xref: /petsc/src/mat/impls/aij/seq/aijfact.c (revision 3c9e8598273a0ead7f0d51888945a303d2e241a0)

#include "src/mat/impls/aij/seq/aij.h"
#include "src/inline/dot.h"
#include "src/inline/spops.h"

#undef __FUNCT__
#define __FUNCT__ "MatOrdering_Flow_SeqAIJ"
PetscErrorCode MatOrdering_Flow_SeqAIJ(Mat mat,const MatOrderingType type,IS *irow,IS *icol)
{
  PetscFunctionBegin;

  SETERRQ(PETSC_ERR_SUP,"Code not written");
#if !defined(PETSC_USE_DEBUG)
  PetscFunctionReturn(0);
#endif
}


EXTERN PetscErrorCode MatMarkDiagonal_SeqAIJ(Mat);
EXTERN PetscErrorCode Mat_AIJ_CheckInode(Mat,PetscTruth);

EXTERN PetscErrorCode SPARSEKIT2dperm(PetscInt*,PetscScalar*,PetscInt*,PetscInt*,PetscScalar*,PetscInt*,PetscInt*,PetscInt*,PetscInt*,PetscInt*);
EXTERN PetscErrorCode SPARSEKIT2ilutp(PetscInt*,PetscScalar*,PetscInt*,PetscInt*,PetscInt*,PetscReal,PetscReal*,PetscInt*,PetscScalar*,PetscInt*,PetscInt*,PetscInt*,PetscScalar*,PetscInt*,PetscInt*,PetscErrorCode*);
EXTERN PetscErrorCode SPARSEKIT2msrcsr(PetscInt*,PetscScalar*,PetscInt*,PetscScalar*,PetscInt*,PetscInt*,PetscScalar*,PetscInt*);

#undef __FUNCT__
#define __FUNCT__ "MatILUDTFactor_SeqAIJ"
  /* ------------------------------------------------------------

          This interface was contribed by Tony Caola

     This routine is an interface to the pivoting drop-tolerance
     ILU routine written by Yousef Saad (saad@cs.umn.edu) as part of
     SPARSEKIT2.

     The SPARSEKIT2 routines used here are covered by the GNU
     copyright; see the file gnu in this directory.

     Thanks to Prof. Saad, Dr. Hysom, and Dr. Smith for their
     help in getting this routine ironed out.

     The major drawback to this routine is that if info->fill is
     not large enough it fails rather than allocating more space;
     this can be fixed by hacking/improving the f2c version of
     Yousef Saad's code.

     ------------------------------------------------------------
*/
PetscErrorCode MatILUDTFactor_SeqAIJ(Mat A,MatFactorInfo *info,IS isrow,IS iscol,Mat *fact)
{
#if defined(PETSC_AVOID_GNUCOPYRIGHT_CODE)
  PetscFunctionBegin;
  SETERRQ(PETSC_ERR_SUP_SYS,"This distribution does not include GNU Copyright code\n\
  You can obtain the drop tolerance routines by installing PETSc from\n\
  www.mcs.anl.gov/petsc\n");
#else
  Mat_SeqAIJ     *a = (Mat_SeqAIJ*)A->data,*b;
  IS             iscolf,isicol,isirow;
  PetscTruth     reorder;
  PetscErrorCode ierr,sierr;
  PetscInt       *c,*r,*ic,i,n = A->m;
  PetscInt       *old_i = a->i,*old_j = a->j,*new_i,*old_i2 = 0,*old_j2 = 0,*new_j;
  PetscInt       *ordcol,*iwk,*iperm,*jw;
  PetscInt       jmax,lfill,job,*o_i,*o_j;
  PetscScalar    *old_a = a->a,*w,*new_a,*old_a2 = 0,*wk,*o_a;
  PetscReal      af;

  PetscFunctionBegin;

  if (info->dt == PETSC_DEFAULT)      info->dt      = .005;
  if (info->dtcount == PETSC_DEFAULT) info->dtcount = (PetscInt)(1.5*a->rmax);
  if (info->dtcol == PETSC_DEFAULT)   info->dtcol   = .01;
  if (info->fill == PETSC_DEFAULT)    info->fill    = ((double)(n*(info->dtcount+1)))/a->nz;
  lfill   = (PetscInt)(info->dtcount/2.0);
  jmax    = (PetscInt)(info->fill*a->nz);


  /* ------------------------------------------------------------
     If reorder=.TRUE., then the original matrix has to be
     reordered to reflect the user selected ordering scheme, and
     then de-reordered so it is in it's original format.
     Because Saad's dperm() is NOT in place, we have to copy
     the original matrix and allocate more storage. . .
     ------------------------------------------------------------
  */

  /* set reorder to true if either isrow or iscol is not identity */
  ierr = ISIdentity(isrow,&reorder);CHKERRQ(ierr);
  if (reorder) {ierr = ISIdentity(iscol,&reorder);CHKERRQ(ierr);}
  reorder = PetscNot(reorder);


  /* storage for ilu factor */
  ierr = PetscMalloc((n+1)*sizeof(PetscInt),&new_i);CHKERRQ(ierr);
  ierr = PetscMalloc(jmax*sizeof(PetscInt),&new_j);CHKERRQ(ierr);
  ierr = PetscMalloc(jmax*sizeof(PetscScalar),&new_a);CHKERRQ(ierr);
  ierr = PetscMalloc(n*sizeof(PetscInt),&ordcol);CHKERRQ(ierr);

  /* ------------------------------------------------------------
     Make sure that everything is Fortran formatted (1-Based)
     ------------------------------------------------------------
  */
  for (i=old_i[0];i<old_i[n];i++) {
    old_j[i]++;
  }
  for(i=0;i<n+1;i++) {
    old_i[i]++;
  };


  if (reorder) {
    ierr = ISGetIndices(iscol,&c);CHKERRQ(ierr);
    ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
    for(i=0;i<n;i++) {
      r[i]  = r[i]+1;
      c[i]  = c[i]+1;
    }
    ierr = PetscMalloc((n+1)*sizeof(PetscInt),&old_i2);CHKERRQ(ierr);
    ierr = PetscMalloc((old_i[n]-old_i[0]+1)*sizeof(PetscInt),&old_j2);CHKERRQ(ierr);
    ierr = PetscMalloc((old_i[n]-old_i[0]+1)*sizeof(PetscScalar),&old_a2);CHKERRQ(ierr);
    job  = 3; SPARSEKIT2dperm(&n,old_a,old_j,old_i,old_a2,old_j2,old_i2,r,c,&job);
    for (i=0;i<n;i++) {
      r[i]  = r[i]-1;
      c[i]  = c[i]-1;
    }
    ierr = ISRestoreIndices(iscol,&c);CHKERRQ(ierr);
    ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
    o_a = old_a2;
    o_j = old_j2;
    o_i = old_i2;
  } else {
    o_a = old_a;
    o_j = old_j;
    o_i = old_i;
  }

  /* ------------------------------------------------------------
     Call Saad's ilutp() routine to generate the factorization
     ------------------------------------------------------------
  */

  ierr = PetscMalloc(2*n*sizeof(PetscInt),&iperm);CHKERRQ(ierr);
  ierr = PetscMalloc(2*n*sizeof(PetscInt),&jw);CHKERRQ(ierr);
  ierr = PetscMalloc(n*sizeof(PetscScalar),&w);CHKERRQ(ierr);

  SPARSEKIT2ilutp(&n,o_a,o_j,o_i,&lfill,(PetscReal)info->dt,&info->dtcol,&n,new_a,new_j,new_i,&jmax,w,jw,iperm,&sierr);
  if (sierr) {
    switch (sierr) {
      case -3: SETERRQ2(PETSC_ERR_LIB,"ilutp(), matrix U overflows, need larger info->fill current fill %g space allocated %D",info->fill,jmax);
      case -2: SETERRQ2(PETSC_ERR_LIB,"ilutp(), matrix L overflows, need larger info->fill current fill %g space allocated %D",info->fill,jmax);
      case -5: SETERRQ(PETSC_ERR_LIB,"ilutp(), zero row encountered");
      case -1: SETERRQ(PETSC_ERR_LIB,"ilutp(), input matrix may be wrong");
      case -4: SETERRQ1(PETSC_ERR_LIB,"ilutp(), illegal info->fill value %D",jmax);
      default: SETERRQ1(PETSC_ERR_LIB,"ilutp(), zero pivot detected on row %D",sierr);
    }
  }

  ierr = PetscFree(w);CHKERRQ(ierr);
  ierr = PetscFree(jw);CHKERRQ(ierr);

  /* ------------------------------------------------------------
     Saad's routine gives the result in Modified Sparse Row (msr)
     Convert to Compressed Sparse Row format (csr)
     ------------------------------------------------------------
  */

  ierr = PetscMalloc(n*sizeof(PetscScalar),&wk);CHKERRQ(ierr);
  ierr = PetscMalloc((n+1)*sizeof(PetscInt),&iwk);CHKERRQ(ierr);

  SPARSEKIT2msrcsr(&n,new_a,new_j,new_a,new_j,new_i,wk,iwk);

  ierr = PetscFree(iwk);CHKERRQ(ierr);
  ierr = PetscFree(wk);CHKERRQ(ierr);

  if (reorder) {
    ierr = PetscFree(old_a2);CHKERRQ(ierr);
    ierr = PetscFree(old_j2);CHKERRQ(ierr);
    ierr = PetscFree(old_i2);CHKERRQ(ierr);
  } else {
    /* fix permutation of old_j that the factorization introduced */
    for (i=old_i[0]; i<old_i[n]; i++) {
      old_j[i-1] = iperm[old_j[i-1]-1];
    }
  }

  /* get rid of the shift to indices starting at 1 */
  for (i=0; i<n+1; i++) {
    old_i[i]--;
  }
  for (i=old_i[0];i<old_i[n];i++) {
    old_j[i]--;
  }

  /* Make the factored matrix 0-based */
  for (i=0; i<n+1; i++) {
    new_i[i]--;
  }
  for (i=new_i[0];i<new_i[n];i++) {
    new_j[i]--;
  }

  /*-- due to the pivoting, we need to reorder iscol to correctly --*/
  /*-- permute the right-hand-side and solution vectors           --*/
  ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
  ierr = ISInvertPermutation(isrow,PETSC_DECIDE,&isirow);CHKERRQ(ierr);
  ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
  for(i=0; i<n; i++) {
    ordcol[i] = ic[iperm[i]-1];
  };
  ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
  ierr = ISDestroy(isicol);CHKERRQ(ierr);

  ierr = PetscFree(iperm);CHKERRQ(ierr);

  ierr = ISCreateGeneral(PETSC_COMM_SELF,n,ordcol,&iscolf);CHKERRQ(ierr);
  ierr = PetscFree(ordcol);CHKERRQ(ierr);

  /*----- put together the new matrix -----*/

  ierr = MatCreate(A->comm,n,n,n,n,fact);CHKERRQ(ierr);
  ierr = MatSetType(*fact,A->type_name);CHKERRQ(ierr);
  ierr = MatSeqAIJSetPreallocation(*fact,0,PETSC_NULL);CHKERRQ(ierr);
  (*fact)->factor    = FACTOR_LU;
  (*fact)->assembled = PETSC_TRUE;

  b = (Mat_SeqAIJ*)(*fact)->data;
  ierr = PetscFree(b->imax);CHKERRQ(ierr);
  b->sorted        = PETSC_FALSE;
  b->singlemalloc  = PETSC_FALSE;
  /* the next line frees the default space generated by the MatCreate() */
  ierr             = PetscFree(b->a);CHKERRQ(ierr);
  ierr             = PetscFree(b->ilen);CHKERRQ(ierr);
  b->a             = new_a;
  b->j             = new_j;
  b->i             = new_i;
  b->ilen          = 0;
  b->imax          = 0;
  /*  I am not sure why these are the inverses of the row and column permutations; but the other way is NO GOOD */
  b->row           = isirow;
  b->col           = iscolf;
  ierr = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
  b->maxnz = b->nz = new_i[n];
  ierr = MatMarkDiagonal_SeqAIJ(*fact);CHKERRQ(ierr);
  (*fact)->info.factor_mallocs = 0;

  ierr = MatMarkDiagonal_SeqAIJ(A);CHKERRQ(ierr);

  /* check out for identical nodes. If found, use inode functions */
  ierr = Mat_AIJ_CheckInode(*fact,PETSC_FALSE);CHKERRQ(ierr);

  af = ((double)b->nz)/((double)a->nz) + .001;
  PetscLogInfo(A,"MatILUDTFactor_SeqAIJ:Fill ratio:given %g needed %g\n",info->fill,af);
  PetscLogInfo(A,"MatILUDTFactor_SeqAIJ:Run with -pc_ilu_fill %g or use \n",af);
  PetscLogInfo(A,"MatILUDTFactor_SeqAIJ:PCILUSetFill(pc,%g);\n",af);
  PetscLogInfo(A,"MatILUDTFactor_SeqAIJ:for best performance.\n");

  PetscFunctionReturn(0);
#endif
}

/*
    Factorization code for AIJ format.
*/
#undef __FUNCT__
#define __FUNCT__ "MatLUFactorSymbolic_SeqAIJ"
PetscErrorCode MatLUFactorSymbolic_SeqAIJ(Mat A,IS isrow,IS iscol,MatFactorInfo *info,Mat *B)
{
  Mat_SeqAIJ     *a = (Mat_SeqAIJ*)A->data,*b;
  IS             isicol;
  PetscErrorCode ierr;
  PetscInt       *r,*ic,i,n = A->m,*ai = a->i,*aj = a->j;
  PetscInt       *ainew,*ajnew,jmax,*fill,*ajtmp,nz;
  PetscInt       *idnew,idx,row,m,fm,nnz,nzi,reallocs = 0,nzbd,*im;
  PetscReal      f;

  PetscFunctionBegin;
  if (A->M != A->N) SETERRQ(PETSC_ERR_ARG_WRONG,"matrix must be square");
  ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
  ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
  ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);

  /* get new row pointers */
  ierr = PetscMalloc((n+1)*sizeof(PetscInt),&ainew);CHKERRQ(ierr);
  ainew[0] = 0;
  /* don't know how many column pointers are needed so estimate */
  f = info->fill;
  jmax  = (PetscInt)(f*ai[n]+1);
  ierr = PetscMalloc((jmax)*sizeof(PetscInt),&ajnew);CHKERRQ(ierr);
  /* fill is a linked list of nonzeros in active row */
  ierr = PetscMalloc((2*n+1)*sizeof(PetscInt),&fill);CHKERRQ(ierr);
  im   = fill + n + 1;
  /* idnew is location of diagonal in factor */
  ierr = PetscMalloc((n+1)*sizeof(PetscInt),&idnew);CHKERRQ(ierr);
  idnew[0] = 0;

  for (i=0; i<n; i++) {
    /* first copy previous fill into linked list */
    nnz     = nz    = ai[r[i]+1] - ai[r[i]];
    if (!nz) SETERRQ(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix");
    ajtmp   = aj + ai[r[i]];
    fill[n] = n;
    while (nz--) {
      fm  = n;
      idx = ic[*ajtmp++];
      do {
        m  = fm;
        fm = fill[m];
      } while (fm < idx);
      fill[m]   = idx;
      fill[idx] = fm;
    }
    row = fill[n];
    while (row < i) {
      ajtmp = ajnew + idnew[row] + 1;
      nzbd  = 1 + idnew[row] - ainew[row];
      nz    = im[row] - nzbd;
      fm    = row;
      while (nz-- > 0) {
        idx = *ajtmp++ ;
        nzbd++;
        if (idx == i) im[row] = nzbd;
        do {
          m  = fm;
          fm = fill[m];
        } while (fm < idx);
        if (fm != idx) {
          fill[m]   = idx;
          fill[idx] = fm;
          fm        = idx;
          nnz++;
        }
      }
      row = fill[row];
    }
    /* copy new filled row into permanent storage */
    ainew[i+1] = ainew[i] + nnz;
    if (ainew[i+1] > jmax) {

      /* 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 */
      PetscInt maxadd = jmax;
      /* maxadd = (int)((f*(ai[n]+(!shift))*(n-i+5))/n); */
      if (maxadd < nnz) maxadd = (n-i)*(nnz+1);
      jmax += maxadd;

      /* allocate a longer ajnew */
      ierr  = PetscMalloc(jmax*sizeof(PetscInt),&ajtmp);CHKERRQ(ierr);
      ierr  = PetscMemcpy(ajtmp,ajnew,(ainew[i])*sizeof(PetscInt));CHKERRQ(ierr);
      ierr  = PetscFree(ajnew);CHKERRQ(ierr);
      ajnew = ajtmp;
      reallocs++; /* count how many times we realloc */
    }
    ajtmp = ajnew + ainew[i];
    fm    = fill[n];
    nzi   = 0;
    im[i] = nnz;
    while (nnz--) {
      if (fm < i) nzi++;
      *ajtmp++ = fm ;
      fm       = fill[fm];
    }
    idnew[i] = ainew[i] + nzi;
  }
  if (ai[n] != 0) {
    PetscReal af = ((PetscReal)ainew[n])/((PetscReal)ai[n]);
    PetscLogInfo(A,"MatLUFactorSymbolic_SeqAIJ:Reallocs %D Fill ratio:given %g needed %g\n",reallocs,f,af);
    PetscLogInfo(A,"MatLUFactorSymbolic_SeqAIJ:Run with -pc_lu_fill %g or use \n",af);
    PetscLogInfo(A,"MatLUFactorSymbolic_SeqAIJ:PCLUSetFill(pc,%g);\n",af);
    PetscLogInfo(A,"MatLUFactorSymbolic_SeqAIJ:for best performance.\n");
  } else {
    PetscLogInfo(A,"MatLUFactorSymbolic_SeqAIJ: Empty matrix\n");
  }

  ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
  ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);

  ierr = PetscFree(fill);CHKERRQ(ierr);

  /* put together the new matrix */
  ierr = MatCreate(A->comm,n,n,n,n,B);CHKERRQ(ierr);
  ierr = MatSetType(*B,A->type_name);CHKERRQ(ierr);
  ierr = MatSeqAIJSetPreallocation(*B,0,PETSC_NULL);CHKERRQ(ierr);
  PetscLogObjectParent(*B,isicol);
  b    = (Mat_SeqAIJ*)(*B)->data;
  ierr = PetscFree(b->imax);CHKERRQ(ierr);
  b->singlemalloc = PETSC_FALSE;
  /* the next line frees the default space generated by the Create() */
  ierr = PetscFree(b->a);CHKERRQ(ierr);
  ierr = PetscFree(b->ilen);CHKERRQ(ierr);
  ierr          = PetscMalloc((ainew[n]+1)*sizeof(PetscScalar),&b->a);CHKERRQ(ierr);
  b->j          = ajnew;
  b->i          = ainew;
  b->diag       = idnew;
  b->ilen       = 0;
  b->imax       = 0;
  b->row        = isrow;
  b->col        = iscol;
  b->lu_damping        = info->damping;
  b->lu_zeropivot      = info->zeropivot;
  b->lu_shift          = info->shift;
  b->lu_shift_fraction = info->shift_fraction;
  ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
  ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
  b->icol       = isicol;
  ierr          = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
  /* In b structure:  Free imax, ilen, old a, old j.
     Allocate idnew, solve_work, new a, new j */
  PetscLogObjectMemory(*B,(ainew[n]-n)*(sizeof(PetscInt)+sizeof(PetscScalar)));
  b->maxnz = b->nz = ainew[n] ;

  (*B)->factor                 =  FACTOR_LU;
  (*B)->info.factor_mallocs    = reallocs;
  (*B)->info.fill_ratio_given  = f;
  ierr = Mat_AIJ_CheckInode(*B,PETSC_FALSE);CHKERRQ(ierr);
  (*B)->ops->lufactornumeric   =  A->ops->lufactornumeric; /* Use Inode variant ONLY if A has inodes */

  if (ai[n] != 0) {
    (*B)->info.fill_ratio_needed = ((PetscReal)ainew[n])/((PetscReal)ai[n]);
  } else {
    (*B)->info.fill_ratio_needed = 0.0;
  }
  PetscFunctionReturn(0);
}
/* ----------------------------------------------------------- */
EXTERN PetscErrorCode Mat_AIJ_CheckInode(Mat,PetscTruth);

#undef __FUNCT__
#define __FUNCT__ "MatLUFactorNumeric_SeqAIJ"
PetscErrorCode MatLUFactorNumeric_SeqAIJ(Mat A,Mat *B)
{
  Mat            C = *B;
  Mat_SeqAIJ     *a = (Mat_SeqAIJ*)A->data,*b = (Mat_SeqAIJ *)C->data;
  IS             isrow = b->row,isicol = b->icol;
  PetscErrorCode ierr;
  PetscInt       *r,*ic,i,j,n = A->m,*ai = b->i,*aj = b->j;
  PetscInt       *ajtmpold,*ajtmp,nz,row,*ics;
  PetscInt       *diag_offset = b->diag,diag,*pj,ndamp = 0, nshift=0;
  PetscScalar    *rtmp,*v,*pc,multiplier,*pv,*rtmps;
  PetscReal      damping = b->lu_damping, zeropivot = b->lu_zeropivot,rs,d;
  PetscReal      row_shift,shift_fraction,shift_amount,shift_lo=0., shift_hi=1., shift_top=0.;
  PetscTruth     damp,lushift;

  PetscFunctionBegin;
  ierr  = ISGetIndices(isrow,&r);CHKERRQ(ierr);
  ierr  = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
  ierr  = PetscMalloc((n+1)*sizeof(PetscScalar),&rtmp);CHKERRQ(ierr);
  ierr  = PetscMemzero(rtmp,(n+1)*sizeof(PetscScalar));CHKERRQ(ierr);
  rtmps = rtmp; ics = ic;

  if (!a->diag) {
    ierr = MatMarkDiagonal_SeqAIJ(A);CHKERRQ(ierr);
  }

  if (b->lu_shift) { /* set max shift */
    PetscInt *aai = a->i,*ddiag = a->diag;
    shift_top = 0;
    for (i=0; i<n; i++) {
      d = PetscAbsScalar((a->a)[ddiag[i]]);
      /* calculate amt of shift needed for this row */
      if (d<=0) {
        row_shift = 0;
      } else {
        row_shift = -2*d;
      }
      v = a->a+aai[i];
      for (j=0; j<aai[i+1]-aai[i]; j++)
	row_shift += PetscAbsScalar(v[j]);
      if (row_shift>shift_top) shift_top = row_shift;
    }
  }

  shift_fraction = 0; shift_amount = 0;
  do {
    damp    = PETSC_FALSE;
    lushift = PETSC_FALSE;
    for (i=0; i<n; i++) {
      nz    = ai[i+1] - ai[i];
      ajtmp = aj + ai[i];
      for  (j=0; j<nz; j++) rtmps[ajtmp[j]] = 0.0;

      /* load in initial (unfactored row) */
      nz       = a->i[r[i]+1] - a->i[r[i]];
      ajtmpold = a->j + a->i[r[i]];
      v        = a->a + a->i[r[i]];
      for (j=0; j<nz; j++) {
        rtmp[ics[ajtmpold[j]]] = v[j];
      }
      rtmp[ics[r[i]]] += damping + shift_amount; /* damp the diagonal of the matrix */

      row = *ajtmp++ ;
      while  (row < i) {
        pc = rtmp + row;
        if (*pc != 0.0) {
          pv         = b->a + diag_offset[row];
          pj         = b->j + diag_offset[row] + 1;
          multiplier = *pc / *pv++;
          *pc        = multiplier;
          nz         = ai[row+1] - diag_offset[row] - 1;
          for (j=0; j<nz; j++) rtmps[pj[j]] -= multiplier * pv[j];
          PetscLogFlops(2*nz);
        }
        row = *ajtmp++;
      }
      /* finished row so stick it into b->a */
      pv   = b->a + ai[i] ;
      pj   = b->j + ai[i] ;
      nz   = ai[i+1] - ai[i];
      diag = diag_offset[i] - ai[i];
      /* 9/13/02 Victor Eijkhout suggested scaling zeropivot by rs for matrices with funny scalings */
      rs   = 0.0;
      for (j=0; j<nz; j++) {
        pv[j] = rtmps[pj[j]];
        if (j != diag) rs += PetscAbsScalar(pv[j]);
      }
#define MAX_NSHIFT 5
      if (PetscRealPart(pv[diag]) <= zeropivot*rs && b->lu_shift) {
	if (nshift>MAX_NSHIFT) {
	  SETERRQ(PETSC_ERR_CONV_FAILED,"Unable to determine shift to enforce positive definite preconditioner");
	} else if (nshift==MAX_NSHIFT) {
	  shift_fraction = shift_hi;
	  lushift      = PETSC_FALSE;
	} else {
	  shift_lo = shift_fraction; shift_fraction = (shift_hi+shift_lo)/2.;
	  lushift      = PETSC_TRUE;
	}
	shift_amount = shift_fraction * shift_top;
        nshift++;
        break;
      }
      if (PetscAbsScalar(pv[diag]) <= zeropivot*rs) {
        if (damping) {
          if (ndamp) damping *= 2.0;
          damp = PETSC_TRUE;
          ndamp++;
          break;
        } else {
          SETERRQ4(PETSC_ERR_MAT_LU_ZRPVT,"Zero pivot row %D value %g tolerance %g * rs %g",i,PetscAbsScalar(pv[diag]),zeropivot,rs);
        }
      }
    }
    if (!lushift && b->lu_shift && shift_fraction>0 && nshift<MAX_NSHIFT) {
      /*
       * if no shift in this attempt & shifting & started shifting & can refine,
       * then try lower shift
       */
      shift_hi       = shift_fraction;
      shift_fraction = (shift_hi+shift_lo)/2.;
      shift_amount   = shift_fraction * shift_top;
      lushift        = PETSC_TRUE;
      nshift++;
    }
  } while (damp || lushift);

  /* invert diagonal entries for simplier triangular solves */
  for (i=0; i<n; i++) {
    b->a[diag_offset[i]] = 1.0/b->a[diag_offset[i]];
  }

  ierr = PetscFree(rtmp);CHKERRQ(ierr);
  ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
  ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
  C->factor = FACTOR_LU;
  (*B)->ops->lufactornumeric   =  A->ops->lufactornumeric; /* Use Inode variant ONLY if A has inodes */
  C->assembled = PETSC_TRUE;
  PetscLogFlops(C->n);
  if (ndamp) {
    PetscLogInfo(0,"MatLUFactorNumerical_SeqAIJ: number of damping tries %D damping value %g\n",ndamp,damping);
  }
  if (nshift) {
    b->lu_shift_fraction = shift_fraction;
    PetscLogInfo(0,"MatLUFactorNumerical_SeqAIJ: diagonal shifted up by %e fraction top_value %e number shifts %D\n",shift_fraction,shift_top,nshift);
  }
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "MatUsePETSc_SeqAIJ"
PetscErrorCode MatUsePETSc_SeqAIJ(Mat A)
{
  PetscFunctionBegin;
  A->ops->lufactorsymbolic = MatLUFactorSymbolic_SeqAIJ;
  A->ops->lufactornumeric  = MatLUFactorNumeric_SeqAIJ;
  PetscFunctionReturn(0);
}


/* ----------------------------------------------------------- */
#undef __FUNCT__
#define __FUNCT__ "MatLUFactor_SeqAIJ"
PetscErrorCode MatLUFactor_SeqAIJ(Mat A,IS row,IS col,MatFactorInfo *info)
{
  PetscErrorCode ierr;
  Mat            C;

  PetscFunctionBegin;
  ierr = MatLUFactorSymbolic(A,row,col,info,&C);CHKERRQ(ierr);
  ierr = MatLUFactorNumeric(A,&C);CHKERRQ(ierr);
  ierr = MatHeaderCopy(A,C);CHKERRQ(ierr);
  PetscLogObjectParent(A,((Mat_SeqAIJ*)(A->data))->icol);
  PetscFunctionReturn(0);
}
/* ----------------------------------------------------------- */
#undef __FUNCT__
#define __FUNCT__ "MatSolve_SeqAIJ"
PetscErrorCode MatSolve_SeqAIJ(Mat A,Vec bb,Vec xx)
{
  Mat_SeqAIJ     *a = (Mat_SeqAIJ*)A->data;
  IS             iscol = a->col,isrow = a->row;
  PetscErrorCode ierr;
  PetscInt       *r,*c,i, n = A->m,*vi,*ai = a->i,*aj = a->j;
  PetscInt       nz,*rout,*cout;
  PetscScalar    *x,*b,*tmp,*tmps,*aa = a->a,sum,*v;

  PetscFunctionBegin;
  if (!n) PetscFunctionReturn(0);

  ierr = VecGetArray(bb,&b);CHKERRQ(ierr);
  ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
  tmp  = a->solve_work;

  ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
  ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout + (n-1);

  /* forward solve the lower triangular */
  tmp[0] = b[*r++];
  tmps   = tmp;
  for (i=1; i<n; i++) {
    v   = aa + ai[i] ;
    vi  = aj + ai[i] ;
    nz  = a->diag[i] - ai[i];
    sum = b[*r++];
    SPARSEDENSEMDOT(sum,tmps,v,vi,nz);
    tmp[i] = sum;
  }

  /* backward solve the upper triangular */
  for (i=n-1; i>=0; i--){
    v   = aa + a->diag[i] + 1;
    vi  = aj + a->diag[i] + 1;
    nz  = ai[i+1] - a->diag[i] - 1;
    sum = tmp[i];
    SPARSEDENSEMDOT(sum,tmps,v,vi,nz);
    x[*c--] = tmp[i] = sum*aa[a->diag[i]];
  }

  ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
  ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
  ierr = VecRestoreArray(bb,&b);CHKERRQ(ierr);
  ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
  PetscLogFlops(2*a->nz - A->n);
  PetscFunctionReturn(0);
}

/* ----------------------------------------------------------- */
#undef __FUNCT__
#define __FUNCT__ "MatSolve_SeqAIJ_NaturalOrdering"
PetscErrorCode MatSolve_SeqAIJ_NaturalOrdering(Mat A,Vec bb,Vec xx)
{
  Mat_SeqAIJ     *a = (Mat_SeqAIJ*)A->data;
  PetscErrorCode ierr;
  PetscInt       n = A->m,*ai = a->i,*aj = a->j,*adiag = a->diag;
  PetscScalar    *x,*b,*aa = a->a;
#if !defined(PETSC_USE_FORTRAN_KERNEL_SOLVEAIJ)
  PetscInt       adiag_i,i,*vi,nz,ai_i;
  PetscScalar    *v,sum;
#endif

  PetscFunctionBegin;
  if (!n) PetscFunctionReturn(0);

  ierr = VecGetArray(bb,&b);CHKERRQ(ierr);
  ierr = VecGetArray(xx,&x);CHKERRQ(ierr);

#if defined(PETSC_USE_FORTRAN_KERNEL_SOLVEAIJ)
  fortransolveaij_(&n,x,ai,aj,adiag,aa,b);
#else
  /* forward solve the lower triangular */
  x[0] = b[0];
  for (i=1; i<n; i++) {
    ai_i = ai[i];
    v    = aa + ai_i;
    vi   = aj + ai_i;
    nz   = adiag[i] - ai_i;
    sum  = b[i];
    while (nz--) sum -= *v++ * x[*vi++];
    x[i] = sum;
  }

  /* backward solve the upper triangular */
  for (i=n-1; i>=0; i--){
    adiag_i = adiag[i];
    v       = aa + adiag_i + 1;
    vi      = aj + adiag_i + 1;
    nz      = ai[i+1] - adiag_i - 1;
    sum     = x[i];
    while (nz--) sum -= *v++ * x[*vi++];
    x[i]    = sum*aa[adiag_i];
  }
#endif
  PetscLogFlops(2*a->nz - A->n);
  ierr = VecRestoreArray(bb,&b);CHKERRQ(ierr);
  ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "MatSolveAdd_SeqAIJ"
PetscErrorCode MatSolveAdd_SeqAIJ(Mat A,Vec bb,Vec yy,Vec xx)
{
  Mat_SeqAIJ     *a = (Mat_SeqAIJ*)A->data;
  IS             iscol = a->col,isrow = a->row;
  PetscErrorCode ierr;
  PetscInt       *r,*c,i, n = A->m,*vi,*ai = a->i,*aj = a->j;
  PetscInt       nz,*rout,*cout;
  PetscScalar    *x,*b,*tmp,*aa = a->a,sum,*v;

  PetscFunctionBegin;
  if (yy != xx) {ierr = VecCopy(yy,xx);CHKERRQ(ierr);}

  ierr = VecGetArray(bb,&b);CHKERRQ(ierr);
  ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
  tmp  = a->solve_work;

  ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
  ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout + (n-1);

  /* forward solve the lower triangular */
  tmp[0] = b[*r++];
  for (i=1; i<n; i++) {
    v   = aa + ai[i] ;
    vi  = aj + ai[i] ;
    nz  = a->diag[i] - ai[i];
    sum = b[*r++];
    while (nz--) sum -= *v++ * tmp[*vi++ ];
    tmp[i] = sum;
  }

  /* backward solve the upper triangular */
  for (i=n-1; i>=0; i--){
    v   = aa + a->diag[i] + 1;
    vi  = aj + a->diag[i] + 1;
    nz  = ai[i+1] - a->diag[i] - 1;
    sum = tmp[i];
    while (nz--) sum -= *v++ * tmp[*vi++ ];
    tmp[i] = sum*aa[a->diag[i]];
    x[*c--] += tmp[i];
  }

  ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
  ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
  ierr = VecRestoreArray(bb,&b);CHKERRQ(ierr);
  ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
  PetscLogFlops(2*a->nz);

  PetscFunctionReturn(0);
}
/* -------------------------------------------------------------------*/
#undef __FUNCT__
#define __FUNCT__ "MatSolveTranspose_SeqAIJ"
PetscErrorCode MatSolveTranspose_SeqAIJ(Mat A,Vec bb,Vec xx)
{
  Mat_SeqAIJ     *a = (Mat_SeqAIJ*)A->data;
  IS             iscol = a->col,isrow = a->row;
  PetscErrorCode ierr;
  PetscInt       *r,*c,i,n = A->m,*vi,*ai = a->i,*aj = a->j;
  PetscInt       nz,*rout,*cout,*diag = a->diag;
  PetscScalar    *x,*b,*tmp,*aa = a->a,*v,s1;

  PetscFunctionBegin;
  ierr = VecGetArray(bb,&b);CHKERRQ(ierr);
  ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
  tmp  = a->solve_work;

  ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
  ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;

  /* copy the b into temp work space according to permutation */
  for (i=0; i<n; i++) tmp[i] = b[c[i]];

  /* forward solve the U^T */
  for (i=0; i<n; i++) {
    v   = aa + diag[i] ;
    vi  = aj + diag[i] + 1;
    nz  = ai[i+1] - diag[i] - 1;
    s1  = tmp[i];
    s1 *= (*v++);  /* multiply by inverse of diagonal entry */
    while (nz--) {
      tmp[*vi++ ] -= (*v++)*s1;
    }
    tmp[i] = s1;
  }

  /* backward solve the L^T */
  for (i=n-1; i>=0; i--){
    v   = aa + diag[i] - 1 ;
    vi  = aj + diag[i] - 1 ;
    nz  = diag[i] - ai[i];
    s1  = tmp[i];
    while (nz--) {
      tmp[*vi-- ] -= (*v--)*s1;
    }
  }

  /* copy tmp into x according to permutation */
  for (i=0; i<n; i++) x[r[i]] = tmp[i];

  ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
  ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
  ierr = VecRestoreArray(bb,&b);CHKERRQ(ierr);
  ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);

  PetscLogFlops(2*a->nz-A->n);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "MatSolveTransposeAdd_SeqAIJ"
PetscErrorCode MatSolveTransposeAdd_SeqAIJ(Mat A,Vec bb,Vec zz,Vec xx)
{
  Mat_SeqAIJ     *a = (Mat_SeqAIJ*)A->data;
  IS             iscol = a->col,isrow = a->row;
  PetscErrorCode ierr;
  PetscInt       *r,*c,i,n = A->m,*vi,*ai = a->i,*aj = a->j;
  PetscInt       nz,*rout,*cout,*diag = a->diag;
  PetscScalar    *x,*b,*tmp,*aa = a->a,*v;

  PetscFunctionBegin;
  if (zz != xx) {ierr = VecCopy(zz,xx);CHKERRQ(ierr);}

  ierr = VecGetArray(bb,&b);CHKERRQ(ierr);
  ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
  tmp = a->solve_work;

  ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
  ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;

  /* copy the b into temp work space according to permutation */
  for (i=0; i<n; i++) tmp[i] = b[c[i]];

  /* forward solve the U^T */
  for (i=0; i<n; i++) {
    v   = aa + diag[i] ;
    vi  = aj + diag[i] + 1;
    nz  = ai[i+1] - diag[i] - 1;
    tmp[i] *= *v++;
    while (nz--) {
      tmp[*vi++ ] -= (*v++)*tmp[i];
    }
  }

  /* backward solve the L^T */
  for (i=n-1; i>=0; i--){
    v   = aa + diag[i] - 1 ;
    vi  = aj + diag[i] - 1 ;
    nz  = diag[i] - ai[i];
    while (nz--) {
      tmp[*vi-- ] -= (*v--)*tmp[i];
    }
  }

  /* copy tmp into x according to permutation */
  for (i=0; i<n; i++) x[r[i]] += tmp[i];

  ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
  ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
  ierr = VecRestoreArray(bb,&b);CHKERRQ(ierr);
  ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);

  PetscLogFlops(2*a->nz);
  PetscFunctionReturn(0);
}
/* ----------------------------------------------------------------*/
EXTERN PetscErrorCode MatMissingDiagonal_SeqAIJ(Mat);

#undef __FUNCT__
#define __FUNCT__ "MatILUFactorSymbolic_SeqAIJ"
PetscErrorCode MatILUFactorSymbolic_SeqAIJ(Mat A,IS isrow,IS iscol,MatFactorInfo *info,Mat *fact)
{
  Mat_SeqAIJ     *a = (Mat_SeqAIJ*)A->data,*b;
  IS             isicol;
  PetscErrorCode ierr;
  PetscInt       *r,*ic,prow,n = A->m,*ai = a->i,*aj = a->j;
  PetscInt       *ainew,*ajnew,jmax,*fill,*xi,nz,*im,*ajfill,*flev;
  PetscInt       *dloc,idx,row,m,fm,nzf,nzi,len, reallocs = 0,dcount = 0;
  PetscInt       incrlev,nnz,i,levels,diagonal_fill;
  PetscTruth     col_identity,row_identity;
  PetscReal      f;

  PetscFunctionBegin;
  f             = info->fill;
  levels        = (PetscInt)info->levels;
  diagonal_fill = (PetscInt)info->diagonal_fill;
  ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);

  /* special case that simply copies fill pattern */
  ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
  ierr = ISIdentity(iscol,&col_identity);CHKERRQ(ierr);
  if (!levels && row_identity && col_identity) {
    ierr = MatDuplicate_SeqAIJ(A,MAT_DO_NOT_COPY_VALUES,fact);CHKERRQ(ierr);
    (*fact)->factor = FACTOR_LU;
    b               = (Mat_SeqAIJ*)(*fact)->data;
    if (!b->diag) {
      ierr = MatMarkDiagonal_SeqAIJ(*fact);CHKERRQ(ierr);
    }
    ierr = MatMissingDiagonal_SeqAIJ(*fact);CHKERRQ(ierr);
    b->row              = isrow;
    b->col              = iscol;
    b->icol             = isicol;
    b->lu_damping       = info->damping;
    b->lu_zeropivot     = info->zeropivot;
    b->lu_shift         = info->shift;
    b->lu_shift_fraction= info->shift_fraction;
    ierr                = PetscMalloc(((*fact)->m+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
    (*fact)->ops->solve = MatSolve_SeqAIJ_NaturalOrdering;
    ierr                = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
    ierr                = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
    PetscFunctionReturn(0);
  }

  ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
  ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);

  /* get new row pointers */
  ierr = PetscMalloc((n+1)*sizeof(PetscInt),&ainew);CHKERRQ(ierr);
  ainew[0] = 0;
  /* don't know how many column pointers are needed so estimate */
  jmax = (PetscInt)(f*(ai[n]+1));
  ierr = PetscMalloc((jmax)*sizeof(PetscInt),&ajnew);CHKERRQ(ierr);
  /* ajfill is level of fill for each fill entry */
  ierr = PetscMalloc((jmax)*sizeof(PetscInt),&ajfill);CHKERRQ(ierr);
  /* fill is a linked list of nonzeros in active row */
  ierr = PetscMalloc((n+1)*sizeof(PetscInt),&fill);CHKERRQ(ierr);
  /* im is level for each filled value */
  ierr = PetscMalloc((n+1)*sizeof(PetscInt),&im);CHKERRQ(ierr);
  /* dloc is location of diagonal in factor */
  ierr = PetscMalloc((n+1)*sizeof(PetscInt),&dloc);CHKERRQ(ierr);
  dloc[0]  = 0;
  for (prow=0; prow<n; prow++) {

    /* copy current row into linked list */
    nzf     = nz  = ai[r[prow]+1] - ai[r[prow]];
    if (!nz) SETERRQ(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix");
    xi      = aj + ai[r[prow]] ;
    fill[n]    = n;
    fill[prow] = -1; /* marker to indicate if diagonal exists */
    while (nz--) {
      fm  = n;
      idx = ic[*xi++ ];
      do {
        m  = fm;
        fm = fill[m];
      } while (fm < idx);
      fill[m]   = idx;
      fill[idx] = fm;
      im[idx]   = 0;
    }

    /* make sure diagonal entry is included */
    if (diagonal_fill && fill[prow] == -1) {
      fm = n;
      while (fill[fm] < prow) fm = fill[fm];
      fill[prow] = fill[fm]; /* insert diagonal into linked list */
      fill[fm]   = prow;
      im[prow]   = 0;
      nzf++;
      dcount++;
    }

    nzi = 0;
    row = fill[n];
    while (row < prow) {
      incrlev = im[row] + 1;
      nz      = dloc[row];
      xi      = ajnew  + ainew[row]  + nz + 1;
      flev    = ajfill + ainew[row]  + nz + 1;
      nnz     = ainew[row+1] - ainew[row] - nz - 1;
      fm      = row;
      while (nnz-- > 0) {
        idx = *xi++ ;
        if (*flev + incrlev > levels) {
          flev++;
          continue;
        }
        do {
          m  = fm;
          fm = fill[m];
        } while (fm < idx);
        if (fm != idx) {
          im[idx]   = *flev + incrlev;
          fill[m]   = idx;
          fill[idx] = fm;
          fm        = idx;
          nzf++;
        } else {
          if (im[idx] > *flev + incrlev) im[idx] = *flev+incrlev;
        }
        flev++;
      }
      row = fill[row];
      nzi++;
    }
    /* copy new filled row into permanent storage */
    ainew[prow+1] = ainew[prow] + nzf;
    if (ainew[prow+1] > jmax) {

      /* 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 = (PetscInt)((f*(ai[n]+!shift)*(n-prow+5))/n); */
      PetscInt maxadd = jmax;
      if (maxadd < nzf) maxadd = (n-prow)*(nzf+1);
      jmax += maxadd;

      /* allocate a longer ajnew and ajfill */
      ierr   = PetscMalloc(jmax*sizeof(PetscInt),&xi);CHKERRQ(ierr);
      ierr   = PetscMemcpy(xi,ajnew,(ainew[prow])*sizeof(PetscInt));CHKERRQ(ierr);
      ierr   = PetscFree(ajnew);CHKERRQ(ierr);
      ajnew  = xi;
      ierr   = PetscMalloc(jmax*sizeof(PetscInt),&xi);CHKERRQ(ierr);
      ierr   = PetscMemcpy(xi,ajfill,(ainew[prow])*sizeof(PetscInt));CHKERRQ(ierr);
      ierr   = PetscFree(ajfill);CHKERRQ(ierr);
      ajfill = xi;
      reallocs++; /* count how many times we realloc */
    }
    xi          = ajnew + ainew[prow] ;
    flev        = ajfill + ainew[prow] ;
    dloc[prow]  = nzi;
    fm          = fill[n];
    while (nzf--) {
      *xi++   = fm ;
      *flev++ = im[fm];
      fm      = fill[fm];
    }
    /* make sure row has diagonal entry */
    if (ajnew[ainew[prow]+dloc[prow]] != prow) {
      SETERRQ1(PETSC_ERR_MAT_LU_ZRPVT,"Row %D has missing diagonal in factored matrix\n\
    try running with -pc_ilu_nonzeros_along_diagonal or -pc_ilu_diagonal_fill",prow);
    }
  }
  ierr = PetscFree(ajfill);CHKERRQ(ierr);
  ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
  ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
  ierr = PetscFree(fill);CHKERRQ(ierr);
  ierr = PetscFree(im);CHKERRQ(ierr);

  {
    PetscReal af = ((PetscReal)ainew[n])/((PetscReal)ai[n]);
    PetscLogInfo(A,"MatILUFactorSymbolic_SeqAIJ:Reallocs %D Fill ratio:given %g needed %g\n",reallocs,f,af);
    PetscLogInfo(A,"MatILUFactorSymbolic_SeqAIJ:Run with -[sub_]pc_ilu_fill %g or use \n",af);
    PetscLogInfo(A,"MatILUFactorSymbolic_SeqAIJ:PCILUSetFill([sub]pc,%g);\n",af);
    PetscLogInfo(A,"MatILUFactorSymbolic_SeqAIJ:for best performance.\n");
    if (diagonal_fill) {
      PetscLogInfo(A,"MatILUFactorSymbolic_SeqAIJ:Detected and replaced %D missing diagonals",dcount);
    }
  }

  /* put together the new matrix */
  ierr = MatCreate(A->comm,n,n,n,n,fact);CHKERRQ(ierr);
  ierr = MatSetType(*fact,A->type_name);CHKERRQ(ierr);
  ierr = MatSeqAIJSetPreallocation(*fact,0,PETSC_NULL);CHKERRQ(ierr);
  PetscLogObjectParent(*fact,isicol);
  b = (Mat_SeqAIJ*)(*fact)->data;
  ierr = PetscFree(b->imax);CHKERRQ(ierr);
  b->singlemalloc = PETSC_FALSE;
  len = (ainew[n] )*sizeof(PetscScalar);
  /* the next line frees the default space generated by the Create() */
  ierr = PetscFree(b->a);CHKERRQ(ierr);
  ierr = PetscFree(b->ilen);CHKERRQ(ierr);
  ierr = PetscMalloc(len+1,&b->a);CHKERRQ(ierr);
  b->j          = ajnew;
  b->i          = ainew;
  for (i=0; i<n; i++) dloc[i] += ainew[i];
  b->diag       = dloc;
  b->ilen       = 0;
  b->imax       = 0;
  b->row        = isrow;
  b->col        = iscol;
  ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
  ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
  b->icol       = isicol;
  ierr = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
  /* In b structure:  Free imax, ilen, old a, old j.
     Allocate dloc, solve_work, new a, new j */
  PetscLogObjectMemory(*fact,(ainew[n]-n) * (sizeof(PetscInt)+sizeof(PetscScalar)));
  b->maxnz             = b->nz = ainew[n] ;
  b->lu_damping        = info->damping;
  b->lu_shift          = info->shift;
  b->lu_shift_fraction = info->shift_fraction;
  b->lu_zeropivot = info->zeropivot;
  (*fact)->factor = FACTOR_LU;
  ierr = Mat_AIJ_CheckInode(*fact,PETSC_FALSE);CHKERRQ(ierr);
  (*fact)->ops->lufactornumeric =  A->ops->lufactornumeric; /* Use Inode variant ONLY if A has inodes */

  (*fact)->info.factor_mallocs    = reallocs;
  (*fact)->info.fill_ratio_given  = f;
  (*fact)->info.fill_ratio_needed = ((PetscReal)ainew[n])/((PetscReal)ai[prow]);
  PetscFunctionReturn(0);
}

#include "src/mat/impls/sbaij/seq/sbaij.h"
#undef __FUNCT__
#define __FUNCT__ "MatCholeskyFactorNumeric_SeqAIJ"
PetscErrorCode MatCholeskyFactorNumeric_SeqAIJ(Mat A,Mat *fact)
{
  Mat_SeqAIJ     *a = (Mat_SeqAIJ*)A->data;
  PetscErrorCode ierr,(*f)(Mat,Mat*);

  PetscFunctionBegin;
  printf(" MatCholeskyFactorNumeric_SeqAIJ ...\n");
  if (!a->sbaijMat){
    ierr = MatConvert_SeqAIJ_SeqSBAIJ(A,MATSEQSBAIJ,&a->sbaijMat);CHKERRQ(ierr);
  }
  ierr = PetscObjectQueryFunction((PetscObject) *fact,"MatCholeskyFactorNumeric",(void (**)(void))&f);CHKERRQ(ierr);
  ierr = (*f)(a->sbaijMat,fact);CHKERRQ(ierr);
  ierr = MatDestroy(a->sbaijMat);CHKERRQ(ierr);
  a->sbaijMat = PETSC_NULL;
  PetscFunctionReturn(0);
}
#undef __FUNCT__
#define __FUNCT__ "MatCholeskyFactorNumeric_SeqAIJ_NaturalOrdering"
PetscErrorCode MatCholeskyFactorNumeric_SeqAIJ_NaturalOrdering(Mat A,Mat *fact)
{
  Mat            C = *fact;
  Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data;
  Mat_SeqSBAIJ   *b=(Mat_SeqSBAIJ*)C->data;
  PetscErrorCode ierr;
  PetscInt       i,j,am = A->m;
  PetscInt       *ai=a->i,*aj=a->j,*bi=b->i,*bj=b->j;
  PetscInt       k,jmin,*jl,*il,nexti,ili,*acol,*bcol,nz,ndamp = 0;
  MatScalar      *rtmp,*ba=b->a,*aa=a->a,dk,uikdi,*aval,*bval;
  PetscReal      damping=b->factor_damping, zeropivot=b->factor_zeropivot,shift_amount;
  PetscTruth     damp,chshift;
  PetscInt       nshift=0;

  PetscFunctionBegin;
  /* initialization */
  /* il and jl record the first nonzero element in each row of the accessing
     window U(0:k, k:am-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) = am indicates the end of a list
     il(i): points to the first nonzero element in U(i,k:am-1)
  */
  nz   = (2*am+1)*sizeof(PetscInt)+am*sizeof(MatScalar);
  ierr = PetscMalloc(nz,&il);CHKERRQ(ierr);
  jl   = il + am;
  rtmp = (MatScalar*)(jl + am);

  shift_amount = 0;
  do {
    damp = PETSC_FALSE;
    chshift = PETSC_FALSE;
    for (i=0; i<am; i++) {
      rtmp[i] = 0.0; jl[i] = am; il[0] = 0;
    }

    for (k = 0; k<am; 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 -- ){
        if (*acol < k) { /* skip lower triangular entries */
          acol++; aval++;
        } else {
          rtmp[*acol++] = *aval++;
          *bval++       = 0.0; /* for in-place factorization */
        }
      }
      /* damp the diagonal of the matrix */
      if (ndamp||nshift) rtmp[k] += damping+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;
          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;
      }

      if (PetscRealPart(dk) < zeropivot && b->factor_shift){
	/* calculate a shift that would make this row diagonally dominant */
	PetscReal rs = PetscAbs(PetscRealPart(dk));
	jmin      = bi[k]+1;
	nz        = bi[k+1] - jmin;
	if (nz){
	  bcol = bj + jmin;
	  bval = ba + jmin;
	  while (nz--){
	    rs += PetscAbsScalar(rtmp[*bcol++]);
	  }
	}
	/* if this shift is less than the previous, just up the previous
	   one by a bit */
	shift_amount = PetscMax(rs,1.1*shift_amount);
	chshift  = PETSC_TRUE;
	/* Unlike in the ILU case there is no exit condition on nshift:
	   we increase the shift until it converges. There is no guarantee that
	   this algorithm converges faster or slower, or is better or worse
	   than the ILU algorithm. */
	nshift++;
	break;
      }
      if (PetscRealPart(dk) < zeropivot){
        if (damping == (PetscReal) PETSC_DECIDE) damping = -PetscRealPart(dk)/(k+1);
        if (damping > 0.0) {
          if (ndamp) damping *= 2.0;
          damp = PETSC_TRUE;
          ndamp++;
          break;
        } else if (PetscAbsScalar(dk) < zeropivot){
          SETERRQ3(PETSC_ERR_MAT_LU_ZRPVT,"Zero pivot row %D value %g tolerance %g",k,PetscRealPart(dk),zeropivot);
        } else {
          PetscLogInfo((PetscObject)A,"Negative pivot %g in row %D of Cholesky factorization\n",PetscRealPart(dk),k);
        }
      }

      /* 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;
      }
    }
  } while (damp||chshift);
  ierr = PetscFree(il);CHKERRQ(ierr);

  C->factor       = FACTOR_CHOLESKY;
  C->assembled    = PETSC_TRUE;
  C->preallocated = PETSC_TRUE;
  PetscLogFlops(C->m);
  if (ndamp) {
    PetscLogInfo(0,"MatCholeskyFactorNumerical_SeqAIJ_NaturalOrdering: number of damping tries %D damping value %g\n",ndamp,damping);
  }
  if (nshift) {
    PetscLogInfo(0,"MatCholeskyFactorNumeric_SeqAIJ_NaturalOrdering diagonal shifted %D shifts\n",nshift);
  }
  PetscFunctionReturn(0);
}

#include "petscbt.h"
#include "src/mat/utils/freespace.h"
#undef __FUNCT__
#define __FUNCT__ "MatICCFactorSymbolic_SeqAIJ"
PetscErrorCode MatICCFactorSymbolic_SeqAIJ(Mat A,IS perm,MatFactorInfo *info,Mat *fact)
{
  Mat_SeqAIJ     *a = (Mat_SeqAIJ*)A->data;
  Mat_SeqSBAIJ   *b;
  Mat            B;
  PetscErrorCode ierr;
  PetscTruth     perm_identity;
  PetscInt       reallocs=0,*rip,i,*ai,*aj,am=A->m,*ui;
  PetscInt       jmin,jmax,nzk,k,j,*jl,prow,*il,nextprow;
  PetscInt       nlnk,*lnk,*lnk_lvl,ncols,*cols,*cols_lvl,*uj,**uj_ptr,**uj_lvl_ptr;
  PetscReal      fill=info->fill,levels=info->levels;
  FreeSpaceList  free_space=PETSC_NULL,current_space=PETSC_NULL;
  FreeSpaceList  free_space_lvl=PETSC_NULL,current_space_lvl=PETSC_NULL;
  PetscBT        lnkbt;

  PetscFunctionBegin;
  ierr = ISIdentity(perm,&perm_identity);CHKERRQ(ierr);
  if (!perm_identity){
    SETERRQ(PETSC_ERR_SUP,"Non-identity permutation is not supported yet");
  }
  ierr = ISGetIndices(perm,&rip);CHKERRQ(ierr);
  ai = a->i; aj = a->j;
  ierr = MatMarkDiagonal_SeqAIJ(A);CHKERRQ(ierr);

  /* levels = 0: simply copies fill pattern */
  if (!levels) {
    ierr = PetscMalloc((am+1)*sizeof(PetscInt),&ui);CHKERRQ(ierr);
    for (i=0; i<am; i++) {
      ui[i] = ai[i+1] - a->diag[i]; /* ui: rowlengths - changes when !perm_identity */
    }
    ierr = MatCreate(PETSC_COMM_SELF,am,am,am,am,fact);CHKERRQ(ierr);
    B = *fact;
    ierr = MatSetType(B,MATSEQSBAIJ);CHKERRQ(ierr);
    ierr = MatSeqSBAIJSetPreallocation(B,1,0,ui);CHKERRQ(ierr);

    b  = (Mat_SeqSBAIJ*)B->data;
    uj = b->j;
    for (i=0; i<am; i++) {
      aj = a->j + a->diag[i];
      for (j=0; j<ui[i]; j++){
        *uj++ = *aj++;
      }
      b->ilen[i] = ui[i];
    }
    ierr = PetscFree(ui);CHKERRQ(ierr);
    ierr = MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
    ierr = MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);

    B->ops->solve                 = MatSolve_SeqSBAIJ_1_NaturalOrdering;
    ierr = PetscObjectComposeFunction((PetscObject)B,"MatCholeskyFactorNumeric_SeqAIJ_NaturalOrdering","dummyname",(FCNVOID)B->ops->choleskyfactornumeric);CHKERRQ(ierr);
    B->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqAIJ_NaturalOrdering;
    PetscFunctionReturn(0);
  }

  /* initialization */
  ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&ui);CHKERRQ(ierr);
  ui[0] = 0;
  ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&cols_lvl);CHKERRQ(ierr);

  /* jl: linked list for storing indices of the pivot rows
     il: il[i] points to the 1st nonzero entry of U(i,k:am-1) */
  ierr = PetscMalloc((2*am+1)*sizeof(PetscInt)+2*am*sizeof(PetscInt*),&jl);CHKERRQ(ierr);
  il         = jl + am;
  uj_ptr     = (PetscInt**)(il + am);
  uj_lvl_ptr = (PetscInt**)(uj_ptr + am);
  for (i=0; i<am; i++){
    jl[i] = am; il[i] = 0;
  }

  /* create and initialize a linked list for storing column indices of the active row k */
  nlnk = am + 1;
  ierr = PetscLLCreate_PermutedLeveled(am,am,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);

  /* initial FreeSpace size is fill*(ai[am]+1) */
  ierr = GetMoreSpace((PetscInt)(fill*(ai[am]+1)),&free_space);CHKERRQ(ierr);
  current_space = free_space;
  ierr = GetMoreSpace((PetscInt)(fill*(ai[am]+1)),&free_space_lvl);CHKERRQ(ierr);
  current_space_lvl = free_space_lvl;

  for (k=0; k<am; 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] - a->diag[rip[k]]; /* changes for !perm_identity */
    cols  = aj + a->diag[rip[k]];
    for (j=0; j<ncols; j++) cols_lvl[j] = -1;  /* initialize level for nonzero entries */
    ierr = PetscLLAdd_PermutedLeveled(ncols,cols,levels,cols_lvl,am,rip,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
    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:am-1) */
      jmax = ui[prow+1];
      ncols = jmax-jmin;
      i     = jmin - ui[prow];
      cols = uj_ptr[prow] + i; /* points to the 2nd nzero entry in U(prow,k:am-1) */
      for (j=0; j<ncols; j++) cols_lvl[j] = *(uj_lvl_ptr[prow] + i + j);
      ierr = PetscLLAdd_PermutedLeveled(ncols,cols,levels,cols_lvl,am,rip,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
      nzk += nlnk;

      /* update il and jl for prow */
      if (jmin < jmax){
        il[prow] = jmin;
        j = *cols; 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 = am - k + 1; /* num of unfactored rows */
      i = PetscMin(i*nzk, i*(i-1)); /* i*nzk, i*(i-1): estimated and max additional space needed */
      ierr = GetMoreSpace(i,&current_space);CHKERRQ(ierr);
      ierr = GetMoreSpace(i,&current_space_lvl);CHKERRQ(ierr);
      reallocs++;
    }

    /* copy data into free_space and free_space_lvl, then initialize lnk */
    ierr = PetscLLClean_PermutedLeveled(am,am,nzk,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);CHKERRQ(ierr);

    /* 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:am-1) */
      jl[k] = jl[i]; jl[i] = k;
      il[k] = ui[k] + 1;
    }
    uj_ptr[k]     = current_space->array;
    uj_lvl_ptr[k] = current_space_lvl->array;

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

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

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

  if (ai[am] != 0) {
    PetscReal af = ((PetscReal)ui[am])/((PetscReal)ai[am]);
    PetscLogInfo(A,"MatICCFactorSymbolic_SeqAIJ:Reallocs %D Fill ratio:given %g needed %g\n",reallocs,fill,af);
    PetscLogInfo(A,"MatICCFactorSymbolic_SeqAIJ:Run with -pc_cholesky_fill %g or use \n",af);
    PetscLogInfo(A,"MatICCFactorSymbolic_SeqAIJ:PCCholeskySetFill(pc,%g) for best performance.\n",af);
  } else {
     PetscLogInfo(A,"MatICCFactorSymbolic_SeqAIJ:Empty matrix.\n");
  }

  ierr = ISRestoreIndices(perm,&rip);CHKERRQ(ierr);
  ierr = PetscFree(jl);CHKERRQ(ierr);
  ierr = PetscFree(cols_lvl);CHKERRQ(ierr);

  /* destroy list of free space and other temporary array(s) */
  ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
  ierr = MakeSpaceContiguous(&free_space,uj);CHKERRQ(ierr);
  ierr = PetscLLDestroy_PermutedLeveled(lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
  ierr = DestroySpace(free_space_lvl);CHKERRQ(ierr);

  /* put together the new matrix in MATSEQSBAIJ format */
  ierr = MatCreate(PETSC_COMM_SELF,am,am,am,am,fact);CHKERRQ(ierr);
  B = *fact;
  ierr = MatSetType(B,MATSEQSBAIJ);CHKERRQ(ierr);
  ierr = MatSeqSBAIJSetPreallocation(B,1,0,PETSC_NULL);CHKERRQ(ierr);

  b = (Mat_SeqSBAIJ*)B->data;
  ierr = PetscFree(b->imax);CHKERRQ(ierr);
  b->singlemalloc = PETSC_FALSE;
  /* the next line frees the default space generated by the Create() */
  ierr = PetscFree(b->a);CHKERRQ(ierr);
  ierr = PetscFree(b->ilen);CHKERRQ(ierr);
  ierr = PetscMalloc((ui[am]+1)*sizeof(MatScalar),&b->a);CHKERRQ(ierr);
  b->j    = uj;
  b->i    = ui;
  b->diag = 0;
  b->ilen = 0;
  b->imax = 0;
  b->row  = perm;
  b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */
  ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
  b->icol = perm;
  ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
  ierr    = PetscMalloc((am+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
  PetscLogObjectMemory(B,(ui[am]-am)*(sizeof(PetscInt)+sizeof(MatScalar)));
  b->maxnz = b->nz = ui[am];

  B->factor                 = FACTOR_CHOLESKY;
  B->info.factor_mallocs    = reallocs;
  B->info.fill_ratio_given  = fill;
  if (ai[am] != 0) {
    B->info.fill_ratio_needed = ((PetscReal)ui[am])/((PetscReal)ai[am]);
  } else {
    B->info.fill_ratio_needed = 0.0;
  }

  B->ops->solve           = MatSolve_SeqSBAIJ_1_NaturalOrdering;
  B->ops->solvetranspose  = MatSolve_SeqSBAIJ_1_NaturalOrdering;
  ierr = PetscObjectComposeFunction((PetscObject) B,"MatCholeskyFactorNumeric_NaturalOrdering","dummyname",(FCNVOID)B->ops->choleskyfactornumeric);CHKERRQ(ierr);
  B->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqAIJ_NaturalOrdering;
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "MatCholeskyFactorSymbolic_SeqAIJ"
PetscErrorCode MatCholeskyFactorSymbolic_SeqAIJ(Mat A,IS perm,MatFactorInfo *info,Mat *fact)
{
  Mat_SeqAIJ     *a = (Mat_SeqAIJ*)A->data;
  Mat_SeqSBAIJ   *b;
  PetscErrorCode ierr;
  PetscTruth     perm_identity;
  PetscReal      fill = info->fill;
  PetscInt       *rip,i,am=A->m,*ai,*aj,reallocs=0,prow;
  PetscInt       *jl,jmin,jmax,nzk,*ui,k,j,*il,nextprow;
  PetscInt       nlnk,*lnk,ncols,*cols,*uj,**uj_ptr;
  FreeSpaceList  free_space=PETSC_NULL,current_space=PETSC_NULL;
  PetscBT        lnkbt;

  PetscFunctionBegin;
  ierr = ISIdentity(perm,&perm_identity);CHKERRQ(ierr);
  if (!perm_identity){
    SETERRQ(PETSC_ERR_SUP,"Non-identity permutation is not supported yet");
  }

  ierr = ISGetIndices(perm,&rip);CHKERRQ(ierr);
  ai = a->i; aj = a->j;
  ierr = MatMarkDiagonal_SeqAIJ(A);CHKERRQ(ierr);

  /* initialization */
  ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&ui);CHKERRQ(ierr);
  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:am-1) */
  ierr = PetscMalloc((2*am+1)*sizeof(PetscInt)+am*sizeof(PetscInt*),&jl);CHKERRQ(ierr);
  il     = jl + am;
  uj_ptr = (PetscInt**)(il + am);
  for (i=0; i<am; i++){
    jl[i] = am; il[i] = 0;
  }

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

  /* initial FreeSpace size is fill*(ai[am]+1) */
  ierr = GetMoreSpace((PetscInt)(fill*(ai[am]+1)),&free_space);CHKERRQ(ierr);
  current_space = free_space;

  for (k=0; k<am; 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] - a->diag[rip[k]]; /* diag would change for !perm_identity */
    cols = aj + a->diag[rip[k]];
    ierr = PetscLLAdd(ncols,cols,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
    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:am-1) */
      jmax = ui[prow+1];
      ncols = jmax-jmin;
      cols = uj_ptr[prow] + jmin - ui[prow]; /* points to the 2nd nzero entry in U(prow,k:am-1) */
      ierr = PetscLLAdd(ncols,cols,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
      nzk += nlnk;

      /* update il and jl for prow */
      if (jmin < jmax){
        il[prow] = jmin;
        j = *cols; 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 = am - k + 1; /* num of unfactored rows */
      i = PetscMin(i*nzk, i*(i-1)); /* i*nzk, i*(i-1): estimated and max additional space needed */
      ierr = GetMoreSpace(i,&current_space);CHKERRQ(ierr);
      reallocs++;
    }

    /* copy data into free space, then initialize lnk */
    ierr = PetscLLClean(am,am,nzk,lnk,current_space->array,lnkbt);CHKERRQ(ierr);

    /* 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:am-1) */
      jl[k] = jl[i]; jl[i] = k;
      il[k] = ui[k] + 1;
    }
    uj_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;
  }

  if (ai[am] != 0) {
    PetscReal af = ((PetscReal)ui[am])/((PetscReal)ai[am]);
    PetscLogInfo(A,"MatCholeskyFactorSymbolic_SeqAIJ:Reallocs %D Fill ratio:given %g needed %g\n",reallocs,fill,af);
    PetscLogInfo(A,"MatCholeskyFactorSymbolic_SeqAIJ:Run with -pc_cholesky_fill %g or use \n",af);
    PetscLogInfo(A,"MatCholeskyFactorSymbolic_SeqAIJ:PCCholeskySetFill(pc,%g) for best performance.\n",af);
  } else {
     PetscLogInfo(A,"MatCholeskyFactorSymbolic_SeqAIJ:Empty matrix.\n");
  }

  ierr = ISRestoreIndices(perm,&rip);CHKERRQ(ierr);
  ierr = PetscFree(jl);CHKERRQ(ierr);

  /* destroy list of free space and other temporary array(s) */
  ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
  ierr = MakeSpaceContiguous(&free_space,uj);CHKERRQ(ierr);
  ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);

  /* put together the new matrix in MATSEQSBAIJ format */
  ierr = MatCreate(PETSC_COMM_SELF,am,am,am,am,fact);CHKERRQ(ierr);
  ierr = MatSetType(*fact,MATSEQSBAIJ);CHKERRQ(ierr);
  ierr = MatSeqSBAIJSetPreallocation(*fact,1,0,PETSC_NULL);CHKERRQ(ierr);

  b = (Mat_SeqSBAIJ*)(*fact)->data;
  ierr = PetscFree(b->imax);CHKERRQ(ierr);
  b->singlemalloc = PETSC_FALSE;
  /* the next line frees the default space generated by the Create() */
  ierr = PetscFree(b->a);CHKERRQ(ierr);
  ierr = PetscFree(b->ilen);CHKERRQ(ierr);
  ierr = PetscMalloc((ui[am]+1)*sizeof(MatScalar),&b->a);CHKERRQ(ierr);
  b->j    = uj;
  b->i    = ui;
  b->diag = 0;
  b->ilen = 0;
  b->imax = 0;
  b->row  = perm;
  b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */
  ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
  b->icol = perm;
  ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
  ierr    = PetscMalloc((am+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
  /* In b structure:  Free imax, ilen, old a, old j.
     Allocate idnew, solve_work, new a, new j */
  PetscLogObjectMemory(*fact,(ui[am]-am)*(sizeof(PetscInt)+sizeof(MatScalar)));
  b->maxnz = b->nz = ui[am];

  (*fact)->factor                 = FACTOR_CHOLESKY;
  (*fact)->info.factor_mallocs    = reallocs;
  (*fact)->info.fill_ratio_given  = fill;
  if (ai[am] != 0) {
    (*fact)->info.fill_ratio_needed = ((PetscReal)ui[am])/((PetscReal)ai[am]);
  } else {
    (*fact)->info.fill_ratio_needed = 0.0;
  }

  (*fact)->ops->solve           = MatSolve_SeqSBAIJ_1_NaturalOrdering;
  (*fact)->ops->solvetranspose  = MatSolve_SeqSBAIJ_1_NaturalOrdering;
  ierr = PetscObjectComposeFunction((PetscObject) *fact,"MatCholeskyFactorNumeric_NaturalOrdering","dummyname",(FCNVOID)(*fact)->ops->choleskyfactornumeric);CHKERRQ(ierr);
  (*fact)->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqAIJ_NaturalOrdering;
  PetscFunctionReturn(0);
}