 = t; \
      else              low1  = t; \
    } \
      for (_i=low1; _i<high1; _i++) { \
        if (rp1[_i] > col) break; \
        if (rp1[_i] == col) { \
          if (addv == ADD_VALUES) { \
            ap1[_i] += value;   \
            /* Not sure LogFlops will slow dow the code or not */ \
            (void)PetscLogFlops(1.0);   \
           } \
          else                    ap1[_i] = value; \
          goto a_noinsert; \
        } \
      }  \
      if (value == 0.0 && ignorezeroentries && row != col) {low1 = 0; high1 = nrow1;goto a_noinsert;} \
      if (nonew == 1) {low1 = 0; high1 = nrow1; goto a_noinsert;}                \
      PetscCheck(nonew != -1,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Inserting a new nonzero at global row/column (%" PetscInt_FMT ", %" PetscInt_FMT ") into matrix", orow, ocol); \
      MatSeqXAIJReallocateAIJ(A,am,1,nrow1,row,col,rmax1,aa,ai,aj,rp1,ap1,aimax,nonew,MatScalar); \
      N = nrow1++ - 1; a->nz++; high1++; \
      /* shift up all the later entries in this row */ \
      PetscCall(PetscArraymove(rp1+_i+1,rp1+_i,N-_i+1));\
      PetscCall(PetscArraymove(ap1+_i+1,ap1+_i,N-_i+1));\
      rp1[_i] = col;  \
      ap1[_i] = value;  \
      A->nonzerostate++;\
      a_noinsert: ; \
      ailen[row] = nrow1; \
}

#define MatSetValues_SeqAIJ_B_Private(row,col,value,addv,orow,ocol) \
  { \
    if (col <= lastcol2) low2 = 0;                        \
    else high2 = nrow2;                                   \
    lastcol2 = col;                                       \
    while (high2-low2 > 5) {                              \
      t = (low2+high2)/2;                                 \
      if (rp2[t] > col) high2 = t;                        \
      else             low2  = t;                         \
    }                                                     \
    for (_i=low2; _i<high2; _i++) {                       \
      if (rp2[_i] > col) break;                           \
      if (rp2[_i] == col) {                               \
        if (addv == ADD_VALUES) {                         \
          ap2[_i] += value;                               \
          (void)PetscLogFlops(1.0);                       \
        }                                                 \
        else                    ap2[_i] = value;          \
        goto b_noinsert;                                  \
      }                                                   \
    }                                                     \
    if (value == 0.0 && ignorezeroentries) {low2 = 0; high2 = nrow2; goto b_noinsert;} \
    if (nonew == 1) {low2 = 0; high2 = nrow2; goto b_noinsert;}                        \
    PetscCheck(nonew != -1,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Inserting a new nonzero at global row/column (%" PetscInt_FMT ", %" PetscInt_FMT ") into matrix", orow, ocol); \
    MatSeqXAIJReallocateAIJ(B,bm,1,nrow2,row,col,rmax2,ba,bi,bj,rp2,ap2,bimax,nonew,MatScalar); \
    N = nrow2++ - 1; b->nz++; high2++;                    \
    /* shift up all the later entries in this row */      \
    PetscCall(PetscArraymove(rp2+_i+1,rp2+_i,N-_i+1));\
    PetscCall(PetscArraymove(ap2+_i+1,ap2+_i,N-_i+1));\
    rp2[_i] = col;                                        \
    ap2[_i] = value;                                      \
    B->nonzerostate++;                                    \
    b_noinsert: ;                                         \
    bilen[row] = nrow2;                                   \
  }

PetscErrorCode MatSetValuesRow_MPIAIJ(Mat A,PetscInt row,const PetscScalar v[])
{
  Mat_MPIAIJ     *mat = (Mat_MPIAIJ*)A->data;
  Mat_SeqAIJ     *a   = (Mat_SeqAIJ*)mat->A->data,*b = (Mat_SeqAIJ*)mat->B->data;
  PetscInt       l,*garray = mat->garray,diag;
  PetscScalar    *aa,*ba;

  PetscFunctionBegin;
  /* code only works for square matrices A */

  /* find size of row to the left of the diagonal part */
  PetscCall(MatGetOwnershipRange(A,&diag,NULL));
  row  = row - diag;
  for (l=0; l<b->i[row+1]-b->i[row]; l++) {
    if (garray[b->j[b->i[row]+l]] > diag) break;
  }
  if (l) {
    PetscCall(MatSeqAIJGetArray(mat->B,&ba));
    PetscCall(PetscArraycpy(ba+b->i[row],v,l));
    PetscCall(MatSeqAIJRestoreArray(mat->B,&ba));
  }

  /* diagonal part */
  if (a->i[row+1]-a->i[row]) {
    PetscCall(MatSeqAIJGetArray(mat->A,&aa));
    PetscCall(PetscArraycpy(aa+a->i[row],v+l,(a->i[row+1]-a->i[row])));
    PetscCall(MatSeqAIJRestoreArray(mat->A,&aa));
  }

  /* right of diagonal part */
  if (b->i[row+1]-b->i[row]-l) {
    PetscCall(MatSeqAIJGetArray(mat->B,&ba));
    PetscCall(PetscArraycpy(ba+b->i[row]+l,v+l+a->i[row+1]-a->i[row],b->i[row+1]-b->i[row]-l));
    PetscCall(MatSeqAIJRestoreArray(mat->B,&ba));
  }
  PetscFunctionReturn(0);
}

PetscErrorCode MatSetValues_MPIAIJ(Mat mat,PetscInt m,const PetscInt im[],PetscInt n,const PetscInt in[],const PetscScalar v[],InsertMode addv)
{
  Mat_MPIAIJ     *aij = (Mat_MPIAIJ*)mat->data;
  PetscScalar    value = 0.0;
  PetscInt       i,j,rstart  = mat->rmap->rstart,rend = mat->rmap->rend;
  PetscInt       cstart      = mat->cmap->rstart,cend = mat->cmap->rend,row,col;
  PetscBool      roworiented = aij->roworiented;

  /* Some Variables required in the macro */
  Mat        A                    = aij->A;
  Mat_SeqAIJ *a                   = (Mat_SeqAIJ*)A->data;
  PetscInt   *aimax               = a->imax,*ai = a->i,*ailen = a->ilen,*aj = a->j;
  PetscBool  ignorezeroentries    = a->ignorezeroentries;
  Mat        B                    = aij->B;
  Mat_SeqAIJ *b                   = (Mat_SeqAIJ*)B->data;
  PetscInt   *bimax               = b->imax,*bi = b->i,*bilen = b->ilen,*bj = b->j,bm = aij->B->rmap->n,am = aij->A->rmap->n;
  MatScalar  *aa,*ba;
  PetscInt   *rp1,*rp2,ii,nrow1,nrow2,_i,rmax1,rmax2,N,low1,high1,low2,high2,t,lastcol1,lastcol2;
  PetscInt   nonew;
  MatScalar  *ap1,*ap2;

  PetscFunctionBegin;
  PetscCall(MatSeqAIJGetArray(A,&aa));
  PetscCall(MatSeqAIJGetArray(B,&ba));
  for (i=0; i<m; i++) {
    if (im[i] < 0) continue;
    PetscCheck(im[i] < mat->rmap->N,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Row too large: row %" PetscInt_FMT " max %" PetscInt_FMT,im[i],mat->rmap->N-1);
    if (im[i] >= rstart && im[i] < rend) {
      row      = im[i] - rstart;
      lastcol1 = -1;
      rp1      = aj + ai[row];
      ap1      = aa + ai[row];
      rmax1    = aimax[row];
      nrow1    = ailen[row];
      low1     = 0;
      high1    = nrow1;
      lastcol2 = -1;
      rp2      = bj + bi[row];
      ap2      = ba + bi[row];
      rmax2    = bimax[row];
      nrow2    = bilen[row];
      low2     = 0;
      high2    = nrow2;

      for (j=0; j<n; j++) {
        if (v)  value = roworiented ? v[i*n+j] : v[i+j*m];
        if (ignorezeroentries && value == 0.0 && (addv == ADD_VALUES) && im[i] != in[j]) continue;
        if (in[j] >= cstart && in[j] < cend) {
          col   = in[j] - cstart;
          nonew = a->nonew;
          MatSetValues_SeqAIJ_A_Private(row,col,value,addv,im[i],in[j]);
        } else if (in[j] < 0) {
          continue;
        } else {
          PetscCheck(in[j] < mat->cmap->N,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Column too large: col %" PetscInt_FMT " max %" PetscInt_FMT,in[j],mat->cmap->N-1);
          if (mat->was_assembled) {
            if (!aij->colmap) {
              PetscCall(MatCreateColmap_MPIAIJ_Private(mat));
            }
#if defined(PETSC_USE_CTABLE)
            PetscCall(PetscTableFind(aij->colmap,in[j]+1,&col)); /* map global col ids to local ones */
            col--;
#else
            col = aij->colmap[in[j]] - 1;
#endif
            if (col < 0 && !((Mat_SeqAIJ*)(aij->B->data))->nonew) { /* col < 0 means in[j] is a new col for B */
              PetscCall(MatDisAssemble_MPIAIJ(mat)); /* Change aij->B from reduced/local format to expanded/global format */
              col  =  in[j];
              /* Reinitialize the variables required by MatSetValues_SeqAIJ_B_Private() */
              B        = aij->B;
              b        = (Mat_SeqAIJ*)B->data;
              bimax    = b->imax; bi = b->i; bilen = b->ilen; bj = b->j; ba = b->a;
              rp2      = bj + bi[row];
              ap2      = ba + bi[row];
              rmax2    = bimax[row];
              nrow2    = bilen[row];
              low2     = 0;
              high2    = nrow2;
              bm       = aij->B->rmap->n;
              ba       = b->a;
            } else if (col < 0 && !(ignorezeroentries && value == 0.0)) {
              if (1 == ((Mat_SeqAIJ*)(aij->B->data))->nonew) {
                PetscCall(PetscInfo(mat,"Skipping of insertion of new nonzero location in off-diagonal portion of matrix %g(%" PetscInt_FMT ",%" PetscInt_FMT ")\n",(double)PetscRealPart(value),im[i],in[j]));
              } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Inserting a new nonzero at global row/column (%" PetscInt_FMT ", %" PetscInt_FMT ") into matrix", im[i], in[j]);
            }
          } else col = in[j];
          nonew = b->nonew;
          MatSetValues_SeqAIJ_B_Private(row,col,value,addv,im[i],in[j]);
        }
      }
    } else {
      PetscCheck(!mat->nooffprocentries,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Setting off process row %" PetscInt_FMT " even though MatSetOption(,MAT_NO_OFF_PROC_ENTRIES,PETSC_TRUE) was set",im[i]);
      if (!aij->donotstash) {
        mat->assembled = PETSC_FALSE;
        if (roworiented) {
          PetscCall(MatStashValuesRow_Private(&mat->stash,im[i],n,in,v+i*n,(PetscBool)(ignorezeroentries && (addv == ADD_VALUES))));
        } else {
          PetscCall(MatStashValuesCol_Private(&mat->stash,im[i],n,in,v+i,m,(PetscBool)(ignorezeroentries && (addv == ADD_VALUES))));
        }
      }
    }
  }
  PetscCall(MatSeqAIJRestoreArray(A,&aa)); /* aa, bb might have been free'd due to reallocation above. But we don't access them here */
  PetscCall(MatSeqAIJRestoreArray(B,&ba));
  PetscFunctionReturn(0);
}

/*
    This function sets the j and ilen arrays (of the diagonal and off-diagonal part) of an MPIAIJ-matrix.
    The values in mat_i have to be sorted and the values in mat_j have to be sorted for each row (CSR-like).
    No off-processor parts off the matrix are allowed here and mat->was_assembled has to be PETSC_FALSE.
*/
PetscErrorCode MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(Mat mat,const PetscInt mat_j[],const PetscInt mat_i[])
{
  Mat_MPIAIJ     *aij        = (Mat_MPIAIJ*)mat->data;
  Mat            A           = aij->A; /* diagonal part of the matrix */
  Mat            B           = aij->B; /* offdiagonal part of the matrix */
  Mat_SeqAIJ     *a          = (Mat_SeqAIJ*)A->data;
  Mat_SeqAIJ     *b          = (Mat_SeqAIJ*)B->data;
  PetscInt       cstart      = mat->cmap->rstart,cend = mat->cmap->rend,col;
  PetscInt       *ailen      = a->ilen,*aj = a->j;
  PetscInt       *bilen      = b->ilen,*bj = b->j;
  PetscInt       am          = aij->A->rmap->n,j;
  PetscInt       diag_so_far = 0,dnz;
  PetscInt       offd_so_far = 0,onz;

  PetscFunctionBegin;
  /* Iterate over all rows of the matrix */
  for (j=0; j<am; j++) {
    dnz = onz = 0;
    /*  Iterate over all non-zero columns of the current row */
    for (col=mat_i[j]; col<mat_i[j+1]; col++) {
      /* If column is in the diagonal */
      if (mat_j[col] >= cstart && mat_j[col] < cend) {
        aj[diag_so_far++] = mat_j[col] - cstart;
        dnz++;
      } else { /* off-diagonal entries */
        bj[offd_so_far++] = mat_j[col];
        onz++;
      }
    }
    ailen[j] = dnz;
    bilen[j] = onz;
  }
  PetscFunctionReturn(0);
}

/*
    This function sets the local j, a and ilen arrays (of the diagonal and off-diagonal part) of an MPIAIJ-matrix.
    The values in mat_i have to be sorted and the values in mat_j have to be sorted for each row (CSR-like).
    No off-processor parts off the matrix are allowed here, they are set at a later point by MatSetValues_MPIAIJ.
    Also, mat->was_assembled has to be false, otherwise the statement aj[rowstart_diag+dnz_row] = mat_j[col] - cstart;
    would not be true and the more complex MatSetValues_MPIAIJ has to be used.
*/
PetscErrorCode MatSetValues_MPIAIJ_CopyFromCSRFormat(Mat mat,const PetscInt mat_j[],const PetscInt mat_i[],const PetscScalar mat_a[])
{
  Mat_MPIAIJ     *aij   = (Mat_MPIAIJ*)mat->data;
  Mat            A      = aij->A; /* diagonal part of the matrix */
  Mat            B      = aij->B; /* offdiagonal part of the matrix */
  Mat_SeqAIJ     *aijd  =(Mat_SeqAIJ*)(aij->A)->data,*aijo=(Mat_SeqAIJ*)(aij->B)->data;
  Mat_SeqAIJ     *a     = (Mat_SeqAIJ*)A->data;
  Mat_SeqAIJ     *b     = (Mat_SeqAIJ*)B->data;
  PetscInt       cstart = mat->cmap->rstart,cend = mat->cmap->rend;
  PetscInt       *ailen = a->ilen,*aj = a->j;
  PetscInt       *bilen = b->ilen,*bj = b->j;
  PetscInt       am     = aij->A->rmap->n,j;
  PetscInt       *full_diag_i=aijd->i,*full_offd_i=aijo->i; /* These variables can also include non-local elements, which are set at a later point. */
  PetscInt       col,dnz_row,onz_row,rowstart_diag,rowstart_offd;
  PetscScalar    *aa = a->a,*ba = b->a;

  PetscFunctionBegin;
  /* Iterate over all rows of the matrix */
  for (j=0; j<am; j++) {
    dnz_row = onz_row = 0;
    rowstart_offd = full_offd_i[j];
    rowstart_diag = full_diag_i[j];
    /*  Iterate over all non-zero columns of the current row */
    for (col=mat_i[j]; col<mat_i[j+1]; col++) {
      /* If column is in the diagonal */
      if (mat_j[col] >= cstart && mat_j[col] < cend) {
        aj[rowstart_diag+dnz_row] = mat_j[col] - cstart;
        aa[rowstart_diag+dnz_row] = mat_a[col];
        dnz_row++;
      } else { /* off-diagonal entries */
        bj[rowstart_offd+onz_row] = mat_j[col];
        ba[rowstart_offd+onz_row] = mat_a[col];
        onz_row++;
      }
    }
    ailen[j] = dnz_row;
    bilen[j] = onz_row;
  }
  PetscFunctionReturn(0);
}

PetscErrorCode MatGetValues_MPIAIJ(Mat mat,PetscInt m,const PetscInt idxm[],PetscInt n,const PetscInt idxn[],PetscScalar v[])
{
  Mat_MPIAIJ     *aij = (Mat_MPIAIJ*)mat->data;
  PetscInt       i,j,rstart = mat->rmap->rstart,rend = mat->rmap->rend;
  PetscInt       cstart = mat->cmap->rstart,cend = mat->cmap->rend,row,col;

  PetscFunctionBegin;
  for (i=0; i<m; i++) {
    if (idxm[i] < 0) continue; /* negative row */
    PetscCheck(idxm[i] < mat->rmap->N,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Row too large: row %" PetscInt_FMT " max %" PetscInt_FMT,idxm[i],mat->rmap->N-1);
    if (idxm[i] >= rstart && idxm[i] < rend) {
      row = idxm[i] - rstart;
      for (j=0; j<n; j++) {
        if (idxn[j] < 0) continue; /* negative column */
        PetscCheck(idxn[j] < mat->cmap->N,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Column too large: col %" PetscInt_FMT " max %" PetscInt_FMT,idxn[j],mat->cmap->N-1);
        if (idxn[j] >= cstart && idxn[j] < cend) {
          col  = idxn[j] - cstart;
          PetscCall(MatGetValues(aij->A,1,&row,1,&col,v+i*n+j));
        } else {
          if (!aij->colmap) {
            PetscCall(MatCreateColmap_MPIAIJ_Private(mat));
          }
#if defined(PETSC_USE_CTABLE)
          PetscCall(PetscTableFind(aij->colmap,idxn[j]+1,&col));
          col--;
#else
          col = aij->colmap[idxn[j]] - 1;
#endif
          if ((col < 0) || (aij->garray[col] != idxn[j])) *(v+i*n+j) = 0.0;
          else {
            PetscCall(MatGetValues(aij->B,1,&row,1,&col,v+i*n+j));
          }
        }
      }
    } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Only local values currently supported");
  }
  PetscFunctionReturn(0);
}

PetscErrorCode MatAssemblyBegin_MPIAIJ(Mat mat,MatAssemblyType mode)
{
  Mat_MPIAIJ     *aij = (Mat_MPIAIJ*)mat->data;
  PetscInt       nstash,reallocs;

  PetscFunctionBegin;
  if (aij->donotstash || mat->nooffprocentries) PetscFunctionReturn(0);

  PetscCall(MatStashScatterBegin_Private(mat,&mat->stash,mat->rmap->range));
  PetscCall(MatStashGetInfo_Private(&mat->stash,&nstash,&reallocs));
  PetscCall(PetscInfo(aij->A,"Stash has %" PetscInt_FMT " entries, uses %" PetscInt_FMT " mallocs.\n",nstash,reallocs));
  PetscFunctionReturn(0);
}

PetscErrorCode MatAssemblyEnd_MPIAIJ(Mat mat,MatAssemblyType mode)
{
  Mat_MPIAIJ     *aij = (Mat_MPIAIJ*)mat->data;
  PetscMPIInt    n;
  PetscInt       i,j,rstart,ncols,flg;
  PetscInt       *row,*col;
  PetscBool      other_disassembled;
  PetscScalar    *val;

  /* do not use 'b = (Mat_SeqAIJ*)aij->B->data' as B can be reset in disassembly */

  PetscFunctionBegin;
  if (!aij->donotstash && !mat->nooffprocentries) {
    while (1) {
      PetscCall(MatStashScatterGetMesg_Private(&mat->stash,&n,&row,&col,&val,&flg));
      if (!flg) break;

      for (i=0; i<n;) {
        /* Now identify the consecutive vals belonging to the same row */
        for (j=i,rstart=row[j]; j<n; j++) {
          if (row[j] != rstart) break;
        }
        if (j < n) ncols = j-i;
        else       ncols = n-i;
        /* Now assemble all these values with a single function call */
        PetscCall(MatSetValues_MPIAIJ(mat,1,row+i,ncols,col+i,val+i,mat->insertmode));
        i    = j;
      }
    }
    PetscCall(MatStashScatterEnd_Private(&mat->stash));
  }
#if defined(PETSC_HAVE_DEVICE)
  if (mat->offloadmask == PETSC_OFFLOAD_CPU) aij->A->offloadmask = PETSC_OFFLOAD_CPU;
  /* We call MatBindToCPU() on aij->A and aij->B here, because if MatBindToCPU_MPIAIJ() is called before assembly, it cannot bind these. */
  if (mat->boundtocpu) {
    PetscCall(MatBindToCPU(aij->A,PETSC_TRUE));
    PetscCall(MatBindToCPU(aij->B,PETSC_TRUE));
  }
#endif
  PetscCall(MatAssemblyBegin(aij->A,mode));
  PetscCall(MatAssemblyEnd(aij->A,mode));

  /* determine if any processor has disassembled, if so we must
     also disassemble ourself, in order that we may reassemble. */
  /*
     if nonzero structure of submatrix B cannot change then we know that
     no processor disassembled thus we can skip this stuff
  */
  if (!((Mat_SeqAIJ*)aij->B->data)->nonew) {
    PetscCall(MPIU_Allreduce(&mat->was_assembled,&other_disassembled,1,MPIU_BOOL,MPI_PROD,PetscObjectComm((PetscObject)mat)));
    if (mat->was_assembled && !other_disassembled) { /* mat on this rank has reduced off-diag B with local col ids, but globaly it does not */
      PetscCall(MatDisAssemble_MPIAIJ(mat));
    }
  }
  if (!mat->was_assembled && mode == MAT_FINAL_ASSEMBLY) {
    PetscCall(MatSetUpMultiply_MPIAIJ(mat));
  }
  PetscCall(MatSetOption(aij->B,MAT_USE_INODES,PETSC_FALSE));
#if defined(PETSC_HAVE_DEVICE)
  if (mat->offloadmask == PETSC_OFFLOAD_CPU && aij->B->offloadmask != PETSC_OFFLOAD_UNALLOCATED) aij->B->offloadmask = PETSC_OFFLOAD_CPU;
#endif
  PetscCall(MatAssemblyBegin(aij->B,mode));
  PetscCall(MatAssemblyEnd(aij->B,mode));

  PetscCall(PetscFree2(aij->rowvalues,aij->rowindices));

  aij->rowvalues = NULL;

  PetscCall(VecDestroy(&aij->diag));

  /* if no new nonzero locations are allowed in matrix then only set the matrix state the first time through */
  if ((!mat->was_assembled && mode == MAT_FINAL_ASSEMBLY) || !((Mat_SeqAIJ*)(aij->A->data))->nonew) {
    PetscObjectState state = aij->A->nonzerostate + aij->B->nonzerostate;
    PetscCall(MPIU_Allreduce(&state,&mat->nonzerostate,1,MPIU_INT64,MPI_SUM,PetscObjectComm((PetscObject)mat)));
  }
#if defined(PETSC_HAVE_DEVICE)
  mat->offloadmask = PETSC_OFFLOAD_BOTH;
#endif
  PetscFunctionReturn(0);
}

PetscErrorCode MatZeroEntries_MPIAIJ(Mat A)
{
  Mat_MPIAIJ     *l = (Mat_MPIAIJ*)A->data;

  PetscFunctionBegin;
  PetscCall(MatZeroEntries(l->A));
  PetscCall(MatZeroEntries(l->B));
  PetscFunctionReturn(0);
}

PetscErrorCode MatZeroRows_MPIAIJ(Mat A,PetscInt N,const PetscInt rows[],PetscScalar diag,Vec x,Vec b)
{
  Mat_MPIAIJ      *mat = (Mat_MPIAIJ *) A->data;
  PetscObjectState sA, sB;
  PetscInt        *lrows;
  PetscInt         r, len;
  PetscBool        cong, lch, gch;

  PetscFunctionBegin;
  /* get locally owned rows */
  PetscCall(MatZeroRowsMapLocal_Private(A,N,rows,&len,&lrows));
  PetscCall(MatHasCongruentLayouts(A,&cong));
  /* fix right hand side if needed */
  if (x && b) {
    const PetscScalar *xx;
    PetscScalar       *bb;

    PetscCheck(cong,PetscObjectComm((PetscObject)A),PETSC_ERR_SUP,"Need matching row/col layout");
    PetscCall(VecGetArrayRead(x, &xx));
    PetscCall(VecGetArray(b, &bb));
    for (r = 0; r < len; ++r) bb[lrows[r]] = diag*xx[lrows[r]];
    PetscCall(VecRestoreArrayRead(x, &xx));
    PetscCall(VecRestoreArray(b, &bb));
  }

  sA = mat->A->nonzerostate;
  sB = mat->B->nonzerostate;

  if (diag != 0.0 && cong) {
    PetscCall(MatZeroRows(mat->A, len, lrows, diag, NULL, NULL));
    PetscCall(MatZeroRows(mat->B, len, lrows, 0.0, NULL, NULL));
  } else if (diag != 0.0) { /* non-square or non congruent layouts -> if keepnonzeropattern is false, we allow for new insertion */
    Mat_SeqAIJ *aijA = (Mat_SeqAIJ*)mat->A->data;
    Mat_SeqAIJ *aijB = (Mat_SeqAIJ*)mat->B->data;
    PetscInt   nnwA, nnwB;
    PetscBool  nnzA, nnzB;

    nnwA = aijA->nonew;
    nnwB = aijB->nonew;
    nnzA = aijA->keepnonzeropattern;
    nnzB = aijB->keepnonzeropattern;
    if (!nnzA) {
      PetscCall(PetscInfo(mat->A,"Requested to not keep the pattern and add a nonzero diagonal; may encounter reallocations on diagonal block.\n"));
      aijA->nonew = 0;
    }
    if (!nnzB) {
      PetscCall(PetscInfo(mat->B,"Requested to not keep the pattern and add a nonzero diagonal; may encounter reallocations on off-diagonal block.\n"));
      aijB->nonew = 0;
    }
    /* Must zero here before the next loop */
    PetscCall(MatZeroRows(mat->A, len, lrows, 0.0, NULL, NULL));
    PetscCall(MatZeroRows(mat->B, len, lrows, 0.0, NULL, NULL));
    for (r = 0; r < len; ++r) {
      const PetscInt row = lrows[r] + A->rmap->rstart;
      if (row >= A->cmap->N) continue;
      PetscCall(MatSetValues(A, 1, &row, 1, &row, &diag, INSERT_VALUES));
    }
    aijA->nonew = nnwA;
    aijB->nonew = nnwB;
  } else {
    PetscCall(MatZeroRows(mat->A, len, lrows, 0.0, NULL, NULL));
    PetscCall(MatZeroRows(mat->B, len, lrows, 0.0, NULL, NULL));
  }
  PetscCall(PetscFree(lrows));
  PetscCall(MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY));

  /* reduce nonzerostate */
  lch = (PetscBool)(sA != mat->A->nonzerostate || sB != mat->B->nonzerostate);
  PetscCall(MPIU_Allreduce(&lch,&gch,1,MPIU_BOOL,MPI_LOR,PetscObjectComm((PetscObject)A)));
  if (gch) A->nonzerostate++;
  PetscFunctionReturn(0);
}

PetscErrorCode MatZeroRowsColumns_MPIAIJ(Mat A,PetscInt N,const PetscInt rows[],PetscScalar diag,Vec x,Vec b)
{
  Mat_MPIAIJ        *l = (Mat_MPIAIJ*)A->data;
  PetscMPIInt       n = A->rmap->n;
  PetscInt          i,j,r,m,len = 0;
  PetscInt          *lrows,*owners = A->rmap->range;
  PetscMPIInt       p = 0;
  PetscSFNode       *rrows;
  PetscSF           sf;
  const PetscScalar *xx;
  PetscScalar       *bb,*mask,*aij_a;
  Vec               xmask,lmask;
  Mat_SeqAIJ        *aij = (Mat_SeqAIJ*)l->B->data;
  const PetscInt    *aj, *ii,*ridx;
  PetscScalar       *aa;

  PetscFunctionBegin;
  /* Create SF where leaves are input rows and roots are owned rows */
  PetscCall(PetscMalloc1(n, &lrows));
  for (r = 0; r < n; ++r) lrows[r] = -1;
  PetscCall(PetscMalloc1(N, &rrows));
  for (r = 0; r < N; ++r) {
    const PetscInt idx   = rows[r];
    PetscCheck(idx >= 0 && A->rmap->N > idx,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Row %" PetscInt_FMT " out of range [0,%" PetscInt_FMT ")",idx,A->rmap->N);
    if (idx < owners[p] || owners[p+1] <= idx) { /* short-circuit the search if the last p owns this row too */
      PetscCall(PetscLayoutFindOwner(A->rmap,idx,&p));
    }
    rrows[r].rank  = p;
    rrows[r].index = rows[r] - owners[p];
  }
  PetscCall(PetscSFCreate(PetscObjectComm((PetscObject) A), &sf));
  PetscCall(PetscSFSetGraph(sf, n, N, NULL, PETSC_OWN_POINTER, rrows, PETSC_OWN_POINTER));
  /* Collect flags for rows to be zeroed */
  PetscCall(PetscSFReduceBegin(sf, MPIU_INT, (PetscInt *) rows, lrows, MPI_LOR));
  PetscCall(PetscSFReduceEnd(sf, MPIU_INT, (PetscInt *) rows, lrows, MPI_LOR));
  PetscCall(PetscSFDestroy(&sf));
  /* Compress and put in row numbers */
  for (r = 0; r < n; ++r) if (lrows[r] >= 0) lrows[len++] = r;
  /* zero diagonal part of matrix */
  PetscCall(MatZeroRowsColumns(l->A,len,lrows,diag,x,b));
  /* handle off diagonal part of matrix */
  PetscCall(MatCreateVecs(A,&xmask,NULL));
  PetscCall(VecDuplicate(l->lvec,&lmask));
  PetscCall(VecGetArray(xmask,&bb));
  for (i=0; i<len; i++) bb[lrows[i]] = 1;
  PetscCall(VecRestoreArray(xmask,&bb));
  PetscCall(VecScatterBegin(l->Mvctx,xmask,lmask,ADD_VALUES,SCATTER_FORWARD));
  PetscCall(VecScatterEnd(l->Mvctx,xmask,lmask,ADD_VALUES,SCATTER_FORWARD));
  PetscCall(VecDestroy(&xmask));
  if (x && b) { /* this code is buggy when the row and column layout don't match */
    PetscBool cong;

    PetscCall(MatHasCongruentLayouts(A,&cong));
    PetscCheck(cong,PetscObjectComm((PetscObject)A),PETSC_ERR_SUP,"Need matching row/col layout");
    PetscCall(VecScatterBegin(l->Mvctx,x,l->lvec,INSERT_VALUES,SCATTER_FORWARD));
    PetscCall(VecScatterEnd(l->Mvctx,x,l->lvec,INSERT_VALUES,SCATTER_FORWARD));
    PetscCall(VecGetArrayRead(l->lvec,&xx));
    PetscCall(VecGetArray(b,&bb));
  }
  PetscCall(VecGetArray(lmask,&mask));
  /* remove zeroed rows of off diagonal matrix */
  PetscCall(MatSeqAIJGetArray(l->B,&aij_a));
  ii = aij->i;
  for (i=0; i<len; i++) {
    PetscCall(PetscArrayzero(aij_a + ii[lrows[i]],ii[lrows[i]+1] - ii[lrows[i]]));
  }
  /* loop over all elements of off process part of matrix zeroing removed columns*/
  if (aij->compressedrow.use) {
    m    = aij->compressedrow.nrows;
    ii   = aij->compressedrow.i;
    ridx = aij->compressedrow.rindex;
    for (i=0; i<m; i++) {
      n  = ii[i+1] - ii[i];
      aj = aij->j + ii[i];
      aa = aij_a + ii[i];

      for (j=0; j<n; j++) {
        if (PetscAbsScalar(mask[*aj])) {
          if (b) bb[*ridx] -= *aa*xx[*aj];
          *aa = 0.0;
        }
        aa++;
        aj++;
      }
      ridx++;
    }
  } else { /* do not use compressed row format */
    m = l->B->rmap->n;
    for (i=0; i<m; i++) {
      n  = ii[i+1] - ii[i];
      aj = aij->j + ii[i];
      aa = aij_a + ii[i];
      for (j=0; j<n; j++) {
        if (PetscAbsScalar(mask[*aj])) {
          if (b) bb[i] -= *aa*xx[*aj];
          *aa = 0.0;
        }
        aa++;
        aj++;
      }
    }
  }
  if (x && b) {
    PetscCall(VecRestoreArray(b,&bb));
    PetscCall(VecRestoreArrayRead(l->lvec,&xx));
  }
  PetscCall(MatSeqAIJRestoreArray(l->B,&aij_a));
  PetscCall(VecRestoreArray(lmask,&mask));
  PetscCall(VecDestroy(&lmask));
  PetscCall(PetscFree(lrows));

  /* only change matrix nonzero state if pattern was allowed to be changed */
  if (!((Mat_SeqAIJ*)(l->A->data))->keepnonzeropattern) {
    PetscObjectState state = l->A->nonzerostate + l->B->nonzerostate;
    PetscCall(MPIU_Allreduce(&state,&A->nonzerostate,1,MPIU_INT64,MPI_SUM,PetscObjectComm((PetscObject)A)));
  }
  PetscFunctionReturn(0);
}

PetscErrorCode MatMult_MPIAIJ(Mat A,Vec xx,Vec yy)
{
  Mat_MPIAIJ     *a = (Mat_MPIAIJ*)A->data;
  PetscInt       nt;
  VecScatter     Mvctx = a->Mvctx;

  PetscFunctionBegin;
  PetscCall(VecGetLocalSize(xx,&nt));
  PetscCheck(nt == A->cmap->n,PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Incompatible partition of A (%" PetscInt_FMT ") and xx (%" PetscInt_FMT ")",A->cmap->n,nt);
  PetscCall(VecScatterBegin(Mvctx,xx,a->lvec,INSERT_VALUES,SCATTER_FORWARD));
  PetscCall((*a->A->ops->mult)(a->A,xx,yy));
  PetscCall(VecScatterEnd(Mvctx,xx,a->lvec,INSERT_VALUES,SCATTER_FORWARD));
  PetscCall((*a->B->ops->multadd)(a->B,a->lvec,yy,yy));
  PetscFunctionReturn(0);
}

PetscErrorCode MatMultDiagonalBlock_MPIAIJ(Mat A,Vec bb,Vec xx)
{
  Mat_MPIAIJ     *a = (Mat_MPIAIJ*)A->data;

  PetscFunctionBegin;
  PetscCall(MatMultDiagonalBlock(a->A,bb,xx));
  PetscFunctionReturn(0);
}

PetscErrorCode MatMultAdd_MPIAIJ(Mat A,Vec xx,Vec yy,Vec zz)
{
  Mat_MPIAIJ     *a = (Mat_MPIAIJ*)A->data;
  VecScatter     Mvctx = a->Mvctx;

  PetscFunctionBegin;
  PetscCall(VecScatterBegin(Mvctx,xx,a->lvec,INSERT_VALUES,SCATTER_FORWARD));
  PetscCall((*a->A->ops->multadd)(a->A,xx,yy,zz));
  PetscCall(VecScatterEnd(Mvctx,xx,a->lvec,INSERT_VALUES,SCATTER_FORWARD));
  PetscCall((*a->B->ops->multadd)(a->B,a->lvec,zz,zz));
  PetscFunctionReturn(0);
}

PetscErrorCode MatMultTranspose_MPIAIJ(Mat A,Vec xx,Vec yy)
{
  Mat_MPIAIJ     *a = (Mat_MPIAIJ*)A->data;

  PetscFunctionBegin;
  /* do nondiagonal part */
  PetscCall((*a->B->ops->multtranspose)(a->B,xx,a->lvec));
  /* do local part */
  PetscCall((*a->A->ops->multtranspose)(a->A,xx,yy));
  /* add partial results together */
  PetscCall(VecScatterBegin(a->Mvctx,a->lvec,yy,ADD_VALUES,SCATTER_REVERSE));
  PetscCall(VecScatterEnd(a->Mvctx,a->lvec,yy,ADD_VALUES,SCATTER_REVERSE));
  PetscFunctionReturn(0);
}

PetscErrorCode MatIsTranspose_MPIAIJ(Mat Amat,Mat Bmat,PetscReal tol,PetscBool  *f)
{
  MPI_Comm       comm;
  Mat_MPIAIJ     *Aij = (Mat_MPIAIJ*) Amat->data, *Bij;
  Mat            Adia = Aij->A, Bdia, Aoff,Boff,*Aoffs,*Boffs;
  IS             Me,Notme;
  PetscInt       M,N,first,last,*notme,i;
  PetscBool      lf;
  PetscMPIInt    size;

  PetscFunctionBegin;
  /* Easy test: symmetric diagonal block */
  Bij  = (Mat_MPIAIJ*) Bmat->data; Bdia = Bij->A;
  PetscCall(MatIsTranspose(Adia,Bdia,tol,&lf));
  PetscCall(MPIU_Allreduce(&lf,f,1,MPIU_BOOL,MPI_LAND,PetscObjectComm((PetscObject)Amat)));
  if (!*f) PetscFunctionReturn(0);
  PetscCall(PetscObjectGetComm((PetscObject)Amat,&comm));
  PetscCallMPI(MPI_Comm_size(comm,&size));
  if (size == 1) PetscFunctionReturn(0);

  /* Hard test: off-diagonal block. This takes a MatCreateSubMatrix. */
  PetscCall(MatGetSize(Amat,&M,&N));
  PetscCall(MatGetOwnershipRange(Amat,&first,&last));
  PetscCall(PetscMalloc1(N-last+first,&notme));
  for (i=0; i<first; i++) notme[i] = i;
  for (i=last; i<M; i++) notme[i-last+first] = i;
  PetscCall(ISCreateGeneral(MPI_COMM_SELF,N-last+first,notme,PETSC_COPY_VALUES,&Notme));
  PetscCall(ISCreateStride(MPI_COMM_SELF,last-first,first,1,&Me));
  PetscCall(MatCreateSubMatrices(Amat,1,&Me,&Notme,MAT_INITIAL_MATRIX,&Aoffs));
  Aoff = Aoffs[0];
  PetscCall(MatCreateSubMatrices(Bmat,1,&Notme,&Me,MAT_INITIAL_MATRIX,&Boffs));
  Boff = Boffs[0];
  PetscCall(MatIsTranspose(Aoff,Boff,tol,f));
  PetscCall(MatDestroyMatrices(1,&Aoffs));
  PetscCall(MatDestroyMatrices(1,&Boffs));
  PetscCall(ISDestroy(&Me));
  PetscCall(ISDestroy(&Notme));
  PetscCall(PetscFree(notme));
  PetscFunctionReturn(0);
}

PetscErrorCode MatIsSymmetric_MPIAIJ(Mat A,PetscReal tol,PetscBool  *f)
{
  PetscFunctionBegin;
  PetscCall(MatIsTranspose_MPIAIJ(A,A,tol,f));
  PetscFunctionReturn(0);
}

PetscErrorCode MatMultTransposeAdd_MPIAIJ(Mat A,Vec xx,Vec yy,Vec zz)
{
  Mat_MPIAIJ     *a = (Mat_MPIAIJ*)A->data;

  PetscFunctionBegin;
  /* do nondiagonal part */
  PetscCall((*a->B->ops->multtranspose)(a->B,xx,a->lvec));
  /* do local part */
  PetscCall((*a->A->ops->multtransposeadd)(a->A,xx,yy,zz));
  /* add partial results together */
  PetscCall(VecScatterBegin(a->Mvctx,a->lvec,zz,ADD_VALUES,SCATTER_REVERSE));
  PetscCall(VecScatterEnd(a->Mvctx,a->lvec,zz,ADD_VALUES,SCATTER_REVERSE));
  PetscFunctionReturn(0);
}

/*
  This only works correctly for square matrices where the subblock A->A is the
   diagonal block
*/
PetscErrorCode MatGetDiagonal_MPIAIJ(Mat A,Vec v)
{
  Mat_MPIAIJ     *a = (Mat_MPIAIJ*)A->data;

  PetscFunctionBegin;
  PetscCheck(A->rmap->N == A->cmap->N,PetscObjectComm((PetscObject)A),PETSC_ERR_SUP,"Supports only square matrix where A->A is diag block");
  PetscCheck(A->rmap->rstart == A->cmap->rstart && A->rmap->rend == A->cmap->rend,PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"row partition must equal col partition");
  PetscCall(MatGetDiagonal(a->A,v));
  PetscFunctionReturn(0);
}

PetscErrorCode MatScale_MPIAIJ(Mat A,PetscScalar aa)
{
  Mat_MPIAIJ     *a = (Mat_MPIAIJ*)A->data;

  PetscFunctionBegin;
  PetscCall(MatScale(a->A,aa));
  PetscCall(MatScale(a->B,aa));
  PetscFunctionReturn(0);
}

/* Free COO stuff; must match allocation methods in MatSetPreallocationCOO_MPIAIJ() */
PETSC_INTERN PetscErrorCode MatResetPreallocationCOO_MPIAIJ(Mat mat)
{
  Mat_MPIAIJ     *aij = (Mat_MPIAIJ*)mat->data;

  PetscFunctionBegin;
  PetscCall(PetscSFDestroy(&aij->coo_sf));
  PetscCall(PetscFree(aij->Aperm1));
  PetscCall(PetscFree(aij->Bperm1));
  PetscCall(PetscFree(aij->Ajmap1));
  PetscCall(PetscFree(aij->Bjmap1));

  PetscCall(PetscFree(aij->Aimap2));
  PetscCall(PetscFree(aij->Bimap2));
  PetscCall(PetscFree(aij->Aperm2));
  PetscCall(PetscFree(aij->Bperm2));
  PetscCall(PetscFree(aij->Ajmap2));
  PetscCall(PetscFree(aij->Bjmap2));

  PetscCall(PetscFree2(aij->sendbuf,aij->recvbuf));
  PetscCall(PetscFree(aij->Cperm1));
  PetscFunctionReturn(0);
}

PetscErrorCode MatDestroy_MPIAIJ(Mat mat)
{
  Mat_MPIAIJ     *aij = (Mat_MPIAIJ*)mat->data;

  PetscFunctionBegin;
#if defined(PETSC_USE_LOG)
  PetscLogObjectState((PetscObject)mat,"Rows=%" PetscInt_FMT ", Cols=%" PetscInt_FMT,mat->rmap->N,mat->cmap->N);
#endif
  PetscCall(MatStashDestroy_Private(&mat->stash));
  PetscCall(VecDestroy(&aij->diag));
  PetscCall(MatDestroy(&aij->A));
  PetscCall(MatDestroy(&aij->B));
#if defined(PETSC_USE_CTABLE)
  PetscCall(PetscTableDestroy(&aij->colmap));
#else
  PetscCall(PetscFree(aij->colmap));
#endif
  PetscCall(PetscFree(aij->garray));
  PetscCall(VecDestroy(&aij->lvec));
  PetscCall(VecScatterDestroy(&aij->Mvctx));
  PetscCall(PetscFree2(aij->rowvalues,aij->rowindices));
  PetscCall(PetscFree(aij->ld));

  /* Free COO */
  PetscCall(MatResetPreallocationCOO_MPIAIJ(mat));

  PetscCall(PetscFree(mat->data));

  /* may be created by MatCreateMPIAIJSumSeqAIJSymbolic */
  PetscCall(PetscObjectCompose((PetscObject)mat,"MatMergeSeqsToMPI",NULL));

  PetscCall(PetscObjectChangeTypeName((PetscObject)mat,NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatStoreValues_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatRetrieveValues_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatIsTranspose_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatMPIAIJSetPreallocation_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatResetPreallocation_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatMPIAIJSetPreallocationCSR_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatDiagonalScaleLocal_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatConvert_mpiaij_mpibaij_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatConvert_mpiaij_mpisbaij_C",NULL));
#if defined(PETSC_HAVE_CUDA)
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatConvert_mpiaij_mpiaijcusparse_C",NULL));
#endif
#if defined(PETSC_HAVE_KOKKOS_KERNELS)
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatConvert_mpiaij_mpiaijkokkos_C",NULL));
#endif
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatConvert_mpiaij_mpidense_C",NULL));
#if defined(PETSC_HAVE_ELEMENTAL)
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatConvert_mpiaij_elemental_C",NULL));
#endif
#if defined(PETSC_HAVE_SCALAPACK)
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatConvert_mpiaij_scalapack_C",NULL));
#endif
#if defined(PETSC_HAVE_HYPRE)
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatConvert_mpiaij_hypre_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatProductSetFromOptions_transpose_mpiaij_mpiaij_C",NULL));
#endif
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatConvert_mpiaij_is_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatProductSetFromOptions_is_mpiaij_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatProductSetFromOptions_mpiaij_mpiaij_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatMPIAIJSetUseScalableIncreaseOverlap_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatConvert_mpiaij_mpiaijperm_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatConvert_mpiaij_mpiaijsell_C",NULL));
#if defined(PETSC_HAVE_MKL_SPARSE)
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatConvert_mpiaij_mpiaijmkl_C",NULL));
#endif
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatConvert_mpiaij_mpiaijcrl_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatConvert_mpiaij_is_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatConvert_mpiaij_mpisell_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatSetPreallocationCOO_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)mat,"MatSetValuesCOO_C",NULL));
  PetscFunctionReturn(0);
}

PetscErrorCode MatView_MPIAIJ_Binary(Mat mat,PetscViewer viewer)
{
  Mat_MPIAIJ        *aij = (Mat_MPIAIJ*)mat->data;
  Mat_SeqAIJ        *A   = (Mat_SeqAIJ*)aij->A->data;
  Mat_SeqAIJ        *B   = (Mat_SeqAIJ*)aij->B->data;
  const PetscInt    *garray = aij->garray;
  const PetscScalar *aa,*ba;
  PetscInt          header[4],M,N,m,rs,cs,nz,cnt,i,ja,jb;
  PetscInt          *rowlens;
  PetscInt          *colidxs;
  PetscScalar       *matvals;

  PetscFunctionBegin;
  PetscCall(PetscViewerSetUp(viewer));

  M  = mat->rmap->N;
  N  = mat->cmap->N;
  m  = mat->rmap->n;
  rs = mat->rmap->rstart;
  cs = mat->cmap->rstart;
  nz = A->nz + B->nz;

  /* write matrix header */
  header[0] = MAT_FILE_CLASSID;
  header[1] = M; header[2] = N; header[3] = nz;
  PetscCallMPI(MPI_Reduce(&nz,&header[3],1,MPIU_INT,MPI_SUM,0,PetscObjectComm((PetscObject)mat)));
  PetscCall(PetscViewerBinaryWrite(viewer,header,4,PETSC_INT));

  /* fill in and store row lengths  */
  PetscCall(PetscMalloc1(m,&rowlens));
  for (i=0; i<m; i++) rowlens[i] = A->i[i+1] - A->i[i] + B->i[i+1] - B->i[i];
  PetscCall(PetscViewerBinaryWriteAll(viewer,rowlens,m,rs,M,PETSC_INT));
  PetscCall(PetscFree(rowlens));

  /* fill in and store column indices */
  PetscCall(PetscMalloc1(nz,&colidxs));
  for (cnt=0, i=0; i<m; i++) {
    for (jb=B->i[i]; jb<B->i[i+1]; jb++) {
      if (garray[B->j[jb]] > cs) break;
      colidxs[cnt++] = garray[B->j[jb]];
    }
    for (ja=A->i[i]; ja<A->i[i+1]; ja++)
      colidxs[cnt++] = A->j[ja] + cs;
    for (; jb<B->i[i+1]; jb++)
      colidxs[cnt++] = garray[B->j[jb]];
  }
  PetscCheck(cnt == nz,PETSC_COMM_SELF,PETSC_ERR_PLIB,"Internal PETSc error: cnt = %" PetscInt_FMT " nz = %" PetscInt_FMT,cnt,nz);
  PetscCall(PetscViewerBinaryWriteAll(viewer,colidxs,nz,PETSC_DETERMINE,PETSC_DETERMINE,PETSC_INT));
  PetscCall(PetscFree(colidxs));

  /* fill in and store nonzero values */
  PetscCall(MatSeqAIJGetArrayRead(aij->A,&aa));
  PetscCall(MatSeqAIJGetArrayRead(aij->B,&ba));
  PetscCall(PetscMalloc1(nz,&matvals));
  for (cnt=0, i=0; i<m; i++) {
    for (jb=B->i[i]; jb<B->i[i+1]; jb++) {
      if (garray[B->j[jb]] > cs) break;
      matvals[cnt++] = ba[jb];
    }
    for (ja=A->i[i]; ja<A->i[i+1]; ja++)
      matvals[cnt++] = aa[ja];
    for (; jb<B->i[i+1]; jb++)
      matvals[cnt++] = ba[jb];
  }
  PetscCall(MatSeqAIJRestoreArrayRead(aij->A,&aa));
  PetscCall(MatSeqAIJRestoreArrayRead(aij->B,&ba));
  PetscCheck(cnt == nz,PETSC_COMM_SELF,PETSC_ERR_LIB,"Internal PETSc error: cnt = %" PetscInt_FMT " nz = %" PetscInt_FMT,cnt,nz);
  PetscCall(PetscViewerBinaryWriteAll(viewer,matvals,nz,PETSC_DETERMINE,PETSC_DETERMINE,PETSC_SCALAR));
  PetscCall(PetscFree(matvals));

  /* write block size option to the viewer's .info file */
  PetscCall(MatView_Binary_BlockSizes(mat,viewer));
  PetscFunctionReturn(0);
}

#include <petscdraw.h>
PetscErrorCode MatView_MPIAIJ_ASCIIorDraworSocket(Mat mat,PetscViewer viewer)
{
  Mat_MPIAIJ        *aij = (Mat_MPIAIJ*)mat->data;
  PetscMPIInt       rank = aij->rank,size = aij->size;
  PetscBool         isdraw,iascii,isbinary;
  PetscViewer       sviewer;
  PetscViewerFormat format;

  PetscFunctionBegin;
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERDRAW,&isdraw));
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii));
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERBINARY,&isbinary));
  if (iascii) {
    PetscCall(PetscViewerGetFormat(viewer,&format));
    if (format == PETSC_VIEWER_LOAD_BALANCE) {
      PetscInt i,nmax = 0,nmin = PETSC_MAX_INT,navg = 0,*nz,nzlocal = ((Mat_SeqAIJ*) (aij->A->data))->nz + ((Mat_SeqAIJ*) (aij->B->data))->nz;
      PetscCall(PetscMalloc1(size,&nz));
      PetscCallMPI(MPI_Allgather(&nzlocal,1,MPIU_INT,nz,1,MPIU_INT,PetscObjectComm((PetscObject)mat)));
      for (i=0; i<(PetscInt)size; i++) {
        nmax = PetscMax(nmax,nz[i]);
        nmin = PetscMin(nmin,nz[i]);
        navg += nz[i];
      }
      PetscCall(PetscFree(nz));
      navg = navg/size;
      PetscCall(PetscViewerASCIIPrintf(viewer,"Load Balance - Nonzeros: Min %" PetscInt_FMT "  avg %" PetscInt_FMT "  max %" PetscInt_FMT "\n",nmin,navg,nmax));
      PetscFunctionReturn(0);
    }
    PetscCall(PetscViewerGetFormat(viewer,&format));
    if (format == PETSC_VIEWER_ASCII_INFO_DETAIL) {
      MatInfo   info;
      PetscInt *inodes=NULL;

      PetscCallMPI(MPI_Comm_rank(PetscObjectComm((PetscObject)mat),&rank));
      PetscCall(MatGetInfo(mat,MAT_LOCAL,&info));
      PetscCall(MatInodeGetInodeSizes(aij->A,NULL,&inodes,NULL));
      PetscCall(PetscViewerASCIIPushSynchronized(viewer));
      if (!inodes) {
        PetscCall(PetscViewerASCIISynchronizedPrintf(viewer,"[%d] Local rows %" PetscInt_FMT " nz %" PetscInt_FMT " nz alloced %" PetscInt_FMT " mem %g, not using I-node routines\n",
                                                   rank,mat->rmap->n,(PetscInt)info.nz_used,(PetscInt)info.nz_allocated,(double)info.memory));
      } else {
        PetscCall(PetscViewerASCIISynchronizedPrintf(viewer,"[%d] Local rows %" PetscInt_FMT " nz %" PetscInt_FMT " nz alloced %" PetscInt_FMT " mem %g, using I-node routines\n",
                                                   rank,mat->rmap->n,(PetscInt)info.nz_used,(PetscInt)info.nz_allocated,(double)info.memory));
      }
      PetscCall(MatGetInfo(aij->A,MAT_LOCAL,&info));
      PetscCall(PetscViewerASCIISynchronizedPrintf(viewer,"[%d] on-diagonal part: nz %" PetscInt_FMT " \n",rank,(PetscInt)info.nz_used));
      PetscCall(MatGetInfo(aij->B,MAT_LOCAL,&info));
      PetscCall(PetscViewerASCIISynchronizedPrintf(viewer,"[%d] off-diagonal part: nz %" PetscInt_FMT " \n",rank,(PetscInt)info.nz_used));
      PetscCall(PetscViewerFlush(viewer));
      PetscCall(PetscViewerASCIIPopSynchronized(viewer));
      PetscCall(PetscViewerASCIIPrintf(viewer,"Information on VecScatter used in matrix-vector product: \n"));
      PetscCall(VecScatterView(aij->Mvctx,viewer));
      PetscFunctionReturn(0);
    } else if (format == PETSC_VIEWER_ASCII_INFO) {
      PetscInt inodecount,inodelimit,*inodes;
      PetscCall(MatInodeGetInodeSizes(aij->A,&inodecount,&inodes,&inodelimit));
      if (inodes) {
        PetscCall(PetscViewerASCIIPrintf(viewer,"using I-node (on process 0) routines: found %" PetscInt_FMT " nodes, limit used is %" PetscInt_FMT "\n",inodecount,inodelimit));
      } else {
        PetscCall(PetscViewerASCIIPrintf(viewer,"not using I-node (on process 0) routines\n"));
      }
      PetscFunctionReturn(0);
    } else if (format == PETSC_VIEWER_ASCII_FACTOR_INFO) {
      PetscFunctionReturn(0);
    }
  } else if (isbinary) {
    if (size == 1) {
      PetscCall(PetscObjectSetName((PetscObject)aij->A,((PetscObject)mat)->name));
      PetscCall(MatView(aij->A,viewer));
    } else {
      PetscCall(MatView_MPIAIJ_Binary(mat,viewer));
    }
    PetscFunctionReturn(0);
  } else if (iascii && size == 1) {
    PetscCall(PetscObjectSetName((PetscObject)aij->A,((PetscObject)mat)->name));
    PetscCall(MatView(aij->A,viewer));
    PetscFunctionReturn(0);
  } else if (isdraw) {
    PetscDraw draw;
    PetscBool isnull;
    PetscCall(PetscViewerDrawGetDraw(viewer,0,&draw));
    PetscCall(PetscDrawIsNull(draw,&isnull));
    if (isnull) PetscFunctionReturn(0);
  }

  { /* assemble the entire matrix onto first processor */
    Mat A = NULL, Av;
    IS  isrow,iscol;

    PetscCall(ISCreateStride(PetscObjectComm((PetscObject)mat),rank == 0 ? mat->rmap->N : 0,0,1,&isrow));
    PetscCall(ISCreateStride(PetscObjectComm((PetscObject)mat),rank == 0 ? mat->cmap->N : 0,0,1,&iscol));
    PetscCall(MatCreateSubMatrix(mat,isrow,iscol,MAT_INITIAL_MATRIX,&A));
    PetscCall(MatMPIAIJGetSeqAIJ(A,&Av,NULL,NULL));
/*  The commented code uses MatCreateSubMatrices instead */
/*
    Mat *AA, A = NULL, Av;
    IS  isrow,iscol;

    PetscCall(ISCreateStride(PetscObjectComm((PetscObject)mat),rank == 0 ? mat->rmap->N : 0,0,1,&isrow));
    PetscCall(ISCreateStride(PetscObjectComm((PetscObject)mat),rank == 0 ? mat->cmap->N : 0,0,1,&iscol));
    PetscCall(MatCreateSubMatrices(mat,1,&isrow,&iscol,MAT_INITIAL_MATRIX,&AA));
    if (rank == 0) {
       PetscCall(PetscObjectReference((PetscObject)AA[0]));
       A    = AA[0];
       Av   = AA[0];
    }
    PetscCall(MatDestroySubMatrices(1,&AA));
*/
    PetscCall(ISDestroy(&iscol));
    PetscCall(ISDestroy(&isrow));
    /*
       Everyone has to call to draw the matrix since the graphics waits are
       synchronized across all processors that share the PetscDraw object
    */
    PetscCall(PetscViewerGetSubViewer(viewer,PETSC_COMM_SELF,&sviewer));
    if (rank == 0) {
      if (((PetscObject)mat)->name) {
        PetscCall(PetscObjectSetName((PetscObject)Av,((PetscObject)mat)->name));
      }
      PetscCall(MatView_SeqAIJ(Av,sviewer));
    }
    PetscCall(PetscViewerRestoreSubViewer(viewer,PETSC_COMM_SELF,&sviewer));
    PetscCall(PetscViewerFlush(viewer));
    PetscCall(MatDestroy(&A));
  }
  PetscFunctionReturn(0);
}

PetscErrorCode MatView_MPIAIJ(Mat mat,PetscViewer viewer)
{
  PetscBool      iascii,isdraw,issocket,isbinary;

  PetscFunctionBegin;
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii));
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERDRAW,&isdraw));
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERBINARY,&isbinary));
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERSOCKET,&issocket));
  if (iascii || isdraw || isbinary || issocket) {
    PetscCall(MatView_MPIAIJ_ASCIIorDraworSocket(mat,viewer));
  }
  PetscFunctionReturn(0);
}

PetscErrorCode MatSOR_MPIAIJ(Mat matin,Vec bb,PetscReal omega,MatSORType flag,PetscReal fshift,PetscInt its,PetscInt lits,Vec xx)
{
  Mat_MPIAIJ     *mat = (Mat_MPIAIJ*)matin->data;
  Vec            bb1 = NULL;
  PetscBool      hasop;

  PetscFunctionBegin;
  if (flag == SOR_APPLY_UPPER) {
    PetscCall((*mat->A->ops->sor)(mat->A,bb,omega,flag,fshift,lits,1,xx));
    PetscFunctionReturn(0);
  }

  if (its > 1 || ~flag & SOR_ZERO_INITIAL_GUESS || flag & SOR_EISENSTAT) {
    PetscCall(VecDuplicate(bb,&bb1));
  }

  if ((flag & SOR_LOCAL_SYMMETRIC_SWEEP) == SOR_LOCAL_SYMMETRIC_SWEEP) {
    if (flag & SOR_ZERO_INITIAL_GUESS) {
      PetscCall((*mat->A->ops->sor)(mat->A,bb,omega,flag,fshift,lits,1,xx));
      its--;
    }

    while (its--) {
      PetscCall(VecScatterBegin(mat->Mvctx,xx,mat->lvec,INSERT_VALUES,SCATTER_FORWARD));
      PetscCall(VecScatterEnd(mat->Mvctx,xx,mat->lvec,INSERT_VALUES,SCATTER_FORWARD));

      /* update rhs: bb1 = bb - B*x */
      PetscCall(VecScale(mat->lvec,-1.0));
      PetscCall((*mat->B->ops->multadd)(mat->B,mat->lvec,bb,bb1));

      /* local sweep */
      PetscCall((*mat->A->ops->sor)(mat->A,bb1,omega,SOR_SYMMETRIC_SWEEP,fshift,lits,1,xx));
    }
  } else if (flag & SOR_LOCAL_FORWARD_SWEEP) {
    if (flag & SOR_ZERO_INITIAL_GUESS) {
      PetscCall((*mat->A->ops->sor)(mat->A,bb,omega,flag,fshift,lits,1,xx));
      its--;
    }
    while (its--) {
      PetscCall(VecScatterBegin(mat->Mvctx,xx,mat->lvec,INSERT_VALUES,SCATTER_FORWARD));
      PetscCall(VecScatterEnd(mat->Mvctx,xx,mat->lvec,INSERT_VALUES,SCATTER_FORWARD));

      /* update rhs: bb1 = bb - B*x */
      PetscCall(VecScale(mat->lvec,-1.0));
      PetscCall((*mat->B->ops->multadd)(mat->B,mat->lvec,bb,bb1));

      /* local sweep */
      PetscCall((*mat->A->ops->sor)(mat->A,bb1,omega,SOR_FORWARD_SWEEP,fshift,lits,1,xx));
    }
  } else if (flag & SOR_LOCAL_BACKWARD_SWEEP) {
    if (flag & SOR_ZERO_INITIAL_GUESS) {
      PetscCall((*mat->A->ops->sor)(mat->A,bb,omega,flag,fshift,lits,1,xx));
      its--;
    }
    while (its--) {
      PetscCall(VecScatterBegin(mat->Mvctx,xx,mat->lvec,INSERT_VALUES,SCATTER_FORWARD));
      PetscCall(VecScatterEnd(mat->Mvctx,xx,mat->lvec,INSERT_VALUES,SCATTER_FORWARD));

      /* update rhs: bb1 = bb - B*x */
      PetscCall(VecScale(mat->lvec,-1.0));
      PetscCall((*mat->B->ops->multadd)(mat->B,mat->lvec,bb,bb1));

      /* local sweep */
      PetscCall((*mat->A->ops->sor)(mat->A,bb1,omega,SOR_BACKWARD_SWEEP,fshift,lits,1,xx));
    }
  } else if (flag & SOR_EISENSTAT) {
    Vec xx1;

    PetscCall(VecDuplicate(bb,&xx1));
    PetscCall((*mat->A->ops->sor)(mat->A,bb,omega,(MatSORType)(SOR_ZERO_INITIAL_GUESS | SOR_LOCAL_BACKWARD_SWEEP),fshift,lits,1,xx));

    PetscCall(VecScatterBegin(mat->Mvctx,xx,mat->lvec,INSERT_VALUES,SCATTER_FORWARD));
    PetscCall(VecScatterEnd(mat->Mvctx,xx,mat->lvec,INSERT_VALUES,SCATTER_FORWARD));
    if (!mat->diag) {
      PetscCall(MatCreateVecs(matin,&mat->diag,NULL));
      PetscCall(MatGetDiagonal(matin,mat->diag));
    }
    PetscCall(MatHasOperation(matin,MATOP_MULT_DIAGONAL_BLOCK,&hasop));
    if (hasop) {
      PetscCall(MatMultDiagonalBlock(matin,xx,bb1));
    } else {
      PetscCall(VecPointwiseMult(bb1,mat->diag,xx));
    }
    PetscCall(VecAYPX(bb1,(omega-2.0)/omega,bb));

    PetscCall(MatMultAdd(mat->B,mat->lvec,bb1,bb1));

    /* local sweep */
    PetscCall((*mat->A->ops->sor)(mat->A,bb1,omega,(MatSORType)(SOR_ZERO_INITIAL_GUESS | SOR_LOCAL_FORWARD_SWEEP),fshift,lits,1,xx1));
    PetscCall(VecAXPY(xx,1.0,xx1));
    PetscCall(VecDestroy(&xx1));
  } else SETERRQ(PetscObjectComm((PetscObject)matin),PETSC_ERR_SUP,"Parallel SOR not supported");

  PetscCall(VecDestroy(&bb1));

  matin->factorerrortype = mat->A->factorerrortype;
  PetscFunctionReturn(0);
}

PetscErrorCode MatPermute_MPIAIJ(Mat A,IS rowp,IS colp,Mat *B)
{
  Mat            aA,aB,Aperm;
  const PetscInt *rwant,*cwant,*gcols,*ai,*bi,*aj,*bj;
  PetscScalar    *aa,*ba;
  PetscInt       i,j,m,n,ng,anz,bnz,*dnnz,*onnz,*tdnnz,*tonnz,*rdest,*cdest,*work,*gcdest;
  PetscSF        rowsf,sf;
  IS             parcolp = NULL;
  PetscBool      done;

  PetscFunctionBegin;
  PetscCall(MatGetLocalSize(A,&m,&n));
  PetscCall(ISGetIndices(rowp,&rwant));
  PetscCall(ISGetIndices(colp,&cwant));
  PetscCall(PetscMalloc3(PetscMax(m,n),&work,m,&rdest,n,&cdest));

  /* Invert row permutation to find out where my rows should go */
  PetscCall(PetscSFCreate(PetscObjectComm((PetscObject)A),&rowsf));
  PetscCall(PetscSFSetGraphLayout(rowsf,A->rmap,A->rmap->n,NULL,PETSC_OWN_POINTER,rwant));
  PetscCall(PetscSFSetFromOptions(rowsf));
  for (i=0; i<m; i++) work[i] = A->rmap->rstart + i;
  PetscCall(PetscSFReduceBegin(rowsf,MPIU_INT,work,rdest,MPI_REPLACE));
  PetscCall(PetscSFReduceEnd(rowsf,MPIU_INT,work,rdest,MPI_REPLACE));

  /* Invert column permutation to find out where my columns should go */
  PetscCall(PetscSFCreate(PetscObjectComm((PetscObject)A),&sf));
  PetscCall(PetscSFSetGraphLayout(sf,A->cmap,A->cmap->n,NULL,PETSC_OWN_POINTER,cwant));
  PetscCall(PetscSFSetFromOptions(sf));
  for (i=0; i<n; i++) work[i] = A->cmap->rstart + i;
  PetscCall(PetscSFReduceBegin(sf,MPIU_INT,work,cdest,MPI_REPLACE));
  PetscCall(PetscSFReduceEnd(sf,MPIU_INT,work,cdest,MPI_REPLACE));
  PetscCall(PetscSFDestroy(&sf));

  PetscCall(ISRestoreIndices(rowp,&rwant));
  PetscCall(ISRestoreIndices(colp,&cwant));
  PetscCall(MatMPIAIJGetSeqAIJ(A,&aA,&aB,&gcols));

  /* Find out where my gcols should go */
  PetscCall(MatGetSize(aB,NULL,&ng));
  PetscCall(PetscMalloc1(ng,&gcdest));
  PetscCall(PetscSFCreate(PetscObjectComm((PetscObject)A),&sf));
  PetscCall(PetscSFSetGraphLayout(sf,A->cmap,ng,NULL,PETSC_OWN_POINTER,gcols));
  PetscCall(PetscSFSetFromOptions(sf));
  PetscCall(PetscSFBcastBegin(sf,MPIU_INT,cdest,gcdest,MPI_REPLACE));
  PetscCall(PetscSFBcastEnd(sf,MPIU_INT,cdest,gcdest,MPI_REPLACE));
  PetscCall(PetscSFDestroy(&sf));

  PetscCall(PetscCalloc4(m,&dnnz,m,&onnz,m,&tdnnz,m,&tonnz));
  PetscCall(MatGetRowIJ(aA,0,PETSC_FALSE,PETSC_FALSE,&anz,&ai,&aj,&done));
  PetscCall(MatGetRowIJ(aB,0,PETSC_FALSE,PETSC_FALSE,&bnz,&bi,&bj,&done));
  for (i=0; i<m; i++) {
    PetscInt    row = rdest[i];
    PetscMPIInt rowner;
    PetscCall(PetscLayoutFindOwner(A->rmap,row,&rowner));
    for (j=ai[i]; j<ai[i+1]; j++) {
      PetscInt    col = cdest[aj[j]];
      PetscMPIInt cowner;
      PetscCall(PetscLayoutFindOwner(A->cmap,col,&cowner)); /* Could build an index for the columns to eliminate this search */
      if (rowner == cowner) dnnz[i]++;
      else onnz[i]++;
    }
    for (j=bi[i]; j<bi[i+1]; j++) {
      PetscInt    col = gcdest[bj[j]];
      PetscMPIInt cowner;
      PetscCall(PetscLayoutFindOwner(A->cmap,col,&cowner));
      if (rowner == cowner) dnnz[i]++;
      else onnz[i]++;
    }
  }
  PetscCall(PetscSFBcastBegin(rowsf,MPIU_INT,dnnz,tdnnz,MPI_REPLACE));
  PetscCall(PetscSFBcastEnd(rowsf,MPIU_INT,dnnz,tdnnz,MPI_REPLACE));
  PetscCall(PetscSFBcastBegin(rowsf,MPIU_INT,onnz,tonnz,MPI_REPLACE));
  PetscCall(PetscSFBcastEnd(rowsf,MPIU_INT,onnz,tonnz,MPI_REPLACE));
  PetscCall(PetscSFDestroy(&rowsf));

  PetscCall(MatCreateAIJ(PetscObjectComm((PetscObject)A),A->rmap->n,A->cmap->n,A->rmap->N,A->cmap->N,0,tdnnz,0,tonnz,&Aperm));
  PetscCall(MatSeqAIJGetArray(aA,&aa));
  PetscCall(MatSeqAIJGetArray(aB,&ba));
  for (i=0; i<m; i++) {
    PetscInt *acols = dnnz,*bcols = onnz; /* Repurpose now-unneeded arrays */
    PetscInt j0,rowlen;
    rowlen = ai[i+1] - ai[i];
    for (j0=j=0; j<rowlen; j0=j) { /* rowlen could be larger than number of rows m, so sum in batches */
      for (; j<PetscMin(rowlen,j0+m); j++) acols[j-j0] = cdest[aj[ai[i]+j]];
      PetscCall(MatSetValues(Aperm,1,&rdest[i],j-j0,acols,aa+ai[i]+j0,INSERT_VALUES));
    }
    rowlen = bi[i+1] - bi[i];
    for (j0=j=0; j<rowlen; j0=j) {
      for (; j<PetscMin(rowlen,j0+m); j++) bcols[j-j0] = gcdest[bj[bi[i]+j]];
      PetscCall(MatSetValues(Aperm,1,&rdest[i],j-j0,bcols,ba+bi[i]+j0,INSERT_VALUES));
    }
  }
  PetscCall(MatAssemblyBegin(Aperm,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(Aperm,MAT_FINAL_ASSEMBLY));
  PetscCall(MatRestoreRowIJ(aA,0,PETSC_FALSE,PETSC_FALSE,&anz,&ai,&aj,&done));
  PetscCall(MatRestoreRowIJ(aB,0,PETSC_FALSE,PETSC_FALSE,&bnz,&bi,&bj,&done));
  PetscCall(MatSeqAIJRestoreArray(aA,&aa));
  PetscCall(MatSeqAIJRestoreArray(aB,&ba));
  PetscCall(PetscFree4(dnnz,onnz,tdnnz,tonnz));
  PetscCall(PetscFree3(work,rdest,cdest));
  PetscCall(PetscFree(gcdest));
  if (parcolp) PetscCall(ISDestroy(&colp));
  *B = Aperm;
  PetscFunctionReturn(0);
}

PetscErrorCode  MatGetGhosts_MPIAIJ(Mat mat,PetscInt *nghosts,const PetscInt *ghosts[])
{
  Mat_MPIAIJ *aij = (Mat_MPIAIJ*)mat->data;

  PetscFunctionBegin;
  PetscCall(MatGetSize(aij->B,NULL,nghosts));
  if (ghosts) *ghosts = aij->garray;
  PetscFunctionReturn(0);
}

PetscErrorCode MatGetInfo_MPIAIJ(Mat matin,MatInfoType flag,MatInfo *info)
{
  Mat_MPIAIJ     *mat = (Mat_MPIAIJ*)matin->data;
  Mat            A    = mat->A,B = mat->B;
  PetscLogDouble isend[5],irecv[5];

  PetscFunctionBegin;
  info->block_size = 1.0;
  PetscCall(MatGetInfo(A,MAT_LOCAL,info));

  isend[0] = info->nz_used; isend[1] = info->nz_allocated; isend[2] = info->nz_unneeded;
  isend[3] = info->memory;  isend[4] = info->mallocs;

  PetscCall(MatGetInfo(B,MAT_LOCAL,info));

  isend[0] += info->nz_used; isend[1] += info->nz_allocated; isend[2] += info->nz_unneeded;
  isend[3] += info->memory;  isend[4] += info->mallocs;
  if (flag == MAT_LOCAL) {
    info->nz_used      = isend[0];
    info->nz_allocated = isend[1];
    info->nz_unneeded  = isend[2];
    info->memory       = isend[3];
    info->mallocs      = isend[4];
  } else if (flag == MAT_GLOBAL_MAX) {
    PetscCall(MPIU_Allreduce(isend,irecv,5,MPIU_PETSCLOGDOUBLE,MPI_MAX,PetscObjectComm((PetscObject)matin)));

    info->nz_used      = irecv[0];
    info->nz_allocated = irecv[1];
    info->nz_unneeded  = irecv[2];
    info->memory       = irecv[3];
    info->mallocs      = irecv[4];
  } else if (flag == MAT_GLOBAL_SUM) {
    PetscCall(MPIU_Allreduce(isend,irecv,5,MPIU_PETSCLOGDOUBLE,MPI_SUM,PetscObjectComm((PetscObject)matin)));

    info->nz_used      = irecv[0];
    info->nz_allocated = irecv[1];
    info->nz_unneeded  = irecv[2];
    info->memory       = irecv[3];
    info->mallocs      = irecv[4];
  }
  info->fill_ratio_given  = 0; /* no parallel LU/ILU/Cholesky */
  info->fill_ratio_needed = 0;
  info->factor_mallocs    = 0;
  PetscFunctionReturn(0);
}

PetscErrorCode MatSetOption_MPIAIJ(Mat A,MatOption op,PetscBool flg)
{
  Mat_MPIAIJ     *a = (Mat_MPIAIJ*)A->data;

  PetscFunctionBegin;
  switch (op) {
  case MAT_NEW_NONZERO_LOCATIONS:
  case MAT_NEW_NONZERO_ALLOCATION_ERR:
  case MAT_UNUSED_NONZERO_LOCATION_ERR:
  case MAT_KEEP_NONZERO_PATTERN:
  case MAT_NEW_NONZERO_LOCATION_ERR:
  case MAT_USE_INODES:
  case MAT_IGNORE_ZERO_ENTRIES:
  case MAT_FORM_EXPLICIT_TRANSPOSE:
    MatCheckPreallocated(A,1);
    PetscCall(MatSetOption(a->A,op,flg));
    PetscCall(MatSetOption(a->B,op,flg));
    break;
  case MAT_ROW_ORIENTED:
    MatCheckPreallocated(A,1);
    a->roworiented = flg;

    PetscCall(MatSetOption(a->A,op,flg));
    PetscCall(MatSetOption(a->B,op,flg));
    break;
  case MAT_FORCE_DIAGONAL_ENTRIES:
  case MAT_SORTED_FULL:
    PetscCall(PetscInfo(A,"Option %s ignored\n",MatOptions[op]));
    break;
  case MAT_IGNORE_OFF_PROC_ENTRIES:
    a->donotstash = flg;
    break;
  /* Symmetry flags are handled directly by MatSetOption() and they don't affect preallocation */
  case MAT_SPD:
  case MAT_SYMMETRIC:
  case MAT_STRUCTURALLY_SYMMETRIC:
  case MAT_HERMITIAN:
  case MAT_SYMMETRY_ETERNAL:
    break;
  case MAT_SUBMAT_SINGLEIS:
    A->submat_singleis = flg;
    break;
  case MAT_STRUCTURE_ONLY:
    /* The option is handled directly by MatSetOption() */
    break;
  default:
    SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"unknown option %d",op);
  }
  PetscFunctionReturn(0);
}

PetscErrorCode MatGetRow_MPIAIJ(Mat matin,PetscInt row,PetscInt *nz,PetscInt **idx,PetscScalar **v)
{
  Mat_MPIAIJ     *mat = (Mat_MPIAIJ*)matin->data;
  PetscScalar    *vworkA,*vworkB,**pvA,**pvB,*v_p;
  PetscInt       i,*cworkA,*cworkB,**pcA,**pcB,cstart = matin->cmap->rstart;
  PetscInt       nztot,nzA,nzB,lrow,rstart = matin->rmap->rstart,rend = matin->rmap->rend;
  PetscInt       *cmap,*idx_p;

  PetscFunctionBegin;
  PetscCheck(!mat->getrowactive,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"Already active");
  mat->getrowactive = PETSC_TRUE;

  if (!mat->rowvalues && (idx || v)) {
    /*
        allocate enough space to hold information from the longest row.
    */
    Mat_SeqAIJ *Aa = (Mat_SeqAIJ*)mat->A->data,*Ba = (Mat_SeqAIJ*)mat->B->data;
    PetscInt   max = 1,tmp;
    for (i=0; i<matin->rmap->n; i++) {
      tmp = Aa->i[i+1] - Aa->i[i] + Ba->i[i+1] - Ba->i[i];
      if (max < tmp) max = tmp;
    }
    PetscCall(PetscMalloc2(max,&mat->rowvalues,max,&mat->rowindices));
  }

  PetscCheck(row >= rstart && row < rend,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Only local rows");
  lrow = row - rstart;

  pvA = &vworkA; pcA = &cworkA; pvB = &vworkB; pcB = &cworkB;
  if (!v)   {pvA = NULL; pvB = NULL;}
  if (!idx) {pcA = NULL; if (!v) pcB = NULL;}
  PetscCall((*mat->A->ops->getrow)(mat->A,lrow,&nzA,pcA,pvA));
  PetscCall((*mat->B->ops->getrow)(mat->B,lrow,&nzB,pcB,pvB));
  nztot = nzA + nzB;

  cmap = mat->garray;
  if (v  || idx) {
    if (nztot) {
      /* Sort by increasing column numbers, assuming A and B already sorted */
      PetscInt imark = -1;
      if (v) {
        *v = v_p = mat->rowvalues;
        for (i=0; i<nzB; i++) {
          if (cmap[cworkB[i]] < cstart) v_p[i] = vworkB[i];
          else break;
        }
        imark = i;
        for (i=0; i<nzA; i++)     v_p[imark+i] = vworkA[i];
        for (i=imark; i<nzB; i++) v_p[nzA+i]   = vworkB[i];
      }
      if (idx) {
        *idx = idx_p = mat->rowindices;
        if (imark > -1) {
          for (i=0; i<imark; i++) {
            idx_p[i] = cmap[cworkB[i]];
          }
        } else {
          for (i=0; i<nzB; i++) {
            if (cmap[cworkB[i]] < cstart) idx_p[i] = cmap[cworkB[i]];
            else break;
          }
          imark = i;
        }
        for (i=0; i<nzA; i++)     idx_p[imark+i] = cstart + cworkA[i];
        for (i=imark; i<nzB; i++) idx_p[nzA+i]   = cmap[cworkB[i]];
      }
    } else {
      if (idx) *idx = NULL;
      if (v)   *v   = NULL;
    }
  }
  *nz  = nztot;
  PetscCall((*mat->A->ops->restorerow)(mat->A,lrow,&nzA,pcA,pvA));
  PetscCall((*mat->B->ops->restorerow)(mat->B,lrow,&nzB,pcB,pvB));
  PetscFunctionReturn(0);
}

PetscErrorCode MatRestoreRow_MPIAIJ(Mat mat,PetscInt row,PetscInt *nz,PetscInt **idx,PetscScalar **v)
{
  Mat_MPIAIJ *aij = (Mat_MPIAIJ*)mat->data;

  PetscFunctionBegin;
  PetscCheck(aij->getrowactive,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"MatGetRow() must be called first");
  aij->getrowactive = PETSC_FALSE;
  PetscFunctionReturn(0);
}

PetscErrorCode MatNorm_MPIAIJ(Mat mat,NormType type,PetscReal *norm)
{
  Mat_MPIAIJ      *aij  = (Mat_MPIAIJ*)mat->data;
  Mat_SeqAIJ      *amat = (Mat_SeqAIJ*)aij->A->data,*bmat = (Mat_SeqAIJ*)aij->B->data;
  PetscInt        i,j,cstart = mat->cmap->rstart;
  PetscReal       sum = 0.0;
  const MatScalar *v,*amata,*bmata;

  PetscFunctionBegin;
  if (aij->size == 1) {
    PetscCall(MatNorm(aij->A,type,norm));
  } else {
    PetscCall(MatSeqAIJGetArrayRead(aij->A,&amata));
    PetscCall(MatSeqAIJGetArrayRead(aij->B,&bmata));
    if (type == NORM_FROBENIUS) {
      v = amata;
      for (i=0; i<amat->nz; i++) {
        sum += PetscRealPart(PetscConj(*v)*(*v)); v++;
      }
      v = bmata;
      for (i=0; i<bmat->nz; i++) {
        sum += PetscRealPart(PetscConj(*v)*(*v)); v++;
      }
      PetscCall(MPIU_Allreduce(&sum,norm,1,MPIU_REAL,MPIU_SUM,PetscObjectComm((PetscObject)mat)));
      *norm = PetscSqrtReal(*norm);
      PetscCall(PetscLogFlops(2.0*amat->nz+2.0*bmat->nz));
    } else if (type == NORM_1) { /* max column norm */
      PetscReal *tmp,*tmp2;
      PetscInt  *jj,*garray = aij->garray;
      PetscCall(PetscCalloc1(mat->cmap->N+1,&tmp));
      PetscCall(PetscMalloc1(mat->cmap->N+1,&tmp2));
      *norm = 0.0;
      v     = amata; jj = amat->j;
      for (j=0; j<amat->nz; j++) {
        tmp[cstart + *jj++] += PetscAbsScalar(*v);  v++;
      }
      v = bmata; jj = bmat->j;
      for (j=0; j<bmat->nz; j++) {
        tmp[garray[*jj++]] += PetscAbsScalar(*v); v++;
      }
      PetscCall(MPIU_Allreduce(tmp,tmp2,mat->cmap->N,MPIU_REAL,MPIU_SUM,PetscObjectComm((PetscObject)mat)));
      for (j=0; j<mat->cmap->N; j++) {
        if (tmp2[j] > *norm) *norm = tmp2[j];
      }
      PetscCall(PetscFree(tmp));
      PetscCall(PetscFree(tmp2));
      PetscCall(PetscLogFlops(PetscMax(amat->nz+bmat->nz-1,0)));
    } else if (type == NORM_INFINITY) { /* max row norm */
      PetscReal ntemp = 0.0;
      for (j=0; j<aij->A->rmap->n; j++) {
        v   = amata + amat->i[j];
        sum = 0.0;
        for (i=0; i<amat->i[j+1]-amat->i[j]; i++) {
          sum += PetscAbsScalar(*v); v++;
        }
        v = bmata + bmat->i[j];
        for (i=0; i<bmat->i[j+1]-bmat->i[j]; i++) {
          sum += PetscAbsScalar(*v); v++;
        }
        if (sum > ntemp) ntemp = sum;
      }
      PetscCall(MPIU_Allreduce(&ntemp,norm,1,MPIU_REAL,MPIU_MAX,PetscObjectComm((PetscObject)mat)));
      PetscCall(PetscLogFlops(PetscMax(amat->nz+bmat->nz-1,0)));
    } else SETERRQ(PetscObjectComm((PetscObject)mat),PETSC_ERR_SUP,"No support for two norm");
    PetscCall(MatSeqAIJRestoreArrayRead(aij->A,&amata));
    PetscCall(MatSeqAIJRestoreArrayRead(aij->B,&bmata));
  }
  PetscFunctionReturn(0);
}

PetscErrorCode MatTranspose_MPIAIJ(Mat A,MatReuse reuse,Mat *matout)
{
  Mat_MPIAIJ      *a    =(Mat_MPIAIJ*)A->data,*b;
  Mat_SeqAIJ      *Aloc =(Mat_SeqAIJ*)a->A->data,*Bloc=(Mat_SeqAIJ*)a->B->data,*sub_B_diag;
  PetscInt        M     = A->rmap->N,N=A->cmap->N,ma,na,mb,nb,row,*cols,*cols_tmp,*B_diag_ilen,i,ncol,A_diag_ncol;
  const PetscInt  *ai,*aj,*bi,*bj,*B_diag_i;
  Mat             B,A_diag,*B_diag;
  const MatScalar *pbv,*bv;

  PetscFunctionBegin;
  ma = A->rmap->n; na = A->cmap->n; mb = a->B->rmap->n; nb = a->B->cmap->n;
  ai = Aloc->i; aj = Aloc->j;
  bi = Bloc->i; bj = Bloc->j;
  if (reuse == MAT_INITIAL_MATRIX || *matout == A) {
    PetscInt             *d_nnz,*g_nnz,*o_nnz;
    PetscSFNode          *oloc;
    PETSC_UNUSED PetscSF sf;

    PetscCall(PetscMalloc4(na,&d_nnz,na,&o_nnz,nb,&g_nnz,nb,&oloc));
    /* compute d_nnz for preallocation */
    PetscCall(PetscArrayzero(d_nnz,na));
    for (i=0; i<ai[ma]; i++) d_nnz[aj[i]]++;
    /* compute local off-diagonal contributions */
    PetscCall(PetscArrayzero(g_nnz,nb));
    for (i=0; i<bi[ma]; i++) g_nnz[bj[i]]++;
    /* map those to global */
    PetscCall(PetscSFCreate(PetscObjectComm((PetscObject)A),&sf));
    PetscCall(PetscSFSetGraphLayout(sf,A->cmap,nb,NULL,PETSC_USE_POINTER,a->garray));
    PetscCall(PetscSFSetFromOptions(sf));
    PetscCall(PetscArrayzero(o_nnz,na));
    PetscCall(PetscSFReduceBegin(sf,MPIU_INT,g_nnz,o_nnz,MPIU_SUM));
    PetscCall(PetscSFReduceEnd(sf,MPIU_INT,g_nnz,o_nnz,MPIU_SUM));
    PetscCall(PetscSFDestroy(&sf));

    PetscCall(MatCreate(PetscObjectComm((PetscObject)A),&B));
    PetscCall(MatSetSizes(B,A->cmap->n,A->rmap->n,N,M));
    PetscCall(MatSetBlockSizes(B,PetscAbs(A->cmap->bs),PetscAbs(A->rmap->bs)));
    PetscCall(MatSetType(B,((PetscObject)A)->type_name));
    PetscCall(MatMPIAIJSetPreallocation(B,0,d_nnz,0,o_nnz));
    PetscCall(PetscFree4(d_nnz,o_nnz,g_nnz,oloc));
  } else {
    B    = *matout;
    PetscCall(MatSetOption(B,MAT_NEW_NONZERO_ALLOCATION_ERR,PETSC_TRUE));
  }

  b           = (Mat_MPIAIJ*)B->data;
  A_diag      = a->A;
  B_diag      = &b->A;
  sub_B_diag  = (Mat_SeqAIJ*)(*B_diag)->data;
  A_diag_ncol = A_diag->cmap->N;
  B_diag_ilen = sub_B_diag->ilen;
  B_diag_i    = sub_B_diag->i;

  /* Set ilen for diagonal of B */
  for (i=0; i<A_diag_ncol; i++) {
    B_diag_ilen[i] = B_diag_i[i+1] - B_diag_i[i];
  }

  /* Transpose the diagonal part of the matrix. In contrast to the offdiagonal part, this can be done
  very quickly (=without using MatSetValues), because all writes are local. */
  PetscCall(MatTranspose(A_diag,MAT_REUSE_MATRIX,B_diag));

  /* copy over the B part */
  PetscCall(PetscMalloc1(bi[mb],&cols));
  PetscCall(MatSeqAIJGetArrayRead(a->B,&bv));
  pbv  = bv;
  row  = A->rmap->rstart;
  for (i=0; i<bi[mb]; i++) cols[i] = a->garray[bj[i]];
  cols_tmp = cols;
  for (i=0; i<mb; i++) {
    ncol = bi[i+1]-bi[i];
    PetscCall(MatSetValues(B,ncol,cols_tmp,1,&row,pbv,INSERT_VALUES));
    row++;
    pbv += ncol; cols_tmp += ncol;
  }
  PetscCall(PetscFree(cols));
  PetscCall(MatSeqAIJRestoreArrayRead(a->B,&bv));

  PetscCall(MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY));
  if (reuse == MAT_INITIAL_MATRIX || reuse == MAT_REUSE_MATRIX) {
    *matout = B;
  } else {
    PetscCall(MatHeaderMerge(A,&B));
  }
  PetscFunctionReturn(0);
}

PetscErrorCode MatDiagonalScale_MPIAIJ(Mat mat,Vec ll,Vec rr)
{
  Mat_MPIAIJ     *aij = (Mat_MPIAIJ*)mat->data;
  Mat            a    = aij->A,b = aij->B;
  PetscInt       s1,s2,s3;

  PetscFunctionBegin;
  PetscCall(MatGetLocalSize(mat,&s2,&s3));
  if (rr) {
    PetscCall(VecGetLocalSize(rr,&s1));
    PetscCheck(s1==s3,PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"right vector non-conforming local size");
    /* Overlap communication with computation. */
    PetscCall(VecScatterBegin(aij->Mvctx,rr,aij->lvec,INSERT_VALUES,SCATTER_FORWARD));
  }
  if (ll) {
    PetscCall(VecGetLocalSize(ll,&s1));
    PetscCheck(s1==s2,PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"left vector non-conforming local size");
    PetscCall((*b->ops->diagonalscale)(b,ll,NULL));
  }
  /* scale  the diagonal block */
  PetscCall((*a->ops->diagonalscale)(a,ll,rr));

  if (rr) {
    /* Do a scatter end and then right scale the off-diagonal block */
    PetscCall(VecScatterEnd(aij->Mvctx,rr,aij->lvec,INSERT_VALUES,SCATTER_FORWARD));
    PetscCall((*b->ops->diagonalscale)(b,NULL,aij->lvec));
  }
  PetscFunctionReturn(0);
}

PetscErrorCode MatSetUnfactored_MPIAIJ(Mat A)
{
  Mat_MPIAIJ     *a = (Mat_MPIAIJ*)A->data;

  PetscFunctionBegin;
  PetscCall(MatSetUnfactored(a->A));
  PetscFunctionReturn(0);
}

PetscErrorCode MatEqual_MPIAIJ(Mat A,Mat B,PetscBool  *flag)
{
  Mat_MPIAIJ     *matB = (Mat_MPIAIJ*)B->data,*matA = (Mat_MPIAIJ*)A->data;
  Mat            a,b,c,d;
  PetscBool      flg;

  PetscFunctionBegin;
  a = matA->A; b = matA->B;
  c = matB->A; d = matB->B;

  PetscCall(MatEqual(a,c,&flg));
  if (flg) {
    PetscCall(MatEqual(b,d,&flg));
  }
  PetscCall(MPIU_Allreduce(&flg,flag,1,MPIU_BOOL,MPI_LAND,PetscObjectComm((PetscObject)A)));
  PetscFunctionReturn(0);
}

PetscErrorCode MatCopy_MPIAIJ(Mat A,Mat B,MatStructure str)
{
  Mat_MPIAIJ     *a = (Mat_MPIAIJ*)A->data;
  Mat_MPIAIJ     *b = (Mat_MPIAIJ*)B->data;

  PetscFunctionBegin;
  /* If the two matrices don't have the same copy implementation, they aren't compatible for fast copy. */
  if ((str != SAME_NONZERO_PATTERN) || (A->ops->copy != B->ops->copy)) {
    /* because of the column compression in the off-processor part of the matrix a->B,
       the number of columns in a->B and b->B may be different, hence we cannot call
       the MatCopy() directly on the two parts. If need be, we can provide a more
       efficient copy than the MatCopy_Basic() by first uncompressing the a->B matrices
       then copying the submatrices */
    PetscCall(MatCopy_Basic(A,B,str));
  } else {
    PetscCall(MatCopy(a->A,b->A,str));
    PetscCall(MatCopy(a->B,b->B,str));
  }
  PetscCall(PetscObjectStateIncrease((PetscObject)B));
  PetscFunctionReturn(0);
}

PetscErrorCode MatSetUp_MPIAIJ(Mat A)
{
  PetscFunctionBegin;
  PetscCall(MatMPIAIJSetPreallocation(A,PETSC_DEFAULT,NULL,PETSC_DEFAULT,NULL));
  PetscFunctionReturn(0);
}

/*
   Computes the number of nonzeros per row needed for preallocation when X and Y
   have different nonzero structure.
*/
PetscErrorCode MatAXPYGetPreallocation_MPIX_private(PetscInt m,const PetscInt *xi,const PetscInt *xj,const PetscInt *xltog,const PetscInt *yi,const PetscInt *yj,const PetscInt *yltog,PetscInt *nnz)
{
  PetscInt       i,j,k,nzx,nzy;

  PetscFunctionBegin;
  /* Set the number of nonzeros in the new matrix */
  for (i=0; i<m; i++) {
    const PetscInt *xjj = xj+xi[i],*yjj = yj+yi[i];
    nzx = xi[i+1] - xi[i];
    nzy = yi[i+1] - yi[i];
    nnz[i] = 0;
    for (j=0,k=0; j<nzx; j++) {                   /* Point in X */
      for (; k<nzy && yltog[yjj[k]]<xltog[xjj[j]]; k++) nnz[i]++; /* Catch up to X */
      if (k<nzy && yltog[yjj[k]]==xltog[xjj[j]]) k++;             /* Skip duplicate */
      nnz[i]++;
    }
    for (; k<nzy; k++) nnz[i]++;
  }
  PetscFunctionReturn(0);
}

/* This is the same as MatAXPYGetPreallocation_SeqAIJ, except that the local-to-global map is provided */
static PetscErrorCode MatAXPYGetPreallocation_MPIAIJ(Mat Y,const PetscInt *yltog,Mat X,const PetscInt *xltog,PetscInt *nnz)
{
  PetscInt       m = Y->rmap->N;
  Mat_SeqAIJ     *x = (Mat_SeqAIJ*)X->data;
  Mat_SeqAIJ     *y = (Mat_SeqAIJ*)Y->data;

  PetscFunctionBegin;
  PetscCall(MatAXPYGetPreallocation_MPIX_private(m,x->i,x->j,xltog,y->i,y->j,yltog,nnz));
  PetscFunctionReturn(0);
}

PetscErrorCode MatAXPY_MPIAIJ(Mat Y,PetscScalar a,Mat X,MatStructure str)
{
  Mat_MPIAIJ     *xx = (Mat_MPIAIJ*)X->data,*yy = (Mat_MPIAIJ*)Y->data;

  PetscFunctionBegin;
  if (str == SAME_NONZERO_PATTERN) {
    PetscCall(MatAXPY(yy->A,a,xx->A,str));
    PetscCall(MatAXPY(yy->B,a,xx->B,str));
  } else if (str == SUBSET_NONZERO_PATTERN) { /* nonzeros of X is a subset of Y's */
    PetscCall(MatAXPY_Basic(Y,a,X,str));
  } else {
    Mat      B;
    PetscInt *nnz_d,*nnz_o;

    PetscCall(PetscMalloc1(yy->A->rmap->N,&nnz_d));
    PetscCall(PetscMalloc1(yy->B->rmap->N,&nnz_o));
    PetscCall(MatCreate(PetscObjectComm((PetscObject)Y),&B));
    PetscCall(PetscObjectSetName((PetscObject)B,((PetscObject)Y)->name));
    PetscCall(MatSetLayouts(B,Y->rmap,Y->cmap));
    PetscCall(MatSetType(B,((PetscObject)Y)->type_name));
    PetscCall(MatAXPYGetPreallocation_SeqAIJ(yy->A,xx->A,nnz_d));
    PetscCall(MatAXPYGetPreallocation_MPIAIJ(yy->B,yy->garray,xx->B,xx->garray,nnz_o));
    PetscCall(MatMPIAIJSetPreallocation(B,0,nnz_d,0,nnz_o));
    PetscCall(MatAXPY_BasicWithPreallocation(B,Y,a,X,str));
    PetscCall(MatHeaderMerge(Y,&B));
    PetscCall(PetscFree(nnz_d));
    PetscCall(PetscFree(nnz_o));
  }
  PetscFunctionReturn(0);
}

PETSC_INTERN PetscErrorCode MatConjugate_SeqAIJ(Mat);

PetscErrorCode MatConjugate_MPIAIJ(Mat mat)
{
  PetscFunctionBegin;
  if (PetscDefined(USE_COMPLEX)) {
    Mat_MPIAIJ *aij = (Mat_MPIAIJ*)mat->data;

    PetscCall(MatConjugate_SeqAIJ(aij->A));
    PetscCall(MatConjugate_SeqAIJ(aij->B));
  }
  PetscFunctionReturn(0);
}

PetscErrorCode MatRealPart_MPIAIJ(Mat A)
{
  Mat_MPIAIJ     *a = (Mat_MPIAIJ*)A->data;

  PetscFunctionBegin;
  PetscCall(MatRealPart(a->A));
  PetscCall(MatRealPart(a->B));
  PetscFunctionReturn(0);
}

PetscErrorCode MatImaginaryPart_MPIAIJ(Mat A)
{
  Mat_MPIAIJ     *a = (Mat_MPIAIJ*)A->data;

  PetscFunctionBegin;
  PetscCall(MatImaginaryPart(a->A));
  PetscCall(MatImaginaryPart(a->B));
  PetscFunctionReturn(0);
}

PetscErrorCode MatGetRowMaxAbs_MPIAIJ(Mat A,Vec v,PetscInt idx[])
{
  Mat_MPIAIJ        *a = (Mat_MPIAIJ*)A->data;
  PetscInt          i,*idxb = NULL,m = A->rmap->n;
  PetscScalar       *va,*vv;
  Vec               vB,vA;
  const PetscScalar *vb;

  PetscFunctionBegin;
  PetscCall(VecCreateSeq(PETSC_COMM_SELF,m,&vA));
  PetscCall(MatGetRowMaxAbs(a->A,vA,idx));

  PetscCall(VecGetArrayWrite(vA,&va));
  if (idx) {
    for (i=0; i<m; i++) {
      if (PetscAbsScalar(va[i])) idx[i] += A->cmap->rstart;
    }
  }

  PetscCall(VecCreateSeq(PETSC_COMM_SELF,m,&vB));
  PetscCall(PetscMalloc1(m,&idxb));
  PetscCall(MatGetRowMaxAbs(a->B,vB,idxb));

  PetscCall(VecGetArrayWrite(v,&vv));
  PetscCall(VecGetArrayRead(vB,&vb));
  for (i=0; i<m; i++) {
    if (PetscAbsScalar(va[i]) < PetscAbsScalar(vb[i])) {
      vv[i] = vb[i];
      if (idx) idx[i] = a->garray[idxb[i]];
    } else {
      vv[i] = va[i];
      if (idx && PetscAbsScalar(va[i]) == PetscAbsScalar(vb[i]) && idxb[i] != -1 && idx[i] > a->garray[idxb[i]])
        idx[i] = a->garray[idxb[i]];
    }
  }
  PetscCall(VecRestoreArrayWrite(vA,&vv));
  PetscCall(VecRestoreArrayWrite(vA,&va));
  PetscCall(VecRestoreArrayRead(vB,&vb));
  PetscCall(PetscFree(idxb));
  PetscCall(VecDestroy(&vA));
  PetscCall(VecDestroy(&vB));
  PetscFunctionReturn(0);
}

PetscErrorCode MatGetRowMinAbs_MPIAIJ(Mat A, Vec v, PetscInt idx[])
{
  Mat_MPIAIJ        *mat   = (Mat_MPIAIJ*) A->data;
  PetscInt          m = A->rmap->n,n = A->cmap->n;
  PetscInt          cstart = A->cmap->rstart,cend = A->cmap->rend;
  PetscInt          *cmap  = mat->garray;
  PetscInt          *diagIdx, *offdiagIdx;
  Vec               diagV, offdiagV;
  PetscScalar       *a, *diagA, *offdiagA;
  const PetscScalar *ba,*bav;
  PetscInt          r,j,col,ncols,*bi,*bj;
  Mat               B = mat->B;
  Mat_SeqAIJ        *b = (Mat_SeqAIJ*)B->data;

  PetscFunctionBegin;
  /* When a process holds entire A and other processes have no entry */
  if (A->cmap->N == n) {
    PetscCall(VecGetArrayWrite(v,&diagA));
    PetscCall(VecCreateSeqWithArray(PETSC_COMM_SELF,1,m,diagA,&diagV));
    PetscCall(MatGetRowMinAbs(mat->A,diagV,idx));
    PetscCall(VecDestroy(&diagV));
    PetscCall(VecRestoreArrayWrite(v,&diagA));
    PetscFunctionReturn(0);
  } else if (n == 0) {
    if (m) {
      PetscCall(VecGetArrayWrite(v,&a));
      for (r = 0; r < m; r++) {a[r] = 0.0; if (idx) idx[r] = -1;}
      PetscCall(VecRestoreArrayWrite(v,&a));
    }
    PetscFunctionReturn(0);
  }

  PetscCall(PetscMalloc2(m,&diagIdx,m,&offdiagIdx));
  PetscCall(VecCreateSeq(PETSC_COMM_SELF, m, &diagV));
  PetscCall(VecCreateSeq(PETSC_COMM_SELF, m, &offdiagV));
  PetscCall(MatGetRowMinAbs(mat->A, diagV, diagIdx));

  /* Get offdiagIdx[] for implicit 0.0 */
  PetscCall(MatSeqAIJGetArrayRead(B,&bav));
  ba   = bav;
  bi   = b->i;
  bj   = b->j;
  PetscCall(VecGetArrayWrite(offdiagV, &offdiagA));
  for (r = 0; r < m; r++) {
    ncols = bi[r+1] - bi[r];
    if (ncols == A->cmap->N - n) { /* Brow is dense */
      offdiagA[r] = *ba; offdiagIdx[r] = cmap[0];
    } else { /* Brow is sparse so already KNOW maximum is 0.0 or higher */
      offdiagA[r] = 0.0;

      /* Find first hole in the cmap */
      for (j=0; j<ncols; j++) {
        col = cmap[bj[j]]; /* global column number = cmap[B column number] */
        if (col > j && j < cstart) {
          offdiagIdx[r] = j; /* global column number of first implicit 0.0 */
          break;
        } else if (col > j + n && j >= cstart) {
          offdiagIdx[r] = j + n; /* global column number of first implicit 0.0 */
          break;
        }
      }
      if (j == ncols && ncols < A->cmap->N - n) {
        /* a hole is outside compressed Bcols */
        if (ncols == 0) {
          if (cstart) {
            offdiagIdx[r] = 0;
          } else offdiagIdx[r] = cend;
        } else { /* ncols > 0 */
          offdiagIdx[r] = cmap[ncols-1] + 1;
          if (offdiagIdx[r] == cstart) offdiagIdx[r] += n;
        }
      }
    }

    for (j=0; j<ncols; j++) {
      if (PetscAbsScalar(offdiagA[r]) > PetscAbsScalar(*ba)) {offdiagA[r] = *ba; offdiagIdx[r] = cmap[*bj];}
      ba++; bj++;
    }
  }

  PetscCall(VecGetArrayWrite(v, &a));
  PetscCall(VecGetArrayRead(diagV, (const PetscScalar**)&diagA));
  for (r = 0; r < m; ++r) {
    if (PetscAbsScalar(diagA[r]) < PetscAbsScalar(offdiagA[r])) {
      a[r]   = diagA[r];
      if (idx) idx[r] = cstart + diagIdx[r];
    } else if (PetscAbsScalar(diagA[r]) == PetscAbsScalar(offdiagA[r])) {
      a[r] = diagA[r];
      if (idx) {
        if (cstart + diagIdx[r] <= offdiagIdx[r]) {
          idx[r] = cstart + diagIdx[r];
        } else idx[r] = offdiagIdx[r];
      }
    } else {
      a[r]   = offdiagA[r];
      if (idx) idx[r] = offdiagIdx[r];
    }
  }
  PetscCall(MatSeqAIJRestoreArrayRead(B,&bav));
  PetscCall(VecRestoreArrayWrite(v, &a));
  PetscCall(VecRestoreArrayRead(diagV, (const PetscScalar**)&diagA));
  PetscCall(VecRestoreArrayWrite(offdiagV, &offdiagA));
  PetscCall(VecDestroy(&diagV));
  PetscCall(VecDestroy(&offdiagV));
  PetscCall(PetscFree2(diagIdx, offdiagIdx));
  PetscFunctionReturn(0);
}

PetscErrorCode MatGetRowMin_MPIAIJ(Mat A,Vec v,PetscInt idx[])
{
  Mat_MPIAIJ        *mat = (Mat_MPIAIJ*) A->data;
  PetscInt          m = A->rmap->n,n = A->cmap->n;
  PetscInt          cstart = A->cmap->rstart,cend = A->cmap->rend;
  PetscInt          *cmap  = mat->garray;
  PetscInt          *diagIdx, *offdiagIdx;
  Vec               diagV, offdiagV;
  PetscScalar       *a, *diagA, *offdiagA;
  const PetscScalar *ba,*bav;
  PetscInt          r,j,col,ncols,*bi,*bj;
  Mat               B = mat->B;
  Mat_SeqAIJ        *b = (Mat_SeqAIJ*)B->data;

  PetscFunctionBegin;
  /* When a process holds entire A and other processes have no entry */
  if (A->cmap->N == n) {
    PetscCall(VecGetArrayWrite(v,&diagA));
    PetscCall(VecCreateSeqWithArray(PETSC_COMM_SELF,1,m,diagA,&diagV));
    PetscCall(MatGetRowMin(mat->A,diagV,idx));
    PetscCall(VecDestroy(&diagV));
    PetscCall(VecRestoreArrayWrite(v,&diagA));
    PetscFunctionReturn(0);
  } else if (n == 0) {
    if (m) {
      PetscCall(VecGetArrayWrite(v,&a));
      for (r = 0; r < m; r++) {a[r] = PETSC_MAX_REAL; if (idx) idx[r] = -1;}
      PetscCall(VecRestoreArrayWrite(v,&a));
    }
    PetscFunctionReturn(0);
  }

  PetscCall(PetscCalloc2(m,&diagIdx,m,&offdiagIdx));
  PetscCall(VecCreateSeq(PETSC_COMM_SELF, m, &diagV));
  PetscCall(VecCreateSeq(PETSC_COMM_SELF, m, &offdiagV));
  PetscCall(MatGetRowMin(mat->A, diagV, diagIdx));

  /* Get offdiagIdx[] for implicit 0.0 */
  PetscCall(MatSeqAIJGetArrayRead(B,&bav));
  ba   = bav;
  bi   = b->i;
  bj   = b->j;
  PetscCall(VecGetArrayWrite(offdiagV, &offdiagA));
  for (r = 0; r < m; r++) {
    ncols = bi[r+1] - bi[r];
    if (ncols == A->cmap->N - n) { /* Brow is dense */
      offdiagA[r] = *ba; offdiagIdx[r] = cmap[0];
    } else { /* Brow is sparse so already KNOW maximum is 0.0 or higher */
      offdiagA[r] = 0.0;

      /* Find first hole in the cmap */
      for (j=0; j<ncols; j++) {
        col = cmap[bj[j]]; /* global column number = cmap[B column number] */
        if (col > j && j < cstart) {
          offdiagIdx[r] = j; /* global column number of first implicit 0.0 */
          break;
        } else if (col > j + n && j >= cstart) {
          offdiagIdx[r] = j + n; /* global column number of first implicit 0.0 */
          break;
        }
      }
      if (j == ncols && ncols < A->cmap->N - n) {
        /* a hole is outside compressed Bcols */
        if (ncols == 0) {
          if (cstart) {
            offdiagIdx[r] = 0;
          } else offdiagIdx[r] = cend;
        } else { /* ncols > 0 */
          offdiagIdx[r] = cmap[ncols-1] + 1;
          if (offdiagIdx[r] == cstart) offdiagIdx[r] += n;
        }
      }
    }

    for (j=0; j<ncols; j++) {
      if (PetscRealPart(offdiagA[r]) > PetscRealPart(*ba)) {offdiagA[r] = *ba; offdiagIdx[r] = cmap[*bj];}
      ba++; bj++;
    }
  }

  PetscCall(VecGetArrayWrite(v, &a));
  PetscCall(VecGetArrayRead(diagV, (const PetscScalar**)&diagA));
  for (r = 0; r < m; ++r) {
    if (PetscRealPart(diagA[r]) < PetscRealPart(offdiagA[r])) {
      a[r]   = diagA[r];
      if (idx) idx[r] = cstart + diagIdx[r];
    } else if (PetscRealPart(diagA[r]) == PetscRealPart(offdiagA[r])) {
      a[r] = diagA[r];
      if (idx) {
        if (cstart + diagIdx[r] <= offdiagIdx[r]) {
          idx[r] = cstart + diagIdx[r];
        } else idx[r] = offdiagIdx[r];
      }
    } else {
      a[r]   = offdiagA[r];
      if (idx) idx[r] = offdiagIdx[r];
    }
  }
  PetscCall(MatSeqAIJRestoreArrayRead(B,&bav));
  PetscCall(VecRestoreArrayWrite(v, &a));
  PetscCall(VecRestoreArrayRead(diagV, (const PetscScalar**)&diagA));
  PetscCall(VecRestoreArrayWrite(offdiagV, &offdiagA));
  PetscCall(VecDestroy(&diagV));
  PetscCall(VecDestroy(&offdiagV));
  PetscCall(PetscFree2(diagIdx, offdiagIdx));
  PetscFunctionReturn(0);
}

PetscErrorCode MatGetRowMax_MPIAIJ(Mat A,Vec v,PetscInt idx[])
{
  Mat_MPIAIJ        *mat = (Mat_MPIAIJ*)A->data;
  PetscInt          m = A->rmap->n,n = A->cmap->n;
  PetscInt          cstart = A->cmap->rstart,cend = A->cmap->rend;
  PetscInt          *cmap  = mat->garray;
  PetscInt          *diagIdx, *offdiagIdx;
  Vec               diagV, offdiagV;
  PetscScalar       *a, *diagA, *offdiagA;
  const PetscScalar *ba,*bav;
  PetscInt          r,j,col,ncols,*bi,*bj;
  Mat               B = mat->B;
  Mat_SeqAIJ        *b = (Mat_SeqAIJ*)B->data;

  PetscFunctionBegin;
  /* When a process holds entire A and other processes have no entry */
  if (A->cmap->N == n) {
    PetscCall(VecGetArrayWrite(v,&diagA));
    PetscCall(VecCreateSeqWithArray(PETSC_COMM_SELF,1,m,diagA,&diagV));
    PetscCall(MatGetRowMax(mat->A,diagV,idx));
    PetscCall(VecDestroy(&diagV));
    PetscCall(VecRestoreArrayWrite(v,&diagA));
    PetscFunctionReturn(0);
  } else if (n == 0) {
    if (m) {
      PetscCall(VecGetArrayWrite(v,&a));
      for (r = 0; r < m; r++) {a[r] = PETSC_MIN_REAL; if (idx) idx[r] = -1;}
      PetscCall(VecRestoreArrayWrite(v,&a));
    }
    PetscFunctionReturn(0);
  }

  PetscCall(PetscMalloc2(m,&diagIdx,m,&offdiagIdx));
  PetscCall(VecCreateSeq(PETSC_COMM_SELF, m, &diagV));
  PetscCall(VecCreateSeq(PETSC_COMM_SELF, m, &offdiagV));
  PetscCall(MatGetRowMax(mat->A, diagV, diagIdx));

  /* Get offdiagIdx[] for implicit 0.0 */
  PetscCall(MatSeqAIJGetArrayRead(B,&bav));
  ba   = bav;
  bi   = b->i;
  bj   = b->j;
  PetscCall(VecGetArrayWrite(offdiagV, &offdiagA));
  for (r = 0; r < m; r++) {
    ncols = bi[r+1] - bi[r];
    if (ncols == A->cmap->N - n) { /* Brow is dense */
      offdiagA[r] = *ba; offdiagIdx[r] = cmap[0];
    } else { /* Brow is sparse so already KNOW maximum is 0.0 or higher */
      offdiagA[r] = 0.0;

      /* Find first hole in the cmap */
      for (j=0; j<ncols; j++) {
        col = cmap[bj[j]]; /* global column number = cmap[B column number] */
        if (col > j && j < cstart) {
          offdiagIdx[r] = j; /* global column number of first implicit 0.0 */
          break;
        } else if (col > j + n && j >= cstart) {
          offdiagIdx[r] = j + n; /* global column number of first implicit 0.0 */
          break;
        }
      }
      if (j == ncols && ncols < A->cmap->N - n) {
        /* a hole is outside compressed Bcols */
        if (ncols == 0) {
          if (cstart) {
            offdiagIdx[r] = 0;
          } else offdiagIdx[r] = cend;
        } else { /* ncols > 0 */
          offdiagIdx[r] = cmap[ncols-1] + 1;
          if (offdiagIdx[r] == cstart) offdiagIdx[r] += n;
        }
      }
    }

    for (j=0; j<ncols; j++) {
      if (PetscRealPart(offdiagA[r]) < PetscRealPart(*ba)) {offdiagA[r] = *ba; offdiagIdx[r] = cmap[*bj];}
      ba++; bj++;
    }
  }

  PetscCall(VecGetArrayWrite(v,    &a));
  PetscCall(VecGetArrayRead(diagV,(const PetscScalar**)&diagA));
  for (r = 0; r < m; ++r) {
    if (PetscRealPart(diagA[r]) > PetscRealPart(offdiagA[r])) {
      a[r] = diagA[r];
      if (idx) idx[r] = cstart + diagIdx[r];
    } else if (PetscRealPart(diagA[r]) == PetscRealPart(offdiagA[r])) {
      a[r] = diagA[r];
      if (idx) {
        if (cstart + diagIdx[r] <= offdiagIdx[r]) {
          idx[r] = cstart + diagIdx[r];
        } else idx[r] = offdiagIdx[r];
      }
    } else {
      a[r] = offdiagA[r];
      if (idx) idx[r] = offdiagIdx[r];
    }
  }
  PetscCall(MatSeqAIJRestoreArrayRead(B,&bav));
  PetscCall(VecRestoreArrayWrite(v,       &a));
  PetscCall(VecRestoreArrayRead(diagV,   (const PetscScalar**)&diagA));
  PetscCall(VecRestoreArrayWrite(offdiagV,&offdiagA));
  PetscCall(VecDestroy(&diagV));
  PetscCall(VecDestroy(&offdiagV));
  PetscCall(PetscFree2(diagIdx, offdiagIdx));
  PetscFunctionReturn(0);
}

PetscErrorCode MatGetSeqNonzeroStructure_MPIAIJ(Mat mat,Mat *newmat)
{
  Mat            *dummy;

  PetscFunctionBegin;
  PetscCall(MatCreateSubMatrix_MPIAIJ_All(mat,MAT_DO_NOT_GET_VALUES,MAT_INITIAL_MATRIX,&dummy));
  *newmat = *dummy;
  PetscCall(PetscFree(dummy));
  PetscFunctionReturn(0);
}

PetscErrorCode  MatInvertBlockDiagonal_MPIAIJ(Mat A,const PetscScalar **values)
{
  Mat_MPIAIJ     *a = (Mat_MPIAIJ*) A->data;

  PetscFunctionBegin;
  PetscCall(MatInvertBlockDiagonal(a->A,values));
  A->factorerrortype = a->A->factorerrortype;
  PetscFunctionReturn(0);
}

static PetscErrorCode  MatSetRandom_MPIAIJ(Mat x,PetscRandom rctx)
{
  Mat_MPIAIJ     *aij = (Mat_MPIAIJ*)x->data;

  PetscFunctionBegin;
  PetscCheck(x->assembled || x->preallocated,PetscObjectComm((PetscObject)x), PETSC_ERR_ARG_WRONGSTATE, "MatSetRandom on an unassembled and unpreallocated MATMPIAIJ is not allowed");
  PetscCall(MatSetRandom(aij->A,rctx));
  if (x->assembled) {
    PetscCall(MatSetRandom(aij->B,rctx));
  } else {
    PetscCall(MatSetRandomSkipColumnRange_SeqAIJ_Private(aij->B,x->cmap->rstart,x->cmap->rend,rctx));
  }
  PetscCall(MatAssemblyBegin(x,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(x,MAT_FINAL_ASSEMBLY));
  PetscFunctionReturn(0);
}

PetscErrorCode MatMPIAIJSetUseScalableIncreaseOverlap_MPIAIJ(Mat A,PetscBool sc)
{
  PetscFunctionBegin;
  if (sc) A->ops->increaseoverlap = MatIncreaseOverlap_MPIAIJ_Scalable;
  else A->ops->increaseoverlap    = MatIncreaseOverlap_MPIAIJ;
  PetscFunctionReturn(0);
}

/*@
   MatMPIAIJGetNumberNonzeros - gets the number of nonzeros in the matrix on this MPI rank

   Not collective

   Input Parameter:
.    A - the matrix

   Output Parameter:
.    nz - the number of nonzeros

 Level: advanced

@*/
PetscErrorCode MatMPIAIJGetNumberNonzeros(Mat A,PetscCount *nz)
{
  Mat_MPIAIJ *maij = (Mat_MPIAIJ*)A->data;
  Mat_SeqAIJ *aaij = (Mat_SeqAIJ*)maij->A->data, *baij = (Mat_SeqAIJ*)maij->B->data;

  PetscFunctionBegin;
  *nz = aaij->i[A->rmap->n] + baij->i[A->rmap->n];
  PetscFunctionReturn(0);
}

/*@
   MatMPIAIJSetUseScalableIncreaseOverlap - Determine if the matrix uses a scalable algorithm to compute the overlap

   Collective on Mat

   Input Parameters:
+    A - the matrix
-    sc - PETSC_TRUE indicates use the scalable algorithm (default is not to use the scalable algorithm)

 Level: advanced

@*/
PetscErrorCode MatMPIAIJSetUseScalableIncreaseOverlap(Mat A,PetscBool sc)
{
  PetscFunctionBegin;
  PetscTryMethod(A,"MatMPIAIJSetUseScalableIncreaseOverlap_C",(Mat,PetscBool),(A,sc));
  PetscFunctionReturn(0);
}

PetscErrorCode MatSetFromOptions_MPIAIJ(PetscOptionItems *PetscOptionsObject,Mat A)
{
  PetscBool            sc = PETSC_FALSE,flg;

  PetscFunctionBegin;
  PetscOptionsHeadBegin(PetscOptionsObject,"MPIAIJ options");
  if (A->ops->increaseoverlap == MatIncreaseOverlap_MPIAIJ_Scalable) sc = PETSC_TRUE;
  PetscCall(PetscOptionsBool("-mat_increase_overlap_scalable","Use a scalable algorithm to compute the overlap","MatIncreaseOverlap",sc,&sc,&flg));
  if (flg) PetscCall(MatMPIAIJSetUseScalableIncreaseOverlap(A,sc));
  PetscOptionsHeadEnd();
  PetscFunctionReturn(0);
}

PetscErrorCode MatShift_MPIAIJ(Mat Y,PetscScalar a)
{
  Mat_MPIAIJ     *maij = (Mat_MPIAIJ*)Y->data;
  Mat_SeqAIJ     *aij = (Mat_SeqAIJ*)maij->A->data;

  PetscFunctionBegin;
  if (!Y->preallocated) {
    PetscCall(MatMPIAIJSetPreallocation(Y,1,NULL,0,NULL));
  } else if (!aij->nz) { /* It does not matter if diagonals of Y only partially lie in maij->A. We just need an estimated preallocation. */
    PetscInt nonew = aij->nonew;
    PetscCall(MatSeqAIJSetPreallocation(maij->A,1,NULL));
    aij->nonew = nonew;
  }
  PetscCall(MatShift_Basic(Y,a));
  PetscFunctionReturn(0);
}

PetscErrorCode MatMissingDiagonal_MPIAIJ(Mat A,PetscBool  *missing,PetscInt *d)
{
  Mat_MPIAIJ     *a = (Mat_MPIAIJ*)A->data;

  PetscFunctionBegin;
  PetscCheck(A->rmap->n == A->cmap->n,PETSC_COMM_SELF,PETSC_ERR_SUP,"Only works for square matrices");
  PetscCall(MatMissingDiagonal(a->A,missing,d));
  if (d) {
    PetscInt rstart;
    PetscCall(MatGetOwnershipRange(A,&rstart,NULL));
    *d += rstart;

  }
  PetscFunctionReturn(0);
}

PetscErrorCode MatInvertVariableBlockDiagonal_MPIAIJ(Mat A,PetscInt nblocks,const PetscInt *bsizes,PetscScalar *diag)
{
  Mat_MPIAIJ     *a = (Mat_MPIAIJ*)A->data;

  PetscFunctionBegin;
  PetscCall(MatInvertVariableBlockDiagonal(a->A,nblocks,bsizes,diag));
  PetscFunctionReturn(0);
}

/* -------------------------------------------------------------------*/
static struct _MatOps MatOps_Values = {MatSetValues_MPIAIJ,
                                       MatGetRow_MPIAIJ,
                                       MatRestoreRow_MPIAIJ,
                                       MatMult_MPIAIJ,
                                /* 4*/ MatMultAdd_MPIAIJ,
                                       MatMultTranspose_MPIAIJ,
                                       MatMultTransposeAdd_MPIAIJ,
                                       NULL,
                                       NULL,
                                       NULL,
                                /*10*/ NULL,
                                       NULL,
                                       NULL,
                                       MatSOR_MPIAIJ,
                                       MatTranspose_MPIAIJ,
                                /*15*/ MatGetInfo_MPIAIJ,
                                       MatEqual_MPIAIJ,
                                       MatGetDiagonal_MPIAIJ,
                                       MatDiagonalScale_MPIAIJ,
                                       MatNorm_MPIAIJ,
                                /*20*/ MatAssemblyBegin_MPIAIJ,
                                       MatAssemblyEnd_MPIAIJ,
                                       MatSetOption_MPIAIJ,
                                       MatZeroEntries_MPIAIJ,
                                /*24*/ MatZeroRows_MPIAIJ,
                                       NULL,
                                       NULL,
                                       NULL,
                                       NULL,
                                /*29*/ MatSetUp_MPIAIJ,
                                       NULL,
                                       NULL,
                                       MatGetDiagonalBlock_MPIAIJ,
                                       NULL,
                                /*34*/ MatDuplicate_MPIAIJ,
                                       NULL,
                                       NULL,
                                       NULL,
                                       NULL,
                                /*39*/ MatAXPY_MPIAIJ,
                                       MatCreateSubMatrices_MPIAIJ,
                                       MatIncreaseOverlap_MPIAIJ,
                                       MatGetValues_MPIAIJ,
                                       MatCopy_MPIAIJ,
                                /*44*/ MatGetRowMax_MPIAIJ,
                                       MatScale_MPIAIJ,
                                       MatShift_MPIAIJ,
                                       MatDiagonalSet_MPIAIJ,
                                       MatZeroRowsColumns_MPIAIJ,
                                /*49*/ MatSetRandom_MPIAIJ,
                                       MatGetRowIJ_MPIAIJ,
                                       MatRestoreRowIJ_MPIAIJ,
                                       NULL,
                                       NULL,
                                /*54*/ MatFDColoringCreate_MPIXAIJ,
                                       NULL,
                                       MatSetUnfactored_MPIAIJ,
                                       MatPermute_MPIAIJ,
                                       NULL,
                                /*59*/ MatCreateSubMatrix_MPIAIJ,
                                       MatDestroy_MPIAIJ,
                                       MatView_MPIAIJ,
                                       NULL,
                                       NULL,
                                /*64*/ NULL,
                                       MatMatMatMultNumeric_MPIAIJ_MPIAIJ_MPIAIJ,
                                       NULL,
                                       NULL,
                                       NULL,
                                /*69*/ MatGetRowMaxAbs_MPIAIJ,
                                       MatGetRowMinAbs_MPIAIJ,
                                       NULL,
                                       NULL,
                                       NULL,
                                       NULL,
                                /*75*/ MatFDColoringApply_AIJ,
                                       MatSetFromOptions_MPIAIJ,
                                       NULL,
                                       NULL,
                                       MatFindZeroDiagonals_MPIAIJ,
                                /*80*/ NULL,
                                       NULL,
                                       NULL,
                                /*83*/ MatLoad_MPIAIJ,
                                       MatIsSymmetric_MPIAIJ,
                                       NULL,
                                       NULL,
                                       NULL,
                                       NULL,
                                /*89*/ NULL,
                                       NULL,
                                       MatMatMultNumeric_MPIAIJ_MPIAIJ,
                                       NULL,
                                       NULL,
                                /*94*/ MatPtAPNumeric_MPIAIJ_MPIAIJ,
                                       NULL,
                                       NULL,
                                       NULL,
                                       MatBindToCPU_MPIAIJ,
                                /*99*/ MatProductSetFromOptions_MPIAIJ,
                                       NULL,
                                       NULL,
                                       MatConjugate_MPIAIJ,
                                       NULL,
                                /*104*/MatSetValuesRow_MPIAIJ,
                                       MatRealPart_MPIAIJ,
                                       MatImaginaryPart_MPIAIJ,
                                       NULL,
                                       NULL,
                                /*109*/NULL,
                                       NULL,
                                       MatGetRowMin_MPIAIJ,
                                       NULL,
                                       MatMissingDiagonal_MPIAIJ,
                                /*114*/MatGetSeqNonzeroStructure_MPIAIJ,
                                       NULL,
                                       MatGetGhosts_MPIAIJ,
                                       NULL,
                                       NULL,
                                /*119*/MatMultDiagonalBlock_MPIAIJ,
                                       NULL,
                                       NULL,
                                       NULL,
                                       MatGetMultiProcBlock_MPIAIJ,
                                /*124*/MatFindNonzeroRows_MPIAIJ,
                                       MatGetColumnReductions_MPIAIJ,
                                       MatInvertBlockDiagonal_MPIAIJ,
                                       MatInvertVariableBlockDiagonal_MPIAIJ,
                                       MatCreateSubMatricesMPI_MPIAIJ,
                                /*129*/NULL,
                                       NULL,
                                       NULL,
                                       MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ,
                                       NULL,
                                /*134*/NULL,
                                       NULL,
                                       NULL,
                                       NULL,
                                       NULL,
                                /*139*/MatSetBlockSizes_MPIAIJ,
                                       NULL,
                                       NULL,
                                       MatFDColoringSetUp_MPIXAIJ,
                                       MatFindOffBlockDiagonalEntries_MPIAIJ,
                                       MatCreateMPIMatConcatenateSeqMat_MPIAIJ,
                                /*145*/NULL,
                                       NULL,
                                       NULL,
                                       MatCreateGraph_Simple_AIJ,
                                       MatFilter_AIJ
};

/* ----------------------------------------------------------------------------------------*/

PetscErrorCode  MatStoreValues_MPIAIJ(Mat mat)
{
  Mat_MPIAIJ     *aij = (Mat_MPIAIJ*)mat->data;

  PetscFunctionBegin;
  PetscCall(MatStoreValues(aij->A));
  PetscCall(MatStoreValues(aij->B));
  PetscFunctionReturn(0);
}

PetscErrorCode  MatRetrieveValues_MPIAIJ(Mat mat)
{
  Mat_MPIAIJ     *aij = (Mat_MPIAIJ*)mat->data;

  PetscFunctionBegin;
  PetscCall(MatRetrieveValues(aij->A));
  PetscCall(MatRetrieveValues(aij->B));
  PetscFunctionReturn(0);
}

PetscErrorCode MatMPIAIJSetPreallocation_MPIAIJ(Mat B,PetscInt d_nz,const PetscInt d_nnz[],PetscInt o_nz,const PetscInt o_nnz[])
{
  Mat_MPIAIJ     *b;
  PetscMPIInt    size;

  PetscFunctionBegin;
  PetscCall(PetscLayoutSetUp(B->rmap));
  PetscCall(PetscLayoutSetUp(B->cmap));
  b = (Mat_MPIAIJ*)B->data;

#if defined(PETSC_USE_CTABLE)
  PetscCall(PetscTableDestroy(&b->colmap));
#else
  PetscCall(PetscFree(b->colmap));
#endif
  PetscCall(PetscFree(b->garray));
  PetscCall(VecDestroy(&b->lvec));
  PetscCall(VecScatterDestroy(&b->Mvctx));

  /* Because the B will have been resized we simply destroy it and create a new one each time */
  PetscCallMPI(MPI_Comm_size(PetscObjectComm((PetscObject)B),&size));
  PetscCall(MatDestroy(&b->B));
  PetscCall(MatCreate(PETSC_COMM_SELF,&b->B));
  PetscCall(MatSetSizes(b->B,B->rmap->n,size > 1 ? B->cmap->N : 0,B->rmap->n,size > 1 ? B->cmap->N : 0));
  PetscCall(MatSetBlockSizesFromMats(b->B,B,B));
  PetscCall(MatSetType(b->B,MATSEQAIJ));
  PetscCall(PetscLogObjectParent((PetscObject)B,(PetscObject)b->B));

  if (!B->preallocated) {
    PetscCall(MatCreate(PETSC_COMM_SELF,&b->A));
    PetscCall(MatSetSizes(b->A,B->rmap->n,B->cmap->n,B->rmap->n,B->cmap->n));
    PetscCall(MatSetBlockSizesFromMats(b->A,B,B));
    PetscCall(MatSetType(b->A,MATSEQAIJ));
    PetscCall(PetscLogObjectParent((PetscObject)B,(PetscObject)b->A));
  }

  PetscCall(MatSeqAIJSetPreallocation(b->A,d_nz,d_nnz));
  PetscCall(MatSeqAIJSetPreallocation(b->B,o_nz,o_nnz));
  B->preallocated  = PETSC_TRUE;
  B->was_assembled = PETSC_FALSE;
  B->assembled     = PETSC_FALSE;
  PetscFunctionReturn(0);
}

PetscErrorCode MatResetPreallocation_MPIAIJ(Mat B)
{
  Mat_MPIAIJ     *b;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(B,MAT_CLASSID,1);
  PetscCall(PetscLayoutSetUp(B->rmap));
  PetscCall(PetscLayoutSetUp(B->cmap));
  b = (Mat_MPIAIJ*)B->data;

#if defined(PETSC_USE_CTABLE)
  PetscCall(PetscTableDestroy(&b->colmap));
#else
  PetscCall(PetscFree(b->colmap));
#endif
  PetscCall(PetscFree(b->garray));
  PetscCall(VecDestroy(&b->lvec));
  PetscCall(VecScatterDestroy(&b->Mvctx));

  PetscCall(MatResetPreallocation(b->A));
  PetscCall(MatResetPreallocation(b->B));
  B->preallocated  = PETSC_TRUE;
  B->was_assembled = PETSC_FALSE;
  B->assembled = PETSC_FALSE;
  PetscFunctionReturn(0);
}

PetscErrorCode MatDuplicate_MPIAIJ(Mat matin,MatDuplicateOption cpvalues,Mat *newmat)
{
  Mat            mat;
  Mat_MPIAIJ     *a,*oldmat = (Mat_MPIAIJ*)matin->data;

  PetscFunctionBegin;
  *newmat = NULL;
  PetscCall(MatCreate(PetscObjectComm((PetscObject)matin),&mat));
  PetscCall(MatSetSizes(mat,matin->rmap->n,matin->cmap->n,matin->rmap->N,matin->cmap->N));
  PetscCall(MatSetBlockSizesFromMats(mat,matin,matin));
  PetscCall(MatSetType(mat,((PetscObject)matin)->type_name));
  a       = (Mat_MPIAIJ*)mat->data;

  mat->factortype   = matin->factortype;
  mat->assembled    = matin->assembled;
  mat->insertmode   = NOT_SET_VALUES;
  mat->preallocated = matin->preallocated;

  a->size         = oldmat->size;
  a->rank         = oldmat->rank;
  a->donotstash   = oldmat->donotstash;
  a->roworiented  = oldmat->roworiented;
  a->rowindices   = NULL;
  a->rowvalues    = NULL;
  a->getrowactive = PETSC_FALSE;

  PetscCall(PetscLayoutReference(matin->rmap,&mat->rmap));
  PetscCall(PetscLayoutReference(matin->cmap,&mat->cmap));

  if (oldmat->colmap) {
#if defined(PETSC_USE_CTABLE)
    PetscCall(PetscTableCreateCopy(oldmat->colmap,&a->colmap));
#else
    PetscCall(PetscMalloc1(mat->cmap->N,&a->colmap));
    PetscCall(PetscLogObjectMemory((PetscObject)mat,(mat->cmap->N)*sizeof(PetscInt)));
    PetscCall(PetscArraycpy(a->colmap,oldmat->colmap,mat->cmap->N));
#endif
  } else a->colmap = NULL;
  if (oldmat->garray) {
    PetscInt len;
    len  = oldmat->B->cmap->n;
    PetscCall(PetscMalloc1(len+1,&a->garray));
    PetscCall(PetscLogObjectMemory((PetscObject)mat,len*sizeof(PetscInt)));
    if (len) PetscCall(PetscArraycpy(a->garray,oldmat->garray,len));
  } else a->garray = NULL;

  /* It may happen MatDuplicate is called with a non-assembled matrix
     In fact, MatDuplicate only requires the matrix to be preallocated
     This may happen inside a DMCreateMatrix_Shell */
  if (oldmat->lvec) {
    PetscCall(VecDuplicate(oldmat->lvec,&a->lvec));
    PetscCall(PetscLogObjectParent((PetscObject)mat,(PetscObject)a->lvec));
  }
  if (oldmat->Mvctx) {
    PetscCall(VecScatterCopy(oldmat->Mvctx,&a->Mvctx));
    PetscCall(PetscLogObjectParent((PetscObject)mat,(PetscObject)a->Mvctx));
  }
  PetscCall(MatDuplicate(oldmat->A,cpvalues,&a->A));
  PetscCall(PetscLogObjectParent((PetscObject)mat,(PetscObject)a->A));
  PetscCall(MatDuplicate(oldmat->B,cpvalues,&a->B));
  PetscCall(PetscLogObjectParent((PetscObject)mat,(PetscObject)a->B));
  PetscCall(PetscFunctionListDuplicate(((PetscObject)matin)->qlist,&((PetscObject)mat)->qlist));
  *newmat = mat;
  PetscFunctionReturn(0);
}

PetscErrorCode MatLoad_MPIAIJ(Mat newMat, PetscViewer viewer)
{
  PetscBool      isbinary, ishdf5;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(newMat,MAT_CLASSID,1);
  PetscValidHeaderSpecific(viewer,PETSC_VIEWER_CLASSID,2);
  /* force binary viewer to load .info file if it has not yet done so */
  PetscCall(PetscViewerSetUp(viewer));
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERBINARY,&isbinary));
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERHDF5,  &ishdf5));
  if (isbinary) {
    PetscCall(MatLoad_MPIAIJ_Binary(newMat,viewer));
  } else if (ishdf5) {
#if defined(PETSC_HAVE_HDF5)
    PetscCall(MatLoad_AIJ_HDF5(newMat,viewer));
#else
    SETERRQ(PetscObjectComm((PetscObject)newMat),PETSC_ERR_SUP,"HDF5 not supported in this build.\nPlease reconfigure using --download-hdf5");
#endif
  } else {
    SETERRQ(PetscObjectComm((PetscObject)newMat),PETSC_ERR_SUP,"Viewer type %s not yet supported for reading %s matrices",((PetscObject)viewer)->type_name,((PetscObject)newMat)->type_name);
  }
  PetscFunctionReturn(0);
}

PetscErrorCode MatLoad_MPIAIJ_Binary(Mat mat, PetscViewer viewer)
{
  PetscInt       header[4],M,N,m,nz,rows,cols,sum,i;
  PetscInt       *rowidxs,*colidxs;
  PetscScalar    *matvals;

  PetscFunctionBegin;
  PetscCall(PetscViewerSetUp(viewer));

  /* read in matrix header */
  PetscCall(PetscViewerBinaryRead(viewer,header,4,NULL,PETSC_INT));
  PetscCheck(header[0] == MAT_FILE_CLASSID,PetscObjectComm((PetscObject)viewer),PETSC_ERR_FILE_UNEXPECTED,"Not a matrix object in file");
  M  = header[1]; N = header[2]; nz = header[3];
  PetscCheck(M >= 0,PetscObjectComm((PetscObject)viewer),PETSC_ERR_FILE_UNEXPECTED,"Matrix row size (%" PetscInt_FMT ") in file is negative",M);
  PetscCheck(N >= 0,PetscObjectComm((PetscObject)viewer),PETSC_ERR_FILE_UNEXPECTED,"Matrix column size (%" PetscInt_FMT ") in file is negative",N);
  PetscCheck(nz >= 0,PETSC_COMM_SELF,PETSC_ERR_FILE_UNEXPECTED,"Matrix stored in special format on disk, cannot load as MPIAIJ");

  /* set block sizes from the viewer's .info file */
  PetscCall(MatLoad_Binary_BlockSizes(mat,viewer));
  /* set global sizes if not set already */
  if (mat->rmap->N < 0) mat->rmap->N = M;
  if (mat->cmap->N < 0) mat->cmap->N = N;
  PetscCall(PetscLayoutSetUp(mat->rmap));
  PetscCall(PetscLayoutSetUp(mat->cmap));

  /* check if the matrix sizes are correct */
  PetscCall(MatGetSize(mat,&rows,&cols));
  PetscCheck(M == rows && N == cols,PETSC_COMM_SELF,PETSC_ERR_FILE_UNEXPECTED, "Matrix in file of different sizes (%" PetscInt_FMT ", %" PetscInt_FMT ") than the input matrix (%" PetscInt_FMT ", %" PetscInt_FMT ")",M,N,rows,cols);

  /* read in row lengths and build row indices */
  PetscCall(MatGetLocalSize(mat,&m,NULL));
  PetscCall(PetscMalloc1(m+1,&rowidxs));
  PetscCall(PetscViewerBinaryReadAll(viewer,rowidxs+1,m,PETSC_DECIDE,M,PETSC_INT));
  rowidxs[0] = 0; for (i=0; i<m; i++) rowidxs[i+1] += rowidxs[i];
  PetscCall(MPIU_Allreduce(&rowidxs[m],&sum,1,MPIU_INT,MPI_SUM,PetscObjectComm((PetscObject)viewer)));
  PetscCheck(sum == nz,PetscObjectComm((PetscObject)viewer),PETSC_ERR_FILE_UNEXPECTED,"Inconsistent matrix data in file: nonzeros = %" PetscInt_FMT ", sum-row-lengths = %" PetscInt_FMT,nz,sum);
  /* read in column indices and matrix values */
  PetscCall(PetscMalloc2(rowidxs[m],&colidxs,rowidxs[m],&matvals));
  PetscCall(PetscViewerBinaryReadAll(viewer,colidxs,rowidxs[m],PETSC_DETERMINE,PETSC_DETERMINE,PETSC_INT));
  PetscCall(PetscViewerBinaryReadAll(viewer,matvals,rowidxs[m],PETSC_DETERMINE,PETSC_DETERMINE,PETSC_SCALAR));
  /* store matrix indices and values */
  PetscCall(MatMPIAIJSetPreallocationCSR(mat,rowidxs,colidxs,matvals));
  PetscCall(PetscFree(rowidxs));
  PetscCall(PetscFree2(colidxs,matvals));
  PetscFunctionReturn(0);
}

/* Not scalable because of ISAllGather() unless getting all columns. */
PetscErrorCode ISGetSeqIS_Private(Mat mat,IS iscol,IS *isseq)
{
  IS             iscol_local;
  PetscBool      isstride;
  PetscMPIInt    lisstride=0,gisstride;

  PetscFunctionBegin;
  /* check if we are grabbing all columns*/
  PetscCall(PetscObjectTypeCompare((PetscObject)iscol,ISSTRIDE,&isstride));

  if (isstride) {
    PetscInt  start,len,mstart,mlen;
    PetscCall(ISStrideGetInfo(iscol,&start,NULL));
    PetscCall(ISGetLocalSize(iscol,&len));
    PetscCall(MatGetOwnershipRangeColumn(mat,&mstart,&mlen));
    if (mstart == start && mlen-mstart == len) lisstride = 1;
  }

  PetscCall(MPIU_Allreduce(&lisstride,&gisstride,1,MPI_INT,MPI_MIN,PetscObjectComm((PetscObject)mat)));
  if (gisstride) {
    PetscInt N;
    PetscCall(MatGetSize(mat,NULL,&N));
    PetscCall(ISCreateStride(PETSC_COMM_SELF,N,0,1,&iscol_local));
    PetscCall(ISSetIdentity(iscol_local));
    PetscCall(PetscInfo(mat,"Optimizing for obtaining all columns of the matrix; skipping ISAllGather()\n"));
  } else {
    PetscInt cbs;
    PetscCall(ISGetBlockSize(iscol,&cbs));
    PetscCall(ISAllGather(iscol,&iscol_local));
    PetscCall(ISSetBlockSize(iscol_local,cbs));
  }

  *isseq = iscol_local;
  PetscFunctionReturn(0);
}

/*
 Used by MatCreateSubMatrix_MPIAIJ_SameRowColDist() to avoid ISAllGather() and global size of iscol_local
 (see MatCreateSubMatrix_MPIAIJ_nonscalable)

 Input Parameters:
   mat - matrix
   isrow - parallel row index set; its local indices are a subset of local columns of mat,
           i.e., mat->rstart <= isrow[i] < mat->rend
   iscol - parallel column index set; its local indices are a subset of local columns of mat,
           i.e., mat->cstart <= iscol[i] < mat->cend
 Output Parameter:
   isrow_d,iscol_d - sequential row and column index sets for retrieving mat->A
   iscol_o - sequential column index set for retrieving mat->B
   garray - column map; garray[i] indicates global location of iscol_o[i] in iscol
 */
PetscErrorCode ISGetSeqIS_SameColDist_Private(Mat mat,IS isrow,IS iscol,IS *isrow_d,IS *iscol_d,IS *iscol_o,const PetscInt *garray[])
{
  Vec            x,cmap;
  const PetscInt *is_idx;
  PetscScalar    *xarray,*cmaparray;
  PetscInt       ncols,isstart,*idx,m,rstart,*cmap1,count;
  Mat_MPIAIJ     *a=(Mat_MPIAIJ*)mat->data;
  Mat            B=a->B;
  Vec            lvec=a->lvec,lcmap;
  PetscInt       i,cstart,cend,Bn=B->cmap->N;
  MPI_Comm       comm;
  VecScatter     Mvctx=a->Mvctx;

  PetscFunctionBegin;
  PetscCall(PetscObjectGetComm((PetscObject)mat,&comm));
  PetscCall(ISGetLocalSize(iscol,&ncols));

  /* (1) iscol is a sub-column vector of mat, pad it with '-1.' to form a full vector x */
  PetscCall(MatCreateVecs(mat,&x,NULL));
  PetscCall(VecSet(x,-1.0));
  PetscCall(VecDuplicate(x,&cmap));
  PetscCall(VecSet(cmap,-1.0));

  /* Get start indices */
  PetscCallMPI(MPI_Scan(&ncols,&isstart,1,MPIU_INT,MPI_SUM,comm));
  isstart -= ncols;
  PetscCall(MatGetOwnershipRangeColumn(mat,&cstart,&cend));

  PetscCall(ISGetIndices(iscol,&is_idx));
  PetscCall(VecGetArray(x,&xarray));
  PetscCall(VecGetArray(cmap,&cmaparray));
  PetscCall(PetscMalloc1(ncols,&idx));
  for (i=0; i<ncols; i++) {
    xarray[is_idx[i]-cstart]    = (PetscScalar)is_idx[i];
    cmaparray[is_idx[i]-cstart] = i + isstart;      /* global index of iscol[i] */
    idx[i]                      = is_idx[i]-cstart; /* local index of iscol[i]  */
  }
  PetscCall(VecRestoreArray(x,&xarray));
  PetscCall(VecRestoreArray(cmap,&cmaparray));
  PetscCall(ISRestoreIndices(iscol,&is_idx));

  /* Get iscol_d */
  PetscCall(ISCreateGeneral(PETSC_COMM_SELF,ncols,idx,PETSC_OWN_POINTER,iscol_d));
  PetscCall(ISGetBlockSize(iscol,&i));
  PetscCall(ISSetBlockSize(*iscol_d,i));

  /* Get isrow_d */
  PetscCall(ISGetLocalSize(isrow,&m));
  rstart = mat->rmap->rstart;
  PetscCall(PetscMalloc1(m,&idx));
  PetscCall(ISGetIndices(isrow,&is_idx));
  for (i=0; i<m; i++) idx[i] = is_idx[i]-rstart;
  PetscCall(ISRestoreIndices(isrow,&is_idx));

  PetscCall(ISCreateGeneral(PETSC_COMM_SELF,m,idx,PETSC_OWN_POINTER,isrow_d));
  PetscCall(ISGetBlockSize(isrow,&i));
  PetscCall(ISSetBlockSize(*isrow_d,i));

  /* (2) Scatter x and cmap using aij->Mvctx to get their off-process portions (see MatMult_MPIAIJ) */
  PetscCall(VecScatterBegin(Mvctx,x,lvec,INSERT_VALUES,SCATTER_FORWARD));
  PetscCall(VecScatterEnd(Mvctx,x,lvec,INSERT_VALUES,SCATTER_FORWARD));

  PetscCall(VecDuplicate(lvec,&lcmap));

  PetscCall(VecScatterBegin(Mvctx,cmap,lcmap,INSERT_VALUES,SCATTER_FORWARD));
  PetscCall(VecScatterEnd(Mvctx,cmap,lcmap,INSERT_VALUES,SCATTER_FORWARD));

  /* (3) create sequential iscol_o (a subset of iscol) and isgarray */
  /* off-process column indices */
  count = 0;
  PetscCall(PetscMalloc1(Bn,&idx));
  PetscCall(PetscMalloc1(Bn,&cmap1));

  PetscCall(VecGetArray(lvec,&xarray));
  PetscCall(VecGetArray(lcmap,&cmaparray));
  for (i=0; i<Bn; i++) {
    if (PetscRealPart(xarray[i]) > -1.0) {
      idx[count]     = i;                   /* local column index in off-diagonal part B */
      cmap1[count] = (PetscInt)PetscRealPart(cmaparray[i]);  /* column index in submat */
      count++;
    }
  }
  PetscCall(VecRestoreArray(lvec,&xarray));
  PetscCall(VecRestoreArray(lcmap,&cmaparray));

  PetscCall(ISCreateGeneral(PETSC_COMM_SELF,count,idx,PETSC_COPY_VALUES,iscol_o));
  /* cannot ensure iscol_o has same blocksize as iscol! */

  PetscCall(PetscFree(idx));
  *garray = cmap1;

  PetscCall(VecDestroy(&x));
  PetscCall(VecDestroy(&cmap));
  PetscCall(VecDestroy(&lcmap));
  PetscFunctionReturn(0);
}

/* isrow and iscol have same processor distribution as mat, output *submat is a submatrix of local mat */
PetscErrorCode MatCreateSubMatrix_MPIAIJ_SameRowColDist(Mat mat,IS isrow,IS iscol,MatReuse call,Mat *submat)
{
  Mat_MPIAIJ     *a = (Mat_MPIAIJ*)mat->data,*asub;
  Mat            M = NULL;
  MPI_Comm       comm;
  IS             iscol_d,isrow_d,iscol_o;
  Mat            Asub = NULL,Bsub = NULL;
  PetscInt       n;

  PetscFunctionBegin;
  PetscCall(PetscObjectGetComm((PetscObject)mat,&comm));

  if (call == MAT_REUSE_MATRIX) {
    /* Retrieve isrow_d, iscol_d and iscol_o from submat */
    PetscCall(PetscObjectQuery((PetscObject)*submat,"isrow_d",(PetscObject*)&isrow_d));
    PetscCheck(isrow_d,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"isrow_d passed in was not used before, cannot reuse");

    PetscCall(PetscObjectQuery((PetscObject)*submat,"iscol_d",(PetscObject*)&iscol_d));
    PetscCheck(iscol_d,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"iscol_d passed in was not used before, cannot reuse");

    PetscCall(PetscObjectQuery((PetscObject)*submat,"iscol_o",(PetscObject*)&iscol_o));
    PetscCheck(iscol_o,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"iscol_o passed in was not used before, cannot reuse");

    /* Update diagonal and off-diagonal portions of submat */
    asub = (Mat_MPIAIJ*)(*submat)->data;
    PetscCall(MatCreateSubMatrix_SeqAIJ(a->A,isrow_d,iscol_d,PETSC_DECIDE,MAT_REUSE_MATRIX,&asub->A));
    PetscCall(ISGetLocalSize(iscol_o,&n));
    if (n) {
      PetscCall(MatCreateSubMatrix_SeqAIJ(a->B,isrow_d,iscol_o,PETSC_DECIDE,MAT_REUSE_MATRIX,&asub->B));
    }
    PetscCall(MatAssemblyBegin(*submat,MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(*submat,MAT_FINAL_ASSEMBLY));

  } else { /* call == MAT_INITIAL_MATRIX) */
    const PetscInt *garray;
    PetscInt        BsubN;

    /* Create isrow_d, iscol_d, iscol_o and isgarray (replace isgarray with array?) */
    PetscCall(ISGetSeqIS_SameColDist_Private(mat,isrow,iscol,&isrow_d,&iscol_d,&iscol_o,&garray));

    /* Create local submatrices Asub and Bsub */
    PetscCall(MatCreateSubMatrix_SeqAIJ(a->A,isrow_d,iscol_d,PETSC_DECIDE,MAT_INITIAL_MATRIX,&Asub));
    PetscCall(MatCreateSubMatrix_SeqAIJ(a->B,isrow_d,iscol_o,PETSC_DECIDE,MAT_INITIAL_MATRIX,&Bsub));

    /* Create submatrix M */
    PetscCall(MatCreateMPIAIJWithSeqAIJ(comm,Asub,Bsub,garray,&M));

    /* If Bsub has empty columns, compress iscol_o such that it will retrieve condensed Bsub from a->B during reuse */
    asub = (Mat_MPIAIJ*)M->data;

    PetscCall(ISGetLocalSize(iscol_o,&BsubN));
    n = asub->B->cmap->N;
    if (BsubN > n) {
      /* This case can be tested using ~petsc/src/tao/bound/tutorials/runplate2_3 */
      const PetscInt *idx;
      PetscInt       i,j,*idx_new,*subgarray = asub->garray;
      PetscCall(PetscInfo(M,"submatrix Bn %" PetscInt_FMT " != BsubN %" PetscInt_FMT ", update iscol_o\n",n,BsubN));

      PetscCall(PetscMalloc1(n,&idx_new));
      j = 0;
      PetscCall(ISGetIndices(iscol_o,&idx));
      for (i=0; i<n; i++) {
        if (j >= BsubN) break;
        while (subgarray[i] > garray[j]) j++;

        if (subgarray[i] == garray[j]) {
          idx_new[i] = idx[j++];
        } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"subgarray[%" PetscInt_FMT "]=%" PetscInt_FMT " cannot < garray[%" PetscInt_FMT "]=%" PetscInt_FMT,i,subgarray[i],j,garray[j]);
      }
      PetscCall(ISRestoreIndices(iscol_o,&idx));

      PetscCall(ISDestroy(&iscol_o));
      PetscCall(ISCreateGeneral(PETSC_COMM_SELF,n,idx_new,PETSC_OWN_POINTER,&iscol_o));

    } else if (BsubN < n) {
      SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"Columns of Bsub (%" PetscInt_FMT ") cannot be smaller than B's (%" PetscInt_FMT ")",BsubN,asub->B->cmap->N);
    }

    PetscCall(PetscFree(garray));
    *submat = M;

    /* Save isrow_d, iscol_d and iscol_o used in processor for next request */
    PetscCall(PetscObjectCompose((PetscObject)M,"isrow_d",(PetscObject)isrow_d));
    PetscCall(ISDestroy(&isrow_d));

    PetscCall(PetscObjectCompose((PetscObject)M,"iscol_d",(PetscObject)iscol_d));
    PetscCall(ISDestroy(&iscol_d));

    PetscCall(PetscObjectCompose((PetscObject)M,"iscol_o",(PetscObject)iscol_o));
    PetscCall(ISDestroy(&iscol_o));
  }
  PetscFunctionReturn(0);
}

PetscErrorCode MatCreateSubMatrix_MPIAIJ(Mat mat,IS isrow,IS iscol,MatReuse call,Mat *newmat)
{
  IS             iscol_local=NULL,isrow_d;
  PetscInt       csize;
  PetscInt       n,i,j,start,end;
  PetscBool      sameRowDist=PETSC_FALSE,sameDist[2],tsameDist[2];
  MPI_Comm       comm;

  PetscFunctionBegin;
  /* If isrow has same processor distribution as mat,
     call MatCreateSubMatrix_MPIAIJ_SameRowDist() to avoid using a hash table with global size of iscol */
  if (call == MAT_REUSE_MATRIX) {
    PetscCall(PetscObjectQuery((PetscObject)*newmat,"isrow_d",(PetscObject*)&isrow_d));
    if (isrow_d) {
      sameRowDist  = PETSC_TRUE;
      tsameDist[1] = PETSC_TRUE; /* sameColDist */
    } else {
      PetscCall(PetscObjectQuery((PetscObject)*newmat,"SubIScol",(PetscObject*)&iscol_local));
      if (iscol_local) {
        sameRowDist  = PETSC_TRUE;
        tsameDist[1] = PETSC_FALSE; /* !sameColDist */
      }
    }
  } else {
    /* Check if isrow has same processor distribution as mat */
    sameDist[0] = PETSC_FALSE;
    PetscCall(ISGetLocalSize(isrow,&n));
    if (!n) {
      sameDist[0] = PETSC_TRUE;
    } else {
      PetscCall(ISGetMinMax(isrow,&i,&j));
      PetscCall(MatGetOwnershipRange(mat,&start,&end));
      if (i >= start && j < end) {
        sameDist[0] = PETSC_TRUE;
      }
    }

    /* Check if iscol has same processor distribution as mat */
    sameDist[1] = PETSC_FALSE;
    PetscCall(ISGetLocalSize(iscol,&n));
    if (!n) {
      sameDist[1] = PETSC_TRUE;
    } else {
      PetscCall(ISGetMinMax(iscol,&i,&j));
      PetscCall(MatGetOwnershipRangeColumn(mat,&start,&end));
      if (i >= start && j < end) sameDist[1] = PETSC_TRUE;
    }

    PetscCall(PetscObjectGetComm((PetscObject)mat,&comm));
    PetscCall(MPIU_Allreduce(&sameDist,&tsameDist,2,MPIU_BOOL,MPI_LAND,comm));
    sameRowDist = tsameDist[0];
  }

  if (sameRowDist) {
    if (tsameDist[1]) { /* sameRowDist & sameColDist */
      /* isrow and iscol have same processor distribution as mat */
      PetscCall(MatCreateSubMatrix_MPIAIJ_SameRowColDist(mat,isrow,iscol,call,newmat));
      PetscFunctionReturn(0);
    } else { /* sameRowDist */
      /* isrow has same processor distribution as mat */
      if (call == MAT_INITIAL_MATRIX) {
        PetscBool sorted;
        PetscCall(ISGetSeqIS_Private(mat,iscol,&iscol_local));
        PetscCall(ISGetLocalSize(iscol_local,&n)); /* local size of iscol_local = global columns of newmat */
        PetscCall(ISGetSize(iscol,&i));
        PetscCheck(n == i,PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"n %" PetscInt_FMT " != size of iscol %" PetscInt_FMT,n,i);

        PetscCall(ISSorted(iscol_local,&sorted));
        if (sorted) {
          /* MatCreateSubMatrix_MPIAIJ_SameRowDist() requires iscol_local be sorted; it can have duplicate indices */
          PetscCall(MatCreateSubMatrix_MPIAIJ_SameRowDist(mat,isrow,iscol,iscol_local,MAT_INITIAL_MATRIX,newmat));
          PetscFunctionReturn(0);
        }
      } else { /* call == MAT_REUSE_MATRIX */
        IS iscol_sub;
        PetscCall(PetscObjectQuery((PetscObject)*newmat,"SubIScol",(PetscObject*)&iscol_sub));
        if (iscol_sub) {
          PetscCall(MatCreateSubMatrix_MPIAIJ_SameRowDist(mat,isrow,iscol,NULL,call,newmat));
          PetscFunctionReturn(0);
        }
      }
    }
  }

  /* General case: iscol -> iscol_local which has global size of iscol */
  if (call == MAT_REUSE_MATRIX) {
    PetscCall(PetscObjectQuery((PetscObject)*newmat,"ISAllGather",(PetscObject*)&iscol_local));
    PetscCheck(iscol_local,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"Submatrix passed in was not used before, cannot reuse");
  } else {
    if (!iscol_local) {
      PetscCall(ISGetSeqIS_Private(mat,iscol,&iscol_local));
    }
  }

  PetscCall(ISGetLocalSize(iscol,&csize));
  PetscCall(MatCreateSubMatrix_MPIAIJ_nonscalable(mat,isrow,iscol_local,csize,call,newmat));

  if (call == MAT_INITIAL_MATRIX) {
    PetscCall(PetscObjectCompose((PetscObject)*newmat,"ISAllGather",(PetscObject)iscol_local));
    PetscCall(ISDestroy(&iscol_local));
  }
  PetscFunctionReturn(0);
}

/*@C
     MatCreateMPIAIJWithSeqAIJ - creates a MPIAIJ matrix using SeqAIJ matrices that contain the "diagonal"
         and "off-diagonal" part of the matrix in CSR format.

   Collective

   Input Parameters:
+  comm - MPI communicator
.  A - "diagonal" portion of matrix
.  B - "off-diagonal" portion of matrix, may have empty columns, will be destroyed by this routine
-  garray - global index of B columns

   Output Parameter:
.   mat - the matrix, with input A as its local diagonal matrix
   Level: advanced

   Notes:
       See MatCreateAIJ() for the definition of "diagonal" and "off-diagonal" portion of the matrix.
       A becomes part of output mat, B is destroyed by this routine. The user cannot use A and B anymore.

.seealso: `MatCreateMPIAIJWithSplitArrays()`
@*/
PetscErrorCode MatCreateMPIAIJWithSeqAIJ(MPI_Comm comm,Mat A,Mat B,const PetscInt garray[],Mat *mat)
{
  Mat_MPIAIJ        *maij;
  Mat_SeqAIJ        *b=(Mat_SeqAIJ*)B->data,*bnew;
  PetscInt          *oi=b->i,*oj=b->j,i,nz,col;
  const PetscScalar *oa;
  Mat               Bnew;
  PetscInt          m,n,N;

  PetscFunctionBegin;
  PetscCall(MatCreate(comm,mat));
  PetscCall(MatGetSize(A,&m,&n));
  PetscCheck(m == B->rmap->N,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"Am %" PetscInt_FMT " != Bm %" PetscInt_FMT,m,B->rmap->N);
  PetscCheck(A->rmap->bs == B->rmap->bs,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"A row bs %" PetscInt_FMT " != B row bs %" PetscInt_FMT,A->rmap->bs,B->rmap->bs);
  /* remove check below; When B is created using iscol_o from ISGetSeqIS_SameColDist_Private(), its bs may not be same as A */
  /* PetscCheck(A->cmap->bs == B->cmap->bs,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"A column bs %" PetscInt_FMT " != B column bs %" PetscInt_FMT,A->cmap->bs,B->cmap->bs); */

  /* Get global columns of mat */
  PetscCall(MPIU_Allreduce(&n,&N,1,MPIU_INT,MPI_SUM,comm));

  PetscCall(MatSetSizes(*mat,m,n,PETSC_DECIDE,N));
  PetscCall(MatSetType(*mat,MATMPIAIJ));
  PetscCall(MatSetBlockSizes(*mat,A->rmap->bs,A->cmap->bs));
  maij = (Mat_MPIAIJ*)(*mat)->data;

  (*mat)->preallocated = PETSC_TRUE;

  PetscCall(PetscLayoutSetUp((*mat)->rmap));
  PetscCall(PetscLayoutSetUp((*mat)->cmap));

  /* Set A as diagonal portion of *mat */
  maij->A = A;

  nz = oi[m];
  for (i=0; i<nz; i++) {
    col   = oj[i];
    oj[i] = garray[col];
  }

  /* Set Bnew as off-diagonal portion of *mat */
  PetscCall(MatSeqAIJGetArrayRead(B,&oa));
  PetscCall(MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,m,N,oi,oj,(PetscScalar*)oa,&Bnew));
  PetscCall(MatSeqAIJRestoreArrayRead(B,&oa));
  bnew        = (Mat_SeqAIJ*)Bnew->data;
  bnew->maxnz = b->maxnz; /* allocated nonzeros of B */
  maij->B     = Bnew;

  PetscCheck(B->rmap->N == Bnew->rmap->N,PETSC_COMM_SELF,PETSC_ERR_PLIB,"BN %" PetscInt_FMT " != BnewN %" PetscInt_FMT,B->rmap->N,Bnew->rmap->N);

  b->singlemalloc = PETSC_FALSE; /* B arrays are shared by Bnew */
  b->free_a       = PETSC_FALSE;
  b->free_ij      = PETSC_FALSE;
  PetscCall(MatDestroy(&B));

  bnew->singlemalloc = PETSC_TRUE; /* arrays will be freed by MatDestroy(&Bnew) */
  bnew->free_a       = PETSC_TRUE;
  bnew->free_ij      = PETSC_TRUE;

  /* condense columns of maij->B */
  PetscCall(MatSetOption(*mat,MAT_NO_OFF_PROC_ENTRIES,PETSC_TRUE));
  PetscCall(MatAssemblyBegin(*mat,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(*mat,MAT_FINAL_ASSEMBLY));
  PetscCall(MatSetOption(*mat,MAT_NO_OFF_PROC_ENTRIES,PETSC_FALSE));
  PetscCall(MatSetOption(*mat,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE));
  PetscFunctionReturn(0);
}

extern PetscErrorCode MatCreateSubMatrices_MPIAIJ_SingleIS_Local(Mat,PetscInt,const IS[],const IS[],MatReuse,PetscBool,Mat*);

PetscErrorCode MatCreateSubMatrix_MPIAIJ_SameRowDist(Mat mat,IS isrow,IS iscol,IS iscol_local,MatReuse call,Mat *newmat)
{
  PetscInt       i,m,n,rstart,row,rend,nz,j,bs,cbs;
  PetscInt       *ii,*jj,nlocal,*dlens,*olens,dlen,olen,jend,mglobal;
  Mat_MPIAIJ     *a=(Mat_MPIAIJ*)mat->data;
  Mat            M,Msub,B=a->B;
  MatScalar      *aa;
  Mat_SeqAIJ     *aij;
  PetscInt       *garray = a->garray,*colsub,Ncols;
  PetscInt       count,Bn=B->cmap->N,cstart=mat->cmap->rstart,cend=mat->cmap->rend;
  IS             iscol_sub,iscmap;
  const PetscInt *is_idx,*cmap;
  PetscBool      allcolumns=PETSC_FALSE;
  MPI_Comm       comm;

  PetscFunctionBegin;
  PetscCall(PetscObjectGetComm((PetscObject)mat,&comm));
  if (call == MAT_REUSE_MATRIX) {
    PetscCall(PetscObjectQuery((PetscObject)*newmat,"SubIScol",(PetscObject*)&iscol_sub));
    PetscCheck(iscol_sub,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"SubIScol passed in was not used before, cannot reuse");
    PetscCall(ISGetLocalSize(iscol_sub,&count));

    PetscCall(PetscObjectQuery((PetscObject)*newmat,"Subcmap",(PetscObject*)&iscmap));
    PetscCheck(iscmap,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"Subcmap passed in was not used before, cannot reuse");

    PetscCall(PetscObjectQuery((PetscObject)*newmat,"SubMatrix",(PetscObject*)&Msub));
    PetscCheck(Msub,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"Submatrix passed in was not used before, cannot reuse");

    PetscCall(MatCreateSubMatrices_MPIAIJ_SingleIS_Local(mat,1,&isrow,&iscol_sub,MAT_REUSE_MATRIX,PETSC_FALSE,&Msub));

  } else { /* call == MAT_INITIAL_MATRIX) */
    PetscBool flg;

    PetscCall(ISGetLocalSize(iscol,&n));
    PetscCall(ISGetSize(iscol,&Ncols));

    /* (1) iscol -> nonscalable iscol_local */
    /* Check for special case: each processor gets entire matrix columns */
    PetscCall(ISIdentity(iscol_local,&flg));
    if (flg && n == mat->cmap->N) allcolumns = PETSC_TRUE;
    PetscCall(MPIU_Allreduce(MPI_IN_PLACE,&allcolumns,1,MPIU_BOOL,MPI_LAND,PetscObjectComm((PetscObject)mat)));
    if (allcolumns) {
      iscol_sub = iscol_local;
      PetscCall(PetscObjectReference((PetscObject)iscol_local));
      PetscCall(ISCreateStride(PETSC_COMM_SELF,n,0,1,&iscmap));

    } else {
      /* (2) iscol_local -> iscol_sub and iscmap. Implementation below requires iscol_local be sorted, it can have duplicate indices */
      PetscInt *idx,*cmap1,k;
      PetscCall(PetscMalloc1(Ncols,&idx));
      PetscCall(PetscMalloc1(Ncols,&cmap1));
      PetscCall(ISGetIndices(iscol_local,&is_idx));
      count = 0;
      k     = 0;
      for (i=0; i<Ncols; i++) {
        j = is_idx[i];
        if (j >= cstart && j < cend) {
          /* diagonal part of mat */
          idx[count]     = j;
          cmap1[count++] = i; /* column index in submat */
        } else if (Bn) {
          /* off-diagonal part of mat */
          if (j == garray[k]) {
            idx[count]     = j;
            cmap1[count++] = i;  /* column index in submat */
          } else if (j > garray[k]) {
            while (j > garray[k] && k < Bn-1) k++;
            if (j == garray[k]) {
              idx[count]     = j;
              cmap1[count++] = i; /* column index in submat */
            }
          }
        }
      }
      PetscCall(ISRestoreIndices(iscol_local,&is_idx));

      PetscCall(ISCreateGeneral(PETSC_COMM_SELF,count,idx,PETSC_OWN_POINTER,&iscol_sub));
      PetscCall(ISGetBlockSize(iscol,&cbs));
      PetscCall(ISSetBlockSize(iscol_sub,cbs));

      PetscCall(ISCreateGeneral(PetscObjectComm((PetscObject)iscol_local),count,cmap1,PETSC_OWN_POINTER,&iscmap));
    }

    /* (3) Create sequential Msub */
    PetscCall(MatCreateSubMatrices_MPIAIJ_SingleIS_Local(mat,1,&isrow,&iscol_sub,MAT_INITIAL_MATRIX,allcolumns,&Msub));
  }

  PetscCall(ISGetLocalSize(iscol_sub,&count));
  aij  = (Mat_SeqAIJ*)(Msub)->data;
  ii   = aij->i;
  PetscCall(ISGetIndices(iscmap,&cmap));

  /*
      m - number of local rows
      Ncols - number of columns (same on all processors)
      rstart - first row in new global matrix generated
  */
  PetscCall(MatGetSize(Msub,&m,NULL));

  if (call == MAT_INITIAL_MATRIX) {
    /* (4) Create parallel newmat */
    PetscMPIInt    rank,size;
    PetscInt       csize;

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

    /*
        Determine the number of non-zeros in the diagonal and off-diagonal
        portions of the matrix in order to do correct preallocation
    */

    /* first get start and end of "diagonal" columns */
    PetscCall(ISGetLocalSize(iscol,&csize));
    if (csize == PETSC_DECIDE) {
      PetscCall(ISGetSize(isrow,&mglobal));
      if (mglobal == Ncols) { /* square matrix */
        nlocal = m;
      } else {
        nlocal = Ncols/size + ((Ncols % size) > rank);
      }
    } else {
      nlocal = csize;
    }
    PetscCallMPI(MPI_Scan(&nlocal,&rend,1,MPIU_INT,MPI_SUM,comm));
    rstart = rend - nlocal;
    PetscCheck(rank != size - 1 || rend == Ncols,PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Local column sizes %" PetscInt_FMT " do not add up to total number of columns %" PetscInt_FMT,rend,Ncols);

    /* next, compute all the lengths */
    jj    = aij->j;
    PetscCall(PetscMalloc1(2*m+1,&dlens));
    olens = dlens + m;
    for (i=0; i<m; i++) {
      jend = ii[i+1] - ii[i];
      olen = 0;
      dlen = 0;
      for (j=0; j<jend; j++) {
        if (cmap[*jj] < rstart || cmap[*jj] >= rend) olen++;
        else dlen++;
        jj++;
      }
      olens[i] = olen;
      dlens[i] = dlen;
    }

    PetscCall(ISGetBlockSize(isrow,&bs));
    PetscCall(ISGetBlockSize(iscol,&cbs));

    PetscCall(MatCreate(comm,&M));
    PetscCall(MatSetSizes(M,m,nlocal,PETSC_DECIDE,Ncols));
    PetscCall(MatSetBlockSizes(M,bs,cbs));
    PetscCall(MatSetType(M,((PetscObject)mat)->type_name));
    PetscCall(MatMPIAIJSetPreallocation(M,0,dlens,0,olens));
    PetscCall(PetscFree(dlens));

  } else { /* call == MAT_REUSE_MATRIX */
    M    = *newmat;
    PetscCall(MatGetLocalSize(M,&i,NULL));
    PetscCheck(i == m,PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Previous matrix must be same size/layout as request");
    PetscCall(MatZeroEntries(M));
    /*
         The next two lines are needed so we may call MatSetValues_MPIAIJ() below directly,
       rather than the slower MatSetValues().
    */
    M->was_assembled = PETSC_TRUE;
    M->assembled     = PETSC_FALSE;
  }

  /* (5) Set values of Msub to *newmat */
  PetscCall(PetscMalloc1(count,&colsub));
  PetscCall(MatGetOwnershipRange(M,&rstart,NULL));

  jj   = aij->j;
  PetscCall(MatSeqAIJGetArrayRead(Msub,(const PetscScalar**)&aa));
  for (i=0; i<m; i++) {
    row = rstart + i;
    nz  = ii[i+1] - ii[i];
    for (j=0; j<nz; j++) colsub[j] = cmap[jj[j]];
    PetscCall(MatSetValues_MPIAIJ(M,1,&row,nz,colsub,aa,INSERT_VALUES));
    jj += nz; aa += nz;
  }
  PetscCall(MatSeqAIJRestoreArrayRead(Msub,(const PetscScalar**)&aa));
  PetscCall(ISRestoreIndices(iscmap,&cmap));

  PetscCall(MatAssemblyBegin(M,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(M,MAT_FINAL_ASSEMBLY));

  PetscCall(PetscFree(colsub));

  /* save Msub, iscol_sub and iscmap used in processor for next request */
  if (call == MAT_INITIAL_MATRIX) {
    *newmat = M;
    PetscCall(PetscObjectCompose((PetscObject)(*newmat),"SubMatrix",(PetscObject)Msub));
    PetscCall(MatDestroy(&Msub));

    PetscCall(PetscObjectCompose((PetscObject)(*newmat),"SubIScol",(PetscObject)iscol_sub));
    PetscCall(ISDestroy(&iscol_sub));

    PetscCall(PetscObjectCompose((PetscObject)(*newmat),"Subcmap",(PetscObject)iscmap));
    PetscCall(ISDestroy(&iscmap));

    if (iscol_local) {
      PetscCall(PetscObjectCompose((PetscObject)(*newmat),"ISAllGather",(PetscObject)iscol_local));
      PetscCall(ISDestroy(&iscol_local));
    }
  }
  PetscFunctionReturn(0);
}

/*
    Not great since it makes two copies of the submatrix, first an SeqAIJ
  in local and then by concatenating the local matrices the end result.
  Writing it directly would be much like MatCreateSubMatrices_MPIAIJ()

  Note: This requires a sequential iscol with all indices.
*/
PetscErrorCode MatCreateSubMatrix_MPIAIJ_nonscalable(Mat mat,IS isrow,IS iscol,PetscInt csize,MatReuse call,Mat *newmat)
{
  PetscMPIInt    rank,size;
  PetscInt       i,m,n,rstart,row,rend,nz,*cwork,j,bs,cbs;
  PetscInt       *ii,*jj,nlocal,*dlens,*olens,dlen,olen,jend,mglobal;
  Mat            M,Mreuse;
  MatScalar      *aa,*vwork;
  MPI_Comm       comm;
  Mat_SeqAIJ     *aij;
  PetscBool      colflag,allcolumns=PETSC_FALSE;

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

  /* Check for special case: each processor gets entire matrix columns */
  PetscCall(ISIdentity(iscol,&colflag));
  PetscCall(ISGetLocalSize(iscol,&n));
  if (colflag && n == mat->cmap->N) allcolumns = PETSC_TRUE;
  PetscCall(MPIU_Allreduce(MPI_IN_PLACE,&allcolumns,1,MPIU_BOOL,MPI_LAND,PetscObjectComm((PetscObject)mat)));

  if (call ==  MAT_REUSE_MATRIX) {
    PetscCall(PetscObjectQuery((PetscObject)*newmat,"SubMatrix",(PetscObject*)&Mreuse));
    PetscCheck(Mreuse,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"Submatrix passed in was not used before, cannot reuse");
    PetscCall(MatCreateSubMatrices_MPIAIJ_SingleIS_Local(mat,1,&isrow,&iscol,MAT_REUSE_MATRIX,allcolumns,&Mreuse));
  } else {
    PetscCall(MatCreateSubMatrices_MPIAIJ_SingleIS_Local(mat,1,&isrow,&iscol,MAT_INITIAL_MATRIX,allcolumns,&Mreuse));
  }

  /*
      m - number of local rows
      n - number of columns (same on all processors)
      rstart - first row in new global matrix generated
  */
  PetscCall(MatGetSize(Mreuse,&m,&n));
  PetscCall(MatGetBlockSizes(Mreuse,&bs,&cbs));
  if (call == MAT_INITIAL_MATRIX) {
    aij = (Mat_SeqAIJ*)(Mreuse)->data;
    ii  = aij->i;
    jj  = aij->j;

    /*
        Determine the number of non-zeros in the diagonal and off-diagonal
        portions of the matrix in order to do correct preallocation
    */

    /* first get start and end of "diagonal" columns */
    if (csize == PETSC_DECIDE) {
      PetscCall(ISGetSize(isrow,&mglobal));
      if (mglobal == n) { /* square matrix */
        nlocal = m;
      } else {
        nlocal = n/size + ((n % size) > rank);
      }
    } else {
      nlocal = csize;
    }
    PetscCallMPI(MPI_Scan(&nlocal,&rend,1,MPIU_INT,MPI_SUM,comm));
    rstart = rend - nlocal;
    PetscCheck(rank != size - 1 || rend == n,PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Local column sizes %" PetscInt_FMT " do not add up to total number of columns %" PetscInt_FMT,rend,n);

    /* next, compute all the lengths */
    PetscCall(PetscMalloc1(2*m+1,&dlens));
    olens = dlens + m;
    for (i=0; i<m; i++) {
      jend = ii[i+1] - ii[i];
      olen = 0;
      dlen = 0;
      for (j=0; j<jend; j++) {
        if (*jj < rstart || *jj >= rend) olen++;
        else dlen++;
        jj++;
      }
      olens[i] = olen;
      dlens[i] = dlen;
    }
    PetscCall(MatCreate(comm,&M));
    PetscCall(MatSetSizes(M,m,nlocal,PETSC_DECIDE,n));
    PetscCall(MatSetBlockSizes(M,bs,cbs));
    PetscCall(MatSetType(M,((PetscObject)mat)->type_name));
    PetscCall(MatMPIAIJSetPreallocation(M,0,dlens,0,olens));
    PetscCall(PetscFree(dlens));
  } else {
    PetscInt ml,nl;

    M    = *newmat;
    PetscCall(MatGetLocalSize(M,&ml,&nl));
    PetscCheck(ml == m,PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Previous matrix must be same size/layout as request");
    PetscCall(MatZeroEntries(M));
    /*
         The next two lines are needed so we may call MatSetValues_MPIAIJ() below directly,
       rather than the slower MatSetValues().
    */
    M->was_assembled = PETSC_TRUE;
    M->assembled     = PETSC_FALSE;
  }
  PetscCall(MatGetOwnershipRange(M,&rstart,&rend));
  aij  = (Mat_SeqAIJ*)(Mreuse)->data;
  ii   = aij->i;
  jj   = aij->j;

  /* trigger copy to CPU if needed */
  PetscCall(MatSeqAIJGetArrayRead(Mreuse,(const PetscScalar**)&aa));
  for (i=0; i<m; i++) {
    row   = rstart + i;
    nz    = ii[i+1] - ii[i];
    cwork = jj; jj += nz;
    vwork = aa; aa += nz;
    PetscCall(MatSetValues_MPIAIJ(M,1,&row,nz,cwork,vwork,INSERT_VALUES));
  }
  PetscCall(MatSeqAIJRestoreArrayRead(Mreuse,(const PetscScalar**)&aa));

  PetscCall(MatAssemblyBegin(M,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(M,MAT_FINAL_ASSEMBLY));
  *newmat = M;

  /* save submatrix used in processor for next request */
  if (call ==  MAT_INITIAL_MATRIX) {
    PetscCall(PetscObjectCompose((PetscObject)M,"SubMatrix",(PetscObject)Mreuse));
    PetscCall(MatDestroy(&Mreuse));
  }
  PetscFunctionReturn(0);
}

PetscErrorCode MatMPIAIJSetPreallocationCSR_MPIAIJ(Mat B,const PetscInt Ii[],const PetscInt J[],const PetscScalar v[])
{
  PetscInt       m,cstart, cend,j,nnz,i,d,*ld;
  PetscInt       *d_nnz,*o_nnz,nnz_max = 0,rstart,ii;
  const PetscInt *JJ;
  PetscBool      nooffprocentries;
  Mat_MPIAIJ     *Aij = (Mat_MPIAIJ*)B->data;

  PetscFunctionBegin;
  PetscCheck(Ii[0] == 0,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Ii[0] must be 0 it is %" PetscInt_FMT,Ii[0]);

  PetscCall(PetscLayoutSetUp(B->rmap));
  PetscCall(PetscLayoutSetUp(B->cmap));
  m      = B->rmap->n;
  cstart = B->cmap->rstart;
  cend   = B->cmap->rend;
  rstart = B->rmap->rstart;

  PetscCall(PetscCalloc2(m,&d_nnz,m,&o_nnz));

  if (PetscDefined(USE_DEBUG)) {
    for (i=0; i<m; i++) {
      nnz = Ii[i+1]- Ii[i];
      JJ  = J + Ii[i];
      PetscCheck(nnz >= 0,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Local row %" PetscInt_FMT " has a negative %" PetscInt_FMT " number of columns",i,nnz);
      PetscCheck(!nnz || !(JJ[0] < 0),PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"Row %" PetscInt_FMT " starts with negative column index %" PetscInt_FMT,i,JJ[0]);
      PetscCheck(!nnz || !(JJ[nnz-1] >= B->cmap->N),PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"Row %" PetscInt_FMT " ends with too large a column index %" PetscInt_FMT " (max allowed %" PetscInt_FMT ")",i,JJ[nnz-1],B->cmap->N);
    }
  }

  for (i=0; i<m; i++) {
    nnz     = Ii[i+1]- Ii[i];
    JJ      = J + Ii[i];
    nnz_max = PetscMax(nnz_max,nnz);
    d       = 0;
    for (j=0; j<nnz; j++) {
      if (cstart <= JJ[j] && JJ[j] < cend) d++;
    }
    d_nnz[i] = d;
    o_nnz[i] = nnz - d;
  }
  PetscCall(MatMPIAIJSetPreallocation(B,0,d_nnz,0,o_nnz));
  PetscCall(PetscFree2(d_nnz,o_nnz));

  for (i=0; i<m; i++) {
    ii   = i + rstart;
    PetscCall(MatSetValues_MPIAIJ(B,1,&ii,Ii[i+1] - Ii[i],J+Ii[i], v ? v + Ii[i] : NULL,INSERT_VALUES));
  }
  nooffprocentries    = B->nooffprocentries;
  B->nooffprocentries = PETSC_TRUE;
  PetscCall(MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY));
  B->nooffprocentries = nooffprocentries;

  /* count number of entries below block diagonal */
  PetscCall(PetscFree(Aij->ld));
  PetscCall(PetscCalloc1(m,&ld));
  Aij->ld = ld;
  for (i=0; i<m; i++) {
    nnz  = Ii[i+1] - Ii[i];
    j     = 0;
    while  (j < nnz && J[j] < cstart) {j++;}
    ld[i] = j;
    J     += nnz;
  }

  PetscCall(MatSetOption(B,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE));
  PetscFunctionReturn(0);
}

/*@
   MatMPIAIJSetPreallocationCSR - Allocates memory for a sparse parallel matrix in AIJ format
   (the default parallel PETSc format).

   Collective

   Input Parameters:
+  B - the matrix
.  i - the indices into j for the start of each local row (starts with zero)
.  j - the column indices for each local row (starts with zero)
-  v - optional values in the matrix

   Level: developer

   Notes:
       The i, j, and v arrays ARE copied by this routine into the internal format used by PETSc;
     thus you CANNOT change the matrix entries by changing the values of v[] after you have
     called this routine. Use MatCreateMPIAIJWithSplitArrays() to avoid needing to copy the arrays.

       The i and j indices are 0 based, and i indices are indices corresponding to the local j array.

       The format which is used for the sparse matrix input, is equivalent to a
    row-major ordering.. i.e for the following matrix, the input data expected is
    as shown

$        1 0 0
$        2 0 3     P0
$       -------
$        4 5 6     P1
$
$     Process0 [P0]: rows_owned=[0,1]
$        i =  {0,1,3}  [size = nrow+1  = 2+1]
$        j =  {0,0,2}  [size = 3]
$        v =  {1,2,3}  [size = 3]
$
$     Process1 [P1]: rows_owned=[2]
$        i =  {0,3}    [size = nrow+1  = 1+1]
$        j =  {0,1,2}  [size = 3]
$        v =  {4,5,6}  [size = 3]

.seealso: `MatCreate()`, `MatCreateSeqAIJ()`, `MatSetValues()`, `MatMPIAIJSetPreallocation()`, `MatCreateAIJ()`, `MATMPIAIJ`,
          `MatCreateSeqAIJWithArrays()`, `MatCreateMPIAIJWithSplitArrays()`
@*/
PetscErrorCode  MatMPIAIJSetPreallocationCSR(Mat B,const PetscInt i[],const PetscInt j[], const PetscScalar v[])
{
  PetscFunctionBegin;
  PetscTryMethod(B,"MatMPIAIJSetPreallocationCSR_C",(Mat,const PetscInt[],const PetscInt[],const PetscScalar[]),(B,i,j,v));
  PetscFunctionReturn(0);
}

/*@C
   MatMPIAIJSetPreallocation - Preallocates memory for a sparse parallel matrix in AIJ format
   (the default parallel PETSc format).  For good matrix assembly performance
   the user should preallocate the matrix storage by setting the parameters
   d_nz (or d_nnz) and o_nz (or o_nnz).  By setting these parameters accurately,
   performance can be increased by more than a factor of 50.

   Collective

   Input Parameters:
+  B - the matrix
.  d_nz  - number of nonzeros per row in DIAGONAL portion of local submatrix
           (same value is used for all local rows)
.  d_nnz - array containing the number of nonzeros in the various rows of the
           DIAGONAL portion of the local submatrix (possibly different for each row)
           or NULL (PETSC_NULL_INTEGER in Fortran), if d_nz is used to specify the nonzero structure.
           The size of this array is equal to the number of local rows, i.e 'm'.
           For matrices that will be factored, you must leave room for (and set)
           the diagonal entry even if it is zero.
.  o_nz  - number of nonzeros per row in the OFF-DIAGONAL portion of local
           submatrix (same value is used for all local rows).
-  o_nnz - array containing the number of nonzeros in the various rows of the
           OFF-DIAGONAL portion of the local submatrix (possibly different for
           each row) or NULL (PETSC_NULL_INTEGER in Fortran), if o_nz is used to specify the nonzero
           structure. The size of this array is equal to the number
           of local rows, i.e 'm'.

   If the *_nnz parameter is given then the *_nz parameter is ignored

   The AIJ format (also called the Yale sparse matrix format or
   compressed row storage (CSR)), is fully compatible with standard Fortran 77
   storage.  The stored row and column indices begin with zero.
   See Users-Manual: ch_mat for details.

   The parallel matrix is partitioned such that the first m0 rows belong to
   process 0, the next m1 rows belong to process 1, the next m2 rows belong
   to process 2 etc.. where m0,m1,m2... are the input parameter 'm'.

   The DIAGONAL portion of the local submatrix of a processor can be defined
   as the submatrix which is obtained by extraction the part corresponding to
   the rows r1-r2 and columns c1-c2 of the global matrix, where r1 is the
   first row that belongs to the processor, r2 is the last row belonging to
   the this processor, and c1-c2 is range of indices of the local part of a
   vector suitable for applying the matrix to.  This is an mxn matrix.  In the
   common case of a square matrix, the row and column ranges are the same and
   the DIAGONAL part is also square. The remaining portion of the local
   submatrix (mxN) constitute the OFF-DIAGONAL portion.

   If o_nnz, d_nnz are specified, then o_nz, and d_nz are ignored.

   You can call MatGetInfo() to get information on how effective the preallocation was;
   for example the fields mallocs,nz_allocated,nz_used,nz_unneeded;
   You can also run with the option -info and look for messages with the string
   malloc in them to see if additional memory allocation was needed.

   Example usage:

   Consider the following 8x8 matrix with 34 non-zero values, that is
   assembled across 3 processors. Lets assume that proc0 owns 3 rows,
   proc1 owns 3 rows, proc2 owns 2 rows. This division can be shown
   as follows:

.vb
            1  2  0  |  0  3  0  |  0  4
    Proc0   0  5  6  |  7  0  0  |  8  0
            9  0 10  | 11  0  0  | 12  0
    -------------------------------------
           13  0 14  | 15 16 17  |  0  0
    Proc1   0 18  0  | 19 20 21  |  0  0
            0  0  0  | 22 23  0  | 24  0
    -------------------------------------
    Proc2  25 26 27  |  0  0 28  | 29  0
           30  0  0  | 31 32 33  |  0 34
.ve

   This can be represented as a collection of submatrices as:

.vb
      A B C
      D E F
      G H I
.ve

   Where the submatrices A,B,C are owned by proc0, D,E,F are
   owned by proc1, G,H,I are owned by proc2.

   The 'm' parameters for proc0,proc1,proc2 are 3,3,2 respectively.
   The 'n' parameters for proc0,proc1,proc2 are 3,3,2 respectively.
   The 'M','N' parameters are 8,8, and have the same values on all procs.

   The DIAGONAL submatrices corresponding to proc0,proc1,proc2 are
   submatrices [A], [E], [I] respectively. The OFF-DIAGONAL submatrices
   corresponding to proc0,proc1,proc2 are [BC], [DF], [GH] respectively.
   Internally, each processor stores the DIAGONAL part, and the OFF-DIAGONAL
   part as SeqAIJ matrices. for eg: proc1 will store [E] as a SeqAIJ
   matrix, ans [DF] as another SeqAIJ matrix.

   When d_nz, o_nz parameters are specified, d_nz storage elements are
   allocated for every row of the local diagonal submatrix, and o_nz
   storage locations are allocated for every row of the OFF-DIAGONAL submat.
   One way to choose d_nz and o_nz is to use the max nonzerors per local
   rows for each of the local DIAGONAL, and the OFF-DIAGONAL submatrices.
   In this case, the values of d_nz,o_nz are:
.vb
     proc0 : dnz = 2, o_nz = 2
     proc1 : dnz = 3, o_nz = 2
     proc2 : dnz = 1, o_nz = 4
.ve
   We are allocating m*(d_nz+o_nz) storage locations for every proc. This
   translates to 3*(2+2)=12 for proc0, 3*(3+2)=15 for proc1, 2*(1+4)=10
   for proc3. i.e we are using 12+15+10=37 storage locations to store
   34 values.

   When d_nnz, o_nnz parameters are specified, the storage is specified
   for every row, corresponding to both DIAGONAL and OFF-DIAGONAL submatrices.
   In the above case the values for d_nnz,o_nnz are:
.vb
     proc0: d_nnz = [2,2,2] and o_nnz = [2,2,2]
     proc1: d_nnz = [3,3,2] and o_nnz = [2,1,1]
     proc2: d_nnz = [1,1]   and o_nnz = [4,4]
.ve
   Here the space allocated is sum of all the above values i.e 34, and
   hence pre-allocation is perfect.

   Level: intermediate

.seealso: `MatCreate()`, `MatCreateSeqAIJ()`, `MatSetValues()`, `MatCreateAIJ()`, `MatMPIAIJSetPreallocationCSR()`,
          `MATMPIAIJ`, `MatGetInfo()`, `PetscSplitOwnership()`
@*/
PetscErrorCode MatMPIAIJSetPreallocation(Mat B,PetscInt d_nz,const PetscInt d_nnz[],PetscInt o_nz,const PetscInt o_nnz[])
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(B,MAT_CLASSID,1);
  PetscValidType(B,1);
  PetscTryMethod(B,"MatMPIAIJSetPreallocation_C",(Mat,PetscInt,const PetscInt[],PetscInt,const PetscInt[]),(B,d_nz,d_nnz,o_nz,o_nnz));
  PetscFunctionReturn(0);
}

/*@
     MatCreateMPIAIJWithArrays - creates a MPI AIJ matrix using arrays that contain in standard
         CSR format for the local rows.

   Collective

   Input Parameters:
+  comm - MPI communicator
.  m - number of local rows (Cannot be PETSC_DECIDE)
.  n - This value should be the same as the local size used in creating the
       x vector for the matrix-vector product y = Ax. (or PETSC_DECIDE to have
       calculated if N is given) For square matrices n is almost always m.
.  M - number of global rows (or PETSC_DETERMINE to have calculated if m is given)
.  N - number of global columns (or PETSC_DETERMINE to have calculated if n is given)
.   i - row indices; that is i[0] = 0, i[row] = i[row-1] + number of elements in that row of the matrix
.   j - column indices
-   a - optional matrix values

   Output Parameter:
.   mat - the matrix

   Level: intermediate

   Notes:
       The i, j, and a arrays ARE copied by this routine into the internal format used by PETSc;
     thus you CANNOT change the matrix entries by changing the values of a[] after you have
     called this routine. Use MatCreateMPIAIJWithSplitArrays() to avoid needing to copy the arrays.

       The i and j indices are 0 based, and i indices are indices corresponding to the local j array.

       The format which is used for the sparse matrix input, is equivalent to a
    row-major ordering.. i.e for the following matrix, the input data expected is
    as shown

       Once you have created the matrix you can update it with new numerical values using MatUpdateMPIAIJWithArrays

$        1 0 0
$        2 0 3     P0
$       -------
$        4 5 6     P1
$
$     Process0 [P0]: rows_owned=[0,1]
$        i =  {0,1,3}  [size = nrow+1  = 2+1]
$        j =  {0,0,2}  [size = 3]
$        v =  {1,2,3}  [size = 3]
$
$     Process1 [P1]: rows_owned=[2]
$        i =  {0,3}    [size = nrow+1  = 1+1]
$        j =  {0,1,2}  [size = 3]
$        v =  {4,5,6}  [size = 3]

.seealso: `MatCreate()`, `MatCreateSeqAIJ()`, `MatSetValues()`, `MatMPIAIJSetPreallocation()`, `MatMPIAIJSetPreallocationCSR()`,
          `MATMPIAIJ`, `MatCreateAIJ()`, `MatCreateMPIAIJWithSplitArrays()`, `MatUpdateMPIAIJWithArrays()`
@*/
PetscErrorCode MatCreateMPIAIJWithArrays(MPI_Comm comm,PetscInt m,PetscInt n,PetscInt M,PetscInt N,const PetscInt i[],const PetscInt j[],const PetscScalar a[],Mat *mat)
{
  PetscFunctionBegin;
  PetscCheck(!i || !i[0],PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"i (row indices) must start with 0");
  PetscCheck(m >= 0,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"local number of rows (m) cannot be PETSC_DECIDE, or negative");
  PetscCall(MatCreate(comm,mat));
  PetscCall(MatSetSizes(*mat,m,n,M,N));
  /* PetscCall(MatSetBlockSizes(M,bs,cbs)); */
  PetscCall(MatSetType(*mat,MATMPIAIJ));
  PetscCall(MatMPIAIJSetPreallocationCSR(*mat,i,j,a));
  PetscFunctionReturn(0);
}

/*@
     MatUpdateMPIAIJWithArrays - updates a MPI AIJ matrix using arrays that contain in standard
         CSR format for the local rows. Only the numerical values are updated the other arrays must be identical to what was passed from MatCreateMPIAIJWithArrays()

     Deprecated: Use `MatUpdateMPIAIJWithArray()`

   Collective

   Input Parameters:
+  mat - the matrix
.  m - number of local rows (Cannot be PETSC_DECIDE)
.  n - This value should be the same as the local size used in creating the
       x vector for the matrix-vector product y = Ax. (or PETSC_DECIDE to have
       calculated if N is given) For square matrices n is almost always m.
.  M - number of global rows (or PETSC_DETERMINE to have calculated if m is given)
.  N - number of global columns (or PETSC_DETERMINE to have calculated if n is given)
.  Ii - row indices; that is Ii[0] = 0, Ii[row] = Ii[row-1] + number of elements in that row of the matrix
.  J - column indices
-  v - matrix values

   Level: intermediate

.seealso: `MatCreate()`, `MatCreateSeqAIJ()`, `MatSetValues()`, `MatMPIAIJSetPreallocation()`, `MatMPIAIJSetPreallocationCSR()`,
          `MATMPIAIJ`, `MatCreateAIJ()`, `MatCreateMPIAIJWithSplitArrays()`, `MatUpdateMPIAIJWithArrays()`, `MatUpdateMPIAIJWithArray()`
@*/
PetscErrorCode MatUpdateMPIAIJWithArrays(Mat mat,PetscInt m,PetscInt n,PetscInt M,PetscInt N,const PetscInt Ii[],const PetscInt J[],const PetscScalar v[])
{
  PetscInt       nnz,i;
  PetscBool      nooffprocentries;
  Mat_MPIAIJ     *Aij = (Mat_MPIAIJ*)mat->data;
  Mat_SeqAIJ     *Ad  = (Mat_SeqAIJ*)Aij->A->data;
  PetscScalar    *ad,*ao;
  PetscInt       ldi,Iii,md;
  const PetscInt *Adi = Ad->i;
  PetscInt       *ld = Aij->ld;

  PetscFunctionBegin;
  PetscCheck(Ii[0] == 0,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"i (row indices) must start with 0");
  PetscCheck(m >= 0,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"local number of rows (m) cannot be PETSC_DECIDE, or negative");
  PetscCheck(m == mat->rmap->n,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Local number of rows cannot change from call to MatUpdateMPIAIJWithArrays()");
  PetscCheck(n == mat->cmap->n,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Local number of columns cannot change from call to MatUpdateMPIAIJWithArrays()");

  PetscCall(MatSeqAIJGetArrayWrite(Aij->A,&ad));
  PetscCall(MatSeqAIJGetArrayWrite(Aij->B,&ao));

  for (i=0; i<m; i++) {
    nnz  = Ii[i+1]- Ii[i];
    Iii  = Ii[i];
    ldi  = ld[i];
    md   = Adi[i+1]-Adi[i];
    PetscCall(PetscArraycpy(ao,v + Iii,ldi));
    PetscCall(PetscArraycpy(ad,v + Iii + ldi,md));
    PetscCall(PetscArraycpy(ao + ldi,v + Iii + ldi + md,nnz - ldi - md));
    ad  += md;
    ao  += nnz - md;
  }
  nooffprocentries      = mat->nooffprocentries;
  mat->nooffprocentries = PETSC_TRUE;
  PetscCall(MatSeqAIJRestoreArrayWrite(Aij->A,&ad));
  PetscCall(MatSeqAIJRestoreArrayWrite(Aij->B,&ao));
  PetscCall(PetscObjectStateIncrease((PetscObject)Aij->A));
  PetscCall(PetscObjectStateIncrease((PetscObject)Aij->B));
  PetscCall(PetscObjectStateIncrease((PetscObject)mat));
  PetscCall(MatAssemblyBegin(mat,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(mat,MAT_FINAL_ASSEMBLY));
  mat->nooffprocentries = nooffprocentries;
  PetscFunctionReturn(0);
}

/*@
     MatUpdateMPIAIJWithArray - updates an MPI AIJ matrix using an array that contains the nonzero values

   Collective

   Input Parameters:
+  mat - the matrix
-  v - matrix values, stored by row

   Level: intermediate

   Notes:
   The matrix must have been obtained with `MatCreateMPIAIJWithArrays()` or `MatMPIAIJSetPreallocationCSR()`

.seealso: `MatCreate()`, `MatCreateSeqAIJ()`, `MatSetValues()`, `MatMPIAIJSetPreallocation()`, `MatMPIAIJSetPreallocationCSR()`,
          `MATMPIAIJ`, `MatCreateAIJ()`, `MatCreateMPIAIJWithSplitArrays()`, `MatUpdateMPIAIJWithArrays()`, `MatUpdateMPIAIJWithArrays()`
@*/
PetscErrorCode MatUpdateMPIAIJWithArray(Mat mat,const PetscScalar v[])
{
  PetscInt       nnz,i,m;
  PetscBool      nooffprocentries;
  Mat_MPIAIJ     *Aij = (Mat_MPIAIJ*)mat->data;
  Mat_SeqAIJ     *Ad  = (Mat_SeqAIJ*)Aij->A->data;
  Mat_SeqAIJ     *Ao  = (Mat_SeqAIJ*)Aij->B->data;
  PetscScalar    *ad,*ao;
  const PetscInt *Adi = Ad->i,*Adj = Ao->i;
  PetscInt       ldi,Iii,md;
  PetscInt       *ld = Aij->ld;

  PetscFunctionBegin;
  m = mat->rmap->n;

  PetscCall(MatSeqAIJGetArrayWrite(Aij->A,&ad));
  PetscCall(MatSeqAIJGetArrayWrite(Aij->B,&ao));
  Iii = 0;
  for (i=0; i<m; i++) {
    nnz  = Adi[i+1]-Adi[i] + Adj[i+1]-Adj[i];
    ldi  = ld[i];
    md   = Adi[i+1]-Adi[i];
    PetscCall(PetscArraycpy(ao,v + Iii,ldi));
    PetscCall(PetscArraycpy(ad,v + Iii + ldi,md));
    PetscCall(PetscArraycpy(ao + ldi,v + Iii + ldi + md,nnz - ldi - md));
    ad  += md;
    ao  += nnz - md;
    Iii += nnz;
  }
  nooffprocentries      = mat->nooffprocentries;
  mat->nooffprocentries = PETSC_TRUE;
  PetscCall(MatSeqAIJRestoreArrayWrite(Aij->A,&ad));
  PetscCall(MatSeqAIJRestoreArrayWrite(Aij->B,&ao));
  PetscCall(PetscObjectStateIncrease((PetscObject)Aij->A));
  PetscCall(PetscObjectStateIncrease((PetscObject)Aij->B));
  PetscCall(PetscObjectStateIncrease((PetscObject)mat));
  PetscCall(MatAssemblyBegin(mat,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(mat,MAT_FINAL_ASSEMBLY));
  mat->nooffprocentries = nooffprocentries;
  PetscFunctionReturn(0);
}

/*@C
   MatCreateAIJ - Creates a sparse parallel matrix in AIJ format
   (the default parallel PETSc format).  For good matrix assembly performance
   the user should preallocate the matrix storage by setting the parameters
   d_nz (or d_nnz) and o_nz (or o_nnz).  By setting these parameters accurately,
   performance can be increased by more than a factor of 50.

   Collective

   Input Parameters:
+  comm - MPI communicator
.  m - number of local rows (or PETSC_DECIDE to have calculated if M is given)
           This value should be the same as the local size used in creating the
           y vector for the matrix-vector product y = Ax.
.  n - This value should be the same as the local size used in creating the
       x vector for the matrix-vector product y = Ax. (or PETSC_DECIDE to have
       calculated if N is given) For square matrices n is almost always m.
.  M - number of global rows (or PETSC_DETERMINE to have calculated if m is given)
.  N - number of global columns (or PETSC_DETERMINE to have calculated if n is given)
.  d_nz  - number of nonzeros per row in DIAGONAL portion of local submatrix
           (same value is used for all local rows)
.  d_nnz - array containing the number of nonzeros in the various rows of the
           DIAGONAL portion of the local submatrix (possibly different for each row)
           or NULL, if d_nz is used to specify the nonzero structure.
           The size of this array is equal to the number of local rows, i.e 'm'.
.  o_nz  - number of nonzeros per row in the OFF-DIAGONAL portion of local
           submatrix (same value is used for all local rows).
-  o_nnz - array containing the number of nonzeros in the various rows of the
           OFF-DIAGONAL portion of the local submatrix (possibly different for
           each row) or NULL, if o_nz is used to specify the nonzero
           structure. The size of this array is equal to the number
           of local rows, i.e 'm'.

   Output Parameter:
.  A - the matrix

   It is recommended that one use the MatCreate(), MatSetType() and/or MatSetFromOptions(),
   MatXXXXSetPreallocation() paradigm instead of this routine directly.
   [MatXXXXSetPreallocation() is, for example, MatSeqAIJSetPreallocation]

   Notes:
   If the *_nnz parameter is given then the *_nz parameter is ignored

   m,n,M,N parameters specify the size of the matrix, and its partitioning across
   processors, while d_nz,d_nnz,o_nz,o_nnz parameters specify the approximate
   storage requirements for this matrix.

   If PETSC_DECIDE or  PETSC_DETERMINE is used for a particular argument on one
   processor than it must be used on all processors that share the object for
   that argument.

   The user MUST specify either the local or global matrix dimensions
   (possibly both).

   The parallel matrix is partitioned across processors such that the
   first m0 rows belong to process 0, the next m1 rows belong to
   process 1, the next m2 rows belong to process 2 etc.. where
   m0,m1,m2,.. are the input parameter 'm'. i.e each processor stores
   values corresponding to [m x N] submatrix.

   The columns are logically partitioned with the n0 columns belonging
   to 0th partition, the next n1 columns belonging to the next
   partition etc.. where n0,n1,n2... are the input parameter 'n'.

   The DIAGONAL portion of the local submatrix on any given processor
   is the submatrix corresponding to the rows and columns m,n
   corresponding to the given processor. i.e diagonal matrix on
   process 0 is [m0 x n0], diagonal matrix on process 1 is [m1 x n1]
   etc. The remaining portion of the local submatrix [m x (N-n)]
   constitute the OFF-DIAGONAL portion. The example below better
   illustrates this concept.

   For a square global matrix we define each processor's diagonal portion
   to be its local rows and the corresponding columns (a square submatrix);
   each processor's off-diagonal portion encompasses the remainder of the
   local matrix (a rectangular submatrix).

   If o_nnz, d_nnz are specified, then o_nz, and d_nz are ignored.

   When calling this routine with a single process communicator, a matrix of
   type SEQAIJ is returned.  If a matrix of type MPIAIJ is desired for this
   type of communicator, use the construction mechanism
.vb
     MatCreate(...,&A); MatSetType(A,MATMPIAIJ); MatSetSizes(A, m,n,M,N); MatMPIAIJSetPreallocation(A,...);
.ve

$     MatCreate(...,&A);
$     MatSetType(A,MATMPIAIJ);
$     MatSetSizes(A, m,n,M,N);
$     MatMPIAIJSetPreallocation(A,...);

   By default, this format uses inodes (identical nodes) when possible.
   We search for consecutive rows with the same nonzero structure, thereby
   reusing matrix information to achieve increased efficiency.

   Options Database Keys:
+  -mat_no_inode  - Do not use inodes
.  -mat_inode_limit <limit> - Sets inode limit (max limit=5)
-  -matmult_vecscatter_view <viewer> - View the vecscatter (i.e., communication pattern) used in MatMult() of sparse parallel matrices.
        See viewer types in manual of MatView(). Of them, ascii_matlab, draw or binary cause the vecscatter be viewed as a matrix.
        Entry (i,j) is the size of message (in bytes) rank i sends to rank j in one MatMult() call.

   Example usage:

   Consider the following 8x8 matrix with 34 non-zero values, that is
   assembled across 3 processors. Lets assume that proc0 owns 3 rows,
   proc1 owns 3 rows, proc2 owns 2 rows. This division can be shown
   as follows

.vb
            1  2  0  |  0  3  0  |  0  4
    Proc0   0  5  6  |  7  0  0  |  8  0
            9  0 10  | 11  0  0  | 12  0
    -------------------------------------
           13  0 14  | 15 16 17  |  0  0
    Proc1   0 18  0  | 19 20 21  |  0  0
            0  0  0  | 22 23  0  | 24  0
    -------------------------------------
    Proc2  25 26 27  |  0  0 28  | 29  0
           30  0  0  | 31 32 33  |  0 34
.ve

   This can be represented as a collection of submatrices as

.vb
      A B C
      D E F
      G H I
.ve

   Where the submatrices A,B,C are owned by proc0, D,E,F are
   owned by proc1, G,H,I are owned by proc2.

   The 'm' parameters for proc0,proc1,proc2 are 3,3,2 respectively.
   The 'n' parameters for proc0,proc1,proc2 are 3,3,2 respectively.
   The 'M','N' parameters are 8,8, and have the same values on all procs.

   The DIAGONAL submatrices corresponding to proc0,proc1,proc2 are
   submatrices [A], [E], [I] respectively. The OFF-DIAGONAL submatrices
   corresponding to proc0,proc1,proc2 are [BC], [DF], [GH] respectively.
   Internally, each processor stores the DIAGONAL part, and the OFF-DIAGONAL
   part as SeqAIJ matrices. for eg: proc1 will store [E] as a SeqAIJ
   matrix, ans [DF] as another SeqAIJ matrix.

   When d_nz, o_nz parameters are specified, d_nz storage elements are
   allocated for every row of the local diagonal submatrix, and o_nz
   storage locations are allocated for every row of the OFF-DIAGONAL submat.
   One way to choose d_nz and o_nz is to use the max nonzerors per local
   rows for each of the local DIAGONAL, and the OFF-DIAGONAL submatrices.
   In this case, the values of d_nz,o_nz are
.vb
     proc0 : dnz = 2, o_nz = 2
     proc1 : dnz = 3, o_nz = 2
     proc2 : dnz = 1, o_nz = 4
.ve
   We are allocating m*(d_nz+o_nz) storage locations for every proc. This
   translates to 3*(2+2)=12 for proc0, 3*(3+2)=15 for proc1, 2*(1+4)=10
   for proc3. i.e we are using 12+15+10=37 storage locations to store
   34 values.

   When d_nnz, o_nnz parameters are specified, the storage is specified
   for every row, corresponding to both DIAGONAL and OFF-DIAGONAL submatrices.
   In the above case the values for d_nnz,o_nnz are
.vb
     proc0: d_nnz = [2,2,2] and o_nnz = [2,2,2]
     proc1: d_nnz = [3,3,2] and o_nnz = [2,1,1]
     proc2: d_nnz = [1,1]   and o_nnz = [4,4]
.ve
   Here the space allocated is sum of all the above values i.e 34, and
   hence pre-allocation is perfect.

   Level: intermediate

.seealso: `MatCreate()`, `MatCreateSeqAIJ()`, `MatSetValues()`, `MatMPIAIJSetPreallocation()`, `MatMPIAIJSetPreallocationCSR()`,
          `MATMPIAIJ`, `MatCreateMPIAIJWithArrays()`
@*/
PetscErrorCode  MatCreateAIJ(MPI_Comm comm,PetscInt m,PetscInt n,PetscInt M,PetscInt N,PetscInt d_nz,const PetscInt d_nnz[],PetscInt o_nz,const PetscInt o_nnz[],Mat *A)
{
  PetscMPIInt    size;

  PetscFunctionBegin;
  PetscCall(MatCreate(comm,A));
  PetscCall(MatSetSizes(*A,m,n,M,N));
  PetscCallMPI(MPI_Comm_size(comm,&size));
  if (size > 1) {
    PetscCall(MatSetType(*A,MATMPIAIJ));
    PetscCall(MatMPIAIJSetPreallocation(*A,d_nz,d_nnz,o_nz,o_nnz));
  } else {
    PetscCall(MatSetType(*A,MATSEQAIJ));
    PetscCall(MatSeqAIJSetPreallocation(*A,d_nz,d_nnz));
  }
  PetscFunctionReturn(0);
}

/*@C
  MatMPIAIJGetSeqAIJ - Returns the local piece of this distributed matrix

  Not collective

  Input Parameter:
. A - The MPIAIJ matrix

  Output Parameters:
+ Ad - The local diagonal block as a SeqAIJ matrix
. Ao - The local off-diagonal block as a SeqAIJ matrix
- colmap - An array mapping local column numbers of Ao to global column numbers of the parallel matrix

  Note: The rows in Ad and Ao are in [0, Nr), where Nr is the number of local rows on this process. The columns
  in Ad are in [0, Nc) where Nc is the number of local columns. The columns are Ao are in [0, Nco), where Nco is
  the number of nonzero columns in the local off-diagonal piece of the matrix A. The array colmap maps these
  local column numbers to global column numbers in the original matrix.

  Level: intermediate

.seealso: `MatMPIAIJGetLocalMat()`, `MatMPIAIJGetLocalMatCondensed()`, `MatCreateAIJ()`, `MATMPIAIJ`, `MATSEQAIJ`
@*/
PetscErrorCode MatMPIAIJGetSeqAIJ(Mat A,Mat *Ad,Mat *Ao,const PetscInt *colmap[])
{
  Mat_MPIAIJ     *a = (Mat_MPIAIJ*)A->data;
  PetscBool      flg;

  PetscFunctionBegin;
  PetscCall(PetscStrbeginswith(((PetscObject)A)->type_name,MATMPIAIJ,&flg));
  PetscCheck(flg,PetscObjectComm((PetscObject)A),PETSC_ERR_SUP,"This function requires a MATMPIAIJ matrix as input");
  if (Ad)     *Ad     = a->A;
  if (Ao)     *Ao     = a->B;
  if (colmap) *colmap = a->garray;
  PetscFunctionReturn(0);
}

PetscErrorCode MatCreateMPIMatConcatenateSeqMat_MPIAIJ(MPI_Comm comm,Mat inmat,PetscInt n,MatReuse scall,Mat *outmat)
{
  PetscInt       m,N,i,rstart,nnz,Ii;
  PetscInt       *indx;
  PetscScalar    *values;
  MatType        rootType;

  PetscFunctionBegin;
  PetscCall(MatGetSize(inmat,&m,&N));
  if (scall == MAT_INITIAL_MATRIX) { /* symbolic phase */
    PetscInt       *dnz,*onz,sum,bs,cbs;

    if (n == PETSC_DECIDE) {
      PetscCall(PetscSplitOwnership(comm,&n,&N));
    }
    /* Check sum(n) = N */
    PetscCall(MPIU_Allreduce(&n,&sum,1,MPIU_INT,MPI_SUM,comm));
    PetscCheck(sum == N,PETSC_COMM_SELF,PETSC_ERR_ARG_INCOMP,"Sum of local columns %" PetscInt_FMT " != global columns %" PetscInt_FMT,sum,N);

    PetscCallMPI(MPI_Scan(&m, &rstart,1,MPIU_INT,MPI_SUM,comm));
    rstart -= m;

    MatPreallocateBegin(comm,m,n,dnz,onz);
    for (i=0; i<m; i++) {
      PetscCall(MatGetRow_SeqAIJ(inmat,i,&nnz,&indx,NULL));
      PetscCall(MatPreallocateSet(i+rstart,nnz,indx,dnz,onz));
      PetscCall(MatRestoreRow_SeqAIJ(inmat,i,&nnz,&indx,NULL));
    }

    PetscCall(MatCreate(comm,outmat));
    PetscCall(MatSetSizes(*outmat,m,n,PETSC_DETERMINE,PETSC_DETERMINE));
    PetscCall(MatGetBlockSizes(inmat,&bs,&cbs));
    PetscCall(MatSetBlockSizes(*outmat,bs,cbs));
    PetscCall(MatGetRootType_Private(inmat,&rootType));
    PetscCall(MatSetType(*outmat,rootType));
    PetscCall(MatSeqAIJSetPreallocation(*outmat,0,dnz));
    PetscCall(MatMPIAIJSetPreallocation(*outmat,0,dnz,0,onz));
    MatPreallocateEnd(dnz,onz);
    PetscCall(MatSetOption(*outmat,MAT_NO_OFF_PROC_ENTRIES,PETSC_TRUE));
  }

  /* numeric phase */
  PetscCall(MatGetOwnershipRange(*outmat,&rstart,NULL));
  for (i=0; i<m; i++) {
    PetscCall(MatGetRow_SeqAIJ(inmat,i,&nnz,&indx,&values));
    Ii   = i + rstart;
    PetscCall(MatSetValues(*outmat,1,&Ii,nnz,indx,values,INSERT_VALUES));
    PetscCall(MatRestoreRow_SeqAIJ(inmat,i,&nnz,&indx,&values));
  }
  PetscCall(MatAssemblyBegin(*outmat,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(*outmat,MAT_FINAL_ASSEMBLY));
  PetscFunctionReturn(0);
}

PetscErrorCode MatFileSplit(Mat A,char *outfile)
{
  PetscMPIInt       rank;
  PetscInt          m,N,i,rstart,nnz;
  size_t            len;
  const PetscInt    *indx;
  PetscViewer       out;
  char              *name;
  Mat               B;
  const PetscScalar *values;

  PetscFunctionBegin;
  PetscCall(MatGetLocalSize(A,&m,NULL));
  PetscCall(MatGetSize(A,NULL,&N));
  /* Should this be the type of the diagonal block of A? */
  PetscCall(MatCreate(PETSC_COMM_SELF,&B));
  PetscCall(MatSetSizes(B,m,N,m,N));
  PetscCall(MatSetBlockSizesFromMats(B,A,A));
  PetscCall(MatSetType(B,MATSEQAIJ));
  PetscCall(MatSeqAIJSetPreallocation(B,0,NULL));
  PetscCall(MatGetOwnershipRange(A,&rstart,NULL));
  for (i=0; i<m; i++) {
    PetscCall(MatGetRow(A,i+rstart,&nnz,&indx,&values));
    PetscCall(MatSetValues(B,1,&i,nnz,indx,values,INSERT_VALUES));
    PetscCall(MatRestoreRow(A,i+rstart,&nnz,&indx,&values));
  }
  PetscCall(MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY));

  PetscCallMPI(MPI_Comm_rank(PetscObjectComm((PetscObject)A),&rank));
  PetscCall(PetscStrlen(outfile,&len));
  PetscCall(PetscMalloc1(len+6,&name));
  PetscCall(PetscSNPrintf(name,len+6,"%s.%d",outfile,rank));
  PetscCall(PetscViewerBinaryOpen(PETSC_COMM_SELF,name,FILE_MODE_APPEND,&out));
  PetscCall(PetscFree(name));
  PetscCall(MatView(B,out));
  PetscCall(PetscViewerDestroy(&out));
  PetscCall(MatDestroy(&B));
  PetscFunctionReturn(0);
}

static PetscErrorCode MatDestroy_MPIAIJ_SeqsToMPI(void *data)
{
  Mat_Merge_SeqsToMPI *merge = (Mat_Merge_SeqsToMPI *)data;

  PetscFunctionBegin;
  if (!merge) PetscFunctionReturn(0);
  PetscCall(PetscFree(merge->id_r));
  PetscCall(PetscFree(merge->len_s));
  PetscCall(PetscFree(merge->len_r));
  PetscCall(PetscFree(merge->bi));
  PetscCall(PetscFree(merge->bj));
  PetscCall(PetscFree(merge->buf_ri[0]));
  PetscCall(PetscFree(merge->buf_ri));
  PetscCall(PetscFree(merge->buf_rj[0]));
  PetscCall(PetscFree(merge->buf_rj));
  PetscCall(PetscFree(merge->coi));
  PetscCall(PetscFree(merge->coj));
  PetscCall(PetscFree(merge->owners_co));
  PetscCall(PetscLayoutDestroy(&merge->rowmap));
  PetscCall(PetscFree(merge));
  PetscFunctionReturn(0);
}

#include <../src/mat/utils/freespace.h>
#include <petscbt.h>

PetscErrorCode MatCreateMPIAIJSumSeqAIJNumeric(Mat seqmat,Mat mpimat)
{
  MPI_Comm            comm;
  Mat_SeqAIJ          *a  =(Mat_SeqAIJ*)seqmat->data;
  PetscMPIInt         size,rank,taga,*len_s;
  PetscInt            N=mpimat->cmap->N,i,j,*owners,*ai=a->i,*aj;
  PetscInt            proc,m;
  PetscInt            **buf_ri,**buf_rj;
  PetscInt            k,anzi,*bj_i,*bi,*bj,arow,bnzi,nextaj;
  PetscInt            nrows,**buf_ri_k,**nextrow,**nextai;
  MPI_Request         *s_waits,*r_waits;
  MPI_Status          *status;
  const MatScalar     *aa,*a_a;
  MatScalar           **abuf_r,*ba_i;
  Mat_Merge_SeqsToMPI *merge;
  PetscContainer      container;

  PetscFunctionBegin;
  PetscCall(PetscObjectGetComm((PetscObject)mpimat,&comm));
  PetscCall(PetscLogEventBegin(MAT_Seqstompinum,seqmat,0,0,0));

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

  PetscCall(PetscObjectQuery((PetscObject)mpimat,"MatMergeSeqsToMPI",(PetscObject*)&container));
  PetscCheck(container,PetscObjectComm((PetscObject)mpimat),PETSC_ERR_PLIB,"Mat not created from MatCreateMPIAIJSumSeqAIJSymbolic");
  PetscCall(PetscContainerGetPointer(container,(void**)&merge));
  PetscCall(MatSeqAIJGetArrayRead(seqmat,&a_a));
  aa   = a_a;

  bi     = merge->bi;
  bj     = merge->bj;
  buf_ri = merge->buf_ri;
  buf_rj = merge->buf_rj;

  PetscCall(PetscMalloc1(size,&status));
  owners = merge->rowmap->range;
  len_s  = merge->len_s;

  /* send and recv matrix values */
  /*-----------------------------*/
  PetscCall(PetscObjectGetNewTag((PetscObject)mpimat,&taga));
  PetscCall(PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits));

  PetscCall(PetscMalloc1(merge->nsend+1,&s_waits));
  for (proc=0,k=0; proc<size; proc++) {
    if (!len_s[proc]) continue;
    i    = owners[proc];
    PetscCallMPI(MPI_Isend(aa+ai[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(PetscFree(status));

  PetscCall(PetscFree(s_waits));
  PetscCall(PetscFree(r_waits));

  /* insert mat values of mpimat */
  /*----------------------------*/
  PetscCall(PetscMalloc1(N,&ba_i));
  PetscCall(PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextai));

  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 */
    nextai[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th recved i-structure  */
  }

  /* set values of ba */
  m    = merge->rowmap->n;
  for (i=0; i<m; i++) {
    arow = owners[rank] + i;
    bj_i = bj+bi[i];  /* col indices of the i-th row of mpimat */
    bnzi = bi[i+1] - bi[i];
    PetscCall(PetscArrayzero(ba_i,bnzi));

    /* add local non-zero vals of this proc's seqmat into ba */
    anzi   = ai[arow+1] - ai[arow];
    aj     = a->j + ai[arow];
    aa     = a_a + ai[arow];
    nextaj = 0;
    for (j=0; nextaj<anzi; j++) {
      if (*(bj_i + j) == aj[nextaj]) { /* bcol == acol */
        ba_i[j] += aa[nextaj++];
      }
    }

    /* add received vals into ba */
    for (k=0; k<merge->nrecv; k++) { /* k-th received message */
      /* i-th row */
      if (i == *nextrow[k]) {
        anzi   = *(nextai[k]+1) - *nextai[k];
        aj     = buf_rj[k] + *(nextai[k]);
        aa     = abuf_r[k] + *(nextai[k]);
        nextaj = 0;
        for (j=0; nextaj<anzi; j++) {
          if (*(bj_i + j) == aj[nextaj]) { /* bcol == acol */
            ba_i[j] += aa[nextaj++];
          }
        }
        nextrow[k]++; nextai[k]++;
      }
    }
    PetscCall(MatSetValues(mpimat,1,&arow,bnzi,bj_i,ba_i,INSERT_VALUES));
  }
  PetscCall(MatSeqAIJRestoreArrayRead(seqmat,&a_a));
  PetscCall(MatAssemblyBegin(mpimat,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(mpimat,MAT_FINAL_ASSEMBLY));

  PetscCall(PetscFree(abuf_r[0]));
  PetscCall(PetscFree(abuf_r));
  PetscCall(PetscFree(ba_i));
  PetscCall(PetscFree3(buf_ri_k,nextrow,nextai));
  PetscCall(PetscLogEventEnd(MAT_Seqstompinum,seqmat,0,0,0));
  PetscFunctionReturn(0);
}

PetscErrorCode  MatCreateMPIAIJSumSeqAIJSymbolic(MPI_Comm comm,Mat seqmat,PetscInt m,PetscInt n,Mat *mpimat)
{
  Mat                 B_mpi;
  Mat_SeqAIJ          *a=(Mat_SeqAIJ*)seqmat->data;
  PetscMPIInt         size,rank,tagi,tagj,*len_s,*len_si,*len_ri;
  PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
  PetscInt            M=seqmat->rmap->n,N=seqmat->cmap->n,i,*owners,*ai=a->i,*aj=a->j;
  PetscInt            len,proc,*dnz,*onz,bs,cbs;
  PetscInt            k,anzi,*bi,*bj,*lnk,nlnk,arow,bnzi,nspacedouble=0;
  PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextai;
  MPI_Request         *si_waits,*sj_waits,*ri_waits,*rj_waits;
  MPI_Status          *status;
  PetscFreeSpaceList  free_space=NULL,current_space=NULL;
  PetscBT             lnkbt;
  Mat_Merge_SeqsToMPI *merge;
  PetscContainer      container;

  PetscFunctionBegin;
  PetscCall(PetscLogEventBegin(MAT_Seqstompisym,seqmat,0,0,0));

  /* make sure it is a PETSc comm */
  PetscCall(PetscCommDuplicate(comm,&comm,NULL));
  PetscCallMPI(MPI_Comm_size(comm,&size));
  PetscCallMPI(MPI_Comm_rank(comm,&rank));

  PetscCall(PetscNew(&merge));
  PetscCall(PetscMalloc1(size,&status));

  /* determine row ownership */
  /*---------------------------------------------------------*/
  PetscCall(PetscLayoutCreate(comm,&merge->rowmap));
  PetscCall(PetscLayoutSetLocalSize(merge->rowmap,m));
  PetscCall(PetscLayoutSetSize(merge->rowmap,M));
  PetscCall(PetscLayoutSetBlockSize(merge->rowmap,1));
  PetscCall(PetscLayoutSetUp(merge->rowmap));
  PetscCall(PetscMalloc1(size,&len_si));
  PetscCall(PetscMalloc1(size,&merge->len_s));

  m      = merge->rowmap->n;
  owners = merge->rowmap->range;

  /* determine the number of messages to send, their lengths */
  /*---------------------------------------------------------*/
  len_s = merge->len_s;

  len          = 0; /* length of buf_si[] */
  merge->nsend = 0;
  for (proc=0; proc<size; proc++) {
    len_si[proc] = 0;
    if (proc == rank) {
      len_s[proc] = 0;
    } else {
      len_si[proc] = owners[proc+1] - owners[proc] + 1;
      len_s[proc]  = ai[owners[proc+1]] - ai[owners[proc]]; /* num of rows to be sent to [proc] */
    }
    if (len_s[proc]) {
      merge->nsend++;
      nrows = 0;
      for (i=owners[proc]; i<owners[proc+1]; i++) {
        if (ai[i+1] > ai[i]) nrows++;
      }
      len_si[proc] = 2*(nrows+1);
      len         += len_si[proc];
    }
  }

  /* determine the number and length of messages to receive for ij-structure */
  /*-------------------------------------------------------------------------*/
  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 of j-structure */
  /*-------------------------------*/
  PetscCall(PetscCommGetNewTag(comm,&tagj));
  PetscCall(PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rj_waits));

  /* post the Isend of j-structure */
  /*--------------------------------*/
  PetscCall(PetscMalloc2(merge->nsend,&si_waits,merge->nsend,&sj_waits));

  for (proc=0, k=0; proc<size; proc++) {
    if (!len_s[proc]) continue;
    i    = owners[proc];
    PetscCallMPI(MPI_Isend(aj+ai[i],len_s[proc],MPIU_INT,proc,tagj,comm,sj_waits+k));
    k++;
  }

  /* receives and sends of j-structure are complete */
  /*------------------------------------------------*/
  if (merge->nrecv) PetscCallMPI(MPI_Waitall(merge->nrecv,rj_waits,status));
  if (merge->nsend) PetscCallMPI(MPI_Waitall(merge->nsend,sj_waits,status));

  /* send and recv i-structure */
  /*---------------------------*/
  PetscCall(PetscCommGetNewTag(comm,&tagi));
  PetscCall(PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&ri_waits));

  PetscCall(PetscMalloc1(len+1,&buf_s));
  buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
  for (proc=0,k=0; proc<size; proc++) {
    if (!len_s[proc]) continue;
    /* form outgoing message for i-structure:
         buf_si[0]:                 nrows to be sent
               [1:nrows]:           row index (global)
               [nrows+1:2*nrows+1]: i-structure index
    */
    /*-------------------------------------------*/
    nrows       = len_si[proc]/2 - 1;
    buf_si_i    = buf_si + nrows+1;
    buf_si[0]   = nrows;
    buf_si_i[0] = 0;
    nrows       = 0;
    for (i=owners[proc]; i<owners[proc+1]; i++) {
      anzi = ai[i+1] - ai[i];
      if (anzi) {
        buf_si_i[nrows+1] = buf_si_i[nrows] + anzi; /* i-structure */
        buf_si[nrows+1]   = i-owners[proc]; /* local row index */
        nrows++;
      }
    }
    PetscCallMPI(MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,si_waits+k));
    k++;
    buf_si += len_si[proc];
  }

  if (merge->nrecv) PetscCallMPI(MPI_Waitall(merge->nrecv,ri_waits,status));
  if (merge->nsend) PetscCallMPI(MPI_Waitall(merge->nsend,si_waits,status));

  PetscCall(PetscInfo(seqmat,"nsend: %d, nrecv: %d\n",merge->nsend,merge->nrecv));
  for (i=0; i<merge->nrecv; i++) {
    PetscCall(PetscInfo(seqmat,"recv len_ri=%d, len_rj=%d from [%d]\n",len_ri[i],merge->len_r[i],merge->id_r[i]));
  }

  PetscCall(PetscFree(len_si));
  PetscCall(PetscFree(len_ri));
  PetscCall(PetscFree(rj_waits));
  PetscCall(PetscFree2(si_waits,sj_waits));
  PetscCall(PetscFree(ri_waits));
  PetscCall(PetscFree(buf_s));
  PetscCall(PetscFree(status));

  /* compute a local seq matrix in each processor */
  /*----------------------------------------------*/
  /* allocate bi array and free space for accumulating nonzero column info */
  PetscCall(PetscMalloc1(m+1,&bi));
  bi[0] = 0;

  /* create and initialize a linked list */
  nlnk = N+1;
  PetscCall(PetscLLCreate(N,N,nlnk,lnk,lnkbt));

  /* initial FreeSpace size is 2*(num of local nnz(seqmat)) */
  len  = ai[owners[rank+1]] - ai[owners[rank]];
  PetscCall(PetscFreeSpaceGet(PetscIntMultTruncate(2,len)+1,&free_space));

  current_space = free_space;

  /* determine symbolic info for each local row */
  PetscCall(PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextai));

  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 */
    nextai[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th recved i-structure  */
  }

  MatPreallocateBegin(comm,m,n,dnz,onz);
  len  = 0;
  for (i=0; i<m; i++) {
    bnzi = 0;
    /* add local non-zero cols of this proc's seqmat into lnk */
    arow  = owners[rank] + i;
    anzi  = ai[arow+1] - ai[arow];
    aj    = a->j + ai[arow];
    PetscCall(PetscLLAddSorted(anzi,aj,N,&nlnk,lnk,lnkbt));
    bnzi += nlnk;
    /* add received col data into lnk */
    for (k=0; k<merge->nrecv; k++) { /* k-th received message */
      if (i == *nextrow[k]) { /* i-th row */
        anzi  = *(nextai[k]+1) - *nextai[k];
        aj    = buf_rj[k] + *nextai[k];
        PetscCall(PetscLLAddSorted(anzi,aj,N,&nlnk,lnk,lnkbt));
        bnzi += nlnk;
        nextrow[k]++; nextai[k]++;
      }
    }
    if (len < bnzi) len = bnzi;  /* =max(bnzi) */

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

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

    bi[i+1] = bi[i] + bnzi;
  }

  PetscCall(PetscFree3(buf_ri_k,nextrow,nextai));

  PetscCall(PetscMalloc1(bi[m]+1,&bj));
  PetscCall(PetscFreeSpaceContiguous(&free_space,bj));
  PetscCall(PetscLLDestroy(lnk,lnkbt));

  /* create symbolic parallel matrix B_mpi */
  /*---------------------------------------*/
  PetscCall(MatGetBlockSizes(seqmat,&bs,&cbs));
  PetscCall(MatCreate(comm,&B_mpi));
  if (n==PETSC_DECIDE) {
    PetscCall(MatSetSizes(B_mpi,m,n,PETSC_DETERMINE,N));
  } else {
    PetscCall(MatSetSizes(B_mpi,m,n,PETSC_DETERMINE,PETSC_DETERMINE));
  }
  PetscCall(MatSetBlockSizes(B_mpi,bs,cbs));
  PetscCall(MatSetType(B_mpi,MATMPIAIJ));
  PetscCall(MatMPIAIJSetPreallocation(B_mpi,0,dnz,0,onz));
  MatPreallocateEnd(dnz,onz);
  PetscCall(MatSetOption(B_mpi,MAT_NEW_NONZERO_ALLOCATION_ERR,PETSC_FALSE));

  /* B_mpi is not ready for use - assembly will be done by MatCreateMPIAIJSumSeqAIJNumeric() */
  B_mpi->assembled  = PETSC_FALSE;
  merge->bi         = bi;
  merge->bj         = bj;
  merge->buf_ri     = buf_ri;
  merge->buf_rj     = buf_rj;
  merge->coi        = NULL;
  merge->coj        = NULL;
  merge->owners_co  = NULL;

  PetscCall(PetscCommDestroy(&comm));

  /* attach the supporting struct to B_mpi for reuse */
  PetscCall(PetscContainerCreate(PETSC_COMM_SELF,&container));
  PetscCall(PetscContainerSetPointer(container,merge));
  PetscCall(PetscContainerSetUserDestroy(container,MatDestroy_MPIAIJ_SeqsToMPI));
  PetscCall(PetscObjectCompose((PetscObject)B_mpi,"MatMergeSeqsToMPI",(PetscObject)container));
  PetscCall(PetscContainerDestroy(&container));
  *mpimat = B_mpi;

  PetscCall(PetscLogEventEnd(MAT_Seqstompisym,seqmat,0,0,0));
  PetscFunctionReturn(0);
}

/*@C
      MatCreateMPIAIJSumSeqAIJ - Creates a MATMPIAIJ matrix by adding sequential
                 matrices from each processor

    Collective

   Input Parameters:
+    comm - the communicators the parallel matrix will live on
.    seqmat - the input sequential matrices
.    m - number of local rows (or PETSC_DECIDE)
.    n - number of local columns (or PETSC_DECIDE)
-    scall - either MAT_INITIAL_MATRIX or MAT_REUSE_MATRIX

   Output Parameter:
.    mpimat - the parallel matrix generated

    Level: advanced

   Notes:
     The dimensions of the sequential matrix in each processor MUST be the same.
     The input seqmat is included into the container "Mat_Merge_SeqsToMPI", and will be
     destroyed when mpimat is destroyed. Call PetscObjectQuery() to access seqmat.
@*/
PetscErrorCode MatCreateMPIAIJSumSeqAIJ(MPI_Comm comm,Mat seqmat,PetscInt m,PetscInt n,MatReuse scall,Mat *mpimat)
{
  PetscMPIInt    size;

  PetscFunctionBegin;
  PetscCallMPI(MPI_Comm_size(comm,&size));
  if (size == 1) {
    PetscCall(PetscLogEventBegin(MAT_Seqstompi,seqmat,0,0,0));
    if (scall == MAT_INITIAL_MATRIX) {
      PetscCall(MatDuplicate(seqmat,MAT_COPY_VALUES,mpimat));
    } else {
      PetscCall(MatCopy(seqmat,*mpimat,SAME_NONZERO_PATTERN));
    }
    PetscCall(PetscLogEventEnd(MAT_Seqstompi,seqmat,0,0,0));
    PetscFunctionReturn(0);
  }
  PetscCall(PetscLogEventBegin(MAT_Seqstompi,seqmat,0,0,0));
  if (scall == MAT_INITIAL_MATRIX) {
    PetscCall(MatCreateMPIAIJSumSeqAIJSymbolic(comm,seqmat,m,n,mpimat));
  }
  PetscCall(MatCreateMPIAIJSumSeqAIJNumeric(seqmat,*mpimat));
  PetscCall(PetscLogEventEnd(MAT_Seqstompi,seqmat,0,0,0));
  PetscFunctionReturn(0);
}

/*@
     MatAIJGetLocalMat - Creates a SeqAIJ from a MATAIJ matrix by taking all its local rows and putting them into a sequential matrix with
          mlocal rows and n columns. Where mlocal is the row count obtained with MatGetLocalSize() and n is the global column count obtained
          with MatGetSize()

    Not Collective

   Input Parameters:
+    A - the matrix
-    scall - either MAT_INITIAL_MATRIX or MAT_REUSE_MATRIX

   Output Parameter:
.    A_loc - the local sequential matrix generated

    Level: developer

   Notes:
     In other words combines the two parts of a parallel MPIAIJ matrix on each process to a single matrix.

     Destroy the matrix with MatDestroy()

.seealso: MatMPIAIJGetLocalMat()

@*/
PetscErrorCode MatAIJGetLocalMat(Mat A,Mat *A_loc)
{
  PetscBool      mpi;

  PetscFunctionBegin;
  PetscCall(PetscObjectTypeCompare((PetscObject)A,MATMPIAIJ,&mpi));
  if (mpi) {
    PetscCall(MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,A_loc));
  } else {
    *A_loc = A;
    PetscCall(PetscObjectReference((PetscObject)*A_loc));
  }
  PetscFunctionReturn(0);
}

/*@
     MatMPIAIJGetLocalMat - Creates a SeqAIJ from a MATMPIAIJ matrix by taking all its local rows and putting them into a sequential matrix with
          mlocal rows and n columns. Where mlocal is the row count obtained with MatGetLocalSize() and n is the global column count obtained
          with MatGetSize()

    Not Collective

   Input Parameters:
+    A - the matrix
-    scall - either MAT_INITIAL_MATRIX or MAT_REUSE_MATRIX

   Output Parameter:
.    A_loc - the local sequential matrix generated

    Level: developer

   Notes:
     In other words combines the two parts of a parallel MPIAIJ matrix on each process to a single matrix.

     When the communicator associated with A has size 1 and MAT_INITIAL_MATRIX is requested, the matrix returned is the diagonal part of A.
     If MAT_REUSE_MATRIX is requested with comm size 1, MatCopy(Adiag,*A_loc,SAME_NONZERO_PATTERN) is called.
     This means that one can preallocate the proper sequential matrix first and then call this routine with MAT_REUSE_MATRIX to safely
     modify the values of the returned A_loc.

.seealso: `MatGetOwnershipRange()`, `MatMPIAIJGetLocalMatCondensed()`, `MatMPIAIJGetLocalMatMerge()`
@*/
PetscErrorCode MatMPIAIJGetLocalMat(Mat A,MatReuse scall,Mat *A_loc)
{
  Mat_MPIAIJ        *mpimat=(Mat_MPIAIJ*)A->data;
  Mat_SeqAIJ        *mat,*a,*b;
  PetscInt          *ai,*aj,*bi,*bj,*cmap=mpimat->garray;
  const PetscScalar *aa,*ba,*aav,*bav;
  PetscScalar       *ca,*cam;
  PetscMPIInt       size;
  PetscInt          am=A->rmap->n,i,j,k,cstart=A->cmap->rstart;
  PetscInt          *ci,*cj,col,ncols_d,ncols_o,jo;
  PetscBool         match;

  PetscFunctionBegin;
  PetscCall(PetscStrbeginswith(((PetscObject)A)->type_name,MATMPIAIJ,&match));
  PetscCheck(match,PetscObjectComm((PetscObject)A), PETSC_ERR_SUP,"Requires MATMPIAIJ matrix as input");
  PetscCallMPI(MPI_Comm_size(PetscObjectComm((PetscObject)A),&size));
  if (size == 1) {
    if (scall == MAT_INITIAL_MATRIX) {
      PetscCall(PetscObjectReference((PetscObject)mpimat->A));
      *A_loc = mpimat->A;
    } else if (scall == MAT_REUSE_MATRIX) {
      PetscCall(MatCopy(mpimat->A,*A_loc,SAME_NONZERO_PATTERN));
    }
    PetscFunctionReturn(0);
  }

  PetscCall(PetscLogEventBegin(MAT_Getlocalmat,A,0,0,0));
  a = (Mat_SeqAIJ*)(mpimat->A)->data;
  b = (Mat_SeqAIJ*)(mpimat->B)->data;
  ai = a->i; aj = a->j; bi = b->i; bj = b->j;
  PetscCall(MatSeqAIJGetArrayRead(mpimat->A,&aav));
  PetscCall(MatSeqAIJGetArrayRead(mpimat->B,&bav));
  aa   = aav;
  ba   = bav;
  if (scall == MAT_INITIAL_MATRIX) {
    PetscCall(PetscMalloc1(1+am,&ci));
    ci[0] = 0;
    for (i=0; i<am; i++) {
      ci[i+1] = ci[i] + (ai[i+1] - ai[i]) + (bi[i+1] - bi[i]);
    }
    PetscCall(PetscMalloc1(1+ci[am],&cj));
    PetscCall(PetscMalloc1(1+ci[am],&ca));
    k    = 0;
    for (i=0; i<am; i++) {
      ncols_o = bi[i+1] - bi[i];
      ncols_d = ai[i+1] - ai[i];
      /* off-diagonal portion of A */
      for (jo=0; jo<ncols_o; jo++) {
        col = cmap[*bj];
        if (col >= cstart) break;
        cj[k]   = col; bj++;
        ca[k++] = *ba++;
      }
      /* diagonal portion of A */
      for (j=0; j<ncols_d; j++) {
        cj[k]   = cstart + *aj++;
        ca[k++] = *aa++;
      }
      /* off-diagonal portion of A */
      for (j=jo; j<ncols_o; j++) {
        cj[k]   = cmap[*bj++];
        ca[k++] = *ba++;
      }
    }
    /* put together the new matrix */
    PetscCall(MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,am,A->cmap->N,ci,cj,ca,A_loc));
    /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
    /* Since these are PETSc arrays, change flags to free them as necessary. */
    mat          = (Mat_SeqAIJ*)(*A_loc)->data;
    mat->free_a  = PETSC_TRUE;
    mat->free_ij = PETSC_TRUE;
    mat->nonew   = 0;
  } else if (scall == MAT_REUSE_MATRIX) {
    mat  =(Mat_SeqAIJ*)(*A_loc)->data;
    ci   = mat->i;
    cj   = mat->j;
    PetscCall(MatSeqAIJGetArrayWrite(*A_loc,&cam));
    for (i=0; i<am; i++) {
      /* off-diagonal portion of A */
      ncols_o = bi[i+1] - bi[i];
      for (jo=0; jo<ncols_o; jo++) {
        col = cmap[*bj];
        if (col >= cstart) break;
        *cam++ = *ba++; bj++;
      }
      /* diagonal portion of A */
      ncols_d = ai[i+1] - ai[i];
      for (j=0; j<ncols_d; j++) *cam++ = *aa++;
      /* off-diagonal portion of A */
      for (j=jo; j<ncols_o; j++) {
        *cam++ = *ba++; bj++;
      }
    }
    PetscCall(MatSeqAIJRestoreArrayWrite(*A_loc,&cam));
  } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Invalid MatReuse %d",(int)scall);
  PetscCall(MatSeqAIJRestoreArrayRead(mpimat->A,&aav));
  PetscCall(MatSeqAIJRestoreArrayRead(mpimat->B,&bav));
  PetscCall(PetscLogEventEnd(MAT_Getlocalmat,A,0,0,0));
  PetscFunctionReturn(0);
}

/*@
     MatMPIAIJGetLocalMatMerge - Creates a SeqAIJ from a MATMPIAIJ matrix by taking all its local rows and putting them into a sequential matrix with
          mlocal rows and n columns. Where n is the sum of the number of columns of the diagonal and offdiagonal part

    Not Collective

   Input Parameters:
+    A - the matrix
-    scall - either MAT_INITIAL_MATRIX or MAT_REUSE_MATRIX

   Output Parameters:
+    glob - sequential IS with global indices associated with the columns of the local sequential matrix generated (can be NULL)
-    A_loc - the local sequential matrix generated

    Level: developer

   Notes:
     This is different from MatMPIAIJGetLocalMat() since the first columns in the returning matrix are those associated with the diagonal part, then those associated with the offdiagonal part (in its local ordering)

.seealso: `MatGetOwnershipRange()`, `MatMPIAIJGetLocalMat()`, `MatMPIAIJGetLocalMatCondensed()`

@*/
PetscErrorCode MatMPIAIJGetLocalMatMerge(Mat A,MatReuse scall,IS *glob,Mat *A_loc)
{
  Mat            Ao,Ad;
  const PetscInt *cmap;
  PetscMPIInt    size;
  PetscErrorCode (*f)(Mat,MatReuse,IS*,Mat*);

  PetscFunctionBegin;
  PetscCall(MatMPIAIJGetSeqAIJ(A,&Ad,&Ao,&cmap));
  PetscCallMPI(MPI_Comm_size(PetscObjectComm((PetscObject)A),&size));
  if (size == 1) {
    if (scall == MAT_INITIAL_MATRIX) {
      PetscCall(PetscObjectReference((PetscObject)Ad));
      *A_loc = Ad;
    } else if (scall == MAT_REUSE_MATRIX) {
      PetscCall(MatCopy(Ad,*A_loc,SAME_NONZERO_PATTERN));
    }
    if (glob) PetscCall(ISCreateStride(PetscObjectComm((PetscObject)Ad),Ad->cmap->n,Ad->cmap->rstart,1,glob));
    PetscFunctionReturn(0);
  }
  PetscCall(PetscObjectQueryFunction((PetscObject)A,"MatMPIAIJGetLocalMatMerge_C",&f));
  PetscCall(PetscLogEventBegin(MAT_Getlocalmat,A,0,0,0));
  if (f) {
    PetscCall((*f)(A,scall,glob,A_loc));
  } else {
    Mat_SeqAIJ        *a = (Mat_SeqAIJ*)Ad->data;
    Mat_SeqAIJ        *b = (Mat_SeqAIJ*)Ao->data;
    Mat_SeqAIJ        *c;
    PetscInt          *ai = a->i, *aj = a->j;
    PetscInt          *bi = b->i, *bj = b->j;
    PetscInt          *ci,*cj;
    const PetscScalar *aa,*ba;
    PetscScalar       *ca;
    PetscInt          i,j,am,dn,on;

    PetscCall(MatGetLocalSize(Ad,&am,&dn));
    PetscCall(MatGetLocalSize(Ao,NULL,&on));
    PetscCall(MatSeqAIJGetArrayRead(Ad,&aa));
    PetscCall(MatSeqAIJGetArrayRead(Ao,&ba));
    if (scall == MAT_INITIAL_MATRIX) {
      PetscInt k;
      PetscCall(PetscMalloc1(1+am,&ci));
      PetscCall(PetscMalloc1(ai[am]+bi[am],&cj));
      PetscCall(PetscMalloc1(ai[am]+bi[am],&ca));
      ci[0] = 0;
      for (i=0,k=0; i<am; i++) {
        const PetscInt ncols_o = bi[i+1] - bi[i];
        const PetscInt ncols_d = ai[i+1] - ai[i];
        ci[i+1] = ci[i] + ncols_o + ncols_d;
        /* diagonal portion of A */
        for (j=0; j<ncols_d; j++,k++) {
          cj[k] = *aj++;
          ca[k] = *aa++;
        }
        /* off-diagonal portion of A */
        for (j=0; j<ncols_o; j++,k++) {
          cj[k] = dn + *bj++;
          ca[k] = *ba++;
        }
      }
      /* put together the new matrix */
      PetscCall(MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,am,dn+on,ci,cj,ca,A_loc));
      /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
      /* Since these are PETSc arrays, change flags to free them as necessary. */
      c          = (Mat_SeqAIJ*)(*A_loc)->data;
      c->free_a  = PETSC_TRUE;
      c->free_ij = PETSC_TRUE;
      c->nonew   = 0;
      PetscCall(MatSetType(*A_loc,((PetscObject)Ad)->type_name));
    } else if (scall == MAT_REUSE_MATRIX) {
      PetscCall(MatSeqAIJGetArrayWrite(*A_loc,&ca));
      for (i=0; i<am; i++) {
        const PetscInt ncols_d = ai[i+1] - ai[i];
        const PetscInt ncols_o = bi[i+1] - bi[i];
        /* diagonal portion of A */
        for (j=0; j<ncols_d; j++) *ca++ = *aa++;
        /* off-diagonal portion of A */
        for (j=0; j<ncols_o; j++) *ca++ = *ba++;
      }
      PetscCall(MatSeqAIJRestoreArrayWrite(*A_loc,&ca));
    } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Invalid MatReuse %d",(int)scall);
    PetscCall(MatSeqAIJRestoreArrayRead(Ad,&aa));
    PetscCall(MatSeqAIJRestoreArrayRead(Ao,&aa));
    if (glob) {
      PetscInt cst, *gidx;

      PetscCall(MatGetOwnershipRangeColumn(A,&cst,NULL));
      PetscCall(PetscMalloc1(dn+on,&gidx));
      for (i=0; i<dn; i++) gidx[i]    = cst + i;
      for (i=0; i<on; i++) gidx[i+dn] = cmap[i];
      PetscCall(ISCreateGeneral(PetscObjectComm((PetscObject)Ad),dn+on,gidx,PETSC_OWN_POINTER,glob));
    }
  }
  PetscCall(PetscLogEventEnd(MAT_Getlocalmat,A,0,0,0));
  PetscFunctionReturn(0);
}

/*@C
     MatMPIAIJGetLocalMatCondensed - Creates a SeqAIJ matrix from an MATMPIAIJ matrix by taking all its local rows and NON-ZERO columns

    Not Collective

   Input Parameters:
+    A - the matrix
.    scall - either MAT_INITIAL_MATRIX or MAT_REUSE_MATRIX
-    row, col - index sets of rows and columns to extract (or NULL)

   Output Parameter:
.    A_loc - the local sequential matrix generated

    Level: developer

.seealso: `MatGetOwnershipRange()`, `MatMPIAIJGetLocalMat()`

@*/
PetscErrorCode MatMPIAIJGetLocalMatCondensed(Mat A,MatReuse scall,IS *row,IS *col,Mat *A_loc)
{
  Mat_MPIAIJ     *a=(Mat_MPIAIJ*)A->data;
  PetscInt       i,start,end,ncols,nzA,nzB,*cmap,imark,*idx;
  IS             isrowa,iscola;
  Mat            *aloc;
  PetscBool      match;

  PetscFunctionBegin;
  PetscCall(PetscObjectTypeCompare((PetscObject)A,MATMPIAIJ,&match));
  PetscCheck(match,PetscObjectComm((PetscObject)A), PETSC_ERR_SUP,"Requires MATMPIAIJ matrix as input");
  PetscCall(PetscLogEventBegin(MAT_Getlocalmatcondensed,A,0,0,0));
  if (!row) {
    start = A->rmap->rstart; end = A->rmap->rend;
    PetscCall(ISCreateStride(PETSC_COMM_SELF,end-start,start,1,&isrowa));
  } else {
    isrowa = *row;
  }
  if (!col) {
    start = A->cmap->rstart;
    cmap  = a->garray;
    nzA   = a->A->cmap->n;
    nzB   = a->B->cmap->n;
    PetscCall(PetscMalloc1(nzA+nzB, &idx));
    ncols = 0;
    for (i=0; i<nzB; i++) {
      if (cmap[i] < start) idx[ncols++] = cmap[i];
      else break;
    }
    imark = i;
    for (i=0; i<nzA; i++) idx[ncols++] = start + i;
    for (i=imark; i<nzB; i++) idx[ncols++] = cmap[i];
    PetscCall(ISCreateGeneral(PETSC_COMM_SELF,ncols,idx,PETSC_OWN_POINTER,&iscola));
  } else {
    iscola = *col;
  }
  if (scall != MAT_INITIAL_MATRIX) {
    PetscCall(PetscMalloc1(1,&aloc));
    aloc[0] = *A_loc;
  }
  PetscCall(MatCreateSubMatrices(A,1,&isrowa,&iscola,scall,&aloc));
  if (!col) { /* attach global id of condensed columns */
    PetscCall(PetscObjectCompose((PetscObject)aloc[0],"_petsc_GetLocalMatCondensed_iscol",(PetscObject)iscola));
  }
  *A_loc = aloc[0];
  PetscCall(PetscFree(aloc));
  if (!row) {
    PetscCall(ISDestroy(&isrowa));
  }
  if (!col) {
    PetscCall(ISDestroy(&iscola));
  }
  PetscCall(PetscLogEventEnd(MAT_Getlocalmatcondensed,A,0,0,0));
  PetscFunctionReturn(0);
}

/*
 * Create a sequential AIJ matrix based on row indices. a whole column is extracted once a row is matched.
 * Row could be local or remote.The routine is designed to be scalable in memory so that nothing is based
 * on a global size.
 * */
PetscErrorCode MatCreateSeqSubMatrixWithRows_Private(Mat P,IS rows,Mat *P_oth)
{
  Mat_MPIAIJ               *p=(Mat_MPIAIJ*)P->data;
  Mat_SeqAIJ               *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data,*p_oth;
  PetscInt                 plocalsize,nrows,*ilocal,*oilocal,i,lidx,*nrcols,*nlcols,ncol;
  PetscMPIInt              owner;
  PetscSFNode              *iremote,*oiremote;
  const PetscInt           *lrowindices;
  PetscSF                  sf,osf;
  PetscInt                 pcstart,*roffsets,*loffsets,*pnnz,j;
  PetscInt                 ontotalcols,dntotalcols,ntotalcols,nout;
  MPI_Comm                 comm;
  ISLocalToGlobalMapping   mapping;
  const PetscScalar        *pd_a,*po_a;

  PetscFunctionBegin;
  PetscCall(PetscObjectGetComm((PetscObject)P,&comm));
  /* plocalsize is the number of roots
   * nrows is the number of leaves
   * */
  PetscCall(MatGetLocalSize(P,&plocalsize,NULL));
  PetscCall(ISGetLocalSize(rows,&nrows));
  PetscCall(PetscCalloc1(nrows,&iremote));
  PetscCall(ISGetIndices(rows,&lrowindices));
  for (i=0;i<nrows;i++) {
    /* Find a remote index and an owner for a row
     * The row could be local or remote
     * */
    owner = 0;
    lidx  = 0;
    PetscCall(PetscLayoutFindOwnerIndex(P->rmap,lrowindices[i],&owner,&lidx));
    iremote[i].index = lidx;
    iremote[i].rank  = owner;
  }
  /* Create SF to communicate how many nonzero columns for each row */
  PetscCall(PetscSFCreate(comm,&sf));
  /* SF will figure out the number of nonzero colunms for each row, and their
   * offsets
   * */
  PetscCall(PetscSFSetGraph(sf,plocalsize,nrows,NULL,PETSC_OWN_POINTER,iremote,PETSC_OWN_POINTER));
  PetscCall(PetscSFSetFromOptions(sf));
  PetscCall(PetscSFSetUp(sf));

  PetscCall(PetscCalloc1(2*(plocalsize+1),&roffsets));
  PetscCall(PetscCalloc1(2*plocalsize,&nrcols));
  PetscCall(PetscCalloc1(nrows,&pnnz));
  roffsets[0] = 0;
  roffsets[1] = 0;
  for (i=0;i<plocalsize;i++) {
    /* diag */
    nrcols[i*2+0] = pd->i[i+1] - pd->i[i];
    /* off diag */
    nrcols[i*2+1] = po->i[i+1] - po->i[i];
    /* compute offsets so that we relative location for each row */
    roffsets[(i+1)*2+0] = roffsets[i*2+0] + nrcols[i*2+0];
    roffsets[(i+1)*2+1] = roffsets[i*2+1] + nrcols[i*2+1];
  }
  PetscCall(PetscCalloc1(2*nrows,&nlcols));
  PetscCall(PetscCalloc1(2*nrows,&loffsets));
  /* 'r' means root, and 'l' means leaf */
  PetscCall(PetscSFBcastBegin(sf,MPIU_2INT,nrcols,nlcols,MPI_REPLACE));
  PetscCall(PetscSFBcastBegin(sf,MPIU_2INT,roffsets,loffsets,MPI_REPLACE));
  PetscCall(PetscSFBcastEnd(sf,MPIU_2INT,nrcols,nlcols,MPI_REPLACE));
  PetscCall(PetscSFBcastEnd(sf,MPIU_2INT,roffsets,loffsets,MPI_REPLACE));
  PetscCall(PetscSFDestroy(&sf));
  PetscCall(PetscFree(roffsets));
  PetscCall(PetscFree(nrcols));
  dntotalcols = 0;
  ontotalcols = 0;
  ncol = 0;
  for (i=0;i<nrows;i++) {
    pnnz[i] = nlcols[i*2+0] + nlcols[i*2+1];
    ncol = PetscMax(pnnz[i],ncol);
    /* diag */
    dntotalcols += nlcols[i*2+0];
    /* off diag */
    ontotalcols += nlcols[i*2+1];
  }
  /* We do not need to figure the right number of columns
   * since all the calculations will be done by going through the raw data
   * */
  PetscCall(MatCreateSeqAIJ(PETSC_COMM_SELF,nrows,ncol,0,pnnz,P_oth));
  PetscCall(MatSetUp(*P_oth));
  PetscCall(PetscFree(pnnz));
  p_oth = (Mat_SeqAIJ*) (*P_oth)->data;
  /* diag */
  PetscCall(PetscCalloc1(dntotalcols,&iremote));
  /* off diag */
  PetscCall(PetscCalloc1(ontotalcols,&oiremote));
  /* diag */
  PetscCall(PetscCalloc1(dntotalcols,&ilocal));
  /* off diag */
  PetscCall(PetscCalloc1(ontotalcols,&oilocal));
  dntotalcols = 0;
  ontotalcols = 0;
  ntotalcols  = 0;
  for (i=0;i<nrows;i++) {
    owner = 0;
    PetscCall(PetscLayoutFindOwnerIndex(P->rmap,lrowindices[i],&owner,NULL));
    /* Set iremote for diag matrix */
    for (j=0;j<nlcols[i*2+0];j++) {
      iremote[dntotalcols].index   = loffsets[i*2+0] + j;
      iremote[dntotalcols].rank    = owner;
      /* P_oth is seqAIJ so that ilocal need to point to the first part of memory */
      ilocal[dntotalcols++]        = ntotalcols++;
    }
    /* off diag */
    for (j=0;j<nlcols[i*2+1];j++) {
      oiremote[ontotalcols].index   = loffsets[i*2+1] + j;
      oiremote[ontotalcols].rank    = owner;
      oilocal[ontotalcols++]        = ntotalcols++;
    }
  }
  PetscCall(ISRestoreIndices(rows,&lrowindices));
  PetscCall(PetscFree(loffsets));
  PetscCall(PetscFree(nlcols));
  PetscCall(PetscSFCreate(comm,&sf));
  /* P serves as roots and P_oth is leaves
   * Diag matrix
   * */
  PetscCall(PetscSFSetGraph(sf,pd->i[plocalsize],dntotalcols,ilocal,PETSC_OWN_POINTER,iremote,PETSC_OWN_POINTER));
  PetscCall(PetscSFSetFromOptions(sf));
  PetscCall(PetscSFSetUp(sf));

  PetscCall(PetscSFCreate(comm,&osf));
  /* Off diag */
  PetscCall(PetscSFSetGraph(osf,po->i[plocalsize],ontotalcols,oilocal,PETSC_OWN_POINTER,oiremote,PETSC_OWN_POINTER));
  PetscCall(PetscSFSetFromOptions(osf));
  PetscCall(PetscSFSetUp(osf));
  PetscCall(MatSeqAIJGetArrayRead(p->A,&pd_a));
  PetscCall(MatSeqAIJGetArrayRead(p->B,&po_a));
  /* We operate on the matrix internal data for saving memory */
  PetscCall(PetscSFBcastBegin(sf,MPIU_SCALAR,pd_a,p_oth->a,MPI_REPLACE));
  PetscCall(PetscSFBcastBegin(osf,MPIU_SCALAR,po_a,p_oth->a,MPI_REPLACE));
  PetscCall(MatGetOwnershipRangeColumn(P,&pcstart,NULL));
  /* Convert to global indices for diag matrix */
  for (i=0;i<pd->i[plocalsize];i++) pd->j[i] += pcstart;
  PetscCall(PetscSFBcastBegin(sf,MPIU_INT,pd->j,p_oth->j,MPI_REPLACE));
  /* We want P_oth store global indices */
  PetscCall(ISLocalToGlobalMappingCreate(comm,1,p->B->cmap->n,p->garray,PETSC_COPY_VALUES,&mapping));
  /* Use memory scalable approach */
  PetscCall(ISLocalToGlobalMappingSetType(mapping,ISLOCALTOGLOBALMAPPINGHASH));
  PetscCall(ISLocalToGlobalMappingApply(mapping,po->i[plocalsize],po->j,po->j));
  PetscCall(PetscSFBcastBegin(osf,MPIU_INT,po->j,p_oth->j,MPI_REPLACE));
  PetscCall(PetscSFBcastEnd(sf,MPIU_INT,pd->j,p_oth->j,MPI_REPLACE));
  /* Convert back to local indices */
  for (i=0;i<pd->i[plocalsize];i++) pd->j[i] -= pcstart;
  PetscCall(PetscSFBcastEnd(osf,MPIU_INT,po->j,p_oth->j,MPI_REPLACE));
  nout = 0;
  PetscCall(ISGlobalToLocalMappingApply(mapping,IS_GTOLM_DROP,po->i[plocalsize],po->j,&nout,po->j));
  PetscCheck(nout == po->i[plocalsize],comm,PETSC_ERR_ARG_INCOMP,"n %" PetscInt_FMT " does not equal to nout %" PetscInt_FMT " ",po->i[plocalsize],nout);
  PetscCall(ISLocalToGlobalMappingDestroy(&mapping));
  /* Exchange values */
  PetscCall(PetscSFBcastEnd(sf,MPIU_SCALAR,pd_a,p_oth->a,MPI_REPLACE));
  PetscCall(PetscSFBcastEnd(osf,MPIU_SCALAR,po_a,p_oth->a,MPI_REPLACE));
  PetscCall(MatSeqAIJRestoreArrayRead(p->A,&pd_a));
  PetscCall(MatSeqAIJRestoreArrayRead(p->B,&po_a));
  /* Stop PETSc from shrinking memory */
  for (i=0;i<nrows;i++) p_oth->ilen[i] = p_oth->imax[i];
  PetscCall(MatAssemblyBegin(*P_oth,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(*P_oth,MAT_FINAL_ASSEMBLY));
  /* Attach PetscSF objects to P_oth so that we can reuse it later */
  PetscCall(PetscObjectCompose((PetscObject)*P_oth,"diagsf",(PetscObject)sf));
  PetscCall(PetscObjectCompose((PetscObject)*P_oth,"offdiagsf",(PetscObject)osf));
  PetscCall(PetscSFDestroy(&sf));
  PetscCall(PetscSFDestroy(&osf));
  PetscFunctionReturn(0);
}

/*
 * Creates a SeqAIJ matrix by taking rows of B that equal to nonzero columns of local A
 * This supports MPIAIJ and MAIJ
 * */
PetscErrorCode MatGetBrowsOfAcols_MPIXAIJ(Mat A,Mat P,PetscInt dof,MatReuse reuse,Mat *P_oth)
{
  Mat_MPIAIJ            *a=(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data;
  Mat_SeqAIJ            *p_oth;
  IS                    rows,map;
  PetscHMapI            hamp;
  PetscInt              i,htsize,*rowindices,off,*mapping,key,count;
  MPI_Comm              comm;
  PetscSF               sf,osf;
  PetscBool             has;

  PetscFunctionBegin;
  PetscCall(PetscObjectGetComm((PetscObject)A,&comm));
  PetscCall(PetscLogEventBegin(MAT_GetBrowsOfAocols,A,P,0,0));
  /* If it is the first time, create an index set of off-diag nonzero columns of A,
   *  and then create a submatrix (that often is an overlapping matrix)
   * */
  if (reuse == MAT_INITIAL_MATRIX) {
    /* Use a hash table to figure out unique keys */
    PetscCall(PetscHMapICreate(&hamp));
    PetscCall(PetscHMapIResize(hamp,a->B->cmap->n));
    PetscCall(PetscCalloc1(a->B->cmap->n,&mapping));
    count = 0;
    /* Assume that  a->g is sorted, otherwise the following does not make sense */
    for (i=0;i<a->B->cmap->n;i++) {
      key  = a->garray[i]/dof;
      PetscCall(PetscHMapIHas(hamp,key,&has));
      if (!has) {
        mapping[i] = count;
        PetscCall(PetscHMapISet(hamp,key,count++));
      } else {
        /* Current 'i' has the same value the previous step */
        mapping[i] = count-1;
      }
    }
    PetscCall(ISCreateGeneral(comm,a->B->cmap->n,mapping,PETSC_OWN_POINTER,&map));
    PetscCall(PetscHMapIGetSize(hamp,&htsize));
    PetscCheck(htsize==count,comm,PETSC_ERR_ARG_INCOMP," Size of hash map %" PetscInt_FMT " is inconsistent with count %" PetscInt_FMT " ",htsize,count);
    PetscCall(PetscCalloc1(htsize,&rowindices));
    off = 0;
    PetscCall(PetscHMapIGetKeys(hamp,&off,rowindices));
    PetscCall(PetscHMapIDestroy(&hamp));
    PetscCall(PetscSortInt(htsize,rowindices));
    PetscCall(ISCreateGeneral(comm,htsize,rowindices,PETSC_OWN_POINTER,&rows));
    /* In case, the matrix was already created but users want to recreate the matrix */
    PetscCall(MatDestroy(P_oth));
    PetscCall(MatCreateSeqSubMatrixWithRows_Private(P,rows,P_oth));
    PetscCall(PetscObjectCompose((PetscObject)*P_oth,"aoffdiagtopothmapping",(PetscObject)map));
    PetscCall(ISDestroy(&map));
    PetscCall(ISDestroy(&rows));
  } else if (reuse == MAT_REUSE_MATRIX) {
    /* If matrix was already created, we simply update values using SF objects
     * that as attached to the matrix ealier.
     */
    const PetscScalar *pd_a,*po_a;

    PetscCall(PetscObjectQuery((PetscObject)*P_oth,"diagsf",(PetscObject*)&sf));
    PetscCall(PetscObjectQuery((PetscObject)*P_oth,"offdiagsf",(PetscObject*)&osf));
    PetscCheck(sf && osf,comm,PETSC_ERR_ARG_NULL,"Matrix is not initialized yet");
    p_oth = (Mat_SeqAIJ*) (*P_oth)->data;
    /* Update values in place */
    PetscCall(MatSeqAIJGetArrayRead(p->A,&pd_a));
    PetscCall(MatSeqAIJGetArrayRead(p->B,&po_a));
    PetscCall(PetscSFBcastBegin(sf,MPIU_SCALAR,pd_a,p_oth->a,MPI_REPLACE));
    PetscCall(PetscSFBcastBegin(osf,MPIU_SCALAR,po_a,p_oth->a,MPI_REPLACE));
    PetscCall(PetscSFBcastEnd(sf,MPIU_SCALAR,pd_a,p_oth->a,MPI_REPLACE));
    PetscCall(PetscSFBcastEnd(osf,MPIU_SCALAR,po_a,p_oth->a,MPI_REPLACE));
    PetscCall(MatSeqAIJRestoreArrayRead(p->A,&pd_a));
    PetscCall(MatSeqAIJRestoreArrayRead(p->B,&po_a));
  } else SETERRQ(comm,PETSC_ERR_ARG_UNKNOWN_TYPE,"Unknown reuse type");
  PetscCall(PetscLogEventEnd(MAT_GetBrowsOfAocols,A,P,0,0));
  PetscFunctionReturn(0);
}

/*@C
  MatGetBrowsOfAcols - Creates a SeqAIJ matrix by taking rows of B that equal to nonzero columns of local A

  Collective on Mat

  Input Parameters:
+ A - the first matrix in mpiaij format
. B - the second matrix in mpiaij format
- scall - either MAT_INITIAL_MATRIX or MAT_REUSE_MATRIX

  Output Parameters:
+ rowb - On input index sets of rows of B to extract (or NULL), modified on output
. colb - On input index sets of columns of B to extract (or NULL), modified on output
- B_seq - the sequential matrix generated

  Level: developer

@*/
PetscErrorCode MatGetBrowsOfAcols(Mat A,Mat B,MatReuse scall,IS *rowb,IS *colb,Mat *B_seq)
{
  Mat_MPIAIJ     *a=(Mat_MPIAIJ*)A->data;
  PetscInt       *idx,i,start,ncols,nzA,nzB,*cmap,imark;
  IS             isrowb,iscolb;
  Mat            *bseq=NULL;

  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);
  }
  PetscCall(PetscLogEventBegin(MAT_GetBrowsOfAcols,A,B,0,0));

  if (scall == MAT_INITIAL_MATRIX) {
    start = A->cmap->rstart;
    cmap  = a->garray;
    nzA   = a->A->cmap->n;
    nzB   = a->B->cmap->n;
    PetscCall(PetscMalloc1(nzA+nzB, &idx));
    ncols = 0;
    for (i=0; i<nzB; i++) {  /* row < local row index */
      if (cmap[i] < start) idx[ncols++] = cmap[i];
      else break;
    }
    imark = i;
    for (i=0; i<nzA; i++) idx[ncols++] = start + i;  /* local rows */
    for (i=imark; i<nzB; i++) idx[ncols++] = cmap[i]; /* row > local row index */
    PetscCall(ISCreateGeneral(PETSC_COMM_SELF,ncols,idx,PETSC_OWN_POINTER,&isrowb));
    PetscCall(ISCreateStride(PETSC_COMM_SELF,B->cmap->N,0,1,&iscolb));
  } else {
    PetscCheck(rowb && colb,PETSC_COMM_SELF,PETSC_ERR_SUP,"IS rowb and colb must be provided for MAT_REUSE_MATRIX");
    isrowb  = *rowb; iscolb = *colb;
    PetscCall(PetscMalloc1(1,&bseq));
    bseq[0] = *B_seq;
  }
  PetscCall(MatCreateSubMatrices(B,1,&isrowb,&iscolb,scall,&bseq));
  *B_seq = bseq[0];
  PetscCall(PetscFree(bseq));
  if (!rowb) {
    PetscCall(ISDestroy(&isrowb));
  } else {
    *rowb = isrowb;
  }
  if (!colb) {
    PetscCall(ISDestroy(&iscolb));
  } else {
    *colb = iscolb;
  }
  PetscCall(PetscLogEventEnd(MAT_GetBrowsOfAcols,A,B,0,0));
  PetscFunctionReturn(0);
}

/*
    MatGetBrowsOfAoCols_MPIAIJ - Creates a SeqAIJ matrix by taking rows of B that equal to nonzero columns
    of the OFF-DIAGONAL portion of local A

    Collective on Mat

   Input Parameters:
+    A,B - the matrices in mpiaij format
-    scall - either MAT_INITIAL_MATRIX or MAT_REUSE_MATRIX

   Output Parameter:
+    startsj_s - starting point in B's sending j-arrays, saved for MAT_REUSE (or NULL)
.    startsj_r - starting point in B's receiving j-arrays, saved for MAT_REUSE (or NULL)
.    bufa_ptr - array for sending matrix values, saved for MAT_REUSE (or NULL)
-    B_oth - the sequential matrix generated with size aBn=a->B->cmap->n by B->cmap->N

    Developer Notes: This directly accesses information inside the VecScatter associated with the matrix-vector product
     for this matrix. This is not desirable..

    Level: developer

*/
PetscErrorCode MatGetBrowsOfAoCols_MPIAIJ(Mat A,Mat B,MatReuse scall,PetscInt **startsj_s,PetscInt **startsj_r,MatScalar **bufa_ptr,Mat *B_oth)
{
  Mat_MPIAIJ             *a=(Mat_MPIAIJ*)A->data;
  Mat_SeqAIJ             *b_oth;
  VecScatter             ctx;
  MPI_Comm               comm;
  const PetscMPIInt      *rprocs,*sprocs;
  const PetscInt         *srow,*rstarts,*sstarts;
  PetscInt               *rowlen,*bufj,*bufJ,ncols = 0,aBn=a->B->cmap->n,row,*b_othi,*b_othj,*rvalues=NULL,*svalues=NULL,*cols,sbs,rbs;
  PetscInt               i,j,k=0,l,ll,nrecvs,nsends,nrows,*rstartsj = NULL,*sstartsj,len;
  PetscScalar            *b_otha,*bufa,*bufA,*vals = NULL;
  MPI_Request            *reqs = NULL,*rwaits = NULL,*swaits = NULL;
  PetscMPIInt            size,tag,rank,nreqs;

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

  if (PetscUnlikely(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);
  }
  PetscCall(PetscLogEventBegin(MAT_GetBrowsOfAocols,A,B,0,0));
  PetscCallMPI(MPI_Comm_rank(comm,&rank));

  if (size == 1) {
    startsj_s = NULL;
    bufa_ptr  = NULL;
    *B_oth    = NULL;
    PetscFunctionReturn(0);
  }

  ctx = a->Mvctx;
  tag = ((PetscObject)ctx)->tag;

  PetscCall(VecScatterGetRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&srow,&sprocs,&sbs));
  /* rprocs[] must be ordered so that indices received from them are ordered in rvalues[], which is key to algorithms used in this subroutine */
  PetscCall(VecScatterGetRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL/*indices not needed*/,&rprocs,&rbs));
  PetscCall(PetscMPIIntCast(nsends+nrecvs,&nreqs));
  PetscCall(PetscMalloc1(nreqs,&reqs));
  rwaits = reqs;
  swaits = reqs + nrecvs;

  if (!startsj_s || !bufa_ptr) scall = MAT_INITIAL_MATRIX;
  if (scall == MAT_INITIAL_MATRIX) {
    /* i-array */
    /*---------*/
    /*  post receives */
    if (nrecvs) PetscCall(PetscMalloc1(rbs*(rstarts[nrecvs] - rstarts[0]),&rvalues)); /* rstarts can be NULL when nrecvs=0 */
    for (i=0; i<nrecvs; i++) {
      rowlen = rvalues + rstarts[i]*rbs;
      nrows  = (rstarts[i+1]-rstarts[i])*rbs; /* num of indices to be received */
      PetscCallMPI(MPI_Irecv(rowlen,nrows,MPIU_INT,rprocs[i],tag,comm,rwaits+i));
    }

    /* pack the outgoing message */
    PetscCall(PetscMalloc2(nsends+1,&sstartsj,nrecvs+1,&rstartsj));

    sstartsj[0] = 0;
    rstartsj[0] = 0;
    len         = 0; /* total length of j or a array to be sent */
    if (nsends) {
      k    = sstarts[0]; /* ATTENTION: sstarts[0] and rstarts[0] are not necessarily zero */
      PetscCall(PetscMalloc1(sbs*(sstarts[nsends]-sstarts[0]),&svalues));
    }
    for (i=0; i<nsends; i++) {
      rowlen = svalues + (sstarts[i]-sstarts[0])*sbs;
      nrows  = sstarts[i+1]-sstarts[i]; /* num of block rows */
      for (j=0; j<nrows; j++) {
        row = srow[k] + B->rmap->range[rank]; /* global row idx */
        for (l=0; l<sbs; l++) {
          PetscCall(MatGetRow_MPIAIJ(B,row+l,&ncols,NULL,NULL)); /* rowlength */

          rowlen[j*sbs+l] = ncols;

          len += ncols;
          PetscCall(MatRestoreRow_MPIAIJ(B,row+l,&ncols,NULL,NULL));
        }
        k++;
      }
      PetscCallMPI(MPI_Isend(rowlen,nrows*sbs,MPIU_INT,sprocs[i],tag,comm,swaits+i));

      sstartsj[i+1] = len;  /* starting point of (i+1)-th outgoing msg in bufj and bufa */
    }
    /* recvs and sends of i-array are completed */
    if (nreqs) PetscCallMPI(MPI_Waitall(nreqs,reqs,MPI_STATUSES_IGNORE));
    PetscCall(PetscFree(svalues));

    /* allocate buffers for sending j and a arrays */
    PetscCall(PetscMalloc1(len+1,&bufj));
    PetscCall(PetscMalloc1(len+1,&bufa));

    /* create i-array of B_oth */
    PetscCall(PetscMalloc1(aBn+2,&b_othi));

    b_othi[0] = 0;
    len       = 0; /* total length of j or a array to be received */
    k         = 0;
    for (i=0; i<nrecvs; i++) {
      rowlen = rvalues + (rstarts[i]-rstarts[0])*rbs;
      nrows  = (rstarts[i+1]-rstarts[i])*rbs; /* num of rows to be received */
      for (j=0; j<nrows; j++) {
        b_othi[k+1] = b_othi[k] + rowlen[j];
        PetscCall(PetscIntSumError(rowlen[j],len,&len));
        k++;
      }
      rstartsj[i+1] = len; /* starting point of (i+1)-th incoming msg in bufj and bufa */
    }
    PetscCall(PetscFree(rvalues));

    /* allocate space for j and a arrays of B_oth */
    PetscCall(PetscMalloc1(b_othi[aBn]+1,&b_othj));
    PetscCall(PetscMalloc1(b_othi[aBn]+1,&b_otha));

    /* j-array */
    /*---------*/
    /*  post receives of j-array */
    for (i=0; i<nrecvs; i++) {
      nrows = rstartsj[i+1]-rstartsj[i]; /* length of the msg received */
      PetscCallMPI(MPI_Irecv(b_othj+rstartsj[i],nrows,MPIU_INT,rprocs[i],tag,comm,rwaits+i));
    }

    /* pack the outgoing message j-array */
    if (nsends) k = sstarts[0];
    for (i=0; i<nsends; i++) {
      nrows = sstarts[i+1]-sstarts[i]; /* num of block rows */
      bufJ  = bufj+sstartsj[i];
      for (j=0; j<nrows; j++) {
        row = srow[k++] + B->rmap->range[rank];  /* global row idx */
        for (ll=0; ll<sbs; ll++) {
          PetscCall(MatGetRow_MPIAIJ(B,row+ll,&ncols,&cols,NULL));
          for (l=0; l<ncols; l++) {
            *bufJ++ = cols[l];
          }
          PetscCall(MatRestoreRow_MPIAIJ(B,row+ll,&ncols,&cols,NULL));
        }
      }
      PetscCallMPI(MPI_Isend(bufj+sstartsj[i],sstartsj[i+1]-sstartsj[i],MPIU_INT,sprocs[i],tag,comm,swaits+i));
    }

    /* recvs and sends of j-array are completed */
    if (nreqs) PetscCallMPI(MPI_Waitall(nreqs,reqs,MPI_STATUSES_IGNORE));
  } else if (scall == MAT_REUSE_MATRIX) {
    sstartsj = *startsj_s;
    rstartsj = *startsj_r;
    bufa     = *bufa_ptr;
    b_oth    = (Mat_SeqAIJ*)(*B_oth)->data;
    PetscCall(MatSeqAIJGetArrayWrite(*B_oth,&b_otha));
  } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE, "Matrix P does not possess an object container");

  /* a-array */
  /*---------*/
  /*  post receives of a-array */
  for (i=0; i<nrecvs; i++) {
    nrows = rstartsj[i+1]-rstartsj[i]; /* length of the msg received */
    PetscCallMPI(MPI_Irecv(b_otha+rstartsj[i],nrows,MPIU_SCALAR,rprocs[i],tag,comm,rwaits+i));
  }

  /* pack the outgoing message a-array */
  if (nsends) k = sstarts[0];
  for (i=0; i<nsends; i++) {
    nrows = sstarts[i+1]-sstarts[i]; /* num of block rows */
    bufA  = bufa+sstartsj[i];
    for (j=0; j<nrows; j++) {
      row = srow[k++] + B->rmap->range[rank];  /* global row idx */
      for (ll=0; ll<sbs; ll++) {
        PetscCall(MatGetRow_MPIAIJ(B,row+ll,&ncols,NULL,&vals));
        for (l=0; l<ncols; l++) {
          *bufA++ = vals[l];
        }
        PetscCall(MatRestoreRow_MPIAIJ(B,row+ll,&ncols,NULL,&vals));
      }
    }
    PetscCallMPI(MPI_Isend(bufa+sstartsj[i],sstartsj[i+1]-sstartsj[i],MPIU_SCALAR,sprocs[i],tag,comm,swaits+i));
  }
  /* recvs and sends of a-array are completed */
  if (nreqs) PetscCallMPI(MPI_Waitall(nreqs,reqs,MPI_STATUSES_IGNORE));
  PetscCall(PetscFree(reqs));

  if (scall == MAT_INITIAL_MATRIX) {
    /* put together the new matrix */
    PetscCall(MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,aBn,B->cmap->N,b_othi,b_othj,b_otha,B_oth));

    /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
    /* Since these are PETSc arrays, change flags to free them as necessary. */
    b_oth          = (Mat_SeqAIJ*)(*B_oth)->data;
    b_oth->free_a  = PETSC_TRUE;
    b_oth->free_ij = PETSC_TRUE;
    b_oth->nonew   = 0;

    PetscCall(PetscFree(bufj));
    if (!startsj_s || !bufa_ptr) {
      PetscCall(PetscFree2(sstartsj,rstartsj));
      PetscCall(PetscFree(bufa_ptr));
    } else {
      *startsj_s = sstartsj;
      *startsj_r = rstartsj;
      *bufa_ptr  = bufa;
    }
  } else if (scall == MAT_REUSE_MATRIX) {
    PetscCall(MatSeqAIJRestoreArrayWrite(*B_oth,&b_otha));
  }

  PetscCall(VecScatterRestoreRemote_Private(ctx,PETSC_TRUE,&nsends,&sstarts,&srow,&sprocs,&sbs));
  PetscCall(VecScatterRestoreRemoteOrdered_Private(ctx,PETSC_FALSE,&nrecvs,&rstarts,NULL,&rprocs,&rbs));
  PetscCall(PetscLogEventEnd(MAT_GetBrowsOfAocols,A,B,0,0));
  PetscFunctionReturn(0);
}

PETSC_INTERN PetscErrorCode MatConvert_MPIAIJ_MPIAIJCRL(Mat,MatType,MatReuse,Mat*);
PETSC_INTERN PetscErrorCode MatConvert_MPIAIJ_MPIAIJPERM(Mat,MatType,MatReuse,Mat*);
PETSC_INTERN PetscErrorCode MatConvert_MPIAIJ_MPIAIJSELL(Mat,MatType,MatReuse,Mat*);
#if defined(PETSC_HAVE_MKL_SPARSE)
PETSC_INTERN PetscErrorCode MatConvert_MPIAIJ_MPIAIJMKL(Mat,MatType,MatReuse,Mat*);
#endif
PETSC_INTERN PetscErrorCode MatConvert_MPIAIJ_MPIBAIJ(Mat,MatType,MatReuse,Mat*);
PETSC_INTERN PetscErrorCode MatConvert_MPIAIJ_MPISBAIJ(Mat,MatType,MatReuse,Mat*);
#if defined(PETSC_HAVE_ELEMENTAL)
PETSC_INTERN PetscErrorCode MatConvert_MPIAIJ_Elemental(Mat,MatType,MatReuse,Mat*);
#endif
#if defined(PETSC_HAVE_SCALAPACK)
PETSC_INTERN PetscErrorCode MatConvert_AIJ_ScaLAPACK(Mat,MatType,MatReuse,Mat*);
#endif
#if defined(PETSC_HAVE_HYPRE)
PETSC_INTERN PetscErrorCode MatConvert_AIJ_HYPRE(Mat,MatType,MatReuse,Mat*);
#endif
#if defined(PETSC_HAVE_CUDA)
PETSC_INTERN PetscErrorCode MatConvert_MPIAIJ_MPIAIJCUSPARSE(Mat,MatType,MatReuse,Mat*);
#endif
#if defined(PETSC_HAVE_KOKKOS_KERNELS)
PETSC_INTERN PetscErrorCode MatConvert_MPIAIJ_MPIAIJKokkos(Mat,MatType,MatReuse,Mat*);
#endif
PETSC_INTERN PetscErrorCode MatConvert_MPIAIJ_MPISELL(Mat,MatType,MatReuse,Mat*);
PETSC_INTERN PetscErrorCode MatConvert_XAIJ_IS(Mat,MatType,MatReuse,Mat*);
PETSC_INTERN PetscErrorCode MatProductSetFromOptions_IS_XAIJ(Mat);

/*
    Computes (B'*A')' since computing B*A directly is untenable

               n                       p                          p
        [             ]       [             ]         [                 ]
      m [      A      ]  *  n [       B     ]   =   m [         C       ]
        [             ]       [             ]         [                 ]

*/
static PetscErrorCode MatMatMultNumeric_MPIDense_MPIAIJ(Mat A,Mat B,Mat C)
{
  Mat            At,Bt,Ct;

  PetscFunctionBegin;
  PetscCall(MatTranspose(A,MAT_INITIAL_MATRIX,&At));
  PetscCall(MatTranspose(B,MAT_INITIAL_MATRIX,&Bt));
  PetscCall(MatMatMult(Bt,At,MAT_INITIAL_MATRIX,PETSC_DEFAULT,&Ct));
  PetscCall(MatDestroy(&At));
  PetscCall(MatDestroy(&Bt));
  PetscCall(MatTranspose(Ct,MAT_REUSE_MATRIX,&C));
  PetscCall(MatDestroy(&Ct));
  PetscFunctionReturn(0);
}

static PetscErrorCode MatMatMultSymbolic_MPIDense_MPIAIJ(Mat A,Mat B,PetscReal fill,Mat C)
{
  PetscBool      cisdense;

  PetscFunctionBegin;
  PetscCheck(A->cmap->n == B->rmap->n,PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"A->cmap->n %" PetscInt_FMT " != B->rmap->n %" PetscInt_FMT,A->cmap->n,B->rmap->n);
  PetscCall(MatSetSizes(C,A->rmap->n,B->cmap->n,A->rmap->N,B->cmap->N));
  PetscCall(MatSetBlockSizesFromMats(C,A,B));
  PetscCall(PetscObjectTypeCompareAny((PetscObject)C,&cisdense,MATMPIDENSE,MATMPIDENSECUDA,""));
  if (!cisdense) {
    PetscCall(MatSetType(C,((PetscObject)A)->type_name));
  }
  PetscCall(MatSetUp(C));

  C->ops->matmultnumeric = MatMatMultNumeric_MPIDense_MPIAIJ;
  PetscFunctionReturn(0);
}

/* ----------------------------------------------------------------*/
static PetscErrorCode MatProductSetFromOptions_MPIDense_MPIAIJ_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_MPIDense_MPIAIJ;
  C->ops->productsymbolic = MatProductSymbolic_AB;
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  if (product->type == MATPRODUCT_AB) {
    PetscCall(MatProductSetFromOptions_MPIDense_MPIAIJ_AB(C));
  }
  PetscFunctionReturn(0);
}

/* Merge two sets of sorted nonzeros and return a CSR for the merged (sequential) matrix

  Input Parameters:

    j1,rowBegin1,rowEnd1,perm1,jmap1: describe the first set of nonzeros (Set1)
    j2,rowBegin2,rowEnd2,perm2,jmap2: describe the second set of nonzeros (Set2)

    mat: both sets' nonzeros are on m rows, where m is the number of local rows of the matrix mat

    For Set1, j1[] contains column indices of the nonzeros.
    For the k-th row (0<=k<m), [rowBegin1[k],rowEnd1[k]) index into j1[] and point to the begin/end nonzero in row k
    respectively (note rowEnd1[k] is not necessarily equal to rwoBegin1[k+1]). Indices in this range of j1[] are sorted,
    but might have repeats. jmap1[t+1] - jmap1[t] is the number of repeats for the t-th unique nonzero in Set1.

    Similar for Set2.

    This routine merges the two sets of nonzeros row by row and removes repeats.

  Output Parameters: (memory is allocated by the caller)

    i[],j[]: the CSR of the merged matrix, which has m rows.
    imap1[]: the k-th unique nonzero in Set1 (k=0,1,...) corresponds to imap1[k]-th unique nonzero in the merged matrix.
    imap2[]: similar to imap1[], but for Set2.
    Note we order nonzeros row-by-row and from left to right.
*/
static PetscErrorCode MatMergeEntries_Internal(Mat mat,const PetscInt j1[],const PetscInt j2[],const PetscCount rowBegin1[],const PetscCount rowEnd1[],
  const PetscCount rowBegin2[],const PetscCount rowEnd2[],const PetscCount jmap1[],const PetscCount jmap2[],
  PetscCount imap1[],PetscCount imap2[],PetscInt i[],PetscInt j[])
{
  PetscInt       r,m; /* Row index of mat */
  PetscCount     t,t1,t2,b1,e1,b2,e2;

  PetscFunctionBegin;
  PetscCall(MatGetLocalSize(mat,&m,NULL));
  t1   = t2 = t = 0; /* Count unique nonzeros of in Set1, Set1 and the merged respectively */
  i[0] = 0;
  for (r=0; r<m; r++) { /* Do row by row merging */
    b1   = rowBegin1[r];
    e1   = rowEnd1[r];
    b2   = rowBegin2[r];
    e2   = rowEnd2[r];
    while (b1 < e1 && b2 < e2) {
      if (j1[b1] == j2[b2]) { /* Same column index and hence same nonzero */
        j[t]      = j1[b1];
        imap1[t1] = t;
        imap2[t2] = t;
        b1       += jmap1[t1+1] - jmap1[t1]; /* Jump to next unique local nonzero */
        b2       += jmap2[t2+1] - jmap2[t2]; /* Jump to next unique remote nonzero */
        t1++; t2++; t++;
      } else if (j1[b1] < j2[b2]) {
        j[t]      = j1[b1];
        imap1[t1] = t;
        b1       += jmap1[t1+1] - jmap1[t1];
        t1++; t++;
      } else {
        j[t]      = j2[b2];
        imap2[t2] = t;
        b2       += jmap2[t2+1] - jmap2[t2];
        t2++; t++;
      }
    }
    /* Merge the remaining in either j1[] or j2[] */
    while (b1 < e1) {
      j[t]      = j1[b1];
      imap1[t1] = t;
      b1       += jmap1[t1+1] - jmap1[t1];
      t1++; t++;
    }
    while (b2 < e2) {
      j[t]      = j2[b2];
      imap2[t2] = t;
      b2       += jmap2[t2+1] - jmap2[t2];
      t2++; t++;
    }
    i[r+1] = t;
  }
  PetscFunctionReturn(0);
}

/* Split nonzeros in a block of local rows into two subsets: those in the diagonal block and those in the off-diagonal block

  Input Parameters:
    mat: an MPI matrix that provides row and column layout information for splitting. Let's say its number of local rows is m.
    n,i[],j[],perm[]: there are n input entries, belonging to m rows. Row/col indices of the entries are stored in i[] and j[]
      respectively, along with a permutation array perm[]. Length of the i[],j[],perm[] arrays is n.

      i[] is already sorted, but within a row, j[] is not sorted and might have repeats.
      i[] might contain negative indices at the beginning, which means the corresponding entries should be ignored in the splitting.

  Output Parameters:
    j[],perm[]: the routine needs to sort j[] within each row along with perm[].
    rowBegin[],rowMid[],rowEnd[]: of length m, and the memory is preallocated and zeroed by the caller.
      They contain indices pointing to j[]. For 0<=r<m, [rowBegin[r],rowMid[r]) point to begin/end entries of row r of the diagonal block,
      and [rowMid[r],rowEnd[r]) point to begin/end entries of row r of the off-diagonal block.

    Aperm[],Ajmap[],Atot,Annz: Arrays are allocated by this routine.
      Atot: number of entries belonging to the diagonal block.
      Annz: number of unique nonzeros belonging to the diagonal block.
      Aperm[Atot] stores values from perm[] for entries belonging to the diagonal block. Length of Aperm[] is Atot, though it may also count
        repeats (i.e., same 'i,j' pair).
      Ajmap[Annz+1] stores the number of repeats of each unique entry belonging to the diagonal block. More precisely, Ajmap[t+1] - Ajmap[t]
        is the number of repeats for the t-th unique entry in the diagonal block. Ajmap[0] is always 0.

      Atot: number of entries belonging to the diagonal block
      Annz: number of unique nonzeros belonging to the diagonal block.

    Bperm[], Bjmap[], Btot, Bnnz are similar but for the off-diagonal block.

    Aperm[],Bperm[],Ajmap[] and Bjmap[] are allocated separately by this routine with PetscMalloc1().
*/
static PetscErrorCode MatSplitEntries_Internal(Mat mat,PetscCount n,const PetscInt i[],PetscInt j[],
  PetscCount perm[],PetscCount rowBegin[],PetscCount rowMid[],PetscCount rowEnd[],
  PetscCount *Atot_,PetscCount **Aperm_,PetscCount *Annz_,PetscCount **Ajmap_,
  PetscCount *Btot_,PetscCount **Bperm_,PetscCount *Bnnz_,PetscCount **Bjmap_)
{
  PetscInt          cstart,cend,rstart,rend,row,col;
  PetscCount        Atot=0,Btot=0; /* Total number of nonzeros in the diagonal and off-diagonal blocks */
  PetscCount        Annz=0,Bnnz=0; /* Number of unique nonzeros in the diagonal and off-diagonal blocks */
  PetscCount        k,m,p,q,r,s,mid;
  PetscCount        *Aperm,*Bperm,*Ajmap,*Bjmap;

  PetscFunctionBegin;
  PetscCall(PetscLayoutGetRange(mat->rmap,&rstart,&rend));
  PetscCall(PetscLayoutGetRange(mat->cmap,&cstart,&cend));
  m    = rend - rstart;

  for (k=0; k<n; k++) {if (i[k]>=0) break;} /* Skip negative rows */

  /* Process [k,n): sort and partition each local row into diag and offdiag portions,
     fill rowBegin[], rowMid[], rowEnd[], and count Atot, Btot, Annz, Bnnz.
  */
  while (k<n) {
    row = i[k];
    /* Entries in [k,s) are in one row. Shift diagonal block col indices so that diag is ahead of offdiag after sorting the row */
    for (s=k; s<n; s++) if (i[s] != row) break;
    for (p=k; p<s; p++) {
      if (j[p] >= cstart && j[p] < cend) j[p] -= PETSC_MAX_INT; /* Shift diag columns to range of [-PETSC_MAX_INT, -1]  */
      else PetscAssert((j[p] >= 0) && (j[p] <= mat->cmap->N),PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Column index %" PetscInt_FMT " is out of range",j[p]);
    }
    PetscCall(PetscSortIntWithCountArray(s-k,j+k,perm+k));
    PetscCall(PetscSortedIntUpperBound(j,k,s,-1,&mid)); /* Separate [k,s) into [k,mid) for diag and [mid,s) for offdiag */
    rowBegin[row-rstart] = k;
    rowMid[row-rstart]   = mid;
    rowEnd[row-rstart]   = s;

    /* Count nonzeros of this diag/offdiag row, which might have repeats */
    Atot += mid - k;
    Btot += s - mid;

    /* Count unique nonzeros of this diag/offdiag row */
    for (p=k; p<mid;) {
      col = j[p];
      do {j[p] += PETSC_MAX_INT; p++;} while (p<mid && j[p] == col); /* Revert the modified diagonal indices */
      Annz++;
    }

    for (p=mid; p<s;) {
      col = j[p];
      do {p++;} while (p<s && j[p] == col);
      Bnnz++;
    }
    k = s;
  }

  /* Allocation according to Atot, Btot, Annz, Bnnz */
  PetscCall(PetscMalloc1(Atot,&Aperm));
  PetscCall(PetscMalloc1(Btot,&Bperm));
  PetscCall(PetscMalloc1(Annz+1,&Ajmap));
  PetscCall(PetscMalloc1(Bnnz+1,&Bjmap));

  /* Re-scan indices and copy diag/offdiag permutation indices to Aperm, Bperm and also fill Ajmap and Bjmap */
  Ajmap[0] = Bjmap[0] = Atot = Btot = Annz = Bnnz = 0;
  for (r=0; r<m; r++) {
    k     = rowBegin[r];
    mid   = rowMid[r];
    s     = rowEnd[r];
    PetscCall(PetscArraycpy(Aperm+Atot,perm+k,  mid-k));
    PetscCall(PetscArraycpy(Bperm+Btot,perm+mid,s-mid));
    Atot += mid - k;
    Btot += s - mid;

    /* Scan column indices in this row and find out how many repeats each unique nonzero has */
    for (p=k; p<mid;) {
      col = j[p];
      q   = p;
      do {p++;} while (p<mid && j[p] == col);
      Ajmap[Annz+1] = Ajmap[Annz] + (p - q);
      Annz++;
    }

    for (p=mid; p<s;) {
      col = j[p];
      q   = p;
      do {p++;} while (p<s && j[p] == col);
      Bjmap[Bnnz+1] = Bjmap[Bnnz] + (p - q);
      Bnnz++;
    }
  }
  /* Output */
  *Aperm_ = Aperm;
  *Annz_  = Annz;
  *Atot_  = Atot;
  *Ajmap_ = Ajmap;
  *Bperm_ = Bperm;
  *Bnnz_  = Bnnz;
  *Btot_  = Btot;
  *Bjmap_ = Bjmap;
  PetscFunctionReturn(0);
}

/* Expand the jmap[] array to make a new one in view of nonzeros in the merged matrix

  Input Parameters:
    nnz1: number of unique nonzeros in a set that was used to produce imap[], jmap[]
    nnz:  number of unique nonzeros in the merged matrix
    imap[nnz1]: i-th nonzero in the set is the imap[i]-th nonzero in the merged matrix
    jmap[nnz1+1]: i-th nonzeron in the set has jmap[i+1] - jmap[i] repeats in the set

  Output Parameter: (memory is allocated by the caller)
    jmap_new[nnz+1]: i-th nonzero in the merged matrix has jmap_new[i+1] - jmap_new[i] repeats in the set

  Example:
    nnz1 = 4
    nnz  = 6
    imap = [1,3,4,5]
    jmap = [0,3,5,6,7]
   then,
    jmap_new = [0,0,3,3,5,6,7]
*/
static PetscErrorCode ExpandJmap_Internal(PetscCount nnz1,PetscCount nnz,const PetscCount imap[],const PetscCount jmap[],PetscCount jmap_new[])
{
  PetscCount k,p;

  PetscFunctionBegin;
  jmap_new[0] = 0;
  p = nnz; /* p loops over jmap_new[] backwards */
  for (k=nnz1-1; k>=0; k--) { /* k loops over imap[] */
    for (; p > imap[k]; p--) jmap_new[p] = jmap[k+1];
  }
  for (; p >= 0; p--) jmap_new[p] = jmap[0];
  PetscFunctionReturn(0);
}

PetscErrorCode MatSetPreallocationCOO_MPIAIJ(Mat mat, PetscCount coo_n, const PetscInt coo_i[], const PetscInt coo_j[])
{
  MPI_Comm                  comm;
  PetscMPIInt               rank,size;
  PetscInt                  m,n,M,N,rstart,rend,cstart,cend; /* Sizes, indices of row/col, therefore with type PetscInt */
  PetscCount                k,p,q,rem; /* Loop variables over coo arrays */
  Mat_MPIAIJ                *mpiaij = (Mat_MPIAIJ*)mat->data;

  PetscFunctionBegin;
  PetscCall(PetscFree(mpiaij->garray));
  PetscCall(VecDestroy(&mpiaij->lvec));
#if defined(PETSC_USE_CTABLE)
  PetscCall(PetscTableDestroy(&mpiaij->colmap));
#else
  PetscCall(PetscFree(mpiaij->colmap));
#endif
  PetscCall(VecScatterDestroy(&mpiaij->Mvctx));
  mat->assembled = PETSC_FALSE;
  mat->was_assembled = PETSC_FALSE;
  PetscCall(MatResetPreallocationCOO_MPIAIJ(mat));

  PetscCall(PetscObjectGetComm((PetscObject)mat,&comm));
  PetscCallMPI(MPI_Comm_size(comm,&size));
  PetscCallMPI(MPI_Comm_rank(comm,&rank));
  PetscCall(PetscLayoutSetUp(mat->rmap));
  PetscCall(PetscLayoutSetUp(mat->cmap));
  PetscCall(PetscLayoutGetRange(mat->rmap,&rstart,&rend));
  PetscCall(PetscLayoutGetRange(mat->cmap,&cstart,&cend));
  PetscCall(MatGetLocalSize(mat,&m,&n));
  PetscCall(MatGetSize(mat,&M,&N));

  /* ---------------------------------------------------------------------------*/
  /* Sort (i,j) by row along with a permutation array, so that the to-be-ignored */
  /* entries come first, then local rows, then remote rows.                     */
  /* ---------------------------------------------------------------------------*/
  PetscCount n1 = coo_n,*perm1;
  PetscInt   *i1,*j1; /* Copies of input COOs along with a permutation array */
  PetscCall(PetscMalloc3(n1,&i1,n1,&j1,n1,&perm1));
  PetscCall(PetscArraycpy(i1,coo_i,n1)); /* Make a copy since we'll modify it */
  PetscCall(PetscArraycpy(j1,coo_j,n1));
  for (k=0; k<n1; k++) perm1[k] = k;

  /* Manipulate indices so that entries with negative row or col indices will have smallest
     row indices, local entries will have greater but negative row indices, and remote entries
     will have positive row indices.
  */
  for (k=0; k<n1; k++) {
    if (i1[k] < 0 || j1[k] < 0) i1[k] = PETSC_MIN_INT; /* e.g., -2^31, minimal to move them ahead */
    else if (i1[k] >= rstart && i1[k] < rend) i1[k] -= PETSC_MAX_INT; /* e.g., minus 2^31-1 to shift local rows to range of [-PETSC_MAX_INT, -1] */
    else {
      PetscCheck(!mat->nooffprocentries,PETSC_COMM_SELF,PETSC_ERR_USER_INPUT,"MAT_NO_OFF_PROC_ENTRIES is set but insert to remote rows");
      if (mpiaij->donotstash) i1[k] = PETSC_MIN_INT; /* Ignore offproc entries as if they had negative indices */
    }
  }

  /* Sort by row; after that, [0,k) have ignored entires, [k,rem) have local rows and [rem,n1) have remote rows */
  PetscCall(PetscSortIntWithIntCountArrayPair(n1,i1,j1,perm1));
  for (k=0; k<n1; k++) {if (i1[k] > PETSC_MIN_INT) break;} /* Advance k to the first entry we need to take care of */
  PetscCall(PetscSortedIntUpperBound(i1,k,n1,rend-1-PETSC_MAX_INT,&rem)); /* rem is upper bound of the last local row */
  for (; k<rem; k++) i1[k] += PETSC_MAX_INT; /* Revert row indices of local rows*/

  /* ---------------------------------------------------------------------------*/
  /*           Split local rows into diag/offdiag portions                      */
  /* ---------------------------------------------------------------------------*/
  PetscCount   *rowBegin1,*rowMid1,*rowEnd1;
  PetscCount   *Ajmap1,*Aperm1,*Bjmap1,*Bperm1,*Cperm1;
  PetscCount   Annz1,Bnnz1,Atot1,Btot1;

  PetscCall(PetscCalloc3(m,&rowBegin1,m,&rowMid1,m,&rowEnd1));
  PetscCall(PetscMalloc1(n1-rem,&Cperm1));
  PetscCall(MatSplitEntries_Internal(mat,rem,i1,j1,perm1,rowBegin1,rowMid1,rowEnd1,&Atot1,&Aperm1,&Annz1,&Ajmap1,&Btot1,&Bperm1,&Bnnz1,&Bjmap1));

  /* ---------------------------------------------------------------------------*/
  /*           Send remote rows to their owner                                  */
  /* ---------------------------------------------------------------------------*/
  /* Find which rows should be sent to which remote ranks*/
  PetscInt       nsend = 0; /* Number of MPI ranks to send data to */
  PetscMPIInt    *sendto; /* [nsend], storing remote ranks */
  PetscInt       *nentries; /* [nsend], storing number of entries sent to remote ranks; Assume PetscInt is big enough for this count, and error if not */
  const PetscInt *ranges;
  PetscInt       maxNsend = size >= 128? 128 : size; /* Assume max 128 neighbors; realloc when needed */

  PetscCall(PetscLayoutGetRanges(mat->rmap,&ranges));
  PetscCall(PetscMalloc2(maxNsend,&sendto,maxNsend,&nentries));
  for (k=rem; k<n1;) {
    PetscMPIInt  owner;
    PetscInt     firstRow,lastRow;

    /* Locate a row range */
    firstRow = i1[k]; /* first row of this owner */
    PetscCall(PetscLayoutFindOwner(mat->rmap,firstRow,&owner));
    lastRow  = ranges[owner+1]-1; /* last row of this owner */

    /* Find the first index 'p' in [k,n) with i[p] belonging to next owner */
    PetscCall(PetscSortedIntUpperBound(i1,k,n1,lastRow,&p));

    /* All entries in [k,p) belong to this remote owner */
    if (nsend >= maxNsend) { /* Double the remote ranks arrays if not long enough */
      PetscMPIInt *sendto2;
      PetscInt    *nentries2;
      PetscInt    maxNsend2 = (maxNsend <= size/2) ? maxNsend*2 : size;

      PetscCall(PetscMalloc2(maxNsend2,&sendto2,maxNsend2,&nentries2));
      PetscCall(PetscArraycpy(sendto2,sendto,maxNsend));
      PetscCall(PetscArraycpy(nentries2,nentries2,maxNsend+1));
      PetscCall(PetscFree2(sendto,nentries2));
      sendto      = sendto2;
      nentries    = nentries2;
      maxNsend    = maxNsend2;
    }
    sendto[nsend]   = owner;
    nentries[nsend] = p - k;
    PetscCall(PetscCountCast(p-k,&nentries[nsend]));
    nsend++;
    k = p;
  }

  /* Build 1st SF to know offsets on remote to send data */
  PetscSF     sf1;
  PetscInt    nroots = 1,nroots2 = 0;
  PetscInt    nleaves = nsend,nleaves2 = 0;
  PetscInt    *offsets;
  PetscSFNode *iremote;

  PetscCall(PetscSFCreate(comm,&sf1));
  PetscCall(PetscMalloc1(nsend,&iremote));
  PetscCall(PetscMalloc1(nsend,&offsets));
  for (k=0; k<nsend; k++) {
    iremote[k].rank  = sendto[k];
    iremote[k].index = 0;
    nleaves2        += nentries[k];
    PetscCheck(nleaves2 >= 0,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Number of SF leaves is too large for PetscInt");
  }
  PetscCall(PetscSFSetGraph(sf1,nroots,nleaves,NULL,PETSC_OWN_POINTER,iremote,PETSC_OWN_POINTER));
  PetscCall(PetscSFFetchAndOpWithMemTypeBegin(sf1,MPIU_INT,PETSC_MEMTYPE_HOST,&nroots2/*rootdata*/,PETSC_MEMTYPE_HOST,nentries/*leafdata*/,PETSC_MEMTYPE_HOST,offsets/*leafupdate*/,MPI_SUM));
  PetscCall(PetscSFFetchAndOpEnd(sf1,MPIU_INT,&nroots2,nentries,offsets,MPI_SUM)); /* Would nroots2 overflow, we check offsets[] below */
  PetscCall(PetscSFDestroy(&sf1));
  PetscAssert(nleaves2 == n1-rem,PETSC_COMM_SELF,PETSC_ERR_PLIB,"nleaves2 %" PetscInt_FMT " != number of remote entries %" PetscCount_FMT "",nleaves2,n1-rem);

  /* Build 2nd SF to send remote COOs to their owner */
  PetscSF sf2;
  nroots  = nroots2;
  nleaves = nleaves2;
  PetscCall(PetscSFCreate(comm,&sf2));
  PetscCall(PetscSFSetFromOptions(sf2));
  PetscCall(PetscMalloc1(nleaves,&iremote));
  p       = 0;
  for (k=0; k<nsend; k++) {
    PetscCheck(offsets[k] >= 0,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Number of SF roots is too large for PetscInt");
    for (q=0; q<nentries[k]; q++,p++) {
      iremote[p].rank  = sendto[k];
      iremote[p].index = offsets[k] + q;
    }
  }
  PetscCall(PetscSFSetGraph(sf2,nroots,nleaves,NULL,PETSC_OWN_POINTER,iremote,PETSC_OWN_POINTER));

  /* sf2 only sends contiguous leafdata to contiguous rootdata. We record the permutation which will be used to fill leafdata */
  PetscCall(PetscArraycpy(Cperm1,perm1+rem,n1-rem));

  /* Send the remote COOs to their owner */
  PetscInt   n2 = nroots,*i2,*j2; /* Buffers for received COOs from other ranks, along with a permutation array */
  PetscCount *perm2; /* Though PetscInt is enough for remote entries, we use PetscCount here as we want to reuse MatSplitEntries_Internal() */
  PetscCall(PetscMalloc3(n2,&i2,n2,&j2,n2,&perm2));
  PetscCall(PetscSFReduceWithMemTypeBegin(sf2,MPIU_INT,PETSC_MEMTYPE_HOST,i1+rem,PETSC_MEMTYPE_HOST,i2,MPI_REPLACE));
  PetscCall(PetscSFReduceEnd(sf2,MPIU_INT,i1+rem,i2,MPI_REPLACE));
  PetscCall(PetscSFReduceWithMemTypeBegin(sf2,MPIU_INT,PETSC_MEMTYPE_HOST,j1+rem,PETSC_MEMTYPE_HOST,j2,MPI_REPLACE));
  PetscCall(PetscSFReduceEnd(sf2,MPIU_INT,j1+rem,j2,MPI_REPLACE));

  PetscCall(PetscFree(offsets));
  PetscCall(PetscFree2(sendto,nentries));

  /* ---------------------------------------------------------------*/
  /* Sort received COOs by row along with the permutation array     */
  /* ---------------------------------------------------------------*/
  for (k=0; k<n2; k++) perm2[k] = k;
  PetscCall(PetscSortIntWithIntCountArrayPair(n2,i2,j2,perm2));

  /* ---------------------------------------------------------------*/
  /* Split received COOs into diag/offdiag portions                 */
  /* ---------------------------------------------------------------*/
  PetscCount  *rowBegin2,*rowMid2,*rowEnd2;
  PetscCount  *Ajmap2,*Aperm2,*Bjmap2,*Bperm2;
  PetscCount  Annz2,Bnnz2,Atot2,Btot2;

  PetscCall(PetscCalloc3(m,&rowBegin2,m,&rowMid2,m,&rowEnd2));
  PetscCall(MatSplitEntries_Internal(mat,n2,i2,j2,perm2,rowBegin2,rowMid2,rowEnd2,&Atot2,&Aperm2,&Annz2,&Ajmap2,&Btot2,&Bperm2,&Bnnz2,&Bjmap2));

  /* --------------------------------------------------------------------------*/
  /* Merge local COOs with received COOs: diag with diag, offdiag with offdiag */
  /* --------------------------------------------------------------------------*/
  PetscInt   *Ai,*Bi;
  PetscInt   *Aj,*Bj;

  PetscCall(PetscMalloc1(m+1,&Ai));
  PetscCall(PetscMalloc1(m+1,&Bi));
  PetscCall(PetscMalloc1(Annz1+Annz2,&Aj)); /* Since local and remote entries might have dups, we might allocate excess memory */
  PetscCall(PetscMalloc1(Bnnz1+Bnnz2,&Bj));

  PetscCount *Aimap1,*Bimap1,*Aimap2,*Bimap2;
  PetscCall(PetscMalloc1(Annz1,&Aimap1));
  PetscCall(PetscMalloc1(Bnnz1,&Bimap1));
  PetscCall(PetscMalloc1(Annz2,&Aimap2));
  PetscCall(PetscMalloc1(Bnnz2,&Bimap2));

  PetscCall(MatMergeEntries_Internal(mat,j1,j2,rowBegin1,rowMid1,rowBegin2,rowMid2,Ajmap1,Ajmap2,Aimap1,Aimap2,Ai,Aj));
  PetscCall(MatMergeEntries_Internal(mat,j1,j2,rowMid1,  rowEnd1,rowMid2,  rowEnd2,Bjmap1,Bjmap2,Bimap1,Bimap2,Bi,Bj));

  /* --------------------------------------------------------------------------*/
  /* Expand Ajmap1/Bjmap1 to make them based off nonzeros in A/B, since we     */
  /* expect nonzeros in A/B most likely have local contributing entries        */
  /* --------------------------------------------------------------------------*/
  PetscInt Annz = Ai[m];
  PetscInt Bnnz = Bi[m];
  PetscCount *Ajmap1_new,*Bjmap1_new;

  PetscCall(PetscMalloc1(Annz+1,&Ajmap1_new));
  PetscCall(PetscMalloc1(Bnnz+1,&Bjmap1_new));

  PetscCall(ExpandJmap_Internal(Annz1,Annz,Aimap1,Ajmap1,Ajmap1_new));
  PetscCall(ExpandJmap_Internal(Bnnz1,Bnnz,Bimap1,Bjmap1,Bjmap1_new));

  PetscCall(PetscFree(Aimap1));
  PetscCall(PetscFree(Ajmap1));
  PetscCall(PetscFree(Bimap1));
  PetscCall(PetscFree(Bjmap1));
  PetscCall(PetscFree3(rowBegin1,rowMid1,rowEnd1));
  PetscCall(PetscFree3(rowBegin2,rowMid2,rowEnd2));
  PetscCall(PetscFree3(i1,j1,perm1));
  PetscCall(PetscFree3(i2,j2,perm2));

  Ajmap1 = Ajmap1_new;
  Bjmap1 = Bjmap1_new;

  /* Reallocate Aj, Bj once we know actual numbers of unique nonzeros in A and B */
  if (Annz < Annz1 + Annz2) {
    PetscInt *Aj_new;
    PetscCall(PetscMalloc1(Annz,&Aj_new));
    PetscCall(PetscArraycpy(Aj_new,Aj,Annz));
    PetscCall(PetscFree(Aj));
    Aj   = Aj_new;
  }

  if (Bnnz < Bnnz1 + Bnnz2) {
    PetscInt *Bj_new;
    PetscCall(PetscMalloc1(Bnnz,&Bj_new));
    PetscCall(PetscArraycpy(Bj_new,Bj,Bnnz));
    PetscCall(PetscFree(Bj));
    Bj   = Bj_new;
  }

  /* --------------------------------------------------------------------------------*/
  /* Create new submatrices for on-process and off-process coupling                  */
  /* --------------------------------------------------------------------------------*/
  PetscScalar   *Aa,*Ba;
  MatType       rtype;
  Mat_SeqAIJ    *a,*b;
  PetscCall(PetscCalloc1(Annz,&Aa)); /* Zero matrix on device */
  PetscCall(PetscCalloc1(Bnnz,&Ba));
  /* make Aj[] local, i.e, based off the start column of the diagonal portion */
  if (cstart) {for (k=0; k<Annz; k++) Aj[k] -= cstart;}
  PetscCall(MatDestroy(&mpiaij->A));
  PetscCall(MatDestroy(&mpiaij->B));
  PetscCall(MatGetRootType_Private(mat,&rtype));
  PetscCall(MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,m,n,Ai,Aj,Aa,&mpiaij->A));
  PetscCall(MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,m,mat->cmap->N,Bi,Bj,Ba,&mpiaij->B));
  PetscCall(MatSetUpMultiply_MPIAIJ(mat));

  a = (Mat_SeqAIJ*)mpiaij->A->data;
  b = (Mat_SeqAIJ*)mpiaij->B->data;
  a->singlemalloc = b->singlemalloc = PETSC_FALSE; /* Let newmat own Ai,Aj,Aa,Bi,Bj,Ba */
  a->free_a       = b->free_a       = PETSC_TRUE;
  a->free_ij      = b->free_ij      = PETSC_TRUE;

  /* conversion must happen AFTER multiply setup */
  PetscCall(MatConvert(mpiaij->A,rtype,MAT_INPLACE_MATRIX,&mpiaij->A));
  PetscCall(MatConvert(mpiaij->B,rtype,MAT_INPLACE_MATRIX,&mpiaij->B));
  PetscCall(VecDestroy(&mpiaij->lvec));
  PetscCall(MatCreateVecs(mpiaij->B,&mpiaij->lvec,NULL));
  PetscCall(PetscLogObjectParent((PetscObject)mat,(PetscObject)mpiaij->lvec));

  mpiaij->coo_n   = coo_n;
  mpiaij->coo_sf  = sf2;
  mpiaij->sendlen = nleaves;
  mpiaij->recvlen = nroots;

  mpiaij->Annz    = Annz;
  mpiaij->Bnnz    = Bnnz;

  mpiaij->Annz2   = Annz2;
  mpiaij->Bnnz2   = Bnnz2;

  mpiaij->Atot1   = Atot1;
  mpiaij->Atot2   = Atot2;
  mpiaij->Btot1   = Btot1;
  mpiaij->Btot2   = Btot2;

  mpiaij->Ajmap1  = Ajmap1;
  mpiaij->Aperm1  = Aperm1;

  mpiaij->Bjmap1  = Bjmap1;
  mpiaij->Bperm1  = Bperm1;

  mpiaij->Aimap2  = Aimap2;
  mpiaij->Ajmap2  = Ajmap2;
  mpiaij->Aperm2  = Aperm2;

  mpiaij->Bimap2  = Bimap2;
  mpiaij->Bjmap2  = Bjmap2;
  mpiaij->Bperm2  = Bperm2;

  mpiaij->Cperm1  = Cperm1;

  /* Allocate in preallocation. If not used, it has zero cost on host */
  PetscCall(PetscMalloc2(mpiaij->sendlen,&mpiaij->sendbuf,mpiaij->recvlen,&mpiaij->recvbuf));
  PetscFunctionReturn(0);
}

static PetscErrorCode MatSetValuesCOO_MPIAIJ(Mat mat,const PetscScalar v[],InsertMode imode)
{
  Mat_MPIAIJ           *mpiaij = (Mat_MPIAIJ*)mat->data;
  Mat                  A = mpiaij->A,B = mpiaij->B;
  PetscCount           Annz = mpiaij->Annz,Annz2 = mpiaij->Annz2,Bnnz = mpiaij->Bnnz,Bnnz2 = mpiaij->Bnnz2;
  PetscScalar          *Aa,*Ba;
  PetscScalar          *sendbuf = mpiaij->sendbuf;
  PetscScalar          *recvbuf = mpiaij->recvbuf;
  const PetscCount     *Ajmap1 = mpiaij->Ajmap1,*Ajmap2 = mpiaij->Ajmap2,*Aimap2 = mpiaij->Aimap2;
  const PetscCount     *Bjmap1 = mpiaij->Bjmap1,*Bjmap2 = mpiaij->Bjmap2,*Bimap2 = mpiaij->Bimap2;
  const PetscCount     *Aperm1 = mpiaij->Aperm1,*Aperm2 = mpiaij->Aperm2,*Bperm1 = mpiaij->Bperm1,*Bperm2 = mpiaij->Bperm2;
  const PetscCount     *Cperm1 = mpiaij->Cperm1;

  PetscFunctionBegin;
  PetscCall(MatSeqAIJGetArray(A,&Aa)); /* Might read and write matrix values */
  PetscCall(MatSeqAIJGetArray(B,&Ba));

  /* Pack entries to be sent to remote */
  for (PetscCount i=0; i<mpiaij->sendlen; i++) sendbuf[i] = v[Cperm1[i]];

  /* Send remote entries to their owner and overlap the communication with local computation */
  PetscCall(PetscSFReduceWithMemTypeBegin(mpiaij->coo_sf,MPIU_SCALAR,PETSC_MEMTYPE_HOST,sendbuf,PETSC_MEMTYPE_HOST,recvbuf,MPI_REPLACE));
  /* Add local entries to A and B */
  for (PetscCount i=0; i<Annz; i++) { /* All nonzeros in A are either zero'ed or added with a value (i.e., initialized) */
    PetscScalar sum = 0.0; /* Do partial summation first to improve numerical stablility */
    for (PetscCount k=Ajmap1[i]; k<Ajmap1[i+1]; k++) sum += v[Aperm1[k]];
    Aa[i] = (imode == INSERT_VALUES? 0.0 : Aa[i]) + sum;
  }
  for (PetscCount i=0; i<Bnnz; i++) {
    PetscScalar sum = 0.0;
    for (PetscCount k=Bjmap1[i]; k<Bjmap1[i+1]; k++) sum += v[Bperm1[k]];
    Ba[i] = (imode == INSERT_VALUES? 0.0 : Ba[i]) + sum;
  }
  PetscCall(PetscSFReduceEnd(mpiaij->coo_sf,MPIU_SCALAR,sendbuf,recvbuf,MPI_REPLACE));

  /* Add received remote entries to A and B */
  for (PetscCount i=0; i<Annz2; i++) {
    for (PetscCount k=Ajmap2[i]; k<Ajmap2[i+1]; k++) Aa[Aimap2[i]] += recvbuf[Aperm2[k]];
  }
  for (PetscCount i=0; i<Bnnz2; i++) {
    for (PetscCount k=Bjmap2[i]; k<Bjmap2[i+1]; k++) Ba[Bimap2[i]] += recvbuf[Bperm2[k]];
  }
  PetscCall(MatSeqAIJRestoreArray(A,&Aa));
  PetscCall(MatSeqAIJRestoreArray(B,&Ba));
  PetscFunctionReturn(0);
}

/* ----------------------------------------------------------------*/

/*MC
   MATMPIAIJ - MATMPIAIJ = "mpiaij" - A matrix type to be used for parallel sparse matrices.

   Options Database Keys:
. -mat_type mpiaij - sets the matrix type to "mpiaij" during a call to MatSetFromOptions()

   Level: beginner

   Notes:
    MatSetValues() may be called for this matrix type with a NULL argument for the numerical values,
    in this case the values associated with the rows and columns one passes in are set to zero
    in the matrix

    MatSetOptions(,MAT_STRUCTURE_ONLY,PETSC_TRUE) may be called for this matrix type. In this no
    space is allocated for the nonzero entries and any entries passed with MatSetValues() are ignored

.seealso: `MatCreateAIJ()`
M*/

PETSC_EXTERN PetscErrorCode MatCreate_MPIAIJ(Mat B)
{
  Mat_MPIAIJ     *b;
  PetscMPIInt    size;

  PetscFunctionBegin;
  PetscCallMPI(MPI_Comm_size(PetscObjectComm((PetscObject)B),&size));

  PetscCall(PetscNewLog(B,&b));
  B->data       = (void*)b;
  PetscCall(PetscMemcpy(B->ops,&MatOps_Values,sizeof(struct _MatOps)));
  B->assembled  = PETSC_FALSE;
  B->insertmode = NOT_SET_VALUES;
  b->size       = size;

  PetscCallMPI(MPI_Comm_rank(PetscObjectComm((PetscObject)B),&b->rank));

  /* build cache for off array entries formed */
  PetscCall(MatStashCreate_Private(PetscObjectComm((PetscObject)B),1,&B->stash));

  b->donotstash  = PETSC_FALSE;
  b->colmap      = NULL;
  b->garray      = NULL;
  b->roworiented = PETSC_TRUE;

  /* stuff used for matrix vector multiply */
  b->lvec  = NULL;
  b->Mvctx = NULL;

  /* stuff for MatGetRow() */
  b->rowindices   = NULL;
  b->rowvalues    = NULL;
  b->getrowactive = PETSC_FALSE;

  /* flexible pointer used in CUSPARSE classes */
  b->spptr = NULL;

  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatMPIAIJSetUseScalableIncreaseOverlap_C",MatMPIAIJSetUseScalableIncreaseOverlap_MPIAIJ));
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatStoreValues_C",MatStoreValues_MPIAIJ));
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatRetrieveValues_C",MatRetrieveValues_MPIAIJ));
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatIsTranspose_C",MatIsTranspose_MPIAIJ));
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatMPIAIJSetPreallocation_C",MatMPIAIJSetPreallocation_MPIAIJ));
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatResetPreallocation_C",MatResetPreallocation_MPIAIJ));
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatMPIAIJSetPreallocationCSR_C",MatMPIAIJSetPreallocationCSR_MPIAIJ));
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatDiagonalScaleLocal_C",MatDiagonalScaleLocal_MPIAIJ));
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatConvert_mpiaij_mpiaijperm_C",MatConvert_MPIAIJ_MPIAIJPERM));
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatConvert_mpiaij_mpiaijsell_C",MatConvert_MPIAIJ_MPIAIJSELL));
#if defined(PETSC_HAVE_CUDA)
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatConvert_mpiaij_mpiaijcusparse_C",MatConvert_MPIAIJ_MPIAIJCUSPARSE));
#endif
#if defined(PETSC_HAVE_KOKKOS_KERNELS)
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatConvert_mpiaij_mpiaijkokkos_C",MatConvert_MPIAIJ_MPIAIJKokkos));
#endif
#if defined(PETSC_HAVE_MKL_SPARSE)
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatConvert_mpiaij_mpiaijmkl_C",MatConvert_MPIAIJ_MPIAIJMKL));
#endif
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatConvert_mpiaij_mpiaijcrl_C",MatConvert_MPIAIJ_MPIAIJCRL));
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatConvert_mpiaij_mpibaij_C",MatConvert_MPIAIJ_MPIBAIJ));
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatConvert_mpiaij_mpisbaij_C",MatConvert_MPIAIJ_MPISBAIJ));
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatConvert_mpiaij_mpidense_C",MatConvert_MPIAIJ_MPIDense));
#if defined(PETSC_HAVE_ELEMENTAL)
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatConvert_mpiaij_elemental_C",MatConvert_MPIAIJ_Elemental));
#endif
#if defined(PETSC_HAVE_SCALAPACK)
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatConvert_mpiaij_scalapack_C",MatConvert_AIJ_ScaLAPACK));
#endif
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatConvert_mpiaij_is_C",MatConvert_XAIJ_IS));
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatConvert_mpiaij_mpisell_C",MatConvert_MPIAIJ_MPISELL));
#if defined(PETSC_HAVE_HYPRE)
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatConvert_mpiaij_hypre_C",MatConvert_AIJ_HYPRE));
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatProductSetFromOptions_transpose_mpiaij_mpiaij_C",MatProductSetFromOptions_Transpose_AIJ_AIJ));
#endif
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatProductSetFromOptions_is_mpiaij_C",MatProductSetFromOptions_IS_XAIJ));
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatProductSetFromOptions_mpiaij_mpiaij_C",MatProductSetFromOptions_MPIAIJ));
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatSetPreallocationCOO_C",MatSetPreallocationCOO_MPIAIJ));
  PetscCall(PetscObjectComposeFunction((PetscObject)B,"MatSetValuesCOO_C",MatSetValuesCOO_MPIAIJ));
  PetscCall(PetscObjectChangeTypeName((PetscObject)B,MATMPIAIJ));
  PetscFunctionReturn(0);
}

/*@C
     MatCreateMPIAIJWithSplitArrays - creates a MPI AIJ matrix using arrays that contain the "diagonal"
         and "off-diagonal" part of the matrix in CSR format.

   Collective

   Input Parameters:
+  comm - MPI communicator
.  m - number of local rows (Cannot be PETSC_DECIDE)
.  n - This value should be the same as the local size used in creating the
       x vector for the matrix-vector product y = Ax. (or PETSC_DECIDE to have
       calculated if N is given) For square matrices n is almost always m.
.  M - number of global rows (or PETSC_DETERMINE to have calculated if m is given)
.  N - number of global columns (or PETSC_DETERMINE to have calculated if n is given)
.   i - row indices for "diagonal" portion of matrix; that is i[0] = 0, i[row] = i[row-1] + number of elements in that row of the matrix
.   j - column indices, which must be local, i.e., based off the start column of the diagonal portion
.   a - matrix values
.   oi - row indices for "off-diagonal" portion of matrix; that is oi[0] = 0, oi[row] = oi[row-1] + number of elements in that row of the matrix
.   oj - column indices, which must be global, representing global columns in the MPIAIJ matrix
-   oa - matrix values

   Output Parameter:
.   mat - the matrix

   Level: advanced

   Notes:
       The i, j, and a arrays ARE NOT copied by this routine into the internal format used by PETSc. The user
       must free the arrays once the matrix has been destroyed and not before.

       The i and j indices are 0 based

       See MatCreateAIJ() for the definition of "diagonal" and "off-diagonal" portion of the matrix

       This sets local rows and cannot be used to set off-processor values.

       Use of this routine is discouraged because it is inflexible and cumbersome to use. It is extremely rare that a
       legacy application natively assembles into exactly this split format. The code to do so is nontrivial and does
       not easily support in-place reassembly. It is recommended to use MatSetValues() (or a variant thereof) because
       the resulting assembly is easier to implement, will work with any matrix format, and the user does not have to
       keep track of the underlying array. Use MatSetOption(A,MAT_NO_OFF_PROC_ENTRIES,PETSC_TRUE) to disable all
       communication if it is known that only local entries will be set.

.seealso: `MatCreate()`, `MatCreateSeqAIJ()`, `MatSetValues()`, `MatMPIAIJSetPreallocation()`, `MatMPIAIJSetPreallocationCSR()`,
          `MATMPIAIJ`, `MatCreateAIJ()`, `MatCreateMPIAIJWithArrays()`
@*/
PetscErrorCode MatCreateMPIAIJWithSplitArrays(MPI_Comm comm,PetscInt m,PetscInt n,PetscInt M,PetscInt N,PetscInt i[],PetscInt j[],PetscScalar a[],PetscInt oi[], PetscInt oj[],PetscScalar oa[],Mat *mat)
{
  Mat_MPIAIJ     *maij;

  PetscFunctionBegin;
  PetscCheck(m >= 0,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"local number of rows (m) cannot be PETSC_DECIDE, or negative");
  PetscCheck(i[0] == 0,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"i (row indices) must start with 0");
  PetscCheck(oi[0] == 0,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"oi (row indices) must start with 0");
  PetscCall(MatCreate(comm,mat));
  PetscCall(MatSetSizes(*mat,m,n,M,N));
  PetscCall(MatSetType(*mat,MATMPIAIJ));
  maij = (Mat_MPIAIJ*) (*mat)->data;

  (*mat)->preallocated = PETSC_TRUE;

  PetscCall(PetscLayoutSetUp((*mat)->rmap));
  PetscCall(PetscLayoutSetUp((*mat)->cmap));

  PetscCall(MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,m,n,i,j,a,&maij->A));
  PetscCall(MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,m,(*mat)->cmap->N,oi,oj,oa,&maij->B));

  PetscCall(MatSetOption(*mat,MAT_NO_OFF_PROC_ENTRIES,PETSC_TRUE));
  PetscCall(MatAssemblyBegin(*mat,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(*mat,MAT_FINAL_ASSEMBLY));
  PetscCall(MatSetOption(*mat,MAT_NO_OFF_PROC_ENTRIES,PETSC_FALSE));
  PetscCall(MatSetOption(*mat,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE));
  PetscFunctionReturn(0);
}

typedef struct {
  Mat       *mp;    /* intermediate products */
  PetscBool *mptmp; /* is the intermediate product temporary ? */
  PetscInt  cp;     /* number of intermediate products */

  /* support for MatGetBrowsOfAoCols_MPIAIJ for P_oth */
  PetscInt    *startsj_s,*startsj_r;
  PetscScalar *bufa;
  Mat         P_oth;

  /* may take advantage of merging product->B */
  Mat Bloc; /* B-local by merging diag and off-diag */

  /* cusparse does not have support to split between symbolic and numeric phases.
     When api_user is true, we don't need to update the numerical values
     of the temporary storage */
  PetscBool reusesym;

  /* support for COO values insertion */
  PetscScalar  *coo_v,*coo_w; /* store on-process and off-process COO scalars, and used as MPI recv/send buffers respectively */
  PetscInt     **own; /* own[i] points to address of on-process COO indices for Mat mp[i] */
  PetscInt     **off; /* off[i] points to address of off-process COO indices for Mat mp[i] */
  PetscBool    hasoffproc; /* if true, have off-process values insertion (i.e. AtB or PtAP) */
  PetscSF      sf; /* used for non-local values insertion and memory malloc */
  PetscMemType mtype;

  /* customization */
  PetscBool abmerge;
  PetscBool P_oth_bind;
} MatMatMPIAIJBACKEND;

PetscErrorCode MatDestroy_MatMatMPIAIJBACKEND(void *data)
{
  MatMatMPIAIJBACKEND *mmdata = (MatMatMPIAIJBACKEND*)data;
  PetscInt            i;

  PetscFunctionBegin;
  PetscCall(PetscFree2(mmdata->startsj_s,mmdata->startsj_r));
  PetscCall(PetscFree(mmdata->bufa));
  PetscCall(PetscSFFree(mmdata->sf,mmdata->mtype,mmdata->coo_v));
  PetscCall(PetscSFFree(mmdata->sf,mmdata->mtype,mmdata->coo_w));
  PetscCall(MatDestroy(&mmdata->P_oth));
  PetscCall(MatDestroy(&mmdata->Bloc));
  PetscCall(PetscSFDestroy(&mmdata->sf));
  for (i = 0; i < mmdata->cp; i++) {
    PetscCall(MatDestroy(&mmdata->mp[i]));
  }
  PetscCall(PetscFree2(mmdata->mp,mmdata->mptmp));
  PetscCall(PetscFree(mmdata->own[0]));
  PetscCall(PetscFree(mmdata->own));
  PetscCall(PetscFree(mmdata->off[0]));
  PetscCall(PetscFree(mmdata->off));
  PetscCall(PetscFree(mmdata));
  PetscFunctionReturn(0);
}

/* Copy selected n entries with indices in idx[] of A to v[].
   If idx is NULL, copy the whole data array of A to v[]
 */
static PetscErrorCode MatSeqAIJCopySubArray(Mat A, PetscInt n, const PetscInt idx[], PetscScalar v[])
{
  PetscErrorCode (*f)(Mat,PetscInt,const PetscInt[],PetscScalar[]);

  PetscFunctionBegin;
  PetscCall(PetscObjectQueryFunction((PetscObject)A,"MatSeqAIJCopySubArray_C",&f));
  if (f) {
    PetscCall((*f)(A,n,idx,v));
  } else {
    const PetscScalar *vv;

    PetscCall(MatSeqAIJGetArrayRead(A,&vv));
    if (n && idx) {
      PetscScalar    *w = v;
      const PetscInt *oi = idx;
      PetscInt       j;

      for (j = 0; j < n; j++) *w++ = vv[*oi++];
    } else {
      PetscCall(PetscArraycpy(v,vv,n));
    }
    PetscCall(MatSeqAIJRestoreArrayRead(A,&vv));
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode MatProductNumeric_MPIAIJBACKEND(Mat C)
{
  MatMatMPIAIJBACKEND *mmdata;
  PetscInt            i,n_d,n_o;

  PetscFunctionBegin;
  MatCheckProduct(C,1);
  PetscCheck(C->product->data,PetscObjectComm((PetscObject)C),PETSC_ERR_PLIB,"Product data empty");
  mmdata = (MatMatMPIAIJBACKEND*)C->product->data;
  if (!mmdata->reusesym) { /* update temporary matrices */
    if (mmdata->P_oth) {
      PetscCall(MatGetBrowsOfAoCols_MPIAIJ(C->product->A,C->product->B,MAT_REUSE_MATRIX,&mmdata->startsj_s,&mmdata->startsj_r,&mmdata->bufa,&mmdata->P_oth));
    }
    if (mmdata->Bloc) {
      PetscCall(MatMPIAIJGetLocalMatMerge(C->product->B,MAT_REUSE_MATRIX,NULL,&mmdata->Bloc));
    }
  }
  mmdata->reusesym = PETSC_FALSE;

  for (i = 0; i < mmdata->cp; i++) {
    PetscCheck(mmdata->mp[i]->ops->productnumeric,PetscObjectComm((PetscObject)mmdata->mp[i]),PETSC_ERR_PLIB,"Missing numeric op for %s",MatProductTypes[mmdata->mp[i]->product->type]);
    PetscCall((*mmdata->mp[i]->ops->productnumeric)(mmdata->mp[i]));
  }
  for (i = 0, n_d = 0, n_o = 0; i < mmdata->cp; i++) {
    PetscInt noff = mmdata->off[i+1] - mmdata->off[i];

    if (mmdata->mptmp[i]) continue;
    if (noff) {
      PetscInt nown = mmdata->own[i+1] - mmdata->own[i];

      PetscCall(MatSeqAIJCopySubArray(mmdata->mp[i],noff,mmdata->off[i],mmdata->coo_w + n_o));
      PetscCall(MatSeqAIJCopySubArray(mmdata->mp[i],nown,mmdata->own[i],mmdata->coo_v + n_d));
      n_o += noff;
      n_d += nown;
    } else {
      Mat_SeqAIJ *mm = (Mat_SeqAIJ*)mmdata->mp[i]->data;

      PetscCall(MatSeqAIJCopySubArray(mmdata->mp[i],mm->nz,NULL,mmdata->coo_v + n_d));
      n_d += mm->nz;
    }
  }
  if (mmdata->hasoffproc) { /* offprocess insertion */
    PetscCall(PetscSFGatherBegin(mmdata->sf,MPIU_SCALAR,mmdata->coo_w,mmdata->coo_v+n_d));
    PetscCall(PetscSFGatherEnd(mmdata->sf,MPIU_SCALAR,mmdata->coo_w,mmdata->coo_v+n_d));
  }
  PetscCall(MatSetValuesCOO(C,mmdata->coo_v,INSERT_VALUES));
  PetscFunctionReturn(0);
}

/* Support for Pt * A, A * P, or Pt * A * P */
#define MAX_NUMBER_INTERMEDIATE 4
PetscErrorCode MatProductSymbolic_MPIAIJBACKEND(Mat C)
{
  Mat_Product            *product = C->product;
  Mat                    A,P,mp[MAX_NUMBER_INTERMEDIATE]; /* A, P and a series of intermediate matrices */
  Mat_MPIAIJ             *a,*p;
  MatMatMPIAIJBACKEND    *mmdata;
  ISLocalToGlobalMapping P_oth_l2g = NULL;
  IS                     glob = NULL;
  const char             *prefix;
  char                   pprefix[256];
  const PetscInt         *globidx,*P_oth_idx;
  PetscInt               i,j,cp,m,n,M,N,*coo_i,*coo_j;
  PetscCount             ncoo,ncoo_d,ncoo_o,ncoo_oown;
  PetscInt               cmapt[MAX_NUMBER_INTERMEDIATE],rmapt[MAX_NUMBER_INTERMEDIATE]; /* col/row map type for each Mat in mp[]. */
                                                                                        /* type-0: consecutive, start from 0; type-1: consecutive with */
                                                                                        /* a base offset; type-2: sparse with a local to global map table */
  const PetscInt         *cmapa[MAX_NUMBER_INTERMEDIATE],*rmapa[MAX_NUMBER_INTERMEDIATE]; /* col/row local to global map array (table) for type-2 map type */

  MatProductType         ptype;
  PetscBool              mptmp[MAX_NUMBER_INTERMEDIATE],hasoffproc = PETSC_FALSE,iscuda,iskokk;
  PetscMPIInt            size;

  PetscFunctionBegin;
  MatCheckProduct(C,1);
  PetscCheck(!product->data,PetscObjectComm((PetscObject)C),PETSC_ERR_PLIB,"Product data not empty");
  ptype = product->type;
  if (product->A->symmetric && ptype == MATPRODUCT_AtB) {
    ptype = MATPRODUCT_AB;
    product->symbolic_used_the_fact_A_is_symmetric = PETSC_TRUE;
  }
  switch (ptype) {
  case MATPRODUCT_AB:
    A = product->A;
    P = product->B;
    m = A->rmap->n;
    n = P->cmap->n;
    M = A->rmap->N;
    N = P->cmap->N;
    hasoffproc = PETSC_FALSE; /* will not scatter mat product values to other processes */
    break;
  case MATPRODUCT_AtB:
    P = product->A;
    A = product->B;
    m = P->cmap->n;
    n = A->cmap->n;
    M = P->cmap->N;
    N = A->cmap->N;
    hasoffproc = PETSC_TRUE;
    break;
  case MATPRODUCT_PtAP:
    A = product->A;
    P = product->B;
    m = P->cmap->n;
    n = P->cmap->n;
    M = P->cmap->N;
    N = P->cmap->N;
    hasoffproc = PETSC_TRUE;
    break;
  default:
    SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_PLIB,"Not for product type %s",MatProductTypes[ptype]);
  }
  PetscCallMPI(MPI_Comm_size(PetscObjectComm((PetscObject)C),&size));
  if (size == 1) hasoffproc = PETSC_FALSE;

  /* defaults */
  for (i=0;i<MAX_NUMBER_INTERMEDIATE;i++) {
    mp[i]    = NULL;
    mptmp[i] = PETSC_FALSE;
    rmapt[i] = -1;
    cmapt[i] = -1;
    rmapa[i] = NULL;
    cmapa[i] = NULL;
  }

  /* customization */
  PetscCall(PetscNew(&mmdata));
  mmdata->reusesym = product->api_user;
  if (ptype == MATPRODUCT_AB) {
    if (product->api_user) {
      PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatMatMult","Mat");
      PetscCall(PetscOptionsBool("-matmatmult_backend_mergeB","Merge product->B local matrices","MatMatMult",mmdata->abmerge,&mmdata->abmerge,NULL));
      PetscCall(PetscOptionsBool("-matmatmult_backend_pothbind","Bind P_oth to CPU","MatBindToCPU",mmdata->P_oth_bind,&mmdata->P_oth_bind,NULL));
      PetscOptionsEnd();
    } else {
      PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_AB","Mat");
      PetscCall(PetscOptionsBool("-mat_product_algorithm_backend_mergeB","Merge product->B local matrices","MatMatMult",mmdata->abmerge,&mmdata->abmerge,NULL));
      PetscCall(PetscOptionsBool("-mat_product_algorithm_backend_pothbind","Bind P_oth to CPU","MatBindToCPU",mmdata->P_oth_bind,&mmdata->P_oth_bind,NULL));
      PetscOptionsEnd();
    }
  } else if (ptype == MATPRODUCT_PtAP) {
    if (product->api_user) {
      PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatPtAP","Mat");
      PetscCall(PetscOptionsBool("-matptap_backend_pothbind","Bind P_oth to CPU","MatBindToCPU",mmdata->P_oth_bind,&mmdata->P_oth_bind,NULL));
      PetscOptionsEnd();
    } else {
      PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_PtAP","Mat");
      PetscCall(PetscOptionsBool("-mat_product_algorithm_backend_pothbind","Bind P_oth to CPU","MatBindToCPU",mmdata->P_oth_bind,&mmdata->P_oth_bind,NULL));
      PetscOptionsEnd();
    }
  }
  a = (Mat_MPIAIJ*)A->data;
  p = (Mat_MPIAIJ*)P->data;
  PetscCall(MatSetSizes(C,m,n,M,N));
  PetscCall(PetscLayoutSetUp(C->rmap));
  PetscCall(PetscLayoutSetUp(C->cmap));
  PetscCall(MatSetType(C,((PetscObject)A)->type_name));
  PetscCall(MatGetOptionsPrefix(C,&prefix));

  cp   = 0;
  switch (ptype) {
  case MATPRODUCT_AB: /* A * P */
    PetscCall(MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&mmdata->startsj_s,&mmdata->startsj_r,&mmdata->bufa,&mmdata->P_oth));

    /* A_diag * P_local (merged or not) */
    if (mmdata->abmerge) { /* P's diagonal and off-diag blocks are merged to one matrix, then multiplied by A_diag */
      /* P is product->B */
      PetscCall(MatMPIAIJGetLocalMatMerge(P,MAT_INITIAL_MATRIX,&glob,&mmdata->Bloc));
      PetscCall(MatProductCreate(a->A,mmdata->Bloc,NULL,&mp[cp]));
      PetscCall(MatProductSetType(mp[cp],MATPRODUCT_AB));
      PetscCall(MatProductSetFill(mp[cp],product->fill));
      PetscCall(PetscSNPrintf(pprefix,sizeof(pprefix),"backend_p%" PetscInt_FMT "_",cp));
      PetscCall(MatSetOptionsPrefix(mp[cp],prefix));
      PetscCall(MatAppendOptionsPrefix(mp[cp],pprefix));
      mp[cp]->product->api_user = product->api_user;
      PetscCall(MatProductSetFromOptions(mp[cp]));
      PetscCheck(mp[cp]->ops->productsymbolic,PetscObjectComm((PetscObject)mp[cp]),PETSC_ERR_PLIB,"Missing symbolic op for %s",MatProductTypes[mp[cp]->product->type]);
      PetscCall((*mp[cp]->ops->productsymbolic)(mp[cp]));
      PetscCall(ISGetIndices(glob,&globidx));
      rmapt[cp] = 1;
      cmapt[cp] = 2;
      cmapa[cp] = globidx;
      mptmp[cp] = PETSC_FALSE;
      cp++;
    } else { /* A_diag * P_diag and A_diag * P_off */
      PetscCall(MatProductCreate(a->A,p->A,NULL,&mp[cp]));
      PetscCall(MatProductSetType(mp[cp],MATPRODUCT_AB));
      PetscCall(MatProductSetFill(mp[cp],product->fill));
      PetscCall(PetscSNPrintf(pprefix,sizeof(pprefix),"backend_p%" PetscInt_FMT "_",cp));
      PetscCall(MatSetOptionsPrefix(mp[cp],prefix));
      PetscCall(MatAppendOptionsPrefix(mp[cp],pprefix));
      mp[cp]->product->api_user = product->api_user;
      PetscCall(MatProductSetFromOptions(mp[cp]));
      PetscCheck(mp[cp]->ops->productsymbolic,PetscObjectComm((PetscObject)mp[cp]),PETSC_ERR_PLIB,"Missing symbolic op for %s",MatProductTypes[mp[cp]->product->type]);
      PetscCall((*mp[cp]->ops->productsymbolic)(mp[cp]));
      rmapt[cp] = 1;
      cmapt[cp] = 1;
      mptmp[cp] = PETSC_FALSE;
      cp++;
      PetscCall(MatProductCreate(a->A,p->B,NULL,&mp[cp]));
      PetscCall(MatProductSetType(mp[cp],MATPRODUCT_AB));
      PetscCall(MatProductSetFill(mp[cp],product->fill));
      PetscCall(PetscSNPrintf(pprefix,sizeof(pprefix),"backend_p%" PetscInt_FMT "_",cp));
      PetscCall(MatSetOptionsPrefix(mp[cp],prefix));
      PetscCall(MatAppendOptionsPrefix(mp[cp],pprefix));
      mp[cp]->product->api_user = product->api_user;
      PetscCall(MatProductSetFromOptions(mp[cp]));
      PetscCheck(mp[cp]->ops->productsymbolic,PetscObjectComm((PetscObject)mp[cp]),PETSC_ERR_PLIB,"Missing symbolic op for %s",MatProductTypes[mp[cp]->product->type]);
      PetscCall((*mp[cp]->ops->productsymbolic)(mp[cp]));
      rmapt[cp] = 1;
      cmapt[cp] = 2;
      cmapa[cp] = p->garray;
      mptmp[cp] = PETSC_FALSE;
      cp++;
    }

    /* A_off * P_other */
    if (mmdata->P_oth) {
      PetscCall(MatSeqAIJCompactOutExtraColumns_SeqAIJ(mmdata->P_oth,&P_oth_l2g)); /* make P_oth use local col ids */
      PetscCall(ISLocalToGlobalMappingGetIndices(P_oth_l2g,&P_oth_idx));
      PetscCall(MatSetType(mmdata->P_oth,((PetscObject)(a->B))->type_name));
      PetscCall(MatBindToCPU(mmdata->P_oth,mmdata->P_oth_bind));
      PetscCall(MatProductCreate(a->B,mmdata->P_oth,NULL,&mp[cp]));
      PetscCall(MatProductSetType(mp[cp],MATPRODUCT_AB));
      PetscCall(MatProductSetFill(mp[cp],product->fill));
      PetscCall(PetscSNPrintf(pprefix,sizeof(pprefix),"backend_p%" PetscInt_FMT "_",cp));
      PetscCall(MatSetOptionsPrefix(mp[cp],prefix));
      PetscCall(MatAppendOptionsPrefix(mp[cp],pprefix));
      mp[cp]->product->api_user = product->api_user;
      PetscCall(MatProductSetFromOptions(mp[cp]));
      PetscCheck(mp[cp]->ops->productsymbolic,PetscObjectComm((PetscObject)mp[cp]),PETSC_ERR_PLIB,"Missing symbolic op for %s",MatProductTypes[mp[cp]->product->type]);
      PetscCall((*mp[cp]->ops->productsymbolic)(mp[cp]));
      rmapt[cp] = 1;
      cmapt[cp] = 2;
      cmapa[cp] = P_oth_idx;
      mptmp[cp] = PETSC_FALSE;
      cp++;
    }
    break;

  case MATPRODUCT_AtB: /* (P^t * A): P_diag * A_loc + P_off * A_loc */
    /* A is product->B */
    PetscCall(MatMPIAIJGetLocalMatMerge(A,MAT_INITIAL_MATRIX,&glob,&mmdata->Bloc));
    if (A == P) { /* when A==P, we can take advantage of the already merged mmdata->Bloc */
      PetscCall(MatProductCreate(mmdata->Bloc,mmdata->Bloc,NULL,&mp[cp]));
      PetscCall(MatProductSetType(mp[cp],MATPRODUCT_AtB));
      PetscCall(MatProductSetFill(mp[cp],product->fill));
      PetscCall(PetscSNPrintf(pprefix,sizeof(pprefix),"backend_p%" PetscInt_FMT "_",cp));
      PetscCall(MatSetOptionsPrefix(mp[cp],prefix));
      PetscCall(MatAppendOptionsPrefix(mp[cp],pprefix));
      mp[cp]->product->api_user = product->api_user;
      PetscCall(MatProductSetFromOptions(mp[cp]));
      PetscCheck(mp[cp]->ops->productsymbolic,PetscObjectComm((PetscObject)mp[cp]),PETSC_ERR_PLIB,"Missing symbolic op for %s",MatProductTypes[mp[cp]->product->type]);
      PetscCall((*mp[cp]->ops->productsymbolic)(mp[cp]));
      PetscCall(ISGetIndices(glob,&globidx));
      rmapt[cp] = 2;
      rmapa[cp] = globidx;
      cmapt[cp] = 2;
      cmapa[cp] = globidx;
      mptmp[cp] = PETSC_FALSE;
      cp++;
    } else {
      PetscCall(MatProductCreate(p->A,mmdata->Bloc,NULL,&mp[cp]));
      PetscCall(MatProductSetType(mp[cp],MATPRODUCT_AtB));
      PetscCall(MatProductSetFill(mp[cp],product->fill));
      PetscCall(PetscSNPrintf(pprefix,sizeof(pprefix),"backend_p%" PetscInt_FMT "_",cp));
      PetscCall(MatSetOptionsPrefix(mp[cp],prefix));
      PetscCall(MatAppendOptionsPrefix(mp[cp],pprefix));
      mp[cp]->product->api_user = product->api_user;
      PetscCall(MatProductSetFromOptions(mp[cp]));
      PetscCheck(mp[cp]->ops->productsymbolic,PetscObjectComm((PetscObject)mp[cp]),PETSC_ERR_PLIB,"Missing symbolic op for %s",MatProductTypes[mp[cp]->product->type]);
      PetscCall((*mp[cp]->ops->productsymbolic)(mp[cp]));
      PetscCall(ISGetIndices(glob,&globidx));
      rmapt[cp] = 1;
      cmapt[cp] = 2;
      cmapa[cp] = globidx;
      mptmp[cp] = PETSC_FALSE;
      cp++;
      PetscCall(MatProductCreate(p->B,mmdata->Bloc,NULL,&mp[cp]));
      PetscCall(MatProductSetType(mp[cp],MATPRODUCT_AtB));
      PetscCall(MatProductSetFill(mp[cp],product->fill));
      PetscCall(PetscSNPrintf(pprefix,sizeof(pprefix),"backend_p%" PetscInt_FMT "_",cp));
      PetscCall(MatSetOptionsPrefix(mp[cp],prefix));
      PetscCall(MatAppendOptionsPrefix(mp[cp],pprefix));
      mp[cp]->product->api_user = product->api_user;
      PetscCall(MatProductSetFromOptions(mp[cp]));
      PetscCheck(mp[cp]->ops->productsymbolic,PetscObjectComm((PetscObject)mp[cp]),PETSC_ERR_PLIB,"Missing symbolic op for %s",MatProductTypes[mp[cp]->product->type]);
      PetscCall((*mp[cp]->ops->productsymbolic)(mp[cp]));
      rmapt[cp] = 2;
      rmapa[cp] = p->garray;
      cmapt[cp] = 2;
      cmapa[cp] = globidx;
      mptmp[cp] = PETSC_FALSE;
      cp++;
    }
    break;
  case MATPRODUCT_PtAP:
    PetscCall(MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&mmdata->startsj_s,&mmdata->startsj_r,&mmdata->bufa,&mmdata->P_oth));
    /* P is product->B */
    PetscCall(MatMPIAIJGetLocalMatMerge(P,MAT_INITIAL_MATRIX,&glob,&mmdata->Bloc));
    PetscCall(MatProductCreate(a->A,mmdata->Bloc,NULL,&mp[cp]));
    PetscCall(MatProductSetType(mp[cp],MATPRODUCT_PtAP));
    PetscCall(MatProductSetFill(mp[cp],product->fill));
    PetscCall(PetscSNPrintf(pprefix,sizeof(pprefix),"backend_p%" PetscInt_FMT "_",cp));
    PetscCall(MatSetOptionsPrefix(mp[cp],prefix));
    PetscCall(MatAppendOptionsPrefix(mp[cp],pprefix));
    mp[cp]->product->api_user = product->api_user;
    PetscCall(MatProductSetFromOptions(mp[cp]));
    PetscCheck(mp[cp]->ops->productsymbolic,PetscObjectComm((PetscObject)mp[cp]),PETSC_ERR_PLIB,"Missing symbolic op for %s",MatProductTypes[mp[cp]->product->type]);
    PetscCall((*mp[cp]->ops->productsymbolic)(mp[cp]));
    PetscCall(ISGetIndices(glob,&globidx));
    rmapt[cp] = 2;
    rmapa[cp] = globidx;
    cmapt[cp] = 2;
    cmapa[cp] = globidx;
    mptmp[cp] = PETSC_FALSE;
    cp++;
    if (mmdata->P_oth) {
      PetscCall(MatSeqAIJCompactOutExtraColumns_SeqAIJ(mmdata->P_oth,&P_oth_l2g));
      PetscCall(ISLocalToGlobalMappingGetIndices(P_oth_l2g,&P_oth_idx));
      PetscCall(MatSetType(mmdata->P_oth,((PetscObject)(a->B))->type_name));
      PetscCall(MatBindToCPU(mmdata->P_oth,mmdata->P_oth_bind));
      PetscCall(MatProductCreate(a->B,mmdata->P_oth,NULL,&mp[cp]));
      PetscCall(MatProductSetType(mp[cp],MATPRODUCT_AB));
      PetscCall(MatProductSetFill(mp[cp],product->fill));
      PetscCall(PetscSNPrintf(pprefix,sizeof(pprefix),"backend_p%" PetscInt_FMT "_",cp));
      PetscCall(MatSetOptionsPrefix(mp[cp],prefix));
      PetscCall(MatAppendOptionsPrefix(mp[cp],pprefix));
      mp[cp]->product->api_user = product->api_user;
      PetscCall(MatProductSetFromOptions(mp[cp]));
      PetscCheck(mp[cp]->ops->productsymbolic,PetscObjectComm((PetscObject)mp[cp]),PETSC_ERR_PLIB,"Missing symbolic op for %s",MatProductTypes[mp[cp]->product->type]);
      PetscCall((*mp[cp]->ops->productsymbolic)(mp[cp]));
      mptmp[cp] = PETSC_TRUE;
      cp++;
      PetscCall(MatProductCreate(mmdata->Bloc,mp[1],NULL,&mp[cp]));
      PetscCall(MatProductSetType(mp[cp],MATPRODUCT_AtB));
      PetscCall(MatProductSetFill(mp[cp],product->fill));
      PetscCall(PetscSNPrintf(pprefix,sizeof(pprefix),"backend_p%" PetscInt_FMT "_",cp));
      PetscCall(MatSetOptionsPrefix(mp[cp],prefix));
      PetscCall(MatAppendOptionsPrefix(mp[cp],pprefix));
      mp[cp]->product->api_user = product->api_user;
      PetscCall(MatProductSetFromOptions(mp[cp]));
      PetscCheck(mp[cp]->ops->productsymbolic,PetscObjectComm((PetscObject)mp[cp]),PETSC_ERR_PLIB,"Missing symbolic op for %s",MatProductTypes[mp[cp]->product->type]);
      PetscCall((*mp[cp]->ops->productsymbolic)(mp[cp]));
      rmapt[cp] = 2;
      rmapa[cp] = globidx;
      cmapt[cp] = 2;
      cmapa[cp] = P_oth_idx;
      mptmp[cp] = PETSC_FALSE;
      cp++;
    }
    break;
  default:
    SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_PLIB,"Not for product type %s",MatProductTypes[ptype]);
  }
  /* sanity check */
  if (size > 1) for (i = 0; i < cp; i++) PetscCheck(rmapt[i] != 2 || hasoffproc,PETSC_COMM_SELF,PETSC_ERR_PLIB,"Unexpected offproc map type for product %" PetscInt_FMT,i);

  PetscCall(PetscMalloc2(cp,&mmdata->mp,cp,&mmdata->mptmp));
  for (i = 0; i < cp; i++) {
    mmdata->mp[i]    = mp[i];
    mmdata->mptmp[i] = mptmp[i];
  }
  mmdata->cp = cp;
  C->product->data       = mmdata;
  C->product->destroy    = MatDestroy_MatMatMPIAIJBACKEND;
  C->ops->productnumeric = MatProductNumeric_MPIAIJBACKEND;

  /* memory type */
  mmdata->mtype = PETSC_MEMTYPE_HOST;
  PetscCall(PetscObjectTypeCompareAny((PetscObject)C,&iscuda,MATSEQAIJCUSPARSE,MATMPIAIJCUSPARSE,""));
  PetscCall(PetscObjectTypeCompareAny((PetscObject)C,&iskokk,MATSEQAIJKOKKOS,MATMPIAIJKOKKOS,""));
  if (iscuda) mmdata->mtype = PETSC_MEMTYPE_CUDA;
  else if (iskokk) mmdata->mtype = PETSC_MEMTYPE_KOKKOS;

  /* prepare coo coordinates for values insertion */

  /* count total nonzeros of those intermediate seqaij Mats
    ncoo_d:    # of nonzeros of matrices that do not have offproc entries
    ncoo_o:    # of nonzeros (of matrices that might have offproc entries) that will be inserted to remote procs
    ncoo_oown: # of nonzeros (of matrices that might have offproc entries) that will be inserted locally
  */
  for (cp = 0, ncoo_d = 0, ncoo_o = 0, ncoo_oown = 0; cp < mmdata->cp; cp++) {
    Mat_SeqAIJ *mm = (Mat_SeqAIJ*)mp[cp]->data;
    if (mptmp[cp]) continue;
    if (rmapt[cp] == 2 && hasoffproc) { /* the rows need to be scatter to all processes (might include self) */
      const PetscInt *rmap = rmapa[cp];
      const PetscInt mr = mp[cp]->rmap->n;
      const PetscInt rs = C->rmap->rstart;
      const PetscInt re = C->rmap->rend;
      const PetscInt *ii  = mm->i;
      for (i = 0; i < mr; i++) {
        const PetscInt gr = rmap[i];
        const PetscInt nz = ii[i+1] - ii[i];
        if (gr < rs || gr >= re) ncoo_o += nz; /* this row is offproc */
        else ncoo_oown += nz; /* this row is local */
      }
    } else ncoo_d += mm->nz;
  }

  /*
    ncoo: total number of nonzeros (including those inserted by remote procs) belonging to this proc

    ncoo = ncoo_d + ncoo_oown + ncoo2, which ncoo2 is number of nonzeros inserted to me by other procs.

    off[0] points to a big index array, which is shared by off[1,2,...]. Similarily, for own[0].

    off[p]: points to the segment for matrix mp[p], storing location of nonzeros that mp[p] will insert to others
    own[p]: points to the segment for matrix mp[p], storing location of nonzeros that mp[p] will insert locally
    so, off[p+1]-off[p] is the number of nonzeros that mp[p] will send to others.

    coo_i/j/v[]: [ncoo] row/col/val of nonzeros belonging to this proc.
    Ex. coo_i[]: the beginning part (of size ncoo_d + ncoo_oown) stores i of local nonzeros, and the remaing part stores i of nonzeros I will receive.
  */
  PetscCall(PetscCalloc1(mmdata->cp+1,&mmdata->off)); /* +1 to make a csr-like data structure */
  PetscCall(PetscCalloc1(mmdata->cp+1,&mmdata->own));

  /* gather (i,j) of nonzeros inserted by remote procs */
  if (hasoffproc) {
    PetscSF  msf;
    PetscInt ncoo2,*coo_i2,*coo_j2;

    PetscCall(PetscMalloc1(ncoo_o,&mmdata->off[0]));
    PetscCall(PetscMalloc1(ncoo_oown,&mmdata->own[0]));
    PetscCall(PetscMalloc2(ncoo_o,&coo_i,ncoo_o,&coo_j)); /* to collect (i,j) of entries to be sent to others */

    for (cp = 0, ncoo_o = 0; cp < mmdata->cp; cp++) {
      Mat_SeqAIJ *mm = (Mat_SeqAIJ*)mp[cp]->data;
      PetscInt   *idxoff = mmdata->off[cp];
      PetscInt   *idxown = mmdata->own[cp];
      if (!mptmp[cp] && rmapt[cp] == 2) { /* row map is sparse */
        const PetscInt *rmap = rmapa[cp];
        const PetscInt *cmap = cmapa[cp];
        const PetscInt *ii  = mm->i;
        PetscInt       *coi = coo_i + ncoo_o;
        PetscInt       *coj = coo_j + ncoo_o;
        const PetscInt mr = mp[cp]->rmap->n;
        const PetscInt rs = C->rmap->rstart;
        const PetscInt re = C->rmap->rend;
        const PetscInt cs = C->cmap->rstart;
        for (i = 0; i < mr; i++) {
          const PetscInt *jj = mm->j + ii[i];
          const PetscInt gr  = rmap[i];
          const PetscInt nz  = ii[i+1] - ii[i];
          if (gr < rs || gr >= re) { /* this is an offproc row */
            for (j = ii[i]; j < ii[i+1]; j++) {
              *coi++ = gr;
              *idxoff++ = j;
            }
            if (!cmapt[cp]) { /* already global */
              for (j = 0; j < nz; j++) *coj++ = jj[j];
            } else if (cmapt[cp] == 1) { /* local to global for owned columns of C */
              for (j = 0; j < nz; j++) *coj++ = jj[j] + cs;
            } else { /* offdiag */
              for (j = 0; j < nz; j++) *coj++ = cmap[jj[j]];
            }
            ncoo_o += nz;
          } else { /* this is a local row */
            for (j = ii[i]; j < ii[i+1]; j++) *idxown++ = j;
          }
        }
      }
      mmdata->off[cp + 1] = idxoff;
      mmdata->own[cp + 1] = idxown;
    }

    PetscCall(PetscSFCreate(PetscObjectComm((PetscObject)C),&mmdata->sf));
    PetscCall(PetscSFSetGraphLayout(mmdata->sf,C->rmap,ncoo_o/*nleaves*/,NULL/*ilocal*/,PETSC_OWN_POINTER,coo_i));
    PetscCall(PetscSFGetMultiSF(mmdata->sf,&msf));
    PetscCall(PetscSFGetGraph(msf,&ncoo2/*nroots*/,NULL,NULL,NULL));
    ncoo = ncoo_d + ncoo_oown + ncoo2;
    PetscCall(PetscMalloc2(ncoo,&coo_i2,ncoo,&coo_j2));
    PetscCall(PetscSFGatherBegin(mmdata->sf,MPIU_INT,coo_i,coo_i2 + ncoo_d + ncoo_oown)); /* put (i,j) of remote nonzeros at back */
    PetscCall(PetscSFGatherEnd(mmdata->sf,MPIU_INT,coo_i,coo_i2 + ncoo_d + ncoo_oown));
    PetscCall(PetscSFGatherBegin(mmdata->sf,MPIU_INT,coo_j,coo_j2 + ncoo_d + ncoo_oown));
    PetscCall(PetscSFGatherEnd(mmdata->sf,MPIU_INT,coo_j,coo_j2 + ncoo_d + ncoo_oown));
    PetscCall(PetscFree2(coo_i,coo_j));
    /* allocate MPI send buffer to collect nonzero values to be sent to remote procs */
    PetscCall(PetscSFMalloc(mmdata->sf,mmdata->mtype,ncoo_o*sizeof(PetscScalar),(void**)&mmdata->coo_w));
    coo_i = coo_i2;
    coo_j = coo_j2;
  } else { /* no offproc values insertion */
    ncoo = ncoo_d;
    PetscCall(PetscMalloc2(ncoo,&coo_i,ncoo,&coo_j));

    PetscCall(PetscSFCreate(PetscObjectComm((PetscObject)C),&mmdata->sf));
    PetscCall(PetscSFSetGraph(mmdata->sf,0,0,NULL,PETSC_OWN_POINTER,NULL,PETSC_OWN_POINTER));
    PetscCall(PetscSFSetUp(mmdata->sf));
  }
  mmdata->hasoffproc = hasoffproc;

  /* gather (i,j) of nonzeros inserted locally */
  for (cp = 0, ncoo_d = 0; cp < mmdata->cp; cp++) {
    Mat_SeqAIJ     *mm = (Mat_SeqAIJ*)mp[cp]->data;
    PetscInt       *coi = coo_i + ncoo_d;
    PetscInt       *coj = coo_j + ncoo_d;
    const PetscInt *jj  = mm->j;
    const PetscInt *ii  = mm->i;
    const PetscInt *cmap = cmapa[cp];
    const PetscInt *rmap = rmapa[cp];
    const PetscInt mr = mp[cp]->rmap->n;
    const PetscInt rs = C->rmap->rstart;
    const PetscInt re = C->rmap->rend;
    const PetscInt cs = C->cmap->rstart;

    if (mptmp[cp]) continue;
    if (rmapt[cp] == 1) { /* consecutive rows */
      /* fill coo_i */
      for (i = 0; i < mr; i++) {
        const PetscInt gr = i + rs;
        for (j = ii[i]; j < ii[i+1]; j++) coi[j] = gr;
      }
      /* fill coo_j */
      if (!cmapt[cp]) { /* type-0, already global */
        PetscCall(PetscArraycpy(coj,jj,mm->nz));
      } else if (cmapt[cp] == 1) { /* type-1, local to global for consecutive columns of C */
        for (j = 0; j < mm->nz; j++) coj[j] = jj[j] + cs; /* lid + col start */
      } else { /* type-2, local to global for sparse columns */
        for (j = 0; j < mm->nz; j++) coj[j] = cmap[jj[j]];
      }
      ncoo_d += mm->nz;
    } else if (rmapt[cp] == 2) { /* sparse rows */
      for (i = 0; i < mr; i++) {
        const PetscInt *jj = mm->j + ii[i];
        const PetscInt gr  = rmap[i];
        const PetscInt nz  = ii[i+1] - ii[i];
        if (gr >= rs && gr < re) { /* local rows */
          for (j = ii[i]; j < ii[i+1]; j++) *coi++ = gr;
          if (!cmapt[cp]) { /* type-0, already global */
            for (j = 0; j < nz; j++) *coj++ = jj[j];
          } else if (cmapt[cp] == 1) { /* local to global for owned columns of C */
            for (j = 0; j < nz; j++) *coj++ = jj[j] + cs;
          } else { /* type-2, local to global for sparse columns */
            for (j = 0; j < nz; j++) *coj++ = cmap[jj[j]];
          }
          ncoo_d += nz;
        }
      }
    }
  }
  if (glob) {
    PetscCall(ISRestoreIndices(glob,&globidx));
  }
  PetscCall(ISDestroy(&glob));
  if (P_oth_l2g) {
    PetscCall(ISLocalToGlobalMappingRestoreIndices(P_oth_l2g,&P_oth_idx));
  }
  PetscCall(ISLocalToGlobalMappingDestroy(&P_oth_l2g));
  /* allocate an array to store all nonzeros (inserted locally or remotely) belonging to this proc */
  PetscCall(PetscSFMalloc(mmdata->sf,mmdata->mtype,ncoo*sizeof(PetscScalar),(void**)&mmdata->coo_v));

  /* preallocate with COO data */
  PetscCall(MatSetPreallocationCOO(C,ncoo,coo_i,coo_j));
  PetscCall(PetscFree2(coo_i,coo_j));
  PetscFunctionReturn(0);
}

PetscErrorCode MatProductSetFromOptions_MPIAIJBACKEND(Mat mat)
{
  Mat_Product *product = mat->product;
#if defined(PETSC_HAVE_DEVICE)
  PetscBool    match   = PETSC_FALSE;
  PetscBool    usecpu  = PETSC_FALSE;
#else
  PetscBool    match   = PETSC_TRUE;
#endif

  PetscFunctionBegin;
  MatCheckProduct(mat,1);
#if defined(PETSC_HAVE_DEVICE)
  if (!product->A->boundtocpu && !product->B->boundtocpu) {
    PetscCall(PetscObjectTypeCompare((PetscObject)product->B,((PetscObject)product->A)->type_name,&match));
  }
  if (match) { /* we can always fallback to the CPU if requested */
    switch (product->type) {
    case MATPRODUCT_AB:
      if (product->api_user) {
        PetscOptionsBegin(PetscObjectComm((PetscObject)mat),((PetscObject)mat)->prefix,"MatMatMult","Mat");
        PetscCall(PetscOptionsBool("-matmatmult_backend_cpu","Use CPU code","MatMatMult",usecpu,&usecpu,NULL));
        PetscOptionsEnd();
      } else {
        PetscOptionsBegin(PetscObjectComm((PetscObject)mat),((PetscObject)mat)->prefix,"MatProduct_AB","Mat");
        PetscCall(PetscOptionsBool("-mat_product_algorithm_backend_cpu","Use CPU code","MatMatMult",usecpu,&usecpu,NULL));
        PetscOptionsEnd();
      }
      break;
    case MATPRODUCT_AtB:
      if (product->api_user) {
        PetscOptionsBegin(PetscObjectComm((PetscObject)mat),((PetscObject)mat)->prefix,"MatTransposeMatMult","Mat");
        PetscCall(PetscOptionsBool("-mattransposematmult_backend_cpu","Use CPU code","MatTransposeMatMult",usecpu,&usecpu,NULL));
        PetscOptionsEnd();
      } else {
        PetscOptionsBegin(PetscObjectComm((PetscObject)mat),((PetscObject)mat)->prefix,"MatProduct_AtB","Mat");
        PetscCall(PetscOptionsBool("-mat_product_algorithm_backend_cpu","Use CPU code","MatTransposeMatMult",usecpu,&usecpu,NULL));
        PetscOptionsEnd();
      }
      break;
    case MATPRODUCT_PtAP:
      if (product->api_user) {
        PetscOptionsBegin(PetscObjectComm((PetscObject)mat),((PetscObject)mat)->prefix,"MatPtAP","Mat");
        PetscCall(PetscOptionsBool("-matptap_backend_cpu","Use CPU code","MatPtAP",usecpu,&usecpu,NULL));
        PetscOptionsEnd();
      } else {
        PetscOptionsBegin(PetscObjectComm((PetscObject)mat),((PetscObject)mat)->prefix,"MatProduct_PtAP","Mat");
        PetscCall(PetscOptionsBool("-mat_product_algorithm_backend_cpu","Use CPU code","MatPtAP",usecpu,&usecpu,NULL));
        PetscOptionsEnd();
      }
      break;
    default:
      break;
    }
    match = (PetscBool)!usecpu;
  }
#endif
  if (match) {
    switch (product->type) {
    case MATPRODUCT_AB:
    case MATPRODUCT_AtB:
    case MATPRODUCT_PtAP:
      mat->ops->productsymbolic = MatProductSymbolic_MPIAIJBACKEND;
      break;
    default:
      break;
    }
  }
  /* fallback to MPIAIJ ops */
  if (!mat->ops->productsymbolic) PetscCall(MatProductSetFromOptions_MPIAIJ(mat));
  PetscFunctionReturn(0);
}

/*
   Produces a set of block column indices of the matrix row, one for each block represented in the original row

   n - the number of block indices in cc[]
   cc - the block indices (must be large enough to contain the indices)
*/
static inline PetscErrorCode MatCollapseRow(Mat Amat,PetscInt row,PetscInt bs,PetscInt *n,PetscInt *cc)
{
  PetscInt       cnt = -1,nidx,j;
  const PetscInt *idx;

  PetscFunctionBegin;
  PetscCall(MatGetRow(Amat,row,&nidx,&idx,NULL));
  if (nidx) {
    cnt = 0;
    cc[cnt] = idx[0]/bs;
    for (j=1; j<nidx; j++) {
      if (cc[cnt] < idx[j]/bs) cc[++cnt] = idx[j]/bs;
    }
  }
  PetscCall(MatRestoreRow(Amat,row,&nidx,&idx,NULL));
  *n = cnt+1;
  PetscFunctionReturn(0);
}

/*
    Produces a set of block column indices of the matrix block row, one for each block represented in the original set of rows

    ncollapsed - the number of block indices
    collapsed - the block indices (must be large enough to contain the indices)
*/
static inline PetscErrorCode MatCollapseRows(Mat Amat,PetscInt start,PetscInt bs,PetscInt *w0,PetscInt *w1,PetscInt *w2,PetscInt *ncollapsed,PetscInt **collapsed)
{
  PetscInt       i,nprev,*cprev = w0,ncur = 0,*ccur = w1,*merged = w2,*cprevtmp;

  PetscFunctionBegin;
  PetscCall(MatCollapseRow(Amat,start,bs,&nprev,cprev));
  for (i=start+1; i<start+bs; i++) {
    PetscCall(MatCollapseRow(Amat,i,bs,&ncur,ccur));
    PetscCall(PetscMergeIntArray(nprev,cprev,ncur,ccur,&nprev,&merged));
    cprevtmp = cprev; cprev = merged; merged = cprevtmp;
  }
  *ncollapsed = nprev;
  if (collapsed) *collapsed  = cprev;
  PetscFunctionReturn(0);
}

/* -------------------------------------------------------------------------- */
/*
 MatCreateGraph_Simple_AIJ - create simple scalar matrix (graph) from potentially blocked matrix

 Input Parameter:
 . Amat - matrix
 - symmetrize - make the result symmetric
 + scale - scale with diagonal

 Output Parameter:
 . a_Gmat - output scalar graph >= 0

 */
PETSC_INTERN PetscErrorCode MatCreateGraph_Simple_AIJ(Mat Amat, PetscBool symmetrize, PetscBool scale, Mat *a_Gmat)
{
  PetscInt       Istart,Iend,Ii,jj,kk,ncols,nloc,NN,MM,bs;
  MPI_Comm       comm;
  Mat            Gmat;
  PetscBool      ismpiaij,isseqaij;
  Mat            a, b, c;
  MatType        jtype;

  PetscFunctionBegin;
  PetscCall(PetscObjectGetComm((PetscObject)Amat,&comm));
  PetscCall(MatGetOwnershipRange(Amat, &Istart, &Iend));
  PetscCall(MatGetSize(Amat, &MM, &NN));
  PetscCall(MatGetBlockSize(Amat, &bs));
  nloc = (Iend-Istart)/bs;

  PetscCall(PetscObjectBaseTypeCompare((PetscObject)Amat,MATSEQAIJ,&isseqaij));
  PetscCall(PetscObjectBaseTypeCompare((PetscObject)Amat,MATMPIAIJ,&ismpiaij));
  PetscCheck(isseqaij || ismpiaij,comm,PETSC_ERR_USER,"Require (MPI)AIJ matrix type");

  /* TODO GPU: these calls are potentially expensive if matrices are large and we want to use the GPU */
  /* A solution consists in providing a new API, MatAIJGetCollapsedAIJ, and each class can provide a fast
     implementation */
  if (bs > 1) {
    PetscCall(MatGetType(Amat,&jtype));
    PetscCall(MatCreate(comm, &Gmat));
    PetscCall(MatSetType(Gmat, jtype));
    PetscCall(MatSetSizes(Gmat,nloc,nloc,PETSC_DETERMINE,PETSC_DETERMINE));
    PetscCall(MatSetBlockSizes(Gmat, 1, 1));
    if (isseqaij || ((Mat_MPIAIJ*)Amat->data)->garray) {
      PetscInt  *d_nnz, *o_nnz;
      MatScalar *aa,val,AA[4096];
      PetscInt  *aj,*ai,AJ[4096],nc;
      if (isseqaij) { a = Amat; b = NULL; }
      else {
        Mat_MPIAIJ *d = (Mat_MPIAIJ*)Amat->data;
        a = d->A; b = d->B;
      }
      PetscCall(PetscInfo(Amat,"New bs>1 Graph. nloc=%" PetscInt_FMT "\n",nloc));
      PetscCall(PetscMalloc2(nloc, &d_nnz,isseqaij ? 0 : nloc, &o_nnz));
      for (c=a, kk=0 ; c && kk<2 ; c=b, kk++){
        PetscInt       *nnz = (c==a) ? d_nnz : o_nnz, nmax=0;
        const PetscInt *cols;
        for (PetscInt brow=0,jj,ok=1,j0; brow < nloc*bs; brow += bs) { // block rows
          PetscCall(MatGetRow(c,brow,&jj,&cols,NULL));
          nnz[brow/bs] = jj/bs;
          if (jj%bs) ok = 0;
          if (cols) j0 = cols[0];
          else j0 = -1;
          PetscCall(MatRestoreRow(c,brow,&jj,&cols,NULL));
          if (nnz[brow/bs]>nmax) nmax = nnz[brow/bs];
          for (PetscInt ii=1; ii < bs && nnz[brow/bs] ; ii++) { // check for non-dense blocks
            PetscCall(MatGetRow(c,brow+ii,&jj,&cols,NULL));
            if (jj%bs) ok = 0;
            if ((cols && j0 != cols[0]) || (!cols && j0 != -1)) ok = 0;
            if (nnz[brow/bs] != jj/bs) ok = 0;
            PetscCall(MatRestoreRow(c,brow+ii,&jj,&cols,NULL));
          }
          if (!ok) {
            PetscCall(PetscFree2(d_nnz,o_nnz));
            goto old_bs;
          }
        }
        PetscCheck(nmax<4096,PETSC_COMM_SELF,PETSC_ERR_USER,"Buffer %" PetscInt_FMT " too small 4096.",nmax);
      }
      PetscCall(MatSeqAIJSetPreallocation(Gmat,0,d_nnz));
      PetscCall(MatMPIAIJSetPreallocation(Gmat,0,d_nnz,0,o_nnz));
      PetscCall(PetscFree2(d_nnz,o_nnz));
      // diag
      for (PetscInt brow=0,n,grow; brow < nloc*bs; brow += bs) { // block rows
        Mat_SeqAIJ *aseq  = (Mat_SeqAIJ*)a->data;
        ai = aseq->i;
        n  = ai[brow+1] - ai[brow];
        aj = aseq->j + ai[brow];
        for (int k=0; k<n; k += bs) { // block columns
          AJ[k/bs] = aj[k]/bs + Istart/bs; // diag starts at (Istart,Istart)
          val = 0;
          for (int ii=0; ii<bs; ii++) { // rows in block
            aa = aseq->a + ai[brow+ii] + k;
            for (int jj=0; jj<bs; jj++) { // columns in block
              val += PetscAbs(PetscRealPart(aa[jj])); // a sort of norm
            }
          }
          AA[k/bs] = val;
        }
        grow = Istart/bs + brow/bs;
        PetscCall(MatSetValues(Gmat,1,&grow,n/bs,AJ,AA,INSERT_VALUES));
      }
      // off-diag
      if (ismpiaij) {
        Mat_MPIAIJ        *aij = (Mat_MPIAIJ*)Amat->data;
        const PetscScalar *vals;
        const PetscInt    *cols, *garray = aij->garray;
        PetscCheck(garray,PETSC_COMM_SELF,PETSC_ERR_USER,"No garray ?");
        for (PetscInt brow=0,grow; brow < nloc*bs; brow += bs) { // block rows
          PetscCall(MatGetRow(b,brow,&ncols,&cols,NULL));
          for (int k=0,cidx=0 ; k < ncols ; k += bs, cidx++) {
            AA[k/bs] = 0;
            AJ[cidx] = garray[cols[k]]/bs;
          }
          nc = ncols/bs;
          PetscCall(MatRestoreRow(b,brow,&ncols,&cols,NULL));
          for (int ii=0; ii<bs; ii++) { // rows in block
            PetscCall(MatGetRow(b,brow+ii,&ncols,&cols,&vals));
            for (int k=0; k<ncols; k += bs) {
              for (int jj=0; jj<bs; jj++) { // cols in block
                AA[k/bs] += PetscAbs(PetscRealPart(vals[k+jj]));
              }
            }
            PetscCall(MatRestoreRow(b,brow+ii,&ncols,&cols,&vals));
          }
          grow = Istart/bs + brow/bs;
          PetscCall(MatSetValues(Gmat,1,&grow,nc,AJ,AA,INSERT_VALUES));
        }
      }
      PetscCall(MatAssemblyBegin(Gmat,MAT_FINAL_ASSEMBLY));
      PetscCall(MatAssemblyEnd(Gmat,MAT_FINAL_ASSEMBLY));
    } else {
      const PetscScalar *vals;
      const PetscInt    *idx;
      PetscInt          *d_nnz, *o_nnz,*w0,*w1,*w2;
      old_bs:
      /*
       Determine the preallocation needed for the scalar matrix derived from the vector matrix.
       */
      PetscCall(PetscInfo(Amat,"OLD bs>1 CreateGraph\n"));
      PetscCall(PetscMalloc2(nloc, &d_nnz,isseqaij ? 0 : nloc, &o_nnz));
      if (isseqaij) {
        PetscInt max_d_nnz;
        /*
         Determine exact preallocation count for (sequential) scalar matrix
         */
        PetscCall(MatSeqAIJGetMaxRowNonzeros(Amat,&max_d_nnz));
        max_d_nnz = PetscMin(nloc,bs*max_d_nnz);
        PetscCall(PetscMalloc3(max_d_nnz, &w0,max_d_nnz, &w1,max_d_nnz, &w2));
        for (Ii = 0, jj = 0; Ii < Iend; Ii += bs, jj++) {
          PetscCall(MatCollapseRows(Amat,Ii,bs,w0,w1,w2,&d_nnz[jj],NULL));
        }
        PetscCall(PetscFree3(w0,w1,w2));
      } else if (ismpiaij) {
        Mat            Daij,Oaij;
        const PetscInt *garray;
        PetscInt       max_d_nnz;
        PetscCall(MatMPIAIJGetSeqAIJ(Amat,&Daij,&Oaij,&garray));
        /*
         Determine exact preallocation count for diagonal block portion of scalar matrix
         */
        PetscCall(MatSeqAIJGetMaxRowNonzeros(Daij,&max_d_nnz));
        max_d_nnz = PetscMin(nloc,bs*max_d_nnz);
        PetscCall(PetscMalloc3(max_d_nnz, &w0,max_d_nnz, &w1,max_d_nnz, &w2));
        for (Ii = 0, jj = 0; Ii < Iend - Istart; Ii += bs, jj++) {
          PetscCall(MatCollapseRows(Daij,Ii,bs,w0,w1,w2,&d_nnz[jj],NULL));
        }
        PetscCall(PetscFree3(w0,w1,w2));
        /*
         Over estimate (usually grossly over), preallocation count for off-diagonal portion of scalar matrix
         */
        for (Ii = 0, jj = 0; Ii < Iend - Istart; Ii += bs, jj++) {
          o_nnz[jj] = 0;
          for (kk=0; kk<bs; kk++) { /* rows that get collapsed to a single row */
            PetscCall(MatGetRow(Oaij,Ii+kk,&ncols,NULL,NULL));
            o_nnz[jj] += ncols;
            PetscCall(MatRestoreRow(Oaij,Ii+kk,&ncols,NULL,NULL));
          }
          if (o_nnz[jj] > (NN/bs-nloc)) o_nnz[jj] = NN/bs-nloc;
        }
      } else SETERRQ(comm,PETSC_ERR_USER,"Require AIJ matrix type");
      /* get scalar copy (norms) of matrix */
      PetscCall(MatSeqAIJSetPreallocation(Gmat,0,d_nnz));
      PetscCall(MatMPIAIJSetPreallocation(Gmat,0,d_nnz,0,o_nnz));
      PetscCall(PetscFree2(d_nnz,o_nnz));
      for (Ii = Istart; Ii < Iend; Ii++) {
        PetscInt dest_row = Ii/bs;
        PetscCall(MatGetRow(Amat,Ii,&ncols,&idx,&vals));
        for (jj=0; jj<ncols; jj++) {
          PetscInt    dest_col = idx[jj]/bs;
          PetscScalar sv       = PetscAbs(PetscRealPart(vals[jj]));
          PetscCall(MatSetValues(Gmat,1,&dest_row,1,&dest_col,&sv,ADD_VALUES));
        }
        PetscCall(MatRestoreRow(Amat,Ii,&ncols,&idx,&vals));
      }
      PetscCall(MatAssemblyBegin(Gmat,MAT_FINAL_ASSEMBLY));
      PetscCall(MatAssemblyEnd(Gmat,MAT_FINAL_ASSEMBLY));
    }
  } else {
    /* TODO GPU: optimization proposal, each class provides fast implementation of this
     procedure via MatAbs API */
    /* just copy scalar matrix & abs() */
    PetscCall(MatDuplicate(Amat, MAT_COPY_VALUES, &Gmat));
    if (isseqaij) { a = Gmat; b = NULL; }
    else {
      Mat_MPIAIJ *d = (Mat_MPIAIJ*)Gmat->data;
      a = d->A; b = d->B;
    }
    /* abs */
    for (c=a, kk=0 ; c && kk<2 ; c=b, kk++){
      MatInfo     info;
      PetscScalar *avals;
      PetscCall(MatGetInfo(c,MAT_LOCAL,&info));
      PetscCall(MatSeqAIJGetArray(c,&avals));
      for (int jj = 0; jj<info.nz_used; jj++) avals[jj] = PetscAbsScalar(avals[jj]);
      PetscCall(MatSeqAIJRestoreArray(c,&avals));
    }
  }
  if (symmetrize) {
    PetscBool issym;
    PetscCall(MatGetOption(Amat,MAT_SYMMETRIC,&issym));
    if (!issym) {
      Mat matTrans;
      PetscCall(MatTranspose(Gmat, MAT_INITIAL_MATRIX, &matTrans));
      PetscCall(MatAXPY(Gmat, 1.0, matTrans, Gmat->structurally_symmetric ? SAME_NONZERO_PATTERN : DIFFERENT_NONZERO_PATTERN));
      PetscCall(MatDestroy(&matTrans));
    }
    PetscCall(MatSetOption(Gmat,MAT_SYMMETRIC,PETSC_TRUE));
  } else {
    PetscCall(MatPropagateSymmetryOptions(Amat, Gmat));
  }
  if (scale) {
    /* scale c for all diagonal values = 1 or -1 */
    Vec               diag;
    PetscCall(MatCreateVecs(Gmat, &diag, NULL));
    PetscCall(MatGetDiagonal(Gmat, diag));
    PetscCall(VecReciprocal(diag));
    PetscCall(VecSqrtAbs(diag));
    PetscCall(MatDiagonalScale(Gmat, diag, diag));
    PetscCall(VecDestroy(&diag));
  }
  PetscCall(MatViewFromOptions(Gmat, NULL, "-mat_graph_view"));
  *a_Gmat = Gmat;
  PetscFunctionReturn(0);
}

/* -------------------------------------------------------------------------- */
/*@C
   MatFilter_AIJ - filter values with small absolute values
     With vfilter < 0 does nothing so should not be called.

   Collective on Mat

   Input Parameters:
+   Gmat - the graph
.   vfilter - threshold parameter [0,1)

 Output Parameter:
 .  filteredG - output filtered scalar graph

   Level: developer

   Notes:
    This is called before graph coarsers are called.
    This could go into Mat, move 'symm' to GAMG

.seealso: `PCGAMGSetThreshold()`
@*/
PETSC_INTERN PetscErrorCode MatFilter_AIJ(Mat Gmat,PetscReal vfilter, Mat *filteredG)
{
  PetscInt          Istart,Iend,ncols,nnz0,nnz1, NN, MM, nloc;
  Mat               tGmat;
  MPI_Comm          comm;
  const PetscScalar *vals;
  const PetscInt    *idx;
  PetscInt          *d_nnz, *o_nnz, kk, *garray = NULL, *AJ, maxcols=0;
  MatScalar         *AA; // this is checked in graph
  PetscBool         isseqaij;
  Mat               a, b, c;
  MatType           jtype;

  PetscFunctionBegin;
  PetscCall(PetscObjectGetComm((PetscObject)Gmat,&comm));
  PetscCall(PetscObjectBaseTypeCompare((PetscObject)Gmat,MATSEQAIJ,&isseqaij));
  PetscCall(MatGetType(Gmat,&jtype));
  PetscCall(MatCreate(comm, &tGmat));
  PetscCall(MatSetType(tGmat, jtype));

  /* TODO GPU: this can be called when filter = 0 -> Probably provide MatAIJThresholdCompress that compresses the entries below a threshold?
               Also, if the matrix is symmetric, can we skip this
               operation? It can be very expensive on large matrices. */

  // global sizes
  PetscCall(MatGetSize(Gmat, &MM, &NN));
  PetscCall(MatGetOwnershipRange(Gmat, &Istart, &Iend));
  nloc = Iend - Istart;
  PetscCall(PetscMalloc2(nloc, &d_nnz,nloc, &o_nnz));
  if (isseqaij) { a = Gmat; b = NULL; }
  else {
    Mat_MPIAIJ *d = (Mat_MPIAIJ*)Gmat->data;
    a = d->A; b = d->B;
    garray = d->garray;
  }
  /* Determine upper bound on non-zeros needed in new filtered matrix */
  for (PetscInt row=0; row < nloc; row++) {
    PetscCall(MatGetRow(a,row,&ncols,NULL,NULL));
    d_nnz[row] = ncols;
    if (ncols>maxcols) maxcols=ncols;
    PetscCall(MatRestoreRow(a,row,&ncols,NULL,NULL));
  }
  if (b) {
    for (PetscInt row=0; row < nloc; row++) {
      PetscCall(MatGetRow(b,row,&ncols,NULL,NULL));
      o_nnz[row] = ncols;
      if (ncols>maxcols) maxcols=ncols;
      PetscCall(MatRestoreRow(b,row,&ncols,NULL,NULL));
    }
  }
  PetscCall(MatSetSizes(tGmat,nloc,nloc,MM,MM));
  PetscCall(MatSetBlockSizes(tGmat, 1, 1));
  PetscCall(MatSeqAIJSetPreallocation(tGmat,0,d_nnz));
  PetscCall(MatMPIAIJSetPreallocation(tGmat,0,d_nnz,0,o_nnz));
  PetscCall(MatSetOption(tGmat,MAT_NO_OFF_PROC_ENTRIES,PETSC_TRUE));
  PetscCall(PetscFree2(d_nnz,o_nnz));
  //
  PetscCall(PetscMalloc2(maxcols, &AA,maxcols, &AJ));
  nnz0 = nnz1 = 0;
  for (c=a, kk=0 ; c && kk<2 ; c=b, kk++){
    for (PetscInt row=0, grow=Istart, ncol_row, jj ; row < nloc; row++,grow++) {
      PetscCall(MatGetRow(c,row,&ncols,&idx,&vals));
      for (ncol_row=jj=0; jj<ncols; jj++,nnz0++) {
        PetscScalar sv = PetscAbs(PetscRealPart(vals[jj]));
        if (PetscRealPart(sv) > vfilter) {
          nnz1++;
          PetscInt cid = idx[jj] + Istart; //diag
          if (c!=a) cid = garray[idx[jj]];
          AA[ncol_row] = vals[jj];
          AJ[ncol_row] = cid;
          ncol_row++;
        }
      }
      PetscCall(MatRestoreRow(c,row,&ncols,&idx,&vals));
      PetscCall(MatSetValues(tGmat,1,&grow,ncol_row,AJ,AA,INSERT_VALUES));
    }
  }
  PetscCall(PetscFree2(AA,AJ));
  PetscCall(MatAssemblyBegin(tGmat,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(tGmat,MAT_FINAL_ASSEMBLY));
  PetscCall(MatPropagateSymmetryOptions(Gmat,tGmat)); /* Normal Mat options are not relevant ? */

  PetscCall(PetscInfo(tGmat,"\t %g%% nnz after filtering, with threshold %g, %g nnz ave. (N=%" PetscInt_FMT ", max row size %d)\n",
                      (!nnz0) ? 1. : 100.*(double)nnz1/(double)nnz0, (double)vfilter,
                      (!nloc) ? 1. : (double)nnz0/(double)nloc,MM,(int)maxcols));

  *filteredG = tGmat;
  PetscCall(MatViewFromOptions(tGmat, NULL, "-mat_filter_graph_view"));
  PetscFunctionReturn(0);
}

/*
    Special version for direct calls from Fortran
*/
#include <petsc/private/fortranimpl.h>

/* Change these macros so can be used in void function */
/* Identical to PetscCallVoid, except it assigns to *_ierr */
#undef  PetscCall
#define PetscCall(...) do {                                                                    \
    PetscErrorCode ierr_msv_mpiaij = __VA_ARGS__;                                              \
    if (PetscUnlikely(ierr_msv_mpiaij)) {                                                      \
      *_ierr = PetscError(PETSC_COMM_SELF,__LINE__,PETSC_FUNCTION_NAME,__FILE__,ierr_msv_mpiaij,PETSC_ERROR_REPEAT," "); \
      return;                                                                                  \
    }                                                                                          \
  } while (0)

#undef SETERRQ
#define SETERRQ(comm,ierr,...) do {                                                            \
    *_ierr = PetscError(comm,__LINE__,PETSC_FUNCTION_NAME,__FILE__,ierr,PETSC_ERROR_INITIAL,__VA_ARGS__); \
    return;                                                                                    \
  } while (0)

#if defined(PETSC_HAVE_FORTRAN_CAPS)
#define matsetvaluesmpiaij_ MATSETVALUESMPIAIJ
#elif !defined(PETSC_HAVE_FORTRAN_UNDERSCORE)
#define matsetvaluesmpiaij_ matsetvaluesmpiaij
#else
#endif
PETSC_EXTERN void matsetvaluesmpiaij_(Mat *mmat,PetscInt *mm,const PetscInt im[],PetscInt *mn,const PetscInt in[],const PetscScalar v[],InsertMode *maddv,PetscErrorCode *_ierr)
{
  Mat          mat  = *mmat;
  PetscInt     m    = *mm, n = *mn;
  InsertMode   addv = *maddv;
  Mat_MPIAIJ  *aij  = (Mat_MPIAIJ*)mat->data;
  PetscScalar  value;

  MatCheckPreallocated(mat,1);
  if (mat->insertmode == NOT_SET_VALUES) mat->insertmode = addv;
  else PetscCheck(mat->insertmode == addv,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"Cannot mix add values and insert values");
  {
    PetscInt  i,j,rstart  = mat->rmap->rstart,rend = mat->rmap->rend;
    PetscInt  cstart      = mat->cmap->rstart,cend = mat->cmap->rend,row,col;
    PetscBool roworiented = aij->roworiented;

    /* Some Variables required in the macro */
    Mat        A                    = aij->A;
    Mat_SeqAIJ *a                   = (Mat_SeqAIJ*)A->data;
    PetscInt   *aimax               = a->imax,*ai = a->i,*ailen = a->ilen,*aj = a->j;
    MatScalar  *aa;
    PetscBool  ignorezeroentries    = (((a->ignorezeroentries)&&(addv==ADD_VALUES)) ? PETSC_TRUE : PETSC_FALSE);
    Mat        B                    = aij->B;
    Mat_SeqAIJ *b                   = (Mat_SeqAIJ*)B->data;
    PetscInt   *bimax               = b->imax,*bi = b->i,*bilen = b->ilen,*bj = b->j,bm = aij->B->rmap->n,am = aij->A->rmap->n;
    MatScalar  *ba;
    /* This variable below is only for the PETSC_HAVE_VIENNACL or PETSC_HAVE_CUDA cases, but we define it in all cases because we
     * cannot use "#if defined" inside a macro. */
    PETSC_UNUSED PetscBool inserted = PETSC_FALSE;

    PetscInt  *rp1,*rp2,ii,nrow1,nrow2,_i,rmax1,rmax2,N,low1,high1,low2,high2,t,lastcol1,lastcol2;
    PetscInt  nonew = a->nonew;
    MatScalar *ap1,*ap2;

    PetscFunctionBegin;
    PetscCall(MatSeqAIJGetArray(A,&aa));
    PetscCall(MatSeqAIJGetArray(B,&ba));
    for (i=0; i<m; i++) {
      if (im[i] < 0) continue;
      PetscCheck(im[i] < mat->rmap->N,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Row too large: row %" PetscInt_FMT " max %" PetscInt_FMT,im[i],mat->rmap->N-1);
      if (im[i] >= rstart && im[i] < rend) {
        row      = im[i] - rstart;
        lastcol1 = -1;
        rp1      = aj + ai[row];
        ap1      = aa + ai[row];
        rmax1    = aimax[row];
        nrow1    = ailen[row];
        low1     = 0;
        high1    = nrow1;
        lastcol2 = -1;
        rp2      = bj + bi[row];
        ap2      = ba + bi[row];
        rmax2    = bimax[row];
        nrow2    = bilen[row];
        low2     = 0;
        high2    = nrow2;

        for (j=0; j<n; j++) {
          if (roworiented) value = v[i*n+j];
          else value = v[i+j*m];
          if (ignorezeroentries && value == 0.0 && (addv == ADD_VALUES) && im[i] != in[j]) continue;
          if (in[j] >= cstart && in[j] < cend) {
            col = in[j] - cstart;
            MatSetValues_SeqAIJ_A_Private(row,col,value,addv,im[i],in[j]);
          } else if (in[j] < 0) continue;
          else if (PetscUnlikelyDebug(in[j] >= mat->cmap->N)) {
            /* extra brace on SETERRQ() is required for --with-errorchecking=0 - due to the next 'else' clause */
            SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Column too large: col %" PetscInt_FMT " max %" PetscInt_FMT,in[j],mat->cmap->N-1);
          } else {
            if (mat->was_assembled) {
              if (!aij->colmap) {
                PetscCall(MatCreateColmap_MPIAIJ_Private(mat));
              }
#if defined(PETSC_USE_CTABLE)
              PetscCall(PetscTableFind(aij->colmap,in[j]+1,&col));
              col--;
#else
              col = aij->colmap[in[j]] - 1;
#endif
              if (col < 0 && !((Mat_SeqAIJ*)(aij->A->data))->nonew) {
                PetscCall(MatDisAssemble_MPIAIJ(mat));
                col  =  in[j];
                /* Reinitialize the variables required by MatSetValues_SeqAIJ_B_Private() */
                B        = aij->B;
                b        = (Mat_SeqAIJ*)B->data;
                bimax    = b->imax; bi = b->i; bilen = b->ilen; bj = b->j;
                rp2      = bj + bi[row];
                ap2      = ba + bi[row];
                rmax2    = bimax[row];
                nrow2    = bilen[row];
                low2     = 0;
                high2    = nrow2;
                bm       = aij->B->rmap->n;
                ba       = b->a;
                inserted = PETSC_FALSE;
              }
            } else col = in[j];
            MatSetValues_SeqAIJ_B_Private(row,col,value,addv,im[i],in[j]);
          }
        }
      } else if (!aij->donotstash) {
        if (roworiented) {
          PetscCall(MatStashValuesRow_Private(&mat->stash,im[i],n,in,v+i*n,(PetscBool)(ignorezeroentries && (addv == ADD_VALUES))));
        } else {
          PetscCall(MatStashValuesCol_Private(&mat->stash,im[i],n,in,v+i,m,(PetscBool)(ignorezeroentries && (addv == ADD_VALUES))));
        }
      }
    }
    PetscCall(MatSeqAIJRestoreArray(A,&aa));
    PetscCall(MatSeqAIJRestoreArray(B,&ba));
  }
  PetscFunctionReturnVoid();
}

/* Undefining these here since they were redefined from their original definition above! No
 * other PETSc functions should be defined past this point, as it is impossible to recover the
 * original definitions */
#undef PetscCall
#undef SETERRQ