xref: /petsc/src/mat/impls/aij/seq/aijfact.c (revision db32a24517e7892d23ded433de58be0568ffb6aa)
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   (B)->ops->solve            = MatSolve_SeqAIJ_inplace;
274   (B)->ops->solvetranspose   = MatSolveTranspose_SeqAIJ_inplace;
275   /* switch to inodes if appropriate */
276   ierr = MatLUFactorSymbolic_SeqAIJ_Inode_inplace(B,A,isrow,iscol,info);CHKERRQ(ierr);
277   PetscFunctionReturn(0);
278 }
279 
280 #undef __FUNCT__
281 #define __FUNCT__ "MatLUFactorSymbolic_SeqAIJ"
282 PetscErrorCode MatLUFactorSymbolic_SeqAIJ(Mat B,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
283 {
284   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
285   IS                 isicol;
286   PetscErrorCode     ierr;
287   const PetscInt     *r,*ic;
288   PetscInt           i,n=A->rmap->n,*ai=a->i,*aj=a->j;
289   PetscInt           *bi,*bj,*ajtmp;
290   PetscInt           *bdiag,row,nnz,nzi,reallocs=0,nzbd,*im;
291   PetscReal          f;
292   PetscInt           nlnk,*lnk,k,**bi_ptr;
293   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
294   PetscBT            lnkbt;
295 
296   PetscFunctionBegin;
297   // Uncomment the oldatastruct part only while testing new data structure for MatSolve()
298   PetscTruth         olddatastruct=PETSC_FALSE;
299   ierr = PetscOptionsGetTruth(PETSC_NULL,"-ilu_old",&olddatastruct,PETSC_NULL);CHKERRQ(ierr);
300   if(olddatastruct){
301     ierr = MatLUFactorSymbolic_SeqAIJ_inplace(B,A,isrow,iscol,info);CHKERRQ(ierr);
302     PetscFunctionReturn(0);
303   }
304 
305 
306   if (A->rmap->N != A->cmap->N) SETERRQ(PETSC_ERR_ARG_WRONG,"matrix must be square");
307   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
308   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
309   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
310 
311   /* get new row and diagonal pointers, must be allocated separately because they will be given to the Mat_SeqAIJ and freed separately */
312   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
313   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);
314   bi[0] = bdiag[0] = 0;
315 
316   /* linked list for storing column indices of the active row */
317   nlnk = n + 1;
318   ierr = PetscLLCreate(n,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
319 
320   ierr = PetscMalloc2(n+1,PetscInt**,&bi_ptr,n+1,PetscInt,&im);CHKERRQ(ierr);
321 
322   /* initial FreeSpace size is f*(ai[n]+1) */
323   f = info->fill;
324   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space);CHKERRQ(ierr);
325   current_space = free_space;
326 
327   for (i=0; i<n; i++) {
328     /* copy previous fill into linked list */
329     nzi = 0;
330     nnz = ai[r[i]+1] - ai[r[i]];
331     if (!nnz) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
332     ajtmp = aj + ai[r[i]];
333     ierr = PetscLLAddPerm(nnz,ajtmp,ic,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
334     nzi += nlnk;
335 
336     /* add pivot rows into linked list */
337     row = lnk[n];
338     while (row < i){
339       nzbd  = bdiag[row] + 1; /* num of entries in the row with column index <= row */
340       ajtmp = bi_ptr[row] + nzbd; /* points to the entry next to the diagonal */
341       ierr  = PetscLLAddSortedLU(ajtmp,row,nlnk,lnk,lnkbt,i,nzbd,im);CHKERRQ(ierr);
342       nzi  += nlnk;
343       row   = lnk[row];
344     }
345     bi[i+1] = bi[i] + nzi;
346     im[i]   = nzi;
347 
348     /* mark bdiag */
349     nzbd = 0;
350     nnz  = nzi;
351     k    = lnk[n];
352     while (nnz-- && k < i){
353       nzbd++;
354       k = lnk[k];
355     }
356     bdiag[i] = nzbd; /* note: bdiag[i] = nnzL as input for PetscFreeSpaceContiguous_LU() */
357 
358     /* if free space is not available, make more free space */
359     if (current_space->local_remaining<nzi) {
360       nnz = 2*(n - i)*nzi; /* estimated and max additional space needed */
361       ierr = PetscFreeSpaceGet(nnz,&current_space);CHKERRQ(ierr);
362       reallocs++;
363     }
364 
365     /* copy data into free space, then initialize lnk */
366     ierr = PetscLLClean(n,n,nzi,lnk,current_space->array,lnkbt);CHKERRQ(ierr);
367     bi_ptr[i] = current_space->array;
368     current_space->array           += nzi;
369     current_space->local_used      += nzi;
370     current_space->local_remaining -= nzi;
371   }
372 #if defined(PETSC_USE_INFO)
373   if (ai[n] != 0) {
374     PetscReal af = ((PetscReal)bi[n])/((PetscReal)ai[n]);
375     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,f,af);CHKERRQ(ierr);
376     ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
377     ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G);\n",af);CHKERRQ(ierr);
378     ierr = PetscInfo(A,"for best performance.\n");CHKERRQ(ierr);
379   } else {
380     ierr = PetscInfo(A,"Empty matrix\n");CHKERRQ(ierr);
381   }
382 #endif
383 
384   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
385   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
386 
387   /* destroy list of free space and other temporary array(s) */
388   ierr = PetscMalloc((bi[n]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
389   ierr = PetscFreeSpaceContiguous_LU(&free_space,bj,n,bi,bdiag);CHKERRQ(ierr);
390   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
391   ierr = PetscFree2(bi_ptr,im);CHKERRQ(ierr);
392 
393   /* put together the new matrix */
394   ierr = MatSeqAIJSetPreallocation_SeqAIJ(B,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
395   ierr = PetscLogObjectParent(B,isicol);CHKERRQ(ierr);
396   b    = (Mat_SeqAIJ*)(B)->data;
397   b->free_a       = PETSC_TRUE;
398   b->free_ij      = PETSC_TRUE;
399   b->singlemalloc = PETSC_FALSE;
400   ierr = PetscMalloc((bdiag[0]+1)*sizeof(PetscScalar),&b->a);CHKERRQ(ierr);
401   b->j          = bj;
402   b->i          = bi;
403   b->diag       = bdiag;
404   b->ilen       = 0;
405   b->imax       = 0;
406   b->row        = isrow;
407   b->col        = iscol;
408   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
409   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
410   b->icol       = isicol;
411   ierr          = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
412 
413   /* In b structure:  Free imax, ilen, old a, old j.  Allocate solve_work, new a, new j */
414   ierr = PetscLogObjectMemory(B,(bdiag[0]+1)*(sizeof(PetscInt)+sizeof(PetscScalar)));CHKERRQ(ierr);
415   b->maxnz = b->nz = bdiag[0]+1;
416   B->factor                = MAT_FACTOR_LU;
417   B->info.factor_mallocs   = reallocs;
418   B->info.fill_ratio_given = f;
419 
420   if (ai[n]) {
421     B->info.fill_ratio_needed = ((PetscReal)(bdiag[0]+1))/((PetscReal)ai[n]);
422   } else {
423     B->info.fill_ratio_needed = 0.0;
424   }
425   B->ops->lufactornumeric = MatLUFactorNumeric_SeqAIJ;
426   /* switch to inodes if appropriate */
427   ierr = Mat_CheckInode_FactorLU(B,PETSC_FALSE);CHKERRQ(ierr);
428   PetscFunctionReturn(0);
429 }
430 
431 /*
432     Trouble in factorization, should we dump the original matrix?
433 */
434 #undef __FUNCT__
435 #define __FUNCT__ "MatFactorDumpMatrix"
436 PetscErrorCode MatFactorDumpMatrix(Mat A)
437 {
438   PetscErrorCode ierr;
439   PetscTruth     flg = PETSC_FALSE;
440 
441   PetscFunctionBegin;
442   ierr = PetscOptionsGetTruth(PETSC_NULL,"-mat_factor_dump_on_error",&flg,PETSC_NULL);CHKERRQ(ierr);
443   if (flg) {
444     PetscViewer viewer;
445     char        filename[PETSC_MAX_PATH_LEN];
446 
447     ierr = PetscSNPrintf(filename,PETSC_MAX_PATH_LEN,"matrix_factor_error.%d",PetscGlobalRank);CHKERRQ(ierr);
448     ierr = PetscViewerBinaryOpen(((PetscObject)A)->comm,filename,FILE_MODE_WRITE,&viewer);CHKERRQ(ierr);
449     ierr = MatView(A,viewer);CHKERRQ(ierr);
450     ierr = PetscViewerDestroy(viewer);CHKERRQ(ierr);
451   }
452   PetscFunctionReturn(0);
453 }
454 
455 #undef __FUNCT__
456 #define __FUNCT__ "MatLUFactorNumeric_SeqAIJ"
457 PetscErrorCode MatLUFactorNumeric_SeqAIJ(Mat B,Mat A,const MatFactorInfo *info)
458 {
459   Mat              C=B;
460   Mat_SeqAIJ       *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ *)C->data;
461   IS               isrow = b->row,isicol = b->icol;
462   PetscErrorCode   ierr;
463   const PetscInt   *r,*ic,*ics;
464   const PetscInt   n=A->rmap->n,*ai=a->i,*aj=a->j,*bi=b->i,*bj=b->j,*bdiag=b->diag;
465   PetscInt         i,j,k,nz,nzL,row,*pj;
466   const PetscInt   *ajtmp,*bjtmp;
467   MatScalar        *rtmp,*pc,multiplier,*pv;
468   const  MatScalar *aa=a->a,*v;
469   PetscTruth       row_identity,col_identity;
470 
471   FactorShiftCtx sctx;
472   PetscInt       *ddiag;
473   PetscReal      rs;
474   MatScalar      d;
475 
476   PetscFunctionBegin;
477   /* MatPivotSetUp(): initialize shift context sctx */
478   ierr = PetscMemzero(&sctx,sizeof(FactorShiftCtx));CHKERRQ(ierr);
479 
480   if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE) { /* set sctx.shift_top=max{rs} */
481     ddiag          = a->diag;
482     sctx.shift_top = info->zeropivot;
483     for (i=0; i<n; i++) {
484       /* calculate sum(|aij|)-RealPart(aii), amt of shift needed for this row */
485       d  = (aa)[ddiag[i]];
486       rs = -PetscAbsScalar(d) - PetscRealPart(d);
487       v  = aa+ai[i];
488       nz = ai[i+1] - ai[i];
489       for (j=0; j<nz; j++)
490 	rs += PetscAbsScalar(v[j]);
491       if (rs>sctx.shift_top) sctx.shift_top = rs;
492     }
493     sctx.shift_top   *= 1.1;
494     sctx.nshift_max   = 5;
495     sctx.shift_lo     = 0.;
496     sctx.shift_hi     = 1.;
497   }
498 
499   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
500   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
501   ierr = PetscMalloc((n+1)*sizeof(MatScalar),&rtmp);CHKERRQ(ierr);
502   ics  = ic;
503 
504   do {
505     sctx.useshift = PETSC_FALSE;
506     for (i=0; i<n; i++){
507       /* zero rtmp */
508       /* L part */
509       nz    = bi[i+1] - bi[i];
510       bjtmp = bj + bi[i];
511       for  (j=0; j<nz; j++) rtmp[bjtmp[j]] = 0.0;
512 
513       /* U part */
514       nz = bdiag[i]-bdiag[i+1];
515       bjtmp = bj + bdiag[i+1]+1;
516       for  (j=0; j<nz; j++) rtmp[bjtmp[j]] = 0.0;
517 
518       /* load in initial (unfactored row) */
519       nz    = ai[r[i]+1] - ai[r[i]];
520       ajtmp = aj + ai[r[i]];
521       v     = aa + ai[r[i]];
522       for (j=0; j<nz; j++) {
523         rtmp[ics[ajtmp[j]]] = v[j];
524       }
525       /* ZeropivotApply() */
526       rtmp[i] += sctx.shift_amount;  /* shift the diagonal of the matrix */
527 
528       /* elimination */
529       bjtmp = bj + bi[i];
530       row   = *bjtmp++;
531       nzL   = bi[i+1] - bi[i];
532       for(k=0; k < nzL;k++) {
533         pc = rtmp + row;
534         if (*pc != 0.0) {
535           pv         = b->a + bdiag[row];
536           multiplier = *pc * (*pv);
537           *pc        = multiplier;
538           pj = b->j + bdiag[row+1]+1; /* beginning of U(row,:) */
539 	  pv = b->a + bdiag[row+1]+1;
540 	  nz = bdiag[row]-bdiag[row+1]-1; /* num of entries in U(row,:) excluding diag */
541           for (j=0; j<nz; j++) rtmp[pj[j]] -= multiplier * pv[j];
542           ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr);
543         }
544         row = *bjtmp++;
545       }
546 
547       /* finished row so stick it into b->a */
548       rs = 0.0;
549       /* L part */
550       pv   = b->a + bi[i] ;
551       pj   = b->j + bi[i] ;
552       nz   = bi[i+1] - bi[i];
553       for (j=0; j<nz; j++) {
554         pv[j] = rtmp[pj[j]]; rs += PetscAbsScalar(pv[j]);
555       }
556 
557       /* U part */
558       pv = b->a + bdiag[i+1]+1;
559       pj = b->j + bdiag[i+1]+1;
560       nz = bdiag[i] - bdiag[i+1]-1;
561       for (j=0; j<nz; j++) {
562         pv[j] = rtmp[pj[j]]; rs += PetscAbsScalar(pv[j]);
563       }
564 
565       /* MatPivotCheck() */
566       sctx.rs  = rs;
567       sctx.pv  = rtmp[i];
568       if (info->shifttype == MAT_SHIFT_NONZERO){
569         ierr = MatPivotCheck_nz(info,sctx,i);CHKERRQ(ierr);
570       } else if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE){
571         ierr = MatPivotCheck_pd(info,sctx,i);CHKERRQ(ierr);
572       } else if (info->shifttype == MAT_SHIFT_INBLOCKS){
573         ierr = MatPivotCheck_inblocks(info,sctx,i);CHKERRQ(ierr);
574       } else {
575         ierr = MatPivotCheck_none(info,sctx,i);CHKERRQ(ierr);
576       }
577       rtmp[i] = sctx.pv;
578 
579       /* Mark diagonal and invert diagonal for simplier triangular solves */
580       pv  = b->a + bdiag[i];
581       *pv = 1.0/rtmp[i];
582 
583     } /* endof for (i=0; i<n; i++){ */
584 
585     /* MatPivotRefine() */
586     if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE && !sctx.useshift && sctx.shift_fraction>0 && sctx.nshift<sctx.nshift_max){
587       /*
588        * if no shift in this attempt & shifting & started shifting & can refine,
589        * then try lower shift
590        */
591       sctx.shift_hi       = sctx.shift_fraction;
592       sctx.shift_fraction = (sctx.shift_hi+sctx.shift_lo)/2.;
593       sctx.shift_amount   = sctx.shift_fraction * sctx.shift_top;
594       sctx.useshift        = PETSC_TRUE;
595       sctx.nshift++;
596     }
597   } while (sctx.useshift);
598 
599   ierr = PetscFree(rtmp);CHKERRQ(ierr);
600   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
601   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
602   if (b->inode.use){
603     C->ops->solve   = MatSolve_SeqAIJ_Inode;
604   } else {
605     ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
606     ierr = ISIdentity(isicol,&col_identity);CHKERRQ(ierr);
607     if (row_identity && col_identity) {
608       C->ops->solve = MatSolve_SeqAIJ_NaturalOrdering;
609     } else {
610       C->ops->solve = MatSolve_SeqAIJ;
611     }
612   }
613   C->ops->solveadd           = MatSolveAdd_SeqAIJ;
614   C->ops->solvetranspose     = MatSolveTranspose_SeqAIJ;
615   C->ops->solvetransposeadd  = MatSolveTransposeAdd_SeqAIJ;
616   C->ops->matsolve           = MatMatSolve_SeqAIJ;
617   C->assembled    = PETSC_TRUE;
618   C->preallocated = PETSC_TRUE;
619   ierr = PetscLogFlops(C->cmap->n);CHKERRQ(ierr);
620 
621   /* MatShiftView(A,info,&sctx) */
622   if (sctx.nshift){
623     if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE) {
624       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);
625     } else if (info->shifttype == MAT_SHIFT_NONZERO) {
626       ierr = PetscInfo2(A,"number of shift_nz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
627     } else if (info->shifttype == MAT_SHIFT_INBLOCKS){
628       ierr = PetscInfo2(A,"number of shift_inblocks applied %D, each shift_amount %G\n",sctx.nshift,info->shiftamount);CHKERRQ(ierr);
629     }
630   }
631   PetscFunctionReturn(0);
632 }
633 
634 #undef __FUNCT__
635 #define __FUNCT__ "MatLUFactorNumeric_SeqAIJ_inplace"
636 PetscErrorCode MatLUFactorNumeric_SeqAIJ_inplace(Mat B,Mat A,const MatFactorInfo *info)
637 {
638   Mat             C=B;
639   Mat_SeqAIJ      *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ *)C->data;
640   IS              isrow = b->row,isicol = b->icol;
641   PetscErrorCode  ierr;
642   const PetscInt   *r,*ic,*ics;
643   PetscInt        nz,row,i,j,n=A->rmap->n,diag;
644   const PetscInt  *ai=a->i,*aj=a->j,*bi=b->i,*bj=b->j;
645   const PetscInt  *ajtmp,*bjtmp,*diag_offset = b->diag,*pj;
646   MatScalar       *pv,*rtmp,*pc,multiplier,d;
647   const MatScalar *v,*aa=a->a;
648   PetscReal       rs=0.0;
649   FactorShiftCtx  sctx;
650   PetscInt        newshift,*ddiag;
651 
652   PetscFunctionBegin;
653   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
654   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
655   ierr = PetscMalloc((n+1)*sizeof(MatScalar),&rtmp);CHKERRQ(ierr);
656   ics  = ic;
657 
658   /* initialize shift context sctx */
659   sctx.nshift         = 0;
660   sctx.nshift_max     = 0;
661   sctx.shift_top      = 0.0;
662   sctx.shift_lo       = 0.0;
663   sctx.shift_hi       = 0.0;
664   sctx.shift_fraction = 0.0;
665   sctx.shift_amount   = 0.0;
666 
667   if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE) { /* set sctx.shift_top=max{rs} */
668     ddiag          = a->diag;
669     sctx.shift_top = info->zeropivot;
670     for (i=0; i<n; i++) {
671       /* calculate sum(|aij|)-RealPart(aii), amt of shift needed for this row */
672       d  = (aa)[ddiag[i]];
673       rs = -PetscAbsScalar(d) - PetscRealPart(d);
674       v  = aa+ai[i];
675       nz = ai[i+1] - ai[i];
676       for (j=0; j<nz; j++)
677 	rs += PetscAbsScalar(v[j]);
678       if (rs>sctx.shift_top) sctx.shift_top = rs;
679     }
680     sctx.shift_top   *= 1.1;
681     sctx.nshift_max   = 5;
682     sctx.shift_lo     = 0.;
683     sctx.shift_hi     = 1.;
684   }
685 
686   do {
687     sctx.useshift = PETSC_FALSE;
688     for (i=0; i<n; i++){
689       nz    = bi[i+1] - bi[i];
690       bjtmp = bj + bi[i];
691       for  (j=0; j<nz; j++) rtmp[bjtmp[j]] = 0.0;
692 
693       /* load in initial (unfactored row) */
694       nz    = ai[r[i]+1] - ai[r[i]];
695       ajtmp = aj + ai[r[i]];
696       v     = aa + ai[r[i]];
697       for (j=0; j<nz; j++) {
698         rtmp[ics[ajtmp[j]]] = v[j];
699       }
700       rtmp[ics[r[i]]] += sctx.shift_amount; /* shift the diagonal of the matrix */
701       /* if (sctx.shift_amount > 0.0) printf("row %d, shift %g\n",i,sctx.shift_amount); */
702 
703       row = *bjtmp++;
704       while  (row < i) {
705         pc = rtmp + row;
706         if (*pc != 0.0) {
707           pv         = b->a + diag_offset[row];
708           pj         = b->j + diag_offset[row] + 1;
709           multiplier = *pc / *pv++;
710           *pc        = multiplier;
711           nz         = bi[row+1] - diag_offset[row] - 1;
712           for (j=0; j<nz; j++) rtmp[pj[j]] -= multiplier * pv[j];
713           ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr);
714         }
715         row = *bjtmp++;
716       }
717       /* finished row so stick it into b->a */
718       pv   = b->a + bi[i] ;
719       pj   = b->j + bi[i] ;
720       nz   = bi[i+1] - bi[i];
721       diag = diag_offset[i] - bi[i];
722       rs   = 0.0;
723       for (j=0; j<nz; j++) {
724         pv[j] = rtmp[pj[j]];
725         rs   += PetscAbsScalar(pv[j]);
726       }
727       rs   -= PetscAbsScalar(pv[diag]);
728 
729       /* 9/13/02 Victor Eijkhout suggested scaling zeropivot by rs for matrices with funny scalings */
730       sctx.rs  = rs;
731       sctx.pv  = pv[diag];
732       ierr = MatLUCheckShift_inline(info,sctx,i,newshift);CHKERRQ(ierr);
733       if (newshift == 1) break;
734     }
735 
736     if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE && !sctx.useshift && sctx.shift_fraction>0 && sctx.nshift<sctx.nshift_max) {
737       /*
738        * if no shift in this attempt & shifting & started shifting & can refine,
739        * then try lower shift
740        */
741       sctx.shift_hi       = sctx.shift_fraction;
742       sctx.shift_fraction = (sctx.shift_hi+sctx.shift_lo)/2.;
743       sctx.shift_amount   = sctx.shift_fraction * sctx.shift_top;
744       sctx.useshift        = PETSC_TRUE;
745       sctx.nshift++;
746     }
747   } while (sctx.useshift);
748 
749   /* invert diagonal entries for simplier triangular solves */
750   for (i=0; i<n; i++) {
751     b->a[diag_offset[i]] = 1.0/b->a[diag_offset[i]];
752   }
753   ierr = PetscFree(rtmp);CHKERRQ(ierr);
754   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
755   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
756   if (b->inode.use) {
757     C->ops->solve   = MatSolve_SeqAIJ_Inode_inplace;
758   } else {
759     PetscTruth row_identity, col_identity;
760     ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
761     ierr = ISIdentity(isicol,&col_identity);CHKERRQ(ierr);
762     if (row_identity && col_identity) {
763       C->ops->solve   = MatSolve_SeqAIJ_NaturalOrdering_inplace;
764     } else {
765       C->ops->solve   = MatSolve_SeqAIJ_inplace;
766     }
767   }
768   C->ops->solveadd           = MatSolveAdd_SeqAIJ_inplace;
769   C->ops->solvetranspose     = MatSolveTranspose_SeqAIJ_inplace;
770   C->ops->solvetransposeadd  = MatSolveTransposeAdd_SeqAIJ_inplace;
771   C->ops->matsolve           = MatMatSolve_SeqAIJ_inplace;
772   C->assembled    = PETSC_TRUE;
773   C->preallocated = PETSC_TRUE;
774   ierr = PetscLogFlops(C->cmap->n);CHKERRQ(ierr);
775   if (sctx.nshift){
776      if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE) {
777       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);
778     } else if (info->shifttype == MAT_SHIFT_NONZERO) {
779       ierr = PetscInfo2(A,"number of shift_nz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
780     }
781   }
782   PetscFunctionReturn(0);
783 }
784 
785 /*
786    This routine implements inplace ILU(0) with row or/and column permutations.
787    Input:
788      A - original matrix
789    Output;
790      A - a->i (rowptr) is same as original rowptr, but factored i-the row is stored in rowperm[i]
791          a->j (col index) is permuted by the inverse of colperm, then sorted
792          a->a reordered accordingly with a->j
793          a->diag (ptr to diagonal elements) is updated.
794 */
795 #undef __FUNCT__
796 #define __FUNCT__ "MatLUFactorNumeric_SeqAIJ_InplaceWithPerm"
797 PetscErrorCode MatLUFactorNumeric_SeqAIJ_InplaceWithPerm(Mat B,Mat A,const MatFactorInfo *info)
798 {
799   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data;
800   IS             isrow = a->row,isicol = a->icol;
801   PetscErrorCode ierr;
802   const PetscInt *r,*ic,*ics;
803   PetscInt       i,j,n=A->rmap->n,*ai=a->i,*aj=a->j;
804   PetscInt       *ajtmp,nz,row;
805   PetscInt       *diag = a->diag,nbdiag,*pj;
806   PetscScalar    *rtmp,*pc,multiplier,d;
807   MatScalar      *v,*pv;
808   PetscReal      rs;
809   FactorShiftCtx sctx;
810   PetscInt       newshift;
811 
812   PetscFunctionBegin;
813   if (A != B) SETERRQ(PETSC_ERR_ARG_INCOMP,"input and output matrix must have same address");
814   ierr  = ISGetIndices(isrow,&r);CHKERRQ(ierr);
815   ierr  = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
816   ierr  = PetscMalloc((n+1)*sizeof(PetscScalar),&rtmp);CHKERRQ(ierr);
817   ierr  = PetscMemzero(rtmp,(n+1)*sizeof(PetscScalar));CHKERRQ(ierr);
818   ics = ic;
819 
820   sctx.shift_top      = 0.;
821   sctx.nshift_max     = 0;
822   sctx.shift_lo       = 0.;
823   sctx.shift_hi       = 0.;
824   sctx.shift_fraction = 0.;
825 
826   if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE) { /* set sctx.shift_top=max{rs} */
827     sctx.shift_top = 0.;
828     for (i=0; i<n; i++) {
829       /* calculate sum(|aij|)-RealPart(aii), amt of shift needed for this row */
830       d  = (a->a)[diag[i]];
831       rs = -PetscAbsScalar(d) - PetscRealPart(d);
832       v  = a->a+ai[i];
833       nz = ai[i+1] - ai[i];
834       for (j=0; j<nz; j++)
835 	rs += PetscAbsScalar(v[j]);
836       if (rs>sctx.shift_top) sctx.shift_top = rs;
837     }
838     if (sctx.shift_top < info->zeropivot) sctx.shift_top = info->zeropivot;
839     sctx.shift_top    *= 1.1;
840     sctx.nshift_max   = 5;
841     sctx.shift_lo     = 0.;
842     sctx.shift_hi     = 1.;
843   }
844 
845   sctx.shift_amount = 0.;
846   sctx.nshift       = 0;
847   do {
848     sctx.useshift = PETSC_FALSE;
849     for (i=0; i<n; i++){
850       /* load in initial unfactored row */
851       nz    = ai[r[i]+1] - ai[r[i]];
852       ajtmp = aj + ai[r[i]];
853       v     = a->a + ai[r[i]];
854       /* sort permuted ajtmp and values v accordingly */
855       for (j=0; j<nz; j++) ajtmp[j] = ics[ajtmp[j]];
856       ierr = PetscSortIntWithScalarArray(nz,ajtmp,v);CHKERRQ(ierr);
857 
858       diag[r[i]] = ai[r[i]];
859       for (j=0; j<nz; j++) {
860         rtmp[ajtmp[j]] = v[j];
861         if (ajtmp[j] < i) diag[r[i]]++; /* update a->diag */
862       }
863       rtmp[r[i]] += sctx.shift_amount; /* shift the diagonal of the matrix */
864 
865       row = *ajtmp++;
866       while  (row < i) {
867         pc = rtmp + row;
868         if (*pc != 0.0) {
869           pv         = a->a + diag[r[row]];
870           pj         = aj + diag[r[row]] + 1;
871 
872           multiplier = *pc / *pv++;
873           *pc        = multiplier;
874           nz         = ai[r[row]+1] - diag[r[row]] - 1;
875           for (j=0; j<nz; j++) rtmp[pj[j]] -= multiplier * pv[j];
876           ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr);
877         }
878         row = *ajtmp++;
879       }
880       /* finished row so overwrite it onto a->a */
881       pv   = a->a + ai[r[i]] ;
882       pj   = aj + ai[r[i]] ;
883       nz   = ai[r[i]+1] - ai[r[i]];
884       nbdiag = diag[r[i]] - ai[r[i]]; /* num of entries before the diagonal */
885 
886       rs   = 0.0;
887       for (j=0; j<nz; j++) {
888         pv[j] = rtmp[pj[j]];
889         if (j != nbdiag) rs += PetscAbsScalar(pv[j]);
890       }
891 
892       /* 9/13/02 Victor Eijkhout suggested scaling zeropivot by rs for matrices with funny scalings */
893       sctx.rs  = rs;
894       sctx.pv  = pv[nbdiag];
895       ierr = MatLUCheckShift_inline(info,sctx,i,newshift);CHKERRQ(ierr);
896       if (newshift == 1) break;
897     }
898 
899     if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE && !sctx.useshift && sctx.shift_fraction>0 && sctx.nshift<sctx.nshift_max) {
900       /*
901        * if no shift in this attempt & shifting & started shifting & can refine,
902        * then try lower shift
903        */
904       sctx.shift_hi        = sctx.shift_fraction;
905       sctx.shift_fraction = (sctx.shift_hi+sctx.shift_lo)/2.;
906       sctx.shift_amount    = sctx.shift_fraction * sctx.shift_top;
907       sctx.useshift         = PETSC_TRUE;
908       sctx.nshift++;
909     }
910   } while (sctx.useshift);
911 
912   /* invert diagonal entries for simplier triangular solves */
913   for (i=0; i<n; i++) {
914     a->a[diag[r[i]]] = 1.0/a->a[diag[r[i]]];
915   }
916 
917   ierr = PetscFree(rtmp);CHKERRQ(ierr);
918   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
919   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
920   A->ops->solve             = MatSolve_SeqAIJ_InplaceWithPerm;
921   A->ops->solveadd          = MatSolveAdd_SeqAIJ_inplace;
922   A->ops->solvetranspose    = MatSolveTranspose_SeqAIJ_inplace;
923   A->ops->solvetransposeadd = MatSolveTransposeAdd_SeqAIJ_inplace;
924   A->assembled = PETSC_TRUE;
925   A->preallocated = PETSC_TRUE;
926   ierr = PetscLogFlops(A->cmap->n);CHKERRQ(ierr);
927   if (sctx.nshift){
928     if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE) {
929       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);
930     } else if (info->shifttype == MAT_SHIFT_NONZERO) {
931       ierr = PetscInfo2(A,"number of shift_nz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
932     }
933   }
934   PetscFunctionReturn(0);
935 }
936 
937 /* ----------------------------------------------------------- */
938 #undef __FUNCT__
939 #define __FUNCT__ "MatLUFactor_SeqAIJ"
940 PetscErrorCode MatLUFactor_SeqAIJ(Mat A,IS row,IS col,const MatFactorInfo *info)
941 {
942   PetscErrorCode ierr;
943   Mat            C;
944 
945   PetscFunctionBegin;
946   ierr = MatGetFactor(A,MAT_SOLVER_PETSC,MAT_FACTOR_LU,&C);CHKERRQ(ierr);
947   ierr = MatLUFactorSymbolic(C,A,row,col,info);CHKERRQ(ierr);
948   ierr = MatLUFactorNumeric(C,A,info);CHKERRQ(ierr);
949   A->ops->solve            = C->ops->solve;
950   A->ops->solvetranspose   = C->ops->solvetranspose;
951   ierr = MatHeaderCopy(A,C);CHKERRQ(ierr);
952   ierr = PetscLogObjectParent(A,((Mat_SeqAIJ*)(A->data))->icol);CHKERRQ(ierr);
953   PetscFunctionReturn(0);
954 }
955 /* ----------------------------------------------------------- */
956 
957 
958 #undef __FUNCT__
959 #define __FUNCT__ "MatSolve_SeqAIJ_inplace"
960 PetscErrorCode MatSolve_SeqAIJ_inplace(Mat A,Vec bb,Vec xx)
961 {
962   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
963   IS                iscol = a->col,isrow = a->row;
964   PetscErrorCode    ierr;
965   PetscInt          i, n = A->rmap->n,*vi,*ai = a->i,*aj = a->j;
966   PetscInt          nz;
967   const PetscInt    *rout,*cout,*r,*c;
968   PetscScalar       *x,*tmp,*tmps,sum;
969   const PetscScalar *b;
970   const MatScalar   *aa = a->a,*v;
971 
972   PetscFunctionBegin;
973   if (!n) PetscFunctionReturn(0);
974 
975   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
976   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
977   tmp  = a->solve_work;
978 
979   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
980   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout + (n-1);
981 
982   /* forward solve the lower triangular */
983   tmp[0] = b[*r++];
984   tmps   = tmp;
985   for (i=1; i<n; i++) {
986     v   = aa + ai[i] ;
987     vi  = aj + ai[i] ;
988     nz  = a->diag[i] - ai[i];
989     sum = b[*r++];
990     PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
991     tmp[i] = sum;
992   }
993 
994   /* backward solve the upper triangular */
995   for (i=n-1; i>=0; i--){
996     v   = aa + a->diag[i] + 1;
997     vi  = aj + a->diag[i] + 1;
998     nz  = ai[i+1] - a->diag[i] - 1;
999     sum = tmp[i];
1000     PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
1001     x[*c--] = tmp[i] = sum*aa[a->diag[i]];
1002   }
1003 
1004   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1005   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1006   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1007   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1008   ierr = PetscLogFlops(2.0*a->nz - A->cmap->n);CHKERRQ(ierr);
1009   PetscFunctionReturn(0);
1010 }
1011 
1012 #undef __FUNCT__
1013 #define __FUNCT__ "MatMatSolve_SeqAIJ_inplace"
1014 PetscErrorCode MatMatSolve_SeqAIJ_inplace(Mat A,Mat B,Mat X)
1015 {
1016   Mat_SeqAIJ      *a = (Mat_SeqAIJ*)A->data;
1017   IS              iscol = a->col,isrow = a->row;
1018   PetscErrorCode  ierr;
1019   PetscInt        i, n = A->rmap->n,*vi,*ai = a->i,*aj = a->j;
1020   PetscInt        nz,neq;
1021   const PetscInt  *rout,*cout,*r,*c;
1022   PetscScalar     *x,*b,*tmp,*tmps,sum;
1023   const MatScalar *aa = a->a,*v;
1024   PetscTruth      bisdense,xisdense;
1025 
1026   PetscFunctionBegin;
1027   if (!n) PetscFunctionReturn(0);
1028 
1029   ierr = PetscTypeCompare((PetscObject)B,MATSEQDENSE,&bisdense);CHKERRQ(ierr);
1030   if (!bisdense) SETERRQ(PETSC_ERR_ARG_INCOMP,"B matrix must be a SeqDense matrix");
1031   ierr = PetscTypeCompare((PetscObject)X,MATSEQDENSE,&xisdense);CHKERRQ(ierr);
1032   if (!xisdense) SETERRQ(PETSC_ERR_ARG_INCOMP,"X matrix must be a SeqDense matrix");
1033 
1034   ierr = MatGetArray(B,&b);CHKERRQ(ierr);
1035   ierr = MatGetArray(X,&x);CHKERRQ(ierr);
1036 
1037   tmp  = a->solve_work;
1038   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1039   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1040 
1041   for (neq=0; neq<B->cmap->n; neq++){
1042     /* forward solve the lower triangular */
1043     tmp[0] = b[r[0]];
1044     tmps   = tmp;
1045     for (i=1; i<n; i++) {
1046       v   = aa + ai[i] ;
1047       vi  = aj + ai[i] ;
1048       nz  = a->diag[i] - ai[i];
1049       sum = b[r[i]];
1050       PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
1051       tmp[i] = sum;
1052     }
1053     /* backward solve the upper triangular */
1054     for (i=n-1; i>=0; i--){
1055       v   = aa + a->diag[i] + 1;
1056       vi  = aj + a->diag[i] + 1;
1057       nz  = ai[i+1] - a->diag[i] - 1;
1058       sum = tmp[i];
1059       PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
1060       x[c[i]] = tmp[i] = sum*aa[a->diag[i]];
1061     }
1062 
1063     b += n;
1064     x += n;
1065   }
1066   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1067   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1068   ierr = MatRestoreArray(B,&b);CHKERRQ(ierr);
1069   ierr = MatRestoreArray(X,&x);CHKERRQ(ierr);
1070   ierr = PetscLogFlops(B->cmap->n*(2.0*a->nz - n));CHKERRQ(ierr);
1071   PetscFunctionReturn(0);
1072 }
1073 
1074 #undef __FUNCT__
1075 #define __FUNCT__ "MatMatSolve_SeqAIJ"
1076 PetscErrorCode MatMatSolve_SeqAIJ(Mat A,Mat B,Mat X)
1077 {
1078   Mat_SeqAIJ      *a = (Mat_SeqAIJ*)A->data;
1079   IS              iscol = a->col,isrow = a->row;
1080   PetscErrorCode  ierr;
1081   PetscInt        i, n = A->rmap->n,*vi,*ai = a->i,*aj = a->j,*adiag = a->diag;
1082   PetscInt        nz,neq;
1083   const PetscInt  *rout,*cout,*r,*c;
1084   PetscScalar     *x,*b,*tmp,sum;
1085   const MatScalar *aa = a->a,*v;
1086   PetscTruth      bisdense,xisdense;
1087 
1088   PetscFunctionBegin;
1089   if (!n) PetscFunctionReturn(0);
1090 
1091   ierr = PetscTypeCompare((PetscObject)B,MATSEQDENSE,&bisdense);CHKERRQ(ierr);
1092   if (!bisdense) SETERRQ(PETSC_ERR_ARG_INCOMP,"B matrix must be a SeqDense matrix");
1093   ierr = PetscTypeCompare((PetscObject)X,MATSEQDENSE,&xisdense);CHKERRQ(ierr);
1094   if (!xisdense) SETERRQ(PETSC_ERR_ARG_INCOMP,"X matrix must be a SeqDense matrix");
1095 
1096   ierr = MatGetArray(B,&b);CHKERRQ(ierr);
1097   ierr = MatGetArray(X,&x);CHKERRQ(ierr);
1098 
1099   tmp  = a->solve_work;
1100   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1101   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1102 
1103   for (neq=0; neq<B->cmap->n; neq++){
1104     /* forward solve the lower triangular */
1105     tmp[0] = b[r[0]];
1106     v      = aa;
1107     vi     = aj;
1108     for (i=1; i<n; i++) {
1109       nz  = ai[i+1] - ai[i];
1110       sum = b[r[i]];
1111       PetscSparseDenseMinusDot(sum,tmp,v,vi,nz);
1112       tmp[i] = sum;
1113       v += nz; vi += nz;
1114     }
1115 
1116     /* backward solve the upper triangular */
1117     for (i=n-1; i>=0; i--){
1118       v   = aa + adiag[i+1]+1;
1119       vi  = aj + adiag[i+1]+1;
1120       nz  = adiag[i]-adiag[i+1]-1;
1121       sum = tmp[i];
1122       PetscSparseDenseMinusDot(sum,tmp,v,vi,nz);
1123       x[c[i]] = tmp[i] = sum*v[nz]; /* v[nz] = aa[adiag[i]] */
1124     }
1125 
1126     b += n;
1127     x += n;
1128   }
1129   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1130   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1131   ierr = MatRestoreArray(B,&b);CHKERRQ(ierr);
1132   ierr = MatRestoreArray(X,&x);CHKERRQ(ierr);
1133   ierr = PetscLogFlops(B->cmap->n*(2.0*a->nz - n));CHKERRQ(ierr);
1134   PetscFunctionReturn(0);
1135 }
1136 
1137 #undef __FUNCT__
1138 #define __FUNCT__ "MatSolve_SeqAIJ_InplaceWithPerm"
1139 PetscErrorCode MatSolve_SeqAIJ_InplaceWithPerm(Mat A,Vec bb,Vec xx)
1140 {
1141   Mat_SeqAIJ      *a = (Mat_SeqAIJ*)A->data;
1142   IS              iscol = a->col,isrow = a->row;
1143   PetscErrorCode  ierr;
1144   const PetscInt  *r,*c,*rout,*cout;
1145   PetscInt        i, n = A->rmap->n,*vi,*ai = a->i,*aj = a->j;
1146   PetscInt        nz,row;
1147   PetscScalar     *x,*b,*tmp,*tmps,sum;
1148   const MatScalar *aa = a->a,*v;
1149 
1150   PetscFunctionBegin;
1151   if (!n) PetscFunctionReturn(0);
1152 
1153   ierr = VecGetArray(bb,&b);CHKERRQ(ierr);
1154   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1155   tmp  = a->solve_work;
1156 
1157   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1158   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout + (n-1);
1159 
1160   /* forward solve the lower triangular */
1161   tmp[0] = b[*r++];
1162   tmps   = tmp;
1163   for (row=1; row<n; row++) {
1164     i   = rout[row]; /* permuted row */
1165     v   = aa + ai[i] ;
1166     vi  = aj + ai[i] ;
1167     nz  = a->diag[i] - ai[i];
1168     sum = b[*r++];
1169     PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
1170     tmp[row] = sum;
1171   }
1172 
1173   /* backward solve the upper triangular */
1174   for (row=n-1; row>=0; row--){
1175     i   = rout[row]; /* permuted row */
1176     v   = aa + a->diag[i] + 1;
1177     vi  = aj + a->diag[i] + 1;
1178     nz  = ai[i+1] - a->diag[i] - 1;
1179     sum = tmp[row];
1180     PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
1181     x[*c--] = tmp[row] = sum*aa[a->diag[i]];
1182   }
1183 
1184   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1185   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1186   ierr = VecRestoreArray(bb,&b);CHKERRQ(ierr);
1187   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1188   ierr = PetscLogFlops(2.0*a->nz - A->cmap->n);CHKERRQ(ierr);
1189   PetscFunctionReturn(0);
1190 }
1191 
1192 /* ----------------------------------------------------------- */
1193 #include "../src/mat/impls/aij/seq/ftn-kernels/fsolve.h"
1194 #undef __FUNCT__
1195 #define __FUNCT__ "MatSolve_SeqAIJ_NaturalOrdering_inplace"
1196 PetscErrorCode MatSolve_SeqAIJ_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
1197 {
1198   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1199   PetscErrorCode    ierr;
1200   PetscInt          n = A->rmap->n;
1201   const PetscInt    *ai = a->i,*aj = a->j,*adiag = a->diag;
1202   PetscScalar       *x;
1203   const PetscScalar *b;
1204   const MatScalar   *aa = a->a;
1205 #if !defined(PETSC_USE_FORTRAN_KERNEL_SOLVEAIJ)
1206   PetscInt          adiag_i,i,nz,ai_i;
1207   const PetscInt    *vi;
1208   const MatScalar   *v;
1209   PetscScalar       sum;
1210 #endif
1211 
1212   PetscFunctionBegin;
1213   if (!n) PetscFunctionReturn(0);
1214 
1215   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1216   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1217 
1218 #if defined(PETSC_USE_FORTRAN_KERNEL_SOLVEAIJ)
1219   fortransolveaij_(&n,x,ai,aj,adiag,aa,b);
1220 #else
1221   /* forward solve the lower triangular */
1222   x[0] = b[0];
1223   for (i=1; i<n; i++) {
1224     ai_i = ai[i];
1225     v    = aa + ai_i;
1226     vi   = aj + ai_i;
1227     nz   = adiag[i] - ai_i;
1228     sum  = b[i];
1229     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
1230     x[i] = sum;
1231   }
1232 
1233   /* backward solve the upper triangular */
1234   for (i=n-1; i>=0; i--){
1235     adiag_i = adiag[i];
1236     v       = aa + adiag_i + 1;
1237     vi      = aj + adiag_i + 1;
1238     nz      = ai[i+1] - adiag_i - 1;
1239     sum     = x[i];
1240     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
1241     x[i]    = sum*aa[adiag_i];
1242   }
1243 #endif
1244   ierr = PetscLogFlops(2.0*a->nz - A->cmap->n);CHKERRQ(ierr);
1245   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1246   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1247   PetscFunctionReturn(0);
1248 }
1249 
1250 #undef __FUNCT__
1251 #define __FUNCT__ "MatSolveAdd_SeqAIJ_inplace"
1252 PetscErrorCode MatSolveAdd_SeqAIJ_inplace(Mat A,Vec bb,Vec yy,Vec xx)
1253 {
1254   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1255   IS                iscol = a->col,isrow = a->row;
1256   PetscErrorCode    ierr;
1257   PetscInt          i, n = A->rmap->n,j;
1258   PetscInt          nz;
1259   const PetscInt    *rout,*cout,*r,*c,*vi,*ai = a->i,*aj = a->j;
1260   PetscScalar       *x,*tmp,sum;
1261   const PetscScalar *b;
1262   const MatScalar   *aa = a->a,*v;
1263 
1264   PetscFunctionBegin;
1265   if (yy != xx) {ierr = VecCopy(yy,xx);CHKERRQ(ierr);}
1266 
1267   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1268   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1269   tmp  = a->solve_work;
1270 
1271   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1272   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout + (n-1);
1273 
1274   /* forward solve the lower triangular */
1275   tmp[0] = b[*r++];
1276   for (i=1; i<n; i++) {
1277     v   = aa + ai[i] ;
1278     vi  = aj + ai[i] ;
1279     nz  = a->diag[i] - ai[i];
1280     sum = b[*r++];
1281     for (j=0; j<nz; j++) sum -= v[j]*tmp[vi[j]];
1282     tmp[i] = sum;
1283   }
1284 
1285   /* backward solve the upper triangular */
1286   for (i=n-1; i>=0; i--){
1287     v   = aa + a->diag[i] + 1;
1288     vi  = aj + a->diag[i] + 1;
1289     nz  = ai[i+1] - a->diag[i] - 1;
1290     sum = tmp[i];
1291     for (j=0; j<nz; j++) sum -= v[j]*tmp[vi[j]];
1292     tmp[i] = sum*aa[a->diag[i]];
1293     x[*c--] += tmp[i];
1294   }
1295 
1296   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1297   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1298   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1299   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1300   ierr = PetscLogFlops(2.0*a->nz);CHKERRQ(ierr);
1301 
1302   PetscFunctionReturn(0);
1303 }
1304 
1305 #undef __FUNCT__
1306 #define __FUNCT__ "MatSolveAdd_SeqAIJ"
1307 PetscErrorCode MatSolveAdd_SeqAIJ(Mat A,Vec bb,Vec yy,Vec xx)
1308 {
1309   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1310   IS                iscol = a->col,isrow = a->row;
1311   PetscErrorCode    ierr;
1312   PetscInt          i, n = A->rmap->n,j;
1313   PetscInt          nz;
1314   const PetscInt    *rout,*cout,*r,*c,*vi,*ai = a->i,*aj = a->j,*adiag = a->diag;
1315   PetscScalar       *x,*tmp,sum;
1316   const PetscScalar *b;
1317   const MatScalar   *aa = a->a,*v;
1318 
1319   PetscFunctionBegin;
1320   if (yy != xx) {ierr = VecCopy(yy,xx);CHKERRQ(ierr);}
1321 
1322   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1323   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1324   tmp  = a->solve_work;
1325 
1326   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1327   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1328 
1329   /* forward solve the lower triangular */
1330   tmp[0] = b[r[0]];
1331   v      = aa;
1332   vi     = aj;
1333   for (i=1; i<n; i++) {
1334     nz  = ai[i+1] - ai[i];
1335     sum = b[r[i]];
1336     for (j=0; j<nz; j++) sum -= v[j]*tmp[vi[j]];
1337     tmp[i] = sum;
1338     v += nz; vi += nz;
1339   }
1340 
1341   /* backward solve the upper triangular */
1342   v  = aa + adiag[n-1];
1343   vi = aj + adiag[n-1];
1344   for (i=n-1; i>=0; i--){
1345     nz  = adiag[i] - adiag[i+1] - 1;
1346     sum = tmp[i];
1347     for (j=0; j<nz; j++) sum -= v[j]*tmp[vi[j]];
1348     tmp[i] = sum*v[nz];
1349     x[c[i]] += tmp[i];
1350     v += nz+1; vi += nz+1;
1351   }
1352 
1353   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1354   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1355   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1356   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1357   ierr = PetscLogFlops(2.0*a->nz);CHKERRQ(ierr);
1358 
1359   PetscFunctionReturn(0);
1360 }
1361 
1362 #undef __FUNCT__
1363 #define __FUNCT__ "MatSolveTranspose_SeqAIJ_inplace"
1364 PetscErrorCode MatSolveTranspose_SeqAIJ_inplace(Mat A,Vec bb,Vec xx)
1365 {
1366   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1367   IS                iscol = a->col,isrow = a->row;
1368   PetscErrorCode    ierr;
1369   const PetscInt    *rout,*cout,*r,*c,*diag = a->diag,*ai = a->i,*aj = a->j,*vi;
1370   PetscInt          i,n = A->rmap->n,j;
1371   PetscInt          nz;
1372   PetscScalar       *x,*tmp,s1;
1373   const MatScalar   *aa = a->a,*v;
1374   const PetscScalar *b;
1375 
1376   PetscFunctionBegin;
1377   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1378   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1379   tmp  = a->solve_work;
1380 
1381   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1382   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1383 
1384   /* copy the b into temp work space according to permutation */
1385   for (i=0; i<n; i++) tmp[i] = b[c[i]];
1386 
1387   /* forward solve the U^T */
1388   for (i=0; i<n; i++) {
1389     v   = aa + diag[i] ;
1390     vi  = aj + diag[i] + 1;
1391     nz  = ai[i+1] - diag[i] - 1;
1392     s1  = tmp[i];
1393     s1 *= (*v++);  /* multiply by inverse of diagonal entry */
1394     for (j=0; j<nz; j++) tmp[vi[j]] -= s1*v[j];
1395     tmp[i] = s1;
1396   }
1397 
1398   /* backward solve the L^T */
1399   for (i=n-1; i>=0; i--){
1400     v   = aa + diag[i] - 1 ;
1401     vi  = aj + diag[i] - 1 ;
1402     nz  = diag[i] - ai[i];
1403     s1  = tmp[i];
1404     for (j=0; j>-nz; j--) tmp[vi[j]] -= s1*v[j];
1405   }
1406 
1407   /* copy tmp into x according to permutation */
1408   for (i=0; i<n; i++) x[r[i]] = tmp[i];
1409 
1410   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1411   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1412   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1413   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1414 
1415   ierr = PetscLogFlops(2.0*a->nz-A->cmap->n);CHKERRQ(ierr);
1416   PetscFunctionReturn(0);
1417 }
1418 
1419 #undef __FUNCT__
1420 #define __FUNCT__ "MatSolveTranspose_SeqAIJ"
1421 PetscErrorCode MatSolveTranspose_SeqAIJ(Mat A,Vec bb,Vec xx)
1422 {
1423   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1424   IS                iscol = a->col,isrow = a->row;
1425   PetscErrorCode    ierr;
1426   const PetscInt    *rout,*cout,*r,*c,*adiag = a->diag,*ai = a->i,*aj = a->j,*vi;
1427   PetscInt          i,n = A->rmap->n,j;
1428   PetscInt          nz;
1429   PetscScalar       *x,*tmp,s1;
1430   const MatScalar   *aa = a->a,*v;
1431   const PetscScalar *b;
1432 
1433   PetscFunctionBegin;
1434   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1435   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1436   tmp  = a->solve_work;
1437 
1438   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1439   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1440 
1441   /* copy the b into temp work space according to permutation */
1442   for (i=0; i<n; i++) tmp[i] = b[c[i]];
1443 
1444   /* forward solve the U^T */
1445   for (i=0; i<n; i++) {
1446     v   = aa + adiag[i+1] + 1;
1447     vi  = aj + adiag[i+1] + 1;
1448     nz  = adiag[i] - adiag[i+1] - 1;
1449     s1  = tmp[i];
1450     s1 *= v[nz];  /* multiply by inverse of diagonal entry */
1451     for (j=0; j<nz; j++) tmp[vi[j]] -= s1*v[j];
1452     tmp[i] = s1;
1453   }
1454 
1455   /* backward solve the L^T */
1456   for (i=n-1; i>=0; i--){
1457     v   = aa + ai[i];
1458     vi  = aj + ai[i];
1459     nz  = ai[i+1] - ai[i];
1460     s1  = tmp[i];
1461     for (j=0; j<nz; j++) tmp[vi[j]] -= s1*v[j];
1462   }
1463 
1464   /* copy tmp into x according to permutation */
1465   for (i=0; i<n; i++) x[r[i]] = tmp[i];
1466 
1467   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1468   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1469   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1470   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1471 
1472   ierr = PetscLogFlops(2.0*a->nz-A->cmap->n);CHKERRQ(ierr);
1473   PetscFunctionReturn(0);
1474 }
1475 
1476 #undef __FUNCT__
1477 #define __FUNCT__ "MatSolveTransposeAdd_SeqAIJ_inplace"
1478 PetscErrorCode MatSolveTransposeAdd_SeqAIJ_inplace(Mat A,Vec bb,Vec zz,Vec xx)
1479 {
1480   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1481   IS                iscol = a->col,isrow = a->row;
1482   PetscErrorCode    ierr;
1483   const PetscInt    *rout,*cout,*r,*c,*diag = a->diag,*ai = a->i,*aj = a->j,*vi;
1484   PetscInt          i,n = A->rmap->n,j;
1485   PetscInt          nz;
1486   PetscScalar       *x,*tmp,s1;
1487   const MatScalar   *aa = a->a,*v;
1488   const PetscScalar *b;
1489 
1490   PetscFunctionBegin;
1491   if (zz != xx) {ierr = VecCopy(zz,xx);CHKERRQ(ierr);}
1492   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1493   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1494   tmp  = a->solve_work;
1495 
1496   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1497   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1498 
1499   /* copy the b into temp work space according to permutation */
1500   for (i=0; i<n; i++) tmp[i] = b[c[i]];
1501 
1502   /* forward solve the U^T */
1503   for (i=0; i<n; i++) {
1504     v   = aa + diag[i] ;
1505     vi  = aj + diag[i] + 1;
1506     nz  = ai[i+1] - diag[i] - 1;
1507     s1  = tmp[i];
1508     s1 *= (*v++);  /* multiply by inverse of diagonal entry */
1509     for (j=0; j<nz; j++) tmp[vi[j]] -= s1*v[j];
1510     tmp[i] = s1;
1511   }
1512 
1513   /* backward solve the L^T */
1514   for (i=n-1; i>=0; i--){
1515     v   = aa + diag[i] - 1 ;
1516     vi  = aj + diag[i] - 1 ;
1517     nz  = diag[i] - ai[i];
1518     s1  = tmp[i];
1519     for (j=0; j>-nz; j--) tmp[vi[j]] -= s1*v[j];
1520   }
1521 
1522   /* copy tmp into x according to permutation */
1523   for (i=0; i<n; i++) x[r[i]] += tmp[i];
1524 
1525   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1526   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1527   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1528   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1529 
1530   ierr = PetscLogFlops(2.0*a->nz-A->cmap->n);CHKERRQ(ierr);
1531   PetscFunctionReturn(0);
1532 }
1533 
1534 #undef __FUNCT__
1535 #define __FUNCT__ "MatSolveTransposeAdd_SeqAIJ"
1536 PetscErrorCode MatSolveTransposeAdd_SeqAIJ(Mat A,Vec bb,Vec zz,Vec xx)
1537 {
1538   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1539   IS                iscol = a->col,isrow = a->row;
1540   PetscErrorCode    ierr;
1541   const PetscInt    *rout,*cout,*r,*c,*adiag = a->diag,*ai = a->i,*aj = a->j,*vi;
1542   PetscInt          i,n = A->rmap->n,j;
1543   PetscInt          nz;
1544   PetscScalar       *x,*tmp,s1;
1545   const MatScalar   *aa = a->a,*v;
1546   const PetscScalar *b;
1547 
1548   PetscFunctionBegin;
1549   if (zz != xx) {ierr = VecCopy(zz,xx);CHKERRQ(ierr);}
1550   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1551   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1552   tmp  = a->solve_work;
1553 
1554   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1555   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1556 
1557   /* copy the b into temp work space according to permutation */
1558   for (i=0; i<n; i++) tmp[i] = b[c[i]];
1559 
1560   /* forward solve the U^T */
1561   for (i=0; i<n; i++) {
1562     v   = aa + adiag[i+1] + 1;
1563     vi  = aj + adiag[i+1] + 1;
1564     nz  = adiag[i] - adiag[i+1] - 1;
1565     s1  = tmp[i];
1566     s1 *= v[nz];  /* multiply by inverse of diagonal entry */
1567     for (j=0; j<nz; j++) tmp[vi[j]] -= s1*v[j];
1568     tmp[i] = s1;
1569   }
1570 
1571 
1572   /* backward solve the L^T */
1573   for (i=n-1; i>=0; i--){
1574     v   = aa + ai[i] ;
1575     vi  = aj + ai[i];
1576     nz  = ai[i+1] - ai[i];
1577     s1  = tmp[i];
1578     for (j=0; j<nz; j++) tmp[vi[j]] -= s1*v[j];
1579   }
1580 
1581   /* copy tmp into x according to permutation */
1582   for (i=0; i<n; i++) x[r[i]] += tmp[i];
1583 
1584   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1585   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1586   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1587   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1588 
1589   ierr = PetscLogFlops(2.0*a->nz-A->cmap->n);CHKERRQ(ierr);
1590   PetscFunctionReturn(0);
1591 }
1592 
1593 /* ----------------------------------------------------------------*/
1594 
1595 EXTERN PetscErrorCode MatDuplicateNoCreate_SeqAIJ(Mat,Mat,MatDuplicateOption,PetscTruth);
1596 
1597 /*
1598    ilu() under revised new data structure.
1599    Factored arrays bj and ba are stored as
1600      L(0,:), L(1,:), ...,L(n-1,:),  U(n-1,:),...,U(i,:),U(i-1,:),...,U(0,:)
1601 
1602    bi=fact->i is an array of size n+1, in which
1603    bi+
1604      bi[i]:  points to 1st entry of L(i,:),i=0,...,n-1
1605      bi[n]:  points to L(n-1,n-1)+1
1606 
1607   bdiag=fact->diag is an array of size n+1,in which
1608      bdiag[i]: points to diagonal of U(i,:), i=0,...,n-1
1609      bdiag[n]: points to entry of U(n-1,0)-1
1610 
1611    U(i,:) contains bdiag[i] as its last entry, i.e.,
1612     U(i,:) = (u[i,i+1],...,u[i,n-1],diag[i])
1613 */
1614 #undef __FUNCT__
1615 #define __FUNCT__ "MatILUFactorSymbolic_SeqAIJ_ilu0"
1616 PetscErrorCode MatILUFactorSymbolic_SeqAIJ_ilu0(Mat fact,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
1617 {
1618 
1619   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
1620   PetscErrorCode     ierr;
1621   const PetscInt     n=A->rmap->n,*ai=a->i,*aj,*adiag=a->diag;
1622   PetscInt           i,j,k=0,nz,*bi,*bj,*bdiag;
1623   PetscTruth         missing;
1624   IS                 isicol;
1625 
1626   PetscFunctionBegin;
1627   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);
1628   ierr = MatMissingDiagonal(A,&missing,&i);CHKERRQ(ierr);
1629   if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",i);
1630   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
1631 
1632   ierr = MatDuplicateNoCreate_SeqAIJ(fact,A,MAT_DO_NOT_COPY_VALUES,PETSC_FALSE);CHKERRQ(ierr);
1633   b    = (Mat_SeqAIJ*)(fact)->data;
1634 
1635   /* allocate matrix arrays for new data structure */
1636   ierr = PetscMalloc3(ai[n]+1,PetscScalar,&b->a,ai[n]+1,PetscInt,&b->j,n+1,PetscInt,&b->i);CHKERRQ(ierr);
1637   ierr = PetscLogObjectMemory(fact,ai[n]*(sizeof(PetscScalar)+sizeof(PetscInt))+(n+1)*sizeof(PetscInt));CHKERRQ(ierr);
1638   b->singlemalloc = PETSC_TRUE;
1639   if (!b->diag){
1640     ierr = PetscMalloc((n+1)*sizeof(PetscInt),&b->diag);CHKERRQ(ierr);
1641     ierr = PetscLogObjectMemory(fact,(n+1)*sizeof(PetscInt));CHKERRQ(ierr);
1642   }
1643   bdiag = b->diag;
1644 
1645   if (n > 0) {
1646     ierr = PetscMemzero(b->a,(ai[n])*sizeof(MatScalar));CHKERRQ(ierr);
1647   }
1648 
1649   /* set bi and bj with new data structure */
1650   bi = b->i;
1651   bj = b->j;
1652 
1653   /* L part */
1654   bi[0] = 0;
1655   for (i=0; i<n; i++){
1656     nz = adiag[i] - ai[i];
1657     bi[i+1] = bi[i] + nz;
1658     aj = a->j + ai[i];
1659     for (j=0; j<nz; j++){
1660       /*   *bj = aj[j]; bj++; */
1661       bj[k++] = aj[j];
1662     }
1663   }
1664 
1665   /* U part */
1666   bdiag[n] = bi[n]-1;
1667   for (i=n-1; i>=0; i--){
1668     nz = ai[i+1] - adiag[i] - 1;
1669     aj = a->j + adiag[i] + 1;
1670     for (j=0; j<nz; j++){
1671       /*      *bj = aj[j]; bj++; */
1672       bj[k++] = aj[j];
1673     }
1674     /* diag[i] */
1675     /*    *bj = i; bj++; */
1676     bj[k++] = i;
1677     bdiag[i] = bdiag[i+1] + nz + 1;
1678   }
1679 
1680   fact->factor                 = MAT_FACTOR_ILU;
1681   fact->info.factor_mallocs    = 0;
1682   fact->info.fill_ratio_given  = info->fill;
1683   fact->info.fill_ratio_needed = 1.0;
1684   fact->ops->lufactornumeric   = MatLUFactorNumeric_SeqAIJ;
1685 
1686   b       = (Mat_SeqAIJ*)(fact)->data;
1687   b->row  = isrow;
1688   b->col  = iscol;
1689   b->icol = isicol;
1690   ierr    = PetscMalloc((fact->rmap->n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1691   ierr    = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
1692   ierr    = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
1693   PetscFunctionReturn(0);
1694 }
1695 
1696 #undef __FUNCT__
1697 #define __FUNCT__ "MatILUFactorSymbolic_SeqAIJ"
1698 PetscErrorCode MatILUFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
1699 {
1700   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
1701   IS                 isicol;
1702   PetscErrorCode     ierr;
1703   const PetscInt     *r,*ic;
1704   PetscInt           n=A->rmap->n,*ai=a->i,*aj=a->j;
1705   PetscInt           *bi,*cols,nnz,*cols_lvl;
1706   PetscInt           *bdiag,prow,fm,nzbd,reallocs=0,dcount=0;
1707   PetscInt           i,levels,diagonal_fill;
1708   PetscTruth         col_identity,row_identity;
1709   PetscReal          f;
1710   PetscInt           nlnk,*lnk,*lnk_lvl=PETSC_NULL;
1711   PetscBT            lnkbt;
1712   PetscInt           nzi,*bj,**bj_ptr,**bjlvl_ptr;
1713   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
1714   PetscFreeSpaceList free_space_lvl=PETSC_NULL,current_space_lvl=PETSC_NULL;
1715 
1716   PetscFunctionBegin;
1717   // Uncomment the old data struct part only while testing new data structure for MatSolve()
1718   PetscTruth         olddatastruct=PETSC_FALSE;
1719   ierr = PetscOptionsGetTruth(PETSC_NULL,"-ilu_old",&olddatastruct,PETSC_NULL);CHKERRQ(ierr);
1720   if(olddatastruct){
1721     ierr = MatILUFactorSymbolic_SeqAIJ_inplace(fact,A,isrow,iscol,info);CHKERRQ(ierr);
1722     PetscFunctionReturn(0);
1723   }
1724 
1725   levels = (PetscInt)info->levels;
1726   ierr   = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
1727   ierr   = ISIdentity(iscol,&col_identity);CHKERRQ(ierr);
1728 
1729   if (!levels && row_identity && col_identity) {
1730     /* special case: ilu(0) with natural ordering */
1731     ierr = MatILUFactorSymbolic_SeqAIJ_ilu0(fact,A,isrow,iscol,info);CHKERRQ(ierr);
1732     ierr = Mat_CheckInode_FactorLU(fact,PETSC_FALSE);CHKERRQ(ierr);
1733     PetscFunctionReturn(0);
1734   }
1735 
1736   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);
1737   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
1738   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
1739   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
1740 
1741   /* get new row and diagonal pointers, must be allocated separately because they will be given to the Mat_SeqAIJ and freed separately */
1742   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
1743   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);
1744   bi[0] = bdiag[0] = 0;
1745 
1746   ierr = PetscMalloc2(n,PetscInt*,&bj_ptr,n,PetscInt*,&bjlvl_ptr);CHKERRQ(ierr);
1747 
1748   /* create a linked list for storing column indices of the active row */
1749   nlnk = n + 1;
1750   ierr = PetscIncompleteLLCreate(n,n,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1751 
1752   /* initial FreeSpace size is f*(ai[n]+1) */
1753   f             = info->fill;
1754   diagonal_fill = (PetscInt)info->diagonal_fill;
1755   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space);CHKERRQ(ierr);
1756   current_space = free_space;
1757   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space_lvl);CHKERRQ(ierr);
1758   current_space_lvl = free_space_lvl;
1759 
1760   for (i=0; i<n; i++) {
1761     nzi = 0;
1762     /* copy current row into linked list */
1763     nnz  = ai[r[i]+1] - ai[r[i]];
1764     if (!nnz) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
1765     cols = aj + ai[r[i]];
1766     lnk[i] = -1; /* marker to indicate if diagonal exists */
1767     ierr = PetscIncompleteLLInit(nnz,cols,n,ic,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1768     nzi += nlnk;
1769 
1770     /* make sure diagonal entry is included */
1771     if (diagonal_fill && lnk[i] == -1) {
1772       fm = n;
1773       while (lnk[fm] < i) fm = lnk[fm];
1774       lnk[i]     = lnk[fm]; /* insert diagonal into linked list */
1775       lnk[fm]    = i;
1776       lnk_lvl[i] = 0;
1777       nzi++; dcount++;
1778     }
1779 
1780     /* add pivot rows into the active row */
1781     nzbd = 0;
1782     prow = lnk[n];
1783     while (prow < i) {
1784       nnz      = bdiag[prow];
1785       cols     = bj_ptr[prow] + nnz + 1;
1786       cols_lvl = bjlvl_ptr[prow] + nnz + 1;
1787       nnz      = bi[prow+1] - bi[prow] - nnz - 1;
1788       ierr = PetscILULLAddSorted(nnz,cols,levels,cols_lvl,prow,nlnk,lnk,lnk_lvl,lnkbt,prow);CHKERRQ(ierr);
1789       nzi += nlnk;
1790       prow = lnk[prow];
1791       nzbd++;
1792     }
1793     bdiag[i] = nzbd;
1794     bi[i+1]  = bi[i] + nzi;
1795 
1796     /* if free space is not available, make more free space */
1797     if (current_space->local_remaining<nzi) {
1798       nnz = 2*nzi*(n - i); /* estimated and max additional space needed */
1799       ierr = PetscFreeSpaceGet(nnz,&current_space);CHKERRQ(ierr);
1800       ierr = PetscFreeSpaceGet(nnz,&current_space_lvl);CHKERRQ(ierr);
1801       reallocs++;
1802     }
1803 
1804     /* copy data into free_space and free_space_lvl, then initialize lnk */
1805     ierr = PetscIncompleteLLClean(n,n,nzi,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);CHKERRQ(ierr);
1806     bj_ptr[i]    = current_space->array;
1807     bjlvl_ptr[i] = current_space_lvl->array;
1808 
1809     /* make sure the active row i has diagonal entry */
1810     if (*(bj_ptr[i]+bdiag[i]) != i) {
1811       SETERRQ1(PETSC_ERR_MAT_LU_ZRPVT,"Row %D has missing diagonal in factored matrix\n\
1812     try running with -pc_factor_nonzeros_along_diagonal or -pc_factor_diagonal_fill",i);
1813     }
1814 
1815     current_space->array           += nzi;
1816     current_space->local_used      += nzi;
1817     current_space->local_remaining -= nzi;
1818     current_space_lvl->array           += nzi;
1819     current_space_lvl->local_used      += nzi;
1820     current_space_lvl->local_remaining -= nzi;
1821   }
1822 
1823   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
1824   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
1825 
1826   /* destroy list of free space and other temporary arrays */
1827   ierr = PetscMalloc((bi[n]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
1828 
1829   /* copy free_space into bj and free free_space; set bi, bj, bdiag in new datastructure; */
1830   ierr = PetscFreeSpaceContiguous_LU(&free_space,bj,n,bi,bdiag);CHKERRQ(ierr);
1831 
1832   ierr = PetscIncompleteLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1833   ierr = PetscFreeSpaceDestroy(free_space_lvl);CHKERRQ(ierr);
1834   ierr = PetscFree2(bj_ptr,bjlvl_ptr);CHKERRQ(ierr);
1835 
1836 #if defined(PETSC_USE_INFO)
1837   {
1838     PetscReal af = ((PetscReal)bi[n])/((PetscReal)ai[n]);
1839     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,f,af);CHKERRQ(ierr);
1840     ierr = PetscInfo1(A,"Run with -[sub_]pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
1841     ierr = PetscInfo1(A,"PCFactorSetFill([sub]pc,%G);\n",af);CHKERRQ(ierr);
1842     ierr = PetscInfo(A,"for best performance.\n");CHKERRQ(ierr);
1843     if (diagonal_fill) {
1844       ierr = PetscInfo1(A,"Detected and replaced %D missing diagonals",dcount);CHKERRQ(ierr);
1845     }
1846   }
1847 #endif
1848 
1849   /* put together the new matrix */
1850   ierr = MatSeqAIJSetPreallocation_SeqAIJ(fact,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
1851   ierr = PetscLogObjectParent(fact,isicol);CHKERRQ(ierr);
1852   b = (Mat_SeqAIJ*)(fact)->data;
1853   b->free_a       = PETSC_TRUE;
1854   b->free_ij      = PETSC_TRUE;
1855   b->singlemalloc = PETSC_FALSE;
1856   ierr = PetscMalloc((bdiag[0]+1)*sizeof(PetscScalar),&b->a);CHKERRQ(ierr);
1857   b->j          = bj;
1858   b->i          = bi;
1859   b->diag       = bdiag;
1860   b->ilen       = 0;
1861   b->imax       = 0;
1862   b->row        = isrow;
1863   b->col        = iscol;
1864   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
1865   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
1866   b->icol       = isicol;
1867   ierr = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1868   /* In b structure:  Free imax, ilen, old a, old j.
1869      Allocate bdiag, solve_work, new a, new j */
1870   ierr = PetscLogObjectMemory(fact,(bdiag[0]+1)*(sizeof(PetscInt)+sizeof(PetscScalar)));CHKERRQ(ierr);
1871   b->maxnz = b->nz = bdiag[0]+1;
1872   (fact)->info.factor_mallocs    = reallocs;
1873   (fact)->info.fill_ratio_given  = f;
1874   (fact)->info.fill_ratio_needed = ((PetscReal)(bdiag[0]+1))/((PetscReal)ai[n]);
1875   (fact)->ops->lufactornumeric = MatLUFactorNumeric_SeqAIJ;
1876   ierr = Mat_CheckInode_FactorLU(fact,PETSC_FALSE);CHKERRQ(ierr);
1877   PetscFunctionReturn(0);
1878 }
1879 
1880 #undef __FUNCT__
1881 #define __FUNCT__ "MatILUFactorSymbolic_SeqAIJ_inplace"
1882 PetscErrorCode MatILUFactorSymbolic_SeqAIJ_inplace(Mat fact,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
1883 {
1884   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
1885   IS                 isicol;
1886   PetscErrorCode     ierr;
1887   const PetscInt     *r,*ic;
1888   PetscInt           n=A->rmap->n,*ai=a->i,*aj=a->j,d;
1889   PetscInt           *bi,*cols,nnz,*cols_lvl;
1890   PetscInt           *bdiag,prow,fm,nzbd,reallocs=0,dcount=0;
1891   PetscInt           i,levels,diagonal_fill;
1892   PetscTruth         col_identity,row_identity;
1893   PetscReal          f;
1894   PetscInt           nlnk,*lnk,*lnk_lvl=PETSC_NULL;
1895   PetscBT            lnkbt;
1896   PetscInt           nzi,*bj,**bj_ptr,**bjlvl_ptr;
1897   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
1898   PetscFreeSpaceList free_space_lvl=PETSC_NULL,current_space_lvl=PETSC_NULL;
1899   PetscTruth         missing;
1900 
1901   PetscFunctionBegin;
1902   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);
1903   f             = info->fill;
1904   levels        = (PetscInt)info->levels;
1905   diagonal_fill = (PetscInt)info->diagonal_fill;
1906   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
1907 
1908   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
1909   ierr = ISIdentity(iscol,&col_identity);CHKERRQ(ierr);
1910   if (!levels && row_identity && col_identity) { /* special case: ilu(0) with natural ordering */
1911     ierr = MatDuplicateNoCreate_SeqAIJ(fact,A,MAT_DO_NOT_COPY_VALUES,PETSC_TRUE);CHKERRQ(ierr);
1912     (fact)->ops->lufactornumeric =  MatLUFactorNumeric_SeqAIJ_inplace;
1913 
1914     fact->factor = MAT_FACTOR_ILU;
1915     (fact)->info.factor_mallocs    = 0;
1916     (fact)->info.fill_ratio_given  = info->fill;
1917     (fact)->info.fill_ratio_needed = 1.0;
1918     b               = (Mat_SeqAIJ*)(fact)->data;
1919     ierr = MatMissingDiagonal(A,&missing,&d);CHKERRQ(ierr);
1920     if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",d);
1921     b->row              = isrow;
1922     b->col              = iscol;
1923     b->icol             = isicol;
1924     ierr                = PetscMalloc(((fact)->rmap->n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1925     ierr                = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
1926     ierr                = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
1927     ierr = MatILUFactorSymbolic_SeqAIJ_Inode_inplace(fact,A,isrow,iscol,info);CHKERRQ(ierr);
1928     PetscFunctionReturn(0);
1929   }
1930 
1931   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
1932   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
1933 
1934   /* get new row and diagonal pointers, must be allocated separately because they will be given to the Mat_SeqAIJ and freed separately */
1935   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
1936   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);
1937   bi[0] = bdiag[0] = 0;
1938 
1939   ierr = PetscMalloc2(n,PetscInt*,&bj_ptr,n,PetscInt*,&bjlvl_ptr);CHKERRQ(ierr);
1940 
1941   /* create a linked list for storing column indices of the active row */
1942   nlnk = n + 1;
1943   ierr = PetscIncompleteLLCreate(n,n,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1944 
1945   /* initial FreeSpace size is f*(ai[n]+1) */
1946   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space);CHKERRQ(ierr);
1947   current_space = free_space;
1948   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space_lvl);CHKERRQ(ierr);
1949   current_space_lvl = free_space_lvl;
1950 
1951   for (i=0; i<n; i++) {
1952     nzi = 0;
1953     /* copy current row into linked list */
1954     nnz  = ai[r[i]+1] - ai[r[i]];
1955     if (!nnz) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
1956     cols = aj + ai[r[i]];
1957     lnk[i] = -1; /* marker to indicate if diagonal exists */
1958     ierr = PetscIncompleteLLInit(nnz,cols,n,ic,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1959     nzi += nlnk;
1960 
1961     /* make sure diagonal entry is included */
1962     if (diagonal_fill && lnk[i] == -1) {
1963       fm = n;
1964       while (lnk[fm] < i) fm = lnk[fm];
1965       lnk[i]     = lnk[fm]; /* insert diagonal into linked list */
1966       lnk[fm]    = i;
1967       lnk_lvl[i] = 0;
1968       nzi++; dcount++;
1969     }
1970 
1971     /* add pivot rows into the active row */
1972     nzbd = 0;
1973     prow = lnk[n];
1974     while (prow < i) {
1975       nnz      = bdiag[prow];
1976       cols     = bj_ptr[prow] + nnz + 1;
1977       cols_lvl = bjlvl_ptr[prow] + nnz + 1;
1978       nnz      = bi[prow+1] - bi[prow] - nnz - 1;
1979       ierr = PetscILULLAddSorted(nnz,cols,levels,cols_lvl,prow,nlnk,lnk,lnk_lvl,lnkbt,prow);CHKERRQ(ierr);
1980       nzi += nlnk;
1981       prow = lnk[prow];
1982       nzbd++;
1983     }
1984     bdiag[i] = nzbd;
1985     bi[i+1]  = bi[i] + nzi;
1986 
1987     /* if free space is not available, make more free space */
1988     if (current_space->local_remaining<nzi) {
1989       nnz = nzi*(n - i); /* estimated and max additional space needed */
1990       ierr = PetscFreeSpaceGet(nnz,&current_space);CHKERRQ(ierr);
1991       ierr = PetscFreeSpaceGet(nnz,&current_space_lvl);CHKERRQ(ierr);
1992       reallocs++;
1993     }
1994 
1995     /* copy data into free_space and free_space_lvl, then initialize lnk */
1996     ierr = PetscIncompleteLLClean(n,n,nzi,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);CHKERRQ(ierr);
1997     bj_ptr[i]    = current_space->array;
1998     bjlvl_ptr[i] = current_space_lvl->array;
1999 
2000     /* make sure the active row i has diagonal entry */
2001     if (*(bj_ptr[i]+bdiag[i]) != i) {
2002       SETERRQ1(PETSC_ERR_MAT_LU_ZRPVT,"Row %D has missing diagonal in factored matrix\n\
2003     try running with -pc_factor_nonzeros_along_diagonal or -pc_factor_diagonal_fill",i);
2004     }
2005 
2006     current_space->array           += nzi;
2007     current_space->local_used      += nzi;
2008     current_space->local_remaining -= nzi;
2009     current_space_lvl->array           += nzi;
2010     current_space_lvl->local_used      += nzi;
2011     current_space_lvl->local_remaining -= nzi;
2012   }
2013 
2014   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
2015   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
2016 
2017   /* destroy list of free space and other temporary arrays */
2018   ierr = PetscMalloc((bi[n]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
2019   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr); /* copy free_space -> bj */
2020   ierr = PetscIncompleteLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
2021   ierr = PetscFreeSpaceDestroy(free_space_lvl);CHKERRQ(ierr);
2022   ierr = PetscFree2(bj_ptr,bjlvl_ptr);CHKERRQ(ierr);
2023 
2024 #if defined(PETSC_USE_INFO)
2025   {
2026     PetscReal af = ((PetscReal)bi[n])/((PetscReal)ai[n]);
2027     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,f,af);CHKERRQ(ierr);
2028     ierr = PetscInfo1(A,"Run with -[sub_]pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
2029     ierr = PetscInfo1(A,"PCFactorSetFill([sub]pc,%G);\n",af);CHKERRQ(ierr);
2030     ierr = PetscInfo(A,"for best performance.\n");CHKERRQ(ierr);
2031     if (diagonal_fill) {
2032       ierr = PetscInfo1(A,"Detected and replaced %D missing diagonals",dcount);CHKERRQ(ierr);
2033     }
2034   }
2035 #endif
2036 
2037   /* put together the new matrix */
2038   ierr = MatSeqAIJSetPreallocation_SeqAIJ(fact,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
2039   ierr = PetscLogObjectParent(fact,isicol);CHKERRQ(ierr);
2040   b = (Mat_SeqAIJ*)(fact)->data;
2041   b->free_a       = PETSC_TRUE;
2042   b->free_ij      = PETSC_TRUE;
2043   b->singlemalloc = PETSC_FALSE;
2044   ierr = PetscMalloc(bi[n]*sizeof(PetscScalar),&b->a);CHKERRQ(ierr);
2045   b->j          = bj;
2046   b->i          = bi;
2047   for (i=0; i<n; i++) bdiag[i] += bi[i];
2048   b->diag       = bdiag;
2049   b->ilen       = 0;
2050   b->imax       = 0;
2051   b->row        = isrow;
2052   b->col        = iscol;
2053   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
2054   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
2055   b->icol       = isicol;
2056   ierr = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
2057   /* In b structure:  Free imax, ilen, old a, old j.
2058      Allocate bdiag, solve_work, new a, new j */
2059   ierr = PetscLogObjectMemory(fact,(bi[n]-n) * (sizeof(PetscInt)+sizeof(PetscScalar)));CHKERRQ(ierr);
2060   b->maxnz             = b->nz = bi[n] ;
2061   (fact)->info.factor_mallocs    = reallocs;
2062   (fact)->info.fill_ratio_given  = f;
2063   (fact)->info.fill_ratio_needed = ((PetscReal)bi[n])/((PetscReal)ai[n]);
2064   (fact)->ops->lufactornumeric =  MatLUFactorNumeric_SeqAIJ_inplace;
2065   ierr = MatILUFactorSymbolic_SeqAIJ_Inode_inplace(fact,A,isrow,iscol,info);CHKERRQ(ierr);
2066   PetscFunctionReturn(0);
2067 }
2068 
2069 #undef __FUNCT__
2070 #define __FUNCT__ "MatCholeskyFactorNumeric_SeqAIJ"
2071 PetscErrorCode MatCholeskyFactorNumeric_SeqAIJ(Mat B,Mat A,const MatFactorInfo *info)
2072 {
2073   Mat            C = B;
2074   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data;
2075   Mat_SeqSBAIJ   *b=(Mat_SeqSBAIJ*)C->data;
2076   IS             ip=b->row,iip = b->icol;
2077   PetscErrorCode ierr;
2078   const PetscInt *rip,*riip;
2079   PetscInt       i,j,mbs=A->rmap->n,*bi=b->i,*bj=b->j,*bdiag=b->diag,*bjtmp;
2080   PetscInt       *ai=a->i,*aj=a->j;
2081   PetscInt       k,jmin,jmax,*c2r,*il,col,nexti,ili,nz;
2082   MatScalar      *rtmp,*ba=b->a,*bval,*aa=a->a,dk,uikdi;
2083   PetscTruth     perm_identity;
2084 
2085   FactorShiftCtx sctx;
2086   PetscReal      rs;
2087   MatScalar      d,*v;
2088 
2089   PetscFunctionBegin;
2090   /* MatPivotSetUp(): initialize shift context sctx */
2091   ierr = PetscMemzero(&sctx,sizeof(FactorShiftCtx));CHKERRQ(ierr);
2092 
2093   if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE) { /* set sctx.shift_top=max{rs} */
2094     sctx.shift_top = info->zeropivot;
2095     for (i=0; i<mbs; i++) {
2096       /* calculate sum(|aij|)-RealPart(aii), amt of shift needed for this row */
2097       d  = (aa)[a->diag[i]];
2098       rs = -PetscAbsScalar(d) - PetscRealPart(d);
2099       v  = aa+ai[i];
2100       nz = ai[i+1] - ai[i];
2101       for (j=0; j<nz; j++)
2102 	rs += PetscAbsScalar(v[j]);
2103       if (rs>sctx.shift_top) sctx.shift_top = rs;
2104     }
2105     sctx.shift_top   *= 1.1;
2106     sctx.nshift_max   = 5;
2107     sctx.shift_lo     = 0.;
2108     sctx.shift_hi     = 1.;
2109   }
2110 
2111   ierr  = ISGetIndices(ip,&rip);CHKERRQ(ierr);
2112   ierr  = ISGetIndices(iip,&riip);CHKERRQ(ierr);
2113 
2114   /* allocate working arrays
2115      c2r: linked list, keep track of pivot rows for a given column. c2r[col]: head of the list for a given col
2116      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
2117   */
2118   ierr = PetscMalloc3(mbs,MatScalar,&rtmp,mbs,PetscInt,&il,mbs,PetscInt,&c2r);CHKERRQ(ierr);
2119 
2120   do {
2121     sctx.useshift = PETSC_FALSE;
2122 
2123     for (i=0; i<mbs; i++) c2r[i] = mbs;
2124     il[0] = 0;
2125 
2126     for (k = 0; k<mbs; k++){
2127       /* zero rtmp */
2128       nz = bi[k+1] - bi[k];
2129       bjtmp = bj + bi[k];
2130       for (j=0; j<nz; j++) rtmp[bjtmp[j]] = 0.0;
2131 
2132       /* load in initial unfactored row */
2133       bval = ba + bi[k];
2134       jmin = ai[rip[k]]; jmax = ai[rip[k]+1];
2135       for (j = jmin; j < jmax; j++){
2136         col = riip[aj[j]];
2137         if (col >= k){ /* only take upper triangular entry */
2138           rtmp[col] = aa[j];
2139           *bval++   = 0.0; /* for in-place factorization */
2140         }
2141       }
2142       /* shift the diagonal of the matrix: ZeropivotApply() */
2143       rtmp[k] += sctx.shift_amount;  /* shift the diagonal of the matrix */
2144 
2145       /* modify k-th row by adding in those rows i with U(i,k)!=0 */
2146       dk = rtmp[k];
2147       i  = c2r[k]; /* first row to be added to k_th row  */
2148 
2149       while (i < k){
2150         nexti = c2r[i]; /* next row to be added to k_th row */
2151 
2152         /* compute multiplier, update diag(k) and U(i,k) */
2153         ili   = il[i];  /* index of first nonzero element in U(i,k:bms-1) */
2154         uikdi = - ba[ili]*ba[bdiag[i]];  /* diagonal(k) */
2155         dk   += uikdi*ba[ili]; /* update diag[k] */
2156         ba[ili] = uikdi; /* -U(i,k) */
2157 
2158         /* add multiple of row i to k-th row */
2159         jmin = ili + 1; jmax = bi[i+1];
2160         if (jmin < jmax){
2161           for (j=jmin; j<jmax; j++) rtmp[bj[j]] += uikdi*ba[j];
2162           /* update il and c2r for row i */
2163           il[i] = jmin;
2164           j = bj[jmin]; c2r[i] = c2r[j]; c2r[j] = i;
2165         }
2166         i = nexti;
2167       }
2168 
2169       /* copy data into U(k,:) */
2170       rs   = 0.0;
2171       jmin = bi[k]; jmax = bi[k+1]-1;
2172       if (jmin < jmax) {
2173         for (j=jmin; j<jmax; j++){
2174           col = bj[j]; ba[j] = rtmp[col]; rs += PetscAbsScalar(ba[j]);
2175         }
2176         /* add the k-th row into il and c2r */
2177         il[k] = jmin;
2178         i = bj[jmin]; c2r[k] = c2r[i]; c2r[i] = k;
2179       }
2180 
2181       /* MatPivotCheck() */
2182       sctx.rs  = rs;
2183       sctx.pv  = dk;
2184       if (info->shifttype == MAT_SHIFT_NONZERO){
2185         ierr = MatPivotCheck_nz(info,sctx,k);CHKERRQ(ierr);
2186       } else if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE){
2187         ierr = MatPivotCheck_pd(info,sctx,k);CHKERRQ(ierr);
2188       } else if (info->shifttype == MAT_SHIFT_INBLOCKS){
2189         ierr = MatPivotCheck_inblocks(info,sctx,k);CHKERRQ(ierr);
2190       } else {
2191         ierr = MatPivotCheck_none(info,sctx,k);CHKERRQ(ierr);
2192       }
2193       dk = sctx.pv;
2194 
2195       ba[bdiag[k]] = 1.0/dk; /* U(k,k) */
2196     }
2197   } while (sctx.useshift);
2198 
2199   ierr = PetscFree3(rtmp,il,c2r);CHKERRQ(ierr);
2200   ierr = ISRestoreIndices(ip,&rip);CHKERRQ(ierr);
2201   ierr = ISRestoreIndices(iip,&riip);CHKERRQ(ierr);
2202 
2203   ierr = ISIdentity(ip,&perm_identity);CHKERRQ(ierr);
2204   if (perm_identity){
2205     B->ops->solve           = MatSolve_SeqSBAIJ_1_NaturalOrdering;
2206     B->ops->solvetranspose  = MatSolve_SeqSBAIJ_1_NaturalOrdering;
2207     B->ops->forwardsolve    = MatForwardSolve_SeqSBAIJ_1_NaturalOrdering;
2208     B->ops->backwardsolve   = MatBackwardSolve_SeqSBAIJ_1_NaturalOrdering;
2209   } else {
2210     B->ops->solve           = MatSolve_SeqSBAIJ_1;
2211     B->ops->solvetranspose  = MatSolve_SeqSBAIJ_1;
2212     B->ops->forwardsolve    = MatForwardSolve_SeqSBAIJ_1;
2213     B->ops->backwardsolve   = MatBackwardSolve_SeqSBAIJ_1;
2214   }
2215 
2216   C->assembled    = PETSC_TRUE;
2217   C->preallocated = PETSC_TRUE;
2218   ierr = PetscLogFlops(C->rmap->n);CHKERRQ(ierr);
2219 
2220   /* MatPivotView() */
2221   if (sctx.nshift){
2222     if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE) {
2223       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);
2224     } else if (info->shifttype == MAT_SHIFT_NONZERO) {
2225       ierr = PetscInfo2(A,"number of shift_nz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
2226     } else if (info->shifttype == MAT_SHIFT_INBLOCKS){
2227       ierr = PetscInfo2(A,"number of shift_inblocks applied %D, each shift_amount %G\n",sctx.nshift,info->shiftamount);CHKERRQ(ierr);
2228     }
2229   }
2230   PetscFunctionReturn(0);
2231 }
2232 
2233 #undef __FUNCT__
2234 #define __FUNCT__ "MatCholeskyFactorNumeric_SeqAIJ_inplace"
2235 PetscErrorCode MatCholeskyFactorNumeric_SeqAIJ_inplace(Mat B,Mat A,const MatFactorInfo *info)
2236 {
2237   Mat            C = B;
2238   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data;
2239   Mat_SeqSBAIJ   *b=(Mat_SeqSBAIJ*)C->data;
2240   IS             ip=b->row,iip = b->icol;
2241   PetscErrorCode ierr;
2242   const PetscInt *rip,*riip;
2243   PetscInt       i,j,mbs=A->rmap->n,*bi=b->i,*bj=b->j,*bcol,*bjtmp;
2244   PetscInt       *ai=a->i,*aj=a->j;
2245   PetscInt       k,jmin,jmax,*jl,*il,col,nexti,ili,nz;
2246   MatScalar      *rtmp,*ba=b->a,*bval,*aa=a->a,dk,uikdi;
2247   PetscReal      zeropivot,rs;
2248   ChShift_Ctx    sctx;
2249   PetscInt       newshift;
2250   PetscTruth     perm_identity;
2251 
2252   PetscFunctionBegin;
2253   zeropivot = info->zeropivot;
2254 
2255   ierr  = ISGetIndices(ip,&rip);CHKERRQ(ierr);
2256   ierr  = ISGetIndices(iip,&riip);CHKERRQ(ierr);
2257 
2258   /* initialization */
2259   ierr = PetscMalloc3(mbs,MatScalar,&rtmp,mbs,PetscInt,&il,mbs,PetscInt,&jl);CHKERRQ(ierr);
2260   sctx.shift_amount = 0;
2261   sctx.nshift       = 0;
2262   do {
2263     sctx.chshift = PETSC_FALSE;
2264     for (i=0; i<mbs; i++) jl[i] = mbs;
2265     il[0] = 0;
2266 
2267     for (k = 0; k<mbs; k++){
2268       /* zero rtmp */
2269       nz = bi[k+1] - bi[k];
2270       bjtmp = bj + bi[k];
2271       for (j=0; j<nz; j++) rtmp[bjtmp[j]] = 0.0;
2272 
2273       bval = ba + bi[k];
2274       /* initialize k-th row by the perm[k]-th row of A */
2275       jmin = ai[rip[k]]; jmax = ai[rip[k]+1];
2276       for (j = jmin; j < jmax; j++){
2277         col = riip[aj[j]];
2278         if (col >= k){ /* only take upper triangular entry */
2279           rtmp[col] = aa[j];
2280           *bval++  = 0.0; /* for in-place factorization */
2281         }
2282       }
2283       /* shift the diagonal of the matrix */
2284       if (sctx.nshift) rtmp[k] += sctx.shift_amount;
2285 
2286       /* modify k-th row by adding in those rows i with U(i,k)!=0 */
2287       dk = rtmp[k];
2288       i = jl[k]; /* first row to be added to k_th row  */
2289 
2290       while (i < k){
2291         nexti = jl[i]; /* next row to be added to k_th row */
2292 
2293         /* compute multiplier, update diag(k) and U(i,k) */
2294         ili = il[i];  /* index of first nonzero element in U(i,k:bms-1) */
2295         uikdi = - ba[ili]*ba[bi[i]];  /* diagonal(k) */
2296         dk += uikdi*ba[ili];
2297         ba[ili] = uikdi; /* -U(i,k) */
2298 
2299         /* add multiple of row i to k-th row */
2300         jmin = ili + 1; jmax = bi[i+1];
2301         if (jmin < jmax){
2302           for (j=jmin; j<jmax; j++) rtmp[bj[j]] += uikdi*ba[j];
2303           /* update il and jl for row i */
2304           il[i] = jmin;
2305           j = bj[jmin]; jl[i] = jl[j]; jl[j] = i;
2306         }
2307         i = nexti;
2308       }
2309 
2310       /* shift the diagonals when zero pivot is detected */
2311       /* compute rs=sum of abs(off-diagonal) */
2312       rs   = 0.0;
2313       jmin = bi[k]+1;
2314       nz   = bi[k+1] - jmin;
2315       bcol = bj + jmin;
2316       for (j=0; j<nz; j++) {
2317         rs += PetscAbsScalar(rtmp[bcol[j]]);
2318       }
2319 
2320       sctx.rs = rs;
2321       sctx.pv = dk;
2322       ierr = MatCholeskyCheckShift_inline(info,sctx,k,newshift);CHKERRQ(ierr);
2323 
2324       if (newshift == 1) {
2325         if (!sctx.shift_amount) {
2326           sctx.shift_amount = 1e-5;
2327         }
2328         break;
2329       }
2330 
2331       /* copy data into U(k,:) */
2332       ba[bi[k]] = 1.0/dk; /* U(k,k) */
2333       jmin = bi[k]+1; jmax = bi[k+1];
2334       if (jmin < jmax) {
2335         for (j=jmin; j<jmax; j++){
2336           col = bj[j]; ba[j] = rtmp[col];
2337         }
2338         /* add the k-th row into il and jl */
2339         il[k] = jmin;
2340         i = bj[jmin]; jl[k] = jl[i]; jl[i] = k;
2341       }
2342     }
2343   } while (sctx.chshift);
2344   ierr = PetscFree3(rtmp,il,jl);CHKERRQ(ierr);
2345   ierr = ISRestoreIndices(ip,&rip);CHKERRQ(ierr);
2346   ierr = ISRestoreIndices(iip,&riip);CHKERRQ(ierr);
2347 
2348   ierr = ISIdentity(ip,&perm_identity);CHKERRQ(ierr);
2349   if (perm_identity){
2350     B->ops->solve           = MatSolve_SeqSBAIJ_1_NaturalOrdering_inplace;
2351     B->ops->solvetranspose  = MatSolve_SeqSBAIJ_1_NaturalOrdering_inplace;
2352     B->ops->forwardsolve    = MatForwardSolve_SeqSBAIJ_1_NaturalOrdering_inplace;
2353     B->ops->backwardsolve   = MatBackwardSolve_SeqSBAIJ_1_NaturalOrdering_inplace;
2354   } else {
2355     B->ops->solve           = MatSolve_SeqSBAIJ_1_inplace;
2356     B->ops->solvetranspose  = MatSolve_SeqSBAIJ_1_inplace;
2357     B->ops->forwardsolve    = MatForwardSolve_SeqSBAIJ_1_inplace;
2358     B->ops->backwardsolve   = MatBackwardSolve_SeqSBAIJ_1_inplace;
2359   }
2360 
2361   C->assembled    = PETSC_TRUE;
2362   C->preallocated = PETSC_TRUE;
2363   ierr = PetscLogFlops(C->rmap->n);CHKERRQ(ierr);
2364   if (sctx.nshift){
2365     if (info->shifttype == MAT_SHIFT_NONZERO) {
2366       ierr = PetscInfo2(A,"number of shiftnz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
2367     } else if (info->shifttype == MAT_SHIFT_POSITIVE_DEFINITE) {
2368       ierr = PetscInfo2(A,"number of shiftpd tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
2369     }
2370   }
2371   PetscFunctionReturn(0);
2372 }
2373 
2374 /*
2375    icc() under revised new data structure.
2376    Factored arrays bj and ba are stored as
2377      U(0,:),...,U(i,:),U(n-1,:)
2378 
2379    ui=fact->i is an array of size n+1, in which
2380    ui+
2381      ui[i]:  points to 1st entry of U(i,:),i=0,...,n-1
2382      ui[n]:  points to U(n-1,n-1)+1
2383 
2384   udiag=fact->diag is an array of size n,in which
2385      udiag[i]: points to diagonal of U(i,:), i=0,...,n-1
2386 
2387    U(i,:) contains udiag[i] as its last entry, i.e.,
2388     U(i,:) = (u[i,i+1],...,u[i,n-1],diag[i])
2389 */
2390 
2391 #undef __FUNCT__
2392 #define __FUNCT__ "MatICCFactorSymbolic_SeqAIJ"
2393 PetscErrorCode MatICCFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
2394 {
2395   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data;
2396   Mat_SeqSBAIJ       *b;
2397   PetscErrorCode     ierr;
2398   PetscTruth         perm_identity,missing;
2399   PetscInt           reallocs=0,i,*ai=a->i,*aj=a->j,am=A->rmap->n,*ui,*udiag;
2400   const PetscInt     *rip,*riip;
2401   PetscInt           jmin,jmax,nzk,k,j,*jl,prow,*il,nextprow;
2402   PetscInt           nlnk,*lnk,*lnk_lvl=PETSC_NULL,d;
2403   PetscInt           ncols,ncols_upper,*cols,*ajtmp,*uj,**uj_ptr,**uj_lvl_ptr;
2404   PetscReal          fill=info->fill,levels=info->levels;
2405   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
2406   PetscFreeSpaceList free_space_lvl=PETSC_NULL,current_space_lvl=PETSC_NULL;
2407   PetscBT            lnkbt;
2408   IS                 iperm;
2409 
2410   PetscFunctionBegin;
2411   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);
2412   ierr = MatMissingDiagonal(A,&missing,&d);CHKERRQ(ierr);
2413   if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",d);
2414   ierr = ISIdentity(perm,&perm_identity);CHKERRQ(ierr);
2415   ierr = ISInvertPermutation(perm,PETSC_DECIDE,&iperm);CHKERRQ(ierr);
2416 
2417   ierr = PetscMalloc((am+1)*sizeof(PetscInt),&ui);CHKERRQ(ierr);
2418   ierr = PetscMalloc((am+1)*sizeof(PetscInt),&udiag);CHKERRQ(ierr);
2419   ui[0] = 0;
2420 
2421   /* ICC(0) without matrix ordering: simply rearrange column indices */
2422   if (!levels && perm_identity) {
2423     for (i=0; i<am; i++) {
2424       ncols    = ai[i+1] - a->diag[i];
2425       ui[i+1]  = ui[i] + ncols;
2426       udiag[i] = ui[i+1] - 1; /* points to the last entry of U(i,:) */
2427     }
2428     ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
2429     cols = uj;
2430     for (i=0; i<am; i++) {
2431       aj    = a->j + a->diag[i] + 1; /* 1st entry of U(i,:) without diagonal */
2432       ncols = ai[i+1] - a->diag[i] -1;
2433       for (j=0; j<ncols; j++) *cols++ = aj[j];
2434       *cols++ = i; /* diagoanl is located as the last entry of U(i,:) */
2435     }
2436   } else { /* case: levels>0 || (levels=0 && !perm_identity) */
2437     ierr = ISGetIndices(iperm,&riip);CHKERRQ(ierr);
2438     ierr = ISGetIndices(perm,&rip);CHKERRQ(ierr);
2439 
2440     /* initialization */
2441     ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&ajtmp);CHKERRQ(ierr);
2442 
2443     /* jl: linked list for storing indices of the pivot rows
2444        il: il[i] points to the 1st nonzero entry of U(i,k:am-1) */
2445     ierr = PetscMalloc4(am,PetscInt*,&uj_ptr,am,PetscInt*,&uj_lvl_ptr,am,PetscInt,&jl,am,PetscInt,&il);CHKERRQ(ierr);
2446     for (i=0; i<am; i++){
2447       jl[i] = am; il[i] = 0;
2448     }
2449 
2450     /* create and initialize a linked list for storing column indices of the active row k */
2451     nlnk = am + 1;
2452     ierr = PetscIncompleteLLCreate(am,am,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
2453 
2454     /* initial FreeSpace size is fill*(ai[am]+1) */
2455     ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space);CHKERRQ(ierr);
2456     current_space = free_space;
2457     ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space_lvl);CHKERRQ(ierr);
2458     current_space_lvl = free_space_lvl;
2459 
2460     for (k=0; k<am; k++){  /* for each active row k */
2461       /* initialize lnk by the column indices of row rip[k] of A */
2462       nzk   = 0;
2463       ncols = ai[rip[k]+1] - ai[rip[k]];
2464       if (!ncols) SETERRQ2(PETSC_ERR_MAT_CH_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",rip[k],k);
2465       ncols_upper = 0;
2466       for (j=0; j<ncols; j++){
2467         i = *(aj + ai[rip[k]] + j); /* unpermuted column index */
2468         if (riip[i] >= k){ /* only take upper triangular entry */
2469           ajtmp[ncols_upper] = i;
2470           ncols_upper++;
2471         }
2472       }
2473       ierr = PetscIncompleteLLInit(ncols_upper,ajtmp,am,riip,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
2474       nzk += nlnk;
2475 
2476       /* update lnk by computing fill-in for each pivot row to be merged in */
2477       prow = jl[k]; /* 1st pivot row */
2478 
2479       while (prow < k){
2480         nextprow = jl[prow];
2481 
2482         /* merge prow into k-th row */
2483         jmin = il[prow] + 1;  /* index of the 2nd nzero entry in U(prow,k:am-1) */
2484         jmax = ui[prow+1];
2485         ncols = jmax-jmin;
2486         i     = jmin - ui[prow];
2487         cols  = uj_ptr[prow] + i; /* points to the 2nd nzero entry in U(prow,k:am-1) */
2488         uj    = uj_lvl_ptr[prow] + i; /* levels of cols */
2489         j     = *(uj - 1);
2490         ierr = PetscICCLLAddSorted(ncols,cols,levels,uj,am,nlnk,lnk,lnk_lvl,lnkbt,j);CHKERRQ(ierr);
2491         nzk += nlnk;
2492 
2493         /* update il and jl for prow */
2494         if (jmin < jmax){
2495           il[prow] = jmin;
2496           j = *cols; jl[prow] = jl[j]; jl[j] = prow;
2497         }
2498         prow = nextprow;
2499       }
2500 
2501       /* if free space is not available, make more free space */
2502       if (current_space->local_remaining<nzk) {
2503         i  = am - k + 1; /* num of unfactored rows */
2504         i *= PetscMin(nzk, i-1); /* i*nzk, i*(i-1): estimated and max additional space needed */
2505         ierr = PetscFreeSpaceGet(i,&current_space);CHKERRQ(ierr);
2506         ierr = PetscFreeSpaceGet(i,&current_space_lvl);CHKERRQ(ierr);
2507         reallocs++;
2508       }
2509 
2510       /* copy data into free_space and free_space_lvl, then initialize lnk */
2511       if (nzk == 0) SETERRQ1(PETSC_ERR_ARG_WRONG,"Empty row %D in ICC matrix factor",k);
2512       ierr = PetscIncompleteLLClean(am,am,nzk,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);CHKERRQ(ierr);
2513 
2514       /* add the k-th row into il and jl */
2515       if (nzk > 1){
2516         i = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:am-1) */
2517         jl[k] = jl[i]; jl[i] = k;
2518         il[k] = ui[k] + 1;
2519       }
2520       uj_ptr[k]     = current_space->array;
2521       uj_lvl_ptr[k] = current_space_lvl->array;
2522 
2523       current_space->array           += nzk;
2524       current_space->local_used      += nzk;
2525       current_space->local_remaining -= nzk;
2526 
2527       current_space_lvl->array           += nzk;
2528       current_space_lvl->local_used      += nzk;
2529       current_space_lvl->local_remaining -= nzk;
2530 
2531       ui[k+1] = ui[k] + nzk;
2532     }
2533 
2534 #if defined(PETSC_USE_INFO)
2535     if (ai[am] != 0) {
2536       PetscReal af = (PetscReal)ui[am]/((PetscReal)ai[am]);
2537       ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,fill,af);CHKERRQ(ierr);
2538       ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
2539       ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G) for best performance.\n",af);CHKERRQ(ierr);
2540     } else {
2541       ierr = PetscInfo(A,"Empty matrix.\n");CHKERRQ(ierr);
2542     }
2543 #endif
2544 
2545     ierr = ISRestoreIndices(perm,&rip);CHKERRQ(ierr);
2546     ierr = ISRestoreIndices(iperm,&riip);CHKERRQ(ierr);
2547     ierr = PetscFree4(uj_ptr,uj_lvl_ptr,jl,il);CHKERRQ(ierr);
2548     ierr = PetscFree(ajtmp);CHKERRQ(ierr);
2549 
2550     /* destroy list of free space and other temporary array(s) */
2551     ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
2552     ierr = PetscFreeSpaceContiguous_Cholesky(&free_space,uj,am,ui,udiag);CHKERRQ(ierr); /* store matrix factor  */
2553     ierr = PetscIncompleteLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
2554     ierr = PetscFreeSpaceDestroy(free_space_lvl);CHKERRQ(ierr);
2555 
2556   } /* end of case: levels>0 || (levels=0 && !perm_identity) */
2557 
2558   /* put together the new matrix in MATSEQSBAIJ format */
2559   b    = (Mat_SeqSBAIJ*)(fact)->data;
2560   b->singlemalloc = PETSC_FALSE;
2561   ierr = PetscMalloc((ui[am]+1)*sizeof(MatScalar),&b->a);CHKERRQ(ierr);
2562   b->j    = uj;
2563   b->i    = ui;
2564   b->diag = udiag;
2565   b->free_diag = PETSC_TRUE;
2566   b->ilen = 0;
2567   b->imax = 0;
2568   b->row  = perm;
2569   b->col  = perm;
2570   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2571   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2572   b->icol = iperm;
2573   b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */
2574   ierr = PetscMalloc((am+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
2575   ierr = PetscLogObjectMemory(fact,ui[am]*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
2576   b->maxnz   = b->nz = ui[am];
2577   b->free_a  = PETSC_TRUE;
2578   b->free_ij = PETSC_TRUE;
2579 
2580   fact->info.factor_mallocs    = reallocs;
2581   fact->info.fill_ratio_given  = fill;
2582   if (ai[am] != 0) {
2583     fact->info.fill_ratio_needed = ((PetscReal)ui[am])/((PetscReal)ai[am]);
2584   } else {
2585     fact->info.fill_ratio_needed = 0.0;
2586   }
2587   fact->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqAIJ;
2588   PetscFunctionReturn(0);
2589 }
2590 
2591 #undef __FUNCT__
2592 #define __FUNCT__ "MatICCFactorSymbolic_SeqAIJ_inplace"
2593 PetscErrorCode MatICCFactorSymbolic_SeqAIJ_inplace(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
2594 {
2595   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data;
2596   Mat_SeqSBAIJ       *b;
2597   PetscErrorCode     ierr;
2598   PetscTruth         perm_identity,missing;
2599   PetscInt           reallocs=0,i,*ai=a->i,*aj=a->j,am=A->rmap->n,*ui,*udiag;
2600   const PetscInt     *rip,*riip;
2601   PetscInt           jmin,jmax,nzk,k,j,*jl,prow,*il,nextprow;
2602   PetscInt           nlnk,*lnk,*lnk_lvl=PETSC_NULL,d;
2603   PetscInt           ncols,ncols_upper,*cols,*ajtmp,*uj,**uj_ptr,**uj_lvl_ptr;
2604   PetscReal          fill=info->fill,levels=info->levels;
2605   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
2606   PetscFreeSpaceList free_space_lvl=PETSC_NULL,current_space_lvl=PETSC_NULL;
2607   PetscBT            lnkbt;
2608   IS                 iperm;
2609 
2610   PetscFunctionBegin;
2611   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);
2612   ierr = MatMissingDiagonal(A,&missing,&d);CHKERRQ(ierr);
2613   if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",d);
2614   ierr = ISIdentity(perm,&perm_identity);CHKERRQ(ierr);
2615   ierr = ISInvertPermutation(perm,PETSC_DECIDE,&iperm);CHKERRQ(ierr);
2616 
2617   ierr = PetscMalloc((am+1)*sizeof(PetscInt),&ui);CHKERRQ(ierr);
2618   ierr = PetscMalloc((am+1)*sizeof(PetscInt),&udiag);CHKERRQ(ierr);
2619   ui[0] = 0;
2620 
2621   /* ICC(0) without matrix ordering: simply copies fill pattern */
2622   if (!levels && perm_identity) {
2623 
2624     for (i=0; i<am; i++) {
2625       ui[i+1]  = ui[i] + ai[i+1] - a->diag[i];
2626       udiag[i] = ui[i];
2627     }
2628     ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
2629     cols = uj;
2630     for (i=0; i<am; i++) {
2631       aj    = a->j + a->diag[i];
2632       ncols = ui[i+1] - ui[i];
2633       for (j=0; j<ncols; j++) *cols++ = *aj++;
2634     }
2635   } else { /* case: levels>0 || (levels=0 && !perm_identity) */
2636     ierr = ISGetIndices(iperm,&riip);CHKERRQ(ierr);
2637     ierr = ISGetIndices(perm,&rip);CHKERRQ(ierr);
2638 
2639     /* initialization */
2640     ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&ajtmp);CHKERRQ(ierr);
2641 
2642     /* jl: linked list for storing indices of the pivot rows
2643        il: il[i] points to the 1st nonzero entry of U(i,k:am-1) */
2644     ierr = PetscMalloc4(am,PetscInt*,&uj_ptr,am,PetscInt*,&uj_lvl_ptr,am,PetscInt,&jl,am,PetscInt,&il);CHKERRQ(ierr);
2645     for (i=0; i<am; i++){
2646       jl[i] = am; il[i] = 0;
2647     }
2648 
2649     /* create and initialize a linked list for storing column indices of the active row k */
2650     nlnk = am + 1;
2651     ierr = PetscIncompleteLLCreate(am,am,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
2652 
2653     /* initial FreeSpace size is fill*(ai[am]+1) */
2654     ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space);CHKERRQ(ierr);
2655     current_space = free_space;
2656     ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space_lvl);CHKERRQ(ierr);
2657     current_space_lvl = free_space_lvl;
2658 
2659     for (k=0; k<am; k++){  /* for each active row k */
2660       /* initialize lnk by the column indices of row rip[k] of A */
2661       nzk   = 0;
2662       ncols = ai[rip[k]+1] - ai[rip[k]];
2663       if (!ncols) SETERRQ2(PETSC_ERR_MAT_CH_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",rip[k],k);
2664       ncols_upper = 0;
2665       for (j=0; j<ncols; j++){
2666         i = *(aj + ai[rip[k]] + j); /* unpermuted column index */
2667         if (riip[i] >= k){ /* only take upper triangular entry */
2668           ajtmp[ncols_upper] = i;
2669           ncols_upper++;
2670         }
2671       }
2672       ierr = PetscIncompleteLLInit(ncols_upper,ajtmp,am,riip,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
2673       nzk += nlnk;
2674 
2675       /* update lnk by computing fill-in for each pivot row to be merged in */
2676       prow = jl[k]; /* 1st pivot row */
2677 
2678       while (prow < k){
2679         nextprow = jl[prow];
2680 
2681         /* merge prow into k-th row */
2682         jmin = il[prow] + 1;  /* index of the 2nd nzero entry in U(prow,k:am-1) */
2683         jmax = ui[prow+1];
2684         ncols = jmax-jmin;
2685         i     = jmin - ui[prow];
2686         cols  = uj_ptr[prow] + i; /* points to the 2nd nzero entry in U(prow,k:am-1) */
2687         uj    = uj_lvl_ptr[prow] + i; /* levels of cols */
2688         j     = *(uj - 1);
2689         ierr = PetscICCLLAddSorted(ncols,cols,levels,uj,am,nlnk,lnk,lnk_lvl,lnkbt,j);CHKERRQ(ierr);
2690         nzk += nlnk;
2691 
2692         /* update il and jl for prow */
2693         if (jmin < jmax){
2694           il[prow] = jmin;
2695           j = *cols; jl[prow] = jl[j]; jl[j] = prow;
2696         }
2697         prow = nextprow;
2698       }
2699 
2700       /* if free space is not available, make more free space */
2701       if (current_space->local_remaining<nzk) {
2702         i = am - k + 1; /* num of unfactored rows */
2703         i *= PetscMin(nzk, (i-1)); /* i*nzk, i*(i-1): estimated and max additional space needed */
2704         ierr = PetscFreeSpaceGet(i,&current_space);CHKERRQ(ierr);
2705         ierr = PetscFreeSpaceGet(i,&current_space_lvl);CHKERRQ(ierr);
2706         reallocs++;
2707       }
2708 
2709       /* copy data into free_space and free_space_lvl, then initialize lnk */
2710       if (nzk == 0) SETERRQ1(PETSC_ERR_ARG_WRONG,"Empty row %D in ICC matrix factor",k);
2711       ierr = PetscIncompleteLLClean(am,am,nzk,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);CHKERRQ(ierr);
2712 
2713       /* add the k-th row into il and jl */
2714       if (nzk > 1){
2715         i = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:am-1) */
2716         jl[k] = jl[i]; jl[i] = k;
2717         il[k] = ui[k] + 1;
2718       }
2719       uj_ptr[k]     = current_space->array;
2720       uj_lvl_ptr[k] = current_space_lvl->array;
2721 
2722       current_space->array           += nzk;
2723       current_space->local_used      += nzk;
2724       current_space->local_remaining -= nzk;
2725 
2726       current_space_lvl->array           += nzk;
2727       current_space_lvl->local_used      += nzk;
2728       current_space_lvl->local_remaining -= nzk;
2729 
2730       ui[k+1] = ui[k] + nzk;
2731     }
2732 
2733 #if defined(PETSC_USE_INFO)
2734     if (ai[am] != 0) {
2735       PetscReal af = (PetscReal)ui[am]/((PetscReal)ai[am]);
2736       ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,fill,af);CHKERRQ(ierr);
2737       ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
2738       ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G) for best performance.\n",af);CHKERRQ(ierr);
2739     } else {
2740       ierr = PetscInfo(A,"Empty matrix.\n");CHKERRQ(ierr);
2741     }
2742 #endif
2743 
2744     ierr = ISRestoreIndices(perm,&rip);CHKERRQ(ierr);
2745     ierr = ISRestoreIndices(iperm,&riip);CHKERRQ(ierr);
2746     ierr = PetscFree4(uj_ptr,uj_lvl_ptr,jl,il);CHKERRQ(ierr);
2747     ierr = PetscFree(ajtmp);CHKERRQ(ierr);
2748 
2749     /* destroy list of free space and other temporary array(s) */
2750     ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
2751     ierr = PetscFreeSpaceContiguous(&free_space,uj);CHKERRQ(ierr);
2752     ierr = PetscIncompleteLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
2753     ierr = PetscFreeSpaceDestroy(free_space_lvl);CHKERRQ(ierr);
2754 
2755   } /* end of case: levels>0 || (levels=0 && !perm_identity) */
2756 
2757   /* put together the new matrix in MATSEQSBAIJ format */
2758 
2759   b    = (Mat_SeqSBAIJ*)fact->data;
2760   b->singlemalloc = PETSC_FALSE;
2761   ierr = PetscMalloc((ui[am]+1)*sizeof(MatScalar),&b->a);CHKERRQ(ierr);
2762   b->j    = uj;
2763   b->i    = ui;
2764   b->diag = udiag;
2765   b->free_diag = PETSC_TRUE;
2766   b->ilen = 0;
2767   b->imax = 0;
2768   b->row  = perm;
2769   b->col  = perm;
2770   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2771   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2772   b->icol = iperm;
2773   b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */
2774   ierr    = PetscMalloc((am+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
2775   ierr = PetscLogObjectMemory(fact,(ui[am]-am)*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
2776   b->maxnz   = b->nz = ui[am];
2777   b->free_a  = PETSC_TRUE;
2778   b->free_ij = PETSC_TRUE;
2779 
2780   fact->info.factor_mallocs    = reallocs;
2781   fact->info.fill_ratio_given  = fill;
2782   if (ai[am] != 0) {
2783     fact->info.fill_ratio_needed = ((PetscReal)ui[am])/((PetscReal)ai[am]);
2784   } else {
2785     fact->info.fill_ratio_needed = 0.0;
2786   }
2787   fact->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqAIJ_inplace;
2788   PetscFunctionReturn(0);
2789 }
2790 
2791 PetscErrorCode MatCholeskyFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
2792 {
2793   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data;
2794   Mat_SeqSBAIJ       *b;
2795   PetscErrorCode     ierr;
2796   PetscTruth         perm_identity;
2797   PetscReal          fill = info->fill;
2798   const PetscInt     *rip,*riip;
2799   PetscInt           i,am=A->rmap->n,*ai=a->i,*aj=a->j,reallocs=0,prow;
2800   PetscInt           *jl,jmin,jmax,nzk,*ui,k,j,*il,nextprow;
2801   PetscInt           nlnk,*lnk,ncols,ncols_upper,*cols,*uj,**ui_ptr,*uj_ptr,*udiag;
2802   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
2803   PetscBT            lnkbt;
2804   IS                 iperm;
2805 
2806   PetscFunctionBegin;
2807   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);
2808   /* check whether perm is the identity mapping */
2809   ierr = ISIdentity(perm,&perm_identity);CHKERRQ(ierr);
2810   ierr = ISInvertPermutation(perm,PETSC_DECIDE,&iperm);CHKERRQ(ierr);
2811   ierr = ISGetIndices(iperm,&riip);CHKERRQ(ierr);
2812   ierr = ISGetIndices(perm,&rip);CHKERRQ(ierr);
2813 
2814   /* initialization */
2815   ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&ui);CHKERRQ(ierr);
2816   ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&udiag);CHKERRQ(ierr);
2817   ui[0] = 0;
2818 
2819   /* jl: linked list for storing indices of the pivot rows
2820      il: il[i] points to the 1st nonzero entry of U(i,k:am-1) */
2821   ierr = PetscMalloc4(am,PetscInt*,&ui_ptr,am,PetscInt,&jl,am,PetscInt,&il,am,PetscInt,&cols);CHKERRQ(ierr);
2822   for (i=0; i<am; i++){
2823     jl[i] = am; il[i] = 0;
2824   }
2825 
2826   /* create and initialize a linked list for storing column indices of the active row k */
2827   nlnk = am + 1;
2828   ierr = PetscLLCreate(am,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2829 
2830   /* initial FreeSpace size is fill*(ai[am]+1) */
2831   ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space);CHKERRQ(ierr);
2832   current_space = free_space;
2833 
2834   for (k=0; k<am; k++){  /* for each active row k */
2835     /* initialize lnk by the column indices of row rip[k] of A */
2836     nzk   = 0;
2837     ncols = ai[rip[k]+1] - ai[rip[k]];
2838     if (!ncols) SETERRQ2(PETSC_ERR_MAT_CH_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",rip[k],k);
2839     ncols_upper = 0;
2840     for (j=0; j<ncols; j++){
2841       i = riip[*(aj + ai[rip[k]] + j)];
2842       if (i >= k){ /* only take upper triangular entry */
2843         cols[ncols_upper] = i;
2844         ncols_upper++;
2845       }
2846     }
2847     ierr = PetscLLAdd(ncols_upper,cols,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2848     nzk += nlnk;
2849 
2850     /* update lnk by computing fill-in for each pivot row to be merged in */
2851     prow = jl[k]; /* 1st pivot row */
2852 
2853     while (prow < k){
2854       nextprow = jl[prow];
2855       /* merge prow into k-th row */
2856       jmin = il[prow] + 1;  /* index of the 2nd nzero entry in U(prow,k:am-1) */
2857       jmax = ui[prow+1];
2858       ncols = jmax-jmin;
2859       uj_ptr = ui_ptr[prow] + jmin - ui[prow]; /* points to the 2nd nzero entry in U(prow,k:am-1) */
2860       ierr = PetscLLAddSorted(ncols,uj_ptr,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2861       nzk += nlnk;
2862 
2863       /* update il and jl for prow */
2864       if (jmin < jmax){
2865         il[prow] = jmin;
2866         j = *uj_ptr; jl[prow] = jl[j]; jl[j] = prow;
2867       }
2868       prow = nextprow;
2869     }
2870 
2871     /* if free space is not available, make more free space */
2872     if (current_space->local_remaining<nzk) {
2873       i  = am - k + 1; /* num of unfactored rows */
2874       i *= PetscMin(nzk,i-1); /* i*nzk, i*(i-1): estimated and max additional space needed */
2875       ierr = PetscFreeSpaceGet(i,&current_space);CHKERRQ(ierr);
2876       reallocs++;
2877     }
2878 
2879     /* copy data into free space, then initialize lnk */
2880     ierr = PetscLLClean(am,am,nzk,lnk,current_space->array,lnkbt);CHKERRQ(ierr);
2881 
2882     /* add the k-th row into il and jl */
2883     if (nzk > 1){
2884       i = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:am-1) */
2885       jl[k] = jl[i]; jl[i] = k;
2886       il[k] = ui[k] + 1;
2887     }
2888     ui_ptr[k] = current_space->array;
2889     current_space->array           += nzk;
2890     current_space->local_used      += nzk;
2891     current_space->local_remaining -= nzk;
2892 
2893     ui[k+1] = ui[k] + nzk;
2894   }
2895 
2896 #if defined(PETSC_USE_INFO)
2897   if (ai[am] != 0) {
2898     PetscReal af = (PetscReal)(ui[am])/((PetscReal)ai[am]);
2899     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,fill,af);CHKERRQ(ierr);
2900     ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
2901     ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G) for best performance.\n",af);CHKERRQ(ierr);
2902   } else {
2903      ierr = PetscInfo(A,"Empty matrix.\n");CHKERRQ(ierr);
2904   }
2905 #endif
2906 
2907   ierr = ISRestoreIndices(perm,&rip);CHKERRQ(ierr);
2908   ierr = ISRestoreIndices(iperm,&riip);CHKERRQ(ierr);
2909   ierr = PetscFree4(ui_ptr,jl,il,cols);CHKERRQ(ierr);
2910 
2911   /* destroy list of free space and other temporary array(s) */
2912   ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
2913   ierr = PetscFreeSpaceContiguous_Cholesky(&free_space,uj,am,ui,udiag);CHKERRQ(ierr); /* store matrix factor */
2914   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
2915 
2916   /* put together the new matrix in MATSEQSBAIJ format */
2917 
2918   b = (Mat_SeqSBAIJ*)fact->data;
2919   b->singlemalloc = PETSC_FALSE;
2920   b->free_a       = PETSC_TRUE;
2921   b->free_ij      = PETSC_TRUE;
2922   ierr = PetscMalloc((ui[am]+1)*sizeof(MatScalar),&b->a);CHKERRQ(ierr);
2923   b->j    = uj;
2924   b->i    = ui;
2925   b->diag = udiag;
2926   b->free_diag = PETSC_TRUE;
2927   b->ilen = 0;
2928   b->imax = 0;
2929   b->row  = perm;
2930   b->col  = perm;
2931   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2932   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2933   b->icol = iperm;
2934   b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */
2935   ierr    = PetscMalloc((am+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
2936   ierr    = PetscLogObjectMemory(fact,ui[am]*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
2937   b->maxnz = b->nz = ui[am];
2938 
2939   fact->info.factor_mallocs    = reallocs;
2940   fact->info.fill_ratio_given  = fill;
2941   if (ai[am] != 0) {
2942     fact->info.fill_ratio_needed = ((PetscReal)ui[am])/((PetscReal)ai[am]);
2943   } else {
2944     fact->info.fill_ratio_needed = 0.0;
2945   }
2946   fact->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqAIJ;
2947   PetscFunctionReturn(0);
2948 }
2949 
2950 #undef __FUNCT__
2951 #define __FUNCT__ "MatCholeskyFactorSymbolic_SeqAIJ_inplace"
2952 PetscErrorCode MatCholeskyFactorSymbolic_SeqAIJ_inplace(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
2953 {
2954   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data;
2955   Mat_SeqSBAIJ       *b;
2956   PetscErrorCode     ierr;
2957   PetscTruth         perm_identity;
2958   PetscReal          fill = info->fill;
2959   const PetscInt     *rip,*riip;
2960   PetscInt           i,am=A->rmap->n,*ai=a->i,*aj=a->j,reallocs=0,prow;
2961   PetscInt           *jl,jmin,jmax,nzk,*ui,k,j,*il,nextprow;
2962   PetscInt           nlnk,*lnk,ncols,ncols_upper,*cols,*uj,**ui_ptr,*uj_ptr;
2963   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
2964   PetscBT            lnkbt;
2965   IS                 iperm;
2966 
2967   PetscFunctionBegin;
2968   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);
2969   /* check whether perm is the identity mapping */
2970   ierr = ISIdentity(perm,&perm_identity);CHKERRQ(ierr);
2971   ierr = ISInvertPermutation(perm,PETSC_DECIDE,&iperm);CHKERRQ(ierr);
2972   ierr = ISGetIndices(iperm,&riip);CHKERRQ(ierr);
2973   ierr = ISGetIndices(perm,&rip);CHKERRQ(ierr);
2974 
2975   /* initialization */
2976   ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&ui);CHKERRQ(ierr);
2977   ui[0] = 0;
2978 
2979   /* jl: linked list for storing indices of the pivot rows
2980      il: il[i] points to the 1st nonzero entry of U(i,k:am-1) */
2981   ierr = PetscMalloc4(am,PetscInt*,&ui_ptr,am,PetscInt,&jl,am,PetscInt,&il,am,PetscInt,&cols);CHKERRQ(ierr);
2982   for (i=0; i<am; i++){
2983     jl[i] = am; il[i] = 0;
2984   }
2985 
2986   /* create and initialize a linked list for storing column indices of the active row k */
2987   nlnk = am + 1;
2988   ierr = PetscLLCreate(am,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2989 
2990   /* initial FreeSpace size is fill*(ai[am]+1) */
2991   ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space);CHKERRQ(ierr);
2992   current_space = free_space;
2993 
2994   for (k=0; k<am; k++){  /* for each active row k */
2995     /* initialize lnk by the column indices of row rip[k] of A */
2996     nzk   = 0;
2997     ncols = ai[rip[k]+1] - ai[rip[k]];
2998     if (!ncols) SETERRQ2(PETSC_ERR_MAT_CH_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",rip[k],k);
2999     ncols_upper = 0;
3000     for (j=0; j<ncols; j++){
3001       i = riip[*(aj + ai[rip[k]] + j)];
3002       if (i >= k){ /* only take upper triangular entry */
3003         cols[ncols_upper] = i;
3004         ncols_upper++;
3005       }
3006     }
3007     ierr = PetscLLAdd(ncols_upper,cols,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
3008     nzk += nlnk;
3009 
3010     /* update lnk by computing fill-in for each pivot row to be merged in */
3011     prow = jl[k]; /* 1st pivot row */
3012 
3013     while (prow < k){
3014       nextprow = jl[prow];
3015       /* merge prow into k-th row */
3016       jmin = il[prow] + 1;  /* index of the 2nd nzero entry in U(prow,k:am-1) */
3017       jmax = ui[prow+1];
3018       ncols = jmax-jmin;
3019       uj_ptr = ui_ptr[prow] + jmin - ui[prow]; /* points to the 2nd nzero entry in U(prow,k:am-1) */
3020       ierr = PetscLLAddSorted(ncols,uj_ptr,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
3021       nzk += nlnk;
3022 
3023       /* update il and jl for prow */
3024       if (jmin < jmax){
3025         il[prow] = jmin;
3026         j = *uj_ptr; jl[prow] = jl[j]; jl[j] = prow;
3027       }
3028       prow = nextprow;
3029     }
3030 
3031     /* if free space is not available, make more free space */
3032     if (current_space->local_remaining<nzk) {
3033       i = am - k + 1; /* num of unfactored rows */
3034       i = PetscMin(i*nzk, i*(i-1)); /* i*nzk, i*(i-1): estimated and max additional space needed */
3035       ierr = PetscFreeSpaceGet(i,&current_space);CHKERRQ(ierr);
3036       reallocs++;
3037     }
3038 
3039     /* copy data into free space, then initialize lnk */
3040     ierr = PetscLLClean(am,am,nzk,lnk,current_space->array,lnkbt);CHKERRQ(ierr);
3041 
3042     /* add the k-th row into il and jl */
3043     if (nzk-1 > 0){
3044       i = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:am-1) */
3045       jl[k] = jl[i]; jl[i] = k;
3046       il[k] = ui[k] + 1;
3047     }
3048     ui_ptr[k] = current_space->array;
3049     current_space->array           += nzk;
3050     current_space->local_used      += nzk;
3051     current_space->local_remaining -= nzk;
3052 
3053     ui[k+1] = ui[k] + nzk;
3054   }
3055 
3056 #if defined(PETSC_USE_INFO)
3057   if (ai[am] != 0) {
3058     PetscReal af = (PetscReal)(ui[am])/((PetscReal)ai[am]);
3059     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,fill,af);CHKERRQ(ierr);
3060     ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
3061     ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G) for best performance.\n",af);CHKERRQ(ierr);
3062   } else {
3063      ierr = PetscInfo(A,"Empty matrix.\n");CHKERRQ(ierr);
3064   }
3065 #endif
3066 
3067   ierr = ISRestoreIndices(perm,&rip);CHKERRQ(ierr);
3068   ierr = ISRestoreIndices(iperm,&riip);CHKERRQ(ierr);
3069   ierr = PetscFree4(ui_ptr,jl,il,cols);CHKERRQ(ierr);
3070 
3071   /* destroy list of free space and other temporary array(s) */
3072   ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
3073   ierr = PetscFreeSpaceContiguous(&free_space,uj);CHKERRQ(ierr);
3074   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
3075 
3076   /* put together the new matrix in MATSEQSBAIJ format */
3077 
3078   b = (Mat_SeqSBAIJ*)fact->data;
3079   b->singlemalloc = PETSC_FALSE;
3080   b->free_a       = PETSC_TRUE;
3081   b->free_ij      = PETSC_TRUE;
3082   ierr = PetscMalloc((ui[am]+1)*sizeof(MatScalar),&b->a);CHKERRQ(ierr);
3083   b->j    = uj;
3084   b->i    = ui;
3085   b->diag = 0;
3086   b->ilen = 0;
3087   b->imax = 0;
3088   b->row  = perm;
3089   b->col  = perm;
3090   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
3091   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
3092   b->icol = iperm;
3093   b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */
3094   ierr    = PetscMalloc((am+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
3095   ierr    = PetscLogObjectMemory(fact,(ui[am]-am)*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
3096   b->maxnz = b->nz = ui[am];
3097 
3098   fact->info.factor_mallocs    = reallocs;
3099   fact->info.fill_ratio_given  = fill;
3100   if (ai[am] != 0) {
3101     fact->info.fill_ratio_needed = ((PetscReal)ui[am])/((PetscReal)ai[am]);
3102   } else {
3103     fact->info.fill_ratio_needed = 0.0;
3104   }
3105   fact->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqAIJ_inplace;
3106   PetscFunctionReturn(0);
3107 }
3108 
3109 #undef __FUNCT__
3110 #define __FUNCT__ "MatSolve_SeqAIJ_NaturalOrdering"
3111 PetscErrorCode MatSolve_SeqAIJ_NaturalOrdering(Mat A,Vec bb,Vec xx)
3112 {
3113   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
3114   PetscErrorCode    ierr;
3115   PetscInt          n = A->rmap->n;
3116   const PetscInt    *ai = a->i,*aj = a->j,*adiag = a->diag,*vi;
3117   PetscScalar       *x,sum;
3118   const PetscScalar *b;
3119   const MatScalar   *aa = a->a,*v;
3120   PetscInt          i,nz;
3121 
3122   PetscFunctionBegin;
3123   if (!n) PetscFunctionReturn(0);
3124 
3125   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
3126   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
3127 
3128   /* forward solve the lower triangular */
3129   x[0] = b[0];
3130   v    = aa;
3131   vi   = aj;
3132   for (i=1; i<n; i++) {
3133     nz  = ai[i+1] - ai[i];
3134     sum = b[i];
3135     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
3136     v  += nz;
3137     vi += nz;
3138     x[i] = sum;
3139   }
3140 
3141   /* backward solve the upper triangular */
3142   for (i=n-1; i>=0; i--){
3143     v   = aa + adiag[i+1] + 1;
3144     vi  = aj + adiag[i+1] + 1;
3145     nz = adiag[i] - adiag[i+1]-1;
3146     sum = x[i];
3147     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
3148     x[i] = sum*v[nz]; /* x[i]=aa[adiag[i]]*sum; v++; */
3149   }
3150 
3151   ierr = PetscLogFlops(2.0*a->nz - A->cmap->n);CHKERRQ(ierr);
3152   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
3153   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
3154   PetscFunctionReturn(0);
3155 }
3156 
3157 #undef __FUNCT__
3158 #define __FUNCT__ "MatSolve_SeqAIJ"
3159 PetscErrorCode MatSolve_SeqAIJ(Mat A,Vec bb,Vec xx)
3160 {
3161   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
3162   IS                iscol = a->col,isrow = a->row;
3163   PetscErrorCode    ierr;
3164   PetscInt          i,n=A->rmap->n,*vi,*ai=a->i,*aj=a->j,*adiag = a->diag,nz;
3165   const PetscInt    *rout,*cout,*r,*c;
3166   PetscScalar       *x,*tmp,sum;
3167   const PetscScalar *b;
3168   const MatScalar   *aa = a->a,*v;
3169 
3170   PetscFunctionBegin;
3171   if (!n) PetscFunctionReturn(0);
3172 
3173   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
3174   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
3175   tmp  = a->solve_work;
3176 
3177   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
3178   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
3179 
3180   /* forward solve the lower triangular */
3181   tmp[0] = b[r[0]];
3182   v      = aa;
3183   vi     = aj;
3184   for (i=1; i<n; i++) {
3185     nz  = ai[i+1] - ai[i];
3186     sum = b[r[i]];
3187     PetscSparseDenseMinusDot(sum,tmp,v,vi,nz);
3188     tmp[i] = sum;
3189     v += nz; vi += nz;
3190   }
3191 
3192   /* backward solve the upper triangular */
3193   for (i=n-1; i>=0; i--){
3194     v   = aa + adiag[i+1]+1;
3195     vi  = aj + adiag[i+1]+1;
3196     nz  = adiag[i]-adiag[i+1]-1;
3197     sum = tmp[i];
3198     PetscSparseDenseMinusDot(sum,tmp,v,vi,nz);
3199     x[c[i]] = tmp[i] = sum*v[nz]; /* v[nz] = aa[adiag[i]] */
3200   }
3201 
3202   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
3203   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
3204   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
3205   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
3206   ierr = PetscLogFlops(2*a->nz - A->cmap->n);CHKERRQ(ierr);
3207   PetscFunctionReturn(0);
3208 }
3209 
3210 #undef __FUNCT__
3211 #define __FUNCT__ "MatILUDTFactor_SeqAIJ"
3212 /*
3213     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
3214 */
3215 PetscErrorCode MatILUDTFactor_SeqAIJ(Mat A,IS isrow,IS iscol,const MatFactorInfo *info,Mat *fact)
3216 {
3217   Mat                B = *fact;
3218   Mat_SeqAIJ         *a=(Mat_SeqAIJ*)A->data,*b;
3219   IS                 isicol;
3220   PetscErrorCode     ierr;
3221   const PetscInt     *r,*ic;
3222   PetscInt           i,n=A->rmap->n,*ai=a->i,*aj=a->j,*ajtmp,*adiag;
3223   PetscInt           *bi,*bj,*bdiag,*bdiag_rev;
3224   PetscInt           row,nzi,nzi_bl,nzi_bu,*im,nzi_al,nzi_au;
3225   PetscInt           nlnk,*lnk;
3226   PetscBT            lnkbt;
3227   PetscTruth         row_identity,icol_identity,both_identity;
3228   MatScalar          *aatmp,*pv,*batmp,*ba,*rtmp,*pc,multiplier,*vtmp,diag_tmp;
3229   const PetscInt     *ics;
3230   PetscInt           j,nz,*pj,*bjtmp,k,ncut,*jtmp;
3231   PetscReal          dt=info->dt,dtcol=info->dtcol,shift=info->shiftamount;
3232   PetscInt           dtcount=(PetscInt)info->dtcount,nnz_max;
3233   PetscTruth         missing;
3234 
3235   PetscFunctionBegin;
3236 
3237   if (dt      == PETSC_DEFAULT) dt      = 0.005;
3238   if (dtcol   == PETSC_DEFAULT) dtcol   = 0.01; /* XXX unused! */
3239   if (dtcount == PETSC_DEFAULT) dtcount = (PetscInt)(1.5*a->rmax);
3240 
3241   /* ------- symbolic factorization, can be reused ---------*/
3242   ierr = MatMissingDiagonal(A,&missing,&i);CHKERRQ(ierr);
3243   if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",i);
3244   adiag=a->diag;
3245 
3246   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
3247 
3248   /* bdiag is location of diagonal in factor */
3249   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);     /* becomes b->diag */
3250   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag_rev);CHKERRQ(ierr); /* temporary */
3251 
3252   /* allocate row pointers bi */
3253   ierr = PetscMalloc((2*n+2)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
3254 
3255   /* allocate bj and ba; max num of nonzero entries is (ai[n]+2*n*dtcount+2) */
3256   if (dtcount > n-1) dtcount = n-1; /* diagonal is excluded */
3257   nnz_max  = ai[n]+2*n*dtcount+2;
3258 
3259   ierr = PetscMalloc((nnz_max+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
3260   ierr = PetscMalloc((nnz_max+1)*sizeof(MatScalar),&ba);CHKERRQ(ierr);
3261 
3262   /* put together the new matrix */
3263   ierr = MatSeqAIJSetPreallocation_SeqAIJ(B,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
3264   ierr = PetscLogObjectParent(B,isicol);CHKERRQ(ierr);
3265   b    = (Mat_SeqAIJ*)B->data;
3266   b->free_a       = PETSC_TRUE;
3267   b->free_ij      = PETSC_TRUE;
3268   b->singlemalloc = PETSC_FALSE;
3269   b->a          = ba;
3270   b->j          = bj;
3271   b->i          = bi;
3272   b->diag       = bdiag;
3273   b->ilen       = 0;
3274   b->imax       = 0;
3275   b->row        = isrow;
3276   b->col        = iscol;
3277   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
3278   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
3279   b->icol       = isicol;
3280   ierr = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
3281 
3282   ierr = PetscLogObjectMemory(B,nnz_max*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
3283   b->maxnz = nnz_max;
3284 
3285   B->factor                = MAT_FACTOR_ILUDT;
3286   B->info.factor_mallocs   = 0;
3287   B->info.fill_ratio_given = ((PetscReal)nnz_max)/((PetscReal)ai[n]);
3288   CHKMEMQ;
3289   /* ------- end of symbolic factorization ---------*/
3290 
3291   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
3292   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
3293   ics  = ic;
3294 
3295   /* linked list for storing column indices of the active row */
3296   nlnk = n + 1;
3297   ierr = PetscLLCreate(n,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
3298 
3299   /* im: used by PetscLLAddSortedLU(); jtmp: working array for column indices of active row */
3300   ierr = PetscMalloc2(n,PetscInt,&im,n,PetscInt,&jtmp);CHKERRQ(ierr);
3301   /* rtmp, vtmp: working arrays for sparse and contiguous row entries of active row */
3302   ierr = PetscMalloc2(n,MatScalar,&rtmp,n,MatScalar,&vtmp);CHKERRQ(ierr);
3303   ierr = PetscMemzero(rtmp,n*sizeof(MatScalar));CHKERRQ(ierr);
3304 
3305   bi[0]    = 0;
3306   bdiag[0] = nnz_max-1; /* location of diag[0] in factor B */
3307   bdiag_rev[n] = bdiag[0];
3308   bi[2*n+1] = bdiag[0]+1; /* endof bj and ba array */
3309   for (i=0; i<n; i++) {
3310     /* copy initial fill into linked list */
3311     nzi = 0; /* nonzeros for active row i */
3312     nzi = ai[r[i]+1] - ai[r[i]];
3313     if (!nzi) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
3314     nzi_al = adiag[r[i]] - ai[r[i]];
3315     nzi_au = ai[r[i]+1] - adiag[r[i]] -1;
3316     ajtmp = aj + ai[r[i]];
3317     ierr = PetscLLAddPerm(nzi,ajtmp,ic,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
3318 
3319     /* load in initial (unfactored row) */
3320     aatmp = a->a + ai[r[i]];
3321     for (j=0; j<nzi; j++) {
3322       rtmp[ics[*ajtmp++]] = *aatmp++;
3323     }
3324 
3325     /* add pivot rows into linked list */
3326     row = lnk[n];
3327     while (row < i ) {
3328       nzi_bl = bi[row+1] - bi[row] + 1;
3329       bjtmp = bj + bdiag[row+1]+1; /* points to 1st column next to the diagonal in U */
3330       ierr  = PetscLLAddSortedLU(bjtmp,row,nlnk,lnk,lnkbt,i,nzi_bl,im);CHKERRQ(ierr);
3331       nzi  += nlnk;
3332       row   = lnk[row];
3333     }
3334 
3335     /* copy data from lnk into jtmp, then initialize lnk */
3336     ierr = PetscLLClean(n,n,nzi,lnk,jtmp,lnkbt);CHKERRQ(ierr);
3337 
3338     /* numerical factorization */
3339     bjtmp = jtmp;
3340     row   = *bjtmp++; /* 1st pivot row */
3341     while  ( row < i ) {
3342       pc         = rtmp + row;
3343       pv         = ba + bdiag[row]; /* 1./(diag of the pivot row) */
3344       multiplier = (*pc) * (*pv);
3345       *pc        = multiplier;
3346       if (PetscAbsScalar(*pc) > dt){ /* apply tolerance dropping rule */
3347         pj         = bj + bdiag[row+1] + 1; /* point to 1st entry of U(row,:) */
3348         pv         = ba + bdiag[row+1] + 1;
3349         /* if (multiplier < -1.0 or multiplier >1.0) printf("row/prow %d, %d, multiplier %g\n",i,row,multiplier); */
3350         nz         = bdiag[row] - bdiag[row+1] - 1; /* num of entries in U(row,:), excluding diagonal */
3351         for (j=0; j<nz; j++) rtmp[*pj++] -= multiplier * (*pv++);
3352         ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr);
3353       }
3354       row = *bjtmp++;
3355     }
3356 
3357     /* copy sparse rtmp into contiguous vtmp; separate L and U part */
3358     diag_tmp = rtmp[i];  /* save diagonal value - may not needed?? */
3359     nzi_bl = 0; j = 0;
3360     while (jtmp[j] < i){ /* Note: jtmp is sorted */
3361       vtmp[j] = rtmp[jtmp[j]]; rtmp[jtmp[j]]=0.0;
3362       nzi_bl++; j++;
3363     }
3364     nzi_bu = nzi - nzi_bl -1;
3365     while (j < nzi){
3366       vtmp[j] = rtmp[jtmp[j]]; rtmp[jtmp[j]]=0.0;
3367       j++;
3368     }
3369 
3370     bjtmp = bj + bi[i];
3371     batmp = ba + bi[i];
3372     /* apply level dropping rule to L part */
3373     ncut = nzi_al + dtcount;
3374     if (ncut < nzi_bl){
3375       ierr = PetscSortSplit(ncut,nzi_bl,vtmp,jtmp);CHKERRQ(ierr);
3376       ierr = PetscSortIntWithScalarArray(ncut,jtmp,vtmp);CHKERRQ(ierr);
3377     } else {
3378       ncut = nzi_bl;
3379     }
3380     for (j=0; j<ncut; j++){
3381       bjtmp[j] = jtmp[j];
3382       batmp[j] = vtmp[j];
3383       /* printf(" (%d,%g),",bjtmp[j],batmp[j]); */
3384     }
3385     bi[i+1] = bi[i] + ncut;
3386     nzi = ncut + 1;
3387 
3388     /* apply level dropping rule to U part */
3389     ncut = nzi_au + dtcount;
3390     if (ncut < nzi_bu){
3391       ierr = PetscSortSplit(ncut,nzi_bu,vtmp+nzi_bl+1,jtmp+nzi_bl+1);CHKERRQ(ierr);
3392       ierr = PetscSortIntWithScalarArray(ncut,jtmp+nzi_bl+1,vtmp+nzi_bl+1);CHKERRQ(ierr);
3393     } else {
3394       ncut = nzi_bu;
3395     }
3396     nzi += ncut;
3397 
3398     /* mark bdiagonal */
3399     bdiag[i+1]       = bdiag[i] - (ncut + 1);
3400     bdiag_rev[n-i-1] = bdiag[i+1];
3401     bi[2*n - i]      = bi[2*n - i +1] - (ncut + 1);
3402     bjtmp = bj + bdiag[i];
3403     batmp = ba + bdiag[i];
3404     *bjtmp = i;
3405     *batmp = diag_tmp; /* rtmp[i]; */
3406     if (*batmp == 0.0) {
3407       *batmp = dt+shift;
3408       /* printf(" row %d add shift %g\n",i,shift); */
3409     }
3410     *batmp = 1.0/(*batmp); /* invert diagonal entries for simplier triangular solves */
3411     /* printf(" (%d,%g),",*bjtmp,*batmp); */
3412 
3413     bjtmp = bj + bdiag[i+1]+1;
3414     batmp = ba + bdiag[i+1]+1;
3415     for (k=0; k<ncut; k++){
3416       bjtmp[k] = jtmp[nzi_bl+1+k];
3417       batmp[k] = vtmp[nzi_bl+1+k];
3418       /* printf(" (%d,%g),",bjtmp[k],batmp[k]); */
3419     }
3420     /* printf("\n"); */
3421 
3422     im[i]   = nzi; /* used by PetscLLAddSortedLU() */
3423     /*
3424     printf("row %d: bi %d, bdiag %d\n",i,bi[i],bdiag[i]);
3425     printf(" ----------------------------\n");
3426     */
3427   } /* for (i=0; i<n; i++) */
3428   /* printf("end of L %d, beginning of U %d\n",bi[n],bdiag[n]); */
3429   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]);
3430 
3431   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
3432   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
3433 
3434   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
3435   ierr = PetscFree2(im,jtmp);CHKERRQ(ierr);
3436   ierr = PetscFree2(rtmp,vtmp);CHKERRQ(ierr);
3437   ierr = PetscFree(bdiag_rev);CHKERRQ(ierr);
3438 
3439   ierr = PetscLogFlops(B->cmap->n);CHKERRQ(ierr);
3440   b->maxnz = b->nz = bi[n] + bdiag[0] - bdiag[n];
3441 
3442   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
3443   ierr = ISIdentity(isicol,&icol_identity);CHKERRQ(ierr);
3444   both_identity = (PetscTruth) (row_identity && icol_identity);
3445   if (row_identity && icol_identity) {
3446     B->ops->solve = MatSolve_SeqAIJ_NaturalOrdering;
3447   } else {
3448     B->ops->solve = MatSolve_SeqAIJ;
3449   }
3450 
3451   B->ops->solveadd          = 0;
3452   B->ops->solvetranspose    = 0;
3453   B->ops->solvetransposeadd = 0;
3454   B->ops->matsolve          = 0;
3455   B->assembled              = PETSC_TRUE;
3456   B->preallocated           = PETSC_TRUE;
3457   PetscFunctionReturn(0);
3458 }
3459 
3460 /* a wraper of MatILUDTFactor_SeqAIJ() */
3461 #undef __FUNCT__
3462 #define __FUNCT__ "MatILUDTFactorSymbolic_SeqAIJ"
3463 /*
3464     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
3465 */
3466 
3467 PetscErrorCode PETSCMAT_DLLEXPORT MatILUDTFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS row,IS col,const MatFactorInfo *info)
3468 {
3469   PetscErrorCode     ierr;
3470 
3471   PetscFunctionBegin;
3472   ierr = MatILUDTFactor_SeqAIJ(A,row,col,info,&fact);CHKERRQ(ierr);
3473   PetscFunctionReturn(0);
3474 }
3475 
3476 /*
3477    same as MatLUFactorNumeric_SeqAIJ(), except using contiguous array matrix factors
3478    - intend to replace existing MatLUFactorNumeric_SeqAIJ()
3479 */
3480 #undef __FUNCT__
3481 #define __FUNCT__ "MatILUDTFactorNumeric_SeqAIJ"
3482 /*
3483     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
3484 */
3485 
3486 PetscErrorCode PETSCMAT_DLLEXPORT MatILUDTFactorNumeric_SeqAIJ(Mat fact,Mat A,const MatFactorInfo *info)
3487 {
3488   Mat            C=fact;
3489   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ *)C->data;
3490   IS             isrow = b->row,isicol = b->icol;
3491   PetscErrorCode ierr;
3492   const PetscInt *r,*ic,*ics;
3493   PetscInt       i,j,k,n=A->rmap->n,*ai=a->i,*aj=a->j,*bi=b->i,*bj=b->j;
3494   PetscInt       *ajtmp,*bjtmp,nz,nzl,nzu,row,*bdiag = b->diag,*pj;
3495   MatScalar      *rtmp,*pc,multiplier,*v,*pv,*aa=a->a;
3496   PetscReal      dt=info->dt,shift=info->shiftamount;
3497   PetscTruth     row_identity, col_identity;
3498 
3499   PetscFunctionBegin;
3500   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
3501   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
3502   ierr = PetscMalloc((n+1)*sizeof(MatScalar),&rtmp);CHKERRQ(ierr);
3503   ics  = ic;
3504 
3505   for (i=0; i<n; i++){
3506     /* initialize rtmp array */
3507     nzl   = bi[i+1] - bi[i];       /* num of nozeros in L(i,:) */
3508     bjtmp = bj + bi[i];
3509     for  (j=0; j<nzl; j++) rtmp[*bjtmp++] = 0.0;
3510     rtmp[i] = 0.0;
3511     nzu   = bdiag[i] - bdiag[i+1]; /* num of nozeros in U(i,:) */
3512     bjtmp = bj + bdiag[i+1] + 1;
3513     for  (j=0; j<nzu; j++) rtmp[*bjtmp++] = 0.0;
3514 
3515     /* load in initial unfactored row of A */
3516     /* printf("row %d\n",i); */
3517     nz    = ai[r[i]+1] - ai[r[i]];
3518     ajtmp = aj + ai[r[i]];
3519     v     = aa + ai[r[i]];
3520     for (j=0; j<nz; j++) {
3521       rtmp[ics[*ajtmp++]] = v[j];
3522       /* printf(" (%d,%g),",ics[ajtmp[j]],rtmp[ics[ajtmp[j]]]); */
3523     }
3524     /* printf("\n"); */
3525 
3526     /* numerical factorization */
3527     bjtmp = bj + bi[i]; /* point to 1st entry of L(i,:) */
3528     nzl   = bi[i+1] - bi[i]; /* num of entries in L(i,:) */
3529     k = 0;
3530     while (k < nzl){
3531       row   = *bjtmp++;
3532       /* printf("  prow %d\n",row); */
3533       pc         = rtmp + row;
3534       pv         = b->a + bdiag[row]; /* 1./(diag of the pivot row) */
3535       multiplier = (*pc) * (*pv);
3536       *pc        = multiplier;
3537       if (PetscAbsScalar(multiplier) > dt){
3538         pj         = bj + bdiag[row+1] + 1; /* point to 1st entry of U(row,:) */
3539         pv         = b->a + bdiag[row+1] + 1;
3540         nz         = bdiag[row] - bdiag[row+1] - 1; /* num of entries in U(row,:), excluding diagonal */
3541         for (j=0; j<nz; j++) rtmp[*pj++] -= multiplier * (*pv++);
3542         /* ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr); */
3543       }
3544       k++;
3545     }
3546 
3547     /* finished row so stick it into b->a */
3548     /* L-part */
3549     pv = b->a + bi[i] ;
3550     pj = bj + bi[i] ;
3551     nzl = bi[i+1] - bi[i];
3552     for (j=0; j<nzl; j++) {
3553       pv[j] = rtmp[pj[j]];
3554       /* printf(" (%d,%g),",pj[j],pv[j]); */
3555     }
3556 
3557     /* diagonal: invert diagonal entries for simplier triangular solves */
3558     if (rtmp[i] == 0.0) rtmp[i] = dt+shift;
3559     b->a[bdiag[i]] = 1.0/rtmp[i];
3560     /* printf(" (%d,%g),",i,b->a[bdiag[i]]); */
3561 
3562     /* U-part */
3563     pv = b->a + bdiag[i+1] + 1;
3564     pj = bj + bdiag[i+1] + 1;
3565     nzu = bdiag[i] - bdiag[i+1] - 1;
3566     for (j=0; j<nzu; j++) {
3567       pv[j] = rtmp[pj[j]];
3568       /* printf(" (%d,%g),",pj[j],pv[j]); */
3569     }
3570     /* printf("\n"); */
3571   }
3572 
3573   ierr = PetscFree(rtmp);CHKERRQ(ierr);
3574   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
3575   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
3576 
3577   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
3578   ierr = ISIdentity(isicol,&col_identity);CHKERRQ(ierr);
3579   if (row_identity && col_identity) {
3580     C->ops->solve   = MatSolve_SeqAIJ_NaturalOrdering;
3581   } else {
3582     C->ops->solve   = MatSolve_SeqAIJ;
3583   }
3584   C->ops->solveadd           = 0;
3585   C->ops->solvetranspose     = 0;
3586   C->ops->solvetransposeadd  = 0;
3587   C->ops->matsolve           = 0;
3588   C->assembled    = PETSC_TRUE;
3589   C->preallocated = PETSC_TRUE;
3590   ierr = PetscLogFlops(C->cmap->n);CHKERRQ(ierr);
3591   PetscFunctionReturn(0);
3592 }
3593