xref: /petsc/src/mat/impls/aij/seq/aijfact.c (revision c7ef6ce2c37faeeecf5bb5fa630bca9d6fdfea17)
1 #define PETSCMAT_DLL
2 
3 #include "../src/mat/impls/aij/seq/aij.h"
4 #include "../src/inline/dot.h"
5 #define PETSC_USE_WHILE_KERNELS
6 #include "../src/inline/spops.h"
7 #include "petscbt.h"
8 #include "../src/mat/utils/freespace.h"
9 
10 EXTERN_C_BEGIN
11 #undef __FUNCT__
12 #define __FUNCT__ "MatOrdering_Flow_SeqAIJ"
13 /*
14       Computes an ordering to get most of the large numerical values in the lower triangular part of the matrix
15 */
16 PetscErrorCode MatOrdering_Flow_SeqAIJ(Mat mat,const MatOrderingType type,IS *irow,IS *icol)
17 {
18   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)mat->data;
19   PetscErrorCode    ierr;
20   PetscInt          i,j,jj,k, kk,n = mat->rmap->n, current = 0, newcurrent = 0,*order;
21   const PetscInt    *ai = a->i, *aj = a->j;
22   const PetscScalar *aa = a->a;
23   PetscTruth        *done;
24   PetscReal         best,past = 0,future;
25 
26   PetscFunctionBegin;
27   /* pick initial row */
28   best = -1;
29   for (i=0; i<n; i++) {
30     future = 0;
31     for (j=ai[i]; j<ai[i+1]; j++) {
32       if (aj[j] != i) future  += PetscAbsScalar(aa[j]); else past = PetscAbsScalar(aa[j]);
33     }
34     if (!future) future = 1.e-10; /* if there is zero in the upper diagonal part want to rank this row high */
35     if (past/future > best) {
36       best = past/future;
37       current = i;
38     }
39   }
40 
41   ierr = PetscMalloc(n*sizeof(PetscTruth),&done);CHKERRQ(ierr);
42   ierr = PetscMalloc(n*sizeof(PetscInt),&order);CHKERRQ(ierr);
43   ierr = PetscMemzero(done,n*sizeof(PetscTruth));CHKERRQ(ierr);
44   order[0] = current;
45   for (i=0; i<n-1; i++) {
46     done[current] = PETSC_TRUE;
47     best          = -1;
48     /* loop over all neighbors of current pivot */
49     for (j=ai[current]; j<ai[current+1]; j++) {
50       jj = aj[j];
51       if (done[jj]) continue;
52       /* loop over columns of potential next row computing weights for below and above diagonal */
53       past = future = 0.0;
54       for (k=ai[jj]; k<ai[jj+1]; k++) {
55         kk = aj[k];
56         if (done[kk]) past += PetscAbsScalar(aa[k]);
57         else if (kk != jj) future  += PetscAbsScalar(aa[k]);
58       }
59       if (!future) future = 1.e-10; /* if there is zero in the upper diagonal part want to rank this row high */
60       if (past/future > best) {
61         best = past/future;
62         newcurrent = jj;
63       }
64     }
65     if (best == -1) { /* no neighbors to select from so select best of all that remain */
66       best = -1;
67       for (k=0; k<n; k++) {
68         if (done[k]) continue;
69         future = 0;
70         past   = 0;
71         for (j=ai[k]; j<ai[k+1]; j++) {
72           kk = aj[j];
73           if (done[kk]) past += PetscAbsScalar(aa[j]);
74           else if (kk != k) future  += PetscAbsScalar(aa[j]);
75         }
76         if (!future) future = 1.e-10; /* if there is zero in the upper diagonal part want to rank this row high */
77         if (past/future > best) {
78           best = past/future;
79           newcurrent = k;
80         }
81       }
82     }
83     if (current == newcurrent) SETERRQ(PETSC_ERR_PLIB,"newcurrent cannot be current");
84     current = newcurrent;
85     order[i+1] = current;
86   }
87   ierr = ISCreateGeneral(PETSC_COMM_SELF,n,order,irow);CHKERRQ(ierr);
88   *icol = *irow;
89   ierr = PetscObjectReference((PetscObject)*irow);CHKERRQ(ierr);
90   ierr = PetscFree(done);CHKERRQ(ierr);
91   ierr = PetscFree(order);CHKERRQ(ierr);
92   PetscFunctionReturn(0);
93 }
94 EXTERN_C_END
95 
96 EXTERN_C_BEGIN
97 #undef __FUNCT__
98 #define __FUNCT__ "MatGetFactorAvailable_seqaij_petsc"
99 PetscErrorCode MatGetFactorAvailable_seqaij_petsc(Mat A,MatFactorType ftype,PetscTruth *flg)
100 {
101   PetscFunctionBegin;
102   *flg = PETSC_TRUE;
103   PetscFunctionReturn(0);
104 }
105 EXTERN_C_END
106 
107 EXTERN_C_BEGIN
108 #undef __FUNCT__
109 #define __FUNCT__ "MatGetFactor_seqaij_petsc"
110 PetscErrorCode MatGetFactor_seqaij_petsc(Mat A,MatFactorType ftype,Mat *B)
111 {
112   PetscInt           n = A->rmap->n;
113   PetscErrorCode     ierr;
114 
115   PetscFunctionBegin;
116   ierr = MatCreate(((PetscObject)A)->comm,B);CHKERRQ(ierr);
117   ierr = MatSetSizes(*B,n,n,n,n);CHKERRQ(ierr);
118   if (ftype == MAT_FACTOR_LU || ftype == MAT_FACTOR_ILU || ftype == MAT_FACTOR_ILUDT){
119     ierr = MatSetType(*B,MATSEQAIJ);CHKERRQ(ierr);
120     (*B)->ops->ilufactorsymbolic = MatILUFactorSymbolic_SeqAIJ;
121     (*B)->ops->lufactorsymbolic  = MatLUFactorSymbolic_SeqAIJ;
122     (*B)->ops->iludtfactor       = MatILUDTFactor_SeqAIJ;
123   } else if (ftype == MAT_FACTOR_CHOLESKY || ftype == MAT_FACTOR_ICC) {
124     ierr = MatSetType(*B,MATSEQSBAIJ);CHKERRQ(ierr);
125     ierr = MatSeqSBAIJSetPreallocation(*B,1,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
126     (*B)->ops->iccfactorsymbolic      = MatICCFactorSymbolic_SeqAIJ;
127     (*B)->ops->choleskyfactorsymbolic = MatCholeskyFactorSymbolic_SeqAIJ;
128   } else SETERRQ(PETSC_ERR_SUP,"Factor type not supported");
129   (*B)->factor = ftype;
130   PetscFunctionReturn(0);
131 }
132 EXTERN_C_END
133 
134 #undef __FUNCT__
135 #define __FUNCT__ "MatLUFactorSymbolic_SeqAIJ"
136 PetscErrorCode MatLUFactorSymbolic_SeqAIJ(Mat B,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
137 {
138   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
139   IS                 isicol;
140   PetscErrorCode     ierr;
141   const PetscInt     *r,*ic;
142   PetscInt           i,n=A->rmap->n,*ai=a->i,*aj=a->j;
143   PetscInt           *bi,*bj,*ajtmp;
144   PetscInt           *bdiag,row,nnz,nzi,reallocs=0,nzbd,*im;
145   PetscReal          f;
146   PetscInt           nlnk,*lnk,k,**bi_ptr;
147   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
148   PetscBT            lnkbt;
149 
150   PetscFunctionBegin;
151   if (A->rmap->N != A->cmap->N) SETERRQ(PETSC_ERR_ARG_WRONG,"matrix must be square");
152   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
153   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
154   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
155 
156   /* get new row pointers */
157   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
158   bi[0] = 0;
159 
160   /* bdiag is location of diagonal in factor */
161   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);
162   bdiag[0] = 0;
163 
164   /* linked list for storing column indices of the active row */
165   nlnk = n + 1;
166   ierr = PetscLLCreate(n,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
167 
168   ierr = PetscMalloc2(n+1,PetscInt**,&bi_ptr,n+1,PetscInt,&im);CHKERRQ(ierr);
169 
170   /* initial FreeSpace size is f*(ai[n]+1) */
171   f = info->fill;
172   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space);CHKERRQ(ierr);
173   current_space = free_space;
174 
175   for (i=0; i<n; i++) {
176     /* copy previous fill into linked list */
177     nzi = 0;
178     nnz = ai[r[i]+1] - ai[r[i]];
179     if (!nnz) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
180     ajtmp = aj + ai[r[i]];
181     ierr = PetscLLAddPerm(nnz,ajtmp,ic,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
182     nzi += nlnk;
183 
184     /* add pivot rows into linked list */
185     row = lnk[n];
186     while (row < i) {
187       nzbd    = bdiag[row] - bi[row] + 1; /* num of entries in the row with column index <= row */
188       ajtmp   = bi_ptr[row] + nzbd; /* points to the entry next to the diagonal */
189       ierr = PetscLLAddSortedLU(ajtmp,row,nlnk,lnk,lnkbt,i,nzbd,im);CHKERRQ(ierr);
190       nzi += nlnk;
191       row  = lnk[row];
192     }
193     bi[i+1] = bi[i] + nzi;
194     im[i]   = nzi;
195 
196     /* mark bdiag */
197     nzbd = 0;
198     nnz  = nzi;
199     k    = lnk[n];
200     while (nnz-- && k < i){
201       nzbd++;
202       k = lnk[k];
203     }
204     bdiag[i] = bi[i] + nzbd;
205 
206     /* if free space is not available, make more free space */
207     if (current_space->local_remaining<nzi) {
208       nnz = (n - i)*nzi; /* estimated and max additional space needed */
209       ierr = PetscFreeSpaceGet(nnz,&current_space);CHKERRQ(ierr);
210       reallocs++;
211     }
212 
213     /* copy data into free space, then initialize lnk */
214     ierr = PetscLLClean(n,n,nzi,lnk,current_space->array,lnkbt);CHKERRQ(ierr);
215     bi_ptr[i] = current_space->array;
216     current_space->array           += nzi;
217     current_space->local_used      += nzi;
218     current_space->local_remaining -= nzi;
219   }
220 #if defined(PETSC_USE_INFO)
221   if (ai[n] != 0) {
222     PetscReal af = ((PetscReal)bi[n])/((PetscReal)ai[n]);
223     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,f,af);CHKERRQ(ierr);
224     ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
225     ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G);\n",af);CHKERRQ(ierr);
226     ierr = PetscInfo(A,"for best performance.\n");CHKERRQ(ierr);
227   } else {
228     ierr = PetscInfo(A,"Empty matrix\n");CHKERRQ(ierr);
229   }
230 #endif
231 
232   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
233   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
234 
235   /* destroy list of free space and other temporary array(s) */
236   ierr = PetscMalloc((bi[n]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
237   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
238   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
239   ierr = PetscFree2(bi_ptr,im);CHKERRQ(ierr);
240 
241   /* put together the new matrix */
242   ierr = MatSeqAIJSetPreallocation_SeqAIJ(B,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
243   ierr = PetscLogObjectParent(B,isicol);CHKERRQ(ierr);
244   b    = (Mat_SeqAIJ*)(B)->data;
245   b->free_a       = PETSC_TRUE;
246   b->free_ij      = PETSC_TRUE;
247   b->singlemalloc = PETSC_FALSE;
248   ierr          = PetscMalloc((bi[n]+1)*sizeof(PetscScalar),&b->a);CHKERRQ(ierr);
249   b->j          = bj;
250   b->i          = bi;
251   b->diag       = bdiag;
252   b->ilen       = 0;
253   b->imax       = 0;
254   b->row        = isrow;
255   b->col        = iscol;
256   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
257   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
258   b->icol       = isicol;
259   ierr          = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
260 
261   /* In b structure:  Free imax, ilen, old a, old j.  Allocate solve_work, new a, new j */
262   ierr = PetscLogObjectMemory(B,(bi[n]-n)*(sizeof(PetscInt)+sizeof(PetscScalar)));CHKERRQ(ierr);
263   b->maxnz = b->nz = bi[n] ;
264 
265   (B)->factor                = MAT_FACTOR_LU;
266   (B)->info.factor_mallocs   = reallocs;
267   (B)->info.fill_ratio_given = f;
268 
269   if (ai[n]) {
270     (B)->info.fill_ratio_needed = ((PetscReal)bi[n])/((PetscReal)ai[n]);
271   } else {
272     (B)->info.fill_ratio_needed = 0.0;
273   }
274   (B)->ops->lufactornumeric  = MatLUFactorNumeric_SeqAIJ;
275   (B)->ops->solve            = MatSolve_SeqAIJ;
276   (B)->ops->solvetranspose   = MatSolveTranspose_SeqAIJ;
277   /* switch to inodes if appropriate */
278   ierr = MatLUFactorSymbolic_Inode(B,A,isrow,iscol,info);CHKERRQ(ierr);
279   PetscFunctionReturn(0);
280 }
281 
282 /*
283     Trouble in factorization, should we dump the original matrix?
284 */
285 #undef __FUNCT__
286 #define __FUNCT__ "MatFactorDumpMatrix"
287 PetscErrorCode MatFactorDumpMatrix(Mat A)
288 {
289   PetscErrorCode ierr;
290   PetscTruth     flg = PETSC_FALSE;
291 
292   PetscFunctionBegin;
293   ierr = PetscOptionsGetTruth(PETSC_NULL,"-mat_factor_dump_on_error",&flg,PETSC_NULL);CHKERRQ(ierr);
294   if (flg) {
295     PetscViewer viewer;
296     char        filename[PETSC_MAX_PATH_LEN];
297 
298     ierr = PetscSNPrintf(filename,PETSC_MAX_PATH_LEN,"matrix_factor_error.%d",PetscGlobalRank);CHKERRQ(ierr);
299     ierr = PetscViewerBinaryOpen(((PetscObject)A)->comm,filename,FILE_MODE_WRITE,&viewer);CHKERRQ(ierr);
300     ierr = MatView(A,viewer);CHKERRQ(ierr);
301     ierr = PetscViewerDestroy(viewer);CHKERRQ(ierr);
302   }
303   PetscFunctionReturn(0);
304 }
305 
306 extern PetscErrorCode MatSolve_Inode(Mat,Vec,Vec);
307 
308 /* ----------------------------------------------------------- */
309 extern PetscErrorCode MatSolve_SeqAIJ_NaturalOrdering_iludt(Mat,Vec,Vec);
310 extern PetscErrorCode MatSolve_SeqAIJ_iludt(Mat,Vec,Vec);
311 
312 #undef __FUNCT__
313 #define __FUNCT__ "MatLUFactorNumeric_SeqAIJ_newdatastruct"
314 PetscErrorCode MatLUFactorNumeric_SeqAIJ_newdatastruct(Mat B,Mat A,const MatFactorInfo *info)
315 {
316   Mat            C=B;
317   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ *)C->data;
318   IS             isrow = b->row,isicol = b->icol;
319   PetscErrorCode ierr;
320   const PetscInt *r,*ic,*ics;
321   PetscInt       i,j,k,n=A->rmap->n,*ai=a->i,*aj=a->j,*bi=b->i,*bj=b->j;
322   PetscInt       *ajtmp,*bjtmp,nz,nzL,row,*bdiag=b->diag,*pj;
323   MatScalar      *rtmp,*pc,multiplier,*v,*pv,*aa=a->a;
324   PetscReal      shift=info->shiftinblocks;
325   PetscTruth     row_identity, col_identity;
326 
327   PetscFunctionBegin;
328   /* printf("MatLUFactorNumeric_SeqAIJ_newdatastruct is called ...\n"); */
329   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
330   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
331   ierr = PetscMalloc((n+1)*sizeof(MatScalar),&rtmp);CHKERRQ(ierr);
332   ics  = ic;
333 
334   for (i=0; i<n; i++){
335     /* zero rtmp */
336     /* L part */
337     nz    = bi[i+1] - bi[i];
338     bjtmp = bj + bi[i];
339     for  (j=0; j<nz; j++) rtmp[bjtmp[j]] = 0.0;
340 
341     /* U part */
342     /* nz = bi[2*n-i] - bi[2*n-i - 1] - 1; */
343     nz = bdiag[i] - bdiag[i+1];
344     bjtmp = bj + bdiag[i+1]+1; /* bi[2*n-i - 1]+1; */
345     for  (j=0; j<nz; j++) rtmp[bjtmp[j]] = 0.0;
346 
347     /* load in initial (unfactored row) */
348     nz    = ai[r[i]+1] - ai[r[i]];
349     ajtmp = aj + ai[r[i]];
350     v     = aa + ai[r[i]];
351     for (j=0; j<nz; j++) {
352       rtmp[ics[ajtmp[j]]] = v[j];
353     }
354     if (rtmp[ics[r[i]]] == 0.0){
355       rtmp[ics[r[i]]] += shift; /* shift the diagonal of the matrix */
356       /* printf("row %d, shift %g\n",i,shift); */
357     }
358 
359     /* elimination */
360     bjtmp = bj + bi[i];
361     row   = *bjtmp++;
362     nzL   = bi[i+1] - bi[i];
363     k   = 0;
364     while  (k < nzL) {
365       pc = rtmp + row;
366       if (*pc != 0.0) {
367         pv         = b->a + bdiag[row];
368         multiplier = *pc * (*pv);
369         *pc        = multiplier;
370         pj         = b->j + bdiag[row+1] + 1;
371         pv         = b->a + bdiag[row+1] + 1;
372         nz         = bdiag[row] - bdiag[row+1] - 1; /* num of entries in U(row,:), excluding diag */
373         for (j=0; j<nz; j++) rtmp[pj[j]] -= multiplier * pv[j];
374         ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr);
375       }
376       row = *bjtmp++; k++;
377     }
378 
379     /* finished row so stick it into b->a */
380     /* L part */
381     pv   = b->a + bi[i] ;
382     pj   = b->j + bi[i] ;
383     nz   = bi[i+1] - bi[i];
384     for (j=0; j<nz; j++) {
385       pv[j] = rtmp[pj[j]];
386     }
387 
388     /* Mark diagonal and invert diagonal for simplier triangular solves */
389     pv  = b->a + bdiag[i];
390     pj  = b->j + bdiag[i];
391     /* if (*pj != i)SETERRQ2(PETSC_ERR_SUP,"row %d != *pj %d",i,*pj) */
392     *pv = 1.0/rtmp[*pj];
393 
394     /* U part */
395     pv   = b->a + bdiag[i+1]+1;
396     pj   = b->j + bdiag[i+1]+1;
397     nz   = bdiag[i] - bdiag[i+1]-1;
398     for (j=0; j<nz; j++) {
399       pv[j] = rtmp[pj[j]];
400     }
401 
402   }
403   ierr = PetscFree(rtmp);CHKERRQ(ierr);
404   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
405   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
406 
407 
408   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
409   ierr = ISIdentity(isicol,&col_identity);CHKERRQ(ierr);
410   if (row_identity && col_identity) {
411     C->ops->solve = MatSolve_SeqAIJ_NaturalOrdering_iludt;
412   } else {
413     C->ops->solve = MatSolve_SeqAIJ_iludt;
414   }
415 
416   C->ops->solveadd           = 0;
417   C->ops->solvetranspose     = 0;
418   C->ops->solvetransposeadd  = 0;
419   C->ops->matsolve           = 0;
420   C->assembled    = PETSC_TRUE;
421   C->preallocated = PETSC_TRUE;
422   ierr = PetscLogFlops(C->cmap->n);CHKERRQ(ierr);
423   PetscFunctionReturn(0);
424 }
425 
426 #undef __FUNCT__
427 #define __FUNCT__ "MatLUFactorNumeric_SeqAIJ"
428 PetscErrorCode MatLUFactorNumeric_SeqAIJ(Mat B,Mat A,const MatFactorInfo *info)
429 {
430   Mat            C=B;
431   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ *)C->data;
432   IS             isrow = b->row,isicol = b->icol;
433   PetscErrorCode ierr;
434   const PetscInt  *r,*ic,*ics;
435   PetscInt       i,j,n=A->rmap->n,*ai=a->i,*aj=a->j,*bi=b->i,*bj=b->j;
436   PetscInt       *ajtmp,*bjtmp,nz,row,*diag_offset = b->diag,diag,*pj;
437   MatScalar      *rtmp,*pc,multiplier,*v,*pv,d,*aa=a->a;
438   PetscReal      rs=0.0;
439   LUShift_Ctx    sctx;
440   PetscInt       newshift,*ddiag;
441 
442   PetscFunctionBegin;
443   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
444   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
445   ierr = PetscMalloc((n+1)*sizeof(MatScalar),&rtmp);CHKERRQ(ierr);
446   ics  = ic;
447 
448   sctx.shift_top      = 0;
449   sctx.nshift_max     = 0;
450   sctx.shift_lo       = 0;
451   sctx.shift_hi       = 0;
452   sctx.shift_fraction = 0;
453 
454   /* if both shift schemes are chosen by user, only use info->shiftpd */
455   if (info->shiftpd) { /* set sctx.shift_top=max{rs} */
456     ddiag          = a->diag;
457     sctx.shift_top = info->zeropivot;
458     for (i=0; i<n; i++) {
459       /* calculate sum(|aij|)-RealPart(aii), amt of shift needed for this row */
460       d  = (aa)[ddiag[i]];
461       rs = -PetscAbsScalar(d) - PetscRealPart(d);
462       v  = aa+ai[i];
463       nz = ai[i+1] - ai[i];
464       for (j=0; j<nz; j++)
465 	rs += PetscAbsScalar(v[j]);
466       if (rs>sctx.shift_top) sctx.shift_top = rs;
467     }
468     sctx.shift_top   *= 1.1;
469     sctx.nshift_max   = 5;
470     sctx.shift_lo     = 0.;
471     sctx.shift_hi     = 1.;
472   }
473 
474   sctx.shift_amount = 0.0;
475   sctx.nshift       = 0;
476   do {
477     sctx.lushift = PETSC_FALSE;
478     for (i=0; i<n; i++){
479       nz    = bi[i+1] - bi[i];
480       bjtmp = bj + bi[i];
481       for  (j=0; j<nz; j++) rtmp[bjtmp[j]] = 0.0;
482 
483       /* load in initial (unfactored row) */
484       nz    = ai[r[i]+1] - ai[r[i]];
485       ajtmp = aj + ai[r[i]];
486       v     = aa + ai[r[i]];
487       for (j=0; j<nz; j++) {
488         rtmp[ics[ajtmp[j]]] = v[j];
489       }
490       rtmp[ics[r[i]]] += sctx.shift_amount; /* shift the diagonal of the matrix */
491       if (sctx.shift_amount > 0.0) printf("row %d, shift %g\n",i,sctx.shift_amount);
492 
493       row = *bjtmp++;
494       while  (row < i) {
495         pc = rtmp + row;
496         if (*pc != 0.0) {
497           pv         = b->a + diag_offset[row];
498           pj         = b->j + diag_offset[row] + 1;
499           multiplier = *pc / *pv++;
500           *pc        = multiplier;
501           nz         = bi[row+1] - diag_offset[row] - 1;
502           for (j=0; j<nz; j++) rtmp[pj[j]] -= multiplier * pv[j];
503           ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr);
504         }
505         row = *bjtmp++;
506       }
507       /* finished row so stick it into b->a */
508       pv   = b->a + bi[i] ;
509       pj   = b->j + bi[i] ;
510       nz   = bi[i+1] - bi[i];
511       diag = diag_offset[i] - bi[i];
512       rs   = -PetscAbsScalar(pv[diag]);
513       for (j=0; j<nz; j++) {
514         pv[j] = rtmp[pj[j]];
515         rs   += PetscAbsScalar(pv[j]);
516       }
517 
518       /* 9/13/02 Victor Eijkhout suggested scaling zeropivot by rs for matrices with funny scalings */
519       sctx.rs  = rs;
520       sctx.pv  = pv[diag];
521       ierr = MatLUCheckShift_inline(info,sctx,i,newshift);CHKERRQ(ierr);
522       if (newshift == 1) break;
523     }
524 
525     if (info->shiftpd && !sctx.lushift && sctx.shift_fraction>0 && sctx.nshift<sctx.nshift_max) {
526       /*
527        * if no shift in this attempt & shifting & started shifting & can refine,
528        * then try lower shift
529        */
530       sctx.shift_hi       = sctx.shift_fraction;
531       sctx.shift_fraction = (sctx.shift_hi+sctx.shift_lo)/2.;
532       sctx.shift_amount   = sctx.shift_fraction * sctx.shift_top;
533       sctx.lushift        = PETSC_TRUE;
534       sctx.nshift++;
535     }
536   } while (sctx.lushift);
537 
538   /* invert diagonal entries for simplier triangular solves */
539   for (i=0; i<n; i++) {
540     b->a[diag_offset[i]] = 1.0/b->a[diag_offset[i]];
541   }
542   ierr = PetscFree(rtmp);CHKERRQ(ierr);
543   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
544   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
545   if (b->inode.use) {
546     C->ops->solve   = MatSolve_Inode;
547   } else {
548     PetscTruth row_identity, col_identity;
549     ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
550     ierr = ISIdentity(isicol,&col_identity);CHKERRQ(ierr);
551     if (row_identity && col_identity) {
552       C->ops->solve   = MatSolve_SeqAIJ_NaturalOrdering;
553     } else {
554       C->ops->solve   = MatSolve_SeqAIJ;
555     }
556   }
557   C->ops->solveadd           = MatSolveAdd_SeqAIJ;
558   C->ops->solvetranspose     = MatSolveTranspose_SeqAIJ;
559   C->ops->solvetransposeadd  = MatSolveTransposeAdd_SeqAIJ;
560   C->ops->matsolve           = MatMatSolve_SeqAIJ;
561   C->assembled    = PETSC_TRUE;
562   C->preallocated = PETSC_TRUE;
563   ierr = PetscLogFlops(C->cmap->n);CHKERRQ(ierr);
564   if (sctx.nshift){
565      if (info->shiftpd) {
566       ierr = PetscInfo4(A,"number of shift_pd tries %D, shift_amount %G, diagonal shifted up by %e fraction top_value %e\n",sctx.nshift,sctx.shift_amount,sctx.shift_fraction,sctx.shift_top);CHKERRQ(ierr);
567     } else if (info->shiftnz) {
568       ierr = PetscInfo2(A,"number of shift_nz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
569     }
570   }
571   PetscFunctionReturn(0);
572 }
573 
574 /*
575    This routine implements inplace ILU(0) with row or/and column permutations.
576    Input:
577      A - original matrix
578    Output;
579      A - a->i (rowptr) is same as original rowptr, but factored i-the row is stored in rowperm[i]
580          a->j (col index) is permuted by the inverse of colperm, then sorted
581          a->a reordered accordingly with a->j
582          a->diag (ptr to diagonal elements) is updated.
583 */
584 #undef __FUNCT__
585 #define __FUNCT__ "MatLUFactorNumeric_SeqAIJ_InplaceWithPerm"
586 PetscErrorCode MatLUFactorNumeric_SeqAIJ_InplaceWithPerm(Mat B,Mat A,const MatFactorInfo *info)
587 {
588   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data;
589   IS             isrow = a->row,isicol = a->icol;
590   PetscErrorCode ierr;
591   const PetscInt *r,*ic,*ics;
592   PetscInt       i,j,n=A->rmap->n,*ai=a->i,*aj=a->j;
593   PetscInt       *ajtmp,nz,row;
594   PetscInt       *diag = a->diag,nbdiag,*pj;
595   PetscScalar    *rtmp,*pc,multiplier,d;
596   MatScalar      *v,*pv;
597   PetscReal      rs;
598   LUShift_Ctx    sctx;
599   PetscInt       newshift;
600 
601   PetscFunctionBegin;
602   if (A != B) SETERRQ(PETSC_ERR_ARG_INCOMP,"input and output matrix must have same address");
603   ierr  = ISGetIndices(isrow,&r);CHKERRQ(ierr);
604   ierr  = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
605   ierr  = PetscMalloc((n+1)*sizeof(PetscScalar),&rtmp);CHKERRQ(ierr);
606   ierr  = PetscMemzero(rtmp,(n+1)*sizeof(PetscScalar));CHKERRQ(ierr);
607   ics = ic;
608 
609   sctx.shift_top      = 0;
610   sctx.nshift_max     = 0;
611   sctx.shift_lo       = 0;
612   sctx.shift_hi       = 0;
613   sctx.shift_fraction = 0;
614 
615   /* if both shift schemes are chosen by user, only use info->shiftpd */
616   if (info->shiftpd) { /* set sctx.shift_top=max{rs} */
617     sctx.shift_top = 0;
618     for (i=0; i<n; i++) {
619       /* calculate sum(|aij|)-RealPart(aii), amt of shift needed for this row */
620       d  = (a->a)[diag[i]];
621       rs = -PetscAbsScalar(d) - PetscRealPart(d);
622       v  = a->a+ai[i];
623       nz = ai[i+1] - ai[i];
624       for (j=0; j<nz; j++)
625 	rs += PetscAbsScalar(v[j]);
626       if (rs>sctx.shift_top) sctx.shift_top = rs;
627     }
628     if (sctx.shift_top < info->zeropivot) sctx.shift_top = info->zeropivot;
629     sctx.shift_top    *= 1.1;
630     sctx.nshift_max   = 5;
631     sctx.shift_lo     = 0.;
632     sctx.shift_hi     = 1.;
633   }
634 
635   sctx.shift_amount = 0;
636   sctx.nshift       = 0;
637   do {
638     sctx.lushift = PETSC_FALSE;
639     for (i=0; i<n; i++){
640       /* load in initial unfactored row */
641       nz    = ai[r[i]+1] - ai[r[i]];
642       ajtmp = aj + ai[r[i]];
643       v     = a->a + ai[r[i]];
644       /* sort permuted ajtmp and values v accordingly */
645       for (j=0; j<nz; j++) ajtmp[j] = ics[ajtmp[j]];
646       ierr = PetscSortIntWithScalarArray(nz,ajtmp,v);CHKERRQ(ierr);
647 
648       diag[r[i]] = ai[r[i]];
649       for (j=0; j<nz; j++) {
650         rtmp[ajtmp[j]] = v[j];
651         if (ajtmp[j] < i) diag[r[i]]++; /* update a->diag */
652       }
653       rtmp[r[i]] += sctx.shift_amount; /* shift the diagonal of the matrix */
654 
655       row = *ajtmp++;
656       while  (row < i) {
657         pc = rtmp + row;
658         if (*pc != 0.0) {
659           pv         = a->a + diag[r[row]];
660           pj         = aj + diag[r[row]] + 1;
661 
662           multiplier = *pc / *pv++;
663           *pc        = multiplier;
664           nz         = ai[r[row]+1] - diag[r[row]] - 1;
665           for (j=0; j<nz; j++) rtmp[pj[j]] -= multiplier * pv[j];
666           ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr);
667         }
668         row = *ajtmp++;
669       }
670       /* finished row so overwrite it onto a->a */
671       pv   = a->a + ai[r[i]] ;
672       pj   = aj + ai[r[i]] ;
673       nz   = ai[r[i]+1] - ai[r[i]];
674       nbdiag = diag[r[i]] - ai[r[i]]; /* num of entries before the diagonal */
675 
676       rs   = 0.0;
677       for (j=0; j<nz; j++) {
678         pv[j] = rtmp[pj[j]];
679         if (j != nbdiag) rs += PetscAbsScalar(pv[j]);
680       }
681 
682       /* 9/13/02 Victor Eijkhout suggested scaling zeropivot by rs for matrices with funny scalings */
683       sctx.rs  = rs;
684       sctx.pv  = pv[nbdiag];
685       ierr = MatLUCheckShift_inline(info,sctx,i,newshift);CHKERRQ(ierr);
686       if (newshift == 1) break;
687     }
688 
689     if (info->shiftpd && !sctx.lushift && sctx.shift_fraction>0 && sctx.nshift<sctx.nshift_max) {
690       /*
691        * if no shift in this attempt & shifting & started shifting & can refine,
692        * then try lower shift
693        */
694       sctx.shift_hi        = sctx.shift_fraction;
695       sctx.shift_fraction = (sctx.shift_hi+sctx.shift_lo)/2.;
696       sctx.shift_amount    = sctx.shift_fraction * sctx.shift_top;
697       sctx.lushift         = PETSC_TRUE;
698       sctx.nshift++;
699     }
700   } while (sctx.lushift);
701 
702   /* invert diagonal entries for simplier triangular solves */
703   for (i=0; i<n; i++) {
704     a->a[diag[r[i]]] = 1.0/a->a[diag[r[i]]];
705   }
706 
707   ierr = PetscFree(rtmp);CHKERRQ(ierr);
708   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
709   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
710   A->ops->solve             = MatSolve_SeqAIJ_InplaceWithPerm;
711   A->ops->solveadd          = MatSolveAdd_SeqAIJ;
712   A->ops->solvetranspose    = MatSolveTranspose_SeqAIJ;
713   A->ops->solvetransposeadd = MatSolveTransposeAdd_SeqAIJ;
714   A->assembled = PETSC_TRUE;
715   A->preallocated = PETSC_TRUE;
716   ierr = PetscLogFlops(A->cmap->n);CHKERRQ(ierr);
717   if (sctx.nshift){
718     if (info->shiftpd) {
719       ierr = PetscInfo4(A,"number of shift_pd tries %D, shift_amount %G, diagonal shifted up by %e fraction top_value %e\n",sctx.nshift,sctx.shift_amount,sctx.shift_fraction,sctx.shift_top);CHKERRQ(ierr);
720     } else if (info->shiftnz) {
721       ierr = PetscInfo2(A,"number of shift_nz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
722     }
723   }
724   PetscFunctionReturn(0);
725 }
726 
727 /* ----------------------------------------------------------- */
728 #undef __FUNCT__
729 #define __FUNCT__ "MatLUFactor_SeqAIJ"
730 PetscErrorCode MatLUFactor_SeqAIJ(Mat A,IS row,IS col,const MatFactorInfo *info)
731 {
732   PetscErrorCode ierr;
733   Mat            C;
734 
735   PetscFunctionBegin;
736   ierr = MatGetFactor(A,MAT_SOLVER_PETSC,MAT_FACTOR_LU,&C);CHKERRQ(ierr);
737   ierr = MatLUFactorSymbolic(C,A,row,col,info);CHKERRQ(ierr);
738   ierr = MatLUFactorNumeric(C,A,info);CHKERRQ(ierr);
739   A->ops->solve            = C->ops->solve;
740   A->ops->solvetranspose   = C->ops->solvetranspose;
741   ierr = MatHeaderCopy(A,C);CHKERRQ(ierr);
742   ierr = PetscLogObjectParent(A,((Mat_SeqAIJ*)(A->data))->icol);CHKERRQ(ierr);
743   PetscFunctionReturn(0);
744 }
745 /* ----------------------------------------------------------- */
746 
747 
748 #undef __FUNCT__
749 #define __FUNCT__ "MatSolve_SeqAIJ"
750 PetscErrorCode MatSolve_SeqAIJ(Mat A,Vec bb,Vec xx)
751 {
752   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
753   IS                iscol = a->col,isrow = a->row;
754   PetscErrorCode    ierr;
755   PetscInt          i, n = A->rmap->n,*vi,*ai = a->i,*aj = a->j;
756   PetscInt          nz;
757   const PetscInt    *rout,*cout,*r,*c;
758   PetscScalar       *x,*tmp,*tmps,sum;
759   const PetscScalar *b;
760   const MatScalar   *aa = a->a,*v;
761 
762   PetscFunctionBegin;
763   if (!n) PetscFunctionReturn(0);
764 
765   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
766   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
767   tmp  = a->solve_work;
768 
769   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
770   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout + (n-1);
771 
772   /* forward solve the lower triangular */
773   tmp[0] = b[*r++];
774   tmps   = tmp;
775   for (i=1; i<n; i++) {
776     v   = aa + ai[i] ;
777     vi  = aj + ai[i] ;
778     nz  = a->diag[i] - ai[i];
779     sum = b[*r++];
780     PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
781     tmp[i] = sum;
782   }
783 
784   /* backward solve the upper triangular */
785   for (i=n-1; i>=0; i--){
786     v   = aa + a->diag[i] + 1;
787     vi  = aj + a->diag[i] + 1;
788     nz  = ai[i+1] - a->diag[i] - 1;
789     sum = tmp[i];
790     PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
791     x[*c--] = tmp[i] = sum*aa[a->diag[i]];
792   }
793 
794   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
795   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
796   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
797   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
798   ierr = PetscLogFlops(2.0*a->nz - A->cmap->n);CHKERRQ(ierr);
799   PetscFunctionReturn(0);
800 }
801 
802 #undef __FUNCT__
803 #define __FUNCT__ "MatMatSolve_SeqAIJ"
804 PetscErrorCode MatMatSolve_SeqAIJ(Mat A,Mat B,Mat X)
805 {
806   Mat_SeqAIJ      *a = (Mat_SeqAIJ*)A->data;
807   IS              iscol = a->col,isrow = a->row;
808   PetscErrorCode  ierr;
809   PetscInt        i, n = A->rmap->n,*vi,*ai = a->i,*aj = a->j;
810   PetscInt        nz,neq;
811   const PetscInt  *rout,*cout,*r,*c;
812   PetscScalar     *x,*b,*tmp,*tmps,sum;
813   const MatScalar *aa = a->a,*v;
814   PetscTruth      bisdense,xisdense;
815 
816   PetscFunctionBegin;
817   if (!n) PetscFunctionReturn(0);
818 
819   ierr = PetscTypeCompare((PetscObject)B,MATSEQDENSE,&bisdense);CHKERRQ(ierr);
820   if (!bisdense) SETERRQ(PETSC_ERR_ARG_INCOMP,"B matrix must be a SeqDense matrix");
821   ierr = PetscTypeCompare((PetscObject)X,MATSEQDENSE,&xisdense);CHKERRQ(ierr);
822   if (!xisdense) SETERRQ(PETSC_ERR_ARG_INCOMP,"X matrix must be a SeqDense matrix");
823 
824   ierr = MatGetArray(B,&b);CHKERRQ(ierr);
825   ierr = MatGetArray(X,&x);CHKERRQ(ierr);
826 
827   tmp  = a->solve_work;
828   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
829   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
830 
831   for (neq=0; neq<B->cmap->n; neq++){
832     /* forward solve the lower triangular */
833     tmp[0] = b[r[0]];
834     tmps   = tmp;
835     for (i=1; i<n; i++) {
836       v   = aa + ai[i] ;
837       vi  = aj + ai[i] ;
838       nz  = a->diag[i] - ai[i];
839       sum = b[r[i]];
840       PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
841       tmp[i] = sum;
842     }
843     /* backward solve the upper triangular */
844     for (i=n-1; i>=0; i--){
845       v   = aa + a->diag[i] + 1;
846       vi  = aj + a->diag[i] + 1;
847       nz  = ai[i+1] - a->diag[i] - 1;
848       sum = tmp[i];
849       PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
850       x[c[i]] = tmp[i] = sum*aa[a->diag[i]];
851     }
852 
853     b += n;
854     x += n;
855   }
856   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
857   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
858   ierr = MatRestoreArray(B,&b);CHKERRQ(ierr);
859   ierr = MatRestoreArray(X,&x);CHKERRQ(ierr);
860   ierr = PetscLogFlops(B->cmap->n*(2.0*a->nz - n));CHKERRQ(ierr);
861   PetscFunctionReturn(0);
862 }
863 
864 #undef __FUNCT__
865 #define __FUNCT__ "MatSolve_SeqAIJ_InplaceWithPerm"
866 PetscErrorCode MatSolve_SeqAIJ_InplaceWithPerm(Mat A,Vec bb,Vec xx)
867 {
868   Mat_SeqAIJ      *a = (Mat_SeqAIJ*)A->data;
869   IS              iscol = a->col,isrow = a->row;
870   PetscErrorCode  ierr;
871   const PetscInt  *r,*c,*rout,*cout;
872   PetscInt        i, n = A->rmap->n,*vi,*ai = a->i,*aj = a->j;
873   PetscInt        nz,row;
874   PetscScalar     *x,*b,*tmp,*tmps,sum;
875   const MatScalar *aa = a->a,*v;
876 
877   PetscFunctionBegin;
878   if (!n) PetscFunctionReturn(0);
879 
880   ierr = VecGetArray(bb,&b);CHKERRQ(ierr);
881   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
882   tmp  = a->solve_work;
883 
884   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
885   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout + (n-1);
886 
887   /* forward solve the lower triangular */
888   tmp[0] = b[*r++];
889   tmps   = tmp;
890   for (row=1; row<n; row++) {
891     i   = rout[row]; /* permuted row */
892     v   = aa + ai[i] ;
893     vi  = aj + ai[i] ;
894     nz  = a->diag[i] - ai[i];
895     sum = b[*r++];
896     PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
897     tmp[row] = sum;
898   }
899 
900   /* backward solve the upper triangular */
901   for (row=n-1; row>=0; row--){
902     i   = rout[row]; /* permuted row */
903     v   = aa + a->diag[i] + 1;
904     vi  = aj + a->diag[i] + 1;
905     nz  = ai[i+1] - a->diag[i] - 1;
906     sum = tmp[row];
907     PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
908     x[*c--] = tmp[row] = sum*aa[a->diag[i]];
909   }
910 
911   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
912   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
913   ierr = VecRestoreArray(bb,&b);CHKERRQ(ierr);
914   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
915   ierr = PetscLogFlops(2.0*a->nz - A->cmap->n);CHKERRQ(ierr);
916   PetscFunctionReturn(0);
917 }
918 
919 /* ----------------------------------------------------------- */
920 #undef __FUNCT__
921 #define __FUNCT__ "MatSolve_SeqAIJ_NaturalOrdering"
922 PetscErrorCode MatSolve_SeqAIJ_NaturalOrdering(Mat A,Vec bb,Vec xx)
923 {
924   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
925   PetscErrorCode    ierr;
926   PetscInt          n = A->rmap->n;
927   const PetscInt    *ai = a->i,*aj = a->j,*adiag = a->diag,*vi;
928   PetscScalar       *x;
929   const PetscScalar *b;
930   const MatScalar   *aa = a->a;
931 #if !defined(PETSC_USE_FORTRAN_KERNEL_SOLVEAIJ)
932   PetscInt          adiag_i,i,nz,ai_i;
933   const MatScalar   *v;
934   PetscScalar       sum;
935 #endif
936 
937   PetscFunctionBegin;
938   if (!n) PetscFunctionReturn(0);
939 
940   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
941   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
942 
943 #if defined(PETSC_USE_FORTRAN_KERNEL_SOLVEAIJ)
944   fortransolveaij_(&n,x,ai,aj,adiag,aa,b);
945 #else
946   /* forward solve the lower triangular */
947   x[0] = b[0];
948   for (i=1; i<n; i++) {
949     ai_i = ai[i];
950     v    = aa + ai_i;
951     vi   = aj + ai_i;
952     nz   = adiag[i] - ai_i;
953     sum  = b[i];
954     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
955     x[i] = sum;
956   }
957 
958   /* backward solve the upper triangular */
959   for (i=n-1; i>=0; i--){
960     adiag_i = adiag[i];
961     v       = aa + adiag_i + 1;
962     vi      = aj + adiag_i + 1;
963     nz      = ai[i+1] - adiag_i - 1;
964     sum     = x[i];
965     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
966     x[i]    = sum*aa[adiag_i];
967   }
968 #endif
969   ierr = PetscLogFlops(2.0*a->nz - A->cmap->n);CHKERRQ(ierr);
970   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
971   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
972   PetscFunctionReturn(0);
973 }
974 
975 #undef __FUNCT__
976 #define __FUNCT__ "MatSolveAdd_SeqAIJ"
977 PetscErrorCode MatSolveAdd_SeqAIJ(Mat A,Vec bb,Vec yy,Vec xx)
978 {
979   Mat_SeqAIJ      *a = (Mat_SeqAIJ*)A->data;
980   IS              iscol = a->col,isrow = a->row;
981   PetscErrorCode  ierr;
982   PetscInt        i, n = A->rmap->n,*vi,*ai = a->i,*aj = a->j;
983   PetscInt        nz;
984   const PetscInt  *rout,*cout,*r,*c;
985   PetscScalar     *x,*b,*tmp,sum;
986   const MatScalar *aa = a->a,*v;
987 
988   PetscFunctionBegin;
989   if (yy != xx) {ierr = VecCopy(yy,xx);CHKERRQ(ierr);}
990 
991   ierr = VecGetArray(bb,&b);CHKERRQ(ierr);
992   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
993   tmp  = a->solve_work;
994 
995   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
996   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout + (n-1);
997 
998   /* forward solve the lower triangular */
999   tmp[0] = b[*r++];
1000   for (i=1; i<n; i++) {
1001     v   = aa + ai[i] ;
1002     vi  = aj + ai[i] ;
1003     nz  = a->diag[i] - ai[i];
1004     sum = b[*r++];
1005     while (nz--) sum -= *v++ * tmp[*vi++ ];
1006     tmp[i] = sum;
1007   }
1008 
1009   /* backward solve the upper triangular */
1010   for (i=n-1; i>=0; i--){
1011     v   = aa + a->diag[i] + 1;
1012     vi  = aj + a->diag[i] + 1;
1013     nz  = ai[i+1] - a->diag[i] - 1;
1014     sum = tmp[i];
1015     while (nz--) sum -= *v++ * tmp[*vi++ ];
1016     tmp[i] = sum*aa[a->diag[i]];
1017     x[*c--] += tmp[i];
1018   }
1019 
1020   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1021   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1022   ierr = VecRestoreArray(bb,&b);CHKERRQ(ierr);
1023   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1024   ierr = PetscLogFlops(2.0*a->nz);CHKERRQ(ierr);
1025 
1026   PetscFunctionReturn(0);
1027 }
1028 /* -------------------------------------------------------------------*/
1029 #undef __FUNCT__
1030 #define __FUNCT__ "MatSolveTranspose_SeqAIJ"
1031 PetscErrorCode MatSolveTranspose_SeqAIJ(Mat A,Vec bb,Vec xx)
1032 {
1033   Mat_SeqAIJ      *a = (Mat_SeqAIJ*)A->data;
1034   IS              iscol = a->col,isrow = a->row;
1035   PetscErrorCode  ierr;
1036   const PetscInt  *rout,*cout,*r,*c;
1037   PetscInt        i,n = A->rmap->n,*vi,*ai = a->i,*aj = a->j;
1038   PetscInt        nz,*diag = a->diag;
1039   PetscScalar     *x,*b,*tmp,s1;
1040   const MatScalar *aa = a->a,*v;
1041 
1042   PetscFunctionBegin;
1043   ierr = VecGetArray(bb,&b);CHKERRQ(ierr);
1044   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1045   tmp  = a->solve_work;
1046 
1047   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1048   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1049 
1050   /* copy the b into temp work space according to permutation */
1051   for (i=0; i<n; i++) tmp[i] = b[c[i]];
1052 
1053   /* forward solve the U^T */
1054   for (i=0; i<n; i++) {
1055     v   = aa + diag[i] ;
1056     vi  = aj + diag[i] + 1;
1057     nz  = ai[i+1] - diag[i] - 1;
1058     s1  = tmp[i];
1059     s1 *= (*v++);  /* multiply by inverse of diagonal entry */
1060     while (nz--) {
1061       tmp[*vi++ ] -= (*v++)*s1;
1062     }
1063     tmp[i] = s1;
1064   }
1065 
1066   /* backward solve the L^T */
1067   for (i=n-1; i>=0; i--){
1068     v   = aa + diag[i] - 1 ;
1069     vi  = aj + diag[i] - 1 ;
1070     nz  = diag[i] - ai[i];
1071     s1  = tmp[i];
1072     while (nz--) {
1073       tmp[*vi-- ] -= (*v--)*s1;
1074     }
1075   }
1076 
1077   /* copy tmp into x according to permutation */
1078   for (i=0; i<n; i++) x[r[i]] = tmp[i];
1079 
1080   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1081   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1082   ierr = VecRestoreArray(bb,&b);CHKERRQ(ierr);
1083   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1084 
1085   ierr = PetscLogFlops(2.0*a->nz-A->cmap->n);CHKERRQ(ierr);
1086   PetscFunctionReturn(0);
1087 }
1088 
1089 #undef __FUNCT__
1090 #define __FUNCT__ "MatSolveTransposeAdd_SeqAIJ"
1091 PetscErrorCode MatSolveTransposeAdd_SeqAIJ(Mat A,Vec bb,Vec zz,Vec xx)
1092 {
1093   Mat_SeqAIJ      *a = (Mat_SeqAIJ*)A->data;
1094   IS              iscol = a->col,isrow = a->row;
1095   PetscErrorCode  ierr;
1096   const PetscInt  *r,*c,*rout,*cout;
1097   PetscInt        i,n = A->rmap->n,*vi,*ai = a->i,*aj = a->j;
1098   PetscInt        nz,*diag = a->diag;
1099   PetscScalar     *x,*b,*tmp;
1100   const MatScalar *aa = a->a,*v;
1101 
1102   PetscFunctionBegin;
1103   if (zz != xx) {ierr = VecCopy(zz,xx);CHKERRQ(ierr);}
1104 
1105   ierr = VecGetArray(bb,&b);CHKERRQ(ierr);
1106   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1107   tmp = a->solve_work;
1108 
1109   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1110   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1111 
1112   /* copy the b into temp work space according to permutation */
1113   for (i=0; i<n; i++) tmp[i] = b[c[i]];
1114 
1115   /* forward solve the U^T */
1116   for (i=0; i<n; i++) {
1117     v   = aa + diag[i] ;
1118     vi  = aj + diag[i] + 1;
1119     nz  = ai[i+1] - diag[i] - 1;
1120     tmp[i] *= *v++;
1121     while (nz--) {
1122       tmp[*vi++ ] -= (*v++)*tmp[i];
1123     }
1124   }
1125 
1126   /* backward solve the L^T */
1127   for (i=n-1; i>=0; i--){
1128     v   = aa + diag[i] - 1 ;
1129     vi  = aj + diag[i] - 1 ;
1130     nz  = diag[i] - ai[i];
1131     while (nz--) {
1132       tmp[*vi-- ] -= (*v--)*tmp[i];
1133     }
1134   }
1135 
1136   /* copy tmp into x according to permutation */
1137   for (i=0; i<n; i++) x[r[i]] += tmp[i];
1138 
1139   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1140   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1141   ierr = VecRestoreArray(bb,&b);CHKERRQ(ierr);
1142   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1143 
1144   ierr = PetscLogFlops(2.0*a->nz);CHKERRQ(ierr);
1145   PetscFunctionReturn(0);
1146 }
1147 /* ----------------------------------------------------------------*/
1148 EXTERN PetscErrorCode Mat_CheckInode(Mat,PetscTruth);
1149 EXTERN PetscErrorCode MatDuplicateNoCreate_SeqAIJ(Mat,Mat,MatDuplicateOption);
1150 
1151 /* ilu(0) with natural ordering under new data structure */
1152 #undef __FUNCT__
1153 #define __FUNCT__ "MatILUFactorSymbolic_SeqAIJ_newdatastruct"
1154 PetscErrorCode MatILUFactorSymbolic_SeqAIJ_newdatastruct(Mat fact,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
1155 {
1156 
1157   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
1158   PetscErrorCode     ierr;
1159   PetscInt           n=A->rmap->n,*ai=a->i,*aj,*adiag=a->diag;
1160   PetscInt           i,j,nz,*bi,*bj,*bdiag,*bdiag_rev;
1161 
1162   PetscFunctionBegin;
1163   ierr = MatDuplicateNoCreate_SeqAIJ(fact,A,MAT_DO_NOT_COPY_VALUES);CHKERRQ(ierr);
1164   b = (Mat_SeqAIJ*)(fact)->data;
1165 
1166   /* replace matrix arrays with single allocations, then reset values */
1167   ierr = PetscFree3(b->a,b->j,b->i);CHKERRQ(ierr);
1168   ierr = PetscFree(b->diag);CHKERRQ(ierr);
1169 
1170   ierr = PetscMalloc((2*n+2)*sizeof(PetscInt),&b->i);CHKERRQ(ierr);
1171   ierr = PetscMalloc((ai[n]+2)*sizeof(PetscInt),&b->j);CHKERRQ(ierr);
1172   ierr = PetscMalloc((ai[n]+2)*sizeof(PetscScalar),&b->a);CHKERRQ(ierr);
1173   b->singlemalloc = PETSC_FALSE;
1174   ierr = PetscMalloc((2*n+2)*sizeof(PetscInt),&b->diag);CHKERRQ(ierr);
1175   bdiag     = b->diag;
1176   bdiag_rev = bdiag + n+1;
1177 
1178   if (n > 0) {
1179     ierr = PetscMemzero(b->a,(ai[n])*sizeof(PetscScalar));CHKERRQ(ierr);
1180   }
1181 
1182   /* set bi and bj with new data structure */
1183   bi = b->i;
1184   bj = b->j;
1185 
1186   /* L part */
1187   bi[0] = 0;
1188   for (i=0; i<n; i++){
1189     nz = adiag[i] - ai[i];
1190     bi[i+1] = bi[i] + nz;
1191     aj = a->j + ai[i];
1192     for (j=0; j<nz; j++){
1193       *bj = aj[j]; bj++;
1194     }
1195   }
1196 
1197   bj++; /* diag[n] */
1198   bdiag[n] = bi[n];
1199 
1200   /* U part */
1201   for (i=n-1; i>=0; i--){
1202     nz = ai[i+1] - adiag[i] - 1;
1203     bi[n+ (n-i)] = bi[n+(n-i)-1] + nz + 1;
1204     aj = a->j + adiag[i] + 1;
1205     for (j=0; j<nz; j++){
1206       *bj = aj[j]; bj++;
1207     }
1208     /* diag[i] */
1209     *bj = i; bj++;
1210     bdiag[i] = bi[n+ (n-i)] ;
1211     bdiag_rev[n-i] = bdiag[i];
1212   }
1213   bdiag_rev[0] = bdiag[n];
1214   PetscFunctionReturn(0);
1215 }
1216 
1217 #undef __FUNCT__
1218 #define __FUNCT__ "MatILUFactorSymbolic_SeqAIJ"
1219 PetscErrorCode MatILUFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
1220 {
1221   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
1222   IS                 isicol;
1223   PetscErrorCode     ierr;
1224   const PetscInt     *r,*ic;
1225   PetscInt           n=A->rmap->n,*ai=a->i,*aj=a->j,d;
1226   PetscInt           *bi,*cols,nnz,*cols_lvl;
1227   PetscInt           *bdiag,prow,fm,nzbd,reallocs=0,dcount=0;
1228   PetscInt           i,levels,diagonal_fill;
1229   PetscTruth         col_identity,row_identity;
1230   PetscReal          f;
1231   PetscInt           nlnk,*lnk,*lnk_lvl=PETSC_NULL;
1232   PetscBT            lnkbt;
1233   PetscInt           nzi,*bj,**bj_ptr,**bjlvl_ptr;
1234   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
1235   PetscFreeSpaceList free_space_lvl=PETSC_NULL,current_space_lvl=PETSC_NULL;
1236   PetscTruth         missing;
1237 
1238   PetscFunctionBegin;
1239   if (A->rmap->n != A->cmap->n) SETERRQ2(PETSC_ERR_ARG_WRONG,"Must be square matrix, rows %D columns %D",A->rmap->n,A->cmap->n);
1240   f             = info->fill;
1241   levels        = (PetscInt)info->levels;
1242   diagonal_fill = (PetscInt)info->diagonal_fill;
1243   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
1244 
1245   /* special case that simply copies fill pattern */
1246   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
1247   ierr = ISIdentity(iscol,&col_identity);CHKERRQ(ierr);
1248   if (!levels && row_identity && col_identity) {
1249 
1250     PetscTruth newdatastruct=PETSC_FALSE;
1251     ierr = PetscOptionsGetTruth(PETSC_NULL,"-ilu_new",&newdatastruct,PETSC_NULL);CHKERRQ(ierr);
1252     if (newdatastruct){
1253       ierr = MatILUFactorSymbolic_SeqAIJ_newdatastruct(fact,A,isrow,iscol,info);CHKERRQ(ierr);
1254       (fact)->ops->lufactornumeric =  MatLUFactorNumeric_SeqAIJ_newdatastruct;
1255     } else {
1256       ierr = MatDuplicateNoCreate_SeqAIJ(fact,A,MAT_DO_NOT_COPY_VALUES);CHKERRQ(ierr);
1257       (fact)->ops->lufactornumeric =  MatLUFactorNumeric_SeqAIJ;
1258     }
1259 
1260     fact->factor = MAT_FACTOR_ILU;
1261     (fact)->info.factor_mallocs    = 0;
1262     (fact)->info.fill_ratio_given  = info->fill;
1263     (fact)->info.fill_ratio_needed = 1.0;
1264     b               = (Mat_SeqAIJ*)(fact)->data;
1265     ierr = MatMissingDiagonal(A,&missing,&d);CHKERRQ(ierr);
1266     if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",d);
1267     b->row              = isrow;
1268     b->col              = iscol;
1269     b->icol             = isicol;
1270     ierr                = PetscMalloc(((fact)->rmap->n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1271     ierr                = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
1272     ierr                = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
1273     ierr = MatILUFactorSymbolic_Inode(fact,A,isrow,iscol,info);CHKERRQ(ierr);
1274     PetscFunctionReturn(0);
1275   }
1276 
1277   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
1278   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
1279 
1280   /* get new row pointers */
1281   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
1282   bi[0] = 0;
1283   /* bdiag is location of diagonal in factor */
1284   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);
1285   bdiag[0]  = 0;
1286 
1287   ierr = PetscMalloc((2*n+1)*sizeof(PetscInt**),&bj_ptr);CHKERRQ(ierr);
1288   bjlvl_ptr = (PetscInt**)(bj_ptr + n);
1289 
1290   /* create a linked list for storing column indices of the active row */
1291   nlnk = n + 1;
1292   ierr = PetscIncompleteLLCreate(n,n,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1293 
1294   /* initial FreeSpace size is f*(ai[n]+1) */
1295   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space);CHKERRQ(ierr);
1296   current_space = free_space;
1297   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space_lvl);CHKERRQ(ierr);
1298   current_space_lvl = free_space_lvl;
1299 
1300   for (i=0; i<n; i++) {
1301     nzi = 0;
1302     /* copy current row into linked list */
1303     nnz  = ai[r[i]+1] - ai[r[i]];
1304     if (!nnz) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
1305     cols = aj + ai[r[i]];
1306     lnk[i] = -1; /* marker to indicate if diagonal exists */
1307     ierr = PetscIncompleteLLInit(nnz,cols,n,ic,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1308     nzi += nlnk;
1309 
1310     /* make sure diagonal entry is included */
1311     if (diagonal_fill && lnk[i] == -1) {
1312       fm = n;
1313       while (lnk[fm] < i) fm = lnk[fm];
1314       lnk[i]     = lnk[fm]; /* insert diagonal into linked list */
1315       lnk[fm]    = i;
1316       lnk_lvl[i] = 0;
1317       nzi++; dcount++;
1318     }
1319 
1320     /* add pivot rows into the active row */
1321     nzbd = 0;
1322     prow = lnk[n];
1323     while (prow < i) {
1324       nnz      = bdiag[prow];
1325       cols     = bj_ptr[prow] + nnz + 1;
1326       cols_lvl = bjlvl_ptr[prow] + nnz + 1;
1327       nnz      = bi[prow+1] - bi[prow] - nnz - 1;
1328       ierr = PetscILULLAddSorted(nnz,cols,levels,cols_lvl,prow,nlnk,lnk,lnk_lvl,lnkbt,prow);CHKERRQ(ierr);
1329       nzi += nlnk;
1330       prow = lnk[prow];
1331       nzbd++;
1332     }
1333     bdiag[i] = nzbd;
1334     bi[i+1]  = bi[i] + nzi;
1335 
1336     /* if free space is not available, make more free space */
1337     if (current_space->local_remaining<nzi) {
1338       nnz = nzi*(n - i); /* estimated and max additional space needed */
1339       ierr = PetscFreeSpaceGet(nnz,&current_space);CHKERRQ(ierr);
1340       ierr = PetscFreeSpaceGet(nnz,&current_space_lvl);CHKERRQ(ierr);
1341       reallocs++;
1342     }
1343 
1344     /* copy data into free_space and free_space_lvl, then initialize lnk */
1345     ierr = PetscIncompleteLLClean(n,n,nzi,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);CHKERRQ(ierr);
1346     bj_ptr[i]    = current_space->array;
1347     bjlvl_ptr[i] = current_space_lvl->array;
1348 
1349     /* make sure the active row i has diagonal entry */
1350     if (*(bj_ptr[i]+bdiag[i]) != i) {
1351       SETERRQ1(PETSC_ERR_MAT_LU_ZRPVT,"Row %D has missing diagonal in factored matrix\n\
1352     try running with -pc_factor_nonzeros_along_diagonal or -pc_factor_diagonal_fill",i);
1353     }
1354 
1355     current_space->array           += nzi;
1356     current_space->local_used      += nzi;
1357     current_space->local_remaining -= nzi;
1358     current_space_lvl->array           += nzi;
1359     current_space_lvl->local_used      += nzi;
1360     current_space_lvl->local_remaining -= nzi;
1361   }
1362 
1363   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
1364   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
1365 
1366   /* destroy list of free space and other temporary arrays */
1367   ierr = PetscMalloc((bi[n]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
1368   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1369   ierr = PetscIncompleteLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1370   ierr = PetscFreeSpaceDestroy(free_space_lvl);CHKERRQ(ierr);
1371   ierr = PetscFree(bj_ptr);CHKERRQ(ierr);
1372 
1373 #if defined(PETSC_USE_INFO)
1374   {
1375     PetscReal af = ((PetscReal)bi[n])/((PetscReal)ai[n]);
1376     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,f,af);CHKERRQ(ierr);
1377     ierr = PetscInfo1(A,"Run with -[sub_]pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
1378     ierr = PetscInfo1(A,"PCFactorSetFill([sub]pc,%G);\n",af);CHKERRQ(ierr);
1379     ierr = PetscInfo(A,"for best performance.\n");CHKERRQ(ierr);
1380     if (diagonal_fill) {
1381       ierr = PetscInfo1(A,"Detected and replaced %D missing diagonals",dcount);CHKERRQ(ierr);
1382     }
1383   }
1384 #endif
1385 
1386   /* put together the new matrix */
1387   ierr = MatSeqAIJSetPreallocation_SeqAIJ(fact,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
1388   ierr = PetscLogObjectParent(fact,isicol);CHKERRQ(ierr);
1389   b = (Mat_SeqAIJ*)(fact)->data;
1390   b->free_a       = PETSC_TRUE;
1391   b->free_ij      = PETSC_TRUE;
1392   b->singlemalloc = PETSC_FALSE;
1393   ierr = PetscMalloc( (bi[n] )*sizeof(PetscScalar),&b->a);CHKERRQ(ierr);
1394   b->j          = bj;
1395   b->i          = bi;
1396   for (i=0; i<n; i++) bdiag[i] += bi[i];
1397   b->diag       = bdiag;
1398   b->ilen       = 0;
1399   b->imax       = 0;
1400   b->row        = isrow;
1401   b->col        = iscol;
1402   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
1403   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
1404   b->icol       = isicol;
1405   ierr = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1406   /* In b structure:  Free imax, ilen, old a, old j.
1407      Allocate bdiag, solve_work, new a, new j */
1408   ierr = PetscLogObjectMemory(fact,(bi[n]-n) * (sizeof(PetscInt)+sizeof(PetscScalar)));CHKERRQ(ierr);
1409   b->maxnz             = b->nz = bi[n] ;
1410   (fact)->info.factor_mallocs    = reallocs;
1411   (fact)->info.fill_ratio_given  = f;
1412   (fact)->info.fill_ratio_needed = ((PetscReal)bi[n])/((PetscReal)ai[n]);
1413   (fact)->ops->lufactornumeric =  MatLUFactorNumeric_SeqAIJ;
1414   ierr = MatILUFactorSymbolic_Inode(fact,A,isrow,iscol,info);CHKERRQ(ierr);
1415   PetscFunctionReturn(0);
1416 }
1417 
1418 #include "../src/mat/impls/sbaij/seq/sbaij.h"
1419 #undef __FUNCT__
1420 #define __FUNCT__ "MatCholeskyFactorNumeric_SeqAIJ"
1421 PetscErrorCode MatCholeskyFactorNumeric_SeqAIJ(Mat B,Mat A,const MatFactorInfo *info)
1422 {
1423   Mat            C = B;
1424   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data;
1425   Mat_SeqSBAIJ   *b=(Mat_SeqSBAIJ*)C->data;
1426   IS             ip=b->row,iip = b->icol;
1427   PetscErrorCode ierr;
1428   const PetscInt *rip,*riip;
1429   PetscInt       i,j,mbs=A->rmap->n,*bi=b->i,*bj=b->j,*bcol;
1430   PetscInt       *ai=a->i,*aj=a->j;
1431   PetscInt       k,jmin,jmax,*jl,*il,col,nexti,ili,nz;
1432   MatScalar      *rtmp,*ba=b->a,*bval,*aa=a->a,dk,uikdi;
1433   PetscReal      zeropivot,rs,shiftnz;
1434   PetscReal      shiftpd;
1435   ChShift_Ctx    sctx;
1436   PetscInt       newshift;
1437   PetscTruth     perm_identity;
1438 
1439   PetscFunctionBegin;
1440 
1441   shiftnz   = info->shiftnz;
1442   shiftpd   = info->shiftpd;
1443   zeropivot = info->zeropivot;
1444 
1445   ierr  = ISGetIndices(ip,&rip);CHKERRQ(ierr);
1446   ierr  = ISGetIndices(iip,&riip);CHKERRQ(ierr);
1447 
1448   /* initialization */
1449   nz   = (2*mbs+1)*sizeof(PetscInt)+mbs*sizeof(MatScalar);
1450   ierr = PetscMalloc(nz,&il);CHKERRQ(ierr);
1451   jl   = il + mbs;
1452   rtmp = (MatScalar*)(jl + mbs);
1453 
1454   sctx.shift_amount = 0;
1455   sctx.nshift       = 0;
1456   do {
1457     sctx.chshift = PETSC_FALSE;
1458     for (i=0; i<mbs; i++) {
1459       rtmp[i] = 0.0; jl[i] = mbs; il[0] = 0;
1460     }
1461 
1462     for (k = 0; k<mbs; k++){
1463       bval = ba + bi[k];
1464       /* initialize k-th row by the perm[k]-th row of A */
1465       jmin = ai[rip[k]]; jmax = ai[rip[k]+1];
1466       for (j = jmin; j < jmax; j++){
1467         col = riip[aj[j]];
1468         if (col >= k){ /* only take upper triangular entry */
1469           rtmp[col] = aa[j];
1470           *bval++  = 0.0; /* for in-place factorization */
1471         }
1472       }
1473       /* shift the diagonal of the matrix */
1474       if (sctx.nshift) rtmp[k] += sctx.shift_amount;
1475 
1476       /* modify k-th row by adding in those rows i with U(i,k)!=0 */
1477       dk = rtmp[k];
1478       i = jl[k]; /* first row to be added to k_th row  */
1479 
1480       while (i < k){
1481         nexti = jl[i]; /* next row to be added to k_th row */
1482 
1483         /* compute multiplier, update diag(k) and U(i,k) */
1484         ili = il[i];  /* index of first nonzero element in U(i,k:bms-1) */
1485         uikdi = - ba[ili]*ba[bi[i]];  /* diagonal(k) */
1486         dk += uikdi*ba[ili];
1487         ba[ili] = uikdi; /* -U(i,k) */
1488 
1489         /* add multiple of row i to k-th row */
1490         jmin = ili + 1; jmax = bi[i+1];
1491         if (jmin < jmax){
1492           for (j=jmin; j<jmax; j++) rtmp[bj[j]] += uikdi*ba[j];
1493           /* update il and jl for row i */
1494           il[i] = jmin;
1495           j = bj[jmin]; jl[i] = jl[j]; jl[j] = i;
1496         }
1497         i = nexti;
1498       }
1499 
1500       /* shift the diagonals when zero pivot is detected */
1501       /* compute rs=sum of abs(off-diagonal) */
1502       rs   = 0.0;
1503       jmin = bi[k]+1;
1504       nz   = bi[k+1] - jmin;
1505       bcol = bj + jmin;
1506       while (nz--){
1507         rs += PetscAbsScalar(rtmp[*bcol]);
1508         bcol++;
1509       }
1510 
1511       sctx.rs = rs;
1512       sctx.pv = dk;
1513       ierr = MatCholeskyCheckShift_inline(info,sctx,k,newshift);CHKERRQ(ierr);
1514 
1515       if (newshift == 1) {
1516         if (!sctx.shift_amount) {
1517           sctx.shift_amount = 1e-5;
1518         }
1519         break;
1520       }
1521 
1522       /* copy data into U(k,:) */
1523       ba[bi[k]] = 1.0/dk; /* U(k,k) */
1524       jmin = bi[k]+1; jmax = bi[k+1];
1525       if (jmin < jmax) {
1526         for (j=jmin; j<jmax; j++){
1527           col = bj[j]; ba[j] = rtmp[col]; rtmp[col] = 0.0;
1528         }
1529         /* add the k-th row into il and jl */
1530         il[k] = jmin;
1531         i = bj[jmin]; jl[k] = jl[i]; jl[i] = k;
1532       }
1533     }
1534   } while (sctx.chshift);
1535   ierr = PetscFree(il);CHKERRQ(ierr);
1536 
1537   ierr = ISRestoreIndices(ip,&rip);CHKERRQ(ierr);
1538   ierr = ISRestoreIndices(iip,&riip);CHKERRQ(ierr);
1539 
1540   ierr = ISIdentity(ip,&perm_identity);CHKERRQ(ierr);
1541   if (perm_identity){
1542     (B)->ops->solve           = MatSolve_SeqSBAIJ_1_NaturalOrdering;
1543     (B)->ops->solvetranspose  = MatSolve_SeqSBAIJ_1_NaturalOrdering;
1544     (B)->ops->forwardsolve    = MatForwardSolve_SeqSBAIJ_1_NaturalOrdering;
1545     (B)->ops->backwardsolve   = MatBackwardSolve_SeqSBAIJ_1_NaturalOrdering;
1546   } else {
1547     (B)->ops->solve           = MatSolve_SeqSBAIJ_1;
1548     (B)->ops->solvetranspose  = MatSolve_SeqSBAIJ_1;
1549     (B)->ops->forwardsolve    = MatForwardSolve_SeqSBAIJ_1;
1550     (B)->ops->backwardsolve   = MatBackwardSolve_SeqSBAIJ_1;
1551   }
1552 
1553   C->assembled    = PETSC_TRUE;
1554   C->preallocated = PETSC_TRUE;
1555   ierr = PetscLogFlops(C->rmap->n);CHKERRQ(ierr);
1556   if (sctx.nshift){
1557     if (shiftnz) {
1558       ierr = PetscInfo2(A,"number of shiftnz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
1559     } else if (shiftpd) {
1560       ierr = PetscInfo2(A,"number of shiftpd tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
1561     }
1562   }
1563   PetscFunctionReturn(0);
1564 }
1565 
1566 #undef __FUNCT__
1567 #define __FUNCT__ "MatICCFactorSymbolic_SeqAIJ"
1568 PetscErrorCode MatICCFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
1569 {
1570   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data;
1571   Mat_SeqSBAIJ       *b;
1572   PetscErrorCode     ierr;
1573   PetscTruth         perm_identity,missing;
1574   PetscInt           reallocs=0,i,*ai=a->i,*aj=a->j,am=A->rmap->n,*ui;
1575   const PetscInt     *rip,*riip;
1576   PetscInt           jmin,jmax,nzk,k,j,*jl,prow,*il,nextprow;
1577   PetscInt           nlnk,*lnk,*lnk_lvl=PETSC_NULL,d;
1578   PetscInt           ncols,ncols_upper,*cols,*ajtmp,*uj,**uj_ptr,**uj_lvl_ptr;
1579   PetscReal          fill=info->fill,levels=info->levels;
1580   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
1581   PetscFreeSpaceList free_space_lvl=PETSC_NULL,current_space_lvl=PETSC_NULL;
1582   PetscBT            lnkbt;
1583   IS                 iperm;
1584 
1585   PetscFunctionBegin;
1586   if (A->rmap->n != A->cmap->n) SETERRQ2(PETSC_ERR_ARG_WRONG,"Must be square matrix, rows %D columns %D",A->rmap->n,A->cmap->n);
1587   ierr = MatMissingDiagonal(A,&missing,&d);CHKERRQ(ierr);
1588   if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",d);
1589   ierr = ISIdentity(perm,&perm_identity);CHKERRQ(ierr);
1590   ierr = ISInvertPermutation(perm,PETSC_DECIDE,&iperm);CHKERRQ(ierr);
1591 
1592   ierr = PetscMalloc((am+1)*sizeof(PetscInt),&ui);CHKERRQ(ierr);
1593   ui[0] = 0;
1594 
1595   /* ICC(0) without matrix ordering: simply copies fill pattern */
1596   if (!levels && perm_identity) {
1597 
1598     for (i=0; i<am; i++) {
1599       ui[i+1] = ui[i] + ai[i+1] - a->diag[i];
1600     }
1601     ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
1602     cols = uj;
1603     for (i=0; i<am; i++) {
1604       aj    = a->j + a->diag[i];
1605       ncols = ui[i+1] - ui[i];
1606       for (j=0; j<ncols; j++) *cols++ = *aj++;
1607     }
1608   } else { /* case: levels>0 || (levels=0 && !perm_identity) */
1609     ierr = ISGetIndices(iperm,&riip);CHKERRQ(ierr);
1610     ierr = ISGetIndices(perm,&rip);CHKERRQ(ierr);
1611 
1612     /* initialization */
1613     ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&ajtmp);CHKERRQ(ierr);
1614 
1615     /* jl: linked list for storing indices of the pivot rows
1616        il: il[i] points to the 1st nonzero entry of U(i,k:am-1) */
1617     ierr = PetscMalloc((2*am+1)*sizeof(PetscInt)+2*am*sizeof(PetscInt**),&jl);CHKERRQ(ierr);
1618     il         = jl + am;
1619     uj_ptr     = (PetscInt**)(il + am);
1620     uj_lvl_ptr = (PetscInt**)(uj_ptr + am);
1621     for (i=0; i<am; i++){
1622       jl[i] = am; il[i] = 0;
1623     }
1624 
1625     /* create and initialize a linked list for storing column indices of the active row k */
1626     nlnk = am + 1;
1627     ierr = PetscIncompleteLLCreate(am,am,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1628 
1629     /* initial FreeSpace size is fill*(ai[am]+1) */
1630     ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space);CHKERRQ(ierr);
1631     current_space = free_space;
1632     ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space_lvl);CHKERRQ(ierr);
1633     current_space_lvl = free_space_lvl;
1634 
1635     for (k=0; k<am; k++){  /* for each active row k */
1636       /* initialize lnk by the column indices of row rip[k] of A */
1637       nzk   = 0;
1638       ncols = ai[rip[k]+1] - ai[rip[k]];
1639       if (!ncols) SETERRQ2(PETSC_ERR_MAT_CH_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",rip[k],k);
1640       ncols_upper = 0;
1641       for (j=0; j<ncols; j++){
1642         i = *(aj + ai[rip[k]] + j); /* unpermuted column index */
1643         if (riip[i] >= k){ /* only take upper triangular entry */
1644           ajtmp[ncols_upper] = i;
1645           ncols_upper++;
1646         }
1647       }
1648       ierr = PetscIncompleteLLInit(ncols_upper,ajtmp,am,riip,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1649       nzk += nlnk;
1650 
1651       /* update lnk by computing fill-in for each pivot row to be merged in */
1652       prow = jl[k]; /* 1st pivot row */
1653 
1654       while (prow < k){
1655         nextprow = jl[prow];
1656 
1657         /* merge prow into k-th row */
1658         jmin = il[prow] + 1;  /* index of the 2nd nzero entry in U(prow,k:am-1) */
1659         jmax = ui[prow+1];
1660         ncols = jmax-jmin;
1661         i     = jmin - ui[prow];
1662         cols  = uj_ptr[prow] + i; /* points to the 2nd nzero entry in U(prow,k:am-1) */
1663         uj    = uj_lvl_ptr[prow] + i; /* levels of cols */
1664         j     = *(uj - 1);
1665         ierr = PetscICCLLAddSorted(ncols,cols,levels,uj,am,nlnk,lnk,lnk_lvl,lnkbt,j);CHKERRQ(ierr);
1666         nzk += nlnk;
1667 
1668         /* update il and jl for prow */
1669         if (jmin < jmax){
1670           il[prow] = jmin;
1671           j = *cols; jl[prow] = jl[j]; jl[j] = prow;
1672         }
1673         prow = nextprow;
1674       }
1675 
1676       /* if free space is not available, make more free space */
1677       if (current_space->local_remaining<nzk) {
1678         i = am - k + 1; /* num of unfactored rows */
1679         i = PetscMin(i*nzk, i*(i-1)); /* i*nzk, i*(i-1): estimated and max additional space needed */
1680         ierr = PetscFreeSpaceGet(i,&current_space);CHKERRQ(ierr);
1681         ierr = PetscFreeSpaceGet(i,&current_space_lvl);CHKERRQ(ierr);
1682         reallocs++;
1683       }
1684 
1685       /* copy data into free_space and free_space_lvl, then initialize lnk */
1686       if (nzk == 0) SETERRQ1(PETSC_ERR_ARG_WRONG,"Empty row %D in ICC matrix factor",k);
1687       ierr = PetscIncompleteLLClean(am,am,nzk,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);CHKERRQ(ierr);
1688 
1689       /* add the k-th row into il and jl */
1690       if (nzk > 1){
1691         i = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:am-1) */
1692         jl[k] = jl[i]; jl[i] = k;
1693         il[k] = ui[k] + 1;
1694       }
1695       uj_ptr[k]     = current_space->array;
1696       uj_lvl_ptr[k] = current_space_lvl->array;
1697 
1698       current_space->array           += nzk;
1699       current_space->local_used      += nzk;
1700       current_space->local_remaining -= nzk;
1701 
1702       current_space_lvl->array           += nzk;
1703       current_space_lvl->local_used      += nzk;
1704       current_space_lvl->local_remaining -= nzk;
1705 
1706       ui[k+1] = ui[k] + nzk;
1707     }
1708 
1709 #if defined(PETSC_USE_INFO)
1710     if (ai[am] != 0) {
1711       PetscReal af = (PetscReal)ui[am]/((PetscReal)ai[am]);
1712       ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,fill,af);CHKERRQ(ierr);
1713       ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
1714       ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G) for best performance.\n",af);CHKERRQ(ierr);
1715     } else {
1716       ierr = PetscInfo(A,"Empty matrix.\n");CHKERRQ(ierr);
1717     }
1718 #endif
1719 
1720     ierr = ISRestoreIndices(perm,&rip);CHKERRQ(ierr);
1721     ierr = ISRestoreIndices(iperm,&riip);CHKERRQ(ierr);
1722     ierr = PetscFree(jl);CHKERRQ(ierr);
1723     ierr = PetscFree(ajtmp);CHKERRQ(ierr);
1724 
1725     /* destroy list of free space and other temporary array(s) */
1726     ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
1727     ierr = PetscFreeSpaceContiguous(&free_space,uj);CHKERRQ(ierr);
1728     ierr = PetscIncompleteLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1729     ierr = PetscFreeSpaceDestroy(free_space_lvl);CHKERRQ(ierr);
1730 
1731   } /* end of case: levels>0 || (levels=0 && !perm_identity) */
1732 
1733   /* put together the new matrix in MATSEQSBAIJ format */
1734 
1735   b    = (Mat_SeqSBAIJ*)(fact)->data;
1736   b->singlemalloc = PETSC_FALSE;
1737   ierr = PetscMalloc((ui[am]+1)*sizeof(MatScalar),&b->a);CHKERRQ(ierr);
1738   b->j    = uj;
1739   b->i    = ui;
1740   b->diag = 0;
1741   b->ilen = 0;
1742   b->imax = 0;
1743   b->row  = perm;
1744   b->col  = perm;
1745   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
1746   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
1747   b->icol = iperm;
1748   b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */
1749   ierr    = PetscMalloc((am+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1750   ierr = PetscLogObjectMemory((fact),(ui[am]-am)*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
1751   b->maxnz   = b->nz = ui[am];
1752   b->free_a  = PETSC_TRUE;
1753   b->free_ij = PETSC_TRUE;
1754 
1755   (fact)->info.factor_mallocs    = reallocs;
1756   (fact)->info.fill_ratio_given  = fill;
1757   if (ai[am] != 0) {
1758     (fact)->info.fill_ratio_needed = ((PetscReal)ui[am])/((PetscReal)ai[am]);
1759   } else {
1760     (fact)->info.fill_ratio_needed = 0.0;
1761   }
1762   (fact)->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqAIJ;
1763   PetscFunctionReturn(0);
1764 }
1765 
1766 #undef __FUNCT__
1767 #define __FUNCT__ "MatCholeskyFactorSymbolic_SeqAIJ"
1768 PetscErrorCode MatCholeskyFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
1769 {
1770   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data;
1771   Mat_SeqSBAIJ       *b;
1772   PetscErrorCode     ierr;
1773   PetscTruth         perm_identity;
1774   PetscReal          fill = info->fill;
1775   const PetscInt     *rip,*riip;
1776   PetscInt           i,am=A->rmap->n,*ai=a->i,*aj=a->j,reallocs=0,prow;
1777   PetscInt           *jl,jmin,jmax,nzk,*ui,k,j,*il,nextprow;
1778   PetscInt           nlnk,*lnk,ncols,ncols_upper,*cols,*uj,**ui_ptr,*uj_ptr;
1779   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
1780   PetscBT            lnkbt;
1781   IS                 iperm;
1782 
1783   PetscFunctionBegin;
1784   if (A->rmap->n != A->cmap->n) SETERRQ2(PETSC_ERR_ARG_WRONG,"Must be square matrix, rows %D columns %D",A->rmap->n,A->cmap->n);
1785   /* check whether perm is the identity mapping */
1786   ierr = ISIdentity(perm,&perm_identity);CHKERRQ(ierr);
1787   ierr = ISInvertPermutation(perm,PETSC_DECIDE,&iperm);CHKERRQ(ierr);
1788   ierr = ISGetIndices(iperm,&riip);CHKERRQ(ierr);
1789   ierr = ISGetIndices(perm,&rip);CHKERRQ(ierr);
1790 
1791   /* initialization */
1792   ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&ui);CHKERRQ(ierr);
1793   ui[0] = 0;
1794 
1795   /* jl: linked list for storing indices of the pivot rows
1796      il: il[i] points to the 1st nonzero entry of U(i,k:am-1) */
1797   ierr = PetscMalloc((3*am+1)*sizeof(PetscInt)+am*sizeof(PetscInt**),&jl);CHKERRQ(ierr);
1798   il     = jl + am;
1799   cols   = il + am;
1800   ui_ptr = (PetscInt**)(cols + am);
1801   for (i=0; i<am; i++){
1802     jl[i] = am; il[i] = 0;
1803   }
1804 
1805   /* create and initialize a linked list for storing column indices of the active row k */
1806   nlnk = am + 1;
1807   ierr = PetscLLCreate(am,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
1808 
1809   /* initial FreeSpace size is fill*(ai[am]+1) */
1810   ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space);CHKERRQ(ierr);
1811   current_space = free_space;
1812 
1813   for (k=0; k<am; k++){  /* for each active row k */
1814     /* initialize lnk by the column indices of row rip[k] of A */
1815     nzk   = 0;
1816     ncols = ai[rip[k]+1] - ai[rip[k]];
1817     if (!ncols) SETERRQ2(PETSC_ERR_MAT_CH_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",rip[k],k);
1818     ncols_upper = 0;
1819     for (j=0; j<ncols; j++){
1820       i = riip[*(aj + ai[rip[k]] + j)];
1821       if (i >= k){ /* only take upper triangular entry */
1822         cols[ncols_upper] = i;
1823         ncols_upper++;
1824       }
1825     }
1826     ierr = PetscLLAdd(ncols_upper,cols,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
1827     nzk += nlnk;
1828 
1829     /* update lnk by computing fill-in for each pivot row to be merged in */
1830     prow = jl[k]; /* 1st pivot row */
1831 
1832     while (prow < k){
1833       nextprow = jl[prow];
1834       /* merge prow into k-th row */
1835       jmin = il[prow] + 1;  /* index of the 2nd nzero entry in U(prow,k:am-1) */
1836       jmax = ui[prow+1];
1837       ncols = jmax-jmin;
1838       uj_ptr = ui_ptr[prow] + jmin - ui[prow]; /* points to the 2nd nzero entry in U(prow,k:am-1) */
1839       ierr = PetscLLAddSorted(ncols,uj_ptr,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
1840       nzk += nlnk;
1841 
1842       /* update il and jl for prow */
1843       if (jmin < jmax){
1844         il[prow] = jmin;
1845         j = *uj_ptr; jl[prow] = jl[j]; jl[j] = prow;
1846       }
1847       prow = nextprow;
1848     }
1849 
1850     /* if free space is not available, make more free space */
1851     if (current_space->local_remaining<nzk) {
1852       i = am - k + 1; /* num of unfactored rows */
1853       i = PetscMin(i*nzk, i*(i-1)); /* i*nzk, i*(i-1): estimated and max additional space needed */
1854       ierr = PetscFreeSpaceGet(i,&current_space);CHKERRQ(ierr);
1855       reallocs++;
1856     }
1857 
1858     /* copy data into free space, then initialize lnk */
1859     ierr = PetscLLClean(am,am,nzk,lnk,current_space->array,lnkbt);CHKERRQ(ierr);
1860 
1861     /* add the k-th row into il and jl */
1862     if (nzk-1 > 0){
1863       i = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:am-1) */
1864       jl[k] = jl[i]; jl[i] = k;
1865       il[k] = ui[k] + 1;
1866     }
1867     ui_ptr[k] = current_space->array;
1868     current_space->array           += nzk;
1869     current_space->local_used      += nzk;
1870     current_space->local_remaining -= nzk;
1871 
1872     ui[k+1] = ui[k] + nzk;
1873   }
1874 
1875 #if defined(PETSC_USE_INFO)
1876   if (ai[am] != 0) {
1877     PetscReal af = (PetscReal)(ui[am])/((PetscReal)ai[am]);
1878     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,fill,af);CHKERRQ(ierr);
1879     ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
1880     ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G) for best performance.\n",af);CHKERRQ(ierr);
1881   } else {
1882      ierr = PetscInfo(A,"Empty matrix.\n");CHKERRQ(ierr);
1883   }
1884 #endif
1885 
1886   ierr = ISRestoreIndices(perm,&rip);CHKERRQ(ierr);
1887   ierr = ISRestoreIndices(iperm,&riip);CHKERRQ(ierr);
1888   ierr = PetscFree(jl);CHKERRQ(ierr);
1889 
1890   /* destroy list of free space and other temporary array(s) */
1891   ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
1892   ierr = PetscFreeSpaceContiguous(&free_space,uj);CHKERRQ(ierr);
1893   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1894 
1895   /* put together the new matrix in MATSEQSBAIJ format */
1896 
1897   b = (Mat_SeqSBAIJ*)(fact)->data;
1898   b->singlemalloc = PETSC_FALSE;
1899   b->free_a       = PETSC_TRUE;
1900   b->free_ij      = PETSC_TRUE;
1901   ierr = PetscMalloc((ui[am]+1)*sizeof(MatScalar),&b->a);CHKERRQ(ierr);
1902   b->j    = uj;
1903   b->i    = ui;
1904   b->diag = 0;
1905   b->ilen = 0;
1906   b->imax = 0;
1907   b->row  = perm;
1908   b->col  = perm;
1909   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
1910   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
1911   b->icol = iperm;
1912   b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */
1913   ierr    = PetscMalloc((am+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1914   ierr    = PetscLogObjectMemory(fact,(ui[am]-am)*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
1915   b->maxnz = b->nz = ui[am];
1916 
1917   (fact)->info.factor_mallocs    = reallocs;
1918   (fact)->info.fill_ratio_given  = fill;
1919   if (ai[am] != 0) {
1920     (fact)->info.fill_ratio_needed = ((PetscReal)ui[am])/((PetscReal)ai[am]);
1921   } else {
1922     (fact)->info.fill_ratio_needed = 0.0;
1923   }
1924   (fact)->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqAIJ;
1925   PetscFunctionReturn(0);
1926 }
1927 
1928 #undef __FUNCT__
1929 #define __FUNCT__ "MatSolve_SeqAIJ_NaturalOrdering_iludt"
1930 PetscErrorCode MatSolve_SeqAIJ_NaturalOrdering_iludt(Mat A,Vec bb,Vec xx)
1931 {
1932   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1933   PetscErrorCode    ierr;
1934   PetscInt          n = A->rmap->n;
1935   const PetscInt    *ai = a->i,*aj = a->j,*vi,*adiag_rev=a->diag+n+1;
1936   PetscScalar       *x,sum;
1937   const PetscScalar *b;
1938   const MatScalar   *aa = a->a,*v;
1939   PetscInt          i,nz,k;
1940 
1941   PetscFunctionBegin;
1942   if (!n) PetscFunctionReturn(0);
1943 
1944   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1945   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1946 
1947   /* forward solve the lower triangular */
1948   x[0] = b[0];
1949   v    = aa;
1950   vi   = aj;
1951   for (i=1; i<n; i++) {
1952     nz  = ai[i+1] - ai[i];
1953     sum = b[i];
1954     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
1955     /*    while (nz--) sum -= *v++ * x[*vi++];*/
1956     v  += nz;
1957     vi += nz;
1958     x[i] = sum;
1959   }
1960 
1961   /* backward solve the upper triangular */
1962   v   = aa + adiag_rev[0] + 1;
1963   vi  = aj + adiag_rev[0] + 1;
1964   k = 0;  /* i + k = n */
1965   for (i=n-1; i>=0; i--){
1966     /* nz  = adiag[i] - adiag[i+1] - 1; */
1967     nz = adiag_rev[k+1] - adiag_rev[k] - 1;
1968     sum = x[i];
1969     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
1970     /* while (nz--) sum -= *v++ * x[*vi++]; */
1971     v   += nz;
1972     vi  += nz;
1973     x[i] = *v++ *sum; /* x[i]=aa[adiag[i]]*sum; v++; */
1974     vi++; k++;
1975   }
1976 
1977   ierr = PetscLogFlops(2.0*a->nz - A->cmap->n);CHKERRQ(ierr);
1978   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1979   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1980   PetscFunctionReturn(0);
1981 }
1982 
1983 #undef __FUNCT__
1984 #define __FUNCT__ "MatSolve_SeqAIJ_iludt"
1985 PetscErrorCode MatSolve_SeqAIJ_iludt(Mat A,Vec bb,Vec xx)
1986 {
1987   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1988   IS                iscol = a->col,isrow = a->row;
1989   PetscErrorCode    ierr;
1990   PetscInt          i,n=A->rmap->n,*vi,*ai = a->i,*aj = a->j,*adiag=a->diag;
1991   PetscInt          nz;
1992   const PetscInt    *rout,*cout,*r,*c;
1993   PetscScalar       *x,*tmp,*tmps;
1994   const PetscScalar *b;
1995   const MatScalar   *aa = a->a,*v;
1996 
1997   PetscFunctionBegin;
1998   if (!n) PetscFunctionReturn(0);
1999 
2000   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
2001   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
2002   tmp  = a->solve_work;
2003 
2004   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
2005   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout + (n-1);
2006 
2007   /* forward solve the lower triangular */
2008   tmp[0] = b[*r++];
2009   tmps   = tmp;
2010   v      = aa;
2011   vi     = aj;
2012   for (i=1; i<n; i++) {
2013     nz  = ai[i+1] - ai[i];
2014     tmp[i] = b[*r++];
2015     PetscSparseDenseMinusDot(tmp[i],tmps,v,vi,nz);
2016     v += nz; vi += nz;
2017   }
2018 
2019   /* backward solve the upper triangular */
2020   v   = aa + adiag[n] + 1;
2021   vi  = aj + adiag[n] + 1;
2022   for (i=n-1; i>=0; i--){
2023     nz  = adiag[i] - adiag[i+1] - 1;
2024     PetscSparseDenseMinusDot(tmp[i],tmps,v,vi,nz);
2025     x[*c--] = tmp[i] = tmp[i]*aa[adiag[i]];
2026     v += nz+1; vi += nz+1;
2027   }
2028 
2029   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
2030   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
2031   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
2032   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
2033   ierr = PetscLogFlops(2*a->nz - A->cmap->n);CHKERRQ(ierr);
2034   PetscFunctionReturn(0);
2035 }
2036 
2037 #undef __FUNCT__
2038 #define __FUNCT__ "MatILUDTFactor_SeqAIJ"
2039 PetscErrorCode MatILUDTFactor_SeqAIJ(Mat A,IS isrow,IS iscol,const MatFactorInfo *info,Mat *fact)
2040 {
2041   Mat                B = *fact;
2042   Mat_SeqAIJ         *a=(Mat_SeqAIJ*)A->data,*b;
2043   IS                 isicol;
2044   PetscErrorCode     ierr;
2045   const PetscInt     *r,*ic;
2046   PetscInt           i,n=A->rmap->n,*ai=a->i,*aj=a->j,*ajtmp,*adiag;
2047   PetscInt           *bi,*bj,*bdiag,*bdiag_rev;
2048   PetscInt           row,nzi,nzi_bl,nzi_bu,*im,dtcount,nzi_al,nzi_au;
2049   PetscInt           nlnk,*lnk;
2050   PetscBT            lnkbt;
2051   PetscTruth         row_identity,icol_identity,both_identity;
2052   MatScalar          *aatmp,*pv,*batmp,*ba,*rtmp,*pc,multiplier,*vtmp,diag_tmp;
2053   const PetscInt     *ics;
2054   PetscInt           j,nz,*pj,*bjtmp,k,ncut,*jtmp;
2055   PetscReal          dt=info->dt,shift=info->shiftinblocks;
2056   PetscInt           nnz_max;
2057   PetscTruth         missing;
2058 
2059   PetscFunctionBegin;
2060   printf("MatILUDTFactor_SeqAIJ is callled ...\n");
2061   /* ------- symbolic factorization, can be reused ---------*/
2062   ierr = MatMissingDiagonal(A,&missing,&i);CHKERRQ(ierr);
2063   if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",i);
2064   adiag=a->diag;
2065 
2066   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
2067 
2068   /* bdiag is location of diagonal in factor */
2069   ierr = PetscMalloc((2*n+2)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);
2070   bdiag_rev = bdiag + n+1;
2071 
2072   /* allocate row pointers bi */
2073   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
2074 
2075   /* allocate bj and ba; max num of nonzero entries is (ai[n]+2*n*dtcount+2) */
2076   dtcount = (PetscInt)info->dtcount;
2077   if (dtcount > n-1) dtcount = n-1; /* diagonal is excluded */
2078   nnz_max  = ai[n]+2*n*dtcount+2;
2079   /* if (nnz_max > n*n) nnz_max = n*n+1; n*n may cause overflow ?? */
2080   ierr = PetscMalloc(nnz_max*sizeof(PetscInt),&bj);CHKERRQ(ierr);
2081   ierr = PetscMalloc(nnz_max*sizeof(MatScalar),&ba);CHKERRQ(ierr);
2082 
2083   /* put together the new matrix */
2084   ierr = MatSeqAIJSetPreallocation_SeqAIJ(B,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
2085   ierr = PetscLogObjectParent(B,isicol);CHKERRQ(ierr);
2086   b    = (Mat_SeqAIJ*)(B)->data;
2087   b->free_a       = PETSC_TRUE;
2088   b->free_ij      = PETSC_TRUE;
2089   b->singlemalloc = PETSC_FALSE;
2090   b->a          = ba;
2091   b->j          = bj;
2092   b->i          = bi;
2093   b->diag       = bdiag;
2094   b->ilen       = 0;
2095   b->imax       = 0;
2096   b->row        = isrow;
2097   b->col        = iscol;
2098   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
2099   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
2100   b->icol       = isicol;
2101   ierr = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
2102 
2103   ierr = PetscLogObjectMemory(B,nnz_max*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
2104   b->maxnz = nnz_max;
2105 
2106   (B)->factor                = MAT_FACTOR_ILUDT;
2107   (B)->info.factor_mallocs   = 0;
2108   (B)->info.fill_ratio_given = ((PetscReal)nnz_max)/((PetscReal)ai[n]);
2109   CHKMEMQ;
2110   /* ------- end of symbolic factorization ---------*/
2111 
2112   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
2113   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
2114   ics  = ic;
2115 
2116   /* linked list for storing column indices of the active row */
2117   nlnk = n + 1;
2118   ierr = PetscLLCreate(n,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2119 
2120   /* im: used by PetscLLAddSortedLU(); jtmp: working array for column indices of active row */
2121   ierr = PetscMalloc((2*n+1)*sizeof(PetscInt),&im);CHKERRQ(ierr);
2122   jtmp = im + n;
2123   /* rtmp, vtmp: working arrays for sparse and contiguous row entries of active row */
2124   ierr = PetscMalloc((2*n+1)*sizeof(MatScalar),&rtmp);CHKERRQ(ierr);
2125   ierr = PetscMemzero(rtmp,(n+1)*sizeof(PetscScalar));CHKERRQ(ierr);
2126   vtmp = rtmp + n;
2127 
2128   bi[0]    = 0;
2129   bdiag[0] = nnz_max-1; /* location of diagonal in factor B */
2130   bdiag_rev[n] = bdiag[0];
2131   for (i=0; i<n; i++) {
2132     /* copy initial fill into linked list */
2133     nzi = 0; /* nonzeros for active row i */
2134     nzi = ai[r[i]+1] - ai[r[i]];
2135     if (!nzi) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
2136     nzi_al = adiag[r[i]] - ai[r[i]];
2137     nzi_au = ai[r[i]+1] - adiag[r[i]] -1;
2138     ajtmp = aj + ai[r[i]];
2139     ierr = PetscLLAddPerm(nzi,ajtmp,ic,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2140 
2141     /* load in initial (unfactored row) */
2142     aatmp = a->a + ai[r[i]];
2143     for (j=0; j<nzi; j++) {
2144       rtmp[ics[*ajtmp++]] = *aatmp++;
2145     }
2146 
2147     /* add pivot rows into linked list */
2148     row = lnk[n];
2149     while (row < i ) {
2150       nzi_bl = bi[row+1] - bi[row] + 1;
2151       bjtmp = bj + bdiag[row+1]+1; /* points to 1st column next to the diagonal in U */
2152       ierr  = PetscLLAddSortedLU(bjtmp,row,nlnk,lnk,lnkbt,i,nzi_bl,im);CHKERRQ(ierr);
2153       nzi  += nlnk;
2154       row   = lnk[row];
2155     }
2156 
2157     /* copy data from lnk into jtmp, then initialize lnk */
2158     ierr = PetscLLClean(n,n,nzi,lnk,jtmp,lnkbt);CHKERRQ(ierr);
2159 
2160     /* numerical factorization */
2161     bjtmp = jtmp;
2162     row   = *bjtmp++; /* 1st pivot row */
2163     while  ( row < i ) {
2164       pc         = rtmp + row;
2165       pv         = ba + bdiag[row]; /* 1./(diag of the pivot row) */
2166       multiplier = (*pc) * (*pv);
2167       *pc        = multiplier;
2168       if (PetscAbsScalar(*pc) > dt){ /* apply tolerance dropping rule */
2169         pj         = bj + bdiag[row+1] + 1; /* point to 1st entry of U(row,:) */
2170         pv         = ba + bdiag[row+1] + 1;
2171         /* if (multiplier < -1.0 or multiplier >1.0) printf("row/prow %d, %d, multiplier %g\n",i,row,multiplier); */
2172         nz         = bdiag[row] - bdiag[row+1] - 1; /* num of entries in U(row,:), excluding diagonal */
2173         for (j=0; j<nz; j++) rtmp[*pj++] -= multiplier * (*pv++);
2174         ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr);
2175       }
2176       row = *bjtmp++;
2177     }
2178 
2179     /* copy sparse rtmp into contiguous vtmp; separate L and U part */
2180     diag_tmp = rtmp[i];  /* save diagonal value - may not needed?? */
2181     nzi_bl = 0; j = 0;
2182     while (jtmp[j] < i){ /* Note: jtmp is sorted */
2183       vtmp[j] = rtmp[jtmp[j]]; rtmp[jtmp[j]]=0.0;
2184       nzi_bl++; j++;
2185     }
2186     nzi_bu = nzi - nzi_bl -1;
2187     while (j < nzi){
2188       vtmp[j] = rtmp[jtmp[j]]; rtmp[jtmp[j]]=0.0;
2189       j++;
2190     }
2191 
2192     bjtmp = bj + bi[i];
2193     batmp = ba + bi[i];
2194     /* apply level dropping rule to L part */
2195     ncut = nzi_al + dtcount;
2196     if (ncut < nzi_bl){
2197       ierr = PetscSortSplit(ncut,nzi_bl,vtmp,jtmp);CHKERRQ(ierr);
2198       ierr = PetscSortIntWithScalarArray(ncut,jtmp,vtmp);CHKERRQ(ierr);
2199     } else {
2200       ncut = nzi_bl;
2201     }
2202     for (j=0; j<ncut; j++){
2203       bjtmp[j] = jtmp[j];
2204       batmp[j] = vtmp[j];
2205       /* printf(" (%d,%g),",bjtmp[j],batmp[j]); */
2206     }
2207     bi[i+1] = bi[i] + ncut;
2208     nzi = ncut + 1;
2209 
2210     /* apply level dropping rule to U part */
2211     ncut = nzi_au + dtcount;
2212     if (ncut < nzi_bu){
2213       ierr = PetscSortSplit(ncut,nzi_bu,vtmp+nzi_bl+1,jtmp+nzi_bl+1);CHKERRQ(ierr);
2214       ierr = PetscSortIntWithScalarArray(ncut,jtmp+nzi_bl+1,vtmp+nzi_bl+1);CHKERRQ(ierr);
2215     } else {
2216       ncut = nzi_bu;
2217     }
2218     nzi += ncut;
2219 
2220     /* mark bdiagonal */
2221     bdiag[i+1]       = bdiag[i] - (ncut + 1);
2222     bdiag_rev[n-i-1] = bdiag[i+1];
2223     bjtmp = bj + bdiag[i];
2224     batmp = ba + bdiag[i];
2225     *bjtmp = i;
2226     *batmp = diag_tmp; /* rtmp[i]; */
2227     if (*batmp == 0.0) {
2228       *batmp = dt+shift;
2229       printf(" row %d add shift %g\n",i,shift);
2230     }
2231     *batmp = 1.0/(*batmp); /* invert diagonal entries for simplier triangular solves */
2232     /* printf(" (%d,%g),",*bjtmp,*batmp); */
2233 
2234     bjtmp = bj + bdiag[i+1]+1;
2235     batmp = ba + bdiag[i+1]+1;
2236     for (k=0; k<ncut; k++){
2237       bjtmp[k] = jtmp[nzi_bl+1+k];
2238       batmp[k] = vtmp[nzi_bl+1+k];
2239       /* printf(" (%d,%g),",bjtmp[k],batmp[k]); */
2240     }
2241     /* printf("\n"); */
2242 
2243     im[i]   = nzi; /* used by PetscLLAddSortedLU() */
2244     /*
2245     printf("row %d: bi %d, bdiag %d\n",i,bi[i],bdiag[i]);
2246     printf(" ----------------------------\n");
2247     */
2248   } /* for (i=0; i<n; i++) */
2249   /* printf("end of L %d, beginning of U %d\n",bi[n],bdiag[n]); */
2250   if (bi[n] >= bdiag[n]) SETERRQ2(PETSC_ERR_ARG_SIZ,"end of L array %d cannot >= the beginning of U array %d",bi[n],bdiag[n]);
2251 
2252   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
2253   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
2254 
2255   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
2256   ierr = PetscFree(im);CHKERRQ(ierr);
2257   ierr = PetscFree(rtmp);CHKERRQ(ierr);
2258 
2259   ierr = PetscLogFlops(B->cmap->n);CHKERRQ(ierr);
2260   b->maxnz = b->nz = bi[n] + bdiag[0] - bdiag[n];
2261 
2262   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
2263   ierr = ISIdentity(isicol,&icol_identity);CHKERRQ(ierr);
2264   both_identity = (PetscTruth) (row_identity && icol_identity);
2265   if (row_identity && icol_identity) {
2266     B->ops->solve = MatSolve_SeqAIJ_NaturalOrdering_iludt;
2267   } else {
2268     B->ops->solve = MatSolve_SeqAIJ_iludt;
2269   }
2270 
2271   B->ops->lufactorsymbolic  = MatILUDTFactorSymbolic_SeqAIJ;
2272   B->ops->lufactornumeric   = MatILUDTFactorNumeric_SeqAIJ;
2273   B->ops->solveadd          = 0;
2274   B->ops->solvetranspose    = 0;
2275   B->ops->solvetransposeadd = 0;
2276   B->ops->matsolve          = 0;
2277   B->assembled              = PETSC_TRUE;
2278   B->preallocated           = PETSC_TRUE;
2279   PetscFunctionReturn(0);
2280 }
2281 
2282 /* a wraper of MatILUDTFactor_SeqAIJ() */
2283 #undef __FUNCT__
2284 #define __FUNCT__ "MatILUDTFactorSymbolic_SeqAIJ"
2285 PetscErrorCode PETSCMAT_DLLEXPORT MatILUDTFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS row,IS col,const MatFactorInfo *info)
2286 {
2287   PetscErrorCode     ierr;
2288 
2289   PetscFunctionBegin;
2290   printf("MatILUDTFactorSymbolic_SeqAIJ is called...\n");
2291   ierr = MatILUDTFactor_SeqAIJ(A,row,col,info,&fact);CHKERRQ(ierr);
2292 
2293   fact->ops->lufactornumeric = MatILUDTFactorNumeric_SeqAIJ;
2294   PetscFunctionReturn(0);
2295 }
2296 
2297 /*
2298    same as MatLUFactorNumeric_SeqAIJ(), except using contiguous array matrix factors
2299    - intend to replace existing MatLUFactorNumeric_SeqAIJ()
2300 */
2301 #undef __FUNCT__
2302 #define __FUNCT__ "MatILUDTFactorNumeric_SeqAIJ"
2303 PetscErrorCode PETSCMAT_DLLEXPORT MatILUDTFactorNumeric_SeqAIJ(Mat fact,Mat A,const MatFactorInfo *info)
2304 {
2305   Mat            C=fact;
2306   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ *)C->data;
2307   IS             isrow = b->row,isicol = b->icol;
2308   PetscErrorCode ierr;
2309   const PetscInt *r,*ic,*ics;
2310   PetscInt       i,j,k,n=A->rmap->n,*ai=a->i,*aj=a->j,*bi=b->i,*bj=b->j;
2311   PetscInt       *ajtmp,*bjtmp,nz,nzl,nzu,row,*bdiag = b->diag,*pj;
2312   MatScalar      *rtmp,*pc,multiplier,*v,*pv,*aa=a->a;
2313   PetscReal      dt=info->dt,shift=info->shiftinblocks;
2314   PetscTruth     row_identity, col_identity;
2315 
2316   PetscFunctionBegin;
2317   printf("MatILUDTFactorNumeric_SeqAIJ is called...\n");
2318   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
2319   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
2320   ierr = PetscMalloc((n+1)*sizeof(MatScalar),&rtmp);CHKERRQ(ierr);
2321   ics  = ic;
2322 
2323   for (i=0; i<n; i++){
2324     /* initialize rtmp array */
2325     nzl   = bi[i+1] - bi[i];       /* num of nozeros in L(i,:) */
2326     bjtmp = bj + bi[i];
2327     for  (j=0; j<nzl; j++) rtmp[*bjtmp++] = 0.0;
2328     rtmp[i] = 0.0;
2329     nzu   = bdiag[i] - bdiag[i+1]; /* num of nozeros in U(i,:) */
2330     bjtmp = bj + bdiag[i+1] + 1;
2331     for  (j=0; j<nzu; j++) rtmp[*bjtmp++] = 0.0;
2332 
2333     /* load in initial unfactored row of A */
2334     /* printf("row %d\n",i); */
2335     nz    = ai[r[i]+1] - ai[r[i]];
2336     ajtmp = aj + ai[r[i]];
2337     v     = aa + ai[r[i]];
2338     for (j=0; j<nz; j++) {
2339       rtmp[ics[*ajtmp++]] = v[j];
2340       /* printf(" (%d,%g),",ics[ajtmp[j]],rtmp[ics[ajtmp[j]]]); */
2341     }
2342     /* printf("\n"); */
2343 
2344     /* numerical factorization */
2345     bjtmp = bj + bi[i]; /* point to 1st entry of L(i,:) */
2346     nzl   = bi[i+1] - bi[i]; /* num of entries in L(i,:) */
2347     k = 0;
2348     while (k < nzl){
2349       row   = *bjtmp++;
2350       /* printf("  prow %d\n",row); */
2351       pc         = rtmp + row;
2352       pv         = b->a + bdiag[row]; /* 1./(diag of the pivot row) */
2353       multiplier = (*pc) * (*pv);
2354       *pc        = multiplier;
2355       if (PetscAbsScalar(multiplier) > dt){
2356         pj         = bj + bdiag[row+1] + 1; /* point to 1st entry of U(row,:) */
2357         pv         = b->a + bdiag[row+1] + 1;
2358         nz         = bdiag[row] - bdiag[row+1] - 1; /* num of entries in U(row,:), excluding diagonal */
2359         for (j=0; j<nz; j++) rtmp[*pj++] -= multiplier * (*pv++);
2360         /* ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr); */
2361       }
2362       k++;
2363     }
2364 
2365     /* finished row so stick it into b->a */
2366     /* L-part */
2367     pv = b->a + bi[i] ;
2368     pj = bj + bi[i] ;
2369     nzl = bi[i+1] - bi[i];
2370     for (j=0; j<nzl; j++) {
2371       pv[j] = rtmp[pj[j]];
2372       /* printf(" (%d,%g),",pj[j],pv[j]); */
2373     }
2374 
2375     /* diagonal: invert diagonal entries for simplier triangular solves */
2376     if (rtmp[i] == 0.0) rtmp[i] = dt+shift;
2377     b->a[bdiag[i]] = 1.0/rtmp[i];
2378     /* printf(" (%d,%g),",i,b->a[bdiag[i]]); */
2379 
2380     /* U-part */
2381     pv = b->a + bdiag[i+1] + 1;
2382     pj = bj + bdiag[i+1] + 1;
2383     nzu = bdiag[i] - bdiag[i+1] - 1;
2384     for (j=0; j<nzu; j++) {
2385       pv[j] = rtmp[pj[j]];
2386       /* printf(" (%d,%g),",pj[j],pv[j]); */
2387     }
2388     /* printf("\n"); */
2389   }
2390 
2391   ierr = PetscFree(rtmp);CHKERRQ(ierr);
2392   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
2393   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
2394 
2395   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
2396   ierr = ISIdentity(isicol,&col_identity);CHKERRQ(ierr);
2397   if (row_identity && col_identity) {
2398     C->ops->solve   = MatSolve_SeqAIJ_NaturalOrdering_iludt;
2399   } else {
2400     C->ops->solve   = MatSolve_SeqAIJ_iludt;
2401   }
2402   C->ops->solveadd           = 0;
2403   C->ops->solvetranspose     = 0;
2404   C->ops->solvetransposeadd  = 0;
2405   C->ops->matsolve           = 0;
2406   C->assembled    = PETSC_TRUE;
2407   C->preallocated = PETSC_TRUE;
2408   ierr = PetscLogFlops(C->cmap->n);CHKERRQ(ierr);
2409   PetscFunctionReturn(0);
2410 }
2411