xref: /petsc/src/mat/impls/aij/seq/aijfact.c (revision d3ac4fa3a036154e7c2cf364e29cf253ab5c80fe)
1 #define PETSCMAT_DLL
2 
3 
4 #include "../src/mat/impls/aij/seq/aij.h"
5 #include "../src/mat/impls/sbaij/seq/sbaij.h"
6 #include "petscbt.h"
7 #include "../src/mat/utils/freespace.h"
8 
9 EXTERN_C_BEGIN
10 #undef __FUNCT__
11 #define __FUNCT__ "MatOrdering_Flow_SeqAIJ"
12 /*
13       Computes an ordering to get most of the large numerical values in the lower triangular part of the matrix
14 */
15 PetscErrorCode MatOrdering_Flow_SeqAIJ(Mat mat,const MatOrderingType type,IS *irow,IS *icol)
16 {
17   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)mat->data;
18   PetscErrorCode    ierr;
19   PetscInt          i,j,jj,k, kk,n = mat->rmap->n, current = 0, newcurrent = 0,*order;
20   const PetscInt    *ai = a->i, *aj = a->j;
21   const PetscScalar *aa = a->a;
22   PetscTruth        *done;
23   PetscReal         best,past = 0,future;
24 
25   PetscFunctionBegin;
26   /* pick initial row */
27   best = -1;
28   for (i=0; i<n; i++) {
29     future = 0.0;
30     for (j=ai[i]; j<ai[i+1]; j++) {
31       if (aj[j] != i) future  += PetscAbsScalar(aa[j]); else past = PetscAbsScalar(aa[j]);
32     }
33     if (!future) future = 1.e-10; /* if there is zero in the upper diagonal part want to rank this row high */
34     if (past/future > best) {
35       best = past/future;
36       current = i;
37     }
38   }
39 
40   ierr = PetscMalloc(n*sizeof(PetscTruth),&done);CHKERRQ(ierr);
41   ierr = PetscMemzero(done,n*sizeof(PetscTruth));CHKERRQ(ierr);
42   ierr = PetscMalloc(n*sizeof(PetscInt),&order);CHKERRQ(ierr);
43   order[0] = current;
44   for (i=0; i<n-1; i++) {
45     done[current] = PETSC_TRUE;
46     best          = -1;
47     /* loop over all neighbors of current pivot */
48     for (j=ai[current]; j<ai[current+1]; j++) {
49       jj = aj[j];
50       if (done[jj]) continue;
51       /* loop over columns of potential next row computing weights for below and above diagonal */
52       past = future = 0.0;
53       for (k=ai[jj]; k<ai[jj+1]; k++) {
54         kk = aj[k];
55         if (done[kk]) past += PetscAbsScalar(aa[k]);
56         else if (kk != jj) future  += PetscAbsScalar(aa[k]);
57       }
58       if (!future) future = 1.e-10; /* if there is zero in the upper diagonal part want to rank this row high */
59       if (past/future > best) {
60         best = past/future;
61         newcurrent = jj;
62       }
63     }
64     if (best == -1) { /* no neighbors to select from so select best of all that remain */
65       best = -1;
66       for (k=0; k<n; k++) {
67         if (done[k]) continue;
68         future = 0.0;
69         past   = 0.0;
70         for (j=ai[k]; j<ai[k+1]; j++) {
71           kk = aj[j];
72           if (done[kk]) past += PetscAbsScalar(aa[j]);
73           else if (kk != k) future  += PetscAbsScalar(aa[j]);
74         }
75         if (!future) future = 1.e-10; /* if there is zero in the upper diagonal part want to rank this row high */
76         if (past/future > best) {
77           best = past/future;
78           newcurrent = k;
79         }
80       }
81     }
82     if (current == newcurrent) SETERRQ(PETSC_ERR_PLIB,"newcurrent cannot be current");
83     current = newcurrent;
84     order[i+1] = current;
85   }
86   ierr = ISCreateGeneral(PETSC_COMM_SELF,n,order,irow);CHKERRQ(ierr);
87   *icol = *irow;
88   ierr = PetscObjectReference((PetscObject)*irow);CHKERRQ(ierr);
89   ierr = PetscFree(done);CHKERRQ(ierr);
90   ierr = PetscFree(order);CHKERRQ(ierr);
91   PetscFunctionReturn(0);
92 }
93 EXTERN_C_END
94 
95 EXTERN_C_BEGIN
96 #undef __FUNCT__
97 #define __FUNCT__ "MatGetFactorAvailable_seqaij_petsc"
98 PetscErrorCode MatGetFactorAvailable_seqaij_petsc(Mat A,MatFactorType ftype,PetscTruth *flg)
99 {
100   PetscFunctionBegin;
101   *flg = PETSC_TRUE;
102   PetscFunctionReturn(0);
103 }
104 EXTERN_C_END
105 
106 EXTERN_C_BEGIN
107 #undef __FUNCT__
108 #define __FUNCT__ "MatGetFactor_seqaij_petsc"
109 PetscErrorCode MatGetFactor_seqaij_petsc(Mat A,MatFactorType ftype,Mat *B)
110 {
111   PetscInt           n = A->rmap->n;
112   PetscErrorCode     ierr;
113 
114   PetscFunctionBegin;
115   ierr = MatCreate(((PetscObject)A)->comm,B);CHKERRQ(ierr);
116   ierr = MatSetSizes(*B,n,n,n,n);CHKERRQ(ierr);
117   if (ftype == MAT_FACTOR_LU || ftype == MAT_FACTOR_ILU || ftype == MAT_FACTOR_ILUDT){
118     ierr = MatSetType(*B,MATSEQAIJ);CHKERRQ(ierr);
119     (*B)->ops->ilufactorsymbolic = MatILUFactorSymbolic_SeqAIJ;
120     (*B)->ops->lufactorsymbolic  = MatLUFactorSymbolic_SeqAIJ;
121   } else if (ftype == MAT_FACTOR_CHOLESKY || ftype == MAT_FACTOR_ICC) {
122     ierr = MatSetType(*B,MATSEQSBAIJ);CHKERRQ(ierr);
123     ierr = MatSeqSBAIJSetPreallocation(*B,1,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
124     (*B)->ops->iccfactorsymbolic      = MatICCFactorSymbolic_SeqAIJ;
125     (*B)->ops->choleskyfactorsymbolic = MatCholeskyFactorSymbolic_SeqAIJ;
126   } else SETERRQ(PETSC_ERR_SUP,"Factor type not supported");
127   (*B)->factor = ftype;
128   PetscFunctionReturn(0);
129 }
130 EXTERN_C_END
131 
132 #undef __FUNCT__
133 #define __FUNCT__ "MatLUFactorSymbolic_SeqAIJ_inplace"
134 PetscErrorCode MatLUFactorSymbolic_SeqAIJ_inplace(Mat B,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
135 {
136   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
137   IS                 isicol;
138   PetscErrorCode     ierr;
139   const PetscInt     *r,*ic;
140   PetscInt           i,n=A->rmap->n,*ai=a->i,*aj=a->j;
141   PetscInt           *bi,*bj,*ajtmp;
142   PetscInt           *bdiag,row,nnz,nzi,reallocs=0,nzbd,*im;
143   PetscReal          f;
144   PetscInt           nlnk,*lnk,k,**bi_ptr;
145   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
146   PetscBT            lnkbt;
147 
148   PetscFunctionBegin;
149   if (A->rmap->N != A->cmap->N) SETERRQ(PETSC_ERR_ARG_WRONG,"matrix must be square");
150   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
151   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
152   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
153 
154   /* get new row pointers */
155   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
156   bi[0] = 0;
157 
158   /* bdiag is location of diagonal in factor */
159   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);
160   bdiag[0] = 0;
161 
162   /* linked list for storing column indices of the active row */
163   nlnk = n + 1;
164   ierr = PetscLLCreate(n,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
165 
166   ierr = PetscMalloc2(n+1,PetscInt**,&bi_ptr,n+1,PetscInt,&im);CHKERRQ(ierr);
167 
168   /* initial FreeSpace size is f*(ai[n]+1) */
169   f = info->fill;
170   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space);CHKERRQ(ierr);
171   current_space = free_space;
172 
173   for (i=0; i<n; i++) {
174     /* copy previous fill into linked list */
175     nzi = 0;
176     nnz = ai[r[i]+1] - ai[r[i]];
177     if (!nnz) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
178     ajtmp = aj + ai[r[i]];
179     ierr = PetscLLAddPerm(nnz,ajtmp,ic,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
180     nzi += nlnk;
181 
182     /* add pivot rows into linked list */
183     row = lnk[n];
184     while (row < i) {
185       nzbd    = bdiag[row] - bi[row] + 1; /* num of entries in the row with column index <= row */
186       ajtmp   = bi_ptr[row] + nzbd; /* points to the entry next to the diagonal */
187       ierr = PetscLLAddSortedLU(ajtmp,row,nlnk,lnk,lnkbt,i,nzbd,im);CHKERRQ(ierr);
188       nzi += nlnk;
189       row  = lnk[row];
190     }
191     bi[i+1] = bi[i] + nzi;
192     im[i]   = nzi;
193 
194     /* mark bdiag */
195     nzbd = 0;
196     nnz  = nzi;
197     k    = lnk[n];
198     while (nnz-- && k < i){
199       nzbd++;
200       k = lnk[k];
201     }
202     bdiag[i] = bi[i] + nzbd;
203 
204     /* if free space is not available, make more free space */
205     if (current_space->local_remaining<nzi) {
206       nnz = (n - i)*nzi; /* estimated and max additional space needed */
207       ierr = PetscFreeSpaceGet(nnz,&current_space);CHKERRQ(ierr);
208       reallocs++;
209     }
210 
211     /* copy data into free space, then initialize lnk */
212     ierr = PetscLLClean(n,n,nzi,lnk,current_space->array,lnkbt);CHKERRQ(ierr);
213     bi_ptr[i] = current_space->array;
214     current_space->array           += nzi;
215     current_space->local_used      += nzi;
216     current_space->local_remaining -= nzi;
217   }
218 #if defined(PETSC_USE_INFO)
219   if (ai[n] != 0) {
220     PetscReal af = ((PetscReal)bi[n])/((PetscReal)ai[n]);
221     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,f,af);CHKERRQ(ierr);
222     ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
223     ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G);\n",af);CHKERRQ(ierr);
224     ierr = PetscInfo(A,"for best performance.\n");CHKERRQ(ierr);
225   } else {
226     ierr = PetscInfo(A,"Empty matrix\n");CHKERRQ(ierr);
227   }
228 #endif
229 
230   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
231   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
232 
233   /* destroy list of free space and other temporary array(s) */
234   ierr = PetscMalloc((bi[n]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
235   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
236   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
237   ierr = PetscFree2(bi_ptr,im);CHKERRQ(ierr);
238 
239   /* put together the new matrix */
240   ierr = MatSeqAIJSetPreallocation_SeqAIJ(B,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
241   ierr = PetscLogObjectParent(B,isicol);CHKERRQ(ierr);
242   b    = (Mat_SeqAIJ*)(B)->data;
243   b->free_a       = PETSC_TRUE;
244   b->free_ij      = PETSC_TRUE;
245   b->singlemalloc = PETSC_FALSE;
246   ierr          = PetscMalloc((bi[n]+1)*sizeof(PetscScalar),&b->a);CHKERRQ(ierr);
247   b->j          = bj;
248   b->i          = bi;
249   b->diag       = bdiag;
250   b->ilen       = 0;
251   b->imax       = 0;
252   b->row        = isrow;
253   b->col        = iscol;
254   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
255   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
256   b->icol       = isicol;
257   ierr          = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
258 
259   /* In b structure:  Free imax, ilen, old a, old j.  Allocate solve_work, new a, new j */
260   ierr = PetscLogObjectMemory(B,(bi[n]-n)*(sizeof(PetscInt)+sizeof(PetscScalar)));CHKERRQ(ierr);
261   b->maxnz = b->nz = bi[n] ;
262 
263   (B)->factor                = MAT_FACTOR_LU;
264   (B)->info.factor_mallocs   = reallocs;
265   (B)->info.fill_ratio_given = f;
266 
267   if (ai[n]) {
268     (B)->info.fill_ratio_needed = ((PetscReal)bi[n])/((PetscReal)ai[n]);
269   } else {
270     (B)->info.fill_ratio_needed = 0.0;
271   }
272   (B)->ops->lufactornumeric  = MatLUFactorNumeric_SeqAIJ_inplace;
273   PetscFunctionReturn(0);
274 }
275 
276 #undef __FUNCT__
277 #define __FUNCT__ "MatLUFactorSymbolic_SeqAIJ"
278 PetscErrorCode MatLUFactorSymbolic_SeqAIJ(Mat B,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
279 {
280   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
281   IS                 isicol;
282   PetscErrorCode     ierr;
283   const PetscInt     *r,*ic;
284   PetscInt           i,n=A->rmap->n,*ai=a->i,*aj=a->j;
285   PetscInt           *bi,*bj,*ajtmp;
286   PetscInt           *bdiag,row,nnz,nzi,reallocs=0,nzbd,*im;
287   PetscReal          f;
288   PetscInt           nlnk,*lnk,k,**bi_ptr;
289   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
290   PetscBT            lnkbt;
291 
292   PetscFunctionBegin;
293   // Uncomment the oldatastruct part only while testing new data structure for MatSolve()
294   PetscTruth         olddatastruct=PETSC_FALSE;
295   ierr = PetscOptionsGetTruth(PETSC_NULL,"-ilu_old",&olddatastruct,PETSC_NULL);CHKERRQ(ierr);
296   if(olddatastruct){
297     ierr = MatLUFactorSymbolic_SeqAIJ_inplace(B,A,isrow,iscol,info);CHKERRQ(ierr);
298     PetscFunctionReturn(0);
299   }
300 
301 
302   if (A->rmap->N != A->cmap->N) SETERRQ(PETSC_ERR_ARG_WRONG,"matrix must be square");
303   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
304   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
305   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
306 
307   /* get new row and diagonal pointers, must be allocated separately because they will be given to the Mat_SeqAIJ and freed separately */
308   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
309   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);
310   bi[0] = bdiag[0] = 0;
311 
312   /* linked list for storing column indices of the active row */
313   nlnk = n + 1;
314   ierr = PetscLLCreate(n,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
315 
316   ierr = PetscMalloc2(n+1,PetscInt**,&bi_ptr,n+1,PetscInt,&im);CHKERRQ(ierr);
317 
318   /* initial FreeSpace size is f*(ai[n]+1) */
319   f = info->fill;
320   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space);CHKERRQ(ierr);
321   current_space = free_space;
322 
323   for (i=0; i<n; i++) {
324     /* copy previous fill into linked list */
325     nzi = 0;
326     nnz = ai[r[i]+1] - ai[r[i]];
327     if (!nnz) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
328     ajtmp = aj + ai[r[i]];
329     ierr = PetscLLAddPerm(nnz,ajtmp,ic,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
330     nzi += nlnk;
331 
332     /* add pivot rows into linked list */
333     row = lnk[n];
334     while (row < i){
335       nzbd  = bdiag[row] + 1; /* num of entries in the row with column index <= row */
336       ajtmp = bi_ptr[row] + nzbd; /* points to the entry next to the diagonal */
337       ierr  = PetscLLAddSortedLU(ajtmp,row,nlnk,lnk,lnkbt,i,nzbd,im);CHKERRQ(ierr);
338       nzi  += nlnk;
339       row   = lnk[row];
340     }
341     bi[i+1] = bi[i] + nzi;
342     im[i]   = nzi;
343 
344     /* mark bdiag */
345     nzbd = 0;
346     nnz  = nzi;
347     k    = lnk[n];
348     while (nnz-- && k < i){
349       nzbd++;
350       k = lnk[k];
351     }
352     bdiag[i] = nzbd; /* note: bdiag[i] = nnzL as input for PetscFreeSpaceContiguous_LU() */
353 
354     /* if free space is not available, make more free space */
355     if (current_space->local_remaining<nzi) {
356       nnz = 2*(n - i)*nzi; /* estimated and max additional space needed */
357       ierr = PetscFreeSpaceGet(nnz,&current_space);CHKERRQ(ierr);
358       reallocs++;
359     }
360 
361     /* copy data into free space, then initialize lnk */
362     ierr = PetscLLClean(n,n,nzi,lnk,current_space->array,lnkbt);CHKERRQ(ierr);
363     bi_ptr[i] = current_space->array;
364     current_space->array           += nzi;
365     current_space->local_used      += nzi;
366     current_space->local_remaining -= nzi;
367   }
368 #if defined(PETSC_USE_INFO)
369   if (ai[n] != 0) {
370     PetscReal af = ((PetscReal)bi[n])/((PetscReal)ai[n]);
371     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,f,af);CHKERRQ(ierr);
372     ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
373     ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G);\n",af);CHKERRQ(ierr);
374     ierr = PetscInfo(A,"for best performance.\n");CHKERRQ(ierr);
375   } else {
376     ierr = PetscInfo(A,"Empty matrix\n");CHKERRQ(ierr);
377   }
378 #endif
379 
380   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
381   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
382 
383   /* destroy list of free space and other temporary array(s) */
384   ierr = PetscMalloc((bi[n]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
385   ierr = PetscFreeSpaceContiguous_LU(&free_space,bj,n,bi,bdiag);CHKERRQ(ierr);
386   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
387   ierr = PetscFree2(bi_ptr,im);CHKERRQ(ierr);
388 
389   /* put together the new matrix */
390   ierr = MatSeqAIJSetPreallocation_SeqAIJ(B,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
391   ierr = PetscLogObjectParent(B,isicol);CHKERRQ(ierr);
392   b    = (Mat_SeqAIJ*)(B)->data;
393   b->free_a       = PETSC_TRUE;
394   b->free_ij      = PETSC_TRUE;
395   b->singlemalloc = PETSC_FALSE;
396   ierr = PetscMalloc((bdiag[0]+1)*sizeof(PetscScalar),&b->a);CHKERRQ(ierr);
397   b->j          = bj;
398   b->i          = bi;
399   b->diag       = bdiag;
400   b->ilen       = 0;
401   b->imax       = 0;
402   b->row        = isrow;
403   b->col        = iscol;
404   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
405   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
406   b->icol       = isicol;
407   ierr          = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
408 
409   /* In b structure:  Free imax, ilen, old a, old j.  Allocate solve_work, new a, new j */
410   ierr = PetscLogObjectMemory(B,(bdiag[0]+1)*(sizeof(PetscInt)+sizeof(PetscScalar)));CHKERRQ(ierr);
411   b->maxnz = b->nz = bdiag[0]+1;
412   B->factor                = MAT_FACTOR_LU;
413   B->info.factor_mallocs   = reallocs;
414   B->info.fill_ratio_given = f;
415 
416   if (ai[n]) {
417     B->info.fill_ratio_needed = ((PetscReal)(bdiag[0]+1))/((PetscReal)ai[n]);
418   } else {
419     B->info.fill_ratio_needed = 0.0;
420   }
421   B->ops->lufactornumeric = MatLUFactorNumeric_SeqAIJ;
422   if (a->inode.use) {
423     B->ops->lufactornumeric = MatLUFactorNumeric_SeqAIJ_Inode;
424   }
425   PetscFunctionReturn(0);
426 }
427 
428 /*
429     Trouble in factorization, should we dump the original matrix?
430 */
431 #undef __FUNCT__
432 #define __FUNCT__ "MatFactorDumpMatrix"
433 PetscErrorCode MatFactorDumpMatrix(Mat A)
434 {
435   PetscErrorCode ierr;
436   PetscTruth     flg = PETSC_FALSE;
437 
438   PetscFunctionBegin;
439   ierr = PetscOptionsGetTruth(PETSC_NULL,"-mat_factor_dump_on_error",&flg,PETSC_NULL);CHKERRQ(ierr);
440   if (flg) {
441     PetscViewer viewer;
442     char        filename[PETSC_MAX_PATH_LEN];
443 
444     ierr = PetscSNPrintf(filename,PETSC_MAX_PATH_LEN,"matrix_factor_error.%d",PetscGlobalRank);CHKERRQ(ierr);
445     ierr = PetscViewerBinaryOpen(((PetscObject)A)->comm,filename,FILE_MODE_WRITE,&viewer);CHKERRQ(ierr);
446     ierr = MatView(A,viewer);CHKERRQ(ierr);
447     ierr = PetscViewerDestroy(viewer);CHKERRQ(ierr);
448   }
449   PetscFunctionReturn(0);
450 }
451 
452 #undef __FUNCT__
453 #define __FUNCT__ "MatLUFactorNumeric_SeqAIJ"
454 PetscErrorCode MatLUFactorNumeric_SeqAIJ(Mat B,Mat A,const MatFactorInfo *info)
455 {
456   Mat              C=B;
457   Mat_SeqAIJ       *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ *)C->data;
458   IS               isrow = b->row,isicol = b->icol;
459   PetscErrorCode   ierr;
460   const PetscInt   *r,*ic,*ics;
461   const PetscInt   n=A->rmap->n,*ai=a->i,*aj=a->j,*bi=b->i,*bj=b->j,*bdiag=b->diag;
462   PetscInt         i,j,k,nz,nzL,row,*pj;
463   const PetscInt   *ajtmp,*bjtmp;
464   MatScalar        *rtmp,*pc,multiplier,*pv;
465   const  MatScalar *aa=a->a,*v;
466   PetscTruth       row_identity,col_identity;
467   FactorShiftCtx   sctx;
468   const PetscInt   *ddiag;
469   PetscReal        rs;
470   MatScalar        d;
471 
472   PetscFunctionBegin;
473   /* MatPivotSetUp(): initialize shift context sctx */
474   ierr = PetscMemzero(&sctx,sizeof(FactorShiftCtx));CHKERRQ(ierr);
475 
476   if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE) { /* set sctx.shift_top=max{rs} */
477     ddiag          = a->diag;
478     sctx.shift_top = info->zeropivot;
479     for (i=0; i<n; i++) {
480       /* calculate sum(|aij|)-RealPart(aii), amt of shift needed for this row */
481       d  = (aa)[ddiag[i]];
482       rs = -PetscAbsScalar(d) - PetscRealPart(d);
483       v  = aa+ai[i];
484       nz = ai[i+1] - ai[i];
485       for (j=0; j<nz; j++)
486 	rs += PetscAbsScalar(v[j]);
487       if (rs>sctx.shift_top) sctx.shift_top = rs;
488     }
489     sctx.shift_top   *= 1.1;
490     sctx.nshift_max   = 5;
491     sctx.shift_lo     = 0.;
492     sctx.shift_hi     = 1.;
493   }
494 
495   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
496   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
497   ierr = PetscMalloc((n+1)*sizeof(MatScalar),&rtmp);CHKERRQ(ierr);
498   ics  = ic;
499 
500   do {
501     sctx.useshift = PETSC_FALSE;
502     for (i=0; i<n; i++){
503       /* zero rtmp */
504       /* L part */
505       nz    = bi[i+1] - bi[i];
506       bjtmp = bj + bi[i];
507       for  (j=0; j<nz; j++) rtmp[bjtmp[j]] = 0.0;
508 
509       /* U part */
510       nz = bdiag[i]-bdiag[i+1];
511       bjtmp = bj + bdiag[i+1]+1;
512       for  (j=0; j<nz; j++) rtmp[bjtmp[j]] = 0.0;
513 
514       /* load in initial (unfactored row) */
515       nz    = ai[r[i]+1] - ai[r[i]];
516       ajtmp = aj + ai[r[i]];
517       v     = aa + ai[r[i]];
518       for (j=0; j<nz; j++) {
519         rtmp[ics[ajtmp[j]]] = v[j];
520       }
521       /* ZeropivotApply() */
522       rtmp[i] += sctx.shift_amount;  /* shift the diagonal of the matrix */
523 
524       /* elimination */
525       bjtmp = bj + bi[i];
526       row   = *bjtmp++;
527       nzL   = bi[i+1] - bi[i];
528       for(k=0; k < nzL;k++) {
529         pc = rtmp + row;
530         if (*pc != 0.0) {
531           pv         = b->a + bdiag[row];
532           multiplier = *pc * (*pv);
533           *pc        = multiplier;
534           pj = b->j + bdiag[row+1]+1; /* beginning of U(row,:) */
535 	  pv = b->a + bdiag[row+1]+1;
536 	  nz = bdiag[row]-bdiag[row+1]-1; /* num of entries in U(row,:) excluding diag */
537           for (j=0; j<nz; j++) rtmp[pj[j]] -= multiplier * pv[j];
538           ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr);
539         }
540         row = *bjtmp++;
541       }
542 
543       /* finished row so stick it into b->a */
544       rs = 0.0;
545       /* L part */
546       pv   = b->a + bi[i] ;
547       pj   = b->j + bi[i] ;
548       nz   = bi[i+1] - bi[i];
549       for (j=0; j<nz; j++) {
550         pv[j] = rtmp[pj[j]]; rs += PetscAbsScalar(pv[j]);
551       }
552 
553       /* U part */
554       pv = b->a + bdiag[i+1]+1;
555       pj = b->j + bdiag[i+1]+1;
556       nz = bdiag[i] - bdiag[i+1]-1;
557       for (j=0; j<nz; j++) {
558         pv[j] = rtmp[pj[j]]; rs += PetscAbsScalar(pv[j]);
559       }
560 
561       /* MatPivotCheck() */
562       sctx.rs  = rs;
563       sctx.pv  = rtmp[i];
564       if (info->shifttype == MAT_SHIFT_NONZERO){
565         ierr = MatPivotCheck_nz(info,sctx,i);CHKERRQ(ierr);
566       } else if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE){
567         ierr = MatPivotCheck_pd(info,sctx,i);CHKERRQ(ierr);
568       } else if (info->shifttype == MAT_SHIFT_INBLOCKS){
569         ierr = MatPivotCheck_inblocks(info,sctx,i);CHKERRQ(ierr);
570       } else {
571         ierr = MatPivotCheck_none(info,sctx,i);CHKERRQ(ierr);
572       }
573       rtmp[i] = sctx.pv;
574 
575       /* Mark diagonal and invert diagonal for simplier triangular solves */
576       pv  = b->a + bdiag[i];
577       *pv = 1.0/rtmp[i];
578 
579     } /* endof for (i=0; i<n; i++){ */
580 
581     /* MatPivotRefine() */
582     if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE && !sctx.useshift && sctx.shift_fraction>0 && sctx.nshift<sctx.nshift_max){
583       /*
584        * if no shift in this attempt & shifting & started shifting & can refine,
585        * then try lower shift
586        */
587       sctx.shift_hi       = sctx.shift_fraction;
588       sctx.shift_fraction = (sctx.shift_hi+sctx.shift_lo)/2.;
589       sctx.shift_amount   = sctx.shift_fraction * sctx.shift_top;
590       sctx.useshift        = PETSC_TRUE;
591       sctx.nshift++;
592     }
593   } while (sctx.useshift);
594 
595   ierr = PetscFree(rtmp);CHKERRQ(ierr);
596   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
597   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
598 
599   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
600   ierr = ISIdentity(isicol,&col_identity);CHKERRQ(ierr);
601   if (row_identity && col_identity) {
602     C->ops->solve = MatSolve_SeqAIJ_NaturalOrdering;
603   } else {
604     C->ops->solve = MatSolve_SeqAIJ;
605   }
606   C->ops->solveadd           = MatSolveAdd_SeqAIJ;
607   C->ops->solvetranspose     = MatSolveTranspose_SeqAIJ;
608   C->ops->solvetransposeadd  = MatSolveTransposeAdd_SeqAIJ;
609   C->ops->matsolve           = MatMatSolve_SeqAIJ;
610   C->assembled    = PETSC_TRUE;
611   C->preallocated = PETSC_TRUE;
612   ierr = PetscLogFlops(C->cmap->n);CHKERRQ(ierr);
613 
614   /* MatShiftView(A,info,&sctx) */
615   if (sctx.nshift){
616     if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE) {
617       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);
618     } else if (info->shifttype == MAT_SHIFT_NONZERO) {
619       ierr = PetscInfo2(A,"number of shift_nz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
620     } else if (info->shifttype == MAT_SHIFT_INBLOCKS){
621       ierr = PetscInfo2(A,"number of shift_inblocks applied %D, each shift_amount %G\n",sctx.nshift,info->shiftamount);CHKERRQ(ierr);
622     }
623   }
624   ierr = Mat_CheckInode_FactorLU(C,PETSC_FALSE);CHKERRQ(ierr);
625   PetscFunctionReturn(0);
626 }
627 
628 #undef __FUNCT__
629 #define __FUNCT__ "MatLUFactorNumeric_SeqAIJ_inplace"
630 PetscErrorCode MatLUFactorNumeric_SeqAIJ_inplace(Mat B,Mat A,const MatFactorInfo *info)
631 {
632   Mat             C=B;
633   Mat_SeqAIJ      *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ *)C->data;
634   IS              isrow = b->row,isicol = b->icol;
635   PetscErrorCode  ierr;
636   const PetscInt   *r,*ic,*ics;
637   PetscInt        nz,row,i,j,n=A->rmap->n,diag;
638   const PetscInt  *ai=a->i,*aj=a->j,*bi=b->i,*bj=b->j;
639   const PetscInt  *ajtmp,*bjtmp,*diag_offset = b->diag,*pj;
640   MatScalar       *pv,*rtmp,*pc,multiplier,d;
641   const MatScalar *v,*aa=a->a;
642   PetscReal       rs=0.0;
643   FactorShiftCtx  sctx;
644   PetscInt        newshift;
645   const PetscInt  *ddiag;
646   PetscTruth      row_identity, col_identity;
647 
648   PetscFunctionBegin;
649   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
650   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
651   ierr = PetscMalloc((n+1)*sizeof(MatScalar),&rtmp);CHKERRQ(ierr);
652   ics  = ic;
653 
654   /* initialize shift context sctx */
655   sctx.nshift         = 0;
656   sctx.nshift_max     = 0;
657   sctx.shift_top      = 0.0;
658   sctx.shift_lo       = 0.0;
659   sctx.shift_hi       = 0.0;
660   sctx.shift_fraction = 0.0;
661   sctx.shift_amount   = 0.0;
662 
663   if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE) { /* set sctx.shift_top=max{rs} */
664     ddiag          = a->diag;
665     sctx.shift_top = info->zeropivot;
666     for (i=0; i<n; i++) {
667       /* calculate sum(|aij|)-RealPart(aii), amt of shift needed for this row */
668       d  = (aa)[ddiag[i]];
669       rs = -PetscAbsScalar(d) - PetscRealPart(d);
670       v  = aa+ai[i];
671       nz = ai[i+1] - ai[i];
672       for (j=0; j<nz; j++)
673 	rs += PetscAbsScalar(v[j]);
674       if (rs>sctx.shift_top) sctx.shift_top = rs;
675     }
676     sctx.shift_top   *= 1.1;
677     sctx.nshift_max   = 5;
678     sctx.shift_lo     = 0.;
679     sctx.shift_hi     = 1.;
680   }
681 
682   do {
683     sctx.useshift = PETSC_FALSE;
684     for (i=0; i<n; i++){
685       nz    = bi[i+1] - bi[i];
686       bjtmp = bj + bi[i];
687       for  (j=0; j<nz; j++) rtmp[bjtmp[j]] = 0.0;
688 
689       /* load in initial (unfactored row) */
690       nz    = ai[r[i]+1] - ai[r[i]];
691       ajtmp = aj + ai[r[i]];
692       v     = aa + ai[r[i]];
693       for (j=0; j<nz; j++) {
694         rtmp[ics[ajtmp[j]]] = v[j];
695       }
696       rtmp[ics[r[i]]] += sctx.shift_amount; /* shift the diagonal of the matrix */
697       /* if (sctx.shift_amount > 0.0) printf("row %d, shift %g\n",i,sctx.shift_amount); */
698 
699       row = *bjtmp++;
700       while  (row < i) {
701         pc = rtmp + row;
702         if (*pc != 0.0) {
703           pv         = b->a + diag_offset[row];
704           pj         = b->j + diag_offset[row] + 1;
705           multiplier = *pc / *pv++;
706           *pc        = multiplier;
707           nz         = bi[row+1] - diag_offset[row] - 1;
708           for (j=0; j<nz; j++) rtmp[pj[j]] -= multiplier * pv[j];
709           ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr);
710         }
711         row = *bjtmp++;
712       }
713       /* finished row so stick it into b->a */
714       pv   = b->a + bi[i] ;
715       pj   = b->j + bi[i] ;
716       nz   = bi[i+1] - bi[i];
717       diag = diag_offset[i] - bi[i];
718       rs   = 0.0;
719       for (j=0; j<nz; j++) {
720         pv[j] = rtmp[pj[j]];
721         rs   += PetscAbsScalar(pv[j]);
722       }
723       rs   -= PetscAbsScalar(pv[diag]);
724 
725       /* 9/13/02 Victor Eijkhout suggested scaling zeropivot by rs for matrices with funny scalings */
726       sctx.rs  = rs;
727       sctx.pv  = pv[diag];
728       ierr = MatLUCheckShift_inline(info,sctx,i,newshift);CHKERRQ(ierr);
729       if (newshift == 1) break;
730     }
731 
732     if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE && !sctx.useshift && sctx.shift_fraction>0 && sctx.nshift<sctx.nshift_max) {
733       /*
734        * if no shift in this attempt & shifting & started shifting & can refine,
735        * then try lower shift
736        */
737       sctx.shift_hi       = sctx.shift_fraction;
738       sctx.shift_fraction = (sctx.shift_hi+sctx.shift_lo)/2.;
739       sctx.shift_amount   = sctx.shift_fraction * sctx.shift_top;
740       sctx.useshift        = PETSC_TRUE;
741       sctx.nshift++;
742     }
743   } while (sctx.useshift);
744 
745   /* invert diagonal entries for simplier triangular solves */
746   for (i=0; i<n; i++) {
747     b->a[diag_offset[i]] = 1.0/b->a[diag_offset[i]];
748   }
749   ierr = PetscFree(rtmp);CHKERRQ(ierr);
750   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
751   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
752 
753   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
754   ierr = ISIdentity(isicol,&col_identity);CHKERRQ(ierr);
755   if (row_identity && col_identity) {
756     C->ops->solve   = MatSolve_SeqAIJ_NaturalOrdering_inplace;
757   } else {
758     C->ops->solve   = MatSolve_SeqAIJ_inplace;
759   }
760   C->ops->solveadd           = MatSolveAdd_SeqAIJ_inplace;
761   C->ops->solvetranspose     = MatSolveTranspose_SeqAIJ_inplace;
762   C->ops->solvetransposeadd  = MatSolveTransposeAdd_SeqAIJ_inplace;
763   C->ops->matsolve           = MatMatSolve_SeqAIJ_inplace;
764   C->assembled    = PETSC_TRUE;
765   C->preallocated = PETSC_TRUE;
766   ierr = PetscLogFlops(C->cmap->n);CHKERRQ(ierr);
767   if (sctx.nshift){
768      if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE) {
769       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);
770     } else if (info->shifttype == MAT_SHIFT_NONZERO) {
771       ierr = PetscInfo2(A,"number of shift_nz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
772     }
773   }
774   (C)->ops->solve            = MatSolve_SeqAIJ_inplace;
775   (C)->ops->solvetranspose   = MatSolveTranspose_SeqAIJ_inplace;
776   ierr = Mat_CheckInode(C,PETSC_FALSE);CHKERRQ(ierr);
777   PetscFunctionReturn(0);
778 }
779 
780 /*
781    This routine implements inplace ILU(0) with row or/and column permutations.
782    Input:
783      A - original matrix
784    Output;
785      A - a->i (rowptr) is same as original rowptr, but factored i-the row is stored in rowperm[i]
786          a->j (col index) is permuted by the inverse of colperm, then sorted
787          a->a reordered accordingly with a->j
788          a->diag (ptr to diagonal elements) is updated.
789 */
790 #undef __FUNCT__
791 #define __FUNCT__ "MatLUFactorNumeric_SeqAIJ_InplaceWithPerm"
792 PetscErrorCode MatLUFactorNumeric_SeqAIJ_InplaceWithPerm(Mat B,Mat A,const MatFactorInfo *info)
793 {
794   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data;
795   IS             isrow = a->row,isicol = a->icol;
796   PetscErrorCode ierr;
797   const PetscInt *r,*ic,*ics;
798   PetscInt       i,j,n=A->rmap->n,*ai=a->i,*aj=a->j;
799   PetscInt       *ajtmp,nz,row;
800   PetscInt       *diag = a->diag,nbdiag,*pj;
801   PetscScalar    *rtmp,*pc,multiplier,d;
802   MatScalar      *v,*pv;
803   PetscReal      rs;
804   FactorShiftCtx sctx;
805   PetscInt       newshift;
806 
807   PetscFunctionBegin;
808   if (A != B) SETERRQ(PETSC_ERR_ARG_INCOMP,"input and output matrix must have same address");
809   ierr  = ISGetIndices(isrow,&r);CHKERRQ(ierr);
810   ierr  = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
811   ierr  = PetscMalloc((n+1)*sizeof(PetscScalar),&rtmp);CHKERRQ(ierr);
812   ierr  = PetscMemzero(rtmp,(n+1)*sizeof(PetscScalar));CHKERRQ(ierr);
813   ics = ic;
814 
815   sctx.shift_top      = 0.;
816   sctx.nshift_max     = 0;
817   sctx.shift_lo       = 0.;
818   sctx.shift_hi       = 0.;
819   sctx.shift_fraction = 0.;
820 
821   if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE) { /* set sctx.shift_top=max{rs} */
822     sctx.shift_top = 0.;
823     for (i=0; i<n; i++) {
824       /* calculate sum(|aij|)-RealPart(aii), amt of shift needed for this row */
825       d  = (a->a)[diag[i]];
826       rs = -PetscAbsScalar(d) - PetscRealPart(d);
827       v  = a->a+ai[i];
828       nz = ai[i+1] - ai[i];
829       for (j=0; j<nz; j++)
830 	rs += PetscAbsScalar(v[j]);
831       if (rs>sctx.shift_top) sctx.shift_top = rs;
832     }
833     if (sctx.shift_top < info->zeropivot) sctx.shift_top = info->zeropivot;
834     sctx.shift_top    *= 1.1;
835     sctx.nshift_max   = 5;
836     sctx.shift_lo     = 0.;
837     sctx.shift_hi     = 1.;
838   }
839 
840   sctx.shift_amount = 0.;
841   sctx.nshift       = 0;
842   do {
843     sctx.useshift = PETSC_FALSE;
844     for (i=0; i<n; i++){
845       /* load in initial unfactored row */
846       nz    = ai[r[i]+1] - ai[r[i]];
847       ajtmp = aj + ai[r[i]];
848       v     = a->a + ai[r[i]];
849       /* sort permuted ajtmp and values v accordingly */
850       for (j=0; j<nz; j++) ajtmp[j] = ics[ajtmp[j]];
851       ierr = PetscSortIntWithScalarArray(nz,ajtmp,v);CHKERRQ(ierr);
852 
853       diag[r[i]] = ai[r[i]];
854       for (j=0; j<nz; j++) {
855         rtmp[ajtmp[j]] = v[j];
856         if (ajtmp[j] < i) diag[r[i]]++; /* update a->diag */
857       }
858       rtmp[r[i]] += sctx.shift_amount; /* shift the diagonal of the matrix */
859 
860       row = *ajtmp++;
861       while  (row < i) {
862         pc = rtmp + row;
863         if (*pc != 0.0) {
864           pv         = a->a + diag[r[row]];
865           pj         = aj + diag[r[row]] + 1;
866 
867           multiplier = *pc / *pv++;
868           *pc        = multiplier;
869           nz         = ai[r[row]+1] - diag[r[row]] - 1;
870           for (j=0; j<nz; j++) rtmp[pj[j]] -= multiplier * pv[j];
871           ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr);
872         }
873         row = *ajtmp++;
874       }
875       /* finished row so overwrite it onto a->a */
876       pv   = a->a + ai[r[i]] ;
877       pj   = aj + ai[r[i]] ;
878       nz   = ai[r[i]+1] - ai[r[i]];
879       nbdiag = diag[r[i]] - ai[r[i]]; /* num of entries before the diagonal */
880 
881       rs   = 0.0;
882       for (j=0; j<nz; j++) {
883         pv[j] = rtmp[pj[j]];
884         if (j != nbdiag) rs += PetscAbsScalar(pv[j]);
885       }
886 
887       /* 9/13/02 Victor Eijkhout suggested scaling zeropivot by rs for matrices with funny scalings */
888       sctx.rs  = rs;
889       sctx.pv  = pv[nbdiag];
890       ierr = MatLUCheckShift_inline(info,sctx,i,newshift);CHKERRQ(ierr);
891       if (newshift == 1) break;
892     }
893 
894     if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE && !sctx.useshift && sctx.shift_fraction>0 && sctx.nshift<sctx.nshift_max) {
895       /*
896        * if no shift in this attempt & shifting & started shifting & can refine,
897        * then try lower shift
898        */
899       sctx.shift_hi        = sctx.shift_fraction;
900       sctx.shift_fraction = (sctx.shift_hi+sctx.shift_lo)/2.;
901       sctx.shift_amount    = sctx.shift_fraction * sctx.shift_top;
902       sctx.useshift         = PETSC_TRUE;
903       sctx.nshift++;
904     }
905   } while (sctx.useshift);
906 
907   /* invert diagonal entries for simplier triangular solves */
908   for (i=0; i<n; i++) {
909     a->a[diag[r[i]]] = 1.0/a->a[diag[r[i]]];
910   }
911 
912   ierr = PetscFree(rtmp);CHKERRQ(ierr);
913   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
914   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
915   A->ops->solve             = MatSolve_SeqAIJ_InplaceWithPerm;
916   A->ops->solveadd          = MatSolveAdd_SeqAIJ_inplace;
917   A->ops->solvetranspose    = MatSolveTranspose_SeqAIJ_inplace;
918   A->ops->solvetransposeadd = MatSolveTransposeAdd_SeqAIJ_inplace;
919   A->assembled = PETSC_TRUE;
920   A->preallocated = PETSC_TRUE;
921   ierr = PetscLogFlops(A->cmap->n);CHKERRQ(ierr);
922   if (sctx.nshift){
923     if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE) {
924       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);
925     } else if (info->shifttype == MAT_SHIFT_NONZERO) {
926       ierr = PetscInfo2(A,"number of shift_nz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
927     }
928   }
929   PetscFunctionReturn(0);
930 }
931 
932 /* ----------------------------------------------------------- */
933 #undef __FUNCT__
934 #define __FUNCT__ "MatLUFactor_SeqAIJ"
935 PetscErrorCode MatLUFactor_SeqAIJ(Mat A,IS row,IS col,const MatFactorInfo *info)
936 {
937   PetscErrorCode ierr;
938   Mat            C;
939 
940   PetscFunctionBegin;
941   ierr = MatGetFactor(A,MAT_SOLVER_PETSC,MAT_FACTOR_LU,&C);CHKERRQ(ierr);
942   ierr = MatLUFactorSymbolic(C,A,row,col,info);CHKERRQ(ierr);
943   ierr = MatLUFactorNumeric(C,A,info);CHKERRQ(ierr);
944   A->ops->solve            = C->ops->solve;
945   A->ops->solvetranspose   = C->ops->solvetranspose;
946   ierr = MatHeaderCopy(A,C);CHKERRQ(ierr);
947   ierr = PetscLogObjectParent(A,((Mat_SeqAIJ*)(A->data))->icol);CHKERRQ(ierr);
948   PetscFunctionReturn(0);
949 }
950 /* ----------------------------------------------------------- */
951 
952 
953 #undef __FUNCT__
954 #define __FUNCT__ "MatSolve_SeqAIJ_inplace"
955 PetscErrorCode MatSolve_SeqAIJ_inplace(Mat A,Vec bb,Vec xx)
956 {
957   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
958   IS                iscol = a->col,isrow = a->row;
959   PetscErrorCode    ierr;
960   PetscInt          i, n = A->rmap->n,*vi,*ai = a->i,*aj = a->j;
961   PetscInt          nz;
962   const PetscInt    *rout,*cout,*r,*c;
963   PetscScalar       *x,*tmp,*tmps,sum;
964   const PetscScalar *b;
965   const MatScalar   *aa = a->a,*v;
966 
967   PetscFunctionBegin;
968   if (!n) PetscFunctionReturn(0);
969 
970   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
971   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
972   tmp  = a->solve_work;
973 
974   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
975   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout + (n-1);
976 
977   /* forward solve the lower triangular */
978   tmp[0] = b[*r++];
979   tmps   = tmp;
980   for (i=1; i<n; i++) {
981     v   = aa + ai[i] ;
982     vi  = aj + ai[i] ;
983     nz  = a->diag[i] - ai[i];
984     sum = b[*r++];
985     PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
986     tmp[i] = sum;
987   }
988 
989   /* backward solve the upper triangular */
990   for (i=n-1; i>=0; i--){
991     v   = aa + a->diag[i] + 1;
992     vi  = aj + a->diag[i] + 1;
993     nz  = ai[i+1] - a->diag[i] - 1;
994     sum = tmp[i];
995     PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
996     x[*c--] = tmp[i] = sum*aa[a->diag[i]];
997   }
998 
999   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1000   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1001   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1002   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1003   ierr = PetscLogFlops(2.0*a->nz - A->cmap->n);CHKERRQ(ierr);
1004   PetscFunctionReturn(0);
1005 }
1006 
1007 #undef __FUNCT__
1008 #define __FUNCT__ "MatMatSolve_SeqAIJ_inplace"
1009 PetscErrorCode MatMatSolve_SeqAIJ_inplace(Mat A,Mat B,Mat X)
1010 {
1011   Mat_SeqAIJ      *a = (Mat_SeqAIJ*)A->data;
1012   IS              iscol = a->col,isrow = a->row;
1013   PetscErrorCode  ierr;
1014   PetscInt        i, n = A->rmap->n,*vi,*ai = a->i,*aj = a->j;
1015   PetscInt        nz,neq;
1016   const PetscInt  *rout,*cout,*r,*c;
1017   PetscScalar     *x,*b,*tmp,*tmps,sum;
1018   const MatScalar *aa = a->a,*v;
1019   PetscTruth      bisdense,xisdense;
1020 
1021   PetscFunctionBegin;
1022   if (!n) PetscFunctionReturn(0);
1023 
1024   ierr = PetscTypeCompare((PetscObject)B,MATSEQDENSE,&bisdense);CHKERRQ(ierr);
1025   if (!bisdense) SETERRQ(PETSC_ERR_ARG_INCOMP,"B matrix must be a SeqDense matrix");
1026   ierr = PetscTypeCompare((PetscObject)X,MATSEQDENSE,&xisdense);CHKERRQ(ierr);
1027   if (!xisdense) SETERRQ(PETSC_ERR_ARG_INCOMP,"X matrix must be a SeqDense matrix");
1028 
1029   ierr = MatGetArray(B,&b);CHKERRQ(ierr);
1030   ierr = MatGetArray(X,&x);CHKERRQ(ierr);
1031 
1032   tmp  = a->solve_work;
1033   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1034   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1035 
1036   for (neq=0; neq<B->cmap->n; neq++){
1037     /* forward solve the lower triangular */
1038     tmp[0] = b[r[0]];
1039     tmps   = tmp;
1040     for (i=1; i<n; i++) {
1041       v   = aa + ai[i] ;
1042       vi  = aj + ai[i] ;
1043       nz  = a->diag[i] - ai[i];
1044       sum = b[r[i]];
1045       PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
1046       tmp[i] = sum;
1047     }
1048     /* backward solve the upper triangular */
1049     for (i=n-1; i>=0; i--){
1050       v   = aa + a->diag[i] + 1;
1051       vi  = aj + a->diag[i] + 1;
1052       nz  = ai[i+1] - a->diag[i] - 1;
1053       sum = tmp[i];
1054       PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
1055       x[c[i]] = tmp[i] = sum*aa[a->diag[i]];
1056     }
1057 
1058     b += n;
1059     x += n;
1060   }
1061   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1062   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1063   ierr = MatRestoreArray(B,&b);CHKERRQ(ierr);
1064   ierr = MatRestoreArray(X,&x);CHKERRQ(ierr);
1065   ierr = PetscLogFlops(B->cmap->n*(2.0*a->nz - n));CHKERRQ(ierr);
1066   PetscFunctionReturn(0);
1067 }
1068 
1069 #undef __FUNCT__
1070 #define __FUNCT__ "MatMatSolve_SeqAIJ"
1071 PetscErrorCode MatMatSolve_SeqAIJ(Mat A,Mat B,Mat X)
1072 {
1073   Mat_SeqAIJ      *a = (Mat_SeqAIJ*)A->data;
1074   IS              iscol = a->col,isrow = a->row;
1075   PetscErrorCode  ierr;
1076   PetscInt        i, n = A->rmap->n,*vi,*ai = a->i,*aj = a->j,*adiag = a->diag;
1077   PetscInt        nz,neq;
1078   const PetscInt  *rout,*cout,*r,*c;
1079   PetscScalar     *x,*b,*tmp,sum;
1080   const MatScalar *aa = a->a,*v;
1081   PetscTruth      bisdense,xisdense;
1082 
1083   PetscFunctionBegin;
1084   if (!n) PetscFunctionReturn(0);
1085 
1086   ierr = PetscTypeCompare((PetscObject)B,MATSEQDENSE,&bisdense);CHKERRQ(ierr);
1087   if (!bisdense) SETERRQ(PETSC_ERR_ARG_INCOMP,"B matrix must be a SeqDense matrix");
1088   ierr = PetscTypeCompare((PetscObject)X,MATSEQDENSE,&xisdense);CHKERRQ(ierr);
1089   if (!xisdense) SETERRQ(PETSC_ERR_ARG_INCOMP,"X matrix must be a SeqDense matrix");
1090 
1091   ierr = MatGetArray(B,&b);CHKERRQ(ierr);
1092   ierr = MatGetArray(X,&x);CHKERRQ(ierr);
1093 
1094   tmp  = a->solve_work;
1095   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1096   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1097 
1098   for (neq=0; neq<B->cmap->n; neq++){
1099     /* forward solve the lower triangular */
1100     tmp[0] = b[r[0]];
1101     v      = aa;
1102     vi     = aj;
1103     for (i=1; i<n; i++) {
1104       nz  = ai[i+1] - ai[i];
1105       sum = b[r[i]];
1106       PetscSparseDenseMinusDot(sum,tmp,v,vi,nz);
1107       tmp[i] = sum;
1108       v += nz; vi += nz;
1109     }
1110 
1111     /* backward solve the upper triangular */
1112     for (i=n-1; i>=0; i--){
1113       v   = aa + adiag[i+1]+1;
1114       vi  = aj + adiag[i+1]+1;
1115       nz  = adiag[i]-adiag[i+1]-1;
1116       sum = tmp[i];
1117       PetscSparseDenseMinusDot(sum,tmp,v,vi,nz);
1118       x[c[i]] = tmp[i] = sum*v[nz]; /* v[nz] = aa[adiag[i]] */
1119     }
1120 
1121     b += n;
1122     x += n;
1123   }
1124   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1125   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1126   ierr = MatRestoreArray(B,&b);CHKERRQ(ierr);
1127   ierr = MatRestoreArray(X,&x);CHKERRQ(ierr);
1128   ierr = PetscLogFlops(B->cmap->n*(2.0*a->nz - n));CHKERRQ(ierr);
1129   PetscFunctionReturn(0);
1130 }
1131 
1132 #undef __FUNCT__
1133 #define __FUNCT__ "MatSolve_SeqAIJ_InplaceWithPerm"
1134 PetscErrorCode MatSolve_SeqAIJ_InplaceWithPerm(Mat A,Vec bb,Vec xx)
1135 {
1136   Mat_SeqAIJ      *a = (Mat_SeqAIJ*)A->data;
1137   IS              iscol = a->col,isrow = a->row;
1138   PetscErrorCode  ierr;
1139   const PetscInt  *r,*c,*rout,*cout;
1140   PetscInt        i, n = A->rmap->n,*vi,*ai = a->i,*aj = a->j;
1141   PetscInt        nz,row;
1142   PetscScalar     *x,*b,*tmp,*tmps,sum;
1143   const MatScalar *aa = a->a,*v;
1144 
1145   PetscFunctionBegin;
1146   if (!n) PetscFunctionReturn(0);
1147 
1148   ierr = VecGetArray(bb,&b);CHKERRQ(ierr);
1149   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1150   tmp  = a->solve_work;
1151 
1152   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1153   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout + (n-1);
1154 
1155   /* forward solve the lower triangular */
1156   tmp[0] = b[*r++];
1157   tmps   = tmp;
1158   for (row=1; row<n; row++) {
1159     i   = rout[row]; /* permuted row */
1160     v   = aa + ai[i] ;
1161     vi  = aj + ai[i] ;
1162     nz  = a->diag[i] - ai[i];
1163     sum = b[*r++];
1164     PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
1165     tmp[row] = sum;
1166   }
1167 
1168   /* backward solve the upper triangular */
1169   for (row=n-1; row>=0; row--){
1170     i   = rout[row]; /* permuted row */
1171     v   = aa + a->diag[i] + 1;
1172     vi  = aj + a->diag[i] + 1;
1173     nz  = ai[i+1] - a->diag[i] - 1;
1174     sum = tmp[row];
1175     PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
1176     x[*c--] = tmp[row] = sum*aa[a->diag[i]];
1177   }
1178 
1179   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1180   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1181   ierr = VecRestoreArray(bb,&b);CHKERRQ(ierr);
1182   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1183   ierr = PetscLogFlops(2.0*a->nz - A->cmap->n);CHKERRQ(ierr);
1184   PetscFunctionReturn(0);
1185 }
1186 
1187 /* ----------------------------------------------------------- */
1188 #include "../src/mat/impls/aij/seq/ftn-kernels/fsolve.h"
1189 #undef __FUNCT__
1190 #define __FUNCT__ "MatSolve_SeqAIJ_NaturalOrdering_inplace"
1191 PetscErrorCode MatSolve_SeqAIJ_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
1192 {
1193   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1194   PetscErrorCode    ierr;
1195   PetscInt          n = A->rmap->n;
1196   const PetscInt    *ai = a->i,*aj = a->j,*adiag = a->diag;
1197   PetscScalar       *x;
1198   const PetscScalar *b;
1199   const MatScalar   *aa = a->a;
1200 #if !defined(PETSC_USE_FORTRAN_KERNEL_SOLVEAIJ)
1201   PetscInt          adiag_i,i,nz,ai_i;
1202   const PetscInt    *vi;
1203   const MatScalar   *v;
1204   PetscScalar       sum;
1205 #endif
1206 
1207   PetscFunctionBegin;
1208   if (!n) PetscFunctionReturn(0);
1209 
1210   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1211   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1212 
1213 #if defined(PETSC_USE_FORTRAN_KERNEL_SOLVEAIJ)
1214   fortransolveaij_(&n,x,ai,aj,adiag,aa,b);
1215 #else
1216   /* forward solve the lower triangular */
1217   x[0] = b[0];
1218   for (i=1; i<n; i++) {
1219     ai_i = ai[i];
1220     v    = aa + ai_i;
1221     vi   = aj + ai_i;
1222     nz   = adiag[i] - ai_i;
1223     sum  = b[i];
1224     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
1225     x[i] = sum;
1226   }
1227 
1228   /* backward solve the upper triangular */
1229   for (i=n-1; i>=0; i--){
1230     adiag_i = adiag[i];
1231     v       = aa + adiag_i + 1;
1232     vi      = aj + adiag_i + 1;
1233     nz      = ai[i+1] - adiag_i - 1;
1234     sum     = x[i];
1235     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
1236     x[i]    = sum*aa[adiag_i];
1237   }
1238 #endif
1239   ierr = PetscLogFlops(2.0*a->nz - A->cmap->n);CHKERRQ(ierr);
1240   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1241   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1242   PetscFunctionReturn(0);
1243 }
1244 
1245 #undef __FUNCT__
1246 #define __FUNCT__ "MatSolveAdd_SeqAIJ_inplace"
1247 PetscErrorCode MatSolveAdd_SeqAIJ_inplace(Mat A,Vec bb,Vec yy,Vec xx)
1248 {
1249   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1250   IS                iscol = a->col,isrow = a->row;
1251   PetscErrorCode    ierr;
1252   PetscInt          i, n = A->rmap->n,j;
1253   PetscInt          nz;
1254   const PetscInt    *rout,*cout,*r,*c,*vi,*ai = a->i,*aj = a->j;
1255   PetscScalar       *x,*tmp,sum;
1256   const PetscScalar *b;
1257   const MatScalar   *aa = a->a,*v;
1258 
1259   PetscFunctionBegin;
1260   if (yy != xx) {ierr = VecCopy(yy,xx);CHKERRQ(ierr);}
1261 
1262   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1263   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1264   tmp  = a->solve_work;
1265 
1266   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1267   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout + (n-1);
1268 
1269   /* forward solve the lower triangular */
1270   tmp[0] = b[*r++];
1271   for (i=1; i<n; i++) {
1272     v   = aa + ai[i] ;
1273     vi  = aj + ai[i] ;
1274     nz  = a->diag[i] - ai[i];
1275     sum = b[*r++];
1276     for (j=0; j<nz; j++) sum -= v[j]*tmp[vi[j]];
1277     tmp[i] = sum;
1278   }
1279 
1280   /* backward solve the upper triangular */
1281   for (i=n-1; i>=0; i--){
1282     v   = aa + a->diag[i] + 1;
1283     vi  = aj + a->diag[i] + 1;
1284     nz  = ai[i+1] - a->diag[i] - 1;
1285     sum = tmp[i];
1286     for (j=0; j<nz; j++) sum -= v[j]*tmp[vi[j]];
1287     tmp[i] = sum*aa[a->diag[i]];
1288     x[*c--] += tmp[i];
1289   }
1290 
1291   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1292   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1293   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1294   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1295   ierr = PetscLogFlops(2.0*a->nz);CHKERRQ(ierr);
1296 
1297   PetscFunctionReturn(0);
1298 }
1299 
1300 #undef __FUNCT__
1301 #define __FUNCT__ "MatSolveAdd_SeqAIJ"
1302 PetscErrorCode MatSolveAdd_SeqAIJ(Mat A,Vec bb,Vec yy,Vec xx)
1303 {
1304   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1305   IS                iscol = a->col,isrow = a->row;
1306   PetscErrorCode    ierr;
1307   PetscInt          i, n = A->rmap->n,j;
1308   PetscInt          nz;
1309   const PetscInt    *rout,*cout,*r,*c,*vi,*ai = a->i,*aj = a->j,*adiag = a->diag;
1310   PetscScalar       *x,*tmp,sum;
1311   const PetscScalar *b;
1312   const MatScalar   *aa = a->a,*v;
1313 
1314   PetscFunctionBegin;
1315   if (yy != xx) {ierr = VecCopy(yy,xx);CHKERRQ(ierr);}
1316 
1317   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1318   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1319   tmp  = a->solve_work;
1320 
1321   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1322   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1323 
1324   /* forward solve the lower triangular */
1325   tmp[0] = b[r[0]];
1326   v      = aa;
1327   vi     = aj;
1328   for (i=1; i<n; i++) {
1329     nz  = ai[i+1] - ai[i];
1330     sum = b[r[i]];
1331     for (j=0; j<nz; j++) sum -= v[j]*tmp[vi[j]];
1332     tmp[i] = sum;
1333     v += nz; vi += nz;
1334   }
1335 
1336   /* backward solve the upper triangular */
1337   v  = aa + adiag[n-1];
1338   vi = aj + adiag[n-1];
1339   for (i=n-1; i>=0; i--){
1340     nz  = adiag[i] - adiag[i+1] - 1;
1341     sum = tmp[i];
1342     for (j=0; j<nz; j++) sum -= v[j]*tmp[vi[j]];
1343     tmp[i] = sum*v[nz];
1344     x[c[i]] += tmp[i];
1345     v += nz+1; vi += nz+1;
1346   }
1347 
1348   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1349   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1350   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1351   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1352   ierr = PetscLogFlops(2.0*a->nz);CHKERRQ(ierr);
1353 
1354   PetscFunctionReturn(0);
1355 }
1356 
1357 #undef __FUNCT__
1358 #define __FUNCT__ "MatSolveTranspose_SeqAIJ_inplace"
1359 PetscErrorCode MatSolveTranspose_SeqAIJ_inplace(Mat A,Vec bb,Vec xx)
1360 {
1361   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1362   IS                iscol = a->col,isrow = a->row;
1363   PetscErrorCode    ierr;
1364   const PetscInt    *rout,*cout,*r,*c,*diag = a->diag,*ai = a->i,*aj = a->j,*vi;
1365   PetscInt          i,n = A->rmap->n,j;
1366   PetscInt          nz;
1367   PetscScalar       *x,*tmp,s1;
1368   const MatScalar   *aa = a->a,*v;
1369   const PetscScalar *b;
1370 
1371   PetscFunctionBegin;
1372   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1373   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1374   tmp  = a->solve_work;
1375 
1376   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1377   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1378 
1379   /* copy the b into temp work space according to permutation */
1380   for (i=0; i<n; i++) tmp[i] = b[c[i]];
1381 
1382   /* forward solve the U^T */
1383   for (i=0; i<n; i++) {
1384     v   = aa + diag[i] ;
1385     vi  = aj + diag[i] + 1;
1386     nz  = ai[i+1] - diag[i] - 1;
1387     s1  = tmp[i];
1388     s1 *= (*v++);  /* multiply by inverse of diagonal entry */
1389     for (j=0; j<nz; j++) tmp[vi[j]] -= s1*v[j];
1390     tmp[i] = s1;
1391   }
1392 
1393   /* backward solve the L^T */
1394   for (i=n-1; i>=0; i--){
1395     v   = aa + diag[i] - 1 ;
1396     vi  = aj + diag[i] - 1 ;
1397     nz  = diag[i] - ai[i];
1398     s1  = tmp[i];
1399     for (j=0; j>-nz; j--) tmp[vi[j]] -= s1*v[j];
1400   }
1401 
1402   /* copy tmp into x according to permutation */
1403   for (i=0; i<n; i++) x[r[i]] = tmp[i];
1404 
1405   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1406   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1407   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1408   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1409 
1410   ierr = PetscLogFlops(2.0*a->nz-A->cmap->n);CHKERRQ(ierr);
1411   PetscFunctionReturn(0);
1412 }
1413 
1414 #undef __FUNCT__
1415 #define __FUNCT__ "MatSolveTranspose_SeqAIJ"
1416 PetscErrorCode MatSolveTranspose_SeqAIJ(Mat A,Vec bb,Vec xx)
1417 {
1418   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1419   IS                iscol = a->col,isrow = a->row;
1420   PetscErrorCode    ierr;
1421   const PetscInt    *rout,*cout,*r,*c,*adiag = a->diag,*ai = a->i,*aj = a->j,*vi;
1422   PetscInt          i,n = A->rmap->n,j;
1423   PetscInt          nz;
1424   PetscScalar       *x,*tmp,s1;
1425   const MatScalar   *aa = a->a,*v;
1426   const PetscScalar *b;
1427 
1428   PetscFunctionBegin;
1429   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1430   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1431   tmp  = a->solve_work;
1432 
1433   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1434   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1435 
1436   /* copy the b into temp work space according to permutation */
1437   for (i=0; i<n; i++) tmp[i] = b[c[i]];
1438 
1439   /* forward solve the U^T */
1440   for (i=0; i<n; i++) {
1441     v   = aa + adiag[i+1] + 1;
1442     vi  = aj + adiag[i+1] + 1;
1443     nz  = adiag[i] - adiag[i+1] - 1;
1444     s1  = tmp[i];
1445     s1 *= v[nz];  /* multiply by inverse of diagonal entry */
1446     for (j=0; j<nz; j++) tmp[vi[j]] -= s1*v[j];
1447     tmp[i] = s1;
1448   }
1449 
1450   /* backward solve the L^T */
1451   for (i=n-1; i>=0; i--){
1452     v   = aa + ai[i];
1453     vi  = aj + ai[i];
1454     nz  = ai[i+1] - ai[i];
1455     s1  = tmp[i];
1456     for (j=0; j<nz; j++) tmp[vi[j]] -= s1*v[j];
1457   }
1458 
1459   /* copy tmp into x according to permutation */
1460   for (i=0; i<n; i++) x[r[i]] = tmp[i];
1461 
1462   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1463   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1464   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1465   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1466 
1467   ierr = PetscLogFlops(2.0*a->nz-A->cmap->n);CHKERRQ(ierr);
1468   PetscFunctionReturn(0);
1469 }
1470 
1471 #undef __FUNCT__
1472 #define __FUNCT__ "MatSolveTransposeAdd_SeqAIJ_inplace"
1473 PetscErrorCode MatSolveTransposeAdd_SeqAIJ_inplace(Mat A,Vec bb,Vec zz,Vec xx)
1474 {
1475   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1476   IS                iscol = a->col,isrow = a->row;
1477   PetscErrorCode    ierr;
1478   const PetscInt    *rout,*cout,*r,*c,*diag = a->diag,*ai = a->i,*aj = a->j,*vi;
1479   PetscInt          i,n = A->rmap->n,j;
1480   PetscInt          nz;
1481   PetscScalar       *x,*tmp,s1;
1482   const MatScalar   *aa = a->a,*v;
1483   const PetscScalar *b;
1484 
1485   PetscFunctionBegin;
1486   if (zz != xx) {ierr = VecCopy(zz,xx);CHKERRQ(ierr);}
1487   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1488   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1489   tmp  = a->solve_work;
1490 
1491   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1492   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1493 
1494   /* copy the b into temp work space according to permutation */
1495   for (i=0; i<n; i++) tmp[i] = b[c[i]];
1496 
1497   /* forward solve the U^T */
1498   for (i=0; i<n; i++) {
1499     v   = aa + diag[i] ;
1500     vi  = aj + diag[i] + 1;
1501     nz  = ai[i+1] - diag[i] - 1;
1502     s1  = tmp[i];
1503     s1 *= (*v++);  /* multiply by inverse of diagonal entry */
1504     for (j=0; j<nz; j++) tmp[vi[j]] -= s1*v[j];
1505     tmp[i] = s1;
1506   }
1507 
1508   /* backward solve the L^T */
1509   for (i=n-1; i>=0; i--){
1510     v   = aa + diag[i] - 1 ;
1511     vi  = aj + diag[i] - 1 ;
1512     nz  = diag[i] - ai[i];
1513     s1  = tmp[i];
1514     for (j=0; j>-nz; j--) tmp[vi[j]] -= s1*v[j];
1515   }
1516 
1517   /* copy tmp into x according to permutation */
1518   for (i=0; i<n; i++) x[r[i]] += tmp[i];
1519 
1520   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1521   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1522   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1523   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1524 
1525   ierr = PetscLogFlops(2.0*a->nz-A->cmap->n);CHKERRQ(ierr);
1526   PetscFunctionReturn(0);
1527 }
1528 
1529 #undef __FUNCT__
1530 #define __FUNCT__ "MatSolveTransposeAdd_SeqAIJ"
1531 PetscErrorCode MatSolveTransposeAdd_SeqAIJ(Mat A,Vec bb,Vec zz,Vec xx)
1532 {
1533   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1534   IS                iscol = a->col,isrow = a->row;
1535   PetscErrorCode    ierr;
1536   const PetscInt    *rout,*cout,*r,*c,*adiag = a->diag,*ai = a->i,*aj = a->j,*vi;
1537   PetscInt          i,n = A->rmap->n,j;
1538   PetscInt          nz;
1539   PetscScalar       *x,*tmp,s1;
1540   const MatScalar   *aa = a->a,*v;
1541   const PetscScalar *b;
1542 
1543   PetscFunctionBegin;
1544   if (zz != xx) {ierr = VecCopy(zz,xx);CHKERRQ(ierr);}
1545   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1546   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1547   tmp  = a->solve_work;
1548 
1549   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1550   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1551 
1552   /* copy the b into temp work space according to permutation */
1553   for (i=0; i<n; i++) tmp[i] = b[c[i]];
1554 
1555   /* forward solve the U^T */
1556   for (i=0; i<n; i++) {
1557     v   = aa + adiag[i+1] + 1;
1558     vi  = aj + adiag[i+1] + 1;
1559     nz  = adiag[i] - adiag[i+1] - 1;
1560     s1  = tmp[i];
1561     s1 *= v[nz];  /* multiply by inverse of diagonal entry */
1562     for (j=0; j<nz; j++) tmp[vi[j]] -= s1*v[j];
1563     tmp[i] = s1;
1564   }
1565 
1566 
1567   /* backward solve the L^T */
1568   for (i=n-1; i>=0; i--){
1569     v   = aa + ai[i] ;
1570     vi  = aj + ai[i];
1571     nz  = ai[i+1] - ai[i];
1572     s1  = tmp[i];
1573     for (j=0; j<nz; j++) tmp[vi[j]] -= s1*v[j];
1574   }
1575 
1576   /* copy tmp into x according to permutation */
1577   for (i=0; i<n; i++) x[r[i]] += tmp[i];
1578 
1579   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1580   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1581   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1582   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1583 
1584   ierr = PetscLogFlops(2.0*a->nz-A->cmap->n);CHKERRQ(ierr);
1585   PetscFunctionReturn(0);
1586 }
1587 
1588 /* ----------------------------------------------------------------*/
1589 
1590 EXTERN PetscErrorCode MatDuplicateNoCreate_SeqAIJ(Mat,Mat,MatDuplicateOption,PetscTruth);
1591 
1592 /*
1593    ilu() under revised new data structure.
1594    Factored arrays bj and ba are stored as
1595      L(0,:), L(1,:), ...,L(n-1,:),  U(n-1,:),...,U(i,:),U(i-1,:),...,U(0,:)
1596 
1597    bi=fact->i is an array of size n+1, in which
1598    bi+
1599      bi[i]:  points to 1st entry of L(i,:),i=0,...,n-1
1600      bi[n]:  points to L(n-1,n-1)+1
1601 
1602   bdiag=fact->diag is an array of size n+1,in which
1603      bdiag[i]: points to diagonal of U(i,:), i=0,...,n-1
1604      bdiag[n]: points to entry of U(n-1,0)-1
1605 
1606    U(i,:) contains bdiag[i] as its last entry, i.e.,
1607     U(i,:) = (u[i,i+1],...,u[i,n-1],diag[i])
1608 */
1609 #undef __FUNCT__
1610 #define __FUNCT__ "MatILUFactorSymbolic_SeqAIJ_ilu0"
1611 PetscErrorCode MatILUFactorSymbolic_SeqAIJ_ilu0(Mat fact,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
1612 {
1613 
1614   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
1615   PetscErrorCode     ierr;
1616   const PetscInt     n=A->rmap->n,*ai=a->i,*aj,*adiag=a->diag;
1617   PetscInt           i,j,k=0,nz,*bi,*bj,*bdiag;
1618   PetscTruth         missing;
1619   IS                 isicol;
1620 
1621   PetscFunctionBegin;
1622   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);
1623   ierr = MatMissingDiagonal(A,&missing,&i);CHKERRQ(ierr);
1624   if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",i);
1625   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
1626 
1627   ierr = MatDuplicateNoCreate_SeqAIJ(fact,A,MAT_DO_NOT_COPY_VALUES,PETSC_FALSE);CHKERRQ(ierr);
1628   b    = (Mat_SeqAIJ*)(fact)->data;
1629 
1630   /* allocate matrix arrays for new data structure */
1631   ierr = PetscMalloc3(ai[n]+1,PetscScalar,&b->a,ai[n]+1,PetscInt,&b->j,n+1,PetscInt,&b->i);CHKERRQ(ierr);
1632   ierr = PetscLogObjectMemory(fact,ai[n]*(sizeof(PetscScalar)+sizeof(PetscInt))+(n+1)*sizeof(PetscInt));CHKERRQ(ierr);
1633   b->singlemalloc = PETSC_TRUE;
1634   if (!b->diag){
1635     ierr = PetscMalloc((n+1)*sizeof(PetscInt),&b->diag);CHKERRQ(ierr);
1636     ierr = PetscLogObjectMemory(fact,(n+1)*sizeof(PetscInt));CHKERRQ(ierr);
1637   }
1638   bdiag = b->diag;
1639 
1640   if (n > 0) {
1641     ierr = PetscMemzero(b->a,(ai[n])*sizeof(MatScalar));CHKERRQ(ierr);
1642   }
1643 
1644   /* set bi and bj with new data structure */
1645   bi = b->i;
1646   bj = b->j;
1647 
1648   /* L part */
1649   bi[0] = 0;
1650   for (i=0; i<n; i++){
1651     nz = adiag[i] - ai[i];
1652     bi[i+1] = bi[i] + nz;
1653     aj = a->j + ai[i];
1654     for (j=0; j<nz; j++){
1655       /*   *bj = aj[j]; bj++; */
1656       bj[k++] = aj[j];
1657     }
1658   }
1659 
1660   /* U part */
1661   bdiag[n] = bi[n]-1;
1662   for (i=n-1; i>=0; i--){
1663     nz = ai[i+1] - adiag[i] - 1;
1664     aj = a->j + adiag[i] + 1;
1665     for (j=0; j<nz; j++){
1666       /*      *bj = aj[j]; bj++; */
1667       bj[k++] = aj[j];
1668     }
1669     /* diag[i] */
1670     /*    *bj = i; bj++; */
1671     bj[k++] = i;
1672     bdiag[i] = bdiag[i+1] + nz + 1;
1673   }
1674 
1675   fact->factor                 = MAT_FACTOR_ILU;
1676   fact->info.factor_mallocs    = 0;
1677   fact->info.fill_ratio_given  = info->fill;
1678   fact->info.fill_ratio_needed = 1.0;
1679   fact->ops->lufactornumeric   = MatLUFactorNumeric_SeqAIJ;
1680 
1681   b       = (Mat_SeqAIJ*)(fact)->data;
1682   b->row  = isrow;
1683   b->col  = iscol;
1684   b->icol = isicol;
1685   ierr    = PetscMalloc((fact->rmap->n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1686   ierr    = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
1687   ierr    = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
1688   PetscFunctionReturn(0);
1689 }
1690 
1691 #undef __FUNCT__
1692 #define __FUNCT__ "MatILUFactorSymbolic_SeqAIJ"
1693 PetscErrorCode MatILUFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
1694 {
1695   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
1696   IS                 isicol;
1697   PetscErrorCode     ierr;
1698   const PetscInt     *r,*ic;
1699   PetscInt           n=A->rmap->n,*ai=a->i,*aj=a->j;
1700   PetscInt           *bi,*cols,nnz,*cols_lvl;
1701   PetscInt           *bdiag,prow,fm,nzbd,reallocs=0,dcount=0;
1702   PetscInt           i,levels,diagonal_fill;
1703   PetscTruth         col_identity,row_identity;
1704   PetscReal          f;
1705   PetscInt           nlnk,*lnk,*lnk_lvl=PETSC_NULL;
1706   PetscBT            lnkbt;
1707   PetscInt           nzi,*bj,**bj_ptr,**bjlvl_ptr;
1708   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
1709   PetscFreeSpaceList free_space_lvl=PETSC_NULL,current_space_lvl=PETSC_NULL;
1710 
1711   PetscFunctionBegin;
1712   // Uncomment the old data struct part only while testing new data structure for MatSolve()
1713   PetscTruth         olddatastruct=PETSC_FALSE;
1714   ierr = PetscOptionsGetTruth(PETSC_NULL,"-ilu_old",&olddatastruct,PETSC_NULL);CHKERRQ(ierr);
1715   if(olddatastruct){
1716     ierr = MatILUFactorSymbolic_SeqAIJ_inplace(fact,A,isrow,iscol,info);CHKERRQ(ierr);
1717     PetscFunctionReturn(0);
1718   }
1719 
1720   levels = (PetscInt)info->levels;
1721   ierr   = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
1722   ierr   = ISIdentity(iscol,&col_identity);CHKERRQ(ierr);
1723 
1724   if (!levels && row_identity && col_identity) {
1725     /* special case: ilu(0) with natural ordering */
1726     ierr = MatILUFactorSymbolic_SeqAIJ_ilu0(fact,A,isrow,iscol,info);CHKERRQ(ierr);
1727     ierr = Mat_CheckInode_FactorLU(fact,PETSC_FALSE);CHKERRQ(ierr);
1728     PetscFunctionReturn(0);
1729   }
1730 
1731   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);
1732   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
1733   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
1734   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
1735 
1736   /* get new row and diagonal pointers, must be allocated separately because they will be given to the Mat_SeqAIJ and freed separately */
1737   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
1738   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);
1739   bi[0] = bdiag[0] = 0;
1740 
1741   ierr = PetscMalloc2(n,PetscInt*,&bj_ptr,n,PetscInt*,&bjlvl_ptr);CHKERRQ(ierr);
1742 
1743   /* create a linked list for storing column indices of the active row */
1744   nlnk = n + 1;
1745   ierr = PetscIncompleteLLCreate(n,n,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1746 
1747   /* initial FreeSpace size is f*(ai[n]+1) */
1748   f             = info->fill;
1749   diagonal_fill = (PetscInt)info->diagonal_fill;
1750   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space);CHKERRQ(ierr);
1751   current_space = free_space;
1752   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space_lvl);CHKERRQ(ierr);
1753   current_space_lvl = free_space_lvl;
1754 
1755   for (i=0; i<n; i++) {
1756     nzi = 0;
1757     /* copy current row into linked list */
1758     nnz  = ai[r[i]+1] - ai[r[i]];
1759     if (!nnz) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
1760     cols = aj + ai[r[i]];
1761     lnk[i] = -1; /* marker to indicate if diagonal exists */
1762     ierr = PetscIncompleteLLInit(nnz,cols,n,ic,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1763     nzi += nlnk;
1764 
1765     /* make sure diagonal entry is included */
1766     if (diagonal_fill && lnk[i] == -1) {
1767       fm = n;
1768       while (lnk[fm] < i) fm = lnk[fm];
1769       lnk[i]     = lnk[fm]; /* insert diagonal into linked list */
1770       lnk[fm]    = i;
1771       lnk_lvl[i] = 0;
1772       nzi++; dcount++;
1773     }
1774 
1775     /* add pivot rows into the active row */
1776     nzbd = 0;
1777     prow = lnk[n];
1778     while (prow < i) {
1779       nnz      = bdiag[prow];
1780       cols     = bj_ptr[prow] + nnz + 1;
1781       cols_lvl = bjlvl_ptr[prow] + nnz + 1;
1782       nnz      = bi[prow+1] - bi[prow] - nnz - 1;
1783       ierr = PetscILULLAddSorted(nnz,cols,levels,cols_lvl,prow,nlnk,lnk,lnk_lvl,lnkbt,prow);CHKERRQ(ierr);
1784       nzi += nlnk;
1785       prow = lnk[prow];
1786       nzbd++;
1787     }
1788     bdiag[i] = nzbd;
1789     bi[i+1]  = bi[i] + nzi;
1790 
1791     /* if free space is not available, make more free space */
1792     if (current_space->local_remaining<nzi) {
1793       nnz = 2*nzi*(n - i); /* estimated and max additional space needed */
1794       ierr = PetscFreeSpaceGet(nnz,&current_space);CHKERRQ(ierr);
1795       ierr = PetscFreeSpaceGet(nnz,&current_space_lvl);CHKERRQ(ierr);
1796       reallocs++;
1797     }
1798 
1799     /* copy data into free_space and free_space_lvl, then initialize lnk */
1800     ierr = PetscIncompleteLLClean(n,n,nzi,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);CHKERRQ(ierr);
1801     bj_ptr[i]    = current_space->array;
1802     bjlvl_ptr[i] = current_space_lvl->array;
1803 
1804     /* make sure the active row i has diagonal entry */
1805     if (*(bj_ptr[i]+bdiag[i]) != i) {
1806       SETERRQ1(PETSC_ERR_MAT_LU_ZRPVT,"Row %D has missing diagonal in factored matrix\n\
1807     try running with -pc_factor_nonzeros_along_diagonal or -pc_factor_diagonal_fill",i);
1808     }
1809 
1810     current_space->array           += nzi;
1811     current_space->local_used      += nzi;
1812     current_space->local_remaining -= nzi;
1813     current_space_lvl->array           += nzi;
1814     current_space_lvl->local_used      += nzi;
1815     current_space_lvl->local_remaining -= nzi;
1816   }
1817 
1818   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
1819   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
1820 
1821   /* destroy list of free space and other temporary arrays */
1822   ierr = PetscMalloc((bi[n]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
1823 
1824   /* copy free_space into bj and free free_space; set bi, bj, bdiag in new datastructure; */
1825   ierr = PetscFreeSpaceContiguous_LU(&free_space,bj,n,bi,bdiag);CHKERRQ(ierr);
1826 
1827   ierr = PetscIncompleteLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1828   ierr = PetscFreeSpaceDestroy(free_space_lvl);CHKERRQ(ierr);
1829   ierr = PetscFree2(bj_ptr,bjlvl_ptr);CHKERRQ(ierr);
1830 
1831 #if defined(PETSC_USE_INFO)
1832   {
1833     PetscReal af = ((PetscReal)bi[n])/((PetscReal)ai[n]);
1834     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,f,af);CHKERRQ(ierr);
1835     ierr = PetscInfo1(A,"Run with -[sub_]pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
1836     ierr = PetscInfo1(A,"PCFactorSetFill([sub]pc,%G);\n",af);CHKERRQ(ierr);
1837     ierr = PetscInfo(A,"for best performance.\n");CHKERRQ(ierr);
1838     if (diagonal_fill) {
1839       ierr = PetscInfo1(A,"Detected and replaced %D missing diagonals",dcount);CHKERRQ(ierr);
1840     }
1841   }
1842 #endif
1843 
1844   /* put together the new matrix */
1845   ierr = MatSeqAIJSetPreallocation_SeqAIJ(fact,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
1846   ierr = PetscLogObjectParent(fact,isicol);CHKERRQ(ierr);
1847   b = (Mat_SeqAIJ*)(fact)->data;
1848   b->free_a       = PETSC_TRUE;
1849   b->free_ij      = PETSC_TRUE;
1850   b->singlemalloc = PETSC_FALSE;
1851   ierr = PetscMalloc((bdiag[0]+1)*sizeof(PetscScalar),&b->a);CHKERRQ(ierr);
1852   b->j          = bj;
1853   b->i          = bi;
1854   b->diag       = bdiag;
1855   b->ilen       = 0;
1856   b->imax       = 0;
1857   b->row        = isrow;
1858   b->col        = iscol;
1859   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
1860   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
1861   b->icol       = isicol;
1862   ierr = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1863   /* In b structure:  Free imax, ilen, old a, old j.
1864      Allocate bdiag, solve_work, new a, new j */
1865   ierr = PetscLogObjectMemory(fact,(bdiag[0]+1)*(sizeof(PetscInt)+sizeof(PetscScalar)));CHKERRQ(ierr);
1866   b->maxnz = b->nz = bdiag[0]+1;
1867   (fact)->info.factor_mallocs    = reallocs;
1868   (fact)->info.fill_ratio_given  = f;
1869   (fact)->info.fill_ratio_needed = ((PetscReal)(bdiag[0]+1))/((PetscReal)ai[n]);
1870   (fact)->ops->lufactornumeric = MatLUFactorNumeric_SeqAIJ;
1871   ierr = Mat_CheckInode_FactorLU(fact,PETSC_FALSE);CHKERRQ(ierr);
1872   PetscFunctionReturn(0);
1873 }
1874 
1875 #undef __FUNCT__
1876 #define __FUNCT__ "MatILUFactorSymbolic_SeqAIJ_inplace"
1877 PetscErrorCode MatILUFactorSymbolic_SeqAIJ_inplace(Mat fact,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
1878 {
1879   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
1880   IS                 isicol;
1881   PetscErrorCode     ierr;
1882   const PetscInt     *r,*ic;
1883   PetscInt           n=A->rmap->n,*ai=a->i,*aj=a->j,d;
1884   PetscInt           *bi,*cols,nnz,*cols_lvl;
1885   PetscInt           *bdiag,prow,fm,nzbd,reallocs=0,dcount=0;
1886   PetscInt           i,levels,diagonal_fill;
1887   PetscTruth         col_identity,row_identity;
1888   PetscReal          f;
1889   PetscInt           nlnk,*lnk,*lnk_lvl=PETSC_NULL;
1890   PetscBT            lnkbt;
1891   PetscInt           nzi,*bj,**bj_ptr,**bjlvl_ptr;
1892   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
1893   PetscFreeSpaceList free_space_lvl=PETSC_NULL,current_space_lvl=PETSC_NULL;
1894   PetscTruth         missing;
1895 
1896   PetscFunctionBegin;
1897   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);
1898   f             = info->fill;
1899   levels        = (PetscInt)info->levels;
1900   diagonal_fill = (PetscInt)info->diagonal_fill;
1901   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
1902 
1903   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
1904   ierr = ISIdentity(iscol,&col_identity);CHKERRQ(ierr);
1905   if (!levels && row_identity && col_identity) { /* special case: ilu(0) with natural ordering */
1906     ierr = MatDuplicateNoCreate_SeqAIJ(fact,A,MAT_DO_NOT_COPY_VALUES,PETSC_TRUE);CHKERRQ(ierr);
1907     (fact)->ops->lufactornumeric =  MatLUFactorNumeric_SeqAIJ_inplace;
1908 
1909     fact->factor = MAT_FACTOR_ILU;
1910     (fact)->info.factor_mallocs    = 0;
1911     (fact)->info.fill_ratio_given  = info->fill;
1912     (fact)->info.fill_ratio_needed = 1.0;
1913     b               = (Mat_SeqAIJ*)(fact)->data;
1914     ierr = MatMissingDiagonal(A,&missing,&d);CHKERRQ(ierr);
1915     if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",d);
1916     b->row              = isrow;
1917     b->col              = iscol;
1918     b->icol             = isicol;
1919     ierr                = PetscMalloc(((fact)->rmap->n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1920     ierr                = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
1921     ierr                = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
1922     PetscFunctionReturn(0);
1923   }
1924 
1925   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
1926   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
1927 
1928   /* get new row and diagonal pointers, must be allocated separately because they will be given to the Mat_SeqAIJ and freed separately */
1929   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
1930   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);
1931   bi[0] = bdiag[0] = 0;
1932 
1933   ierr = PetscMalloc2(n,PetscInt*,&bj_ptr,n,PetscInt*,&bjlvl_ptr);CHKERRQ(ierr);
1934 
1935   /* create a linked list for storing column indices of the active row */
1936   nlnk = n + 1;
1937   ierr = PetscIncompleteLLCreate(n,n,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1938 
1939   /* initial FreeSpace size is f*(ai[n]+1) */
1940   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space);CHKERRQ(ierr);
1941   current_space = free_space;
1942   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space_lvl);CHKERRQ(ierr);
1943   current_space_lvl = free_space_lvl;
1944 
1945   for (i=0; i<n; i++) {
1946     nzi = 0;
1947     /* copy current row into linked list */
1948     nnz  = ai[r[i]+1] - ai[r[i]];
1949     if (!nnz) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
1950     cols = aj + ai[r[i]];
1951     lnk[i] = -1; /* marker to indicate if diagonal exists */
1952     ierr = PetscIncompleteLLInit(nnz,cols,n,ic,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1953     nzi += nlnk;
1954 
1955     /* make sure diagonal entry is included */
1956     if (diagonal_fill && lnk[i] == -1) {
1957       fm = n;
1958       while (lnk[fm] < i) fm = lnk[fm];
1959       lnk[i]     = lnk[fm]; /* insert diagonal into linked list */
1960       lnk[fm]    = i;
1961       lnk_lvl[i] = 0;
1962       nzi++; dcount++;
1963     }
1964 
1965     /* add pivot rows into the active row */
1966     nzbd = 0;
1967     prow = lnk[n];
1968     while (prow < i) {
1969       nnz      = bdiag[prow];
1970       cols     = bj_ptr[prow] + nnz + 1;
1971       cols_lvl = bjlvl_ptr[prow] + nnz + 1;
1972       nnz      = bi[prow+1] - bi[prow] - nnz - 1;
1973       ierr = PetscILULLAddSorted(nnz,cols,levels,cols_lvl,prow,nlnk,lnk,lnk_lvl,lnkbt,prow);CHKERRQ(ierr);
1974       nzi += nlnk;
1975       prow = lnk[prow];
1976       nzbd++;
1977     }
1978     bdiag[i] = nzbd;
1979     bi[i+1]  = bi[i] + nzi;
1980 
1981     /* if free space is not available, make more free space */
1982     if (current_space->local_remaining<nzi) {
1983       nnz = nzi*(n - i); /* estimated and max additional space needed */
1984       ierr = PetscFreeSpaceGet(nnz,&current_space);CHKERRQ(ierr);
1985       ierr = PetscFreeSpaceGet(nnz,&current_space_lvl);CHKERRQ(ierr);
1986       reallocs++;
1987     }
1988 
1989     /* copy data into free_space and free_space_lvl, then initialize lnk */
1990     ierr = PetscIncompleteLLClean(n,n,nzi,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);CHKERRQ(ierr);
1991     bj_ptr[i]    = current_space->array;
1992     bjlvl_ptr[i] = current_space_lvl->array;
1993 
1994     /* make sure the active row i has diagonal entry */
1995     if (*(bj_ptr[i]+bdiag[i]) != i) {
1996       SETERRQ1(PETSC_ERR_MAT_LU_ZRPVT,"Row %D has missing diagonal in factored matrix\n\
1997     try running with -pc_factor_nonzeros_along_diagonal or -pc_factor_diagonal_fill",i);
1998     }
1999 
2000     current_space->array           += nzi;
2001     current_space->local_used      += nzi;
2002     current_space->local_remaining -= nzi;
2003     current_space_lvl->array           += nzi;
2004     current_space_lvl->local_used      += nzi;
2005     current_space_lvl->local_remaining -= nzi;
2006   }
2007 
2008   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
2009   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
2010 
2011   /* destroy list of free space and other temporary arrays */
2012   ierr = PetscMalloc((bi[n]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
2013   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr); /* copy free_space -> bj */
2014   ierr = PetscIncompleteLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
2015   ierr = PetscFreeSpaceDestroy(free_space_lvl);CHKERRQ(ierr);
2016   ierr = PetscFree2(bj_ptr,bjlvl_ptr);CHKERRQ(ierr);
2017 
2018 #if defined(PETSC_USE_INFO)
2019   {
2020     PetscReal af = ((PetscReal)bi[n])/((PetscReal)ai[n]);
2021     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,f,af);CHKERRQ(ierr);
2022     ierr = PetscInfo1(A,"Run with -[sub_]pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
2023     ierr = PetscInfo1(A,"PCFactorSetFill([sub]pc,%G);\n",af);CHKERRQ(ierr);
2024     ierr = PetscInfo(A,"for best performance.\n");CHKERRQ(ierr);
2025     if (diagonal_fill) {
2026       ierr = PetscInfo1(A,"Detected and replaced %D missing diagonals",dcount);CHKERRQ(ierr);
2027     }
2028   }
2029 #endif
2030 
2031   /* put together the new matrix */
2032   ierr = MatSeqAIJSetPreallocation_SeqAIJ(fact,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
2033   ierr = PetscLogObjectParent(fact,isicol);CHKERRQ(ierr);
2034   b = (Mat_SeqAIJ*)(fact)->data;
2035   b->free_a       = PETSC_TRUE;
2036   b->free_ij      = PETSC_TRUE;
2037   b->singlemalloc = PETSC_FALSE;
2038   ierr = PetscMalloc(bi[n]*sizeof(PetscScalar),&b->a);CHKERRQ(ierr);
2039   b->j          = bj;
2040   b->i          = bi;
2041   for (i=0; i<n; i++) bdiag[i] += bi[i];
2042   b->diag       = bdiag;
2043   b->ilen       = 0;
2044   b->imax       = 0;
2045   b->row        = isrow;
2046   b->col        = iscol;
2047   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
2048   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
2049   b->icol       = isicol;
2050   ierr = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
2051   /* In b structure:  Free imax, ilen, old a, old j.
2052      Allocate bdiag, solve_work, new a, new j */
2053   ierr = PetscLogObjectMemory(fact,(bi[n]-n) * (sizeof(PetscInt)+sizeof(PetscScalar)));CHKERRQ(ierr);
2054   b->maxnz             = b->nz = bi[n] ;
2055   (fact)->info.factor_mallocs    = reallocs;
2056   (fact)->info.fill_ratio_given  = f;
2057   (fact)->info.fill_ratio_needed = ((PetscReal)bi[n])/((PetscReal)ai[n]);
2058   (fact)->ops->lufactornumeric =  MatLUFactorNumeric_SeqAIJ_inplace;
2059   if (a->inode.use) {
2060     (fact)->ops->lufactornumeric = MatLUFactorNumeric_SeqAIJ_Inode;
2061   }
2062   PetscFunctionReturn(0);
2063 }
2064 
2065 #undef __FUNCT__
2066 #define __FUNCT__ "MatCholeskyFactorNumeric_SeqAIJ"
2067 PetscErrorCode MatCholeskyFactorNumeric_SeqAIJ(Mat B,Mat A,const MatFactorInfo *info)
2068 {
2069   Mat            C = B;
2070   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data;
2071   Mat_SeqSBAIJ   *b=(Mat_SeqSBAIJ*)C->data;
2072   IS             ip=b->row,iip = b->icol;
2073   PetscErrorCode ierr;
2074   const PetscInt *rip,*riip;
2075   PetscInt       i,j,mbs=A->rmap->n,*bi=b->i,*bj=b->j,*bdiag=b->diag,*bjtmp;
2076   PetscInt       *ai=a->i,*aj=a->j;
2077   PetscInt       k,jmin,jmax,*c2r,*il,col,nexti,ili,nz;
2078   MatScalar      *rtmp,*ba=b->a,*bval,*aa=a->a,dk,uikdi;
2079   PetscTruth     perm_identity;
2080 
2081   FactorShiftCtx sctx;
2082   PetscReal      rs;
2083   MatScalar      d,*v;
2084 
2085   PetscFunctionBegin;
2086   /* MatPivotSetUp(): initialize shift context sctx */
2087   ierr = PetscMemzero(&sctx,sizeof(FactorShiftCtx));CHKERRQ(ierr);
2088 
2089   if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE) { /* set sctx.shift_top=max{rs} */
2090     sctx.shift_top = info->zeropivot;
2091     for (i=0; i<mbs; i++) {
2092       /* calculate sum(|aij|)-RealPart(aii), amt of shift needed for this row */
2093       d  = (aa)[a->diag[i]];
2094       rs = -PetscAbsScalar(d) - PetscRealPart(d);
2095       v  = aa+ai[i];
2096       nz = ai[i+1] - ai[i];
2097       for (j=0; j<nz; j++)
2098 	rs += PetscAbsScalar(v[j]);
2099       if (rs>sctx.shift_top) sctx.shift_top = rs;
2100     }
2101     sctx.shift_top   *= 1.1;
2102     sctx.nshift_max   = 5;
2103     sctx.shift_lo     = 0.;
2104     sctx.shift_hi     = 1.;
2105   }
2106 
2107   ierr  = ISGetIndices(ip,&rip);CHKERRQ(ierr);
2108   ierr  = ISGetIndices(iip,&riip);CHKERRQ(ierr);
2109 
2110   /* allocate working arrays
2111      c2r: linked list, keep track of pivot rows for a given column. c2r[col]: head of the list for a given col
2112      il:  for active k row, il[i] gives the index of the 1st nonzero entry in U[i,k:n-1] in bj and ba arrays
2113   */
2114   ierr = PetscMalloc3(mbs,MatScalar,&rtmp,mbs,PetscInt,&il,mbs,PetscInt,&c2r);CHKERRQ(ierr);
2115 
2116   do {
2117     sctx.useshift = PETSC_FALSE;
2118 
2119     for (i=0; i<mbs; i++) c2r[i] = mbs;
2120     il[0] = 0;
2121 
2122     for (k = 0; k<mbs; k++){
2123       /* zero rtmp */
2124       nz = bi[k+1] - bi[k];
2125       bjtmp = bj + bi[k];
2126       for (j=0; j<nz; j++) rtmp[bjtmp[j]] = 0.0;
2127 
2128       /* load in initial unfactored row */
2129       bval = ba + bi[k];
2130       jmin = ai[rip[k]]; jmax = ai[rip[k]+1];
2131       for (j = jmin; j < jmax; j++){
2132         col = riip[aj[j]];
2133         if (col >= k){ /* only take upper triangular entry */
2134           rtmp[col] = aa[j];
2135           *bval++   = 0.0; /* for in-place factorization */
2136         }
2137       }
2138       /* shift the diagonal of the matrix: ZeropivotApply() */
2139       rtmp[k] += sctx.shift_amount;  /* shift the diagonal of the matrix */
2140 
2141       /* modify k-th row by adding in those rows i with U(i,k)!=0 */
2142       dk = rtmp[k];
2143       i  = c2r[k]; /* first row to be added to k_th row  */
2144 
2145       while (i < k){
2146         nexti = c2r[i]; /* next row to be added to k_th row */
2147 
2148         /* compute multiplier, update diag(k) and U(i,k) */
2149         ili   = il[i];  /* index of first nonzero element in U(i,k:bms-1) */
2150         uikdi = - ba[ili]*ba[bdiag[i]];  /* diagonal(k) */
2151         dk   += uikdi*ba[ili]; /* update diag[k] */
2152         ba[ili] = uikdi; /* -U(i,k) */
2153 
2154         /* add multiple of row i to k-th row */
2155         jmin = ili + 1; jmax = bi[i+1];
2156         if (jmin < jmax){
2157           for (j=jmin; j<jmax; j++) rtmp[bj[j]] += uikdi*ba[j];
2158           /* update il and c2r for row i */
2159           il[i] = jmin;
2160           j = bj[jmin]; c2r[i] = c2r[j]; c2r[j] = i;
2161         }
2162         i = nexti;
2163       }
2164 
2165       /* copy data into U(k,:) */
2166       rs   = 0.0;
2167       jmin = bi[k]; jmax = bi[k+1]-1;
2168       if (jmin < jmax) {
2169         for (j=jmin; j<jmax; j++){
2170           col = bj[j]; ba[j] = rtmp[col]; rs += PetscAbsScalar(ba[j]);
2171         }
2172         /* add the k-th row into il and c2r */
2173         il[k] = jmin;
2174         i = bj[jmin]; c2r[k] = c2r[i]; c2r[i] = k;
2175       }
2176 
2177       /* MatPivotCheck() */
2178       sctx.rs  = rs;
2179       sctx.pv  = dk;
2180       if (info->shifttype == MAT_SHIFT_NONZERO){
2181         ierr = MatPivotCheck_nz(info,sctx,k);CHKERRQ(ierr);
2182       } else if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE){
2183         ierr = MatPivotCheck_pd(info,sctx,k);CHKERRQ(ierr);
2184       } else if (info->shifttype == MAT_SHIFT_INBLOCKS){
2185         ierr = MatPivotCheck_inblocks(info,sctx,k);CHKERRQ(ierr);
2186       } else {
2187         ierr = MatPivotCheck_none(info,sctx,k);CHKERRQ(ierr);
2188       }
2189       dk = sctx.pv;
2190 
2191       ba[bdiag[k]] = 1.0/dk; /* U(k,k) */
2192     }
2193   } while (sctx.useshift);
2194 
2195   ierr = PetscFree3(rtmp,il,c2r);CHKERRQ(ierr);
2196   ierr = ISRestoreIndices(ip,&rip);CHKERRQ(ierr);
2197   ierr = ISRestoreIndices(iip,&riip);CHKERRQ(ierr);
2198 
2199   ierr = ISIdentity(ip,&perm_identity);CHKERRQ(ierr);
2200   if (perm_identity){
2201     B->ops->solve           = MatSolve_SeqSBAIJ_1_NaturalOrdering;
2202     B->ops->solvetranspose  = MatSolve_SeqSBAIJ_1_NaturalOrdering;
2203     B->ops->forwardsolve    = MatForwardSolve_SeqSBAIJ_1_NaturalOrdering;
2204     B->ops->backwardsolve   = MatBackwardSolve_SeqSBAIJ_1_NaturalOrdering;
2205   } else {
2206     B->ops->solve           = MatSolve_SeqSBAIJ_1;
2207     B->ops->solvetranspose  = MatSolve_SeqSBAIJ_1;
2208     B->ops->forwardsolve    = MatForwardSolve_SeqSBAIJ_1;
2209     B->ops->backwardsolve   = MatBackwardSolve_SeqSBAIJ_1;
2210   }
2211 
2212   C->assembled    = PETSC_TRUE;
2213   C->preallocated = PETSC_TRUE;
2214   ierr = PetscLogFlops(C->rmap->n);CHKERRQ(ierr);
2215 
2216   /* MatPivotView() */
2217   if (sctx.nshift){
2218     if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE) {
2219       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);
2220     } else if (info->shifttype == MAT_SHIFT_NONZERO) {
2221       ierr = PetscInfo2(A,"number of shift_nz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
2222     } else if (info->shifttype == MAT_SHIFT_INBLOCKS){
2223       ierr = PetscInfo2(A,"number of shift_inblocks applied %D, each shift_amount %G\n",sctx.nshift,info->shiftamount);CHKERRQ(ierr);
2224     }
2225   }
2226   PetscFunctionReturn(0);
2227 }
2228 
2229 #undef __FUNCT__
2230 #define __FUNCT__ "MatCholeskyFactorNumeric_SeqAIJ_inplace"
2231 PetscErrorCode MatCholeskyFactorNumeric_SeqAIJ_inplace(Mat B,Mat A,const MatFactorInfo *info)
2232 {
2233   Mat            C = B;
2234   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data;
2235   Mat_SeqSBAIJ   *b=(Mat_SeqSBAIJ*)C->data;
2236   IS             ip=b->row,iip = b->icol;
2237   PetscErrorCode ierr;
2238   const PetscInt *rip,*riip;
2239   PetscInt       i,j,mbs=A->rmap->n,*bi=b->i,*bj=b->j,*bcol,*bjtmp;
2240   PetscInt       *ai=a->i,*aj=a->j;
2241   PetscInt       k,jmin,jmax,*jl,*il,col,nexti,ili,nz;
2242   MatScalar      *rtmp,*ba=b->a,*bval,*aa=a->a,dk,uikdi;
2243   PetscReal      zeropivot,rs;
2244   ChShift_Ctx    sctx;
2245   PetscInt       newshift;
2246   PetscTruth     perm_identity;
2247 
2248   PetscFunctionBegin;
2249   zeropivot = info->zeropivot;
2250 
2251   ierr  = ISGetIndices(ip,&rip);CHKERRQ(ierr);
2252   ierr  = ISGetIndices(iip,&riip);CHKERRQ(ierr);
2253 
2254   /* initialization */
2255   ierr = PetscMalloc3(mbs,MatScalar,&rtmp,mbs,PetscInt,&il,mbs,PetscInt,&jl);CHKERRQ(ierr);
2256   sctx.shift_amount = 0;
2257   sctx.nshift       = 0;
2258   do {
2259     sctx.chshift = PETSC_FALSE;
2260     for (i=0; i<mbs; i++) jl[i] = mbs;
2261     il[0] = 0;
2262 
2263     for (k = 0; k<mbs; k++){
2264       /* zero rtmp */
2265       nz = bi[k+1] - bi[k];
2266       bjtmp = bj + bi[k];
2267       for (j=0; j<nz; j++) rtmp[bjtmp[j]] = 0.0;
2268 
2269       bval = ba + bi[k];
2270       /* initialize k-th row by the perm[k]-th row of A */
2271       jmin = ai[rip[k]]; jmax = ai[rip[k]+1];
2272       for (j = jmin; j < jmax; j++){
2273         col = riip[aj[j]];
2274         if (col >= k){ /* only take upper triangular entry */
2275           rtmp[col] = aa[j];
2276           *bval++  = 0.0; /* for in-place factorization */
2277         }
2278       }
2279       /* shift the diagonal of the matrix */
2280       if (sctx.nshift) rtmp[k] += sctx.shift_amount;
2281 
2282       /* modify k-th row by adding in those rows i with U(i,k)!=0 */
2283       dk = rtmp[k];
2284       i = jl[k]; /* first row to be added to k_th row  */
2285 
2286       while (i < k){
2287         nexti = jl[i]; /* next row to be added to k_th row */
2288 
2289         /* compute multiplier, update diag(k) and U(i,k) */
2290         ili = il[i];  /* index of first nonzero element in U(i,k:bms-1) */
2291         uikdi = - ba[ili]*ba[bi[i]];  /* diagonal(k) */
2292         dk += uikdi*ba[ili];
2293         ba[ili] = uikdi; /* -U(i,k) */
2294 
2295         /* add multiple of row i to k-th row */
2296         jmin = ili + 1; jmax = bi[i+1];
2297         if (jmin < jmax){
2298           for (j=jmin; j<jmax; j++) rtmp[bj[j]] += uikdi*ba[j];
2299           /* update il and jl for row i */
2300           il[i] = jmin;
2301           j = bj[jmin]; jl[i] = jl[j]; jl[j] = i;
2302         }
2303         i = nexti;
2304       }
2305 
2306       /* shift the diagonals when zero pivot is detected */
2307       /* compute rs=sum of abs(off-diagonal) */
2308       rs   = 0.0;
2309       jmin = bi[k]+1;
2310       nz   = bi[k+1] - jmin;
2311       bcol = bj + jmin;
2312       for (j=0; j<nz; j++) {
2313         rs += PetscAbsScalar(rtmp[bcol[j]]);
2314       }
2315 
2316       sctx.rs = rs;
2317       sctx.pv = dk;
2318       ierr = MatCholeskyCheckShift_inline(info,sctx,k,newshift);CHKERRQ(ierr);
2319 
2320       if (newshift == 1) {
2321         if (!sctx.shift_amount) {
2322           sctx.shift_amount = 1e-5;
2323         }
2324         break;
2325       }
2326 
2327       /* copy data into U(k,:) */
2328       ba[bi[k]] = 1.0/dk; /* U(k,k) */
2329       jmin = bi[k]+1; jmax = bi[k+1];
2330       if (jmin < jmax) {
2331         for (j=jmin; j<jmax; j++){
2332           col = bj[j]; ba[j] = rtmp[col];
2333         }
2334         /* add the k-th row into il and jl */
2335         il[k] = jmin;
2336         i = bj[jmin]; jl[k] = jl[i]; jl[i] = k;
2337       }
2338     }
2339   } while (sctx.chshift);
2340   ierr = PetscFree3(rtmp,il,jl);CHKERRQ(ierr);
2341   ierr = ISRestoreIndices(ip,&rip);CHKERRQ(ierr);
2342   ierr = ISRestoreIndices(iip,&riip);CHKERRQ(ierr);
2343 
2344   ierr = ISIdentity(ip,&perm_identity);CHKERRQ(ierr);
2345   if (perm_identity){
2346     B->ops->solve           = MatSolve_SeqSBAIJ_1_NaturalOrdering_inplace;
2347     B->ops->solvetranspose  = MatSolve_SeqSBAIJ_1_NaturalOrdering_inplace;
2348     B->ops->forwardsolve    = MatForwardSolve_SeqSBAIJ_1_NaturalOrdering_inplace;
2349     B->ops->backwardsolve   = MatBackwardSolve_SeqSBAIJ_1_NaturalOrdering_inplace;
2350   } else {
2351     B->ops->solve           = MatSolve_SeqSBAIJ_1_inplace;
2352     B->ops->solvetranspose  = MatSolve_SeqSBAIJ_1_inplace;
2353     B->ops->forwardsolve    = MatForwardSolve_SeqSBAIJ_1_inplace;
2354     B->ops->backwardsolve   = MatBackwardSolve_SeqSBAIJ_1_inplace;
2355   }
2356 
2357   C->assembled    = PETSC_TRUE;
2358   C->preallocated = PETSC_TRUE;
2359   ierr = PetscLogFlops(C->rmap->n);CHKERRQ(ierr);
2360   if (sctx.nshift){
2361     if (info->shifttype == MAT_SHIFT_NONZERO) {
2362       ierr = PetscInfo2(A,"number of shiftnz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
2363     } else if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE) {
2364       ierr = PetscInfo2(A,"number of shiftpd tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
2365     }
2366   }
2367   PetscFunctionReturn(0);
2368 }
2369 
2370 /*
2371    icc() under revised new data structure.
2372    Factored arrays bj and ba are stored as
2373      U(0,:),...,U(i,:),U(n-1,:)
2374 
2375    ui=fact->i is an array of size n+1, in which
2376    ui+
2377      ui[i]:  points to 1st entry of U(i,:),i=0,...,n-1
2378      ui[n]:  points to U(n-1,n-1)+1
2379 
2380   udiag=fact->diag is an array of size n,in which
2381      udiag[i]: points to diagonal of U(i,:), i=0,...,n-1
2382 
2383    U(i,:) contains udiag[i] as its last entry, i.e.,
2384     U(i,:) = (u[i,i+1],...,u[i,n-1],diag[i])
2385 */
2386 
2387 #undef __FUNCT__
2388 #define __FUNCT__ "MatICCFactorSymbolic_SeqAIJ"
2389 PetscErrorCode MatICCFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
2390 {
2391   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data;
2392   Mat_SeqSBAIJ       *b;
2393   PetscErrorCode     ierr;
2394   PetscTruth         perm_identity,missing;
2395   PetscInt           reallocs=0,i,*ai=a->i,*aj=a->j,am=A->rmap->n,*ui,*udiag;
2396   const PetscInt     *rip,*riip;
2397   PetscInt           jmin,jmax,nzk,k,j,*jl,prow,*il,nextprow;
2398   PetscInt           nlnk,*lnk,*lnk_lvl=PETSC_NULL,d;
2399   PetscInt           ncols,ncols_upper,*cols,*ajtmp,*uj,**uj_ptr,**uj_lvl_ptr;
2400   PetscReal          fill=info->fill,levels=info->levels;
2401   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
2402   PetscFreeSpaceList free_space_lvl=PETSC_NULL,current_space_lvl=PETSC_NULL;
2403   PetscBT            lnkbt;
2404   IS                 iperm;
2405 
2406   PetscFunctionBegin;
2407   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);
2408   ierr = MatMissingDiagonal(A,&missing,&d);CHKERRQ(ierr);
2409   if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",d);
2410   ierr = ISIdentity(perm,&perm_identity);CHKERRQ(ierr);
2411   ierr = ISInvertPermutation(perm,PETSC_DECIDE,&iperm);CHKERRQ(ierr);
2412 
2413   ierr = PetscMalloc((am+1)*sizeof(PetscInt),&ui);CHKERRQ(ierr);
2414   ierr = PetscMalloc((am+1)*sizeof(PetscInt),&udiag);CHKERRQ(ierr);
2415   ui[0] = 0;
2416 
2417   /* ICC(0) without matrix ordering: simply rearrange column indices */
2418   if (!levels && perm_identity) {
2419     for (i=0; i<am; i++) {
2420       ncols    = ai[i+1] - a->diag[i];
2421       ui[i+1]  = ui[i] + ncols;
2422       udiag[i] = ui[i+1] - 1; /* points to the last entry of U(i,:) */
2423     }
2424     ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
2425     cols = uj;
2426     for (i=0; i<am; i++) {
2427       aj    = a->j + a->diag[i] + 1; /* 1st entry of U(i,:) without diagonal */
2428       ncols = ai[i+1] - a->diag[i] -1;
2429       for (j=0; j<ncols; j++) *cols++ = aj[j];
2430       *cols++ = i; /* diagoanl is located as the last entry of U(i,:) */
2431     }
2432   } else { /* case: levels>0 || (levels=0 && !perm_identity) */
2433     ierr = ISGetIndices(iperm,&riip);CHKERRQ(ierr);
2434     ierr = ISGetIndices(perm,&rip);CHKERRQ(ierr);
2435 
2436     /* initialization */
2437     ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&ajtmp);CHKERRQ(ierr);
2438 
2439     /* jl: linked list for storing indices of the pivot rows
2440        il: il[i] points to the 1st nonzero entry of U(i,k:am-1) */
2441     ierr = PetscMalloc4(am,PetscInt*,&uj_ptr,am,PetscInt*,&uj_lvl_ptr,am,PetscInt,&jl,am,PetscInt,&il);CHKERRQ(ierr);
2442     for (i=0; i<am; i++){
2443       jl[i] = am; il[i] = 0;
2444     }
2445 
2446     /* create and initialize a linked list for storing column indices of the active row k */
2447     nlnk = am + 1;
2448     ierr = PetscIncompleteLLCreate(am,am,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
2449 
2450     /* initial FreeSpace size is fill*(ai[am]+1) */
2451     ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space);CHKERRQ(ierr);
2452     current_space = free_space;
2453     ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space_lvl);CHKERRQ(ierr);
2454     current_space_lvl = free_space_lvl;
2455 
2456     for (k=0; k<am; k++){  /* for each active row k */
2457       /* initialize lnk by the column indices of row rip[k] of A */
2458       nzk   = 0;
2459       ncols = ai[rip[k]+1] - ai[rip[k]];
2460       if (!ncols) SETERRQ2(PETSC_ERR_MAT_CH_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",rip[k],k);
2461       ncols_upper = 0;
2462       for (j=0; j<ncols; j++){
2463         i = *(aj + ai[rip[k]] + j); /* unpermuted column index */
2464         if (riip[i] >= k){ /* only take upper triangular entry */
2465           ajtmp[ncols_upper] = i;
2466           ncols_upper++;
2467         }
2468       }
2469       ierr = PetscIncompleteLLInit(ncols_upper,ajtmp,am,riip,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
2470       nzk += nlnk;
2471 
2472       /* update lnk by computing fill-in for each pivot row to be merged in */
2473       prow = jl[k]; /* 1st pivot row */
2474 
2475       while (prow < k){
2476         nextprow = jl[prow];
2477 
2478         /* merge prow into k-th row */
2479         jmin = il[prow] + 1;  /* index of the 2nd nzero entry in U(prow,k:am-1) */
2480         jmax = ui[prow+1];
2481         ncols = jmax-jmin;
2482         i     = jmin - ui[prow];
2483         cols  = uj_ptr[prow] + i; /* points to the 2nd nzero entry in U(prow,k:am-1) */
2484         uj    = uj_lvl_ptr[prow] + i; /* levels of cols */
2485         j     = *(uj - 1);
2486         ierr = PetscICCLLAddSorted(ncols,cols,levels,uj,am,nlnk,lnk,lnk_lvl,lnkbt,j);CHKERRQ(ierr);
2487         nzk += nlnk;
2488 
2489         /* update il and jl for prow */
2490         if (jmin < jmax){
2491           il[prow] = jmin;
2492           j = *cols; jl[prow] = jl[j]; jl[j] = prow;
2493         }
2494         prow = nextprow;
2495       }
2496 
2497       /* if free space is not available, make more free space */
2498       if (current_space->local_remaining<nzk) {
2499         i  = am - k + 1; /* num of unfactored rows */
2500         i *= PetscMin(nzk, i-1); /* i*nzk, i*(i-1): estimated and max additional space needed */
2501         ierr = PetscFreeSpaceGet(i,&current_space);CHKERRQ(ierr);
2502         ierr = PetscFreeSpaceGet(i,&current_space_lvl);CHKERRQ(ierr);
2503         reallocs++;
2504       }
2505 
2506       /* copy data into free_space and free_space_lvl, then initialize lnk */
2507       if (nzk == 0) SETERRQ1(PETSC_ERR_ARG_WRONG,"Empty row %D in ICC matrix factor",k);
2508       ierr = PetscIncompleteLLClean(am,am,nzk,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);CHKERRQ(ierr);
2509 
2510       /* add the k-th row into il and jl */
2511       if (nzk > 1){
2512         i = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:am-1) */
2513         jl[k] = jl[i]; jl[i] = k;
2514         il[k] = ui[k] + 1;
2515       }
2516       uj_ptr[k]     = current_space->array;
2517       uj_lvl_ptr[k] = current_space_lvl->array;
2518 
2519       current_space->array           += nzk;
2520       current_space->local_used      += nzk;
2521       current_space->local_remaining -= nzk;
2522 
2523       current_space_lvl->array           += nzk;
2524       current_space_lvl->local_used      += nzk;
2525       current_space_lvl->local_remaining -= nzk;
2526 
2527       ui[k+1] = ui[k] + nzk;
2528     }
2529 
2530 #if defined(PETSC_USE_INFO)
2531     if (ai[am] != 0) {
2532       PetscReal af = (PetscReal)ui[am]/((PetscReal)ai[am]);
2533       ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,fill,af);CHKERRQ(ierr);
2534       ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
2535       ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G) for best performance.\n",af);CHKERRQ(ierr);
2536     } else {
2537       ierr = PetscInfo(A,"Empty matrix.\n");CHKERRQ(ierr);
2538     }
2539 #endif
2540 
2541     ierr = ISRestoreIndices(perm,&rip);CHKERRQ(ierr);
2542     ierr = ISRestoreIndices(iperm,&riip);CHKERRQ(ierr);
2543     ierr = PetscFree4(uj_ptr,uj_lvl_ptr,jl,il);CHKERRQ(ierr);
2544     ierr = PetscFree(ajtmp);CHKERRQ(ierr);
2545 
2546     /* destroy list of free space and other temporary array(s) */
2547     ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
2548     ierr = PetscFreeSpaceContiguous_Cholesky(&free_space,uj,am,ui,udiag);CHKERRQ(ierr); /* store matrix factor  */
2549     ierr = PetscIncompleteLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
2550     ierr = PetscFreeSpaceDestroy(free_space_lvl);CHKERRQ(ierr);
2551 
2552   } /* end of case: levels>0 || (levels=0 && !perm_identity) */
2553 
2554   /* put together the new matrix in MATSEQSBAIJ format */
2555   b    = (Mat_SeqSBAIJ*)(fact)->data;
2556   b->singlemalloc = PETSC_FALSE;
2557   ierr = PetscMalloc((ui[am]+1)*sizeof(MatScalar),&b->a);CHKERRQ(ierr);
2558   b->j    = uj;
2559   b->i    = ui;
2560   b->diag = udiag;
2561   b->free_diag = PETSC_TRUE;
2562   b->ilen = 0;
2563   b->imax = 0;
2564   b->row  = perm;
2565   b->col  = perm;
2566   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2567   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2568   b->icol = iperm;
2569   b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */
2570   ierr = PetscMalloc((am+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
2571   ierr = PetscLogObjectMemory(fact,ui[am]*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
2572   b->maxnz   = b->nz = ui[am];
2573   b->free_a  = PETSC_TRUE;
2574   b->free_ij = PETSC_TRUE;
2575 
2576   fact->info.factor_mallocs    = reallocs;
2577   fact->info.fill_ratio_given  = fill;
2578   if (ai[am] != 0) {
2579     fact->info.fill_ratio_needed = ((PetscReal)ui[am])/((PetscReal)ai[am]);
2580   } else {
2581     fact->info.fill_ratio_needed = 0.0;
2582   }
2583   fact->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqAIJ;
2584   PetscFunctionReturn(0);
2585 }
2586 
2587 #undef __FUNCT__
2588 #define __FUNCT__ "MatICCFactorSymbolic_SeqAIJ_inplace"
2589 PetscErrorCode MatICCFactorSymbolic_SeqAIJ_inplace(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
2590 {
2591   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data;
2592   Mat_SeqSBAIJ       *b;
2593   PetscErrorCode     ierr;
2594   PetscTruth         perm_identity,missing;
2595   PetscInt           reallocs=0,i,*ai=a->i,*aj=a->j,am=A->rmap->n,*ui,*udiag;
2596   const PetscInt     *rip,*riip;
2597   PetscInt           jmin,jmax,nzk,k,j,*jl,prow,*il,nextprow;
2598   PetscInt           nlnk,*lnk,*lnk_lvl=PETSC_NULL,d;
2599   PetscInt           ncols,ncols_upper,*cols,*ajtmp,*uj,**uj_ptr,**uj_lvl_ptr;
2600   PetscReal          fill=info->fill,levels=info->levels;
2601   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
2602   PetscFreeSpaceList free_space_lvl=PETSC_NULL,current_space_lvl=PETSC_NULL;
2603   PetscBT            lnkbt;
2604   IS                 iperm;
2605 
2606   PetscFunctionBegin;
2607   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);
2608   ierr = MatMissingDiagonal(A,&missing,&d);CHKERRQ(ierr);
2609   if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",d);
2610   ierr = ISIdentity(perm,&perm_identity);CHKERRQ(ierr);
2611   ierr = ISInvertPermutation(perm,PETSC_DECIDE,&iperm);CHKERRQ(ierr);
2612 
2613   ierr = PetscMalloc((am+1)*sizeof(PetscInt),&ui);CHKERRQ(ierr);
2614   ierr = PetscMalloc((am+1)*sizeof(PetscInt),&udiag);CHKERRQ(ierr);
2615   ui[0] = 0;
2616 
2617   /* ICC(0) without matrix ordering: simply copies fill pattern */
2618   if (!levels && perm_identity) {
2619 
2620     for (i=0; i<am; i++) {
2621       ui[i+1]  = ui[i] + ai[i+1] - a->diag[i];
2622       udiag[i] = ui[i];
2623     }
2624     ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
2625     cols = uj;
2626     for (i=0; i<am; i++) {
2627       aj    = a->j + a->diag[i];
2628       ncols = ui[i+1] - ui[i];
2629       for (j=0; j<ncols; j++) *cols++ = *aj++;
2630     }
2631   } else { /* case: levels>0 || (levels=0 && !perm_identity) */
2632     ierr = ISGetIndices(iperm,&riip);CHKERRQ(ierr);
2633     ierr = ISGetIndices(perm,&rip);CHKERRQ(ierr);
2634 
2635     /* initialization */
2636     ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&ajtmp);CHKERRQ(ierr);
2637 
2638     /* jl: linked list for storing indices of the pivot rows
2639        il: il[i] points to the 1st nonzero entry of U(i,k:am-1) */
2640     ierr = PetscMalloc4(am,PetscInt*,&uj_ptr,am,PetscInt*,&uj_lvl_ptr,am,PetscInt,&jl,am,PetscInt,&il);CHKERRQ(ierr);
2641     for (i=0; i<am; i++){
2642       jl[i] = am; il[i] = 0;
2643     }
2644 
2645     /* create and initialize a linked list for storing column indices of the active row k */
2646     nlnk = am + 1;
2647     ierr = PetscIncompleteLLCreate(am,am,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
2648 
2649     /* initial FreeSpace size is fill*(ai[am]+1) */
2650     ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space);CHKERRQ(ierr);
2651     current_space = free_space;
2652     ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space_lvl);CHKERRQ(ierr);
2653     current_space_lvl = free_space_lvl;
2654 
2655     for (k=0; k<am; k++){  /* for each active row k */
2656       /* initialize lnk by the column indices of row rip[k] of A */
2657       nzk   = 0;
2658       ncols = ai[rip[k]+1] - ai[rip[k]];
2659       if (!ncols) SETERRQ2(PETSC_ERR_MAT_CH_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",rip[k],k);
2660       ncols_upper = 0;
2661       for (j=0; j<ncols; j++){
2662         i = *(aj + ai[rip[k]] + j); /* unpermuted column index */
2663         if (riip[i] >= k){ /* only take upper triangular entry */
2664           ajtmp[ncols_upper] = i;
2665           ncols_upper++;
2666         }
2667       }
2668       ierr = PetscIncompleteLLInit(ncols_upper,ajtmp,am,riip,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
2669       nzk += nlnk;
2670 
2671       /* update lnk by computing fill-in for each pivot row to be merged in */
2672       prow = jl[k]; /* 1st pivot row */
2673 
2674       while (prow < k){
2675         nextprow = jl[prow];
2676 
2677         /* merge prow into k-th row */
2678         jmin = il[prow] + 1;  /* index of the 2nd nzero entry in U(prow,k:am-1) */
2679         jmax = ui[prow+1];
2680         ncols = jmax-jmin;
2681         i     = jmin - ui[prow];
2682         cols  = uj_ptr[prow] + i; /* points to the 2nd nzero entry in U(prow,k:am-1) */
2683         uj    = uj_lvl_ptr[prow] + i; /* levels of cols */
2684         j     = *(uj - 1);
2685         ierr = PetscICCLLAddSorted(ncols,cols,levels,uj,am,nlnk,lnk,lnk_lvl,lnkbt,j);CHKERRQ(ierr);
2686         nzk += nlnk;
2687 
2688         /* update il and jl for prow */
2689         if (jmin < jmax){
2690           il[prow] = jmin;
2691           j = *cols; jl[prow] = jl[j]; jl[j] = prow;
2692         }
2693         prow = nextprow;
2694       }
2695 
2696       /* if free space is not available, make more free space */
2697       if (current_space->local_remaining<nzk) {
2698         i = am - k + 1; /* num of unfactored rows */
2699         i *= PetscMin(nzk, (i-1)); /* i*nzk, i*(i-1): estimated and max additional space needed */
2700         ierr = PetscFreeSpaceGet(i,&current_space);CHKERRQ(ierr);
2701         ierr = PetscFreeSpaceGet(i,&current_space_lvl);CHKERRQ(ierr);
2702         reallocs++;
2703       }
2704 
2705       /* copy data into free_space and free_space_lvl, then initialize lnk */
2706       if (nzk == 0) SETERRQ1(PETSC_ERR_ARG_WRONG,"Empty row %D in ICC matrix factor",k);
2707       ierr = PetscIncompleteLLClean(am,am,nzk,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);CHKERRQ(ierr);
2708 
2709       /* add the k-th row into il and jl */
2710       if (nzk > 1){
2711         i = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:am-1) */
2712         jl[k] = jl[i]; jl[i] = k;
2713         il[k] = ui[k] + 1;
2714       }
2715       uj_ptr[k]     = current_space->array;
2716       uj_lvl_ptr[k] = current_space_lvl->array;
2717 
2718       current_space->array           += nzk;
2719       current_space->local_used      += nzk;
2720       current_space->local_remaining -= nzk;
2721 
2722       current_space_lvl->array           += nzk;
2723       current_space_lvl->local_used      += nzk;
2724       current_space_lvl->local_remaining -= nzk;
2725 
2726       ui[k+1] = ui[k] + nzk;
2727     }
2728 
2729 #if defined(PETSC_USE_INFO)
2730     if (ai[am] != 0) {
2731       PetscReal af = (PetscReal)ui[am]/((PetscReal)ai[am]);
2732       ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,fill,af);CHKERRQ(ierr);
2733       ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
2734       ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G) for best performance.\n",af);CHKERRQ(ierr);
2735     } else {
2736       ierr = PetscInfo(A,"Empty matrix.\n");CHKERRQ(ierr);
2737     }
2738 #endif
2739 
2740     ierr = ISRestoreIndices(perm,&rip);CHKERRQ(ierr);
2741     ierr = ISRestoreIndices(iperm,&riip);CHKERRQ(ierr);
2742     ierr = PetscFree4(uj_ptr,uj_lvl_ptr,jl,il);CHKERRQ(ierr);
2743     ierr = PetscFree(ajtmp);CHKERRQ(ierr);
2744 
2745     /* destroy list of free space and other temporary array(s) */
2746     ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
2747     ierr = PetscFreeSpaceContiguous(&free_space,uj);CHKERRQ(ierr);
2748     ierr = PetscIncompleteLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
2749     ierr = PetscFreeSpaceDestroy(free_space_lvl);CHKERRQ(ierr);
2750 
2751   } /* end of case: levels>0 || (levels=0 && !perm_identity) */
2752 
2753   /* put together the new matrix in MATSEQSBAIJ format */
2754 
2755   b    = (Mat_SeqSBAIJ*)fact->data;
2756   b->singlemalloc = PETSC_FALSE;
2757   ierr = PetscMalloc((ui[am]+1)*sizeof(MatScalar),&b->a);CHKERRQ(ierr);
2758   b->j    = uj;
2759   b->i    = ui;
2760   b->diag = udiag;
2761   b->free_diag = PETSC_TRUE;
2762   b->ilen = 0;
2763   b->imax = 0;
2764   b->row  = perm;
2765   b->col  = perm;
2766   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2767   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2768   b->icol = iperm;
2769   b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */
2770   ierr    = PetscMalloc((am+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
2771   ierr = PetscLogObjectMemory(fact,(ui[am]-am)*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
2772   b->maxnz   = b->nz = ui[am];
2773   b->free_a  = PETSC_TRUE;
2774   b->free_ij = PETSC_TRUE;
2775 
2776   fact->info.factor_mallocs    = reallocs;
2777   fact->info.fill_ratio_given  = fill;
2778   if (ai[am] != 0) {
2779     fact->info.fill_ratio_needed = ((PetscReal)ui[am])/((PetscReal)ai[am]);
2780   } else {
2781     fact->info.fill_ratio_needed = 0.0;
2782   }
2783   fact->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqAIJ_inplace;
2784   PetscFunctionReturn(0);
2785 }
2786 
2787 PetscErrorCode MatCholeskyFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
2788 {
2789   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data;
2790   Mat_SeqSBAIJ       *b;
2791   PetscErrorCode     ierr;
2792   PetscTruth         perm_identity;
2793   PetscReal          fill = info->fill;
2794   const PetscInt     *rip,*riip;
2795   PetscInt           i,am=A->rmap->n,*ai=a->i,*aj=a->j,reallocs=0,prow;
2796   PetscInt           *jl,jmin,jmax,nzk,*ui,k,j,*il,nextprow;
2797   PetscInt           nlnk,*lnk,ncols,ncols_upper,*cols,*uj,**ui_ptr,*uj_ptr,*udiag;
2798   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
2799   PetscBT            lnkbt;
2800   IS                 iperm;
2801 
2802   PetscFunctionBegin;
2803   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);
2804   /* check whether perm is the identity mapping */
2805   ierr = ISIdentity(perm,&perm_identity);CHKERRQ(ierr);
2806   ierr = ISInvertPermutation(perm,PETSC_DECIDE,&iperm);CHKERRQ(ierr);
2807   ierr = ISGetIndices(iperm,&riip);CHKERRQ(ierr);
2808   ierr = ISGetIndices(perm,&rip);CHKERRQ(ierr);
2809 
2810   /* initialization */
2811   ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&ui);CHKERRQ(ierr);
2812   ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&udiag);CHKERRQ(ierr);
2813   ui[0] = 0;
2814 
2815   /* jl: linked list for storing indices of the pivot rows
2816      il: il[i] points to the 1st nonzero entry of U(i,k:am-1) */
2817   ierr = PetscMalloc4(am,PetscInt*,&ui_ptr,am,PetscInt,&jl,am,PetscInt,&il,am,PetscInt,&cols);CHKERRQ(ierr);
2818   for (i=0; i<am; i++){
2819     jl[i] = am; il[i] = 0;
2820   }
2821 
2822   /* create and initialize a linked list for storing column indices of the active row k */
2823   nlnk = am + 1;
2824   ierr = PetscLLCreate(am,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2825 
2826   /* initial FreeSpace size is fill*(ai[am]+1) */
2827   ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space);CHKERRQ(ierr);
2828   current_space = free_space;
2829 
2830   for (k=0; k<am; k++){  /* for each active row k */
2831     /* initialize lnk by the column indices of row rip[k] of A */
2832     nzk   = 0;
2833     ncols = ai[rip[k]+1] - ai[rip[k]];
2834     if (!ncols) SETERRQ2(PETSC_ERR_MAT_CH_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",rip[k],k);
2835     ncols_upper = 0;
2836     for (j=0; j<ncols; j++){
2837       i = riip[*(aj + ai[rip[k]] + j)];
2838       if (i >= k){ /* only take upper triangular entry */
2839         cols[ncols_upper] = i;
2840         ncols_upper++;
2841       }
2842     }
2843     ierr = PetscLLAdd(ncols_upper,cols,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2844     nzk += nlnk;
2845 
2846     /* update lnk by computing fill-in for each pivot row to be merged in */
2847     prow = jl[k]; /* 1st pivot row */
2848 
2849     while (prow < k){
2850       nextprow = jl[prow];
2851       /* merge prow into k-th row */
2852       jmin = il[prow] + 1;  /* index of the 2nd nzero entry in U(prow,k:am-1) */
2853       jmax = ui[prow+1];
2854       ncols = jmax-jmin;
2855       uj_ptr = ui_ptr[prow] + jmin - ui[prow]; /* points to the 2nd nzero entry in U(prow,k:am-1) */
2856       ierr = PetscLLAddSorted(ncols,uj_ptr,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2857       nzk += nlnk;
2858 
2859       /* update il and jl for prow */
2860       if (jmin < jmax){
2861         il[prow] = jmin;
2862         j = *uj_ptr; jl[prow] = jl[j]; jl[j] = prow;
2863       }
2864       prow = nextprow;
2865     }
2866 
2867     /* if free space is not available, make more free space */
2868     if (current_space->local_remaining<nzk) {
2869       i  = am - k + 1; /* num of unfactored rows */
2870       i *= PetscMin(nzk,i-1); /* i*nzk, i*(i-1): estimated and max additional space needed */
2871       ierr = PetscFreeSpaceGet(i,&current_space);CHKERRQ(ierr);
2872       reallocs++;
2873     }
2874 
2875     /* copy data into free space, then initialize lnk */
2876     ierr = PetscLLClean(am,am,nzk,lnk,current_space->array,lnkbt);CHKERRQ(ierr);
2877 
2878     /* add the k-th row into il and jl */
2879     if (nzk > 1){
2880       i = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:am-1) */
2881       jl[k] = jl[i]; jl[i] = k;
2882       il[k] = ui[k] + 1;
2883     }
2884     ui_ptr[k] = current_space->array;
2885     current_space->array           += nzk;
2886     current_space->local_used      += nzk;
2887     current_space->local_remaining -= nzk;
2888 
2889     ui[k+1] = ui[k] + nzk;
2890   }
2891 
2892 #if defined(PETSC_USE_INFO)
2893   if (ai[am] != 0) {
2894     PetscReal af = (PetscReal)(ui[am])/((PetscReal)ai[am]);
2895     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,fill,af);CHKERRQ(ierr);
2896     ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
2897     ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G) for best performance.\n",af);CHKERRQ(ierr);
2898   } else {
2899      ierr = PetscInfo(A,"Empty matrix.\n");CHKERRQ(ierr);
2900   }
2901 #endif
2902 
2903   ierr = ISRestoreIndices(perm,&rip);CHKERRQ(ierr);
2904   ierr = ISRestoreIndices(iperm,&riip);CHKERRQ(ierr);
2905   ierr = PetscFree4(ui_ptr,jl,il,cols);CHKERRQ(ierr);
2906 
2907   /* destroy list of free space and other temporary array(s) */
2908   ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
2909   ierr = PetscFreeSpaceContiguous_Cholesky(&free_space,uj,am,ui,udiag);CHKERRQ(ierr); /* store matrix factor */
2910   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
2911 
2912   /* put together the new matrix in MATSEQSBAIJ format */
2913 
2914   b = (Mat_SeqSBAIJ*)fact->data;
2915   b->singlemalloc = PETSC_FALSE;
2916   b->free_a       = PETSC_TRUE;
2917   b->free_ij      = PETSC_TRUE;
2918   ierr = PetscMalloc((ui[am]+1)*sizeof(MatScalar),&b->a);CHKERRQ(ierr);
2919   b->j    = uj;
2920   b->i    = ui;
2921   b->diag = udiag;
2922   b->free_diag = PETSC_TRUE;
2923   b->ilen = 0;
2924   b->imax = 0;
2925   b->row  = perm;
2926   b->col  = perm;
2927   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2928   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2929   b->icol = iperm;
2930   b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */
2931   ierr    = PetscMalloc((am+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
2932   ierr    = PetscLogObjectMemory(fact,ui[am]*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
2933   b->maxnz = b->nz = ui[am];
2934 
2935   fact->info.factor_mallocs    = reallocs;
2936   fact->info.fill_ratio_given  = fill;
2937   if (ai[am] != 0) {
2938     fact->info.fill_ratio_needed = ((PetscReal)ui[am])/((PetscReal)ai[am]);
2939   } else {
2940     fact->info.fill_ratio_needed = 0.0;
2941   }
2942   fact->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqAIJ;
2943   PetscFunctionReturn(0);
2944 }
2945 
2946 #undef __FUNCT__
2947 #define __FUNCT__ "MatCholeskyFactorSymbolic_SeqAIJ_inplace"
2948 PetscErrorCode MatCholeskyFactorSymbolic_SeqAIJ_inplace(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
2949 {
2950   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data;
2951   Mat_SeqSBAIJ       *b;
2952   PetscErrorCode     ierr;
2953   PetscTruth         perm_identity;
2954   PetscReal          fill = info->fill;
2955   const PetscInt     *rip,*riip;
2956   PetscInt           i,am=A->rmap->n,*ai=a->i,*aj=a->j,reallocs=0,prow;
2957   PetscInt           *jl,jmin,jmax,nzk,*ui,k,j,*il,nextprow;
2958   PetscInt           nlnk,*lnk,ncols,ncols_upper,*cols,*uj,**ui_ptr,*uj_ptr;
2959   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
2960   PetscBT            lnkbt;
2961   IS                 iperm;
2962 
2963   PetscFunctionBegin;
2964   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);
2965   /* check whether perm is the identity mapping */
2966   ierr = ISIdentity(perm,&perm_identity);CHKERRQ(ierr);
2967   ierr = ISInvertPermutation(perm,PETSC_DECIDE,&iperm);CHKERRQ(ierr);
2968   ierr = ISGetIndices(iperm,&riip);CHKERRQ(ierr);
2969   ierr = ISGetIndices(perm,&rip);CHKERRQ(ierr);
2970 
2971   /* initialization */
2972   ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&ui);CHKERRQ(ierr);
2973   ui[0] = 0;
2974 
2975   /* jl: linked list for storing indices of the pivot rows
2976      il: il[i] points to the 1st nonzero entry of U(i,k:am-1) */
2977   ierr = PetscMalloc4(am,PetscInt*,&ui_ptr,am,PetscInt,&jl,am,PetscInt,&il,am,PetscInt,&cols);CHKERRQ(ierr);
2978   for (i=0; i<am; i++){
2979     jl[i] = am; il[i] = 0;
2980   }
2981 
2982   /* create and initialize a linked list for storing column indices of the active row k */
2983   nlnk = am + 1;
2984   ierr = PetscLLCreate(am,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2985 
2986   /* initial FreeSpace size is fill*(ai[am]+1) */
2987   ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space);CHKERRQ(ierr);
2988   current_space = free_space;
2989 
2990   for (k=0; k<am; k++){  /* for each active row k */
2991     /* initialize lnk by the column indices of row rip[k] of A */
2992     nzk   = 0;
2993     ncols = ai[rip[k]+1] - ai[rip[k]];
2994     if (!ncols) SETERRQ2(PETSC_ERR_MAT_CH_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",rip[k],k);
2995     ncols_upper = 0;
2996     for (j=0; j<ncols; j++){
2997       i = riip[*(aj + ai[rip[k]] + j)];
2998       if (i >= k){ /* only take upper triangular entry */
2999         cols[ncols_upper] = i;
3000         ncols_upper++;
3001       }
3002     }
3003     ierr = PetscLLAdd(ncols_upper,cols,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
3004     nzk += nlnk;
3005 
3006     /* update lnk by computing fill-in for each pivot row to be merged in */
3007     prow = jl[k]; /* 1st pivot row */
3008 
3009     while (prow < k){
3010       nextprow = jl[prow];
3011       /* merge prow into k-th row */
3012       jmin = il[prow] + 1;  /* index of the 2nd nzero entry in U(prow,k:am-1) */
3013       jmax = ui[prow+1];
3014       ncols = jmax-jmin;
3015       uj_ptr = ui_ptr[prow] + jmin - ui[prow]; /* points to the 2nd nzero entry in U(prow,k:am-1) */
3016       ierr = PetscLLAddSorted(ncols,uj_ptr,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
3017       nzk += nlnk;
3018 
3019       /* update il and jl for prow */
3020       if (jmin < jmax){
3021         il[prow] = jmin;
3022         j = *uj_ptr; jl[prow] = jl[j]; jl[j] = prow;
3023       }
3024       prow = nextprow;
3025     }
3026 
3027     /* if free space is not available, make more free space */
3028     if (current_space->local_remaining<nzk) {
3029       i = am - k + 1; /* num of unfactored rows */
3030       i = PetscMin(i*nzk, i*(i-1)); /* i*nzk, i*(i-1): estimated and max additional space needed */
3031       ierr = PetscFreeSpaceGet(i,&current_space);CHKERRQ(ierr);
3032       reallocs++;
3033     }
3034 
3035     /* copy data into free space, then initialize lnk */
3036     ierr = PetscLLClean(am,am,nzk,lnk,current_space->array,lnkbt);CHKERRQ(ierr);
3037 
3038     /* add the k-th row into il and jl */
3039     if (nzk-1 > 0){
3040       i = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:am-1) */
3041       jl[k] = jl[i]; jl[i] = k;
3042       il[k] = ui[k] + 1;
3043     }
3044     ui_ptr[k] = current_space->array;
3045     current_space->array           += nzk;
3046     current_space->local_used      += nzk;
3047     current_space->local_remaining -= nzk;
3048 
3049     ui[k+1] = ui[k] + nzk;
3050   }
3051 
3052 #if defined(PETSC_USE_INFO)
3053   if (ai[am] != 0) {
3054     PetscReal af = (PetscReal)(ui[am])/((PetscReal)ai[am]);
3055     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,fill,af);CHKERRQ(ierr);
3056     ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
3057     ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G) for best performance.\n",af);CHKERRQ(ierr);
3058   } else {
3059      ierr = PetscInfo(A,"Empty matrix.\n");CHKERRQ(ierr);
3060   }
3061 #endif
3062 
3063   ierr = ISRestoreIndices(perm,&rip);CHKERRQ(ierr);
3064   ierr = ISRestoreIndices(iperm,&riip);CHKERRQ(ierr);
3065   ierr = PetscFree4(ui_ptr,jl,il,cols);CHKERRQ(ierr);
3066 
3067   /* destroy list of free space and other temporary array(s) */
3068   ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
3069   ierr = PetscFreeSpaceContiguous(&free_space,uj);CHKERRQ(ierr);
3070   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
3071 
3072   /* put together the new matrix in MATSEQSBAIJ format */
3073 
3074   b = (Mat_SeqSBAIJ*)fact->data;
3075   b->singlemalloc = PETSC_FALSE;
3076   b->free_a       = PETSC_TRUE;
3077   b->free_ij      = PETSC_TRUE;
3078   ierr = PetscMalloc((ui[am]+1)*sizeof(MatScalar),&b->a);CHKERRQ(ierr);
3079   b->j    = uj;
3080   b->i    = ui;
3081   b->diag = 0;
3082   b->ilen = 0;
3083   b->imax = 0;
3084   b->row  = perm;
3085   b->col  = perm;
3086   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
3087   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
3088   b->icol = iperm;
3089   b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */
3090   ierr    = PetscMalloc((am+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
3091   ierr    = PetscLogObjectMemory(fact,(ui[am]-am)*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
3092   b->maxnz = b->nz = ui[am];
3093 
3094   fact->info.factor_mallocs    = reallocs;
3095   fact->info.fill_ratio_given  = fill;
3096   if (ai[am] != 0) {
3097     fact->info.fill_ratio_needed = ((PetscReal)ui[am])/((PetscReal)ai[am]);
3098   } else {
3099     fact->info.fill_ratio_needed = 0.0;
3100   }
3101   fact->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqAIJ_inplace;
3102   PetscFunctionReturn(0);
3103 }
3104 
3105 #undef __FUNCT__
3106 #define __FUNCT__ "MatSolve_SeqAIJ_NaturalOrdering"
3107 PetscErrorCode MatSolve_SeqAIJ_NaturalOrdering(Mat A,Vec bb,Vec xx)
3108 {
3109   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
3110   PetscErrorCode    ierr;
3111   PetscInt          n = A->rmap->n;
3112   const PetscInt    *ai = a->i,*aj = a->j,*adiag = a->diag,*vi;
3113   PetscScalar       *x,sum;
3114   const PetscScalar *b;
3115   const MatScalar   *aa = a->a,*v;
3116   PetscInt          i,nz;
3117 
3118   PetscFunctionBegin;
3119   if (!n) PetscFunctionReturn(0);
3120 
3121   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
3122   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
3123 
3124   /* forward solve the lower triangular */
3125   x[0] = b[0];
3126   v    = aa;
3127   vi   = aj;
3128   for (i=1; i<n; i++) {
3129     nz  = ai[i+1] - ai[i];
3130     sum = b[i];
3131     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
3132     v  += nz;
3133     vi += nz;
3134     x[i] = sum;
3135   }
3136 
3137   /* backward solve the upper triangular */
3138   for (i=n-1; i>=0; i--){
3139     v   = aa + adiag[i+1] + 1;
3140     vi  = aj + adiag[i+1] + 1;
3141     nz = adiag[i] - adiag[i+1]-1;
3142     sum = x[i];
3143     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
3144     x[i] = sum*v[nz]; /* x[i]=aa[adiag[i]]*sum; v++; */
3145   }
3146 
3147   ierr = PetscLogFlops(2.0*a->nz - A->cmap->n);CHKERRQ(ierr);
3148   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
3149   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
3150   PetscFunctionReturn(0);
3151 }
3152 
3153 #undef __FUNCT__
3154 #define __FUNCT__ "MatSolve_SeqAIJ"
3155 PetscErrorCode MatSolve_SeqAIJ(Mat A,Vec bb,Vec xx)
3156 {
3157   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
3158   IS                iscol = a->col,isrow = a->row;
3159   PetscErrorCode    ierr;
3160   PetscInt          i,n=A->rmap->n,*vi,*ai=a->i,*aj=a->j,*adiag = a->diag,nz;
3161   const PetscInt    *rout,*cout,*r,*c;
3162   PetscScalar       *x,*tmp,sum;
3163   const PetscScalar *b;
3164   const MatScalar   *aa = a->a,*v;
3165 
3166   PetscFunctionBegin;
3167   if (!n) PetscFunctionReturn(0);
3168 
3169   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
3170   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
3171   tmp  = a->solve_work;
3172 
3173   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
3174   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
3175 
3176   /* forward solve the lower triangular */
3177   tmp[0] = b[r[0]];
3178   v      = aa;
3179   vi     = aj;
3180   for (i=1; i<n; i++) {
3181     nz  = ai[i+1] - ai[i];
3182     sum = b[r[i]];
3183     PetscSparseDenseMinusDot(sum,tmp,v,vi,nz);
3184     tmp[i] = sum;
3185     v += nz; vi += nz;
3186   }
3187 
3188   /* backward solve the upper triangular */
3189   for (i=n-1; i>=0; i--){
3190     v   = aa + adiag[i+1]+1;
3191     vi  = aj + adiag[i+1]+1;
3192     nz  = adiag[i]-adiag[i+1]-1;
3193     sum = tmp[i];
3194     PetscSparseDenseMinusDot(sum,tmp,v,vi,nz);
3195     x[c[i]] = tmp[i] = sum*v[nz]; /* v[nz] = aa[adiag[i]] */
3196   }
3197 
3198   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
3199   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
3200   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
3201   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
3202   ierr = PetscLogFlops(2*a->nz - A->cmap->n);CHKERRQ(ierr);
3203   PetscFunctionReturn(0);
3204 }
3205 
3206 #undef __FUNCT__
3207 #define __FUNCT__ "MatILUDTFactor_SeqAIJ"
3208 /*
3209     This will get a new name and become a varient of MatILUFactor_SeqAIJ() there is no longer seperate functions in the matrix function table for dt factors
3210 */
3211 PetscErrorCode MatILUDTFactor_SeqAIJ(Mat A,IS isrow,IS iscol,const MatFactorInfo *info,Mat *fact)
3212 {
3213   Mat                B = *fact;
3214   Mat_SeqAIJ         *a=(Mat_SeqAIJ*)A->data,*b;
3215   IS                 isicol;
3216   PetscErrorCode     ierr;
3217   const PetscInt     *r,*ic;
3218   PetscInt           i,n=A->rmap->n,*ai=a->i,*aj=a->j,*ajtmp,*adiag;
3219   PetscInt           *bi,*bj,*bdiag,*bdiag_rev;
3220   PetscInt           row,nzi,nzi_bl,nzi_bu,*im,nzi_al,nzi_au;
3221   PetscInt           nlnk,*lnk;
3222   PetscBT            lnkbt;
3223   PetscTruth         row_identity,icol_identity,both_identity;
3224   MatScalar          *aatmp,*pv,*batmp,*ba,*rtmp,*pc,multiplier,*vtmp,diag_tmp;
3225   const PetscInt     *ics;
3226   PetscInt           j,nz,*pj,*bjtmp,k,ncut,*jtmp;
3227   PetscReal          dt=info->dt,dtcol=info->dtcol,shift=info->shiftamount;
3228   PetscInt           dtcount=(PetscInt)info->dtcount,nnz_max;
3229   PetscTruth         missing;
3230 
3231   PetscFunctionBegin;
3232 
3233   if (dt      == PETSC_DEFAULT) dt      = 0.005;
3234   if (dtcol   == PETSC_DEFAULT) dtcol   = 0.01; /* XXX unused! */
3235   if (dtcount == PETSC_DEFAULT) dtcount = (PetscInt)(1.5*a->rmax);
3236 
3237   /* ------- symbolic factorization, can be reused ---------*/
3238   ierr = MatMissingDiagonal(A,&missing,&i);CHKERRQ(ierr);
3239   if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",i);
3240   adiag=a->diag;
3241 
3242   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
3243 
3244   /* bdiag is location of diagonal in factor */
3245   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);     /* becomes b->diag */
3246   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag_rev);CHKERRQ(ierr); /* temporary */
3247 
3248   /* allocate row pointers bi */
3249   ierr = PetscMalloc((2*n+2)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
3250 
3251   /* allocate bj and ba; max num of nonzero entries is (ai[n]+2*n*dtcount+2) */
3252   if (dtcount > n-1) dtcount = n-1; /* diagonal is excluded */
3253   nnz_max  = ai[n]+2*n*dtcount+2;
3254 
3255   ierr = PetscMalloc((nnz_max+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
3256   ierr = PetscMalloc((nnz_max+1)*sizeof(MatScalar),&ba);CHKERRQ(ierr);
3257 
3258   /* put together the new matrix */
3259   ierr = MatSeqAIJSetPreallocation_SeqAIJ(B,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
3260   ierr = PetscLogObjectParent(B,isicol);CHKERRQ(ierr);
3261   b    = (Mat_SeqAIJ*)B->data;
3262   b->free_a       = PETSC_TRUE;
3263   b->free_ij      = PETSC_TRUE;
3264   b->singlemalloc = PETSC_FALSE;
3265   b->a          = ba;
3266   b->j          = bj;
3267   b->i          = bi;
3268   b->diag       = bdiag;
3269   b->ilen       = 0;
3270   b->imax       = 0;
3271   b->row        = isrow;
3272   b->col        = iscol;
3273   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
3274   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
3275   b->icol       = isicol;
3276   ierr = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
3277 
3278   ierr = PetscLogObjectMemory(B,nnz_max*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
3279   b->maxnz = nnz_max;
3280 
3281   B->factor                = MAT_FACTOR_ILUDT;
3282   B->info.factor_mallocs   = 0;
3283   B->info.fill_ratio_given = ((PetscReal)nnz_max)/((PetscReal)ai[n]);
3284   CHKMEMQ;
3285   /* ------- end of symbolic factorization ---------*/
3286 
3287   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
3288   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
3289   ics  = ic;
3290 
3291   /* linked list for storing column indices of the active row */
3292   nlnk = n + 1;
3293   ierr = PetscLLCreate(n,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
3294 
3295   /* im: used by PetscLLAddSortedLU(); jtmp: working array for column indices of active row */
3296   ierr = PetscMalloc2(n,PetscInt,&im,n,PetscInt,&jtmp);CHKERRQ(ierr);
3297   /* rtmp, vtmp: working arrays for sparse and contiguous row entries of active row */
3298   ierr = PetscMalloc2(n,MatScalar,&rtmp,n,MatScalar,&vtmp);CHKERRQ(ierr);
3299   ierr = PetscMemzero(rtmp,n*sizeof(MatScalar));CHKERRQ(ierr);
3300 
3301   bi[0]    = 0;
3302   bdiag[0] = nnz_max-1; /* location of diag[0] in factor B */
3303   bdiag_rev[n] = bdiag[0];
3304   bi[2*n+1] = bdiag[0]+1; /* endof bj and ba array */
3305   for (i=0; i<n; i++) {
3306     /* copy initial fill into linked list */
3307     nzi = 0; /* nonzeros for active row i */
3308     nzi = ai[r[i]+1] - ai[r[i]];
3309     if (!nzi) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
3310     nzi_al = adiag[r[i]] - ai[r[i]];
3311     nzi_au = ai[r[i]+1] - adiag[r[i]] -1;
3312     ajtmp = aj + ai[r[i]];
3313     ierr = PetscLLAddPerm(nzi,ajtmp,ic,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
3314 
3315     /* load in initial (unfactored row) */
3316     aatmp = a->a + ai[r[i]];
3317     for (j=0; j<nzi; j++) {
3318       rtmp[ics[*ajtmp++]] = *aatmp++;
3319     }
3320 
3321     /* add pivot rows into linked list */
3322     row = lnk[n];
3323     while (row < i ) {
3324       nzi_bl = bi[row+1] - bi[row] + 1;
3325       bjtmp = bj + bdiag[row+1]+1; /* points to 1st column next to the diagonal in U */
3326       ierr  = PetscLLAddSortedLU(bjtmp,row,nlnk,lnk,lnkbt,i,nzi_bl,im);CHKERRQ(ierr);
3327       nzi  += nlnk;
3328       row   = lnk[row];
3329     }
3330 
3331     /* copy data from lnk into jtmp, then initialize lnk */
3332     ierr = PetscLLClean(n,n,nzi,lnk,jtmp,lnkbt);CHKERRQ(ierr);
3333 
3334     /* numerical factorization */
3335     bjtmp = jtmp;
3336     row   = *bjtmp++; /* 1st pivot row */
3337     while  ( row < i ) {
3338       pc         = rtmp + row;
3339       pv         = ba + bdiag[row]; /* 1./(diag of the pivot row) */
3340       multiplier = (*pc) * (*pv);
3341       *pc        = multiplier;
3342       if (PetscAbsScalar(*pc) > dt){ /* apply tolerance dropping rule */
3343         pj         = bj + bdiag[row+1] + 1; /* point to 1st entry of U(row,:) */
3344         pv         = ba + bdiag[row+1] + 1;
3345         /* if (multiplier < -1.0 or multiplier >1.0) printf("row/prow %d, %d, multiplier %g\n",i,row,multiplier); */
3346         nz         = bdiag[row] - bdiag[row+1] - 1; /* num of entries in U(row,:), excluding diagonal */
3347         for (j=0; j<nz; j++) rtmp[*pj++] -= multiplier * (*pv++);
3348         ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr);
3349       }
3350       row = *bjtmp++;
3351     }
3352 
3353     /* copy sparse rtmp into contiguous vtmp; separate L and U part */
3354     diag_tmp = rtmp[i];  /* save diagonal value - may not needed?? */
3355     nzi_bl = 0; j = 0;
3356     while (jtmp[j] < i){ /* Note: jtmp is sorted */
3357       vtmp[j] = rtmp[jtmp[j]]; rtmp[jtmp[j]]=0.0;
3358       nzi_bl++; j++;
3359     }
3360     nzi_bu = nzi - nzi_bl -1;
3361     while (j < nzi){
3362       vtmp[j] = rtmp[jtmp[j]]; rtmp[jtmp[j]]=0.0;
3363       j++;
3364     }
3365 
3366     bjtmp = bj + bi[i];
3367     batmp = ba + bi[i];
3368     /* apply level dropping rule to L part */
3369     ncut = nzi_al + dtcount;
3370     if (ncut < nzi_bl){
3371       ierr = PetscSortSplit(ncut,nzi_bl,vtmp,jtmp);CHKERRQ(ierr);
3372       ierr = PetscSortIntWithScalarArray(ncut,jtmp,vtmp);CHKERRQ(ierr);
3373     } else {
3374       ncut = nzi_bl;
3375     }
3376     for (j=0; j<ncut; j++){
3377       bjtmp[j] = jtmp[j];
3378       batmp[j] = vtmp[j];
3379       /* printf(" (%d,%g),",bjtmp[j],batmp[j]); */
3380     }
3381     bi[i+1] = bi[i] + ncut;
3382     nzi = ncut + 1;
3383 
3384     /* apply level dropping rule to U part */
3385     ncut = nzi_au + dtcount;
3386     if (ncut < nzi_bu){
3387       ierr = PetscSortSplit(ncut,nzi_bu,vtmp+nzi_bl+1,jtmp+nzi_bl+1);CHKERRQ(ierr);
3388       ierr = PetscSortIntWithScalarArray(ncut,jtmp+nzi_bl+1,vtmp+nzi_bl+1);CHKERRQ(ierr);
3389     } else {
3390       ncut = nzi_bu;
3391     }
3392     nzi += ncut;
3393 
3394     /* mark bdiagonal */
3395     bdiag[i+1]       = bdiag[i] - (ncut + 1);
3396     bdiag_rev[n-i-1] = bdiag[i+1];
3397     bi[2*n - i]      = bi[2*n - i +1] - (ncut + 1);
3398     bjtmp = bj + bdiag[i];
3399     batmp = ba + bdiag[i];
3400     *bjtmp = i;
3401     *batmp = diag_tmp; /* rtmp[i]; */
3402     if (*batmp == 0.0) {
3403       *batmp = dt+shift;
3404       /* printf(" row %d add shift %g\n",i,shift); */
3405     }
3406     *batmp = 1.0/(*batmp); /* invert diagonal entries for simplier triangular solves */
3407     /* printf(" (%d,%g),",*bjtmp,*batmp); */
3408 
3409     bjtmp = bj + bdiag[i+1]+1;
3410     batmp = ba + bdiag[i+1]+1;
3411     for (k=0; k<ncut; k++){
3412       bjtmp[k] = jtmp[nzi_bl+1+k];
3413       batmp[k] = vtmp[nzi_bl+1+k];
3414       /* printf(" (%d,%g),",bjtmp[k],batmp[k]); */
3415     }
3416     /* printf("\n"); */
3417 
3418     im[i]   = nzi; /* used by PetscLLAddSortedLU() */
3419     /*
3420     printf("row %d: bi %d, bdiag %d\n",i,bi[i],bdiag[i]);
3421     printf(" ----------------------------\n");
3422     */
3423   } /* for (i=0; i<n; i++) */
3424   /* printf("end of L %d, beginning of U %d\n",bi[n],bdiag[n]); */
3425   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]);
3426 
3427   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
3428   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
3429 
3430   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
3431   ierr = PetscFree2(im,jtmp);CHKERRQ(ierr);
3432   ierr = PetscFree2(rtmp,vtmp);CHKERRQ(ierr);
3433   ierr = PetscFree(bdiag_rev);CHKERRQ(ierr);
3434 
3435   ierr = PetscLogFlops(B->cmap->n);CHKERRQ(ierr);
3436   b->maxnz = b->nz = bi[n] + bdiag[0] - bdiag[n];
3437 
3438   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
3439   ierr = ISIdentity(isicol,&icol_identity);CHKERRQ(ierr);
3440   both_identity = (PetscTruth) (row_identity && icol_identity);
3441   if (row_identity && icol_identity) {
3442     B->ops->solve = MatSolve_SeqAIJ_NaturalOrdering;
3443   } else {
3444     B->ops->solve = MatSolve_SeqAIJ;
3445   }
3446 
3447   B->ops->solveadd          = 0;
3448   B->ops->solvetranspose    = 0;
3449   B->ops->solvetransposeadd = 0;
3450   B->ops->matsolve          = 0;
3451   B->assembled              = PETSC_TRUE;
3452   B->preallocated           = PETSC_TRUE;
3453   PetscFunctionReturn(0);
3454 }
3455 
3456 /* a wraper of MatILUDTFactor_SeqAIJ() */
3457 #undef __FUNCT__
3458 #define __FUNCT__ "MatILUDTFactorSymbolic_SeqAIJ"
3459 /*
3460     This will get a new name and become a varient of MatILUFactor_SeqAIJ() there is no longer seperate functions in the matrix function table for dt factors
3461 */
3462 
3463 PetscErrorCode PETSCMAT_DLLEXPORT MatILUDTFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS row,IS col,const MatFactorInfo *info)
3464 {
3465   PetscErrorCode     ierr;
3466 
3467   PetscFunctionBegin;
3468   ierr = MatILUDTFactor_SeqAIJ(A,row,col,info,&fact);CHKERRQ(ierr);
3469   PetscFunctionReturn(0);
3470 }
3471 
3472 /*
3473    same as MatLUFactorNumeric_SeqAIJ(), except using contiguous array matrix factors
3474    - intend to replace existing MatLUFactorNumeric_SeqAIJ()
3475 */
3476 #undef __FUNCT__
3477 #define __FUNCT__ "MatILUDTFactorNumeric_SeqAIJ"
3478 /*
3479     This will get a new name and become a varient of MatILUFactor_SeqAIJ() there is no longer seperate functions in the matrix function table for dt factors
3480 */
3481 
3482 PetscErrorCode PETSCMAT_DLLEXPORT MatILUDTFactorNumeric_SeqAIJ(Mat fact,Mat A,const MatFactorInfo *info)
3483 {
3484   Mat            C=fact;
3485   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ *)C->data;
3486   IS             isrow = b->row,isicol = b->icol;
3487   PetscErrorCode ierr;
3488   const PetscInt *r,*ic,*ics;
3489   PetscInt       i,j,k,n=A->rmap->n,*ai=a->i,*aj=a->j,*bi=b->i,*bj=b->j;
3490   PetscInt       *ajtmp,*bjtmp,nz,nzl,nzu,row,*bdiag = b->diag,*pj;
3491   MatScalar      *rtmp,*pc,multiplier,*v,*pv,*aa=a->a;
3492   PetscReal      dt=info->dt,shift=info->shiftamount;
3493   PetscTruth     row_identity, col_identity;
3494 
3495   PetscFunctionBegin;
3496   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
3497   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
3498   ierr = PetscMalloc((n+1)*sizeof(MatScalar),&rtmp);CHKERRQ(ierr);
3499   ics  = ic;
3500 
3501   for (i=0; i<n; i++){
3502     /* initialize rtmp array */
3503     nzl   = bi[i+1] - bi[i];       /* num of nozeros in L(i,:) */
3504     bjtmp = bj + bi[i];
3505     for  (j=0; j<nzl; j++) rtmp[*bjtmp++] = 0.0;
3506     rtmp[i] = 0.0;
3507     nzu   = bdiag[i] - bdiag[i+1]; /* num of nozeros in U(i,:) */
3508     bjtmp = bj + bdiag[i+1] + 1;
3509     for  (j=0; j<nzu; j++) rtmp[*bjtmp++] = 0.0;
3510 
3511     /* load in initial unfactored row of A */
3512     /* printf("row %d\n",i); */
3513     nz    = ai[r[i]+1] - ai[r[i]];
3514     ajtmp = aj + ai[r[i]];
3515     v     = aa + ai[r[i]];
3516     for (j=0; j<nz; j++) {
3517       rtmp[ics[*ajtmp++]] = v[j];
3518       /* printf(" (%d,%g),",ics[ajtmp[j]],rtmp[ics[ajtmp[j]]]); */
3519     }
3520     /* printf("\n"); */
3521 
3522     /* numerical factorization */
3523     bjtmp = bj + bi[i]; /* point to 1st entry of L(i,:) */
3524     nzl   = bi[i+1] - bi[i]; /* num of entries in L(i,:) */
3525     k = 0;
3526     while (k < nzl){
3527       row   = *bjtmp++;
3528       /* printf("  prow %d\n",row); */
3529       pc         = rtmp + row;
3530       pv         = b->a + bdiag[row]; /* 1./(diag of the pivot row) */
3531       multiplier = (*pc) * (*pv);
3532       *pc        = multiplier;
3533       if (PetscAbsScalar(multiplier) > dt){
3534         pj         = bj + bdiag[row+1] + 1; /* point to 1st entry of U(row,:) */
3535         pv         = b->a + bdiag[row+1] + 1;
3536         nz         = bdiag[row] - bdiag[row+1] - 1; /* num of entries in U(row,:), excluding diagonal */
3537         for (j=0; j<nz; j++) rtmp[*pj++] -= multiplier * (*pv++);
3538         /* ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr); */
3539       }
3540       k++;
3541     }
3542 
3543     /* finished row so stick it into b->a */
3544     /* L-part */
3545     pv = b->a + bi[i] ;
3546     pj = bj + bi[i] ;
3547     nzl = bi[i+1] - bi[i];
3548     for (j=0; j<nzl; j++) {
3549       pv[j] = rtmp[pj[j]];
3550       /* printf(" (%d,%g),",pj[j],pv[j]); */
3551     }
3552 
3553     /* diagonal: invert diagonal entries for simplier triangular solves */
3554     if (rtmp[i] == 0.0) rtmp[i] = dt+shift;
3555     b->a[bdiag[i]] = 1.0/rtmp[i];
3556     /* printf(" (%d,%g),",i,b->a[bdiag[i]]); */
3557 
3558     /* U-part */
3559     pv = b->a + bdiag[i+1] + 1;
3560     pj = bj + bdiag[i+1] + 1;
3561     nzu = bdiag[i] - bdiag[i+1] - 1;
3562     for (j=0; j<nzu; j++) {
3563       pv[j] = rtmp[pj[j]];
3564       /* printf(" (%d,%g),",pj[j],pv[j]); */
3565     }
3566     /* printf("\n"); */
3567   }
3568 
3569   ierr = PetscFree(rtmp);CHKERRQ(ierr);
3570   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
3571   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
3572 
3573   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
3574   ierr = ISIdentity(isicol,&col_identity);CHKERRQ(ierr);
3575   if (row_identity && col_identity) {
3576     C->ops->solve   = MatSolve_SeqAIJ_NaturalOrdering;
3577   } else {
3578     C->ops->solve   = MatSolve_SeqAIJ;
3579   }
3580   C->ops->solveadd           = 0;
3581   C->ops->solvetranspose     = 0;
3582   C->ops->solvetransposeadd  = 0;
3583   C->ops->matsolve           = 0;
3584   C->assembled    = PETSC_TRUE;
3585   C->preallocated = PETSC_TRUE;
3586   ierr = PetscLogFlops(C->cmap->n);CHKERRQ(ierr);
3587   PetscFunctionReturn(0);
3588 }
3589