Date: 2016-07-24T16:56:07+00:00

Actual source code: sbaijfact.c
petsc-3.7.3 2016-07-24
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      2: #include <../src/mat/impls/baij/seq/baij.h>
      3: #include <../src/mat/impls/sbaij/seq/sbaij.h>
      4: #include <petsc/private/kernels/blockinvert.h>
      5: #include <petscis.h>

      7: /*
      8:   input:
      9:    F -- numeric factor
     10:   output:
     11:    nneg, nzero, npos: matrix inertia
     12: */

     16: PetscErrorCode MatGetInertia_SeqSBAIJ(Mat F,PetscInt *nneig,PetscInt *nzero,PetscInt *npos)
     17: {
     18:   Mat_SeqSBAIJ *fact_ptr = (Mat_SeqSBAIJ*)F->data;
     19:   MatScalar    *dd       = fact_ptr->a;
     20:   PetscInt     mbs       =fact_ptr->mbs,bs=F->rmap->bs,i,nneig_tmp,npos_tmp,*fi = fact_ptr->diag;

     23:   if (bs != 1) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_SUP,"No support for bs: %D >1 yet",bs);
     24:   nneig_tmp = 0; npos_tmp = 0;
     25:   for (i=0; i<mbs; i++) {
     26:     if (PetscRealPart(dd[*fi]) > 0.0) npos_tmp++;
     27:     else if (PetscRealPart(dd[*fi]) < 0.0) nneig_tmp++;
     28:     fi++;
     29:   }
     30:   if (nneig) *nneig = nneig_tmp;
     31:   if (npos)  *npos  = npos_tmp;
     32:   if (nzero) *nzero = mbs - nneig_tmp - npos_tmp;
     33:   return(0);
     34: }

     36: /*
     37:   Symbolic U^T*D*U factorization for SBAIJ format. Modified from SSF of YSMP.
     38:   Use Modified Sparse Row (MSR) storage for u and ju. See page 85, "Iterative Methods ..." by Saad.
     39: */
     42: PetscErrorCode MatCholeskyFactorSymbolic_SeqSBAIJ_MSR(Mat F,Mat A,IS perm,const MatFactorInfo *info)
     43: {
     44:   Mat_SeqSBAIJ   *a = (Mat_SeqSBAIJ*)A->data,*b;
     46:   const PetscInt *rip,*ai,*aj;
     47:   PetscInt       i,mbs = a->mbs,*jutmp,bs = A->rmap->bs,bs2=a->bs2;
     48:   PetscInt       m,reallocs = 0,prow;
     49:   PetscInt       *jl,*q,jmin,jmax,juidx,nzk,qm,*iu,*ju,k,j,vj,umax,maxadd;
     50:   PetscReal      f = info->fill;
     51:   PetscBool      perm_identity;

     54:   /* check whether perm is the identity mapping */
     55:   ISIdentity(perm,&perm_identity);
     56:   ISGetIndices(perm,&rip);

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

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

     65:     MatReorderingSeqSBAIJ(A,perm);

     67:     ai = a->inew; aj = a->jnew;
     68:   }

     70:   /* initialization */
     71:   PetscMalloc1(mbs+1,&iu);
     72:   umax  = (PetscInt)(f*ai[mbs] + 1); umax += mbs + 1;
     73:   PetscMalloc1(umax,&ju);
     74:   iu[0] = mbs+1;
     75:   juidx = mbs + 1; /* index for ju */
     76:   /* jl linked list for pivot row -- linked list for col index */
     77:   PetscMalloc2(mbs,&jl,mbs,&q);
     78:   for (i=0; i<mbs; i++) {
     79:     jl[i] = mbs;
     80:     q[i]  = 0;
     81:   }

     83:   /* for each row k */
     84:   for (k=0; k<mbs; k++) {
     85:     for (i=0; i<mbs; i++) q[i] = 0;  /* to be removed! */
     86:     nzk  = 0; /* num. of nz blocks in k-th block row with diagonal block excluded */
     87:     q[k] = mbs;
     88:     /* initialize nonzero structure of k-th row to row rip[k] of A */
     89:     jmin = ai[rip[k]] +1; /* exclude diag[k] */
     90:     jmax = ai[rip[k]+1];
     91:     for (j=jmin; j<jmax; j++) {
     92:       vj = rip[aj[j]]; /* col. value */
     93:       if (vj > k) {
     94:         qm = k;
     95:         do {
     96:           m = qm; qm = q[m];
     97:         } while (qm < vj);
     98:         if (qm == vj) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Duplicate entry in A\n");
     99:         nzk++;
    100:         q[m]  = vj;
    101:         q[vj] = qm;
    102:       } /* if (vj > k) */
    103:     } /* for (j=jmin; j<jmax; j++) */

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

    110:     while (prow < k) {
    111:       /* merge row prow into k-th row */
    112:       jmin = iu[prow] + 1; jmax = iu[prow+1];
    113:       qm   = k;
    114:       for (j=jmin; j<jmax; j++) {
    115:         vj = ju[j];
    116:         do {
    117:           m = qm; qm = q[m];
    118:         } while (qm < vj);
    119:         if (qm != vj) {
    120:           nzk++; q[m] = vj; q[vj] = qm; qm = vj;
    121:         }
    122:       }
    123:       prow = jl[prow]; /* next pivot row */
    124:     }

    126:     /* add k to row list for first nonzero element in k-th row */
    127:     if (nzk > 0) {
    128:       i     = q[k]; /* col value of first nonzero element in U(k, k+1:mbs-1) */
    129:       jl[k] = jl[i]; jl[i] = k;
    130:     }
    131:     iu[k+1] = iu[k] + nzk;

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

    142:       /* allocate a longer ju */
    143:       PetscMalloc1(umax,&jutmp);
    144:       PetscMemcpy(jutmp,ju,iu[k]*sizeof(PetscInt));
    145:       PetscFree(ju);
    146:       ju   = jutmp;
    147:       reallocs++; /* count how many times we realloc */
    148:     }

    150:     /* save nonzero structure of k-th row in ju */
    151:     i=k;
    152:     while (nzk--) {
    153:       i           = q[i];
    154:       ju[juidx++] = i;
    155:     }
    156:   }

    158: #if defined(PETSC_USE_INFO)
    159:   if (ai[mbs] != 0) {
    160:     PetscReal af = ((PetscReal)iu[mbs])/((PetscReal)ai[mbs]);
    161:     PetscInfo3(A,"Reallocs %D Fill ratio:given %g needed %g\n",reallocs,(double)f,(double)af);
    162:     PetscInfo1(A,"Run with -pc_factor_fill %g or use \n",(double)af);
    163:     PetscInfo1(A,"PCFactorSetFill(pc,%g);\n",(double)af);
    164:     PetscInfo(A,"for best performance.\n");
    165:   } else {
    166:     PetscInfo(A,"Empty matrix.\n");
    167:   }
    168: #endif

    170:   ISRestoreIndices(perm,&rip);
    171:   PetscFree2(jl,q);

    173:   /* put together the new matrix */
    174:   MatSeqSBAIJSetPreallocation_SeqSBAIJ(F,bs,MAT_SKIP_ALLOCATION,NULL);

    176:   /* PetscLogObjectParent((PetscObject)B,(PetscObject)iperm); */
    177:   b                = (Mat_SeqSBAIJ*)(F)->data;
    178:   b->singlemalloc  = PETSC_FALSE;
    179:   b->free_a        = PETSC_TRUE;
    180:   b->free_ij       = PETSC_TRUE;

    182:   PetscMalloc1((iu[mbs]+1)*bs2,&b->a);
    183:   b->j    = ju;
    184:   b->i    = iu;
    185:   b->diag = 0;
    186:   b->ilen = 0;
    187:   b->imax = 0;
    188:   b->row  = perm;

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

    192:   PetscObjectReference((PetscObject)perm);

    194:   b->icol = perm;
    195:   PetscObjectReference((PetscObject)perm);
    196:   PetscMalloc1(bs*mbs+bs,&b->solve_work);
    197:   /* In b structure:  Free imax, ilen, old a, old j.
    198:      Allocate idnew, solve_work, new a, new j */
    199:   PetscLogObjectMemory((PetscObject)F,(iu[mbs]-mbs)*(sizeof(PetscInt)+sizeof(MatScalar)));
    200:   b->maxnz = b->nz = iu[mbs];

    202:   (F)->info.factor_mallocs   = reallocs;
    203:   (F)->info.fill_ratio_given = f;
    204:   if (ai[mbs] != 0) {
    205:     (F)->info.fill_ratio_needed = ((PetscReal)iu[mbs])/((PetscReal)ai[mbs]);
    206:   } else {
    207:     (F)->info.fill_ratio_needed = 0.0;
    208:   }
    209:   MatSeqSBAIJSetNumericFactorization_inplace(F,perm_identity);
    210:   return(0);
    211: }
    212: /*
    213:     Symbolic U^T*D*U factorization for SBAIJ format.
    214:     See MatICCFactorSymbolic_SeqAIJ() for description of its data structure.
    215: */
    216: #include <petscbt.h>
    217: #include <../src/mat/utils/freespace.h>
    220: PetscErrorCode MatCholeskyFactorSymbolic_SeqSBAIJ(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
    221: {
    222:   Mat_SeqSBAIJ       *a = (Mat_SeqSBAIJ*)A->data;
    223:   Mat_SeqSBAIJ       *b;
    224:   PetscErrorCode     ierr;
    225:   PetscBool          perm_identity,missing;
    226:   PetscReal          fill = info->fill;
    227:   const PetscInt     *rip,*ai=a->i,*aj=a->j;
    228:   PetscInt           i,mbs=a->mbs,bs=A->rmap->bs,reallocs=0,prow;
    229:   PetscInt           *jl,jmin,jmax,nzk,*ui,k,j,*il,nextprow;
    230:   PetscInt           nlnk,*lnk,ncols,*cols,*uj,**ui_ptr,*uj_ptr,*udiag;
    231:   PetscFreeSpaceList free_space=NULL,current_space=NULL;
    232:   PetscBT            lnkbt;

    235:   if (A->rmap->n != A->cmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Must be square matrix, rows %D columns %D",A->rmap->n,A->cmap->n);
    236:   MatMissingDiagonal(A,&missing,&i);
    237:   if (missing) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",i);
    238:   if (bs > 1) {
    239:     MatCholeskyFactorSymbolic_SeqSBAIJ_inplace(fact,A,perm,info);
    240:     return(0);
    241:   }

    243:   /* check whether perm is the identity mapping */
    244:   ISIdentity(perm,&perm_identity);
    245:   if (!perm_identity) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Matrix reordering is not supported for sbaij matrix. Use aij format");
    246:   a->permute = PETSC_FALSE;
    247:   ISGetIndices(perm,&rip);

    249:   /* initialization */
    250:   PetscMalloc1(mbs+1,&ui);
    251:   PetscMalloc1(mbs+1,&udiag);
    252:   ui[0] = 0;

    254:   /* jl: linked list for storing indices of the pivot rows
    255:      il: il[i] points to the 1st nonzero entry of U(i,k:mbs-1) */
    256:   PetscMalloc4(mbs,&ui_ptr,mbs,&il,mbs,&jl,mbs,&cols);
    257:   for (i=0; i<mbs; i++) {
    258:     jl[i] = mbs; il[i] = 0;
    259:   }

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

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

    269:   for (k=0; k<mbs; k++) {  /* for each active row k */
    270:     /* initialize lnk by the column indices of row rip[k] of A */
    271:     nzk   = 0;
    272:     ncols = ai[k+1] - ai[k];
    273:     if (!ncols) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_MAT_CH_ZRPVT,"Empty row %D in matrix ",k);
    274:     for (j=0; j<ncols; j++) {
    275:       i       = *(aj + ai[k] + j);
    276:       cols[j] = i;
    277:     }
    278:     PetscLLAdd(ncols,cols,mbs,nlnk,lnk,lnkbt);
    279:     nzk += nlnk;

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

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

    294:       /* update il and jl for prow */
    295:       if (jmin < jmax) {
    296:         il[prow] = jmin;
    297:         j        = *uj_ptr; jl[prow] = jl[j]; jl[j] = prow;
    298:       }
    299:       prow = nextprow;
    300:     }

    302:     /* if free space is not available, make more free space */
    303:     if (current_space->local_remaining<nzk) {
    304:       i    = mbs - k + 1; /* num of unfactored rows */
    305:       i    = PetscIntMultTruncate(i,PetscMin(nzk, i-1)); /* i*nzk, i*(i-1): estimated and max additional space needed */
    306:       PetscFreeSpaceGet(i,&current_space);
    307:       reallocs++;
    308:     }

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

    313:     /* add the k-th row into il and jl */
    314:     if (nzk > 1) {
    315:       i     = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:mbs-1) */
    316:       jl[k] = jl[i]; jl[i] = k;
    317:       il[k] = ui[k] + 1;
    318:     }
    319:     ui_ptr[k] = current_space->array;

    321:     current_space->array           += nzk;
    322:     current_space->local_used      += nzk;
    323:     current_space->local_remaining -= nzk;

    325:     ui[k+1] = ui[k] + nzk;
    326:   }

    328:   ISRestoreIndices(perm,&rip);
    329:   PetscFree4(ui_ptr,il,jl,cols);

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

    336:   /* put together the new matrix in MATSEQSBAIJ format */
    337:   MatSeqSBAIJSetPreallocation_SeqSBAIJ(fact,bs,MAT_SKIP_ALLOCATION,NULL);

    339:   b               = (Mat_SeqSBAIJ*)fact->data;
    340:   b->singlemalloc = PETSC_FALSE;
    341:   b->free_a       = PETSC_TRUE;
    342:   b->free_ij      = PETSC_TRUE;

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

    346:   b->j         = uj;
    347:   b->i         = ui;
    348:   b->diag      = udiag;
    349:   b->free_diag = PETSC_TRUE;
    350:   b->ilen      = 0;
    351:   b->imax      = 0;
    352:   b->row       = perm;
    353:   b->icol      = perm;

    355:   PetscObjectReference((PetscObject)perm);
    356:   PetscObjectReference((PetscObject)perm);

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

    360:   PetscMalloc1(mbs+1,&b->solve_work);
    361:   PetscLogObjectMemory((PetscObject)fact,ui[mbs]*(sizeof(PetscInt)+sizeof(MatScalar)));

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

    365:   fact->info.factor_mallocs   = reallocs;
    366:   fact->info.fill_ratio_given = fill;
    367:   if (ai[mbs] != 0) {
    368:     fact->info.fill_ratio_needed = ((PetscReal)ui[mbs])/ai[mbs];
    369:   } else {
    370:     fact->info.fill_ratio_needed = 0.0;
    371:   }
    372: #if defined(PETSC_USE_INFO)
    373:   if (ai[mbs] != 0) {
    374:     PetscReal af = fact->info.fill_ratio_needed;
    375:     PetscInfo3(A,"Reallocs %D Fill ratio:given %g needed %g\n",reallocs,(double)fill,(double)af);
    376:     PetscInfo1(A,"Run with -pc_factor_fill %g or use \n",(double)af);
    377:     PetscInfo1(A,"PCFactorSetFill(pc,%g) for best performance.\n",(double)af);
    378:   } else {
    379:     PetscInfo(A,"Empty matrix.\n");
    380:   }
    381: #endif
    382:   fact->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqSBAIJ_1_NaturalOrdering;
    383:   return(0);
    384: }

    388: PetscErrorCode MatCholeskyFactorSymbolic_SeqSBAIJ_inplace(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
    389: {
    390:   Mat_SeqSBAIJ       *a = (Mat_SeqSBAIJ*)A->data;
    391:   Mat_SeqSBAIJ       *b;
    392:   PetscErrorCode     ierr;
    393:   PetscBool          perm_identity,missing;
    394:   PetscReal          fill = info->fill;
    395:   const PetscInt     *rip,*ai,*aj;
    396:   PetscInt           i,mbs=a->mbs,bs=A->rmap->bs,reallocs=0,prow,d;
    397:   PetscInt           *jl,jmin,jmax,nzk,*ui,k,j,*il,nextprow;
    398:   PetscInt           nlnk,*lnk,ncols,*cols,*uj,**ui_ptr,*uj_ptr;
    399:   PetscFreeSpaceList free_space=NULL,current_space=NULL;
    400:   PetscBT            lnkbt;

    403:   MatMissingDiagonal(A,&missing,&d);
    404:   if (missing) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",d);

    406:   /*
    407:    This code originally uses Modified Sparse Row (MSR) storage
    408:    (see page 85, "Iterative Methods ..." by Saad) for the output matrix B - bad choise!
    409:    Then it is rewritten so the factor B takes seqsbaij format. However the associated
    410:    MatCholeskyFactorNumeric_() have not been modified for the cases of bs>1 or !perm_identity,
    411:    thus the original code in MSR format is still used for these cases.
    412:    The code below should replace MatCholeskyFactorSymbolic_SeqSBAIJ_MSR() whenever
    413:    MatCholeskyFactorNumeric_() is modified for using sbaij symbolic factor.
    414:   */
    415:   if (bs > 1) {
    416:     MatCholeskyFactorSymbolic_SeqSBAIJ_MSR(fact,A,perm,info);
    417:     return(0);
    418:   }

    420:   /* check whether perm is the identity mapping */
    421:   ISIdentity(perm,&perm_identity);

    423:   if (perm_identity) {
    424:     a->permute = PETSC_FALSE;

    426:     ai = a->i; aj = a->j;
    427:   } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Matrix reordering is not supported for sbaij matrix. Use aij format");
    428:   ISGetIndices(perm,&rip);

    430:   /* initialization */
    431:   PetscMalloc1(mbs+1,&ui);
    432:   ui[0] = 0;

    434:   /* jl: linked list for storing indices of the pivot rows
    435:      il: il[i] points to the 1st nonzero entry of U(i,k:mbs-1) */
    436:   PetscMalloc4(mbs,&ui_ptr,mbs,&il,mbs,&jl,mbs,&cols);
    437:   for (i=0; i<mbs; i++) {
    438:     jl[i] = mbs; il[i] = 0;
    439:   }

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

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

    449:   for (k=0; k<mbs; k++) {  /* for each active row k */
    450:     /* initialize lnk by the column indices of row rip[k] of A */
    451:     nzk   = 0;
    452:     ncols = ai[rip[k]+1] - ai[rip[k]];
    453:     for (j=0; j<ncols; j++) {
    454:       i       = *(aj + ai[rip[k]] + j);
    455:       cols[j] = rip[i];
    456:     }
    457:     PetscLLAdd(ncols,cols,mbs,nlnk,lnk,lnkbt);
    458:     nzk += nlnk;

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

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

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

    477:         j = *uj_ptr; jl[prow] = jl[j]; jl[j] = prow;
    478:       }
    479:       prow = nextprow;
    480:     }

    482:     /* if free space is not available, make more free space */
    483:     if (current_space->local_remaining<nzk) {
    484:       i    = mbs - k + 1; /* num of unfactored rows */
    485:       i    = PetscMin(PetscIntMultTruncate(i,nzk), PetscIntMultTruncate(i,i-1)); /* i*nzk, i*(i-1): estimated and max additional space needed */
    486:       PetscFreeSpaceGet(i,&current_space);
    487:       reallocs++;
    488:     }

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

    493:     /* add the k-th row into il and jl */
    494:     if (nzk-1 > 0) {
    495:       i     = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:mbs-1) */
    496:       jl[k] = jl[i]; jl[i] = k;
    497:       il[k] = ui[k] + 1;
    498:     }
    499:     ui_ptr[k] = current_space->array;

    501:     current_space->array           += nzk;
    502:     current_space->local_used      += nzk;
    503:     current_space->local_remaining -= nzk;

    505:     ui[k+1] = ui[k] + nzk;
    506:   }

    508:   ISRestoreIndices(perm,&rip);
    509:   PetscFree4(ui_ptr,il,jl,cols);

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

    516:   /* put together the new matrix in MATSEQSBAIJ format */
    517:   MatSeqSBAIJSetPreallocation_SeqSBAIJ(fact,bs,MAT_SKIP_ALLOCATION,NULL);

    519:   b               = (Mat_SeqSBAIJ*)fact->data;
    520:   b->singlemalloc = PETSC_FALSE;
    521:   b->free_a       = PETSC_TRUE;
    522:   b->free_ij      = PETSC_TRUE;

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

    526:   b->j    = uj;
    527:   b->i    = ui;
    528:   b->diag = 0;
    529:   b->ilen = 0;
    530:   b->imax = 0;
    531:   b->row  = perm;

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

    535:   PetscObjectReference((PetscObject)perm);
    536:   b->icol  = perm;
    537:   PetscObjectReference((PetscObject)perm);
    538:   PetscMalloc1(mbs+1,&b->solve_work);
    539:   PetscLogObjectMemory((PetscObject)fact,(ui[mbs]-mbs)*(sizeof(PetscInt)+sizeof(MatScalar)));
    540:   b->maxnz = b->nz = ui[mbs];

    542:   fact->info.factor_mallocs   = reallocs;
    543:   fact->info.fill_ratio_given = fill;
    544:   if (ai[mbs] != 0) {
    545:     fact->info.fill_ratio_needed = ((PetscReal)ui[mbs])/ai[mbs];
    546:   } else {
    547:     fact->info.fill_ratio_needed = 0.0;
    548:   }
    549: #if defined(PETSC_USE_INFO)
    550:   if (ai[mbs] != 0) {
    551:     PetscReal af = fact->info.fill_ratio_needed;
    552:     PetscInfo3(A,"Reallocs %D Fill ratio:given %g needed %g\n",reallocs,(double)fill,(double)af);
    553:     PetscInfo1(A,"Run with -pc_factor_fill %g or use \n",(double)af);
    554:     PetscInfo1(A,"PCFactorSetFill(pc,%g) for best performance.\n",(double)af);
    555:   } else {
    556:     PetscInfo(A,"Empty matrix.\n");
    557:   }
    558: #endif
    559:   MatSeqSBAIJSetNumericFactorization_inplace(fact,perm_identity);
    560:   return(0);
    561: }

    565: PetscErrorCode MatCholeskyFactorNumeric_SeqSBAIJ_N(Mat C,Mat A,const MatFactorInfo *info)
    566: {
    567:   Mat_SeqSBAIJ   *a   = (Mat_SeqSBAIJ*)A->data,*b = (Mat_SeqSBAIJ*)C->data;
    568:   IS             perm = b->row;
    570:   const PetscInt *ai,*aj,*perm_ptr,mbs=a->mbs,*bi=b->i,*bj=b->j;
    571:   PetscInt       i,j;
    572:   PetscInt       *a2anew,k,k1,jmin,jmax,*jl,*il,vj,nexti,ili;
    573:   PetscInt       bs  =A->rmap->bs,bs2 = a->bs2;
    574:   MatScalar      *ba = b->a,*aa,*ap,*dk,*uik;
    575:   MatScalar      *u,*diag,*rtmp,*rtmp_ptr;
    576:   MatScalar      *work;
    577:   PetscInt       *pivots;
    578:   PetscBool      allowzeropivot,zeropivotdetected;

    581:   /* initialization */
    582:   PetscCalloc1(bs2*mbs,&rtmp);
    583:   PetscMalloc2(mbs,&il,mbs,&jl);
    584:   allowzeropivot = PetscNot(A->erroriffailure);

    586:   il[0] = 0;
    587:   for (i=0; i<mbs; i++) jl[i] = mbs;
    588:
    589:   PetscMalloc3(bs2,&dk,bs2,&uik,bs,&work);
    590:   PetscMalloc1(bs,&pivots);

    592:   ISGetIndices(perm,&perm_ptr);

    594:   /* check permutation */
    595:   if (!a->permute) {
    596:     ai = a->i; aj = a->j; aa = a->a;
    597:   } else {
    598:     ai   = a->inew; aj = a->jnew;
    599:     PetscMalloc1(bs2*ai[mbs],&aa);
    600:     PetscMemcpy(aa,a->a,bs2*ai[mbs]*sizeof(MatScalar));
    601:     PetscMalloc1(ai[mbs],&a2anew);
    602:     PetscMemcpy(a2anew,a->a2anew,(ai[mbs])*sizeof(PetscInt));

    604:     for (i=0; i<mbs; i++) {
    605:       jmin = ai[i]; jmax = ai[i+1];
    606:       for (j=jmin; j<jmax; j++) {
    607:         while (a2anew[j] != j) {
    608:           k = a2anew[j]; a2anew[j] = a2anew[k]; a2anew[k] = k;
    609:           for (k1=0; k1<bs2; k1++) {
    610:             dk[k1]       = aa[k*bs2+k1];
    611:             aa[k*bs2+k1] = aa[j*bs2+k1];
    612:             aa[j*bs2+k1] = dk[k1];
    613:           }
    614:         }
    615:         /* transform columnoriented blocks that lie in the lower triangle to roworiented blocks */
    616:         if (i > aj[j]) {
    617:           /* printf("change orientation, row: %d, col: %d\n",i,aj[j]); */
    618:           ap = aa + j*bs2;                     /* ptr to the beginning of j-th block of aa */
    619:           for (k=0; k<bs2; k++) dk[k] = ap[k]; /* dk <- j-th block of aa */
    620:           for (k=0; k<bs; k++) {               /* j-th block of aa <- dk^T */
    621:             for (k1=0; k1<bs; k1++) *ap++ = dk[k + bs*k1];
    622:           }
    623:         }
    624:       }
    625:     }
    626:     PetscFree(a2anew);
    627:   }

    629:   /* for each row k */
    630:   for (k = 0; k<mbs; k++) {

    632:     /*initialize k-th row with elements nonzero in row perm(k) of A */
    633:     jmin = ai[perm_ptr[k]]; jmax = ai[perm_ptr[k]+1];

    635:     ap = aa + jmin*bs2;
    636:     for (j = jmin; j < jmax; j++) {
    637:       vj       = perm_ptr[aj[j]];   /* block col. index */
    638:       rtmp_ptr = rtmp + vj*bs2;
    639:       for (i=0; i<bs2; i++) *rtmp_ptr++ = *ap++;
    640:     }

    642:     /* modify k-th row by adding in those rows i with U(i,k) != 0 */
    643:     PetscMemcpy(dk,rtmp+k*bs2,bs2*sizeof(MatScalar));
    644:     i    = jl[k]; /* first row to be added to k_th row  */

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

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

    652:       /* uik = -inv(Di)*U_bar(i,k) */
    653:       diag = ba + i*bs2;
    654:       u    = ba + ili*bs2;
    655:       PetscMemzero(uik,bs2*sizeof(MatScalar));
    656:       PetscKernel_A_gets_A_minus_B_times_C(bs,uik,diag,u);

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

    662:       /* update -U(i,k) */
    663:       PetscMemcpy(ba+ili*bs2,uik,bs2*sizeof(MatScalar));

    665:       /* add multiple of row i to k-th row ... */
    666:       jmin = ili + 1; jmax = bi[i+1];
    667:       if (jmin < jmax) {
    668:         for (j=jmin; j<jmax; j++) {
    669:           /* rtmp += -U(i,k)^T * U_bar(i,j) */
    670:           rtmp_ptr = rtmp + bj[j]*bs2;
    671:           u        = ba + j*bs2;
    672:           PetscKernel_A_gets_A_plus_Btranspose_times_C(bs,rtmp_ptr,uik,u);
    673:         }
    674:         PetscLogFlops(2.0*bs*bs2*(jmax-jmin));

    676:         /* ... add i to row list for next nonzero entry */
    677:         il[i] = jmin;             /* update il(i) in column k+1, ... mbs-1 */
    678:         j     = bj[jmin];
    679:         jl[i] = jl[j]; jl[j] = i; /* update jl */
    680:       }
    681:       i = nexti;
    682:     }

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

    686:     /* invert diagonal block */
    687:     diag = ba+k*bs2;
    688:     PetscMemcpy(diag,dk,bs2*sizeof(MatScalar));

    690:     PetscKernel_A_gets_inverse_A(bs,diag,pivots,work,allowzeropivot,&zeropivotdetected);
    691:     if (zeropivotdetected) C->errortype = MAT_FACTOR_NUMERIC_ZEROPIVOT;

    693:     jmin = bi[k]; jmax = bi[k+1];
    694:     if (jmin < jmax) {
    695:       for (j=jmin; j<jmax; j++) {
    696:         vj       = bj[j];      /* block col. index of U */
    697:         u        = ba + j*bs2;
    698:         rtmp_ptr = rtmp + vj*bs2;
    699:         for (k1=0; k1<bs2; k1++) {
    700:           *u++        = *rtmp_ptr;
    701:           *rtmp_ptr++ = 0.0;
    702:         }
    703:       }

    705:       /* ... add k to row list for first nonzero entry in k-th row */
    706:       il[k] = jmin;
    707:       i     = bj[jmin];
    708:       jl[k] = jl[i]; jl[i] = k;
    709:     }
    710:   }

    712:   PetscFree(rtmp);
    713:   PetscFree2(il,jl);
    714:   PetscFree3(dk,uik,work);
    715:   PetscFree(pivots);
    716:   if (a->permute) {
    717:     PetscFree(aa);
    718:   }

    720:   ISRestoreIndices(perm,&perm_ptr);

    722:   C->ops->solve          = MatSolve_SeqSBAIJ_N_inplace;
    723:   C->ops->solvetranspose = MatSolve_SeqSBAIJ_N_inplace;
    724:   C->ops->forwardsolve   = MatForwardSolve_SeqSBAIJ_N_inplace;
    725:   C->ops->backwardsolve  = MatBackwardSolve_SeqSBAIJ_N_inplace;

    727:   C->assembled    = PETSC_TRUE;
    728:   C->preallocated = PETSC_TRUE;

    730:   PetscLogFlops(1.3333*bs*bs2*b->mbs); /* from inverting diagonal blocks */
    731:   return(0);
    732: }

    736: PetscErrorCode MatCholeskyFactorNumeric_SeqSBAIJ_N_NaturalOrdering(Mat C,Mat A,const MatFactorInfo *info)
    737: {
    738:   Mat_SeqSBAIJ   *a = (Mat_SeqSBAIJ*)A->data,*b = (Mat_SeqSBAIJ*)C->data;
    740:   PetscInt       i,j,mbs=a->mbs,*bi=b->i,*bj=b->j;
    741:   PetscInt       *ai,*aj,k,k1,jmin,jmax,*jl,*il,vj,nexti,ili;
    742:   PetscInt       bs  =A->rmap->bs,bs2 = a->bs2;
    743:   MatScalar      *ba = b->a,*aa,*ap,*dk,*uik;
    744:   MatScalar      *u,*diag,*rtmp,*rtmp_ptr;
    745:   MatScalar      *work;
    746:   PetscInt       *pivots;
    747:   PetscBool      allowzeropivot,zeropivotdetected;

    750:   PetscCalloc1(bs2*mbs,&rtmp);
    751:   PetscMalloc2(mbs,&il,mbs,&jl);
    752:   il[0] = 0;
    753:   for (i=0; i<mbs; i++) jl[i] = mbs;
    754:
    755:   PetscMalloc3(bs2,&dk,bs2,&uik,bs,&work);
    756:   PetscMalloc1(bs,&pivots);
    757:   allowzeropivot = PetscNot(A->erroriffailure);

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

    761:   /* for each row k */
    762:   for (k = 0; k<mbs; k++) {

    764:     /*initialize k-th row with elements nonzero in row k of A */
    765:     jmin = ai[k]; jmax = ai[k+1];
    766:     ap   = aa + jmin*bs2;
    767:     for (j = jmin; j < jmax; j++) {
    768:       vj       = aj[j];   /* block col. index */
    769:       rtmp_ptr = rtmp + vj*bs2;
    770:       for (i=0; i<bs2; i++) *rtmp_ptr++ = *ap++;
    771:     }

    773:     /* modify k-th row by adding in those rows i with U(i,k) != 0 */
    774:     PetscMemcpy(dk,rtmp+k*bs2,bs2*sizeof(MatScalar));
    775:     i    = jl[k]; /* first row to be added to k_th row  */

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

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

    783:       /* uik = -inv(Di)*U_bar(i,k) */
    784:       diag = ba + i*bs2;
    785:       u    = ba + ili*bs2;
    786:       PetscMemzero(uik,bs2*sizeof(MatScalar));
    787:       PetscKernel_A_gets_A_minus_B_times_C(bs,uik,diag,u);

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

    793:       /* update -U(i,k) */
    794:       PetscMemcpy(ba+ili*bs2,uik,bs2*sizeof(MatScalar));

    796:       /* add multiple of row i to k-th row ... */
    797:       jmin = ili + 1; jmax = bi[i+1];
    798:       if (jmin < jmax) {
    799:         for (j=jmin; j<jmax; j++) {
    800:           /* rtmp += -U(i,k)^T * U_bar(i,j) */
    801:           rtmp_ptr = rtmp + bj[j]*bs2;
    802:           u        = ba + j*bs2;
    803:           PetscKernel_A_gets_A_plus_Btranspose_times_C(bs,rtmp_ptr,uik,u);
    804:         }
    805:         PetscLogFlops(2.0*bs*bs2*(jmax-jmin));

    807:         /* ... add i to row list for next nonzero entry */
    808:         il[i] = jmin;             /* update il(i) in column k+1, ... mbs-1 */
    809:         j     = bj[jmin];
    810:         jl[i] = jl[j]; jl[j] = i; /* update jl */
    811:       }
    812:       i = nexti;
    813:     }

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

    817:     /* invert diagonal block */
    818:     diag = ba+k*bs2;
    819:     PetscMemcpy(diag,dk,bs2*sizeof(MatScalar));

    821:     PetscKernel_A_gets_inverse_A(bs,diag,pivots,work,allowzeropivot,&zeropivotdetected);
    822:     if (zeropivotdetected) C->errortype = MAT_FACTOR_NUMERIC_ZEROPIVOT;

    824:     jmin = bi[k]; jmax = bi[k+1];
    825:     if (jmin < jmax) {
    826:       for (j=jmin; j<jmax; j++) {
    827:         vj       = bj[j];      /* block col. index of U */
    828:         u        = ba + j*bs2;
    829:         rtmp_ptr = rtmp + vj*bs2;
    830:         for (k1=0; k1<bs2; k1++) {
    831:           *u++        = *rtmp_ptr;
    832:           *rtmp_ptr++ = 0.0;
    833:         }
    834:       }

    836:       /* ... add k to row list for first nonzero entry in k-th row */
    837:       il[k] = jmin;
    838:       i     = bj[jmin];
    839:       jl[k] = jl[i]; jl[i] = k;
    840:     }
    841:   }

    843:   PetscFree(rtmp);
    844:   PetscFree2(il,jl);
    845:   PetscFree3(dk,uik,work);
    846:   PetscFree(pivots);

    848:   C->ops->solve          = MatSolve_SeqSBAIJ_N_NaturalOrdering_inplace;
    849:   C->ops->solvetranspose = MatSolve_SeqSBAIJ_N_NaturalOrdering_inplace;
    850:   C->ops->forwardsolve   = MatForwardSolve_SeqSBAIJ_N_NaturalOrdering_inplace;
    851:   C->ops->backwardsolve  = MatBackwardSolve_SeqSBAIJ_N_NaturalOrdering_inplace;
    852:   C->assembled           = PETSC_TRUE;
    853:   C->preallocated        = PETSC_TRUE;

    855:   PetscLogFlops(1.3333*bs*bs2*b->mbs); /* from inverting diagonal blocks */
    856:   return(0);
    857: }

    859: /*
    860:     Numeric U^T*D*U factorization for SBAIJ format. Modified from SNF of YSMP.
    861:     Version for blocks 2 by 2.
    862: */
    865: PetscErrorCode MatCholeskyFactorNumeric_SeqSBAIJ_2(Mat C,Mat A,const MatFactorInfo *info)
    866: {
    867:   Mat_SeqSBAIJ   *a   = (Mat_SeqSBAIJ*)A->data,*b = (Mat_SeqSBAIJ*)C->data;
    868:   IS             perm = b->row;
    870:   const PetscInt *ai,*aj,*perm_ptr;
    871:   PetscInt       i,j,mbs=a->mbs,*bi=b->i,*bj=b->j;
    872:   PetscInt       *a2anew,k,k1,jmin,jmax,*jl,*il,vj,nexti,ili;
    873:   MatScalar      *ba = b->a,*aa,*ap;
    874:   MatScalar      *u,*diag,*rtmp,*rtmp_ptr,dk[4],uik[4];
    875:   PetscReal      shift = info->shiftamount;
    876:   PetscBool      allowzeropivot,zeropivotdetected;

    879:   allowzeropivot = PetscNot(A->erroriffailure);

    881:   /* initialization */
    882:   /* il and jl record the first nonzero element in each row of the accessing
    883:      window U(0:k, k:mbs-1).
    884:      jl:    list of rows to be added to uneliminated rows
    885:             i>= k: jl(i) is the first row to be added to row i
    886:             i<  k: jl(i) is the row following row i in some list of rows
    887:             jl(i) = mbs indicates the end of a list
    888:      il(i): points to the first nonzero element in columns k,...,mbs-1 of
    889:             row i of U */
    890:   PetscCalloc1(4*mbs,&rtmp);
    891:   PetscMalloc2(mbs,&il,mbs,&jl);
    892:   il[0] = 0;
    893:   for (i=0; i<mbs; i++) jl[i] = mbs;
    894:
    895:   ISGetIndices(perm,&perm_ptr);

    897:   /* check permutation */
    898:   if (!a->permute) {
    899:     ai = a->i; aj = a->j; aa = a->a;
    900:   } else {
    901:     ai   = a->inew; aj = a->jnew;
    902:     PetscMalloc1(4*ai[mbs],&aa);
    903:     PetscMemcpy(aa,a->a,4*ai[mbs]*sizeof(MatScalar));
    904:     PetscMalloc1(ai[mbs],&a2anew);
    905:     PetscMemcpy(a2anew,a->a2anew,(ai[mbs])*sizeof(PetscInt));

    907:     for (i=0; i<mbs; i++) {
    908:       jmin = ai[i]; jmax = ai[i+1];
    909:       for (j=jmin; j<jmax; j++) {
    910:         while (a2anew[j] != j) {
    911:           k = a2anew[j]; a2anew[j] = a2anew[k]; a2anew[k] = k;
    912:           for (k1=0; k1<4; k1++) {
    913:             dk[k1]     = aa[k*4+k1];
    914:             aa[k*4+k1] = aa[j*4+k1];
    915:             aa[j*4+k1] = dk[k1];
    916:           }
    917:         }
    918:         /* transform columnoriented blocks that lie in the lower triangle to roworiented blocks */
    919:         if (i > aj[j]) {
    920:           /* printf("change orientation, row: %d, col: %d\n",i,aj[j]); */
    921:           ap    = aa + j*4;  /* ptr to the beginning of the block */
    922:           dk[1] = ap[1];     /* swap ap[1] and ap[2] */
    923:           ap[1] = ap[2];
    924:           ap[2] = dk[1];
    925:         }
    926:       }
    927:     }
    928:     PetscFree(a2anew);
    929:   }

    931:   /* for each row k */
    932:   for (k = 0; k<mbs; k++) {

    934:     /*initialize k-th row with elements nonzero in row perm(k) of A */
    935:     jmin = ai[perm_ptr[k]]; jmax = ai[perm_ptr[k]+1];
    936:     ap   = aa + jmin*4;
    937:     for (j = jmin; j < jmax; j++) {
    938:       vj       = perm_ptr[aj[j]];   /* block col. index */
    939:       rtmp_ptr = rtmp + vj*4;
    940:       for (i=0; i<4; i++) *rtmp_ptr++ = *ap++;
    941:     }

    943:     /* modify k-th row by adding in those rows i with U(i,k) != 0 */
    944:     PetscMemcpy(dk,rtmp+k*4,4*sizeof(MatScalar));
    945:     i    = jl[k]; /* first row to be added to k_th row  */

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

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

    953:       /* uik = -inv(Di)*U_bar(i,k): - ba[ili]*ba[i] */
    954:       diag   = ba + i*4;
    955:       u      = ba + ili*4;
    956:       uik[0] = -(diag[0]*u[0] + diag[2]*u[1]);
    957:       uik[1] = -(diag[1]*u[0] + diag[3]*u[1]);
    958:       uik[2] = -(diag[0]*u[2] + diag[2]*u[3]);
    959:       uik[3] = -(diag[1]*u[2] + diag[3]*u[3]);

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

    967:       PetscLogFlops(16.0*2.0);

    969:       /* update -U(i,k): ba[ili] = uik */
    970:       PetscMemcpy(ba+ili*4,uik,4*sizeof(MatScalar));

    972:       /* add multiple of row i to k-th row ... */
    973:       jmin = ili + 1; jmax = bi[i+1];
    974:       if (jmin < jmax) {
    975:         for (j=jmin; j<jmax; j++) {
    976:           /* rtmp += -U(i,k)^T * U_bar(i,j): rtmp[bj[j]] += uik*ba[j]; */
    977:           rtmp_ptr     = rtmp + bj[j]*4;
    978:           u            = ba + j*4;
    979:           rtmp_ptr[0] += uik[0]*u[0] + uik[1]*u[1];
    980:           rtmp_ptr[1] += uik[2]*u[0] + uik[3]*u[1];
    981:           rtmp_ptr[2] += uik[0]*u[2] + uik[1]*u[3];
    982:           rtmp_ptr[3] += uik[2]*u[2] + uik[3]*u[3];
    983:         }
    984:         PetscLogFlops(16.0*(jmax-jmin));

    986:         /* ... add i to row list for next nonzero entry */
    987:         il[i] = jmin;             /* update il(i) in column k+1, ... mbs-1 */
    988:         j     = bj[jmin];
    989:         jl[i] = jl[j]; jl[j] = i; /* update jl */
    990:       }
    991:       i = nexti;
    992:     }

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

    996:     /* invert diagonal block */
    997:     diag = ba+k*4;
    998:     PetscMemcpy(diag,dk,4*sizeof(MatScalar));
    999:     PetscKernel_A_gets_inverse_A_2(diag,shift,allowzeropivot,&zeropivotdetected);
    1000:     if (zeropivotdetected) C->errortype = MAT_FACTOR_NUMERIC_ZEROPIVOT;

    1002:     jmin = bi[k]; jmax = bi[k+1];
    1003:     if (jmin < jmax) {
    1004:       for (j=jmin; j<jmax; j++) {
    1005:         vj       = bj[j];      /* block col. index of U */
    1006:         u        = ba + j*4;
    1007:         rtmp_ptr = rtmp + vj*4;
    1008:         for (k1=0; k1<4; k1++) {
    1009:           *u++        = *rtmp_ptr;
    1010:           *rtmp_ptr++ = 0.0;
    1011:         }
    1012:       }

    1014:       /* ... add k to row list for first nonzero entry in k-th row */
    1015:       il[k] = jmin;
    1016:       i     = bj[jmin];
    1017:       jl[k] = jl[i]; jl[i] = k;
    1018:     }
    1019:   }

    1021:   PetscFree(rtmp);
    1022:   PetscFree2(il,jl);
    1023:   if (a->permute) {
    1024:     PetscFree(aa);
    1025:   }
    1026:   ISRestoreIndices(perm,&perm_ptr);

    1028:   C->ops->solve          = MatSolve_SeqSBAIJ_2_inplace;
    1029:   C->ops->solvetranspose = MatSolve_SeqSBAIJ_2_inplace;
    1030:   C->assembled           = PETSC_TRUE;
    1031:   C->preallocated        = PETSC_TRUE;

    1033:   PetscLogFlops(1.3333*8*b->mbs); /* from inverting diagonal blocks */
    1034:   return(0);
    1035: }

    1037: /*
    1038:       Version for when blocks are 2 by 2 Using natural ordering
    1039: */
    1042: PetscErrorCode MatCholeskyFactorNumeric_SeqSBAIJ_2_NaturalOrdering(Mat C,Mat A,const MatFactorInfo *info)
    1043: {
    1044:   Mat_SeqSBAIJ   *a = (Mat_SeqSBAIJ*)A->data,*b = (Mat_SeqSBAIJ*)C->data;
    1046:   PetscInt       i,j,mbs=a->mbs,*bi=b->i,*bj=b->j;
    1047:   PetscInt       *ai,*aj,k,k1,jmin,jmax,*jl,*il,vj,nexti,ili;
    1048:   MatScalar      *ba = b->a,*aa,*ap,dk[8],uik[8];
    1049:   MatScalar      *u,*diag,*rtmp,*rtmp_ptr;
    1050:   PetscReal      shift = info->shiftamount;
    1051:   PetscBool      allowzeropivot,zeropivotdetected;

    1054:   allowzeropivot = PetscNot(A->erroriffailure);

    1056:   /* initialization */
    1057:   /* il and jl record the first nonzero element in each row of the accessing
    1058:      window U(0:k, k:mbs-1).
    1059:      jl:    list of rows to be added to uneliminated rows
    1060:             i>= k: jl(i) is the first row to be added to row i
    1061:             i<  k: jl(i) is the row following row i in some list of rows
    1062:             jl(i) = mbs indicates the end of a list
    1063:      il(i): points to the first nonzero element in columns k,...,mbs-1 of
    1064:             row i of U */
    1065:   PetscCalloc1(4*mbs,&rtmp);
    1066:   PetscMalloc2(mbs,&il,mbs,&jl);
    1067:   il[0] = 0;
    1068:   for (i=0; i<mbs; i++) jl[i] = mbs;
    1069:
    1070:   ai = a->i; aj = a->j; aa = a->a;

    1072:   /* for each row k */
    1073:   for (k = 0; k<mbs; k++) {

    1075:     /*initialize k-th row with elements nonzero in row k of A */
    1076:     jmin = ai[k]; jmax = ai[k+1];
    1077:     ap   = aa + jmin*4;
    1078:     for (j = jmin; j < jmax; j++) {
    1079:       vj       = aj[j];   /* block col. index */
    1080:       rtmp_ptr = rtmp + vj*4;
    1081:       for (i=0; i<4; i++) *rtmp_ptr++ = *ap++;
    1082:     }

    1084:     /* modify k-th row by adding in those rows i with U(i,k) != 0 */
    1085:     PetscMemcpy(dk,rtmp+k*4,4*sizeof(MatScalar));
    1086:     i    = jl[k]; /* first row to be added to k_th row  */

    1088:     while (i < k) {
    1089:       nexti = jl[i]; /* next row to be added to k_th row */

    1091:       /* compute multiplier */
    1092:       ili = il[i];  /* index of first nonzero element in U(i,k:bms-1) */

    1094:       /* uik = -inv(Di)*U_bar(i,k): - ba[ili]*ba[i] */
    1095:       diag   = ba + i*4;
    1096:       u      = ba + ili*4;
    1097:       uik[0] = -(diag[0]*u[0] + diag[2]*u[1]);
    1098:       uik[1] = -(diag[1]*u[0] + diag[3]*u[1]);
    1099:       uik[2] = -(diag[0]*u[2] + diag[2]*u[3]);
    1100:       uik[3] = -(diag[1]*u[2] + diag[3]*u[3]);

    1102:       /* update D(k) += -U(i,k)^T * U_bar(i,k): dk += uik*ba[ili] */
    1103:       dk[0] += uik[0]*u[0] + uik[1]*u[1];
    1104:       dk[1] += uik[2]*u[0] + uik[3]*u[1];
    1105:       dk[2] += uik[0]*u[2] + uik[1]*u[3];
    1106:       dk[3] += uik[2]*u[2] + uik[3]*u[3];

    1108:       PetscLogFlops(16.0*2.0);

    1110:       /* update -U(i,k): ba[ili] = uik */
    1111:       PetscMemcpy(ba+ili*4,uik,4*sizeof(MatScalar));

    1113:       /* add multiple of row i to k-th row ... */
    1114:       jmin = ili + 1; jmax = bi[i+1];
    1115:       if (jmin < jmax) {
    1116:         for (j=jmin; j<jmax; j++) {
    1117:           /* rtmp += -U(i,k)^T * U_bar(i,j): rtmp[bj[j]] += uik*ba[j]; */
    1118:           rtmp_ptr     = rtmp + bj[j]*4;
    1119:           u            = ba + j*4;
    1120:           rtmp_ptr[0] += uik[0]*u[0] + uik[1]*u[1];
    1121:           rtmp_ptr[1] += uik[2]*u[0] + uik[3]*u[1];
    1122:           rtmp_ptr[2] += uik[0]*u[2] + uik[1]*u[3];
    1123:           rtmp_ptr[3] += uik[2]*u[2] + uik[3]*u[3];
    1124:         }
    1125:         PetscLogFlops(16.0*(jmax-jmin));

    1127:         /* ... add i to row list for next nonzero entry */
    1128:         il[i] = jmin;             /* update il(i) in column k+1, ... mbs-1 */
    1129:         j     = bj[jmin];
    1130:         jl[i] = jl[j]; jl[j] = i; /* update jl */
    1131:       }
    1132:       i = nexti;
    1133:     }

    1135:     /* save nonzero entries in k-th row of U ... */

    1137:     /* invert diagonal block */
    1138:     diag = ba+k*4;
    1139:     PetscMemcpy(diag,dk,4*sizeof(MatScalar));
    1140:     PetscKernel_A_gets_inverse_A_2(diag,shift,allowzeropivot,&zeropivotdetected);
    1141:     if (zeropivotdetected) C->errortype = MAT_FACTOR_NUMERIC_ZEROPIVOT;

    1143:     jmin = bi[k]; jmax = bi[k+1];
    1144:     if (jmin < jmax) {
    1145:       for (j=jmin; j<jmax; j++) {
    1146:         vj       = bj[j];      /* block col. index of U */
    1147:         u        = ba + j*4;
    1148:         rtmp_ptr = rtmp + vj*4;
    1149:         for (k1=0; k1<4; k1++) {
    1150:           *u++        = *rtmp_ptr;
    1151:           *rtmp_ptr++ = 0.0;
    1152:         }
    1153:       }

    1155:       /* ... add k to row list for first nonzero entry in k-th row */
    1156:       il[k] = jmin;
    1157:       i     = bj[jmin];
    1158:       jl[k] = jl[i]; jl[i] = k;
    1159:     }
    1160:   }

    1162:   PetscFree(rtmp);
    1163:   PetscFree2(il,jl);

    1165:   C->ops->solve          = MatSolve_SeqSBAIJ_2_NaturalOrdering_inplace;
    1166:   C->ops->solvetranspose = MatSolve_SeqSBAIJ_2_NaturalOrdering_inplace;
    1167:   C->ops->forwardsolve   = MatForwardSolve_SeqSBAIJ_2_NaturalOrdering_inplace;
    1168:   C->ops->backwardsolve  = MatBackwardSolve_SeqSBAIJ_2_NaturalOrdering_inplace;
    1169:   C->assembled           = PETSC_TRUE;
    1170:   C->preallocated        = PETSC_TRUE;

    1172:   PetscLogFlops(1.3333*8*b->mbs); /* from inverting diagonal blocks */
    1173:   return(0);
    1174: }

    1176: /*
    1177:     Numeric U^T*D*U factorization for SBAIJ format.
    1178:     Version for blocks are 1 by 1.
    1179: */
    1182: PetscErrorCode MatCholeskyFactorNumeric_SeqSBAIJ_1_inplace(Mat C,Mat A,const MatFactorInfo *info)
    1183: {
    1184:   Mat_SeqSBAIJ   *a=(Mat_SeqSBAIJ*)A->data,*b=(Mat_SeqSBAIJ*)C->data;
    1185:   IS             ip=b->row;
    1187:   const PetscInt *ai,*aj,*rip;
    1188:   PetscInt       *a2anew,i,j,mbs=a->mbs,*bi=b->i,*bj=b->j,*bcol;
    1189:   PetscInt       k,jmin,jmax,*jl,*il,col,nexti,ili,nz;
    1190:   MatScalar      *rtmp,*ba=b->a,*bval,*aa,dk,uikdi;
    1191:   PetscReal      rs;
    1192:   FactorShiftCtx sctx;

    1195:   /* MatPivotSetUp(): initialize shift context sctx */
    1196:   PetscMemzero(&sctx,sizeof(FactorShiftCtx));

    1198:   ISGetIndices(ip,&rip);
    1199:   if (!a->permute) {
    1200:     ai = a->i; aj = a->j; aa = a->a;
    1201:   } else {
    1202:     ai     = a->inew; aj = a->jnew;
    1203:     nz     = ai[mbs];
    1204:     PetscMalloc1(nz,&aa);
    1205:     a2anew = a->a2anew;
    1206:     bval   = a->a;
    1207:     for (j=0; j<nz; j++) {
    1208:       aa[a2anew[j]] = *(bval++);
    1209:     }
    1210:   }

    1212:   /* initialization */
    1213:   /* il and jl record the first nonzero element in each row of the accessing
    1214:      window U(0:k, k:mbs-1).
    1215:      jl:    list of rows to be added to uneliminated rows
    1216:             i>= k: jl(i) is the first row to be added to row i
    1217:             i<  k: jl(i) is the row following row i in some list of rows
    1218:             jl(i) = mbs indicates the end of a list
    1219:      il(i): points to the first nonzero element in columns k,...,mbs-1 of
    1220:             row i of U */
    1221:   PetscMalloc3(mbs,&rtmp,mbs,&il,mbs,&jl);

    1223:   do {
    1224:     sctx.newshift = PETSC_FALSE;
    1225:     il[0] = 0;
    1226:     for (i=0; i<mbs; i++) {
    1227:       rtmp[i] = 0.0; jl[i] = mbs;
    1228:     }

    1230:     for (k = 0; k<mbs; k++) {
    1231:       /*initialize k-th row by the perm[k]-th row of A */
    1232:       jmin = ai[rip[k]]; jmax = ai[rip[k]+1];
    1233:       bval = ba + bi[k];
    1234:       for (j = jmin; j < jmax; j++) {
    1235:         col       = rip[aj[j]];
    1236:         rtmp[col] = aa[j];
    1237:         *bval++   = 0.0; /* for in-place factorization */
    1238:       }

    1240:       /* shift the diagonal of the matrix */
    1241:       if (sctx.nshift) rtmp[k] += sctx.shift_amount;

    1243:       /* modify k-th row by adding in those rows i with U(i,k)!=0 */
    1244:       dk = rtmp[k];
    1245:       i  = jl[k]; /* first row to be added to k_th row  */

    1247:       while (i < k) {
    1248:         nexti = jl[i]; /* next row to be added to k_th row */

    1250:         /* compute multiplier, update diag(k) and U(i,k) */
    1251:         ili     = il[i]; /* index of first nonzero element in U(i,k:bms-1) */
    1252:         uikdi   = -ba[ili]*ba[bi[i]]; /* diagonal(k) */
    1253:         dk     += uikdi*ba[ili];
    1254:         ba[ili] = uikdi; /* -U(i,k) */

    1256:         /* add multiple of row i to k-th row */
    1257:         jmin = ili + 1; jmax = bi[i+1];
    1258:         if (jmin < jmax) {
    1259:           for (j=jmin; j<jmax; j++) rtmp[bj[j]] += uikdi*ba[j];
    1260:           PetscLogFlops(2.0*(jmax-jmin));

    1262:           /* update il and jl for row i */
    1263:           il[i] = jmin;
    1264:           j     = bj[jmin]; jl[i] = jl[j]; jl[j] = i;
    1265:         }
    1266:         i = nexti;
    1267:       }

    1269:       /* shift the diagonals when zero pivot is detected */
    1270:       /* compute rs=sum of abs(off-diagonal) */
    1271:       rs   = 0.0;
    1272:       jmin = bi[k]+1;
    1273:       nz   = bi[k+1] - jmin;
    1274:       if (nz) {
    1275:         bcol = bj + jmin;
    1276:         while (nz--) {
    1277:           rs += PetscAbsScalar(rtmp[*bcol]);
    1278:           bcol++;
    1279:         }
    1280:       }

    1282:       sctx.rs = rs;
    1283:       sctx.pv = dk;
    1284:       MatPivotCheck(C,A,info,&sctx,k);
    1285:       if (sctx.newshift) break;    /* sctx.shift_amount is updated */
    1286:       dk = sctx.pv;

    1288:       /* copy data into U(k,:) */
    1289:       ba[bi[k]] = 1.0/dk; /* U(k,k) */
    1290:       jmin      = bi[k]+1; jmax = bi[k+1];
    1291:       if (jmin < jmax) {
    1292:         for (j=jmin; j<jmax; j++) {
    1293:           col = bj[j]; ba[j] = rtmp[col]; rtmp[col] = 0.0;
    1294:         }
    1295:         /* add the k-th row into il and jl */
    1296:         il[k] = jmin;
    1297:         i     = bj[jmin]; jl[k] = jl[i]; jl[i] = k;
    1298:       }
    1299:     }
    1300:   } while (sctx.newshift);
    1301:   PetscFree3(rtmp,il,jl);
    1302:   if (a->permute) {PetscFree(aa);}

    1304:   ISRestoreIndices(ip,&rip);

    1306:   C->ops->solve          = MatSolve_SeqSBAIJ_1_inplace;
    1307:   C->ops->solves         = MatSolves_SeqSBAIJ_1_inplace;
    1308:   C->ops->solvetranspose = MatSolve_SeqSBAIJ_1_inplace;
    1309:   C->ops->forwardsolve   = MatForwardSolve_SeqSBAIJ_1_inplace;
    1310:   C->ops->backwardsolve  = MatBackwardSolve_SeqSBAIJ_1_inplace;
    1311:   C->assembled           = PETSC_TRUE;
    1312:   C->preallocated        = PETSC_TRUE;

    1314:   PetscLogFlops(C->rmap->N);
    1315:   if (sctx.nshift) {
    1316:     if (info->shifttype == (PetscReal)MAT_SHIFT_NONZERO) {
    1317:       PetscInfo2(A,"number of shiftnz tries %D, shift_amount %g\n",sctx.nshift,(double)sctx.shift_amount);
    1318:     } else if (info->shifttype == (PetscReal)MAT_SHIFT_POSITIVE_DEFINITE) {
    1319:       PetscInfo2(A,"number of shiftpd tries %D, shift_amount %g\n",sctx.nshift,(double)sctx.shift_amount);
    1320:     }
    1321:   }
    1322:   return(0);
    1323: }

    1325: /*
    1326:   Version for when blocks are 1 by 1 Using natural ordering under new datastructure
    1327:   Modified from MatCholeskyFactorNumeric_SeqAIJ()
    1328: */
    1331: PetscErrorCode MatCholeskyFactorNumeric_SeqSBAIJ_1_NaturalOrdering(Mat B,Mat A,const MatFactorInfo *info)
    1332: {
    1333:   Mat_SeqSBAIJ   *a=(Mat_SeqSBAIJ*)A->data;
    1334:   Mat_SeqSBAIJ   *b=(Mat_SeqSBAIJ*)B->data;
    1336:   PetscInt       i,j,mbs=A->rmap->n,*bi=b->i,*bj=b->j,*bdiag=b->diag,*bjtmp;
    1337:   PetscInt       *ai=a->i,*aj=a->j,*ajtmp;
    1338:   PetscInt       k,jmin,jmax,*c2r,*il,col,nexti,ili,nz;
    1339:   MatScalar      *rtmp,*ba=b->a,*bval,*aa=a->a,dk,uikdi;
    1340:   FactorShiftCtx sctx;
    1341:   PetscReal      rs;
    1342:   MatScalar      d,*v;

    1345:   PetscMalloc3(mbs,&rtmp,mbs,&il,mbs,&c2r);

    1347:   /* MatPivotSetUp(): initialize shift context sctx */
    1348:   PetscMemzero(&sctx,sizeof(FactorShiftCtx));

    1350:   if (info->shifttype == (PetscReal)MAT_SHIFT_POSITIVE_DEFINITE) { /* set sctx.shift_top=max{rs} */
    1351:     sctx.shift_top = info->zeropivot;

    1353:     PetscMemzero(rtmp,mbs*sizeof(MatScalar));

    1355:     for (i=0; i<mbs; i++) {
    1356:       /* calculate sum(|aij|)-RealPart(aii), amt of shift needed for this row */
    1357:       d        = (aa)[a->diag[i]];
    1358:       rtmp[i] += -PetscRealPart(d);  /* diagonal entry */
    1359:       ajtmp    = aj + ai[i] + 1;     /* exclude diagonal */
    1360:       v        = aa + ai[i] + 1;
    1361:       nz       = ai[i+1] - ai[i] - 1;
    1362:       for (j=0; j<nz; j++) {
    1363:         rtmp[i]        += PetscAbsScalar(v[j]);
    1364:         rtmp[ajtmp[j]] += PetscAbsScalar(v[j]);
    1365:       }
    1366:       if (PetscRealPart(rtmp[i]) > sctx.shift_top) sctx.shift_top = PetscRealPart(rtmp[i]);
    1367:     }
    1368:     sctx.shift_top *= 1.1;
    1369:     sctx.nshift_max = 5;
    1370:     sctx.shift_lo   = 0.;
    1371:     sctx.shift_hi   = 1.;
    1372:   }

    1374:   /* allocate working arrays
    1375:      c2r: linked list, keep track of pivot rows for a given column. c2r[col]: head of the list for a given col
    1376:      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
    1377:   */
    1378:   do {
    1379:     sctx.newshift = PETSC_FALSE;

    1381:     for (i=0; i<mbs; i++) c2r[i] = mbs;
    1382:     if (mbs) il[0] = 0;

    1384:     for (k = 0; k<mbs; k++) {
    1385:       /* zero rtmp */
    1386:       nz    = bi[k+1] - bi[k];
    1387:       bjtmp = bj + bi[k];
    1388:       for (j=0; j<nz; j++) rtmp[bjtmp[j]] = 0.0;

    1390:       /* load in initial unfactored row */
    1391:       bval = ba + bi[k];
    1392:       jmin = ai[k]; jmax = ai[k+1];
    1393:       for (j = jmin; j < jmax; j++) {
    1394:         col       = aj[j];
    1395:         rtmp[col] = aa[j];
    1396:         *bval++   = 0.0; /* for in-place factorization */
    1397:       }
    1398:       /* shift the diagonal of the matrix: ZeropivotApply() */
    1399:       rtmp[k] += sctx.shift_amount;  /* shift the diagonal of the matrix */

    1401:       /* modify k-th row by adding in those rows i with U(i,k)!=0 */
    1402:       dk = rtmp[k];
    1403:       i  = c2r[k]; /* first row to be added to k_th row  */

    1405:       while (i < k) {
    1406:         nexti = c2r[i]; /* next row to be added to k_th row */

    1408:         /* compute multiplier, update diag(k) and U(i,k) */
    1409:         ili     = il[i]; /* index of first nonzero element in U(i,k:bms-1) */
    1410:         uikdi   = -ba[ili]*ba[bdiag[i]]; /* diagonal(k) */
    1411:         dk     += uikdi*ba[ili]; /* update diag[k] */
    1412:         ba[ili] = uikdi; /* -U(i,k) */

    1414:         /* add multiple of row i to k-th row */
    1415:         jmin = ili + 1; jmax = bi[i+1];
    1416:         if (jmin < jmax) {
    1417:           for (j=jmin; j<jmax; j++) rtmp[bj[j]] += uikdi*ba[j];
    1418:           /* update il and c2r for row i */
    1419:           il[i] = jmin;
    1420:           j     = bj[jmin]; c2r[i] = c2r[j]; c2r[j] = i;
    1421:         }
    1422:         i = nexti;
    1423:       }

    1425:       /* copy data into U(k,:) */
    1426:       rs   = 0.0;
    1427:       jmin = bi[k]; jmax = bi[k+1]-1;
    1428:       if (jmin < jmax) {
    1429:         for (j=jmin; j<jmax; j++) {
    1430:           col = bj[j]; ba[j] = rtmp[col]; rs += PetscAbsScalar(ba[j]);
    1431:         }
    1432:         /* add the k-th row into il and c2r */
    1433:         il[k] = jmin;
    1434:         i     = bj[jmin]; c2r[k] = c2r[i]; c2r[i] = k;
    1435:       }

    1437:       sctx.rs = rs;
    1438:       sctx.pv = dk;
    1439:       MatPivotCheck(B,A,info,&sctx,k);
    1440:       if (sctx.newshift) break;
    1441:       dk = sctx.pv;

    1443:       ba[bdiag[k]] = 1.0/dk; /* U(k,k) */
    1444:     }
    1445:   } while (sctx.newshift);

    1447:   PetscFree3(rtmp,il,c2r);

    1449:   B->ops->solve          = MatSolve_SeqSBAIJ_1_NaturalOrdering;
    1450:   B->ops->solves         = MatSolves_SeqSBAIJ_1;
    1451:   B->ops->solvetranspose = MatSolve_SeqSBAIJ_1_NaturalOrdering;
    1452:   B->ops->forwardsolve   = MatForwardSolve_SeqSBAIJ_1_NaturalOrdering;
    1453:   B->ops->backwardsolve  = MatBackwardSolve_SeqSBAIJ_1_NaturalOrdering;

    1455:   B->assembled    = PETSC_TRUE;
    1456:   B->preallocated = PETSC_TRUE;

    1458:   PetscLogFlops(B->rmap->n);

    1460:   /* MatPivotView() */
    1461:   if (sctx.nshift) {
    1462:     if (info->shifttype == (PetscReal)MAT_SHIFT_POSITIVE_DEFINITE) {
    1463:       PetscInfo4(A,"number of shift_pd tries %D, 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);
    1464:     } else if (info->shifttype == (PetscReal)MAT_SHIFT_NONZERO) {
    1465:       PetscInfo2(A,"number of shift_nz tries %D, shift_amount %g\n",sctx.nshift,(double)sctx.shift_amount);
    1466:     } else if (info->shifttype == (PetscReal)MAT_SHIFT_INBLOCKS) {
    1467:       PetscInfo2(A,"number of shift_inblocks applied %D, each shift_amount %g\n",sctx.nshift,(double)info->shiftamount);
    1468:     }
    1469:   }
    1470:   return(0);
    1471: }

    1475: PetscErrorCode MatCholeskyFactorNumeric_SeqSBAIJ_1_NaturalOrdering_inplace(Mat C,Mat A,const MatFactorInfo *info)
    1476: {
    1477:   Mat_SeqSBAIJ   *a=(Mat_SeqSBAIJ*)A->data,*b=(Mat_SeqSBAIJ*)C->data;
    1479:   PetscInt       i,j,mbs = a->mbs;
    1480:   PetscInt       *ai=a->i,*aj=a->j,*bi=b->i,*bj=b->j;
    1481:   PetscInt       k,jmin,*jl,*il,nexti,ili,*acol,*bcol,nz;
    1482:   MatScalar      *rtmp,*ba=b->a,*aa=a->a,dk,uikdi,*aval,*bval;
    1483:   PetscReal      rs;
    1484:   FactorShiftCtx sctx;

    1487:   /* MatPivotSetUp(): initialize shift context sctx */
    1488:   PetscMemzero(&sctx,sizeof(FactorShiftCtx));

    1490:   /* initialization */
    1491:   /* il and jl record the first nonzero element in each row of the accessing
    1492:      window U(0:k, k:mbs-1).
    1493:      jl:    list of rows to be added to uneliminated rows
    1494:             i>= k: jl(i) is the first row to be added to row i
    1495:             i<  k: jl(i) is the row following row i in some list of rows
    1496:             jl(i) = mbs indicates the end of a list
    1497:      il(i): points to the first nonzero element in U(i,k:mbs-1)
    1498:   */
    1499:   PetscMalloc1(mbs,&rtmp);
    1500:   PetscMalloc2(mbs,&il,mbs,&jl);

    1502:   do {
    1503:     sctx.newshift = PETSC_FALSE;
    1504:     il[0] = 0;
    1505:     for (i=0; i<mbs; i++) {
    1506:       rtmp[i] = 0.0; jl[i] = mbs;
    1507:     }

    1509:     for (k = 0; k<mbs; k++) {
    1510:       /*initialize k-th row with elements nonzero in row perm(k) of A */
    1511:       nz   = ai[k+1] - ai[k];
    1512:       acol = aj + ai[k];
    1513:       aval = aa + ai[k];
    1514:       bval = ba + bi[k];
    1515:       while (nz--) {
    1516:         rtmp[*acol++] = *aval++;
    1517:         *bval++       = 0.0; /* for in-place factorization */
    1518:       }

    1520:       /* shift the diagonal of the matrix */
    1521:       if (sctx.nshift) rtmp[k] += sctx.shift_amount;

    1523:       /* modify k-th row by adding in those rows i with U(i,k)!=0 */
    1524:       dk = rtmp[k];
    1525:       i  = jl[k]; /* first row to be added to k_th row  */

    1527:       while (i < k) {
    1528:         nexti = jl[i]; /* next row to be added to k_th row */
    1529:         /* compute multiplier, update D(k) and U(i,k) */
    1530:         ili     = il[i]; /* index of first nonzero element in U(i,k:bms-1) */
    1531:         uikdi   = -ba[ili]*ba[bi[i]];
    1532:         dk     += uikdi*ba[ili];
    1533:         ba[ili] = uikdi; /* -U(i,k) */

    1535:         /* add multiple of row i to k-th row ... */
    1536:         jmin = ili + 1;
    1537:         nz   = bi[i+1] - jmin;
    1538:         if (nz > 0) {
    1539:           bcol = bj + jmin;
    1540:           bval = ba + jmin;
    1541:           PetscLogFlops(2.0*nz);
    1542:           while (nz--) rtmp[*bcol++] += uikdi*(*bval++);

    1544:           /* update il and jl for i-th row */
    1545:           il[i] = jmin;
    1546:           j     = bj[jmin]; jl[i] = jl[j]; jl[j] = i;
    1547:         }
    1548:         i = nexti;
    1549:       }

    1551:       /* shift the diagonals when zero pivot is detected */
    1552:       /* compute rs=sum of abs(off-diagonal) */
    1553:       rs   = 0.0;
    1554:       jmin = bi[k]+1;
    1555:       nz   = bi[k+1] - jmin;
    1556:       if (nz) {
    1557:         bcol = bj + jmin;
    1558:         while (nz--) {
    1559:           rs += PetscAbsScalar(rtmp[*bcol]);
    1560:           bcol++;
    1561:         }
    1562:       }

    1564:       sctx.rs = rs;
    1565:       sctx.pv = dk;
    1566:       MatPivotCheck(C,A,info,&sctx,k);
    1567:       if (sctx.newshift) break;    /* sctx.shift_amount is updated */
    1568:       dk = sctx.pv;

    1570:       /* copy data into U(k,:) */
    1571:       ba[bi[k]] = 1.0/dk;
    1572:       jmin      = bi[k]+1;
    1573:       nz        = bi[k+1] - jmin;
    1574:       if (nz) {
    1575:         bcol = bj + jmin;
    1576:         bval = ba + jmin;
    1577:         while (nz--) {
    1578:           *bval++       = rtmp[*bcol];
    1579:           rtmp[*bcol++] = 0.0;
    1580:         }
    1581:         /* add k-th row into il and jl */
    1582:         il[k] = jmin;
    1583:         i     = bj[jmin]; jl[k] = jl[i]; jl[i] = k;
    1584:       }
    1585:     } /* end of for (k = 0; k<mbs; k++) */
    1586:   } while (sctx.newshift);
    1587:   PetscFree(rtmp);
    1588:   PetscFree2(il,jl);

    1590:   C->ops->solve          = MatSolve_SeqSBAIJ_1_NaturalOrdering_inplace;
    1591:   C->ops->solves         = MatSolves_SeqSBAIJ_1_inplace;
    1592:   C->ops->solvetranspose = MatSolve_SeqSBAIJ_1_NaturalOrdering_inplace;
    1593:   C->ops->forwardsolve   = MatForwardSolve_SeqSBAIJ_1_NaturalOrdering_inplace;
    1594:   C->ops->backwardsolve  = MatBackwardSolve_SeqSBAIJ_1_NaturalOrdering_inplace;

    1596:   C->assembled    = PETSC_TRUE;
    1597:   C->preallocated = PETSC_TRUE;

    1599:   PetscLogFlops(C->rmap->N);
    1600:   if (sctx.nshift) {
    1601:     if (info->shifttype == (PetscReal)MAT_SHIFT_NONZERO) {
    1602:       PetscInfo2(A,"number of shiftnz tries %D, shift_amount %g\n",sctx.nshift,(double)sctx.shift_amount);
    1603:     } else if (info->shifttype == (PetscReal)MAT_SHIFT_POSITIVE_DEFINITE) {
    1604:       PetscInfo2(A,"number of shiftpd tries %D, shift_amount %g\n",sctx.nshift,(double)sctx.shift_amount);
    1605:     }
    1606:   }
    1607:   return(0);
    1608: }

    1612: PetscErrorCode MatCholeskyFactor_SeqSBAIJ(Mat A,IS perm,const MatFactorInfo *info)
    1613: {
    1615:   Mat            C;

    1618:   MatGetFactor(A,"petsc",MAT_FACTOR_CHOLESKY,&C);
    1619:   MatCholeskyFactorSymbolic(C,A,perm,info);
    1620:   MatCholeskyFactorNumeric(C,A,info);

    1622:   A->ops->solve          = C->ops->solve;
    1623:   A->ops->solvetranspose = C->ops->solvetranspose;

    1625:   MatHeaderMerge(A,&C);
    1626:   return(0);
    1627: }