xref: /petsc/src/mat/impls/sbaij/seq/sbaijfact.c (revision 66976f2f44dcc61d86a452a70219fb23b45d00f0)

#include <../src/mat/impls/baij/seq/baij.h>
#include <../src/mat/impls/sbaij/seq/sbaij.h>
#include <petsc/private/kernels/blockinvert.h>
#include <petscis.h>

PetscErrorCode MatGetInertia_SeqSBAIJ(Mat F, PetscInt *nneg, PetscInt *nzero, PetscInt *npos)
{
  Mat_SeqSBAIJ *fact = (Mat_SeqSBAIJ *)F->data;
  MatScalar    *dd   = fact->a;
  PetscInt      mbs = fact->mbs, bs = F->rmap->bs, i, nneg_tmp, npos_tmp, *fi = fact->diag;

  PetscFunctionBegin;
  PetscCheck(bs == 1, PETSC_COMM_SELF, PETSC_ERR_SUP, "No support for bs: %" PetscInt_FMT " >1 yet", bs);

  nneg_tmp = 0;
  npos_tmp = 0;
  for (i = 0; i < mbs; i++) {
    if (PetscRealPart(dd[*fi]) > 0.0) npos_tmp++;
    else if (PetscRealPart(dd[*fi]) < 0.0) nneg_tmp++;
    fi++;
  }
  if (nneg) *nneg = nneg_tmp;
  if (npos) *npos = npos_tmp;
  if (nzero) *nzero = mbs - nneg_tmp - npos_tmp;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
  Symbolic U^T*D*U factorization for SBAIJ format. Modified from SSF of YSMP.
  Use Modified Sparse Row (MSR) storage for u and ju. See page 85, "Iterative Methods ..." by Saad.
*/
static PetscErrorCode MatCholeskyFactorSymbolic_SeqSBAIJ_MSR(Mat F, Mat A, IS perm, const MatFactorInfo *info)
{
  Mat_SeqSBAIJ   *a = (Mat_SeqSBAIJ *)A->data, *b;
  const PetscInt *rip, *ai, *aj;
  PetscInt        i, mbs = a->mbs, *jutmp, bs = A->rmap->bs, bs2 = a->bs2;
  PetscInt        m, reallocs = 0, prow;
  PetscInt       *jl, *q, jmin, jmax, juidx, nzk, qm, *iu, *ju, k, j, vj, umax, maxadd;
  PetscReal       f = info->fill;
  PetscBool       perm_identity;

  PetscFunctionBegin;
  /* check whether perm is the identity mapping */
  PetscCall(ISIdentity(perm, &perm_identity));
  PetscCall(ISGetIndices(perm, &rip));

  if (perm_identity) { /* without permutation */
    a->permute = PETSC_FALSE;

    ai = a->i;
    aj = a->j;
  } else { /* non-trivial permutation */
    a->permute = PETSC_TRUE;

    PetscCall(MatReorderingSeqSBAIJ(A, perm));

    ai = a->inew;
    aj = a->jnew;
  }

  /* initialization */
  PetscCall(PetscMalloc1(mbs + 1, &iu));
  umax = (PetscInt)(f * ai[mbs] + 1);
  umax += mbs + 1;
  PetscCall(PetscMalloc1(umax, &ju));
  iu[0] = mbs + 1;
  juidx = mbs + 1; /* index for ju */
  /* jl linked list for pivot row -- linked list for col index */
  PetscCall(PetscMalloc2(mbs, &jl, mbs, &q));
  for (i = 0; i < mbs; i++) {
    jl[i] = mbs;
    q[i]  = 0;
  }

  /* for each row k */
  for (k = 0; k < mbs; k++) {
    for (i = 0; i < mbs; i++) q[i] = 0; /* to be removed! */
    nzk  = 0;                           /* num. of nz blocks in k-th block row with diagonal block excluded */
    q[k] = mbs;
    /* initialize nonzero structure of k-th row to row rip[k] of A */
    jmin = ai[rip[k]] + 1; /* exclude diag[k] */
    jmax = ai[rip[k] + 1];
    for (j = jmin; j < jmax; j++) {
      vj = rip[aj[j]]; /* col. value */
      if (vj > k) {
        qm = k;
        do {
          m  = qm;
          qm = q[m];
        } while (qm < vj);
        PetscCheck(qm != vj, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Duplicate entry in A");
        nzk++;
        q[m]  = vj;
        q[vj] = qm;
      } /* if (vj > k) */
    }   /* for (j=jmin; j<jmax; j++) */

    /* modify nonzero structure of k-th row by computing fill-in
       for each row i to be merged in */
    prow = k;
    prow = jl[prow]; /* next pivot row (== mbs for symbolic factorization) */

    while (prow < k) {
      /* merge row prow into k-th row */
      jmin = iu[prow] + 1;
      jmax = iu[prow + 1];
      qm   = k;
      for (j = jmin; j < jmax; j++) {
        vj = ju[j];
        do {
          m  = qm;
          qm = q[m];
        } while (qm < vj);
        if (qm != vj) {
          nzk++;
          q[m]  = vj;
          q[vj] = qm;
          qm    = vj;
        }
      }
      prow = jl[prow]; /* next pivot row */
    }

    /* add k to row list for first nonzero element in k-th row */
    if (nzk > 0) {
      i     = q[k]; /* col value of first nonzero element in U(k, k+1:mbs-1) */
      jl[k] = jl[i];
      jl[i] = k;
    }
    iu[k + 1] = iu[k] + nzk;

    /* allocate more space to ju if needed */
    if (iu[k + 1] > umax) {
      /* estimate how much additional space we will need */
      /* use the strategy suggested by David Hysom <hysom@perch-t.icase.edu> */
      /* just double the memory each time */
      maxadd = umax;
      if (maxadd < nzk) maxadd = (mbs - k) * (nzk + 1) / 2;
      umax += maxadd;

      /* allocate a longer ju */
      PetscCall(PetscMalloc1(umax, &jutmp));
      PetscCall(PetscArraycpy(jutmp, ju, iu[k]));
      PetscCall(PetscFree(ju));
      ju = jutmp;
      reallocs++; /* count how many times we realloc */
    }

    /* save nonzero structure of k-th row in ju */
    i = k;
    while (nzk--) {
      i           = q[i];
      ju[juidx++] = i;
    }
  }

#if defined(PETSC_USE_INFO)
  if (ai[mbs] != 0) {
    PetscReal af = ((PetscReal)iu[mbs]) / ((PetscReal)ai[mbs]);
    PetscCall(PetscInfo(A, "Reallocs %" PetscInt_FMT " Fill ratio:given %g needed %g\n", reallocs, (double)f, (double)af));
    PetscCall(PetscInfo(A, "Run with -pc_factor_fill %g or use \n", (double)af));
    PetscCall(PetscInfo(A, "PCFactorSetFill(pc,%g);\n", (double)af));
    PetscCall(PetscInfo(A, "for best performance.\n"));
  } else {
    PetscCall(PetscInfo(A, "Empty matrix\n"));
  }
#endif

  PetscCall(ISRestoreIndices(perm, &rip));
  PetscCall(PetscFree2(jl, q));

  /* put together the new matrix */
  PetscCall(MatSeqSBAIJSetPreallocation(F, bs, MAT_SKIP_ALLOCATION, NULL));

  b               = (Mat_SeqSBAIJ *)(F)->data;
  b->singlemalloc = PETSC_FALSE;
  b->free_a       = PETSC_TRUE;
  b->free_ij      = PETSC_TRUE;

  PetscCall(PetscMalloc1((iu[mbs] + 1) * bs2, &b->a));
  b->j    = ju;
  b->i    = iu;
  b->diag = NULL;
  b->ilen = NULL;
  b->imax = NULL;
  b->row  = perm;

  b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */

  PetscCall(PetscObjectReference((PetscObject)perm));

  b->icol = perm;
  PetscCall(PetscObjectReference((PetscObject)perm));
  PetscCall(PetscMalloc1(bs * mbs + bs, &b->solve_work));
  /* In b structure:  Free imax, ilen, old a, old j.
     Allocate idnew, solve_work, new a, new j */
  b->maxnz = b->nz = iu[mbs];

  (F)->info.factor_mallocs   = reallocs;
  (F)->info.fill_ratio_given = f;
  if (ai[mbs] != 0) {
    (F)->info.fill_ratio_needed = ((PetscReal)iu[mbs]) / ((PetscReal)ai[mbs]);
  } else {
    (F)->info.fill_ratio_needed = 0.0;
  }
  PetscCall(MatSeqSBAIJSetNumericFactorization_inplace(F, perm_identity));
  PetscFunctionReturn(PETSC_SUCCESS);
}
/*
    Symbolic U^T*D*U factorization for SBAIJ format.
    See MatICCFactorSymbolic_SeqAIJ() for description of its data structure.
*/
#include <petscbt.h>
#include <../src/mat/utils/freespace.h>
PetscErrorCode MatCholeskyFactorSymbolic_SeqSBAIJ(Mat fact, Mat A, IS perm, const MatFactorInfo *info)
{
  Mat_SeqSBAIJ      *a = (Mat_SeqSBAIJ *)A->data;
  Mat_SeqSBAIJ      *b;
  PetscBool          perm_identity, missing;
  PetscReal          fill = info->fill;
  const PetscInt    *rip, *ai = a->i, *aj = a->j;
  PetscInt           i, mbs = a->mbs, bs = A->rmap->bs, reallocs = 0, prow;
  PetscInt          *jl, jmin, jmax, nzk, *ui, k, j, *il, nextprow;
  PetscInt           nlnk, *lnk, ncols, *cols, *uj, **ui_ptr, *uj_ptr, *udiag;
  PetscFreeSpaceList free_space = NULL, current_space = NULL;
  PetscBT            lnkbt;

  PetscFunctionBegin;
  PetscCheck(A->rmap->n == A->cmap->n, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Must be square matrix, rows %" PetscInt_FMT " columns %" PetscInt_FMT, A->rmap->n, A->cmap->n);
  PetscCall(MatMissingDiagonal(A, &missing, &i));
  PetscCheck(!missing, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONGSTATE, "Matrix is missing diagonal entry %" PetscInt_FMT, i);
  if (bs > 1) {
    PetscCall(MatCholeskyFactorSymbolic_SeqSBAIJ_inplace(fact, A, perm, info));
    PetscFunctionReturn(PETSC_SUCCESS);
  }

  /* check whether perm is the identity mapping */
  PetscCall(ISIdentity(perm, &perm_identity));
  PetscCheck(perm_identity, PETSC_COMM_SELF, PETSC_ERR_SUP, "Matrix reordering is not supported for sbaij matrix. Use aij format");
  a->permute = PETSC_FALSE;
  PetscCall(ISGetIndices(perm, &rip));

  /* initialization */
  PetscCall(PetscMalloc1(mbs + 1, &ui));
  PetscCall(PetscMalloc1(mbs + 1, &udiag));
  ui[0] = 0;

  /* jl: linked list for storing indices of the pivot rows
     il: il[i] points to the 1st nonzero entry of U(i,k:mbs-1) */
  PetscCall(PetscMalloc4(mbs, &ui_ptr, mbs, &il, mbs, &jl, mbs, &cols));
  for (i = 0; i < mbs; i++) {
    jl[i] = mbs;
    il[i] = 0;
  }

  /* create and initialize a linked list for storing column indices of the active row k */
  nlnk = mbs + 1;
  PetscCall(PetscLLCreate(mbs, mbs, nlnk, lnk, lnkbt));

  /* initial FreeSpace size is fill*(ai[mbs]+1) */
  PetscCall(PetscFreeSpaceGet(PetscRealIntMultTruncate(fill, ai[mbs] + 1), &free_space));
  current_space = free_space;

  for (k = 0; k < mbs; k++) { /* for each active row k */
    /* initialize lnk by the column indices of row rip[k] of A */
    nzk   = 0;
    ncols = ai[k + 1] - ai[k];
    PetscCheck(ncols, PETSC_COMM_SELF, PETSC_ERR_MAT_CH_ZRPVT, "Empty row %" PetscInt_FMT " in matrix ", k);
    for (j = 0; j < ncols; j++) {
      i       = *(aj + ai[k] + j);
      cols[j] = i;
    }
    PetscCall(PetscLLAdd(ncols, cols, mbs, &nlnk, lnk, lnkbt));
    nzk += nlnk;

    /* update lnk by computing fill-in for each pivot row to be merged in */
    prow = jl[k]; /* 1st pivot row */

    while (prow < k) {
      nextprow = jl[prow];
      /* merge prow into k-th row */
      jmin   = il[prow] + 1; /* index of the 2nd nzero entry in U(prow,k:mbs-1) */
      jmax   = ui[prow + 1];
      ncols  = jmax - jmin;
      uj_ptr = ui_ptr[prow] + jmin - ui[prow]; /* points to the 2nd nzero entry in U(prow,k:mbs-1) */
      PetscCall(PetscLLAddSorted(ncols, uj_ptr, mbs, &nlnk, lnk, lnkbt));
      nzk += nlnk;

      /* update il and jl for prow */
      if (jmin < jmax) {
        il[prow] = jmin;
        j        = *uj_ptr;
        jl[prow] = jl[j];
        jl[j]    = prow;
      }
      prow = nextprow;
    }

    /* if free space is not available, make more free space */
    if (current_space->local_remaining < nzk) {
      i = mbs - k + 1;                                   /* num of unfactored rows */
      i = PetscIntMultTruncate(i, PetscMin(nzk, i - 1)); /* i*nzk, i*(i-1): estimated and max additional space needed */
      PetscCall(PetscFreeSpaceGet(i, &current_space));
      reallocs++;
    }

    /* copy data into free space, then initialize lnk */
    PetscCall(PetscLLClean(mbs, mbs, nzk, lnk, current_space->array, lnkbt));

    /* add the k-th row into il and jl */
    if (nzk > 1) {
      i     = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:mbs-1) */
      jl[k] = jl[i];
      jl[i] = k;
      il[k] = ui[k] + 1;
    }
    ui_ptr[k] = current_space->array;

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

    ui[k + 1] = ui[k] + nzk;
  }

  PetscCall(ISRestoreIndices(perm, &rip));
  PetscCall(PetscFree4(ui_ptr, il, jl, cols));

  /* destroy list of free space and other temporary array(s) */
  PetscCall(PetscMalloc1(ui[mbs] + 1, &uj));
  PetscCall(PetscFreeSpaceContiguous_Cholesky(&free_space, uj, mbs, ui, udiag)); /* store matrix factor */
  PetscCall(PetscLLDestroy(lnk, lnkbt));

  /* put together the new matrix in MATSEQSBAIJ format */
  PetscCall(MatSeqSBAIJSetPreallocation(fact, bs, MAT_SKIP_ALLOCATION, NULL));

  b               = (Mat_SeqSBAIJ *)fact->data;
  b->singlemalloc = PETSC_FALSE;
  b->free_a       = PETSC_TRUE;
  b->free_ij      = PETSC_TRUE;

  PetscCall(PetscMalloc1(ui[mbs] + 1, &b->a));

  b->j         = uj;
  b->i         = ui;
  b->diag      = udiag;
  b->free_diag = PETSC_TRUE;
  b->ilen      = NULL;
  b->imax      = NULL;
  b->row       = perm;
  b->icol      = perm;

  PetscCall(PetscObjectReference((PetscObject)perm));
  PetscCall(PetscObjectReference((PetscObject)perm));

  b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */

  PetscCall(PetscMalloc1(mbs + 1, &b->solve_work));

  b->maxnz = b->nz = ui[mbs];

  fact->info.factor_mallocs   = reallocs;
  fact->info.fill_ratio_given = fill;
  if (ai[mbs] != 0) {
    fact->info.fill_ratio_needed = ((PetscReal)ui[mbs]) / ai[mbs];
  } else {
    fact->info.fill_ratio_needed = 0.0;
  }
#if defined(PETSC_USE_INFO)
  if (ai[mbs] != 0) {
    PetscReal af = fact->info.fill_ratio_needed;
    PetscCall(PetscInfo(A, "Reallocs %" PetscInt_FMT " Fill ratio:given %g needed %g\n", reallocs, (double)fill, (double)af));
    PetscCall(PetscInfo(A, "Run with -pc_factor_fill %g or use \n", (double)af));
    PetscCall(PetscInfo(A, "PCFactorSetFill(pc,%g) for best performance.\n", (double)af));
  } else {
    PetscCall(PetscInfo(A, "Empty matrix\n"));
  }
#endif
  fact->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqSBAIJ_1_NaturalOrdering;
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode MatCholeskyFactorSymbolic_SeqSBAIJ_inplace(Mat fact, Mat A, IS perm, const MatFactorInfo *info)
{
  Mat_SeqSBAIJ      *a = (Mat_SeqSBAIJ *)A->data;
  Mat_SeqSBAIJ      *b;
  PetscBool          perm_identity, missing;
  PetscReal          fill = info->fill;
  const PetscInt    *rip, *ai, *aj;
  PetscInt           i, mbs = a->mbs, bs = A->rmap->bs, reallocs = 0, prow, d;
  PetscInt          *jl, jmin, jmax, nzk, *ui, k, j, *il, nextprow;
  PetscInt           nlnk, *lnk, ncols, *cols, *uj, **ui_ptr, *uj_ptr;
  PetscFreeSpaceList free_space = NULL, current_space = NULL;
  PetscBT            lnkbt;

  PetscFunctionBegin;
  PetscCall(MatMissingDiagonal(A, &missing, &d));
  PetscCheck(!missing, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONGSTATE, "Matrix is missing diagonal entry %" PetscInt_FMT, d);

  /*
   This code originally uses Modified Sparse Row (MSR) storage
   (see page 85, "Iterative Methods ..." by Saad) for the output matrix B - bad choice!
   Then it is rewritten so the factor B takes seqsbaij format. However the associated
   MatCholeskyFactorNumeric_() have not been modified for the cases of bs>1 or !perm_identity,
   thus the original code in MSR format is still used for these cases.
   The code below should replace MatCholeskyFactorSymbolic_SeqSBAIJ_MSR() whenever
   MatCholeskyFactorNumeric_() is modified for using sbaij symbolic factor.
  */
  if (bs > 1) {
    PetscCall(MatCholeskyFactorSymbolic_SeqSBAIJ_MSR(fact, A, perm, info));
    PetscFunctionReturn(PETSC_SUCCESS);
  }

  /* check whether perm is the identity mapping */
  PetscCall(ISIdentity(perm, &perm_identity));
  PetscCheck(perm_identity, PETSC_COMM_SELF, PETSC_ERR_SUP, "Matrix reordering is not supported for sbaij matrix. Use aij format");
  a->permute = PETSC_FALSE;
  ai         = a->i;
  aj         = a->j;
  PetscCall(ISGetIndices(perm, &rip));

  /* initialization */
  PetscCall(PetscMalloc1(mbs + 1, &ui));
  ui[0] = 0;

  /* jl: linked list for storing indices of the pivot rows
     il: il[i] points to the 1st nonzero entry of U(i,k:mbs-1) */
  PetscCall(PetscMalloc4(mbs, &ui_ptr, mbs, &il, mbs, &jl, mbs, &cols));
  for (i = 0; i < mbs; i++) {
    jl[i] = mbs;
    il[i] = 0;
  }

  /* create and initialize a linked list for storing column indices of the active row k */
  nlnk = mbs + 1;
  PetscCall(PetscLLCreate(mbs, mbs, nlnk, lnk, lnkbt));

  /* initial FreeSpace size is fill*(ai[mbs]+1) */
  PetscCall(PetscFreeSpaceGet(PetscRealIntMultTruncate(fill, ai[mbs] + 1), &free_space));
  current_space = free_space;

  for (k = 0; k < mbs; k++) { /* for each active row k */
    /* initialize lnk by the column indices of row rip[k] of A */
    nzk   = 0;
    ncols = ai[rip[k] + 1] - ai[rip[k]];
    for (j = 0; j < ncols; j++) {
      i       = *(aj + ai[rip[k]] + j);
      cols[j] = rip[i];
    }
    PetscCall(PetscLLAdd(ncols, cols, mbs, &nlnk, lnk, lnkbt));
    nzk += nlnk;

    /* update lnk by computing fill-in for each pivot row to be merged in */
    prow = jl[k]; /* 1st pivot row */

    while (prow < k) {
      nextprow = jl[prow];
      /* merge prow into k-th row */
      jmin   = il[prow] + 1; /* index of the 2nd nzero entry in U(prow,k:mbs-1) */
      jmax   = ui[prow + 1];
      ncols  = jmax - jmin;
      uj_ptr = ui_ptr[prow] + jmin - ui[prow]; /* points to the 2nd nzero entry in U(prow,k:mbs-1) */
      PetscCall(PetscLLAddSorted(ncols, uj_ptr, mbs, &nlnk, lnk, lnkbt));
      nzk += nlnk;

      /* update il and jl for prow */
      if (jmin < jmax) {
        il[prow] = jmin;

        j        = *uj_ptr;
        jl[prow] = jl[j];
        jl[j]    = prow;
      }
      prow = nextprow;
    }

    /* if free space is not available, make more free space */
    if (current_space->local_remaining < nzk) {
      i = mbs - k + 1;                                                            /* num of unfactored rows */
      i = PetscMin(PetscIntMultTruncate(i, nzk), PetscIntMultTruncate(i, i - 1)); /* i*nzk, i*(i-1): estimated and max additional space needed */
      PetscCall(PetscFreeSpaceGet(i, &current_space));
      reallocs++;
    }

    /* copy data into free space, then initialize lnk */
    PetscCall(PetscLLClean(mbs, mbs, nzk, lnk, current_space->array, lnkbt));

    /* add the k-th row into il and jl */
    if (nzk - 1 > 0) {
      i     = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:mbs-1) */
      jl[k] = jl[i];
      jl[i] = k;
      il[k] = ui[k] + 1;
    }
    ui_ptr[k] = current_space->array;

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

    ui[k + 1] = ui[k] + nzk;
  }

  PetscCall(ISRestoreIndices(perm, &rip));
  PetscCall(PetscFree4(ui_ptr, il, jl, cols));

  /* destroy list of free space and other temporary array(s) */
  PetscCall(PetscMalloc1(ui[mbs] + 1, &uj));
  PetscCall(PetscFreeSpaceContiguous(&free_space, uj));
  PetscCall(PetscLLDestroy(lnk, lnkbt));

  /* put together the new matrix in MATSEQSBAIJ format */
  PetscCall(MatSeqSBAIJSetPreallocation(fact, bs, MAT_SKIP_ALLOCATION, NULL));

  b               = (Mat_SeqSBAIJ *)fact->data;
  b->singlemalloc = PETSC_FALSE;
  b->free_a       = PETSC_TRUE;
  b->free_ij      = PETSC_TRUE;

  PetscCall(PetscMalloc1(ui[mbs] + 1, &b->a));

  b->j    = uj;
  b->i    = ui;
  b->diag = NULL;
  b->ilen = NULL;
  b->imax = NULL;
  b->row  = perm;

  b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */

  PetscCall(PetscObjectReference((PetscObject)perm));
  b->icol = perm;
  PetscCall(PetscObjectReference((PetscObject)perm));
  PetscCall(PetscMalloc1(mbs + 1, &b->solve_work));
  b->maxnz = b->nz = ui[mbs];

  fact->info.factor_mallocs   = reallocs;
  fact->info.fill_ratio_given = fill;
  if (ai[mbs] != 0) {
    fact->info.fill_ratio_needed = ((PetscReal)ui[mbs]) / ai[mbs];
  } else {
    fact->info.fill_ratio_needed = 0.0;
  }
#if defined(PETSC_USE_INFO)
  if (ai[mbs] != 0) {
    PetscReal af = fact->info.fill_ratio_needed;
    PetscCall(PetscInfo(A, "Reallocs %" PetscInt_FMT " Fill ratio:given %g needed %g\n", reallocs, (double)fill, (double)af));
    PetscCall(PetscInfo(A, "Run with -pc_factor_fill %g or use \n", (double)af));
    PetscCall(PetscInfo(A, "PCFactorSetFill(pc,%g) for best performance.\n", (double)af));
  } else {
    PetscCall(PetscInfo(A, "Empty matrix\n"));
  }
#endif
  PetscCall(MatSeqSBAIJSetNumericFactorization_inplace(fact, perm_identity));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode MatCholeskyFactorNumeric_SeqSBAIJ_N(Mat C, Mat A, const MatFactorInfo *info)
{
  Mat_SeqSBAIJ   *a = (Mat_SeqSBAIJ *)A->data, *b = (Mat_SeqSBAIJ *)C->data;
  IS              perm = b->row;
  const PetscInt *ai, *aj, *perm_ptr, mbs = a->mbs, *bi = b->i, *bj = b->j;
  PetscInt        i, j;
  PetscInt       *a2anew, k, k1, jmin, jmax, *jl, *il, vj, nexti, ili;
  PetscInt        bs = A->rmap->bs, bs2 = a->bs2;
  MatScalar      *ba = b->a, *aa, *ap, *dk, *uik;
  MatScalar      *u, *diag, *rtmp, *rtmp_ptr;
  MatScalar      *work;
  PetscInt       *pivots;
  PetscBool       allowzeropivot, zeropivotdetected;

  PetscFunctionBegin;
  /* initialization */
  PetscCall(PetscCalloc1(bs2 * mbs, &rtmp));
  PetscCall(PetscMalloc2(mbs, &il, mbs, &jl));
  allowzeropivot = PetscNot(A->erroriffailure);

  il[0] = 0;
  for (i = 0; i < mbs; i++) jl[i] = mbs;

  PetscCall(PetscMalloc3(bs2, &dk, bs2, &uik, bs, &work));
  PetscCall(PetscMalloc1(bs, &pivots));

  PetscCall(ISGetIndices(perm, &perm_ptr));

  /* check permutation */
  if (!a->permute) {
    ai = a->i;
    aj = a->j;
    aa = a->a;
  } else {
    ai = a->inew;
    aj = a->jnew;
    PetscCall(PetscMalloc1(bs2 * ai[mbs], &aa));
    PetscCall(PetscArraycpy(aa, a->a, bs2 * ai[mbs]));
    PetscCall(PetscMalloc1(ai[mbs], &a2anew));
    PetscCall(PetscArraycpy(a2anew, a->a2anew, ai[mbs]));

    for (i = 0; i < mbs; i++) {
      jmin = ai[i];
      jmax = ai[i + 1];
      for (j = jmin; j < jmax; j++) {
        while (a2anew[j] != j) {
          k         = a2anew[j];
          a2anew[j] = a2anew[k];
          a2anew[k] = k;
          for (k1 = 0; k1 < bs2; k1++) {
            dk[k1]           = aa[k * bs2 + k1];
            aa[k * bs2 + k1] = aa[j * bs2 + k1];
            aa[j * bs2 + k1] = dk[k1];
          }
        }
        /* transform columnoriented blocks that lie in the lower triangle to roworiented blocks */
        if (i > aj[j]) {
          ap = aa + j * bs2;                       /* ptr to the beginning of j-th block of aa */
          for (k = 0; k < bs2; k++) dk[k] = ap[k]; /* dk <- j-th block of aa */
          for (k = 0; k < bs; k++) {               /* j-th block of aa <- dk^T */
            for (k1 = 0; k1 < bs; k1++) *ap++ = dk[k + bs * k1];
          }
        }
      }
    }
    PetscCall(PetscFree(a2anew));
  }

  /* for each row k */
  for (k = 0; k < mbs; k++) {
    /*initialize k-th row with elements nonzero in row perm(k) of A */
    jmin = ai[perm_ptr[k]];
    jmax = ai[perm_ptr[k] + 1];

    ap = aa + jmin * bs2;
    for (j = jmin; j < jmax; j++) {
      vj       = perm_ptr[aj[j]]; /* block col. index */
      rtmp_ptr = rtmp + vj * bs2;
      for (i = 0; i < bs2; i++) *rtmp_ptr++ = *ap++;
    }

    /* modify k-th row by adding in those rows i with U(i,k) != 0 */
    PetscCall(PetscArraycpy(dk, rtmp + k * bs2, bs2));
    i = jl[k]; /* first row to be added to k_th row  */

    while (i < k) {
      nexti = jl[i]; /* next row to be added to k_th row */

      /* compute multiplier */
      ili = il[i]; /* index of first nonzero element in U(i,k:bms-1) */

      /* uik = -inv(Di)*U_bar(i,k) */
      diag = ba + i * bs2;
      u    = ba + ili * bs2;
      PetscCall(PetscArrayzero(uik, bs2));
      PetscKernel_A_gets_A_minus_B_times_C(bs, uik, diag, u);

      /* update D(k) += -U(i,k)^T * U_bar(i,k) */
      PetscKernel_A_gets_A_plus_Btranspose_times_C(bs, dk, uik, u);
      PetscCall(PetscLogFlops(4.0 * bs * bs2));

      /* update -U(i,k) */
      PetscCall(PetscArraycpy(ba + ili * bs2, uik, bs2));

      /* add multiple of row i to k-th row ... */
      jmin = ili + 1;
      jmax = bi[i + 1];
      if (jmin < jmax) {
        for (j = jmin; j < jmax; j++) {
          /* rtmp += -U(i,k)^T * U_bar(i,j) */
          rtmp_ptr = rtmp + bj[j] * bs2;
          u        = ba + j * bs2;
          PetscKernel_A_gets_A_plus_Btranspose_times_C(bs, rtmp_ptr, uik, u);
        }
        PetscCall(PetscLogFlops(2.0 * bs * bs2 * (jmax - jmin)));

        /* ... add i to row list for next nonzero entry */
        il[i] = jmin; /* update il(i) in column k+1, ... mbs-1 */
        j     = bj[jmin];
        jl[i] = jl[j];
        jl[j] = i; /* update jl */
      }
      i = nexti;
    }

    /* save nonzero entries in k-th row of U ... */

    /* invert diagonal block */
    diag = ba + k * bs2;
    PetscCall(PetscArraycpy(diag, dk, bs2));

    PetscCall(PetscKernel_A_gets_inverse_A(bs, diag, pivots, work, allowzeropivot, &zeropivotdetected));
    if (zeropivotdetected) C->factorerrortype = MAT_FACTOR_NUMERIC_ZEROPIVOT;

    jmin = bi[k];
    jmax = bi[k + 1];
    if (jmin < jmax) {
      for (j = jmin; j < jmax; j++) {
        vj       = bj[j]; /* block col. index of U */
        u        = ba + j * bs2;
        rtmp_ptr = rtmp + vj * bs2;
        for (k1 = 0; k1 < bs2; k1++) {
          *u++        = *rtmp_ptr;
          *rtmp_ptr++ = 0.0;
        }
      }

      /* ... add k to row list for first nonzero entry in k-th row */
      il[k] = jmin;
      i     = bj[jmin];
      jl[k] = jl[i];
      jl[i] = k;
    }
  }

  PetscCall(PetscFree(rtmp));
  PetscCall(PetscFree2(il, jl));
  PetscCall(PetscFree3(dk, uik, work));
  PetscCall(PetscFree(pivots));
  if (a->permute) PetscCall(PetscFree(aa));

  PetscCall(ISRestoreIndices(perm, &perm_ptr));

  C->ops->solve          = MatSolve_SeqSBAIJ_N_inplace;
  C->ops->solvetranspose = MatSolve_SeqSBAIJ_N_inplace;
  C->ops->forwardsolve   = MatForwardSolve_SeqSBAIJ_N_inplace;
  C->ops->backwardsolve  = MatBackwardSolve_SeqSBAIJ_N_inplace;

  C->assembled    = PETSC_TRUE;
  C->preallocated = PETSC_TRUE;

  PetscCall(PetscLogFlops(1.3333 * bs * bs2 * b->mbs)); /* from inverting diagonal blocks */
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode MatCholeskyFactorNumeric_SeqSBAIJ_N_NaturalOrdering(Mat C, Mat A, const MatFactorInfo *info)
{
  Mat_SeqSBAIJ *a = (Mat_SeqSBAIJ *)A->data, *b = (Mat_SeqSBAIJ *)C->data;
  PetscInt      i, j, mbs = a->mbs, *bi = b->i, *bj = b->j;
  PetscInt     *ai, *aj, k, k1, jmin, jmax, *jl, *il, vj, nexti, ili;
  PetscInt      bs = A->rmap->bs, bs2 = a->bs2;
  MatScalar    *ba = b->a, *aa, *ap, *dk, *uik;
  MatScalar    *u, *diag, *rtmp, *rtmp_ptr;
  MatScalar    *work;
  PetscInt     *pivots;
  PetscBool     allowzeropivot, zeropivotdetected;

  PetscFunctionBegin;
  PetscCall(PetscCalloc1(bs2 * mbs, &rtmp));
  PetscCall(PetscMalloc2(mbs, &il, mbs, &jl));
  il[0] = 0;
  for (i = 0; i < mbs; i++) jl[i] = mbs;

  PetscCall(PetscMalloc3(bs2, &dk, bs2, &uik, bs, &work));
  PetscCall(PetscMalloc1(bs, &pivots));
  allowzeropivot = PetscNot(A->erroriffailure);

  ai = a->i;
  aj = a->j;
  aa = a->a;

  /* for each row k */
  for (k = 0; k < mbs; k++) {
    /*initialize k-th row with elements nonzero in row k of A */
    jmin = ai[k];
    jmax = ai[k + 1];
    ap   = aa + jmin * bs2;
    for (j = jmin; j < jmax; j++) {
      vj       = aj[j]; /* block col. index */
      rtmp_ptr = rtmp + vj * bs2;
      for (i = 0; i < bs2; i++) *rtmp_ptr++ = *ap++;
    }

    /* modify k-th row by adding in those rows i with U(i,k) != 0 */
    PetscCall(PetscArraycpy(dk, rtmp + k * bs2, bs2));
    i = jl[k]; /* first row to be added to k_th row  */

    while (i < k) {
      nexti = jl[i]; /* next row to be added to k_th row */

      /* compute multiplier */
      ili = il[i]; /* index of first nonzero element in U(i,k:bms-1) */

      /* uik = -inv(Di)*U_bar(i,k) */
      diag = ba + i * bs2;
      u    = ba + ili * bs2;
      PetscCall(PetscArrayzero(uik, bs2));
      PetscKernel_A_gets_A_minus_B_times_C(bs, uik, diag, u);

      /* update D(k) += -U(i,k)^T * U_bar(i,k) */
      PetscKernel_A_gets_A_plus_Btranspose_times_C(bs, dk, uik, u);
      PetscCall(PetscLogFlops(2.0 * bs * bs2));

      /* update -U(i,k) */
      PetscCall(PetscArraycpy(ba + ili * bs2, uik, bs2));

      /* add multiple of row i to k-th row ... */
      jmin = ili + 1;
      jmax = bi[i + 1];
      if (jmin < jmax) {
        for (j = jmin; j < jmax; j++) {
          /* rtmp += -U(i,k)^T * U_bar(i,j) */
          rtmp_ptr = rtmp + bj[j] * bs2;
          u        = ba + j * bs2;
          PetscKernel_A_gets_A_plus_Btranspose_times_C(bs, rtmp_ptr, uik, u);
        }
        PetscCall(PetscLogFlops(2.0 * bs * bs2 * (jmax - jmin)));

        /* ... add i to row list for next nonzero entry */
        il[i] = jmin; /* update il(i) in column k+1, ... mbs-1 */
        j     = bj[jmin];
        jl[i] = jl[j];
        jl[j] = i; /* update jl */
      }
      i = nexti;
    }

    /* save nonzero entries in k-th row of U ... */

    /* invert diagonal block */
    diag = ba + k * bs2;
    PetscCall(PetscArraycpy(diag, dk, bs2));

    PetscCall(PetscKernel_A_gets_inverse_A(bs, diag, pivots, work, allowzeropivot, &zeropivotdetected));
    if (zeropivotdetected) C->factorerrortype = MAT_FACTOR_NUMERIC_ZEROPIVOT;

    jmin = bi[k];
    jmax = bi[k + 1];
    if (jmin < jmax) {
      for (j = jmin; j < jmax; j++) {
        vj       = bj[j]; /* block col. index of U */
        u        = ba + j * bs2;
        rtmp_ptr = rtmp + vj * bs2;
        for (k1 = 0; k1 < bs2; k1++) {
          *u++        = *rtmp_ptr;
          *rtmp_ptr++ = 0.0;
        }
      }

      /* ... add k to row list for first nonzero entry in k-th row */
      il[k] = jmin;
      i     = bj[jmin];
      jl[k] = jl[i];
      jl[i] = k;
    }
  }

  PetscCall(PetscFree(rtmp));
  PetscCall(PetscFree2(il, jl));
  PetscCall(PetscFree3(dk, uik, work));
  PetscCall(PetscFree(pivots));

  C->ops->solve          = MatSolve_SeqSBAIJ_N_NaturalOrdering_inplace;
  C->ops->solvetranspose = MatSolve_SeqSBAIJ_N_NaturalOrdering_inplace;
  C->ops->forwardsolve   = MatForwardSolve_SeqSBAIJ_N_NaturalOrdering_inplace;
  C->ops->backwardsolve  = MatBackwardSolve_SeqSBAIJ_N_NaturalOrdering_inplace;
  C->assembled           = PETSC_TRUE;
  C->preallocated        = PETSC_TRUE;

  PetscCall(PetscLogFlops(1.3333 * bs * bs2 * b->mbs)); /* from inverting diagonal blocks */
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
    Numeric U^T*D*U factorization for SBAIJ format. Modified from SNF of YSMP.
    Version for blocks 2 by 2.
*/
PetscErrorCode MatCholeskyFactorNumeric_SeqSBAIJ_2(Mat C, Mat A, const MatFactorInfo *info)
{
  Mat_SeqSBAIJ   *a = (Mat_SeqSBAIJ *)A->data, *b = (Mat_SeqSBAIJ *)C->data;
  IS              perm = b->row;
  const PetscInt *ai, *aj, *perm_ptr;
  PetscInt        i, j, mbs = a->mbs, *bi = b->i, *bj = b->j;
  PetscInt       *a2anew, k, k1, jmin, jmax, *jl, *il, vj, nexti, ili;
  MatScalar      *ba = b->a, *aa, *ap;
  MatScalar      *u, *diag, *rtmp, *rtmp_ptr, dk[4], uik[4];
  PetscReal       shift = info->shiftamount;
  PetscBool       allowzeropivot, zeropivotdetected;

  PetscFunctionBegin;
  allowzeropivot = PetscNot(A->erroriffailure);

  /* initialization */
  /* il and jl record the first nonzero element in each row of the accessing
     window U(0:k, k:mbs-1).
     jl:    list of rows to be added to uneliminated rows
            i>= k: jl(i) is the first row to be added to row i
            i<  k: jl(i) is the row following row i in some list of rows
            jl(i) = mbs indicates the end of a list
     il(i): points to the first nonzero element in columns k,...,mbs-1 of
            row i of U */
  PetscCall(PetscCalloc1(4 * mbs, &rtmp));
  PetscCall(PetscMalloc2(mbs, &il, mbs, &jl));
  il[0] = 0;
  for (i = 0; i < mbs; i++) jl[i] = mbs;

  PetscCall(ISGetIndices(perm, &perm_ptr));

  /* check permutation */
  if (!a->permute) {
    ai = a->i;
    aj = a->j;
    aa = a->a;
  } else {
    ai = a->inew;
    aj = a->jnew;
    PetscCall(PetscMalloc1(4 * ai[mbs], &aa));
    PetscCall(PetscArraycpy(aa, a->a, 4 * ai[mbs]));
    PetscCall(PetscMalloc1(ai[mbs], &a2anew));
    PetscCall(PetscArraycpy(a2anew, a->a2anew, ai[mbs]));

    for (i = 0; i < mbs; i++) {
      jmin = ai[i];
      jmax = ai[i + 1];
      for (j = jmin; j < jmax; j++) {
        while (a2anew[j] != j) {
          k         = a2anew[j];
          a2anew[j] = a2anew[k];
          a2anew[k] = k;
          for (k1 = 0; k1 < 4; k1++) {
            dk[k1]         = aa[k * 4 + k1];
            aa[k * 4 + k1] = aa[j * 4 + k1];
            aa[j * 4 + k1] = dk[k1];
          }
        }
        /* transform columnoriented blocks that lie in the lower triangle to roworiented blocks */
        if (i > aj[j]) {
          ap    = aa + j * 4; /* ptr to the beginning of the block */
          dk[1] = ap[1];      /* swap ap[1] and ap[2] */
          ap[1] = ap[2];
          ap[2] = dk[1];
        }
      }
    }
    PetscCall(PetscFree(a2anew));
  }

  /* for each row k */
  for (k = 0; k < mbs; k++) {
    /*initialize k-th row with elements nonzero in row perm(k) of A */
    jmin = ai[perm_ptr[k]];
    jmax = ai[perm_ptr[k] + 1];
    ap   = aa + jmin * 4;
    for (j = jmin; j < jmax; j++) {
      vj       = perm_ptr[aj[j]]; /* block col. index */
      rtmp_ptr = rtmp + vj * 4;
      for (i = 0; i < 4; i++) *rtmp_ptr++ = *ap++;
    }

    /* modify k-th row by adding in those rows i with U(i,k) != 0 */
    PetscCall(PetscArraycpy(dk, rtmp + k * 4, 4));
    i = jl[k]; /* first row to be added to k_th row  */

    while (i < k) {
      nexti = jl[i]; /* next row to be added to k_th row */

      /* compute multiplier */
      ili = il[i]; /* index of first nonzero element in U(i,k:bms-1) */

      /* uik = -inv(Di)*U_bar(i,k): - ba[ili]*ba[i] */
      diag   = ba + i * 4;
      u      = ba + ili * 4;
      uik[0] = -(diag[0] * u[0] + diag[2] * u[1]);
      uik[1] = -(diag[1] * u[0] + diag[3] * u[1]);
      uik[2] = -(diag[0] * u[2] + diag[2] * u[3]);
      uik[3] = -(diag[1] * u[2] + diag[3] * u[3]);

      /* update D(k) += -U(i,k)^T * U_bar(i,k): dk += uik*ba[ili] */
      dk[0] += uik[0] * u[0] + uik[1] * u[1];
      dk[1] += uik[2] * u[0] + uik[3] * u[1];
      dk[2] += uik[0] * u[2] + uik[1] * u[3];
      dk[3] += uik[2] * u[2] + uik[3] * u[3];

      PetscCall(PetscLogFlops(16.0 * 2.0));

      /* update -U(i,k): ba[ili] = uik */
      PetscCall(PetscArraycpy(ba + ili * 4, uik, 4));

      /* add multiple of row i to k-th row ... */
      jmin = ili + 1;
      jmax = bi[i + 1];
      if (jmin < jmax) {
        for (j = jmin; j < jmax; j++) {
          /* rtmp += -U(i,k)^T * U_bar(i,j): rtmp[bj[j]] += uik*ba[j]; */
          rtmp_ptr = rtmp + bj[j] * 4;
          u        = ba + j * 4;
          rtmp_ptr[0] += uik[0] * u[0] + uik[1] * u[1];
          rtmp_ptr[1] += uik[2] * u[0] + uik[3] * u[1];
          rtmp_ptr[2] += uik[0] * u[2] + uik[1] * u[3];
          rtmp_ptr[3] += uik[2] * u[2] + uik[3] * u[3];
        }
        PetscCall(PetscLogFlops(16.0 * (jmax - jmin)));

        /* ... add i to row list for next nonzero entry */
        il[i] = jmin; /* update il(i) in column k+1, ... mbs-1 */
        j     = bj[jmin];
        jl[i] = jl[j];
        jl[j] = i; /* update jl */
      }
      i = nexti;
    }

    /* save nonzero entries in k-th row of U ... */

    /* invert diagonal block */
    diag = ba + k * 4;
    PetscCall(PetscArraycpy(diag, dk, 4));
    PetscCall(PetscKernel_A_gets_inverse_A_2(diag, shift, allowzeropivot, &zeropivotdetected));
    if (zeropivotdetected) C->factorerrortype = MAT_FACTOR_NUMERIC_ZEROPIVOT;

    jmin = bi[k];
    jmax = bi[k + 1];
    if (jmin < jmax) {
      for (j = jmin; j < jmax; j++) {
        vj       = bj[j]; /* block col. index of U */
        u        = ba + j * 4;
        rtmp_ptr = rtmp + vj * 4;
        for (k1 = 0; k1 < 4; k1++) {
          *u++        = *rtmp_ptr;
          *rtmp_ptr++ = 0.0;
        }
      }

      /* ... add k to row list for first nonzero entry in k-th row */
      il[k] = jmin;
      i     = bj[jmin];
      jl[k] = jl[i];
      jl[i] = k;
    }
  }

  PetscCall(PetscFree(rtmp));
  PetscCall(PetscFree2(il, jl));
  if (a->permute) PetscCall(PetscFree(aa));
  PetscCall(ISRestoreIndices(perm, &perm_ptr));

  C->ops->solve          = MatSolve_SeqSBAIJ_2_inplace;
  C->ops->solvetranspose = MatSolve_SeqSBAIJ_2_inplace;
  C->assembled           = PETSC_TRUE;
  C->preallocated        = PETSC_TRUE;

  PetscCall(PetscLogFlops(1.3333 * 8 * b->mbs)); /* from inverting diagonal blocks */
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
      Version for when blocks are 2 by 2 Using natural ordering
*/
PetscErrorCode MatCholeskyFactorNumeric_SeqSBAIJ_2_NaturalOrdering(Mat C, Mat A, const MatFactorInfo *info)
{
  Mat_SeqSBAIJ *a = (Mat_SeqSBAIJ *)A->data, *b = (Mat_SeqSBAIJ *)C->data;
  PetscInt      i, j, mbs = a->mbs, *bi = b->i, *bj = b->j;
  PetscInt     *ai, *aj, k, k1, jmin, jmax, *jl, *il, vj, nexti, ili;
  MatScalar    *ba = b->a, *aa, *ap, dk[8], uik[8];
  MatScalar    *u, *diag, *rtmp, *rtmp_ptr;
  PetscReal     shift = info->shiftamount;
  PetscBool     allowzeropivot, zeropivotdetected;

  PetscFunctionBegin;
  allowzeropivot = PetscNot(A->erroriffailure);

  /* initialization */
  /* il and jl record the first nonzero element in each row of the accessing
     window U(0:k, k:mbs-1).
     jl:    list of rows to be added to uneliminated rows
            i>= k: jl(i) is the first row to be added to row i
            i<  k: jl(i) is the row following row i in some list of rows
            jl(i) = mbs indicates the end of a list
     il(i): points to the first nonzero element in columns k,...,mbs-1 of
            row i of U */
  PetscCall(PetscCalloc1(4 * mbs, &rtmp));
  PetscCall(PetscMalloc2(mbs, &il, mbs, &jl));
  il[0] = 0;
  for (i = 0; i < mbs; i++) jl[i] = mbs;

  ai = a->i;
  aj = a->j;
  aa = a->a;

  /* for each row k */
  for (k = 0; k < mbs; k++) {
    /*initialize k-th row with elements nonzero in row k of A */
    jmin = ai[k];
    jmax = ai[k + 1];
    ap   = aa + jmin * 4;
    for (j = jmin; j < jmax; j++) {
      vj       = aj[j]; /* block col. index */
      rtmp_ptr = rtmp + vj * 4;
      for (i = 0; i < 4; i++) *rtmp_ptr++ = *ap++;
    }

    /* modify k-th row by adding in those rows i with U(i,k) != 0 */
    PetscCall(PetscArraycpy(dk, rtmp + k * 4, 4));
    i = jl[k]; /* first row to be added to k_th row  */

    while (i < k) {
      nexti = jl[i]; /* next row to be added to k_th row */

      /* compute multiplier */
      ili = il[i]; /* index of first nonzero element in U(i,k:bms-1) */

      /* uik = -inv(Di)*U_bar(i,k): - ba[ili]*ba[i] */
      diag   = ba + i * 4;
      u      = ba + ili * 4;
      uik[0] = -(diag[0] * u[0] + diag[2] * u[1]);
      uik[1] = -(diag[1] * u[0] + diag[3] * u[1]);
      uik[2] = -(diag[0] * u[2] + diag[2] * u[3]);
      uik[3] = -(diag[1] * u[2] + diag[3] * u[3]);

      /* update D(k) += -U(i,k)^T * U_bar(i,k): dk += uik*ba[ili] */
      dk[0] += uik[0] * u[0] + uik[1] * u[1];
      dk[1] += uik[2] * u[0] + uik[3] * u[1];
      dk[2] += uik[0] * u[2] + uik[1] * u[3];
      dk[3] += uik[2] * u[2] + uik[3] * u[3];

      PetscCall(PetscLogFlops(16.0 * 2.0));

      /* update -U(i,k): ba[ili] = uik */
      PetscCall(PetscArraycpy(ba + ili * 4, uik, 4));

      /* add multiple of row i to k-th row ... */
      jmin = ili + 1;
      jmax = bi[i + 1];
      if (jmin < jmax) {
        for (j = jmin; j < jmax; j++) {
          /* rtmp += -U(i,k)^T * U_bar(i,j): rtmp[bj[j]] += uik*ba[j]; */
          rtmp_ptr = rtmp + bj[j] * 4;
          u        = ba + j * 4;
          rtmp_ptr[0] += uik[0] * u[0] + uik[1] * u[1];
          rtmp_ptr[1] += uik[2] * u[0] + uik[3] * u[1];
          rtmp_ptr[2] += uik[0] * u[2] + uik[1] * u[3];
          rtmp_ptr[3] += uik[2] * u[2] + uik[3] * u[3];
        }
        PetscCall(PetscLogFlops(16.0 * (jmax - jmin)));

        /* ... add i to row list for next nonzero entry */
        il[i] = jmin; /* update il(i) in column k+1, ... mbs-1 */
        j     = bj[jmin];
        jl[i] = jl[j];
        jl[j] = i; /* update jl */
      }
      i = nexti;
    }

    /* save nonzero entries in k-th row of U ... */

    /* invert diagonal block */
    diag = ba + k * 4;
    PetscCall(PetscArraycpy(diag, dk, 4));
    PetscCall(PetscKernel_A_gets_inverse_A_2(diag, shift, allowzeropivot, &zeropivotdetected));
    if (zeropivotdetected) C->factorerrortype = MAT_FACTOR_NUMERIC_ZEROPIVOT;

    jmin = bi[k];
    jmax = bi[k + 1];
    if (jmin < jmax) {
      for (j = jmin; j < jmax; j++) {
        vj       = bj[j]; /* block col. index of U */
        u        = ba + j * 4;
        rtmp_ptr = rtmp + vj * 4;
        for (k1 = 0; k1 < 4; k1++) {
          *u++        = *rtmp_ptr;
          *rtmp_ptr++ = 0.0;
        }
      }

      /* ... add k to row list for first nonzero entry in k-th row */
      il[k] = jmin;
      i     = bj[jmin];
      jl[k] = jl[i];
      jl[i] = k;
    }
  }

  PetscCall(PetscFree(rtmp));
  PetscCall(PetscFree2(il, jl));

  C->ops->solve          = MatSolve_SeqSBAIJ_2_NaturalOrdering_inplace;
  C->ops->solvetranspose = MatSolve_SeqSBAIJ_2_NaturalOrdering_inplace;
  C->ops->forwardsolve   = MatForwardSolve_SeqSBAIJ_2_NaturalOrdering_inplace;
  C->ops->backwardsolve  = MatBackwardSolve_SeqSBAIJ_2_NaturalOrdering_inplace;
  C->assembled           = PETSC_TRUE;
  C->preallocated        = PETSC_TRUE;

  PetscCall(PetscLogFlops(1.3333 * 8 * b->mbs)); /* from inverting diagonal blocks */
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
    Numeric U^T*D*U factorization for SBAIJ format.
    Version for blocks are 1 by 1.
*/
PetscErrorCode MatCholeskyFactorNumeric_SeqSBAIJ_1_inplace(Mat C, Mat A, const MatFactorInfo *info)
{
  Mat_SeqSBAIJ   *a = (Mat_SeqSBAIJ *)A->data, *b = (Mat_SeqSBAIJ *)C->data;
  IS              ip = b->row;
  const PetscInt *ai, *aj, *rip;
  PetscInt       *a2anew, i, j, mbs = a->mbs, *bi = b->i, *bj = b->j, *bcol;
  PetscInt        k, jmin, jmax, *jl, *il, col, nexti, ili, nz;
  MatScalar      *rtmp, *ba = b->a, *bval, *aa, dk, uikdi;
  PetscReal       rs;
  FactorShiftCtx  sctx;

  PetscFunctionBegin;
  /* MatPivotSetUp(): initialize shift context sctx */
  PetscCall(PetscMemzero(&sctx, sizeof(FactorShiftCtx)));

  PetscCall(ISGetIndices(ip, &rip));
  if (!a->permute) {
    ai = a->i;
    aj = a->j;
    aa = a->a;
  } else {
    ai = a->inew;
    aj = a->jnew;
    nz = ai[mbs];
    PetscCall(PetscMalloc1(nz, &aa));
    a2anew = a->a2anew;
    bval   = a->a;
    for (j = 0; j < nz; j++) aa[a2anew[j]] = *(bval++);
  }

  /* initialization */
  /* il and jl record the first nonzero element in each row of the accessing
     window U(0:k, k:mbs-1).
     jl:    list of rows to be added to uneliminated rows
            i>= k: jl(i) is the first row to be added to row i
            i<  k: jl(i) is the row following row i in some list of rows
            jl(i) = mbs indicates the end of a list
     il(i): points to the first nonzero element in columns k,...,mbs-1 of
            row i of U */
  PetscCall(PetscMalloc3(mbs, &rtmp, mbs, &il, mbs, &jl));

  do {
    sctx.newshift = PETSC_FALSE;
    il[0]         = 0;
    for (i = 0; i < mbs; i++) {
      rtmp[i] = 0.0;
      jl[i]   = mbs;
    }

    for (k = 0; k < mbs; k++) {
      /*initialize k-th row by the perm[k]-th row of A */
      jmin = ai[rip[k]];
      jmax = ai[rip[k] + 1];
      bval = ba + bi[k];
      for (j = jmin; j < jmax; j++) {
        col       = rip[aj[j]];
        rtmp[col] = aa[j];
        *bval++   = 0.0; /* for in-place factorization */
      }

      /* shift the diagonal of the matrix */
      if (sctx.nshift) rtmp[k] += sctx.shift_amount;

      /* modify k-th row by adding in those rows i with U(i,k)!=0 */
      dk = rtmp[k];
      i  = jl[k]; /* first row to be added to k_th row  */

      while (i < k) {
        nexti = jl[i]; /* next row to be added to k_th row */

        /* compute multiplier, update diag(k) and U(i,k) */
        ili   = il[i];                /* index of first nonzero element in U(i,k:bms-1) */
        uikdi = -ba[ili] * ba[bi[i]]; /* diagonal(k) */
        dk += uikdi * ba[ili];
        ba[ili] = uikdi; /* -U(i,k) */

        /* add multiple of row i to k-th row */
        jmin = ili + 1;
        jmax = bi[i + 1];
        if (jmin < jmax) {
          for (j = jmin; j < jmax; j++) rtmp[bj[j]] += uikdi * ba[j];
          PetscCall(PetscLogFlops(2.0 * (jmax - jmin)));

          /* update il and jl for row i */
          il[i] = jmin;
          j     = bj[jmin];
          jl[i] = jl[j];
          jl[j] = i;
        }
        i = nexti;
      }

      /* shift the diagonals when zero pivot is detected */
      /* compute rs=sum of abs(off-diagonal) */
      rs   = 0.0;
      jmin = bi[k] + 1;
      nz   = bi[k + 1] - jmin;
      if (nz) {
        bcol = bj + jmin;
        while (nz--) {
          rs += PetscAbsScalar(rtmp[*bcol]);
          bcol++;
        }
      }

      sctx.rs = rs;
      sctx.pv = dk;
      PetscCall(MatPivotCheck(C, A, info, &sctx, k));
      if (sctx.newshift) break; /* sctx.shift_amount is updated */
      dk = sctx.pv;

      /* copy data into U(k,:) */
      ba[bi[k]] = 1.0 / dk; /* U(k,k) */
      jmin      = bi[k] + 1;
      jmax      = bi[k + 1];
      if (jmin < jmax) {
        for (j = jmin; j < jmax; j++) {
          col       = bj[j];
          ba[j]     = rtmp[col];
          rtmp[col] = 0.0;
        }
        /* add the k-th row into il and jl */
        il[k] = jmin;
        i     = bj[jmin];
        jl[k] = jl[i];
        jl[i] = k;
      }
    }
  } while (sctx.newshift);
  PetscCall(PetscFree3(rtmp, il, jl));
  if (a->permute) PetscCall(PetscFree(aa));

  PetscCall(ISRestoreIndices(ip, &rip));

  C->ops->solve          = MatSolve_SeqSBAIJ_1_inplace;
  C->ops->solves         = MatSolves_SeqSBAIJ_1_inplace;
  C->ops->solvetranspose = MatSolve_SeqSBAIJ_1_inplace;
  C->ops->forwardsolve   = MatForwardSolve_SeqSBAIJ_1_inplace;
  C->ops->backwardsolve  = MatBackwardSolve_SeqSBAIJ_1_inplace;
  C->assembled           = PETSC_TRUE;
  C->preallocated        = PETSC_TRUE;

  PetscCall(PetscLogFlops(C->rmap->N));
  if (sctx.nshift) {
    if (info->shifttype == (PetscReal)MAT_SHIFT_NONZERO) {
      PetscCall(PetscInfo(A, "number of shiftnz tries %" PetscInt_FMT ", shift_amount %g\n", sctx.nshift, (double)sctx.shift_amount));
    } else if (info->shifttype == (PetscReal)MAT_SHIFT_POSITIVE_DEFINITE) {
      PetscCall(PetscInfo(A, "number of shiftpd tries %" PetscInt_FMT ", shift_amount %g\n", sctx.nshift, (double)sctx.shift_amount));
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
  Version for when blocks are 1 by 1 Using natural ordering under new datastructure
  Modified from MatCholeskyFactorNumeric_SeqAIJ()
*/
PetscErrorCode MatCholeskyFactorNumeric_SeqSBAIJ_1_NaturalOrdering(Mat B, Mat A, const MatFactorInfo *info)
{
  Mat_SeqSBAIJ  *a = (Mat_SeqSBAIJ *)A->data;
  Mat_SeqSBAIJ  *b = (Mat_SeqSBAIJ *)B->data;
  PetscInt       i, j, mbs = A->rmap->n, *bi = b->i, *bj = b->j, *bdiag = b->diag, *bjtmp;
  PetscInt      *ai = a->i, *aj = a->j, *ajtmp;
  PetscInt       k, jmin, jmax, *c2r, *il, col, nexti, ili, nz;
  MatScalar     *rtmp, *ba = b->a, *bval, *aa = a->a, dk, uikdi;
  FactorShiftCtx sctx;
  PetscReal      rs;
  MatScalar      d, *v;

  PetscFunctionBegin;
  PetscCall(PetscMalloc3(mbs, &rtmp, mbs, &il, mbs, &c2r));

  /* MatPivotSetUp(): initialize shift context sctx */
  PetscCall(PetscMemzero(&sctx, sizeof(FactorShiftCtx)));

  if (info->shifttype == (PetscReal)MAT_SHIFT_POSITIVE_DEFINITE) { /* set sctx.shift_top=max{rs} */
    sctx.shift_top = info->zeropivot;

    PetscCall(PetscArrayzero(rtmp, mbs));

    for (i = 0; i < mbs; i++) {
      /* calculate sum(|aij|)-RealPart(aii), amt of shift needed for this row */
      d = (aa)[a->diag[i]];
      rtmp[i] += -PetscRealPart(d); /* diagonal entry */
      ajtmp = aj + ai[i] + 1;       /* exclude diagonal */
      v     = aa + ai[i] + 1;
      nz    = ai[i + 1] - ai[i] - 1;
      for (j = 0; j < nz; j++) {
        rtmp[i] += PetscAbsScalar(v[j]);
        rtmp[ajtmp[j]] += PetscAbsScalar(v[j]);
      }
      if (PetscRealPart(rtmp[i]) > sctx.shift_top) sctx.shift_top = PetscRealPart(rtmp[i]);
    }
    sctx.shift_top *= 1.1;
    sctx.nshift_max = 5;
    sctx.shift_lo   = 0.;
    sctx.shift_hi   = 1.;
  }

  /* allocate working arrays
     c2r: linked list, keep track of pivot rows for a given column. c2r[col]: head of the list for a given col
     il:  for active k row, il[i] gives the index of the 1st nonzero entry in U[i,k:n-1] in bj and ba arrays
  */
  do {
    sctx.newshift = PETSC_FALSE;

    for (i = 0; i < mbs; i++) c2r[i] = mbs;
    if (mbs) il[0] = 0;

    for (k = 0; k < mbs; k++) {
      /* zero rtmp */
      nz    = bi[k + 1] - bi[k];
      bjtmp = bj + bi[k];
      for (j = 0; j < nz; j++) rtmp[bjtmp[j]] = 0.0;

      /* load in initial unfactored row */
      bval = ba + bi[k];
      jmin = ai[k];
      jmax = ai[k + 1];
      for (j = jmin; j < jmax; j++) {
        col       = aj[j];
        rtmp[col] = aa[j];
        *bval++   = 0.0; /* for in-place factorization */
      }
      /* shift the diagonal of the matrix: ZeropivotApply() */
      rtmp[k] += sctx.shift_amount; /* shift the diagonal of the matrix */

      /* modify k-th row by adding in those rows i with U(i,k)!=0 */
      dk = rtmp[k];
      i  = c2r[k]; /* first row to be added to k_th row  */

      while (i < k) {
        nexti = c2r[i]; /* next row to be added to k_th row */

        /* compute multiplier, update diag(k) and U(i,k) */
        ili   = il[i];                   /* index of first nonzero element in U(i,k:bms-1) */
        uikdi = -ba[ili] * ba[bdiag[i]]; /* diagonal(k) */
        dk += uikdi * ba[ili];           /* update diag[k] */
        ba[ili] = uikdi;                 /* -U(i,k) */

        /* add multiple of row i to k-th row */
        jmin = ili + 1;
        jmax = bi[i + 1];
        if (jmin < jmax) {
          for (j = jmin; j < jmax; j++) rtmp[bj[j]] += uikdi * ba[j];
          /* update il and c2r for row i */
          il[i]  = jmin;
          j      = bj[jmin];
          c2r[i] = c2r[j];
          c2r[j] = i;
        }
        i = nexti;
      }

      /* copy data into U(k,:) */
      rs   = 0.0;
      jmin = bi[k];
      jmax = bi[k + 1] - 1;
      if (jmin < jmax) {
        for (j = jmin; j < jmax; j++) {
          col   = bj[j];
          ba[j] = rtmp[col];
          rs += PetscAbsScalar(ba[j]);
        }
        /* add the k-th row into il and c2r */
        il[k]  = jmin;
        i      = bj[jmin];
        c2r[k] = c2r[i];
        c2r[i] = k;
      }

      sctx.rs = rs;
      sctx.pv = dk;
      PetscCall(MatPivotCheck(B, A, info, &sctx, k));
      if (sctx.newshift) break;
      dk = sctx.pv;

      ba[bdiag[k]] = 1.0 / dk; /* U(k,k) */
    }
  } while (sctx.newshift);

  PetscCall(PetscFree3(rtmp, il, c2r));

  B->ops->solve          = MatSolve_SeqSBAIJ_1_NaturalOrdering;
  B->ops->solves         = MatSolves_SeqSBAIJ_1;
  B->ops->solvetranspose = MatSolve_SeqSBAIJ_1_NaturalOrdering;
  B->ops->matsolve       = MatMatSolve_SeqSBAIJ_1_NaturalOrdering;
  B->ops->forwardsolve   = MatForwardSolve_SeqSBAIJ_1_NaturalOrdering;
  B->ops->backwardsolve  = MatBackwardSolve_SeqSBAIJ_1_NaturalOrdering;

  B->assembled    = PETSC_TRUE;
  B->preallocated = PETSC_TRUE;

  PetscCall(PetscLogFlops(B->rmap->n));

  /* MatPivotView() */
  if (sctx.nshift) {
    if (info->shifttype == (PetscReal)MAT_SHIFT_POSITIVE_DEFINITE) {
      PetscCall(PetscInfo(A, "number of shift_pd tries %" PetscInt_FMT ", shift_amount %g, diagonal shifted up by %e fraction top_value %e\n", sctx.nshift, (double)sctx.shift_amount, (double)sctx.shift_fraction, (double)sctx.shift_top));
    } else if (info->shifttype == (PetscReal)MAT_SHIFT_NONZERO) {
      PetscCall(PetscInfo(A, "number of shift_nz tries %" PetscInt_FMT ", shift_amount %g\n", sctx.nshift, (double)sctx.shift_amount));
    } else if (info->shifttype == (PetscReal)MAT_SHIFT_INBLOCKS) {
      PetscCall(PetscInfo(A, "number of shift_inblocks applied %" PetscInt_FMT ", each shift_amount %g\n", sctx.nshift, (double)info->shiftamount));
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode MatCholeskyFactorNumeric_SeqSBAIJ_1_NaturalOrdering_inplace(Mat C, Mat A, const MatFactorInfo *info)
{
  Mat_SeqSBAIJ  *a = (Mat_SeqSBAIJ *)A->data, *b = (Mat_SeqSBAIJ *)C->data;
  PetscInt       i, j, mbs = a->mbs;
  PetscInt      *ai = a->i, *aj = a->j, *bi = b->i, *bj = b->j;
  PetscInt       k, jmin, *jl, *il, nexti, ili, *acol, *bcol, nz;
  MatScalar     *rtmp, *ba = b->a, *aa = a->a, dk, uikdi, *aval, *bval;
  PetscReal      rs;
  FactorShiftCtx sctx;

  PetscFunctionBegin;
  /* MatPivotSetUp(): initialize shift context sctx */
  PetscCall(PetscMemzero(&sctx, sizeof(FactorShiftCtx)));

  /* initialization */
  /* il and jl record the first nonzero element in each row of the accessing
     window U(0:k, k:mbs-1).
     jl:    list of rows to be added to uneliminated rows
            i>= k: jl(i) is the first row to be added to row i
            i<  k: jl(i) is the row following row i in some list of rows
            jl(i) = mbs indicates the end of a list
     il(i): points to the first nonzero element in U(i,k:mbs-1)
  */
  PetscCall(PetscMalloc1(mbs, &rtmp));
  PetscCall(PetscMalloc2(mbs, &il, mbs, &jl));

  do {
    sctx.newshift = PETSC_FALSE;
    il[0]         = 0;
    for (i = 0; i < mbs; i++) {
      rtmp[i] = 0.0;
      jl[i]   = mbs;
    }

    for (k = 0; k < mbs; k++) {
      /*initialize k-th row with elements nonzero in row perm(k) of A */
      nz   = ai[k + 1] - ai[k];
      acol = aj + ai[k];
      aval = aa + ai[k];
      bval = ba + bi[k];
      while (nz--) {
        rtmp[*acol++] = *aval++;
        *bval++       = 0.0; /* for in-place factorization */
      }

      /* shift the diagonal of the matrix */
      if (sctx.nshift) rtmp[k] += sctx.shift_amount;

      /* modify k-th row by adding in those rows i with U(i,k)!=0 */
      dk = rtmp[k];
      i  = jl[k]; /* first row to be added to k_th row  */

      while (i < k) {
        nexti = jl[i]; /* next row to be added to k_th row */
        /* compute multiplier, update D(k) and U(i,k) */
        ili   = il[i]; /* index of first nonzero element in U(i,k:bms-1) */
        uikdi = -ba[ili] * ba[bi[i]];
        dk += uikdi * ba[ili];
        ba[ili] = uikdi; /* -U(i,k) */

        /* add multiple of row i to k-th row ... */
        jmin = ili + 1;
        nz   = bi[i + 1] - jmin;
        if (nz > 0) {
          bcol = bj + jmin;
          bval = ba + jmin;
          PetscCall(PetscLogFlops(2.0 * nz));
          while (nz--) rtmp[*bcol++] += uikdi * (*bval++);

          /* update il and jl for i-th row */
          il[i] = jmin;
          j     = bj[jmin];
          jl[i] = jl[j];
          jl[j] = i;
        }
        i = nexti;
      }

      /* shift the diagonals when zero pivot is detected */
      /* compute rs=sum of abs(off-diagonal) */
      rs   = 0.0;
      jmin = bi[k] + 1;
      nz   = bi[k + 1] - jmin;
      if (nz) {
        bcol = bj + jmin;
        while (nz--) {
          rs += PetscAbsScalar(rtmp[*bcol]);
          bcol++;
        }
      }

      sctx.rs = rs;
      sctx.pv = dk;
      PetscCall(MatPivotCheck(C, A, info, &sctx, k));
      if (sctx.newshift) break; /* sctx.shift_amount is updated */
      dk = sctx.pv;

      /* copy data into U(k,:) */
      ba[bi[k]] = 1.0 / dk;
      jmin      = bi[k] + 1;
      nz        = bi[k + 1] - jmin;
      if (nz) {
        bcol = bj + jmin;
        bval = ba + jmin;
        while (nz--) {
          *bval++       = rtmp[*bcol];
          rtmp[*bcol++] = 0.0;
        }
        /* add k-th row into il and jl */
        il[k] = jmin;
        i     = bj[jmin];
        jl[k] = jl[i];
        jl[i] = k;
      }
    } /* end of for (k = 0; k<mbs; k++) */
  } while (sctx.newshift);
  PetscCall(PetscFree(rtmp));
  PetscCall(PetscFree2(il, jl));

  C->ops->solve          = MatSolve_SeqSBAIJ_1_NaturalOrdering_inplace;
  C->ops->solves         = MatSolves_SeqSBAIJ_1_inplace;
  C->ops->solvetranspose = MatSolve_SeqSBAIJ_1_NaturalOrdering_inplace;
  C->ops->forwardsolve   = MatForwardSolve_SeqSBAIJ_1_NaturalOrdering_inplace;
  C->ops->backwardsolve  = MatBackwardSolve_SeqSBAIJ_1_NaturalOrdering_inplace;

  C->assembled    = PETSC_TRUE;
  C->preallocated = PETSC_TRUE;

  PetscCall(PetscLogFlops(C->rmap->N));
  if (sctx.nshift) {
    if (info->shifttype == (PetscReal)MAT_SHIFT_NONZERO) {
      PetscCall(PetscInfo(A, "number of shiftnz tries %" PetscInt_FMT ", shift_amount %g\n", sctx.nshift, (double)sctx.shift_amount));
    } else if (info->shifttype == (PetscReal)MAT_SHIFT_POSITIVE_DEFINITE) {
      PetscCall(PetscInfo(A, "number of shiftpd tries %" PetscInt_FMT ", shift_amount %g\n", sctx.nshift, (double)sctx.shift_amount));
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode MatCholeskyFactor_SeqSBAIJ(Mat A, IS perm, const MatFactorInfo *info)
{
  Mat C;

  PetscFunctionBegin;
  PetscCall(MatGetFactor(A, "petsc", MAT_FACTOR_CHOLESKY, &C));
  PetscCall(MatCholeskyFactorSymbolic(C, A, perm, info));
  PetscCall(MatCholeskyFactorNumeric(C, A, info));

  A->ops->solve          = C->ops->solve;
  A->ops->solvetranspose = C->ops->solvetranspose;

  PetscCall(MatHeaderMerge(A, &C));
  PetscFunctionReturn(PETSC_SUCCESS);
}