xref: /petsc/src/mat/impls/aij/seq/aijfact.c (revision 6bd79f970d52324ea4a48ec7fbdd5fbabbcf960c)
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)->factortype = 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)->factortype            = 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   if (a->inode.size) {
274     (B)->ops->lufactornumeric  = MatLUFactorNumeric_SeqAIJ_Inode_inplace;
275   }
276   PetscFunctionReturn(0);
277 }
278 
279 #undef __FUNCT__
280 #define __FUNCT__ "MatLUFactorSymbolic_SeqAIJ"
281 PetscErrorCode MatLUFactorSymbolic_SeqAIJ(Mat B,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
282 {
283   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
284   IS                 isicol;
285   PetscErrorCode     ierr;
286   const PetscInt     *r,*ic;
287   PetscInt           i,n=A->rmap->n,*ai=a->i,*aj=a->j;
288   PetscInt           *bi,*bj,*ajtmp;
289   PetscInt           *bdiag,row,nnz,nzi,reallocs=0,nzbd,*im;
290   PetscReal          f;
291   PetscInt           nlnk,*lnk,k,**bi_ptr;
292   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
293   PetscBT            lnkbt;
294   PetscTruth         olddatastruct=PETSC_FALSE;
295 
296   PetscFunctionBegin;
297   /* Uncomment the oldatastruct part only while testing new data structure for MatSolve() */
298   ierr = PetscOptionsGetTruth(PETSC_NULL,"-lu_old",&olddatastruct,PETSC_NULL);CHKERRQ(ierr);
299   if(olddatastruct){
300     ierr = MatLUFactorSymbolic_SeqAIJ_inplace(B,A,isrow,iscol,info);CHKERRQ(ierr);
301     PetscFunctionReturn(0);
302   }
303 
304 
305   if (A->rmap->N != A->cmap->N) SETERRQ(PETSC_ERR_ARG_WRONG,"matrix must be square");
306   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
307   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
308   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
309 
310   /* get new row and diagonal pointers, must be allocated separately because they will be given to the Mat_SeqAIJ and freed separately */
311   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
312   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);
313   bi[0] = bdiag[0] = 0;
314 
315   /* linked list for storing column indices of the active row */
316   nlnk = n + 1;
317   ierr = PetscLLCreate(n,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
318 
319   ierr = PetscMalloc2(n+1,PetscInt**,&bi_ptr,n+1,PetscInt,&im);CHKERRQ(ierr);
320 
321   /* initial FreeSpace size is f*(ai[n]+1) */
322   f = info->fill;
323   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space);CHKERRQ(ierr);
324   current_space = free_space;
325 
326   for (i=0; i<n; i++) {
327     /* copy previous fill into linked list */
328     nzi = 0;
329     nnz = ai[r[i]+1] - ai[r[i]];
330     if (!nnz) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
331     ajtmp = aj + ai[r[i]];
332     ierr = PetscLLAddPerm(nnz,ajtmp,ic,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
333     nzi += nlnk;
334 
335     /* add pivot rows into linked list */
336     row = lnk[n];
337     while (row < i){
338       nzbd  = bdiag[row] + 1; /* num of entries in the row with column index <= row */
339       ajtmp = bi_ptr[row] + nzbd; /* points to the entry next to the diagonal */
340       ierr  = PetscLLAddSortedLU(ajtmp,row,nlnk,lnk,lnkbt,i,nzbd,im);CHKERRQ(ierr);
341       nzi  += nlnk;
342       row   = lnk[row];
343     }
344     bi[i+1] = bi[i] + nzi;
345     im[i]   = nzi;
346 
347     /* mark bdiag */
348     nzbd = 0;
349     nnz  = nzi;
350     k    = lnk[n];
351     while (nnz-- && k < i){
352       nzbd++;
353       k = lnk[k];
354     }
355     bdiag[i] = nzbd; /* note: bdiag[i] = nnzL as input for PetscFreeSpaceContiguous_LU() */
356 
357     /* if free space is not available, make more free space */
358     if (current_space->local_remaining<nzi) {
359       nnz = 2*(n - i)*nzi; /* estimated and max additional space needed */
360       ierr = PetscFreeSpaceGet(nnz,&current_space);CHKERRQ(ierr);
361       reallocs++;
362     }
363 
364     /* copy data into free space, then initialize lnk */
365     ierr = PetscLLClean(n,n,nzi,lnk,current_space->array,lnkbt);CHKERRQ(ierr);
366     bi_ptr[i] = current_space->array;
367     current_space->array           += nzi;
368     current_space->local_used      += nzi;
369     current_space->local_remaining -= nzi;
370   }
371 #if defined(PETSC_USE_INFO)
372   if (ai[n] != 0) {
373     PetscReal af = ((PetscReal)bi[n])/((PetscReal)ai[n]);
374     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,f,af);CHKERRQ(ierr);
375     ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
376     ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G);\n",af);CHKERRQ(ierr);
377     ierr = PetscInfo(A,"for best performance.\n");CHKERRQ(ierr);
378   } else {
379     ierr = PetscInfo(A,"Empty matrix\n");CHKERRQ(ierr);
380   }
381 #endif
382 
383   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
384   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
385 
386   /* destroy list of free space and other temporary array(s) */
387   ierr = PetscMalloc((bi[n]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
388   ierr = PetscFreeSpaceContiguous_LU(&free_space,bj,n,bi,bdiag);CHKERRQ(ierr);
389   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
390   ierr = PetscFree2(bi_ptr,im);CHKERRQ(ierr);
391 
392   /* put together the new matrix */
393   ierr = MatSeqAIJSetPreallocation_SeqAIJ(B,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
394   ierr = PetscLogObjectParent(B,isicol);CHKERRQ(ierr);
395   b    = (Mat_SeqAIJ*)(B)->data;
396   b->free_a       = PETSC_TRUE;
397   b->free_ij      = PETSC_TRUE;
398   b->singlemalloc = PETSC_FALSE;
399   ierr = PetscMalloc((bdiag[0]+1)*sizeof(PetscScalar),&b->a);CHKERRQ(ierr);
400   b->j          = bj;
401   b->i          = bi;
402   b->diag       = bdiag;
403   b->ilen       = 0;
404   b->imax       = 0;
405   b->row        = isrow;
406   b->col        = iscol;
407   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
408   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
409   b->icol       = isicol;
410   ierr          = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
411 
412   /* In b structure:  Free imax, ilen, old a, old j.  Allocate solve_work, new a, new j */
413   ierr = PetscLogObjectMemory(B,(bdiag[0]+1)*(sizeof(PetscInt)+sizeof(PetscScalar)));CHKERRQ(ierr);
414   b->maxnz = b->nz = bdiag[0]+1;
415   B->factortype            = MAT_FACTOR_LU;
416   B->info.factor_mallocs   = reallocs;
417   B->info.fill_ratio_given = f;
418 
419   if (ai[n]) {
420     B->info.fill_ratio_needed = ((PetscReal)(bdiag[0]+1))/((PetscReal)ai[n]);
421   } else {
422     B->info.fill_ratio_needed = 0.0;
423   }
424   B->ops->lufactornumeric = MatLUFactorNumeric_SeqAIJ;
425   if (a->inode.size) {
426     B->ops->lufactornumeric = MatLUFactorNumeric_SeqAIJ_Inode;
427   }
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   FactorShiftCtx   sctx;
471   const PetscInt   *ddiag;
472   PetscReal        rs;
473   MatScalar        d;
474   PetscInt         newshift = 0;
475 
476   PetscFunctionBegin;
477   /* MatPivotSetUp(): initialize shift context sctx */
478   ierr = PetscMemzero(&sctx,sizeof(FactorShiftCtx));CHKERRQ(ierr);
479 
480   if (info->shifttype == (PetscReal) 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       sctx.rs  = rs;
566       sctx.pv  = rtmp[i];
567       ierr = MatPivotCheck(info,&sctx,i,&newshift);CHKERRQ(ierr);
568       if(newshift == 1) break;
569       rtmp[i] = sctx.pv;
570 
571       /* Mark diagonal and invert diagonal for simplier triangular solves */
572       pv  = b->a + bdiag[i];
573       *pv = 1.0/rtmp[i];
574 
575     } /* endof for (i=0; i<n; i++){ */
576 
577     /* MatPivotRefine() */
578     if (info->shifttype == (PetscReal)MAT_SHIFT_POSITIVE_DEFINITE && !sctx.useshift && sctx.shift_fraction>0 && sctx.nshift<sctx.nshift_max){
579       /*
580        * if no shift in this attempt & shifting & started shifting & can refine,
581        * then try lower shift
582        */
583       sctx.shift_hi       = sctx.shift_fraction;
584       sctx.shift_fraction = (sctx.shift_hi+sctx.shift_lo)/2.;
585       sctx.shift_amount   = sctx.shift_fraction * sctx.shift_top;
586       sctx.useshift        = PETSC_TRUE;
587       sctx.nshift++;
588     }
589   } while (sctx.useshift);
590 
591   ierr = PetscFree(rtmp);CHKERRQ(ierr);
592   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
593   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
594 
595   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
596   ierr = ISIdentity(isicol,&col_identity);CHKERRQ(ierr);
597   if (row_identity && col_identity) {
598     C->ops->solve = MatSolve_SeqAIJ_NaturalOrdering;
599   } else {
600     C->ops->solve = MatSolve_SeqAIJ;
601   }
602   C->ops->solveadd           = MatSolveAdd_SeqAIJ;
603   C->ops->solvetranspose     = MatSolveTranspose_SeqAIJ;
604   C->ops->solvetransposeadd  = MatSolveTransposeAdd_SeqAIJ;
605   C->ops->matsolve           = MatMatSolve_SeqAIJ;
606   C->assembled    = PETSC_TRUE;
607   C->preallocated = PETSC_TRUE;
608   ierr = PetscLogFlops(C->cmap->n);CHKERRQ(ierr);
609 
610   /* MatShiftView(A,info,&sctx) */
611   if (sctx.nshift){
612     if (info->shifttype == (PetscReal)MAT_SHIFT_POSITIVE_DEFINITE) {
613       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);
614     } else if (info->shifttype == (PetscReal)MAT_SHIFT_NONZERO) {
615       ierr = PetscInfo2(A,"number of shift_nz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
616     } else if (info->shifttype == (PetscReal)MAT_SHIFT_INBLOCKS){
617       ierr = PetscInfo2(A,"number of shift_inblocks applied %D, each shift_amount %G\n",sctx.nshift,info->shiftamount);CHKERRQ(ierr);
618     }
619   }
620   ierr = Mat_CheckInode_FactorLU(C,PETSC_FALSE);CHKERRQ(ierr);
621   PetscFunctionReturn(0);
622 }
623 
624 #undef __FUNCT__
625 #define __FUNCT__ "MatLUFactorNumeric_SeqAIJ_inplace"
626 PetscErrorCode MatLUFactorNumeric_SeqAIJ_inplace(Mat B,Mat A,const MatFactorInfo *info)
627 {
628   Mat             C=B;
629   Mat_SeqAIJ      *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ *)C->data;
630   IS              isrow = b->row,isicol = b->icol;
631   PetscErrorCode  ierr;
632   const PetscInt   *r,*ic,*ics;
633   PetscInt        nz,row,i,j,n=A->rmap->n,diag;
634   const PetscInt  *ai=a->i,*aj=a->j,*bi=b->i,*bj=b->j;
635   const PetscInt  *ajtmp,*bjtmp,*diag_offset = b->diag,*pj;
636   MatScalar       *pv,*rtmp,*pc,multiplier,d;
637   const MatScalar *v,*aa=a->a;
638   PetscReal       rs=0.0;
639   FactorShiftCtx  sctx;
640   PetscInt        newshift;
641   const PetscInt  *ddiag;
642   PetscTruth      row_identity, col_identity;
643 
644   PetscFunctionBegin;
645   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
646   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
647   ierr = PetscMalloc((n+1)*sizeof(MatScalar),&rtmp);CHKERRQ(ierr);
648   ics  = ic;
649 
650   /* initialize shift context sctx */
651   sctx.nshift         = 0;
652   sctx.nshift_max     = 0;
653   sctx.shift_top      = 0.0;
654   sctx.shift_lo       = 0.0;
655   sctx.shift_hi       = 0.0;
656   sctx.shift_fraction = 0.0;
657   sctx.shift_amount   = 0.0;
658 
659   if (info->shifttype == (PetscReal)MAT_SHIFT_POSITIVE_DEFINITE) { /* set sctx.shift_top=max{rs} */
660     ddiag          = a->diag;
661     sctx.shift_top = info->zeropivot;
662     for (i=0; i<n; i++) {
663       /* calculate sum(|aij|)-RealPart(aii), amt of shift needed for this row */
664       d  = (aa)[ddiag[i]];
665       rs = -PetscAbsScalar(d) - PetscRealPart(d);
666       v  = aa+ai[i];
667       nz = ai[i+1] - ai[i];
668       for (j=0; j<nz; j++)
669 	rs += PetscAbsScalar(v[j]);
670       if (rs>sctx.shift_top) sctx.shift_top = rs;
671     }
672     sctx.shift_top   *= 1.1;
673     sctx.nshift_max   = 5;
674     sctx.shift_lo     = 0.;
675     sctx.shift_hi     = 1.;
676   }
677 
678   do {
679     sctx.useshift = PETSC_FALSE;
680     for (i=0; i<n; i++){
681       nz    = bi[i+1] - bi[i];
682       bjtmp = bj + bi[i];
683       for  (j=0; j<nz; j++) rtmp[bjtmp[j]] = 0.0;
684 
685       /* load in initial (unfactored row) */
686       nz    = ai[r[i]+1] - ai[r[i]];
687       ajtmp = aj + ai[r[i]];
688       v     = aa + ai[r[i]];
689       for (j=0; j<nz; j++) {
690         rtmp[ics[ajtmp[j]]] = v[j];
691       }
692       rtmp[ics[r[i]]] += sctx.shift_amount; /* shift the diagonal of the matrix */
693       /* if (sctx.shift_amount > 0.0) printf("row %d, shift %g\n",i,sctx.shift_amount); */
694 
695       row = *bjtmp++;
696       while  (row < i) {
697         pc = rtmp + row;
698         if (*pc != 0.0) {
699           pv         = b->a + diag_offset[row];
700           pj         = b->j + diag_offset[row] + 1;
701           multiplier = *pc / *pv++;
702           *pc        = multiplier;
703           nz         = bi[row+1] - diag_offset[row] - 1;
704           for (j=0; j<nz; j++) rtmp[pj[j]] -= multiplier * pv[j];
705           ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr);
706         }
707         row = *bjtmp++;
708       }
709       /* finished row so stick it into b->a */
710       pv   = b->a + bi[i] ;
711       pj   = b->j + bi[i] ;
712       nz   = bi[i+1] - bi[i];
713       diag = diag_offset[i] - bi[i];
714       rs   = 0.0;
715       for (j=0; j<nz; j++) {
716         pv[j] = rtmp[pj[j]];
717         rs   += PetscAbsScalar(pv[j]);
718       }
719       rs   -= PetscAbsScalar(pv[diag]);
720 
721       /* 9/13/02 Victor Eijkhout suggested scaling zeropivot by rs for matrices with funny scalings */
722       sctx.rs  = rs;
723       sctx.pv  = pv[diag];
724       ierr = MatLUCheckShift_inline(info,sctx,i,newshift);CHKERRQ(ierr);
725       if (newshift == 1) break;
726     }
727 
728     if (info->shifttype == (PetscReal)MAT_SHIFT_POSITIVE_DEFINITE && !sctx.useshift && sctx.shift_fraction>0 && sctx.nshift<sctx.nshift_max) {
729       /*
730        * if no shift in this attempt & shifting & started shifting & can refine,
731        * then try lower shift
732        */
733       sctx.shift_hi       = sctx.shift_fraction;
734       sctx.shift_fraction = (sctx.shift_hi+sctx.shift_lo)/2.;
735       sctx.shift_amount   = sctx.shift_fraction * sctx.shift_top;
736       sctx.useshift        = PETSC_TRUE;
737       sctx.nshift++;
738     }
739   } while (sctx.useshift);
740 
741   /* invert diagonal entries for simplier triangular solves */
742   for (i=0; i<n; i++) {
743     b->a[diag_offset[i]] = 1.0/b->a[diag_offset[i]];
744   }
745   ierr = PetscFree(rtmp);CHKERRQ(ierr);
746   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
747   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
748 
749   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
750   ierr = ISIdentity(isicol,&col_identity);CHKERRQ(ierr);
751   if (row_identity && col_identity) {
752     C->ops->solve   = MatSolve_SeqAIJ_NaturalOrdering_inplace;
753   } else {
754     C->ops->solve   = MatSolve_SeqAIJ_inplace;
755   }
756   C->ops->solveadd           = MatSolveAdd_SeqAIJ_inplace;
757   C->ops->solvetranspose     = MatSolveTranspose_SeqAIJ_inplace;
758   C->ops->solvetransposeadd  = MatSolveTransposeAdd_SeqAIJ_inplace;
759   C->ops->matsolve           = MatMatSolve_SeqAIJ_inplace;
760   C->assembled    = PETSC_TRUE;
761   C->preallocated = PETSC_TRUE;
762   ierr = PetscLogFlops(C->cmap->n);CHKERRQ(ierr);
763   if (sctx.nshift){
764      if (info->shifttype == (PetscReal)MAT_SHIFT_POSITIVE_DEFINITE) {
765       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);
766     } else if (info->shifttype == (PetscReal)MAT_SHIFT_NONZERO) {
767       ierr = PetscInfo2(A,"number of shift_nz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
768     }
769   }
770   (C)->ops->solve            = MatSolve_SeqAIJ_inplace;
771   (C)->ops->solvetranspose   = MatSolveTranspose_SeqAIJ_inplace;
772   ierr = Mat_CheckInode(C,PETSC_FALSE);CHKERRQ(ierr);
773   PetscFunctionReturn(0);
774 }
775 
776 /*
777    This routine implements inplace ILU(0) with row or/and column permutations.
778    Input:
779      A - original matrix
780    Output;
781      A - a->i (rowptr) is same as original rowptr, but factored i-the row is stored in rowperm[i]
782          a->j (col index) is permuted by the inverse of colperm, then sorted
783          a->a reordered accordingly with a->j
784          a->diag (ptr to diagonal elements) is updated.
785 */
786 #undef __FUNCT__
787 #define __FUNCT__ "MatLUFactorNumeric_SeqAIJ_InplaceWithPerm"
788 PetscErrorCode MatLUFactorNumeric_SeqAIJ_InplaceWithPerm(Mat B,Mat A,const MatFactorInfo *info)
789 {
790   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data;
791   IS             isrow = a->row,isicol = a->icol;
792   PetscErrorCode ierr;
793   const PetscInt *r,*ic,*ics;
794   PetscInt       i,j,n=A->rmap->n,*ai=a->i,*aj=a->j;
795   PetscInt       *ajtmp,nz,row;
796   PetscInt       *diag = a->diag,nbdiag,*pj;
797   PetscScalar    *rtmp,*pc,multiplier,d;
798   MatScalar      *v,*pv;
799   PetscReal      rs;
800   FactorShiftCtx sctx;
801   PetscInt       newshift = 0;
802 
803   PetscFunctionBegin;
804   if (A != B) SETERRQ(PETSC_ERR_ARG_INCOMP,"input and output matrix must have same address");
805   ierr  = ISGetIndices(isrow,&r);CHKERRQ(ierr);
806   ierr  = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
807   ierr  = PetscMalloc((n+1)*sizeof(PetscScalar),&rtmp);CHKERRQ(ierr);
808   ierr  = PetscMemzero(rtmp,(n+1)*sizeof(PetscScalar));CHKERRQ(ierr);
809   ics = ic;
810 
811   sctx.shift_top      = 0.;
812   sctx.nshift_max     = 0;
813   sctx.shift_lo       = 0.;
814   sctx.shift_hi       = 0.;
815   sctx.shift_fraction = 0.;
816 
817   if (info->shifttype == (PetscReal)MAT_SHIFT_POSITIVE_DEFINITE) { /* set sctx.shift_top=max{rs} */
818     sctx.shift_top = 0.;
819     for (i=0; i<n; i++) {
820       /* calculate sum(|aij|)-RealPart(aii), amt of shift needed for this row */
821       d  = (a->a)[diag[i]];
822       rs = -PetscAbsScalar(d) - PetscRealPart(d);
823       v  = a->a+ai[i];
824       nz = ai[i+1] - ai[i];
825       for (j=0; j<nz; j++)
826 	rs += PetscAbsScalar(v[j]);
827       if (rs>sctx.shift_top) sctx.shift_top = rs;
828     }
829     if (sctx.shift_top < info->zeropivot) sctx.shift_top = info->zeropivot;
830     sctx.shift_top    *= 1.1;
831     sctx.nshift_max   = 5;
832     sctx.shift_lo     = 0.;
833     sctx.shift_hi     = 1.;
834   }
835 
836   sctx.shift_amount = 0.;
837   sctx.nshift       = 0;
838   do {
839     sctx.useshift = PETSC_FALSE;
840     for (i=0; i<n; i++){
841       /* load in initial unfactored row */
842       nz    = ai[r[i]+1] - ai[r[i]];
843       ajtmp = aj + ai[r[i]];
844       v     = a->a + ai[r[i]];
845       /* sort permuted ajtmp and values v accordingly */
846       for (j=0; j<nz; j++) ajtmp[j] = ics[ajtmp[j]];
847       ierr = PetscSortIntWithScalarArray(nz,ajtmp,v);CHKERRQ(ierr);
848 
849       diag[r[i]] = ai[r[i]];
850       for (j=0; j<nz; j++) {
851         rtmp[ajtmp[j]] = v[j];
852         if (ajtmp[j] < i) diag[r[i]]++; /* update a->diag */
853       }
854       rtmp[r[i]] += sctx.shift_amount; /* shift the diagonal of the matrix */
855 
856       row = *ajtmp++;
857       while  (row < i) {
858         pc = rtmp + row;
859         if (*pc != 0.0) {
860           pv         = a->a + diag[r[row]];
861           pj         = aj + diag[r[row]] + 1;
862 
863           multiplier = *pc / *pv++;
864           *pc        = multiplier;
865           nz         = ai[r[row]+1] - diag[r[row]] - 1;
866           for (j=0; j<nz; j++) rtmp[pj[j]] -= multiplier * pv[j];
867           ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr);
868         }
869         row = *ajtmp++;
870       }
871       /* finished row so overwrite it onto a->a */
872       pv   = a->a + ai[r[i]] ;
873       pj   = aj + ai[r[i]] ;
874       nz   = ai[r[i]+1] - ai[r[i]];
875       nbdiag = diag[r[i]] - ai[r[i]]; /* num of entries before the diagonal */
876 
877       rs   = 0.0;
878       for (j=0; j<nz; j++) {
879         pv[j] = rtmp[pj[j]];
880         if (j != nbdiag) rs += PetscAbsScalar(pv[j]);
881       }
882 
883       /* 9/13/02 Victor Eijkhout suggested scaling zeropivot by rs for matrices with funny scalings */
884       sctx.rs  = rs;
885       sctx.pv  = pv[nbdiag];
886       ierr = MatLUCheckShift_inline(info,sctx,i,newshift);CHKERRQ(ierr);
887       if (newshift == 1) break;
888     }
889 
890     if (info->shifttype == (PetscReal)MAT_SHIFT_POSITIVE_DEFINITE && !sctx.useshift && sctx.shift_fraction>0 && sctx.nshift<sctx.nshift_max) {
891       /*
892        * if no shift in this attempt & shifting & started shifting & can refine,
893        * then try lower shift
894        */
895       sctx.shift_hi        = sctx.shift_fraction;
896       sctx.shift_fraction = (sctx.shift_hi+sctx.shift_lo)/2.;
897       sctx.shift_amount    = sctx.shift_fraction * sctx.shift_top;
898       sctx.useshift         = PETSC_TRUE;
899       sctx.nshift++;
900     }
901   } while (sctx.useshift);
902 
903   /* invert diagonal entries for simplier triangular solves */
904   for (i=0; i<n; i++) {
905     a->a[diag[r[i]]] = 1.0/a->a[diag[r[i]]];
906   }
907 
908   ierr = PetscFree(rtmp);CHKERRQ(ierr);
909   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
910   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
911   A->ops->solve             = MatSolve_SeqAIJ_InplaceWithPerm;
912   A->ops->solveadd          = MatSolveAdd_SeqAIJ_inplace;
913   A->ops->solvetranspose    = MatSolveTranspose_SeqAIJ_inplace;
914   A->ops->solvetransposeadd = MatSolveTransposeAdd_SeqAIJ_inplace;
915   A->assembled = PETSC_TRUE;
916   A->preallocated = PETSC_TRUE;
917   ierr = PetscLogFlops(A->cmap->n);CHKERRQ(ierr);
918   if (sctx.nshift){
919     if (info->shifttype == (PetscReal)MAT_SHIFT_POSITIVE_DEFINITE) {
920       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);
921     } else if (info->shifttype == (PetscReal)MAT_SHIFT_NONZERO) {
922       ierr = PetscInfo2(A,"number of shift_nz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
923     }
924   }
925   PetscFunctionReturn(0);
926 }
927 
928 /* ----------------------------------------------------------- */
929 #undef __FUNCT__
930 #define __FUNCT__ "MatLUFactor_SeqAIJ"
931 PetscErrorCode MatLUFactor_SeqAIJ(Mat A,IS row,IS col,const MatFactorInfo *info)
932 {
933   PetscErrorCode ierr;
934   Mat            C;
935 
936   PetscFunctionBegin;
937   ierr = MatGetFactor(A,MAT_SOLVER_PETSC,MAT_FACTOR_LU,&C);CHKERRQ(ierr);
938   ierr = MatLUFactorSymbolic(C,A,row,col,info);CHKERRQ(ierr);
939   ierr = MatLUFactorNumeric(C,A,info);CHKERRQ(ierr);
940   A->ops->solve            = C->ops->solve;
941   A->ops->solvetranspose   = C->ops->solvetranspose;
942   ierr = MatHeaderCopy(A,C);CHKERRQ(ierr);
943   ierr = PetscLogObjectParent(A,((Mat_SeqAIJ*)(A->data))->icol);CHKERRQ(ierr);
944   PetscFunctionReturn(0);
945 }
946 /* ----------------------------------------------------------- */
947 
948 
949 #undef __FUNCT__
950 #define __FUNCT__ "MatSolve_SeqAIJ_inplace"
951 PetscErrorCode MatSolve_SeqAIJ_inplace(Mat A,Vec bb,Vec xx)
952 {
953   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
954   IS                iscol = a->col,isrow = a->row;
955   PetscErrorCode    ierr;
956   PetscInt          i, n = A->rmap->n,*vi,*ai = a->i,*aj = a->j;
957   PetscInt          nz;
958   const PetscInt    *rout,*cout,*r,*c;
959   PetscScalar       *x,*tmp,*tmps,sum;
960   const PetscScalar *b;
961   const MatScalar   *aa = a->a,*v;
962 
963   PetscFunctionBegin;
964   if (!n) PetscFunctionReturn(0);
965 
966   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
967   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
968   tmp  = a->solve_work;
969 
970   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
971   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout + (n-1);
972 
973   /* forward solve the lower triangular */
974   tmp[0] = b[*r++];
975   tmps   = tmp;
976   for (i=1; i<n; i++) {
977     v   = aa + ai[i] ;
978     vi  = aj + ai[i] ;
979     nz  = a->diag[i] - ai[i];
980     sum = b[*r++];
981     PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
982     tmp[i] = sum;
983   }
984 
985   /* backward solve the upper triangular */
986   for (i=n-1; i>=0; i--){
987     v   = aa + a->diag[i] + 1;
988     vi  = aj + a->diag[i] + 1;
989     nz  = ai[i+1] - a->diag[i] - 1;
990     sum = tmp[i];
991     PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
992     x[*c--] = tmp[i] = sum*aa[a->diag[i]];
993   }
994 
995   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
996   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
997   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
998   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
999   ierr = PetscLogFlops(2.0*a->nz - A->cmap->n);CHKERRQ(ierr);
1000   PetscFunctionReturn(0);
1001 }
1002 
1003 #undef __FUNCT__
1004 #define __FUNCT__ "MatMatSolve_SeqAIJ_inplace"
1005 PetscErrorCode MatMatSolve_SeqAIJ_inplace(Mat A,Mat B,Mat X)
1006 {
1007   Mat_SeqAIJ      *a = (Mat_SeqAIJ*)A->data;
1008   IS              iscol = a->col,isrow = a->row;
1009   PetscErrorCode  ierr;
1010   PetscInt        i, n = A->rmap->n,*vi,*ai = a->i,*aj = a->j;
1011   PetscInt        nz,neq;
1012   const PetscInt  *rout,*cout,*r,*c;
1013   PetscScalar     *x,*b,*tmp,*tmps,sum;
1014   const MatScalar *aa = a->a,*v;
1015   PetscTruth      bisdense,xisdense;
1016 
1017   PetscFunctionBegin;
1018   if (!n) PetscFunctionReturn(0);
1019 
1020   ierr = PetscTypeCompare((PetscObject)B,MATSEQDENSE,&bisdense);CHKERRQ(ierr);
1021   if (!bisdense) SETERRQ(PETSC_ERR_ARG_INCOMP,"B matrix must be a SeqDense matrix");
1022   ierr = PetscTypeCompare((PetscObject)X,MATSEQDENSE,&xisdense);CHKERRQ(ierr);
1023   if (!xisdense) SETERRQ(PETSC_ERR_ARG_INCOMP,"X matrix must be a SeqDense matrix");
1024 
1025   ierr = MatGetArray(B,&b);CHKERRQ(ierr);
1026   ierr = MatGetArray(X,&x);CHKERRQ(ierr);
1027 
1028   tmp  = a->solve_work;
1029   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1030   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1031 
1032   for (neq=0; neq<B->cmap->n; neq++){
1033     /* forward solve the lower triangular */
1034     tmp[0] = b[r[0]];
1035     tmps   = tmp;
1036     for (i=1; i<n; i++) {
1037       v   = aa + ai[i] ;
1038       vi  = aj + ai[i] ;
1039       nz  = a->diag[i] - ai[i];
1040       sum = b[r[i]];
1041       PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
1042       tmp[i] = sum;
1043     }
1044     /* backward solve the upper triangular */
1045     for (i=n-1; i>=0; i--){
1046       v   = aa + a->diag[i] + 1;
1047       vi  = aj + a->diag[i] + 1;
1048       nz  = ai[i+1] - a->diag[i] - 1;
1049       sum = tmp[i];
1050       PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
1051       x[c[i]] = tmp[i] = sum*aa[a->diag[i]];
1052     }
1053 
1054     b += n;
1055     x += n;
1056   }
1057   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1058   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1059   ierr = MatRestoreArray(B,&b);CHKERRQ(ierr);
1060   ierr = MatRestoreArray(X,&x);CHKERRQ(ierr);
1061   ierr = PetscLogFlops(B->cmap->n*(2.0*a->nz - n));CHKERRQ(ierr);
1062   PetscFunctionReturn(0);
1063 }
1064 
1065 #undef __FUNCT__
1066 #define __FUNCT__ "MatMatSolve_SeqAIJ"
1067 PetscErrorCode MatMatSolve_SeqAIJ(Mat A,Mat B,Mat X)
1068 {
1069   Mat_SeqAIJ      *a = (Mat_SeqAIJ*)A->data;
1070   IS              iscol = a->col,isrow = a->row;
1071   PetscErrorCode  ierr;
1072   PetscInt        i, n = A->rmap->n,*vi,*ai = a->i,*aj = a->j,*adiag = a->diag;
1073   PetscInt        nz,neq;
1074   const PetscInt  *rout,*cout,*r,*c;
1075   PetscScalar     *x,*b,*tmp,sum;
1076   const MatScalar *aa = a->a,*v;
1077   PetscTruth      bisdense,xisdense;
1078 
1079   PetscFunctionBegin;
1080   if (!n) PetscFunctionReturn(0);
1081 
1082   ierr = PetscTypeCompare((PetscObject)B,MATSEQDENSE,&bisdense);CHKERRQ(ierr);
1083   if (!bisdense) SETERRQ(PETSC_ERR_ARG_INCOMP,"B matrix must be a SeqDense matrix");
1084   ierr = PetscTypeCompare((PetscObject)X,MATSEQDENSE,&xisdense);CHKERRQ(ierr);
1085   if (!xisdense) SETERRQ(PETSC_ERR_ARG_INCOMP,"X matrix must be a SeqDense matrix");
1086 
1087   ierr = MatGetArray(B,&b);CHKERRQ(ierr);
1088   ierr = MatGetArray(X,&x);CHKERRQ(ierr);
1089 
1090   tmp  = a->solve_work;
1091   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1092   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1093 
1094   for (neq=0; neq<B->cmap->n; neq++){
1095     /* forward solve the lower triangular */
1096     tmp[0] = b[r[0]];
1097     v      = aa;
1098     vi     = aj;
1099     for (i=1; i<n; i++) {
1100       nz  = ai[i+1] - ai[i];
1101       sum = b[r[i]];
1102       PetscSparseDenseMinusDot(sum,tmp,v,vi,nz);
1103       tmp[i] = sum;
1104       v += nz; vi += nz;
1105     }
1106 
1107     /* backward solve the upper triangular */
1108     for (i=n-1; i>=0; i--){
1109       v   = aa + adiag[i+1]+1;
1110       vi  = aj + adiag[i+1]+1;
1111       nz  = adiag[i]-adiag[i+1]-1;
1112       sum = tmp[i];
1113       PetscSparseDenseMinusDot(sum,tmp,v,vi,nz);
1114       x[c[i]] = tmp[i] = sum*v[nz]; /* v[nz] = aa[adiag[i]] */
1115     }
1116 
1117     b += n;
1118     x += n;
1119   }
1120   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1121   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1122   ierr = MatRestoreArray(B,&b);CHKERRQ(ierr);
1123   ierr = MatRestoreArray(X,&x);CHKERRQ(ierr);
1124   ierr = PetscLogFlops(B->cmap->n*(2.0*a->nz - n));CHKERRQ(ierr);
1125   PetscFunctionReturn(0);
1126 }
1127 
1128 #undef __FUNCT__
1129 #define __FUNCT__ "MatSolve_SeqAIJ_InplaceWithPerm"
1130 PetscErrorCode MatSolve_SeqAIJ_InplaceWithPerm(Mat A,Vec bb,Vec xx)
1131 {
1132   Mat_SeqAIJ      *a = (Mat_SeqAIJ*)A->data;
1133   IS              iscol = a->col,isrow = a->row;
1134   PetscErrorCode  ierr;
1135   const PetscInt  *r,*c,*rout,*cout;
1136   PetscInt        i, n = A->rmap->n,*vi,*ai = a->i,*aj = a->j;
1137   PetscInt        nz,row;
1138   PetscScalar     *x,*b,*tmp,*tmps,sum;
1139   const MatScalar *aa = a->a,*v;
1140 
1141   PetscFunctionBegin;
1142   if (!n) PetscFunctionReturn(0);
1143 
1144   ierr = VecGetArray(bb,&b);CHKERRQ(ierr);
1145   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1146   tmp  = a->solve_work;
1147 
1148   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1149   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout + (n-1);
1150 
1151   /* forward solve the lower triangular */
1152   tmp[0] = b[*r++];
1153   tmps   = tmp;
1154   for (row=1; row<n; row++) {
1155     i   = rout[row]; /* permuted row */
1156     v   = aa + ai[i] ;
1157     vi  = aj + ai[i] ;
1158     nz  = a->diag[i] - ai[i];
1159     sum = b[*r++];
1160     PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
1161     tmp[row] = sum;
1162   }
1163 
1164   /* backward solve the upper triangular */
1165   for (row=n-1; row>=0; row--){
1166     i   = rout[row]; /* permuted row */
1167     v   = aa + a->diag[i] + 1;
1168     vi  = aj + a->diag[i] + 1;
1169     nz  = ai[i+1] - a->diag[i] - 1;
1170     sum = tmp[row];
1171     PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
1172     x[*c--] = tmp[row] = sum*aa[a->diag[i]];
1173   }
1174 
1175   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1176   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1177   ierr = VecRestoreArray(bb,&b);CHKERRQ(ierr);
1178   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1179   ierr = PetscLogFlops(2.0*a->nz - A->cmap->n);CHKERRQ(ierr);
1180   PetscFunctionReturn(0);
1181 }
1182 
1183 /* ----------------------------------------------------------- */
1184 #include "../src/mat/impls/aij/seq/ftn-kernels/fsolve.h"
1185 #undef __FUNCT__
1186 #define __FUNCT__ "MatSolve_SeqAIJ_NaturalOrdering_inplace"
1187 PetscErrorCode MatSolve_SeqAIJ_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
1188 {
1189   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1190   PetscErrorCode    ierr;
1191   PetscInt          n = A->rmap->n;
1192   const PetscInt    *ai = a->i,*aj = a->j,*adiag = a->diag;
1193   PetscScalar       *x;
1194   const PetscScalar *b;
1195   const MatScalar   *aa = a->a;
1196 #if !defined(PETSC_USE_FORTRAN_KERNEL_SOLVEAIJ)
1197   PetscInt          adiag_i,i,nz,ai_i;
1198   const PetscInt    *vi;
1199   const MatScalar   *v;
1200   PetscScalar       sum;
1201 #endif
1202 
1203   PetscFunctionBegin;
1204   if (!n) PetscFunctionReturn(0);
1205 
1206   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1207   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1208 
1209 #if defined(PETSC_USE_FORTRAN_KERNEL_SOLVEAIJ)
1210   fortransolveaij_(&n,x,ai,aj,adiag,aa,b);
1211 #else
1212   /* forward solve the lower triangular */
1213   x[0] = b[0];
1214   for (i=1; i<n; i++) {
1215     ai_i = ai[i];
1216     v    = aa + ai_i;
1217     vi   = aj + ai_i;
1218     nz   = adiag[i] - ai_i;
1219     sum  = b[i];
1220     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
1221     x[i] = sum;
1222   }
1223 
1224   /* backward solve the upper triangular */
1225   for (i=n-1; i>=0; i--){
1226     adiag_i = adiag[i];
1227     v       = aa + adiag_i + 1;
1228     vi      = aj + adiag_i + 1;
1229     nz      = ai[i+1] - adiag_i - 1;
1230     sum     = x[i];
1231     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
1232     x[i]    = sum*aa[adiag_i];
1233   }
1234 #endif
1235   ierr = PetscLogFlops(2.0*a->nz - A->cmap->n);CHKERRQ(ierr);
1236   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1237   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1238   PetscFunctionReturn(0);
1239 }
1240 
1241 #undef __FUNCT__
1242 #define __FUNCT__ "MatSolveAdd_SeqAIJ_inplace"
1243 PetscErrorCode MatSolveAdd_SeqAIJ_inplace(Mat A,Vec bb,Vec yy,Vec xx)
1244 {
1245   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1246   IS                iscol = a->col,isrow = a->row;
1247   PetscErrorCode    ierr;
1248   PetscInt          i, n = A->rmap->n,j;
1249   PetscInt          nz;
1250   const PetscInt    *rout,*cout,*r,*c,*vi,*ai = a->i,*aj = a->j;
1251   PetscScalar       *x,*tmp,sum;
1252   const PetscScalar *b;
1253   const MatScalar   *aa = a->a,*v;
1254 
1255   PetscFunctionBegin;
1256   if (yy != xx) {ierr = VecCopy(yy,xx);CHKERRQ(ierr);}
1257 
1258   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1259   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1260   tmp  = a->solve_work;
1261 
1262   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1263   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout + (n-1);
1264 
1265   /* forward solve the lower triangular */
1266   tmp[0] = b[*r++];
1267   for (i=1; i<n; i++) {
1268     v   = aa + ai[i] ;
1269     vi  = aj + ai[i] ;
1270     nz  = a->diag[i] - ai[i];
1271     sum = b[*r++];
1272     for (j=0; j<nz; j++) sum -= v[j]*tmp[vi[j]];
1273     tmp[i] = sum;
1274   }
1275 
1276   /* backward solve the upper triangular */
1277   for (i=n-1; i>=0; i--){
1278     v   = aa + a->diag[i] + 1;
1279     vi  = aj + a->diag[i] + 1;
1280     nz  = ai[i+1] - a->diag[i] - 1;
1281     sum = tmp[i];
1282     for (j=0; j<nz; j++) sum -= v[j]*tmp[vi[j]];
1283     tmp[i] = sum*aa[a->diag[i]];
1284     x[*c--] += tmp[i];
1285   }
1286 
1287   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1288   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1289   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1290   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1291   ierr = PetscLogFlops(2.0*a->nz);CHKERRQ(ierr);
1292 
1293   PetscFunctionReturn(0);
1294 }
1295 
1296 #undef __FUNCT__
1297 #define __FUNCT__ "MatSolveAdd_SeqAIJ"
1298 PetscErrorCode MatSolveAdd_SeqAIJ(Mat A,Vec bb,Vec yy,Vec xx)
1299 {
1300   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1301   IS                iscol = a->col,isrow = a->row;
1302   PetscErrorCode    ierr;
1303   PetscInt          i, n = A->rmap->n,j;
1304   PetscInt          nz;
1305   const PetscInt    *rout,*cout,*r,*c,*vi,*ai = a->i,*aj = a->j,*adiag = a->diag;
1306   PetscScalar       *x,*tmp,sum;
1307   const PetscScalar *b;
1308   const MatScalar   *aa = a->a,*v;
1309 
1310   PetscFunctionBegin;
1311   if (yy != xx) {ierr = VecCopy(yy,xx);CHKERRQ(ierr);}
1312 
1313   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1314   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1315   tmp  = a->solve_work;
1316 
1317   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1318   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1319 
1320   /* forward solve the lower triangular */
1321   tmp[0] = b[r[0]];
1322   v      = aa;
1323   vi     = aj;
1324   for (i=1; i<n; i++) {
1325     nz  = ai[i+1] - ai[i];
1326     sum = b[r[i]];
1327     for (j=0; j<nz; j++) sum -= v[j]*tmp[vi[j]];
1328     tmp[i] = sum;
1329     v += nz; vi += nz;
1330   }
1331 
1332   /* backward solve the upper triangular */
1333   v  = aa + adiag[n-1];
1334   vi = aj + adiag[n-1];
1335   for (i=n-1; i>=0; i--){
1336     nz  = adiag[i] - adiag[i+1] - 1;
1337     sum = tmp[i];
1338     for (j=0; j<nz; j++) sum -= v[j]*tmp[vi[j]];
1339     tmp[i] = sum*v[nz];
1340     x[c[i]] += tmp[i];
1341     v += nz+1; vi += nz+1;
1342   }
1343 
1344   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1345   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1346   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1347   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1348   ierr = PetscLogFlops(2.0*a->nz);CHKERRQ(ierr);
1349 
1350   PetscFunctionReturn(0);
1351 }
1352 
1353 #undef __FUNCT__
1354 #define __FUNCT__ "MatSolveTranspose_SeqAIJ_inplace"
1355 PetscErrorCode MatSolveTranspose_SeqAIJ_inplace(Mat A,Vec bb,Vec xx)
1356 {
1357   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1358   IS                iscol = a->col,isrow = a->row;
1359   PetscErrorCode    ierr;
1360   const PetscInt    *rout,*cout,*r,*c,*diag = a->diag,*ai = a->i,*aj = a->j,*vi;
1361   PetscInt          i,n = A->rmap->n,j;
1362   PetscInt          nz;
1363   PetscScalar       *x,*tmp,s1;
1364   const MatScalar   *aa = a->a,*v;
1365   const PetscScalar *b;
1366 
1367   PetscFunctionBegin;
1368   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1369   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1370   tmp  = a->solve_work;
1371 
1372   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1373   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1374 
1375   /* copy the b into temp work space according to permutation */
1376   for (i=0; i<n; i++) tmp[i] = b[c[i]];
1377 
1378   /* forward solve the U^T */
1379   for (i=0; i<n; i++) {
1380     v   = aa + diag[i] ;
1381     vi  = aj + diag[i] + 1;
1382     nz  = ai[i+1] - diag[i] - 1;
1383     s1  = tmp[i];
1384     s1 *= (*v++);  /* multiply by inverse of diagonal entry */
1385     for (j=0; j<nz; j++) tmp[vi[j]] -= s1*v[j];
1386     tmp[i] = s1;
1387   }
1388 
1389   /* backward solve the L^T */
1390   for (i=n-1; i>=0; i--){
1391     v   = aa + diag[i] - 1 ;
1392     vi  = aj + diag[i] - 1 ;
1393     nz  = diag[i] - ai[i];
1394     s1  = tmp[i];
1395     for (j=0; j>-nz; j--) tmp[vi[j]] -= s1*v[j];
1396   }
1397 
1398   /* copy tmp into x according to permutation */
1399   for (i=0; i<n; i++) x[r[i]] = tmp[i];
1400 
1401   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1402   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1403   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1404   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1405 
1406   ierr = PetscLogFlops(2.0*a->nz-A->cmap->n);CHKERRQ(ierr);
1407   PetscFunctionReturn(0);
1408 }
1409 
1410 #undef __FUNCT__
1411 #define __FUNCT__ "MatSolveTranspose_SeqAIJ"
1412 PetscErrorCode MatSolveTranspose_SeqAIJ(Mat A,Vec bb,Vec xx)
1413 {
1414   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1415   IS                iscol = a->col,isrow = a->row;
1416   PetscErrorCode    ierr;
1417   const PetscInt    *rout,*cout,*r,*c,*adiag = a->diag,*ai = a->i,*aj = a->j,*vi;
1418   PetscInt          i,n = A->rmap->n,j;
1419   PetscInt          nz;
1420   PetscScalar       *x,*tmp,s1;
1421   const MatScalar   *aa = a->a,*v;
1422   const PetscScalar *b;
1423 
1424   PetscFunctionBegin;
1425   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1426   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1427   tmp  = a->solve_work;
1428 
1429   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1430   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1431 
1432   /* copy the b into temp work space according to permutation */
1433   for (i=0; i<n; i++) tmp[i] = b[c[i]];
1434 
1435   /* forward solve the U^T */
1436   for (i=0; i<n; i++) {
1437     v   = aa + adiag[i+1] + 1;
1438     vi  = aj + adiag[i+1] + 1;
1439     nz  = adiag[i] - adiag[i+1] - 1;
1440     s1  = tmp[i];
1441     s1 *= v[nz];  /* multiply by inverse of diagonal entry */
1442     for (j=0; j<nz; j++) tmp[vi[j]] -= s1*v[j];
1443     tmp[i] = s1;
1444   }
1445 
1446   /* backward solve the L^T */
1447   for (i=n-1; i>=0; i--){
1448     v   = aa + ai[i];
1449     vi  = aj + ai[i];
1450     nz  = ai[i+1] - ai[i];
1451     s1  = tmp[i];
1452     for (j=0; j<nz; j++) tmp[vi[j]] -= s1*v[j];
1453   }
1454 
1455   /* copy tmp into x according to permutation */
1456   for (i=0; i<n; i++) x[r[i]] = tmp[i];
1457 
1458   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1459   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1460   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1461   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1462 
1463   ierr = PetscLogFlops(2.0*a->nz-A->cmap->n);CHKERRQ(ierr);
1464   PetscFunctionReturn(0);
1465 }
1466 
1467 #undef __FUNCT__
1468 #define __FUNCT__ "MatSolveTransposeAdd_SeqAIJ_inplace"
1469 PetscErrorCode MatSolveTransposeAdd_SeqAIJ_inplace(Mat A,Vec bb,Vec zz,Vec xx)
1470 {
1471   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1472   IS                iscol = a->col,isrow = a->row;
1473   PetscErrorCode    ierr;
1474   const PetscInt    *rout,*cout,*r,*c,*diag = a->diag,*ai = a->i,*aj = a->j,*vi;
1475   PetscInt          i,n = A->rmap->n,j;
1476   PetscInt          nz;
1477   PetscScalar       *x,*tmp,s1;
1478   const MatScalar   *aa = a->a,*v;
1479   const PetscScalar *b;
1480 
1481   PetscFunctionBegin;
1482   if (zz != xx) {ierr = VecCopy(zz,xx);CHKERRQ(ierr);}
1483   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1484   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1485   tmp  = a->solve_work;
1486 
1487   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1488   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1489 
1490   /* copy the b into temp work space according to permutation */
1491   for (i=0; i<n; i++) tmp[i] = b[c[i]];
1492 
1493   /* forward solve the U^T */
1494   for (i=0; i<n; i++) {
1495     v   = aa + diag[i] ;
1496     vi  = aj + diag[i] + 1;
1497     nz  = ai[i+1] - diag[i] - 1;
1498     s1  = tmp[i];
1499     s1 *= (*v++);  /* multiply by inverse of diagonal entry */
1500     for (j=0; j<nz; j++) tmp[vi[j]] -= s1*v[j];
1501     tmp[i] = s1;
1502   }
1503 
1504   /* backward solve the L^T */
1505   for (i=n-1; i>=0; i--){
1506     v   = aa + diag[i] - 1 ;
1507     vi  = aj + diag[i] - 1 ;
1508     nz  = diag[i] - ai[i];
1509     s1  = tmp[i];
1510     for (j=0; j>-nz; j--) tmp[vi[j]] -= s1*v[j];
1511   }
1512 
1513   /* copy tmp into x according to permutation */
1514   for (i=0; i<n; i++) x[r[i]] += tmp[i];
1515 
1516   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1517   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1518   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1519   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1520 
1521   ierr = PetscLogFlops(2.0*a->nz-A->cmap->n);CHKERRQ(ierr);
1522   PetscFunctionReturn(0);
1523 }
1524 
1525 #undef __FUNCT__
1526 #define __FUNCT__ "MatSolveTransposeAdd_SeqAIJ"
1527 PetscErrorCode MatSolveTransposeAdd_SeqAIJ(Mat A,Vec bb,Vec zz,Vec xx)
1528 {
1529   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
1530   IS                iscol = a->col,isrow = a->row;
1531   PetscErrorCode    ierr;
1532   const PetscInt    *rout,*cout,*r,*c,*adiag = a->diag,*ai = a->i,*aj = a->j,*vi;
1533   PetscInt          i,n = A->rmap->n,j;
1534   PetscInt          nz;
1535   PetscScalar       *x,*tmp,s1;
1536   const MatScalar   *aa = a->a,*v;
1537   const PetscScalar *b;
1538 
1539   PetscFunctionBegin;
1540   if (zz != xx) {ierr = VecCopy(zz,xx);CHKERRQ(ierr);}
1541   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1542   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1543   tmp  = a->solve_work;
1544 
1545   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1546   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1547 
1548   /* copy the b into temp work space according to permutation */
1549   for (i=0; i<n; i++) tmp[i] = b[c[i]];
1550 
1551   /* forward solve the U^T */
1552   for (i=0; i<n; i++) {
1553     v   = aa + adiag[i+1] + 1;
1554     vi  = aj + adiag[i+1] + 1;
1555     nz  = adiag[i] - adiag[i+1] - 1;
1556     s1  = tmp[i];
1557     s1 *= v[nz];  /* multiply by inverse of diagonal entry */
1558     for (j=0; j<nz; j++) tmp[vi[j]] -= s1*v[j];
1559     tmp[i] = s1;
1560   }
1561 
1562 
1563   /* backward solve the L^T */
1564   for (i=n-1; i>=0; i--){
1565     v   = aa + ai[i] ;
1566     vi  = aj + ai[i];
1567     nz  = ai[i+1] - ai[i];
1568     s1  = tmp[i];
1569     for (j=0; j<nz; j++) tmp[vi[j]] -= s1*v[j];
1570   }
1571 
1572   /* copy tmp into x according to permutation */
1573   for (i=0; i<n; i++) x[r[i]] += tmp[i];
1574 
1575   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1576   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1577   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
1578   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1579 
1580   ierr = PetscLogFlops(2.0*a->nz-A->cmap->n);CHKERRQ(ierr);
1581   PetscFunctionReturn(0);
1582 }
1583 
1584 /* ----------------------------------------------------------------*/
1585 
1586 EXTERN PetscErrorCode MatDuplicateNoCreate_SeqAIJ(Mat,Mat,MatDuplicateOption,PetscTruth);
1587 
1588 /*
1589    ilu() under revised new data structure.
1590    Factored arrays bj and ba are stored as
1591      L(0,:), L(1,:), ...,L(n-1,:),  U(n-1,:),...,U(i,:),U(i-1,:),...,U(0,:)
1592 
1593    bi=fact->i is an array of size n+1, in which
1594    bi+
1595      bi[i]:  points to 1st entry of L(i,:),i=0,...,n-1
1596      bi[n]:  points to L(n-1,n-1)+1
1597 
1598   bdiag=fact->diag is an array of size n+1,in which
1599      bdiag[i]: points to diagonal of U(i,:), i=0,...,n-1
1600      bdiag[n]: points to entry of U(n-1,0)-1
1601 
1602    U(i,:) contains bdiag[i] as its last entry, i.e.,
1603     U(i,:) = (u[i,i+1],...,u[i,n-1],diag[i])
1604 */
1605 #undef __FUNCT__
1606 #define __FUNCT__ "MatILUFactorSymbolic_SeqAIJ_ilu0"
1607 PetscErrorCode MatILUFactorSymbolic_SeqAIJ_ilu0(Mat fact,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
1608 {
1609 
1610   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
1611   PetscErrorCode     ierr;
1612   const PetscInt     n=A->rmap->n,*ai=a->i,*aj,*adiag=a->diag;
1613   PetscInt           i,j,k=0,nz,*bi,*bj,*bdiag;
1614   PetscTruth         missing;
1615   IS                 isicol;
1616 
1617   PetscFunctionBegin;
1618   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);
1619   ierr = MatMissingDiagonal(A,&missing,&i);CHKERRQ(ierr);
1620   if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",i);
1621   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
1622 
1623   ierr = MatDuplicateNoCreate_SeqAIJ(fact,A,MAT_DO_NOT_COPY_VALUES,PETSC_FALSE);CHKERRQ(ierr);
1624   b    = (Mat_SeqAIJ*)(fact)->data;
1625 
1626   /* allocate matrix arrays for new data structure */
1627   ierr = PetscMalloc3(ai[n]+1,PetscScalar,&b->a,ai[n]+1,PetscInt,&b->j,n+1,PetscInt,&b->i);CHKERRQ(ierr);
1628   ierr = PetscLogObjectMemory(fact,ai[n]*(sizeof(PetscScalar)+sizeof(PetscInt))+(n+1)*sizeof(PetscInt));CHKERRQ(ierr);
1629   b->singlemalloc = PETSC_TRUE;
1630   if (!b->diag){
1631     ierr = PetscMalloc((n+1)*sizeof(PetscInt),&b->diag);CHKERRQ(ierr);
1632     ierr = PetscLogObjectMemory(fact,(n+1)*sizeof(PetscInt));CHKERRQ(ierr);
1633   }
1634   bdiag = b->diag;
1635 
1636   if (n > 0) {
1637     ierr = PetscMemzero(b->a,(ai[n])*sizeof(MatScalar));CHKERRQ(ierr);
1638   }
1639 
1640   /* set bi and bj with new data structure */
1641   bi = b->i;
1642   bj = b->j;
1643 
1644   /* L part */
1645   bi[0] = 0;
1646   for (i=0; i<n; i++){
1647     nz = adiag[i] - ai[i];
1648     bi[i+1] = bi[i] + nz;
1649     aj = a->j + ai[i];
1650     for (j=0; j<nz; j++){
1651       /*   *bj = aj[j]; bj++; */
1652       bj[k++] = aj[j];
1653     }
1654   }
1655 
1656   /* U part */
1657   bdiag[n] = bi[n]-1;
1658   for (i=n-1; i>=0; i--){
1659     nz = ai[i+1] - adiag[i] - 1;
1660     aj = a->j + adiag[i] + 1;
1661     for (j=0; j<nz; j++){
1662       /*      *bj = aj[j]; bj++; */
1663       bj[k++] = aj[j];
1664     }
1665     /* diag[i] */
1666     /*    *bj = i; bj++; */
1667     bj[k++] = i;
1668     bdiag[i] = bdiag[i+1] + nz + 1;
1669   }
1670 
1671   fact->factortype             = MAT_FACTOR_ILU;
1672   fact->info.factor_mallocs    = 0;
1673   fact->info.fill_ratio_given  = info->fill;
1674   fact->info.fill_ratio_needed = 1.0;
1675   fact->ops->lufactornumeric   = MatLUFactorNumeric_SeqAIJ;
1676 
1677   b       = (Mat_SeqAIJ*)(fact)->data;
1678   b->row  = isrow;
1679   b->col  = iscol;
1680   b->icol = isicol;
1681   ierr    = PetscMalloc((fact->rmap->n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1682   ierr    = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
1683   ierr    = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
1684   PetscFunctionReturn(0);
1685 }
1686 
1687 #undef __FUNCT__
1688 #define __FUNCT__ "MatILUFactorSymbolic_SeqAIJ"
1689 PetscErrorCode MatILUFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
1690 {
1691   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
1692   IS                 isicol;
1693   PetscErrorCode     ierr;
1694   const PetscInt     *r,*ic;
1695   PetscInt           n=A->rmap->n,*ai=a->i,*aj=a->j;
1696   PetscInt           *bi,*cols,nnz,*cols_lvl;
1697   PetscInt           *bdiag,prow,fm,nzbd,reallocs=0,dcount=0;
1698   PetscInt           i,levels,diagonal_fill;
1699   PetscTruth         col_identity,row_identity;
1700   PetscReal          f;
1701   PetscInt           nlnk,*lnk,*lnk_lvl=PETSC_NULL;
1702   PetscBT            lnkbt;
1703   PetscInt           nzi,*bj,**bj_ptr,**bjlvl_ptr;
1704   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
1705   PetscFreeSpaceList free_space_lvl=PETSC_NULL,current_space_lvl=PETSC_NULL;
1706 
1707   PetscFunctionBegin;
1708   /* Uncomment the old data struct part only while testing new data structure for MatSolve() */
1709   /*
1710   PetscTruth         olddatastruct=PETSC_FALSE;
1711   ierr = PetscOptionsGetTruth(PETSC_NULL,"-ilu_old",&olddatastruct,PETSC_NULL);CHKERRQ(ierr);
1712   if(olddatastruct){
1713     ierr = MatILUFactorSymbolic_SeqAIJ_inplace(fact,A,isrow,iscol,info);CHKERRQ(ierr);
1714     PetscFunctionReturn(0);
1715   }
1716   */
1717 
1718   levels = (PetscInt)info->levels;
1719   ierr   = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
1720   ierr   = ISIdentity(iscol,&col_identity);CHKERRQ(ierr);
1721 
1722   if (!levels && row_identity && col_identity) {
1723     /* special case: ilu(0) with natural ordering */
1724     ierr = MatILUFactorSymbolic_SeqAIJ_ilu0(fact,A,isrow,iscol,info);CHKERRQ(ierr);
1725     if (a->inode.size) {
1726       fact->ops->lufactornumeric  = MatLUFactorNumeric_SeqAIJ_Inode;
1727     }
1728     PetscFunctionReturn(0);
1729   }
1730 
1731   if (A->rmap->n != A->cmap->n) SETERRQ2(PETSC_ERR_ARG_WRONG,"Must be square matrix, rows %D columns %D",A->rmap->n,A->cmap->n);
1732   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
1733   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
1734   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
1735 
1736   /* get new row and diagonal pointers, must be allocated separately because they will be given to the Mat_SeqAIJ and freed separately */
1737   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
1738   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);
1739   bi[0] = bdiag[0] = 0;
1740 
1741   ierr = PetscMalloc2(n,PetscInt*,&bj_ptr,n,PetscInt*,&bjlvl_ptr);CHKERRQ(ierr);
1742 
1743   /* create a linked list for storing column indices of the active row */
1744   nlnk = n + 1;
1745   ierr = PetscIncompleteLLCreate(n,n,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1746 
1747   /* initial FreeSpace size is f*(ai[n]+1) */
1748   f             = info->fill;
1749   diagonal_fill = (PetscInt)info->diagonal_fill;
1750   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space);CHKERRQ(ierr);
1751   current_space = free_space;
1752   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space_lvl);CHKERRQ(ierr);
1753   current_space_lvl = free_space_lvl;
1754 
1755   for (i=0; i<n; i++) {
1756     nzi = 0;
1757     /* copy current row into linked list */
1758     nnz  = ai[r[i]+1] - ai[r[i]];
1759     if (!nnz) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
1760     cols = aj + ai[r[i]];
1761     lnk[i] = -1; /* marker to indicate if diagonal exists */
1762     ierr = PetscIncompleteLLInit(nnz,cols,n,ic,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1763     nzi += nlnk;
1764 
1765     /* make sure diagonal entry is included */
1766     if (diagonal_fill && lnk[i] == -1) {
1767       fm = n;
1768       while (lnk[fm] < i) fm = lnk[fm];
1769       lnk[i]     = lnk[fm]; /* insert diagonal into linked list */
1770       lnk[fm]    = i;
1771       lnk_lvl[i] = 0;
1772       nzi++; dcount++;
1773     }
1774 
1775     /* add pivot rows into the active row */
1776     nzbd = 0;
1777     prow = lnk[n];
1778     while (prow < i) {
1779       nnz      = bdiag[prow];
1780       cols     = bj_ptr[prow] + nnz + 1;
1781       cols_lvl = bjlvl_ptr[prow] + nnz + 1;
1782       nnz      = bi[prow+1] - bi[prow] - nnz - 1;
1783       ierr = PetscILULLAddSorted(nnz,cols,levels,cols_lvl,prow,nlnk,lnk,lnk_lvl,lnkbt,prow);CHKERRQ(ierr);
1784       nzi += nlnk;
1785       prow = lnk[prow];
1786       nzbd++;
1787     }
1788     bdiag[i] = nzbd;
1789     bi[i+1]  = bi[i] + nzi;
1790 
1791     /* if free space is not available, make more free space */
1792     if (current_space->local_remaining<nzi) {
1793       nnz = 2*nzi*(n - i); /* estimated and max additional space needed */
1794       ierr = PetscFreeSpaceGet(nnz,&current_space);CHKERRQ(ierr);
1795       ierr = PetscFreeSpaceGet(nnz,&current_space_lvl);CHKERRQ(ierr);
1796       reallocs++;
1797     }
1798 
1799     /* copy data into free_space and free_space_lvl, then initialize lnk */
1800     ierr = PetscIncompleteLLClean(n,n,nzi,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);CHKERRQ(ierr);
1801     bj_ptr[i]    = current_space->array;
1802     bjlvl_ptr[i] = current_space_lvl->array;
1803 
1804     /* make sure the active row i has diagonal entry */
1805     if (*(bj_ptr[i]+bdiag[i]) != i) {
1806       SETERRQ1(PETSC_ERR_MAT_LU_ZRPVT,"Row %D has missing diagonal in factored matrix\n\
1807     try running with -pc_factor_nonzeros_along_diagonal or -pc_factor_diagonal_fill",i);
1808     }
1809 
1810     current_space->array           += nzi;
1811     current_space->local_used      += nzi;
1812     current_space->local_remaining -= nzi;
1813     current_space_lvl->array           += nzi;
1814     current_space_lvl->local_used      += nzi;
1815     current_space_lvl->local_remaining -= nzi;
1816   }
1817 
1818   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
1819   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
1820 
1821   /* destroy list of free space and other temporary arrays */
1822   ierr = PetscMalloc((bi[n]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
1823 
1824   /* copy free_space into bj and free free_space; set bi, bj, bdiag in new datastructure; */
1825   ierr = PetscFreeSpaceContiguous_LU(&free_space,bj,n,bi,bdiag);CHKERRQ(ierr);
1826 
1827   ierr = PetscIncompleteLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1828   ierr = PetscFreeSpaceDestroy(free_space_lvl);CHKERRQ(ierr);
1829   ierr = PetscFree2(bj_ptr,bjlvl_ptr);CHKERRQ(ierr);
1830 
1831 #if defined(PETSC_USE_INFO)
1832   {
1833     PetscReal af = ((PetscReal)bi[n])/((PetscReal)ai[n]);
1834     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,f,af);CHKERRQ(ierr);
1835     ierr = PetscInfo1(A,"Run with -[sub_]pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
1836     ierr = PetscInfo1(A,"PCFactorSetFill([sub]pc,%G);\n",af);CHKERRQ(ierr);
1837     ierr = PetscInfo(A,"for best performance.\n");CHKERRQ(ierr);
1838     if (diagonal_fill) {
1839       ierr = PetscInfo1(A,"Detected and replaced %D missing diagonals",dcount);CHKERRQ(ierr);
1840     }
1841   }
1842 #endif
1843 
1844   /* put together the new matrix */
1845   ierr = MatSeqAIJSetPreallocation_SeqAIJ(fact,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
1846   ierr = PetscLogObjectParent(fact,isicol);CHKERRQ(ierr);
1847   b = (Mat_SeqAIJ*)(fact)->data;
1848   b->free_a       = PETSC_TRUE;
1849   b->free_ij      = PETSC_TRUE;
1850   b->singlemalloc = PETSC_FALSE;
1851   ierr = PetscMalloc((bdiag[0]+1)*sizeof(PetscScalar),&b->a);CHKERRQ(ierr);
1852   b->j          = bj;
1853   b->i          = bi;
1854   b->diag       = bdiag;
1855   b->ilen       = 0;
1856   b->imax       = 0;
1857   b->row        = isrow;
1858   b->col        = iscol;
1859   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
1860   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
1861   b->icol       = isicol;
1862   ierr = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1863   /* In b structure:  Free imax, ilen, old a, old j.
1864      Allocate bdiag, solve_work, new a, new j */
1865   ierr = PetscLogObjectMemory(fact,(bdiag[0]+1)*(sizeof(PetscInt)+sizeof(PetscScalar)));CHKERRQ(ierr);
1866   b->maxnz = b->nz = bdiag[0]+1;
1867   (fact)->info.factor_mallocs    = reallocs;
1868   (fact)->info.fill_ratio_given  = f;
1869   (fact)->info.fill_ratio_needed = ((PetscReal)(bdiag[0]+1))/((PetscReal)ai[n]);
1870   (fact)->ops->lufactornumeric = MatLUFactorNumeric_SeqAIJ;
1871   if (a->inode.size) {
1872     (fact)->ops->lufactornumeric  = MatLUFactorNumeric_SeqAIJ_Inode;
1873   }
1874   PetscFunctionReturn(0);
1875 }
1876 
1877 #undef __FUNCT__
1878 #define __FUNCT__ "MatILUFactorSymbolic_SeqAIJ_inplace"
1879 PetscErrorCode MatILUFactorSymbolic_SeqAIJ_inplace(Mat fact,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
1880 {
1881   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
1882   IS                 isicol;
1883   PetscErrorCode     ierr;
1884   const PetscInt     *r,*ic;
1885   PetscInt           n=A->rmap->n,*ai=a->i,*aj=a->j,d;
1886   PetscInt           *bi,*cols,nnz,*cols_lvl;
1887   PetscInt           *bdiag,prow,fm,nzbd,reallocs=0,dcount=0;
1888   PetscInt           i,levels,diagonal_fill;
1889   PetscTruth         col_identity,row_identity;
1890   PetscReal          f;
1891   PetscInt           nlnk,*lnk,*lnk_lvl=PETSC_NULL;
1892   PetscBT            lnkbt;
1893   PetscInt           nzi,*bj,**bj_ptr,**bjlvl_ptr;
1894   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
1895   PetscFreeSpaceList free_space_lvl=PETSC_NULL,current_space_lvl=PETSC_NULL;
1896   PetscTruth         missing;
1897 
1898   PetscFunctionBegin;
1899   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);
1900   f             = info->fill;
1901   levels        = (PetscInt)info->levels;
1902   diagonal_fill = (PetscInt)info->diagonal_fill;
1903   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
1904 
1905   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
1906   ierr = ISIdentity(iscol,&col_identity);CHKERRQ(ierr);
1907   if (!levels && row_identity && col_identity) { /* special case: ilu(0) with natural ordering */
1908     ierr = MatDuplicateNoCreate_SeqAIJ(fact,A,MAT_DO_NOT_COPY_VALUES,PETSC_TRUE);CHKERRQ(ierr);
1909     (fact)->ops->lufactornumeric =  MatLUFactorNumeric_SeqAIJ_inplace;
1910     if (a->inode.size) {
1911       (fact)->ops->lufactornumeric  = MatLUFactorNumeric_SeqAIJ_Inode_inplace;
1912     }
1913     fact->factortype = MAT_FACTOR_ILU;
1914     (fact)->info.factor_mallocs    = 0;
1915     (fact)->info.fill_ratio_given  = info->fill;
1916     (fact)->info.fill_ratio_needed = 1.0;
1917     b               = (Mat_SeqAIJ*)(fact)->data;
1918     ierr = MatMissingDiagonal(A,&missing,&d);CHKERRQ(ierr);
1919     if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",d);
1920     b->row              = isrow;
1921     b->col              = iscol;
1922     b->icol             = isicol;
1923     ierr                = PetscMalloc(((fact)->rmap->n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1924     ierr                = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
1925     ierr                = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
1926     PetscFunctionReturn(0);
1927   }
1928 
1929   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
1930   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
1931 
1932   /* get new row and diagonal pointers, must be allocated separately because they will be given to the Mat_SeqAIJ and freed separately */
1933   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
1934   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);
1935   bi[0] = bdiag[0] = 0;
1936 
1937   ierr = PetscMalloc2(n,PetscInt*,&bj_ptr,n,PetscInt*,&bjlvl_ptr);CHKERRQ(ierr);
1938 
1939   /* create a linked list for storing column indices of the active row */
1940   nlnk = n + 1;
1941   ierr = PetscIncompleteLLCreate(n,n,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1942 
1943   /* initial FreeSpace size is f*(ai[n]+1) */
1944   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space);CHKERRQ(ierr);
1945   current_space = free_space;
1946   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space_lvl);CHKERRQ(ierr);
1947   current_space_lvl = free_space_lvl;
1948 
1949   for (i=0; i<n; i++) {
1950     nzi = 0;
1951     /* copy current row into linked list */
1952     nnz  = ai[r[i]+1] - ai[r[i]];
1953     if (!nnz) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
1954     cols = aj + ai[r[i]];
1955     lnk[i] = -1; /* marker to indicate if diagonal exists */
1956     ierr = PetscIncompleteLLInit(nnz,cols,n,ic,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1957     nzi += nlnk;
1958 
1959     /* make sure diagonal entry is included */
1960     if (diagonal_fill && lnk[i] == -1) {
1961       fm = n;
1962       while (lnk[fm] < i) fm = lnk[fm];
1963       lnk[i]     = lnk[fm]; /* insert diagonal into linked list */
1964       lnk[fm]    = i;
1965       lnk_lvl[i] = 0;
1966       nzi++; dcount++;
1967     }
1968 
1969     /* add pivot rows into the active row */
1970     nzbd = 0;
1971     prow = lnk[n];
1972     while (prow < i) {
1973       nnz      = bdiag[prow];
1974       cols     = bj_ptr[prow] + nnz + 1;
1975       cols_lvl = bjlvl_ptr[prow] + nnz + 1;
1976       nnz      = bi[prow+1] - bi[prow] - nnz - 1;
1977       ierr = PetscILULLAddSorted(nnz,cols,levels,cols_lvl,prow,nlnk,lnk,lnk_lvl,lnkbt,prow);CHKERRQ(ierr);
1978       nzi += nlnk;
1979       prow = lnk[prow];
1980       nzbd++;
1981     }
1982     bdiag[i] = nzbd;
1983     bi[i+1]  = bi[i] + nzi;
1984 
1985     /* if free space is not available, make more free space */
1986     if (current_space->local_remaining<nzi) {
1987       nnz = nzi*(n - i); /* estimated and max additional space needed */
1988       ierr = PetscFreeSpaceGet(nnz,&current_space);CHKERRQ(ierr);
1989       ierr = PetscFreeSpaceGet(nnz,&current_space_lvl);CHKERRQ(ierr);
1990       reallocs++;
1991     }
1992 
1993     /* copy data into free_space and free_space_lvl, then initialize lnk */
1994     ierr = PetscIncompleteLLClean(n,n,nzi,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);CHKERRQ(ierr);
1995     bj_ptr[i]    = current_space->array;
1996     bjlvl_ptr[i] = current_space_lvl->array;
1997 
1998     /* make sure the active row i has diagonal entry */
1999     if (*(bj_ptr[i]+bdiag[i]) != i) {
2000       SETERRQ1(PETSC_ERR_MAT_LU_ZRPVT,"Row %D has missing diagonal in factored matrix\n\
2001     try running with -pc_factor_nonzeros_along_diagonal or -pc_factor_diagonal_fill",i);
2002     }
2003 
2004     current_space->array           += nzi;
2005     current_space->local_used      += nzi;
2006     current_space->local_remaining -= nzi;
2007     current_space_lvl->array           += nzi;
2008     current_space_lvl->local_used      += nzi;
2009     current_space_lvl->local_remaining -= nzi;
2010   }
2011 
2012   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
2013   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
2014 
2015   /* destroy list of free space and other temporary arrays */
2016   ierr = PetscMalloc((bi[n]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
2017   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr); /* copy free_space -> bj */
2018   ierr = PetscIncompleteLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
2019   ierr = PetscFreeSpaceDestroy(free_space_lvl);CHKERRQ(ierr);
2020   ierr = PetscFree2(bj_ptr,bjlvl_ptr);CHKERRQ(ierr);
2021 
2022 #if defined(PETSC_USE_INFO)
2023   {
2024     PetscReal af = ((PetscReal)bi[n])/((PetscReal)ai[n]);
2025     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,f,af);CHKERRQ(ierr);
2026     ierr = PetscInfo1(A,"Run with -[sub_]pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
2027     ierr = PetscInfo1(A,"PCFactorSetFill([sub]pc,%G);\n",af);CHKERRQ(ierr);
2028     ierr = PetscInfo(A,"for best performance.\n");CHKERRQ(ierr);
2029     if (diagonal_fill) {
2030       ierr = PetscInfo1(A,"Detected and replaced %D missing diagonals",dcount);CHKERRQ(ierr);
2031     }
2032   }
2033 #endif
2034 
2035   /* put together the new matrix */
2036   ierr = MatSeqAIJSetPreallocation_SeqAIJ(fact,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
2037   ierr = PetscLogObjectParent(fact,isicol);CHKERRQ(ierr);
2038   b = (Mat_SeqAIJ*)(fact)->data;
2039   b->free_a       = PETSC_TRUE;
2040   b->free_ij      = PETSC_TRUE;
2041   b->singlemalloc = PETSC_FALSE;
2042   ierr = PetscMalloc(bi[n]*sizeof(PetscScalar),&b->a);CHKERRQ(ierr);
2043   b->j          = bj;
2044   b->i          = bi;
2045   for (i=0; i<n; i++) bdiag[i] += bi[i];
2046   b->diag       = bdiag;
2047   b->ilen       = 0;
2048   b->imax       = 0;
2049   b->row        = isrow;
2050   b->col        = iscol;
2051   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
2052   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
2053   b->icol       = isicol;
2054   ierr = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
2055   /* In b structure:  Free imax, ilen, old a, old j.
2056      Allocate bdiag, solve_work, new a, new j */
2057   ierr = PetscLogObjectMemory(fact,(bi[n]-n) * (sizeof(PetscInt)+sizeof(PetscScalar)));CHKERRQ(ierr);
2058   b->maxnz             = b->nz = bi[n] ;
2059   (fact)->info.factor_mallocs    = reallocs;
2060   (fact)->info.fill_ratio_given  = f;
2061   (fact)->info.fill_ratio_needed = ((PetscReal)bi[n])/((PetscReal)ai[n]);
2062   (fact)->ops->lufactornumeric =  MatLUFactorNumeric_SeqAIJ_inplace;
2063   if (a->inode.size) {
2064     (fact)->ops->lufactornumeric = MatLUFactorNumeric_SeqAIJ_Inode_inplace;
2065   }
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   PetscInt       newshift = 0;
2085 
2086   FactorShiftCtx sctx;
2087   PetscReal      rs;
2088   MatScalar      d,*v;
2089 
2090   PetscFunctionBegin;
2091   /* MatPivotSetUp(): initialize shift context sctx */
2092   ierr = PetscMemzero(&sctx,sizeof(FactorShiftCtx));CHKERRQ(ierr);
2093 
2094   if (info->shifttype == (PetscReal)MAT_SHIFT_POSITIVE_DEFINITE) { /* set sctx.shift_top=max{rs} */
2095     sctx.shift_top = info->zeropivot;
2096     for (i=0; i<mbs; i++) {
2097       /* calculate sum(|aij|)-RealPart(aii), amt of shift needed for this row */
2098       d  = (aa)[a->diag[i]];
2099       rs = -PetscAbsScalar(d) - PetscRealPart(d);
2100       v  = aa+ai[i];
2101       nz = ai[i+1] - ai[i];
2102       for (j=0; j<nz; j++)
2103 	rs += PetscAbsScalar(v[j]);
2104       if (rs>sctx.shift_top) sctx.shift_top = rs;
2105     }
2106     sctx.shift_top   *= 1.1;
2107     sctx.nshift_max   = 5;
2108     sctx.shift_lo     = 0.;
2109     sctx.shift_hi     = 1.;
2110   }
2111 
2112   ierr  = ISGetIndices(ip,&rip);CHKERRQ(ierr);
2113   ierr  = ISGetIndices(iip,&riip);CHKERRQ(ierr);
2114 
2115   /* allocate working arrays
2116      c2r: linked list, keep track of pivot rows for a given column. c2r[col]: head of the list for a given col
2117      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
2118   */
2119   ierr = PetscMalloc3(mbs,MatScalar,&rtmp,mbs,PetscInt,&il,mbs,PetscInt,&c2r);CHKERRQ(ierr);
2120 
2121   do {
2122     sctx.useshift = PETSC_FALSE;
2123 
2124     for (i=0; i<mbs; i++) c2r[i] = mbs;
2125     il[0] = 0;
2126 
2127     for (k = 0; k<mbs; k++){
2128       /* zero rtmp */
2129       nz = bi[k+1] - bi[k];
2130       bjtmp = bj + bi[k];
2131       for (j=0; j<nz; j++) rtmp[bjtmp[j]] = 0.0;
2132 
2133       /* load in initial unfactored row */
2134       bval = ba + bi[k];
2135       jmin = ai[rip[k]]; jmax = ai[rip[k]+1];
2136       for (j = jmin; j < jmax; j++){
2137         col = riip[aj[j]];
2138         if (col >= k){ /* only take upper triangular entry */
2139           rtmp[col] = aa[j];
2140           *bval++   = 0.0; /* for in-place factorization */
2141         }
2142       }
2143       /* shift the diagonal of the matrix: ZeropivotApply() */
2144       rtmp[k] += sctx.shift_amount;  /* shift the diagonal of the matrix */
2145 
2146       /* modify k-th row by adding in those rows i with U(i,k)!=0 */
2147       dk = rtmp[k];
2148       i  = c2r[k]; /* first row to be added to k_th row  */
2149 
2150       while (i < k){
2151         nexti = c2r[i]; /* next row to be added to k_th row */
2152 
2153         /* compute multiplier, update diag(k) and U(i,k) */
2154         ili   = il[i];  /* index of first nonzero element in U(i,k:bms-1) */
2155         uikdi = - ba[ili]*ba[bdiag[i]];  /* diagonal(k) */
2156         dk   += uikdi*ba[ili]; /* update diag[k] */
2157         ba[ili] = uikdi; /* -U(i,k) */
2158 
2159         /* add multiple of row i to k-th row */
2160         jmin = ili + 1; jmax = bi[i+1];
2161         if (jmin < jmax){
2162           for (j=jmin; j<jmax; j++) rtmp[bj[j]] += uikdi*ba[j];
2163           /* update il and c2r for row i */
2164           il[i] = jmin;
2165           j = bj[jmin]; c2r[i] = c2r[j]; c2r[j] = i;
2166         }
2167         i = nexti;
2168       }
2169 
2170       /* copy data into U(k,:) */
2171       rs   = 0.0;
2172       jmin = bi[k]; jmax = bi[k+1]-1;
2173       if (jmin < jmax) {
2174         for (j=jmin; j<jmax; j++){
2175           col = bj[j]; ba[j] = rtmp[col]; rs += PetscAbsScalar(ba[j]);
2176         }
2177         /* add the k-th row into il and c2r */
2178         il[k] = jmin;
2179         i = bj[jmin]; c2r[k] = c2r[i]; c2r[i] = k;
2180       }
2181 
2182       /* MatPivotCheck() */
2183       sctx.rs  = rs;
2184       sctx.pv  = dk;
2185       ierr = MatPivotCheck(info,&sctx,i,&newshift);CHKERRQ(ierr);
2186       if(newshift == 1) break;
2187       dk = sctx.pv;
2188 
2189       ba[bdiag[k]] = 1.0/dk; /* U(k,k) */
2190     }
2191   } while (sctx.useshift);
2192 
2193   ierr = PetscFree3(rtmp,il,c2r);CHKERRQ(ierr);
2194   ierr = ISRestoreIndices(ip,&rip);CHKERRQ(ierr);
2195   ierr = ISRestoreIndices(iip,&riip);CHKERRQ(ierr);
2196 
2197   ierr = ISIdentity(ip,&perm_identity);CHKERRQ(ierr);
2198   if (perm_identity){
2199     B->ops->solve           = MatSolve_SeqSBAIJ_1_NaturalOrdering;
2200     B->ops->solvetranspose  = MatSolve_SeqSBAIJ_1_NaturalOrdering;
2201     B->ops->forwardsolve    = MatForwardSolve_SeqSBAIJ_1_NaturalOrdering;
2202     B->ops->backwardsolve   = MatBackwardSolve_SeqSBAIJ_1_NaturalOrdering;
2203   } else {
2204     B->ops->solve           = MatSolve_SeqSBAIJ_1;
2205     B->ops->solvetranspose  = MatSolve_SeqSBAIJ_1;
2206     B->ops->forwardsolve    = MatForwardSolve_SeqSBAIJ_1;
2207     B->ops->backwardsolve   = MatBackwardSolve_SeqSBAIJ_1;
2208   }
2209 
2210   C->assembled    = PETSC_TRUE;
2211   C->preallocated = PETSC_TRUE;
2212   ierr = PetscLogFlops(C->rmap->n);CHKERRQ(ierr);
2213 
2214   /* MatPivotView() */
2215   if (sctx.nshift){
2216     if (info->shifttype == (PetscReal)MAT_SHIFT_POSITIVE_DEFINITE) {
2217       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);
2218     } else if (info->shifttype == (PetscReal)MAT_SHIFT_NONZERO) {
2219       ierr = PetscInfo2(A,"number of shift_nz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
2220     } else if (info->shifttype == (PetscReal)MAT_SHIFT_INBLOCKS){
2221       ierr = PetscInfo2(A,"number of shift_inblocks applied %D, each shift_amount %G\n",sctx.nshift,info->shiftamount);CHKERRQ(ierr);
2222     }
2223   }
2224   PetscFunctionReturn(0);
2225 }
2226 
2227 #undef __FUNCT__
2228 #define __FUNCT__ "MatCholeskyFactorNumeric_SeqAIJ_inplace"
2229 PetscErrorCode MatCholeskyFactorNumeric_SeqAIJ_inplace(Mat B,Mat A,const MatFactorInfo *info)
2230 {
2231   Mat            C = B;
2232   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data;
2233   Mat_SeqSBAIJ   *b=(Mat_SeqSBAIJ*)C->data;
2234   IS             ip=b->row,iip = b->icol;
2235   PetscErrorCode ierr;
2236   const PetscInt *rip,*riip;
2237   PetscInt       i,j,mbs=A->rmap->n,*bi=b->i,*bj=b->j,*bcol,*bjtmp;
2238   PetscInt       *ai=a->i,*aj=a->j;
2239   PetscInt       k,jmin,jmax,*jl,*il,col,nexti,ili,nz;
2240   MatScalar      *rtmp,*ba=b->a,*bval,*aa=a->a,dk,uikdi;
2241   PetscReal      zeropivot,rs;
2242   ChShift_Ctx    sctx;
2243   PetscInt       newshift;
2244   PetscTruth     perm_identity;
2245 
2246   PetscFunctionBegin;
2247   zeropivot = info->zeropivot;
2248 
2249   ierr  = ISGetIndices(ip,&rip);CHKERRQ(ierr);
2250   ierr  = ISGetIndices(iip,&riip);CHKERRQ(ierr);
2251 
2252   /* initialization */
2253   ierr = PetscMalloc3(mbs,MatScalar,&rtmp,mbs,PetscInt,&il,mbs,PetscInt,&jl);CHKERRQ(ierr);
2254   sctx.shift_amount = 0;
2255   sctx.nshift       = 0;
2256   do {
2257     sctx.chshift = PETSC_FALSE;
2258     for (i=0; i<mbs; i++) jl[i] = mbs;
2259     il[0] = 0;
2260 
2261     for (k = 0; k<mbs; k++){
2262       /* zero rtmp */
2263       nz = bi[k+1] - bi[k];
2264       bjtmp = bj + bi[k];
2265       for (j=0; j<nz; j++) rtmp[bjtmp[j]] = 0.0;
2266 
2267       bval = ba + bi[k];
2268       /* initialize k-th row by the perm[k]-th row of A */
2269       jmin = ai[rip[k]]; jmax = ai[rip[k]+1];
2270       for (j = jmin; j < jmax; j++){
2271         col = riip[aj[j]];
2272         if (col >= k){ /* only take upper triangular entry */
2273           rtmp[col] = aa[j];
2274           *bval++  = 0.0; /* for in-place factorization */
2275         }
2276       }
2277       /* shift the diagonal of the matrix */
2278       if (sctx.nshift) rtmp[k] += sctx.shift_amount;
2279 
2280       /* modify k-th row by adding in those rows i with U(i,k)!=0 */
2281       dk = rtmp[k];
2282       i = jl[k]; /* first row to be added to k_th row  */
2283 
2284       while (i < k){
2285         nexti = jl[i]; /* next row to be added to k_th row */
2286 
2287         /* compute multiplier, update diag(k) and U(i,k) */
2288         ili = il[i];  /* index of first nonzero element in U(i,k:bms-1) */
2289         uikdi = - ba[ili]*ba[bi[i]];  /* diagonal(k) */
2290         dk += uikdi*ba[ili];
2291         ba[ili] = uikdi; /* -U(i,k) */
2292 
2293         /* add multiple of row i to k-th row */
2294         jmin = ili + 1; jmax = bi[i+1];
2295         if (jmin < jmax){
2296           for (j=jmin; j<jmax; j++) rtmp[bj[j]] += uikdi*ba[j];
2297           /* update il and jl for row i */
2298           il[i] = jmin;
2299           j = bj[jmin]; jl[i] = jl[j]; jl[j] = i;
2300         }
2301         i = nexti;
2302       }
2303 
2304       /* shift the diagonals when zero pivot is detected */
2305       /* compute rs=sum of abs(off-diagonal) */
2306       rs   = 0.0;
2307       jmin = bi[k]+1;
2308       nz   = bi[k+1] - jmin;
2309       bcol = bj + jmin;
2310       for (j=0; j<nz; j++) {
2311         rs += PetscAbsScalar(rtmp[bcol[j]]);
2312       }
2313 
2314       sctx.rs = rs;
2315       sctx.pv = dk;
2316       ierr = MatCholeskyCheckShift_inline(info,sctx,k,newshift);CHKERRQ(ierr);
2317 
2318       if (newshift == 1) {
2319         if (!sctx.shift_amount) {
2320           sctx.shift_amount = 1e-5;
2321         }
2322         break;
2323       }
2324 
2325       /* copy data into U(k,:) */
2326       ba[bi[k]] = 1.0/dk; /* U(k,k) */
2327       jmin = bi[k]+1; jmax = bi[k+1];
2328       if (jmin < jmax) {
2329         for (j=jmin; j<jmax; j++){
2330           col = bj[j]; ba[j] = rtmp[col];
2331         }
2332         /* add the k-th row into il and jl */
2333         il[k] = jmin;
2334         i = bj[jmin]; jl[k] = jl[i]; jl[i] = k;
2335       }
2336     }
2337   } while (sctx.chshift);
2338   ierr = PetscFree3(rtmp,il,jl);CHKERRQ(ierr);
2339   ierr = ISRestoreIndices(ip,&rip);CHKERRQ(ierr);
2340   ierr = ISRestoreIndices(iip,&riip);CHKERRQ(ierr);
2341 
2342   ierr = ISIdentity(ip,&perm_identity);CHKERRQ(ierr);
2343   if (perm_identity){
2344     B->ops->solve           = MatSolve_SeqSBAIJ_1_NaturalOrdering_inplace;
2345     B->ops->solvetranspose  = MatSolve_SeqSBAIJ_1_NaturalOrdering_inplace;
2346     B->ops->forwardsolve    = MatForwardSolve_SeqSBAIJ_1_NaturalOrdering_inplace;
2347     B->ops->backwardsolve   = MatBackwardSolve_SeqSBAIJ_1_NaturalOrdering_inplace;
2348   } else {
2349     B->ops->solve           = MatSolve_SeqSBAIJ_1_inplace;
2350     B->ops->solvetranspose  = MatSolve_SeqSBAIJ_1_inplace;
2351     B->ops->forwardsolve    = MatForwardSolve_SeqSBAIJ_1_inplace;
2352     B->ops->backwardsolve   = MatBackwardSolve_SeqSBAIJ_1_inplace;
2353   }
2354 
2355   C->assembled    = PETSC_TRUE;
2356   C->preallocated = PETSC_TRUE;
2357   ierr = PetscLogFlops(C->rmap->n);CHKERRQ(ierr);
2358   if (sctx.nshift){
2359     if (info->shifttype == (PetscReal)MAT_SHIFT_NONZERO) {
2360       ierr = PetscInfo2(A,"number of shiftnz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
2361     } else if (info->shifttype == (PetscReal)MAT_SHIFT_POSITIVE_DEFINITE) {
2362       ierr = PetscInfo2(A,"number of shiftpd tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
2363     }
2364   }
2365   PetscFunctionReturn(0);
2366 }
2367 
2368 /*
2369    icc() under revised new data structure.
2370    Factored arrays bj and ba are stored as
2371      U(0,:),...,U(i,:),U(n-1,:)
2372 
2373    ui=fact->i is an array of size n+1, in which
2374    ui+
2375      ui[i]:  points to 1st entry of U(i,:),i=0,...,n-1
2376      ui[n]:  points to U(n-1,n-1)+1
2377 
2378   udiag=fact->diag is an array of size n,in which
2379      udiag[i]: points to diagonal of U(i,:), i=0,...,n-1
2380 
2381    U(i,:) contains udiag[i] as its last entry, i.e.,
2382     U(i,:) = (u[i,i+1],...,u[i,n-1],diag[i])
2383 */
2384 
2385 #undef __FUNCT__
2386 #define __FUNCT__ "MatICCFactorSymbolic_SeqAIJ"
2387 PetscErrorCode MatICCFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
2388 {
2389   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data;
2390   Mat_SeqSBAIJ       *b;
2391   PetscErrorCode     ierr;
2392   PetscTruth         perm_identity,missing;
2393   PetscInt           reallocs=0,i,*ai=a->i,*aj=a->j,am=A->rmap->n,*ui,*udiag;
2394   const PetscInt     *rip,*riip;
2395   PetscInt           jmin,jmax,nzk,k,j,*jl,prow,*il,nextprow;
2396   PetscInt           nlnk,*lnk,*lnk_lvl=PETSC_NULL,d;
2397   PetscInt           ncols,ncols_upper,*cols,*ajtmp,*uj,**uj_ptr,**uj_lvl_ptr;
2398   PetscReal          fill=info->fill,levels=info->levels;
2399   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
2400   PetscFreeSpaceList free_space_lvl=PETSC_NULL,current_space_lvl=PETSC_NULL;
2401   PetscBT            lnkbt;
2402   IS                 iperm;
2403 
2404   PetscFunctionBegin;
2405   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);
2406   ierr = MatMissingDiagonal(A,&missing,&d);CHKERRQ(ierr);
2407   if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",d);
2408   ierr = ISIdentity(perm,&perm_identity);CHKERRQ(ierr);
2409   ierr = ISInvertPermutation(perm,PETSC_DECIDE,&iperm);CHKERRQ(ierr);
2410 
2411   ierr = PetscMalloc((am+1)*sizeof(PetscInt),&ui);CHKERRQ(ierr);
2412   ierr = PetscMalloc((am+1)*sizeof(PetscInt),&udiag);CHKERRQ(ierr);
2413   ui[0] = 0;
2414 
2415   /* ICC(0) without matrix ordering: simply rearrange column indices */
2416   if (!levels && perm_identity) {
2417     for (i=0; i<am; i++) {
2418       ncols    = ai[i+1] - a->diag[i];
2419       ui[i+1]  = ui[i] + ncols;
2420       udiag[i] = ui[i+1] - 1; /* points to the last entry of U(i,:) */
2421     }
2422     ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
2423     cols = uj;
2424     for (i=0; i<am; i++) {
2425       aj    = a->j + a->diag[i] + 1; /* 1st entry of U(i,:) without diagonal */
2426       ncols = ai[i+1] - a->diag[i] -1;
2427       for (j=0; j<ncols; j++) *cols++ = aj[j];
2428       *cols++ = i; /* diagoanl is located as the last entry of U(i,:) */
2429     }
2430   } else { /* case: levels>0 || (levels=0 && !perm_identity) */
2431     ierr = ISGetIndices(iperm,&riip);CHKERRQ(ierr);
2432     ierr = ISGetIndices(perm,&rip);CHKERRQ(ierr);
2433 
2434     /* initialization */
2435     ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&ajtmp);CHKERRQ(ierr);
2436 
2437     /* jl: linked list for storing indices of the pivot rows
2438        il: il[i] points to the 1st nonzero entry of U(i,k:am-1) */
2439     ierr = PetscMalloc4(am,PetscInt*,&uj_ptr,am,PetscInt*,&uj_lvl_ptr,am,PetscInt,&jl,am,PetscInt,&il);CHKERRQ(ierr);
2440     for (i=0; i<am; i++){
2441       jl[i] = am; il[i] = 0;
2442     }
2443 
2444     /* create and initialize a linked list for storing column indices of the active row k */
2445     nlnk = am + 1;
2446     ierr = PetscIncompleteLLCreate(am,am,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
2447 
2448     /* initial FreeSpace size is fill*(ai[am]+1) */
2449     ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space);CHKERRQ(ierr);
2450     current_space = free_space;
2451     ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space_lvl);CHKERRQ(ierr);
2452     current_space_lvl = free_space_lvl;
2453 
2454     for (k=0; k<am; k++){  /* for each active row k */
2455       /* initialize lnk by the column indices of row rip[k] of A */
2456       nzk   = 0;
2457       ncols = ai[rip[k]+1] - ai[rip[k]];
2458       if (!ncols) SETERRQ2(PETSC_ERR_MAT_CH_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",rip[k],k);
2459       ncols_upper = 0;
2460       for (j=0; j<ncols; j++){
2461         i = *(aj + ai[rip[k]] + j); /* unpermuted column index */
2462         if (riip[i] >= k){ /* only take upper triangular entry */
2463           ajtmp[ncols_upper] = i;
2464           ncols_upper++;
2465         }
2466       }
2467       ierr = PetscIncompleteLLInit(ncols_upper,ajtmp,am,riip,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
2468       nzk += nlnk;
2469 
2470       /* update lnk by computing fill-in for each pivot row to be merged in */
2471       prow = jl[k]; /* 1st pivot row */
2472 
2473       while (prow < k){
2474         nextprow = jl[prow];
2475 
2476         /* merge prow into k-th row */
2477         jmin = il[prow] + 1;  /* index of the 2nd nzero entry in U(prow,k:am-1) */
2478         jmax = ui[prow+1];
2479         ncols = jmax-jmin;
2480         i     = jmin - ui[prow];
2481         cols  = uj_ptr[prow] + i; /* points to the 2nd nzero entry in U(prow,k:am-1) */
2482         uj    = uj_lvl_ptr[prow] + i; /* levels of cols */
2483         j     = *(uj - 1);
2484         ierr = PetscICCLLAddSorted(ncols,cols,levels,uj,am,nlnk,lnk,lnk_lvl,lnkbt,j);CHKERRQ(ierr);
2485         nzk += nlnk;
2486 
2487         /* update il and jl for prow */
2488         if (jmin < jmax){
2489           il[prow] = jmin;
2490           j = *cols; jl[prow] = jl[j]; jl[j] = prow;
2491         }
2492         prow = nextprow;
2493       }
2494 
2495       /* if free space is not available, make more free space */
2496       if (current_space->local_remaining<nzk) {
2497         i  = am - k + 1; /* num of unfactored rows */
2498         i *= PetscMin(nzk, i-1); /* i*nzk, i*(i-1): estimated and max additional space needed */
2499         ierr = PetscFreeSpaceGet(i,&current_space);CHKERRQ(ierr);
2500         ierr = PetscFreeSpaceGet(i,&current_space_lvl);CHKERRQ(ierr);
2501         reallocs++;
2502       }
2503 
2504       /* copy data into free_space and free_space_lvl, then initialize lnk */
2505       if (nzk == 0) SETERRQ1(PETSC_ERR_ARG_WRONG,"Empty row %D in ICC matrix factor",k);
2506       ierr = PetscIncompleteLLClean(am,am,nzk,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);CHKERRQ(ierr);
2507 
2508       /* add the k-th row into il and jl */
2509       if (nzk > 1){
2510         i = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:am-1) */
2511         jl[k] = jl[i]; jl[i] = k;
2512         il[k] = ui[k] + 1;
2513       }
2514       uj_ptr[k]     = current_space->array;
2515       uj_lvl_ptr[k] = current_space_lvl->array;
2516 
2517       current_space->array           += nzk;
2518       current_space->local_used      += nzk;
2519       current_space->local_remaining -= nzk;
2520 
2521       current_space_lvl->array           += nzk;
2522       current_space_lvl->local_used      += nzk;
2523       current_space_lvl->local_remaining -= nzk;
2524 
2525       ui[k+1] = ui[k] + nzk;
2526     }
2527 
2528 #if defined(PETSC_USE_INFO)
2529     if (ai[am] != 0) {
2530       PetscReal af = (PetscReal)ui[am]/((PetscReal)ai[am]);
2531       ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,fill,af);CHKERRQ(ierr);
2532       ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
2533       ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G) for best performance.\n",af);CHKERRQ(ierr);
2534     } else {
2535       ierr = PetscInfo(A,"Empty matrix.\n");CHKERRQ(ierr);
2536     }
2537 #endif
2538 
2539     ierr = ISRestoreIndices(perm,&rip);CHKERRQ(ierr);
2540     ierr = ISRestoreIndices(iperm,&riip);CHKERRQ(ierr);
2541     ierr = PetscFree4(uj_ptr,uj_lvl_ptr,jl,il);CHKERRQ(ierr);
2542     ierr = PetscFree(ajtmp);CHKERRQ(ierr);
2543 
2544     /* destroy list of free space and other temporary array(s) */
2545     ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
2546     ierr = PetscFreeSpaceContiguous_Cholesky(&free_space,uj,am,ui,udiag);CHKERRQ(ierr); /* store matrix factor  */
2547     ierr = PetscIncompleteLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
2548     ierr = PetscFreeSpaceDestroy(free_space_lvl);CHKERRQ(ierr);
2549 
2550   } /* end of case: levels>0 || (levels=0 && !perm_identity) */
2551 
2552   /* put together the new matrix in MATSEQSBAIJ format */
2553   b    = (Mat_SeqSBAIJ*)(fact)->data;
2554   b->singlemalloc = PETSC_FALSE;
2555   ierr = PetscMalloc((ui[am]+1)*sizeof(MatScalar),&b->a);CHKERRQ(ierr);
2556   b->j    = uj;
2557   b->i    = ui;
2558   b->diag = udiag;
2559   b->free_diag = PETSC_TRUE;
2560   b->ilen = 0;
2561   b->imax = 0;
2562   b->row  = perm;
2563   b->col  = perm;
2564   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2565   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2566   b->icol = iperm;
2567   b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */
2568   ierr = PetscMalloc((am+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
2569   ierr = PetscLogObjectMemory(fact,ui[am]*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
2570   b->maxnz   = b->nz = ui[am];
2571   b->free_a  = PETSC_TRUE;
2572   b->free_ij = PETSC_TRUE;
2573 
2574   fact->info.factor_mallocs    = reallocs;
2575   fact->info.fill_ratio_given  = fill;
2576   if (ai[am] != 0) {
2577     fact->info.fill_ratio_needed = ((PetscReal)ui[am])/((PetscReal)ai[am]);
2578   } else {
2579     fact->info.fill_ratio_needed = 0.0;
2580   }
2581   fact->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqAIJ;
2582   PetscFunctionReturn(0);
2583 }
2584 
2585 #undef __FUNCT__
2586 #define __FUNCT__ "MatICCFactorSymbolic_SeqAIJ_inplace"
2587 PetscErrorCode MatICCFactorSymbolic_SeqAIJ_inplace(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
2588 {
2589   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data;
2590   Mat_SeqSBAIJ       *b;
2591   PetscErrorCode     ierr;
2592   PetscTruth         perm_identity,missing;
2593   PetscInt           reallocs=0,i,*ai=a->i,*aj=a->j,am=A->rmap->n,*ui,*udiag;
2594   const PetscInt     *rip,*riip;
2595   PetscInt           jmin,jmax,nzk,k,j,*jl,prow,*il,nextprow;
2596   PetscInt           nlnk,*lnk,*lnk_lvl=PETSC_NULL,d;
2597   PetscInt           ncols,ncols_upper,*cols,*ajtmp,*uj,**uj_ptr,**uj_lvl_ptr;
2598   PetscReal          fill=info->fill,levels=info->levels;
2599   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
2600   PetscFreeSpaceList free_space_lvl=PETSC_NULL,current_space_lvl=PETSC_NULL;
2601   PetscBT            lnkbt;
2602   IS                 iperm;
2603 
2604   PetscFunctionBegin;
2605   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);
2606   ierr = MatMissingDiagonal(A,&missing,&d);CHKERRQ(ierr);
2607   if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",d);
2608   ierr = ISIdentity(perm,&perm_identity);CHKERRQ(ierr);
2609   ierr = ISInvertPermutation(perm,PETSC_DECIDE,&iperm);CHKERRQ(ierr);
2610 
2611   ierr = PetscMalloc((am+1)*sizeof(PetscInt),&ui);CHKERRQ(ierr);
2612   ierr = PetscMalloc((am+1)*sizeof(PetscInt),&udiag);CHKERRQ(ierr);
2613   ui[0] = 0;
2614 
2615   /* ICC(0) without matrix ordering: simply copies fill pattern */
2616   if (!levels && perm_identity) {
2617 
2618     for (i=0; i<am; i++) {
2619       ui[i+1]  = ui[i] + ai[i+1] - a->diag[i];
2620       udiag[i] = ui[i];
2621     }
2622     ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
2623     cols = uj;
2624     for (i=0; i<am; i++) {
2625       aj    = a->j + a->diag[i];
2626       ncols = ui[i+1] - ui[i];
2627       for (j=0; j<ncols; j++) *cols++ = *aj++;
2628     }
2629   } else { /* case: levels>0 || (levels=0 && !perm_identity) */
2630     ierr = ISGetIndices(iperm,&riip);CHKERRQ(ierr);
2631     ierr = ISGetIndices(perm,&rip);CHKERRQ(ierr);
2632 
2633     /* initialization */
2634     ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&ajtmp);CHKERRQ(ierr);
2635 
2636     /* jl: linked list for storing indices of the pivot rows
2637        il: il[i] points to the 1st nonzero entry of U(i,k:am-1) */
2638     ierr = PetscMalloc4(am,PetscInt*,&uj_ptr,am,PetscInt*,&uj_lvl_ptr,am,PetscInt,&jl,am,PetscInt,&il);CHKERRQ(ierr);
2639     for (i=0; i<am; i++){
2640       jl[i] = am; il[i] = 0;
2641     }
2642 
2643     /* create and initialize a linked list for storing column indices of the active row k */
2644     nlnk = am + 1;
2645     ierr = PetscIncompleteLLCreate(am,am,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
2646 
2647     /* initial FreeSpace size is fill*(ai[am]+1) */
2648     ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space);CHKERRQ(ierr);
2649     current_space = free_space;
2650     ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space_lvl);CHKERRQ(ierr);
2651     current_space_lvl = free_space_lvl;
2652 
2653     for (k=0; k<am; k++){  /* for each active row k */
2654       /* initialize lnk by the column indices of row rip[k] of A */
2655       nzk   = 0;
2656       ncols = ai[rip[k]+1] - ai[rip[k]];
2657       if (!ncols) SETERRQ2(PETSC_ERR_MAT_CH_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",rip[k],k);
2658       ncols_upper = 0;
2659       for (j=0; j<ncols; j++){
2660         i = *(aj + ai[rip[k]] + j); /* unpermuted column index */
2661         if (riip[i] >= k){ /* only take upper triangular entry */
2662           ajtmp[ncols_upper] = i;
2663           ncols_upper++;
2664         }
2665       }
2666       ierr = PetscIncompleteLLInit(ncols_upper,ajtmp,am,riip,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
2667       nzk += nlnk;
2668 
2669       /* update lnk by computing fill-in for each pivot row to be merged in */
2670       prow = jl[k]; /* 1st pivot row */
2671 
2672       while (prow < k){
2673         nextprow = jl[prow];
2674 
2675         /* merge prow into k-th row */
2676         jmin = il[prow] + 1;  /* index of the 2nd nzero entry in U(prow,k:am-1) */
2677         jmax = ui[prow+1];
2678         ncols = jmax-jmin;
2679         i     = jmin - ui[prow];
2680         cols  = uj_ptr[prow] + i; /* points to the 2nd nzero entry in U(prow,k:am-1) */
2681         uj    = uj_lvl_ptr[prow] + i; /* levels of cols */
2682         j     = *(uj - 1);
2683         ierr = PetscICCLLAddSorted(ncols,cols,levels,uj,am,nlnk,lnk,lnk_lvl,lnkbt,j);CHKERRQ(ierr);
2684         nzk += nlnk;
2685 
2686         /* update il and jl for prow */
2687         if (jmin < jmax){
2688           il[prow] = jmin;
2689           j = *cols; jl[prow] = jl[j]; jl[j] = prow;
2690         }
2691         prow = nextprow;
2692       }
2693 
2694       /* if free space is not available, make more free space */
2695       if (current_space->local_remaining<nzk) {
2696         i = am - k + 1; /* num of unfactored rows */
2697         i *= PetscMin(nzk, (i-1)); /* i*nzk, i*(i-1): estimated and max additional space needed */
2698         ierr = PetscFreeSpaceGet(i,&current_space);CHKERRQ(ierr);
2699         ierr = PetscFreeSpaceGet(i,&current_space_lvl);CHKERRQ(ierr);
2700         reallocs++;
2701       }
2702 
2703       /* copy data into free_space and free_space_lvl, then initialize lnk */
2704       if (nzk == 0) SETERRQ1(PETSC_ERR_ARG_WRONG,"Empty row %D in ICC matrix factor",k);
2705       ierr = PetscIncompleteLLClean(am,am,nzk,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);CHKERRQ(ierr);
2706 
2707       /* add the k-th row into il and jl */
2708       if (nzk > 1){
2709         i = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:am-1) */
2710         jl[k] = jl[i]; jl[i] = k;
2711         il[k] = ui[k] + 1;
2712       }
2713       uj_ptr[k]     = current_space->array;
2714       uj_lvl_ptr[k] = current_space_lvl->array;
2715 
2716       current_space->array           += nzk;
2717       current_space->local_used      += nzk;
2718       current_space->local_remaining -= nzk;
2719 
2720       current_space_lvl->array           += nzk;
2721       current_space_lvl->local_used      += nzk;
2722       current_space_lvl->local_remaining -= nzk;
2723 
2724       ui[k+1] = ui[k] + nzk;
2725     }
2726 
2727 #if defined(PETSC_USE_INFO)
2728     if (ai[am] != 0) {
2729       PetscReal af = (PetscReal)ui[am]/((PetscReal)ai[am]);
2730       ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,fill,af);CHKERRQ(ierr);
2731       ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
2732       ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G) for best performance.\n",af);CHKERRQ(ierr);
2733     } else {
2734       ierr = PetscInfo(A,"Empty matrix.\n");CHKERRQ(ierr);
2735     }
2736 #endif
2737 
2738     ierr = ISRestoreIndices(perm,&rip);CHKERRQ(ierr);
2739     ierr = ISRestoreIndices(iperm,&riip);CHKERRQ(ierr);
2740     ierr = PetscFree4(uj_ptr,uj_lvl_ptr,jl,il);CHKERRQ(ierr);
2741     ierr = PetscFree(ajtmp);CHKERRQ(ierr);
2742 
2743     /* destroy list of free space and other temporary array(s) */
2744     ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
2745     ierr = PetscFreeSpaceContiguous(&free_space,uj);CHKERRQ(ierr);
2746     ierr = PetscIncompleteLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
2747     ierr = PetscFreeSpaceDestroy(free_space_lvl);CHKERRQ(ierr);
2748 
2749   } /* end of case: levels>0 || (levels=0 && !perm_identity) */
2750 
2751   /* put together the new matrix in MATSEQSBAIJ format */
2752 
2753   b    = (Mat_SeqSBAIJ*)fact->data;
2754   b->singlemalloc = PETSC_FALSE;
2755   ierr = PetscMalloc((ui[am]+1)*sizeof(MatScalar),&b->a);CHKERRQ(ierr);
2756   b->j    = uj;
2757   b->i    = ui;
2758   b->diag = udiag;
2759   b->free_diag = PETSC_TRUE;
2760   b->ilen = 0;
2761   b->imax = 0;
2762   b->row  = perm;
2763   b->col  = perm;
2764   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2765   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2766   b->icol = iperm;
2767   b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */
2768   ierr    = PetscMalloc((am+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
2769   ierr = PetscLogObjectMemory(fact,(ui[am]-am)*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
2770   b->maxnz   = b->nz = ui[am];
2771   b->free_a  = PETSC_TRUE;
2772   b->free_ij = PETSC_TRUE;
2773 
2774   fact->info.factor_mallocs    = reallocs;
2775   fact->info.fill_ratio_given  = fill;
2776   if (ai[am] != 0) {
2777     fact->info.fill_ratio_needed = ((PetscReal)ui[am])/((PetscReal)ai[am]);
2778   } else {
2779     fact->info.fill_ratio_needed = 0.0;
2780   }
2781   fact->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqAIJ_inplace;
2782   PetscFunctionReturn(0);
2783 }
2784 
2785 PetscErrorCode MatCholeskyFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
2786 {
2787   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data;
2788   Mat_SeqSBAIJ       *b;
2789   PetscErrorCode     ierr;
2790   PetscTruth         perm_identity;
2791   PetscReal          fill = info->fill;
2792   const PetscInt     *rip,*riip;
2793   PetscInt           i,am=A->rmap->n,*ai=a->i,*aj=a->j,reallocs=0,prow;
2794   PetscInt           *jl,jmin,jmax,nzk,*ui,k,j,*il,nextprow;
2795   PetscInt           nlnk,*lnk,ncols,ncols_upper,*cols,*uj,**ui_ptr,*uj_ptr,*udiag;
2796   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
2797   PetscBT            lnkbt;
2798   IS                 iperm;
2799 
2800   PetscFunctionBegin;
2801   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);
2802   /* check whether perm is the identity mapping */
2803   ierr = ISIdentity(perm,&perm_identity);CHKERRQ(ierr);
2804   ierr = ISInvertPermutation(perm,PETSC_DECIDE,&iperm);CHKERRQ(ierr);
2805   ierr = ISGetIndices(iperm,&riip);CHKERRQ(ierr);
2806   ierr = ISGetIndices(perm,&rip);CHKERRQ(ierr);
2807 
2808   /* initialization */
2809   ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&ui);CHKERRQ(ierr);
2810   ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&udiag);CHKERRQ(ierr);
2811   ui[0] = 0;
2812 
2813   /* jl: linked list for storing indices of the pivot rows
2814      il: il[i] points to the 1st nonzero entry of U(i,k:am-1) */
2815   ierr = PetscMalloc4(am,PetscInt*,&ui_ptr,am,PetscInt,&jl,am,PetscInt,&il,am,PetscInt,&cols);CHKERRQ(ierr);
2816   for (i=0; i<am; i++){
2817     jl[i] = am; il[i] = 0;
2818   }
2819 
2820   /* create and initialize a linked list for storing column indices of the active row k */
2821   nlnk = am + 1;
2822   ierr = PetscLLCreate(am,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2823 
2824   /* initial FreeSpace size is fill*(ai[am]+1) */
2825   ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space);CHKERRQ(ierr);
2826   current_space = free_space;
2827 
2828   for (k=0; k<am; k++){  /* for each active row k */
2829     /* initialize lnk by the column indices of row rip[k] of A */
2830     nzk   = 0;
2831     ncols = ai[rip[k]+1] - ai[rip[k]];
2832     if (!ncols) SETERRQ2(PETSC_ERR_MAT_CH_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",rip[k],k);
2833     ncols_upper = 0;
2834     for (j=0; j<ncols; j++){
2835       i = riip[*(aj + ai[rip[k]] + j)];
2836       if (i >= k){ /* only take upper triangular entry */
2837         cols[ncols_upper] = i;
2838         ncols_upper++;
2839       }
2840     }
2841     ierr = PetscLLAdd(ncols_upper,cols,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2842     nzk += nlnk;
2843 
2844     /* update lnk by computing fill-in for each pivot row to be merged in */
2845     prow = jl[k]; /* 1st pivot row */
2846 
2847     while (prow < k){
2848       nextprow = jl[prow];
2849       /* merge prow into k-th row */
2850       jmin = il[prow] + 1;  /* index of the 2nd nzero entry in U(prow,k:am-1) */
2851       jmax = ui[prow+1];
2852       ncols = jmax-jmin;
2853       uj_ptr = ui_ptr[prow] + jmin - ui[prow]; /* points to the 2nd nzero entry in U(prow,k:am-1) */
2854       ierr = PetscLLAddSorted(ncols,uj_ptr,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2855       nzk += nlnk;
2856 
2857       /* update il and jl for prow */
2858       if (jmin < jmax){
2859         il[prow] = jmin;
2860         j = *uj_ptr; jl[prow] = jl[j]; jl[j] = prow;
2861       }
2862       prow = nextprow;
2863     }
2864 
2865     /* if free space is not available, make more free space */
2866     if (current_space->local_remaining<nzk) {
2867       i  = am - k + 1; /* num of unfactored rows */
2868       i *= PetscMin(nzk,i-1); /* i*nzk, i*(i-1): estimated and max additional space needed */
2869       ierr = PetscFreeSpaceGet(i,&current_space);CHKERRQ(ierr);
2870       reallocs++;
2871     }
2872 
2873     /* copy data into free space, then initialize lnk */
2874     ierr = PetscLLClean(am,am,nzk,lnk,current_space->array,lnkbt);CHKERRQ(ierr);
2875 
2876     /* add the k-th row into il and jl */
2877     if (nzk > 1){
2878       i = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:am-1) */
2879       jl[k] = jl[i]; jl[i] = k;
2880       il[k] = ui[k] + 1;
2881     }
2882     ui_ptr[k] = current_space->array;
2883     current_space->array           += nzk;
2884     current_space->local_used      += nzk;
2885     current_space->local_remaining -= nzk;
2886 
2887     ui[k+1] = ui[k] + nzk;
2888   }
2889 
2890 #if defined(PETSC_USE_INFO)
2891   if (ai[am] != 0) {
2892     PetscReal af = (PetscReal)(ui[am])/((PetscReal)ai[am]);
2893     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,fill,af);CHKERRQ(ierr);
2894     ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
2895     ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G) for best performance.\n",af);CHKERRQ(ierr);
2896   } else {
2897      ierr = PetscInfo(A,"Empty matrix.\n");CHKERRQ(ierr);
2898   }
2899 #endif
2900 
2901   ierr = ISRestoreIndices(perm,&rip);CHKERRQ(ierr);
2902   ierr = ISRestoreIndices(iperm,&riip);CHKERRQ(ierr);
2903   ierr = PetscFree4(ui_ptr,jl,il,cols);CHKERRQ(ierr);
2904 
2905   /* destroy list of free space and other temporary array(s) */
2906   ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
2907   ierr = PetscFreeSpaceContiguous_Cholesky(&free_space,uj,am,ui,udiag);CHKERRQ(ierr); /* store matrix factor */
2908   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
2909 
2910   /* put together the new matrix in MATSEQSBAIJ format */
2911 
2912   b = (Mat_SeqSBAIJ*)fact->data;
2913   b->singlemalloc = PETSC_FALSE;
2914   b->free_a       = PETSC_TRUE;
2915   b->free_ij      = PETSC_TRUE;
2916   ierr = PetscMalloc((ui[am]+1)*sizeof(MatScalar),&b->a);CHKERRQ(ierr);
2917   b->j    = uj;
2918   b->i    = ui;
2919   b->diag = udiag;
2920   b->free_diag = PETSC_TRUE;
2921   b->ilen = 0;
2922   b->imax = 0;
2923   b->row  = perm;
2924   b->col  = perm;
2925   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2926   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2927   b->icol = iperm;
2928   b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */
2929   ierr    = PetscMalloc((am+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
2930   ierr    = PetscLogObjectMemory(fact,ui[am]*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
2931   b->maxnz = b->nz = ui[am];
2932 
2933   fact->info.factor_mallocs    = reallocs;
2934   fact->info.fill_ratio_given  = fill;
2935   if (ai[am] != 0) {
2936     fact->info.fill_ratio_needed = ((PetscReal)ui[am])/((PetscReal)ai[am]);
2937   } else {
2938     fact->info.fill_ratio_needed = 0.0;
2939   }
2940   fact->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqAIJ;
2941   PetscFunctionReturn(0);
2942 }
2943 
2944 #undef __FUNCT__
2945 #define __FUNCT__ "MatCholeskyFactorSymbolic_SeqAIJ_inplace"
2946 PetscErrorCode MatCholeskyFactorSymbolic_SeqAIJ_inplace(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
2947 {
2948   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data;
2949   Mat_SeqSBAIJ       *b;
2950   PetscErrorCode     ierr;
2951   PetscTruth         perm_identity;
2952   PetscReal          fill = info->fill;
2953   const PetscInt     *rip,*riip;
2954   PetscInt           i,am=A->rmap->n,*ai=a->i,*aj=a->j,reallocs=0,prow;
2955   PetscInt           *jl,jmin,jmax,nzk,*ui,k,j,*il,nextprow;
2956   PetscInt           nlnk,*lnk,ncols,ncols_upper,*cols,*uj,**ui_ptr,*uj_ptr;
2957   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
2958   PetscBT            lnkbt;
2959   IS                 iperm;
2960 
2961   PetscFunctionBegin;
2962   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);
2963   /* check whether perm is the identity mapping */
2964   ierr = ISIdentity(perm,&perm_identity);CHKERRQ(ierr);
2965   ierr = ISInvertPermutation(perm,PETSC_DECIDE,&iperm);CHKERRQ(ierr);
2966   ierr = ISGetIndices(iperm,&riip);CHKERRQ(ierr);
2967   ierr = ISGetIndices(perm,&rip);CHKERRQ(ierr);
2968 
2969   /* initialization */
2970   ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&ui);CHKERRQ(ierr);
2971   ui[0] = 0;
2972 
2973   /* jl: linked list for storing indices of the pivot rows
2974      il: il[i] points to the 1st nonzero entry of U(i,k:am-1) */
2975   ierr = PetscMalloc4(am,PetscInt*,&ui_ptr,am,PetscInt,&jl,am,PetscInt,&il,am,PetscInt,&cols);CHKERRQ(ierr);
2976   for (i=0; i<am; i++){
2977     jl[i] = am; il[i] = 0;
2978   }
2979 
2980   /* create and initialize a linked list for storing column indices of the active row k */
2981   nlnk = am + 1;
2982   ierr = PetscLLCreate(am,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2983 
2984   /* initial FreeSpace size is fill*(ai[am]+1) */
2985   ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space);CHKERRQ(ierr);
2986   current_space = free_space;
2987 
2988   for (k=0; k<am; k++){  /* for each active row k */
2989     /* initialize lnk by the column indices of row rip[k] of A */
2990     nzk   = 0;
2991     ncols = ai[rip[k]+1] - ai[rip[k]];
2992     if (!ncols) SETERRQ2(PETSC_ERR_MAT_CH_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",rip[k],k);
2993     ncols_upper = 0;
2994     for (j=0; j<ncols; j++){
2995       i = riip[*(aj + ai[rip[k]] + j)];
2996       if (i >= k){ /* only take upper triangular entry */
2997         cols[ncols_upper] = i;
2998         ncols_upper++;
2999       }
3000     }
3001     ierr = PetscLLAdd(ncols_upper,cols,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
3002     nzk += nlnk;
3003 
3004     /* update lnk by computing fill-in for each pivot row to be merged in */
3005     prow = jl[k]; /* 1st pivot row */
3006 
3007     while (prow < k){
3008       nextprow = jl[prow];
3009       /* merge prow into k-th row */
3010       jmin = il[prow] + 1;  /* index of the 2nd nzero entry in U(prow,k:am-1) */
3011       jmax = ui[prow+1];
3012       ncols = jmax-jmin;
3013       uj_ptr = ui_ptr[prow] + jmin - ui[prow]; /* points to the 2nd nzero entry in U(prow,k:am-1) */
3014       ierr = PetscLLAddSorted(ncols,uj_ptr,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
3015       nzk += nlnk;
3016 
3017       /* update il and jl for prow */
3018       if (jmin < jmax){
3019         il[prow] = jmin;
3020         j = *uj_ptr; jl[prow] = jl[j]; jl[j] = prow;
3021       }
3022       prow = nextprow;
3023     }
3024 
3025     /* if free space is not available, make more free space */
3026     if (current_space->local_remaining<nzk) {
3027       i = am - k + 1; /* num of unfactored rows */
3028       i = PetscMin(i*nzk, i*(i-1)); /* i*nzk, i*(i-1): estimated and max additional space needed */
3029       ierr = PetscFreeSpaceGet(i,&current_space);CHKERRQ(ierr);
3030       reallocs++;
3031     }
3032 
3033     /* copy data into free space, then initialize lnk */
3034     ierr = PetscLLClean(am,am,nzk,lnk,current_space->array,lnkbt);CHKERRQ(ierr);
3035 
3036     /* add the k-th row into il and jl */
3037     if (nzk-1 > 0){
3038       i = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:am-1) */
3039       jl[k] = jl[i]; jl[i] = k;
3040       il[k] = ui[k] + 1;
3041     }
3042     ui_ptr[k] = current_space->array;
3043     current_space->array           += nzk;
3044     current_space->local_used      += nzk;
3045     current_space->local_remaining -= nzk;
3046 
3047     ui[k+1] = ui[k] + nzk;
3048   }
3049 
3050 #if defined(PETSC_USE_INFO)
3051   if (ai[am] != 0) {
3052     PetscReal af = (PetscReal)(ui[am])/((PetscReal)ai[am]);
3053     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,fill,af);CHKERRQ(ierr);
3054     ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
3055     ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G) for best performance.\n",af);CHKERRQ(ierr);
3056   } else {
3057      ierr = PetscInfo(A,"Empty matrix.\n");CHKERRQ(ierr);
3058   }
3059 #endif
3060 
3061   ierr = ISRestoreIndices(perm,&rip);CHKERRQ(ierr);
3062   ierr = ISRestoreIndices(iperm,&riip);CHKERRQ(ierr);
3063   ierr = PetscFree4(ui_ptr,jl,il,cols);CHKERRQ(ierr);
3064 
3065   /* destroy list of free space and other temporary array(s) */
3066   ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
3067   ierr = PetscFreeSpaceContiguous(&free_space,uj);CHKERRQ(ierr);
3068   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
3069 
3070   /* put together the new matrix in MATSEQSBAIJ format */
3071 
3072   b = (Mat_SeqSBAIJ*)fact->data;
3073   b->singlemalloc = PETSC_FALSE;
3074   b->free_a       = PETSC_TRUE;
3075   b->free_ij      = PETSC_TRUE;
3076   ierr = PetscMalloc((ui[am]+1)*sizeof(MatScalar),&b->a);CHKERRQ(ierr);
3077   b->j    = uj;
3078   b->i    = ui;
3079   b->diag = 0;
3080   b->ilen = 0;
3081   b->imax = 0;
3082   b->row  = perm;
3083   b->col  = perm;
3084   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
3085   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
3086   b->icol = iperm;
3087   b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */
3088   ierr    = PetscMalloc((am+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
3089   ierr    = PetscLogObjectMemory(fact,(ui[am]-am)*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
3090   b->maxnz = b->nz = ui[am];
3091 
3092   fact->info.factor_mallocs    = reallocs;
3093   fact->info.fill_ratio_given  = fill;
3094   if (ai[am] != 0) {
3095     fact->info.fill_ratio_needed = ((PetscReal)ui[am])/((PetscReal)ai[am]);
3096   } else {
3097     fact->info.fill_ratio_needed = 0.0;
3098   }
3099   fact->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqAIJ_inplace;
3100   PetscFunctionReturn(0);
3101 }
3102 
3103 #undef __FUNCT__
3104 #define __FUNCT__ "MatSolve_SeqAIJ_NaturalOrdering"
3105 PetscErrorCode MatSolve_SeqAIJ_NaturalOrdering(Mat A,Vec bb,Vec xx)
3106 {
3107   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
3108   PetscErrorCode    ierr;
3109   PetscInt          n = A->rmap->n;
3110   const PetscInt    *ai = a->i,*aj = a->j,*adiag = a->diag,*vi;
3111   PetscScalar       *x,sum;
3112   const PetscScalar *b;
3113   const MatScalar   *aa = a->a,*v;
3114   PetscInt          i,nz;
3115 
3116   PetscFunctionBegin;
3117   if (!n) PetscFunctionReturn(0);
3118 
3119   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
3120   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
3121 
3122   /* forward solve the lower triangular */
3123   x[0] = b[0];
3124   v    = aa;
3125   vi   = aj;
3126   for (i=1; i<n; i++) {
3127     nz  = ai[i+1] - ai[i];
3128     sum = b[i];
3129     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
3130     v  += nz;
3131     vi += nz;
3132     x[i] = sum;
3133   }
3134 
3135   /* backward solve the upper triangular */
3136   for (i=n-1; i>=0; i--){
3137     v   = aa + adiag[i+1] + 1;
3138     vi  = aj + adiag[i+1] + 1;
3139     nz = adiag[i] - adiag[i+1]-1;
3140     sum = x[i];
3141     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
3142     x[i] = sum*v[nz]; /* x[i]=aa[adiag[i]]*sum; v++; */
3143   }
3144 
3145   ierr = PetscLogFlops(2.0*a->nz - A->cmap->n);CHKERRQ(ierr);
3146   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
3147   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
3148   PetscFunctionReturn(0);
3149 }
3150 
3151 #undef __FUNCT__
3152 #define __FUNCT__ "MatSolve_SeqAIJ"
3153 PetscErrorCode MatSolve_SeqAIJ(Mat A,Vec bb,Vec xx)
3154 {
3155   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
3156   IS                iscol = a->col,isrow = a->row;
3157   PetscErrorCode    ierr;
3158   PetscInt          i,n=A->rmap->n,*vi,*ai=a->i,*aj=a->j,*adiag = a->diag,nz;
3159   const PetscInt    *rout,*cout,*r,*c;
3160   PetscScalar       *x,*tmp,sum;
3161   const PetscScalar *b;
3162   const MatScalar   *aa = a->a,*v;
3163 
3164   PetscFunctionBegin;
3165   if (!n) PetscFunctionReturn(0);
3166 
3167   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
3168   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
3169   tmp  = a->solve_work;
3170 
3171   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
3172   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
3173 
3174   /* forward solve the lower triangular */
3175   tmp[0] = b[r[0]];
3176   v      = aa;
3177   vi     = aj;
3178   for (i=1; i<n; i++) {
3179     nz  = ai[i+1] - ai[i];
3180     sum = b[r[i]];
3181     PetscSparseDenseMinusDot(sum,tmp,v,vi,nz);
3182     tmp[i] = sum;
3183     v += nz; vi += nz;
3184   }
3185 
3186   /* backward solve the upper triangular */
3187   for (i=n-1; i>=0; i--){
3188     v   = aa + adiag[i+1]+1;
3189     vi  = aj + adiag[i+1]+1;
3190     nz  = adiag[i]-adiag[i+1]-1;
3191     sum = tmp[i];
3192     PetscSparseDenseMinusDot(sum,tmp,v,vi,nz);
3193     x[c[i]] = tmp[i] = sum*v[nz]; /* v[nz] = aa[adiag[i]] */
3194   }
3195 
3196   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
3197   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
3198   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
3199   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
3200   ierr = PetscLogFlops(2*a->nz - A->cmap->n);CHKERRQ(ierr);
3201   PetscFunctionReturn(0);
3202 }
3203 
3204 #undef __FUNCT__
3205 #define __FUNCT__ "MatILUDTFactor_SeqAIJ"
3206 /*
3207     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
3208 */
3209 PetscErrorCode MatILUDTFactor_SeqAIJ(Mat A,IS isrow,IS iscol,const MatFactorInfo *info,Mat *fact)
3210 {
3211   Mat                B = *fact;
3212   Mat_SeqAIJ         *a=(Mat_SeqAIJ*)A->data,*b;
3213   IS                 isicol;
3214   PetscErrorCode     ierr;
3215   const PetscInt     *r,*ic;
3216   PetscInt           i,n=A->rmap->n,*ai=a->i,*aj=a->j,*ajtmp,*adiag;
3217   PetscInt           *bi,*bj,*bdiag,*bdiag_rev;
3218   PetscInt           row,nzi,nzi_bl,nzi_bu,*im,nzi_al,nzi_au;
3219   PetscInt           nlnk,*lnk;
3220   PetscBT            lnkbt;
3221   PetscTruth         row_identity,icol_identity,both_identity;
3222   MatScalar          *aatmp,*pv,*batmp,*ba,*rtmp,*pc,multiplier,*vtmp,diag_tmp;
3223   const PetscInt     *ics;
3224   PetscInt           j,nz,*pj,*bjtmp,k,ncut,*jtmp;
3225   PetscReal          dt=info->dt,dtcol=info->dtcol,shift=info->shiftamount;
3226   PetscInt           dtcount=(PetscInt)info->dtcount,nnz_max;
3227   PetscTruth         missing;
3228 
3229   PetscFunctionBegin;
3230 
3231   if (dt      == PETSC_DEFAULT) dt      = 0.005;
3232   if (dtcol   == PETSC_DEFAULT) dtcol   = 0.01; /* XXX unused! */
3233   if (dtcount == PETSC_DEFAULT) dtcount = (PetscInt)(1.5*a->rmax);
3234 
3235   /* ------- symbolic factorization, can be reused ---------*/
3236   ierr = MatMissingDiagonal(A,&missing,&i);CHKERRQ(ierr);
3237   if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",i);
3238   adiag=a->diag;
3239 
3240   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
3241 
3242   /* bdiag is location of diagonal in factor */
3243   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);     /* becomes b->diag */
3244   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag_rev);CHKERRQ(ierr); /* temporary */
3245 
3246   /* allocate row pointers bi */
3247   ierr = PetscMalloc((2*n+2)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
3248 
3249   /* allocate bj and ba; max num of nonzero entries is (ai[n]+2*n*dtcount+2) */
3250   if (dtcount > n-1) dtcount = n-1; /* diagonal is excluded */
3251   nnz_max  = ai[n]+2*n*dtcount+2;
3252 
3253   ierr = PetscMalloc((nnz_max+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
3254   ierr = PetscMalloc((nnz_max+1)*sizeof(MatScalar),&ba);CHKERRQ(ierr);
3255 
3256   /* put together the new matrix */
3257   ierr = MatSeqAIJSetPreallocation_SeqAIJ(B,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
3258   ierr = PetscLogObjectParent(B,isicol);CHKERRQ(ierr);
3259   b    = (Mat_SeqAIJ*)B->data;
3260   b->free_a       = PETSC_TRUE;
3261   b->free_ij      = PETSC_TRUE;
3262   b->singlemalloc = PETSC_FALSE;
3263   b->a          = ba;
3264   b->j          = bj;
3265   b->i          = bi;
3266   b->diag       = bdiag;
3267   b->ilen       = 0;
3268   b->imax       = 0;
3269   b->row        = isrow;
3270   b->col        = iscol;
3271   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
3272   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
3273   b->icol       = isicol;
3274   ierr = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
3275 
3276   ierr = PetscLogObjectMemory(B,nnz_max*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
3277   b->maxnz = nnz_max;
3278 
3279   B->factortype            = MAT_FACTOR_ILUDT;
3280   B->info.factor_mallocs   = 0;
3281   B->info.fill_ratio_given = ((PetscReal)nnz_max)/((PetscReal)ai[n]);
3282   CHKMEMQ;
3283   /* ------- end of symbolic factorization ---------*/
3284 
3285   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
3286   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
3287   ics  = ic;
3288 
3289   /* linked list for storing column indices of the active row */
3290   nlnk = n + 1;
3291   ierr = PetscLLCreate(n,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
3292 
3293   /* im: used by PetscLLAddSortedLU(); jtmp: working array for column indices of active row */
3294   ierr = PetscMalloc2(n,PetscInt,&im,n,PetscInt,&jtmp);CHKERRQ(ierr);
3295   /* rtmp, vtmp: working arrays for sparse and contiguous row entries of active row */
3296   ierr = PetscMalloc2(n,MatScalar,&rtmp,n,MatScalar,&vtmp);CHKERRQ(ierr);
3297   ierr = PetscMemzero(rtmp,n*sizeof(MatScalar));CHKERRQ(ierr);
3298 
3299   bi[0]    = 0;
3300   bdiag[0] = nnz_max-1; /* location of diag[0] in factor B */
3301   bdiag_rev[n] = bdiag[0];
3302   bi[2*n+1] = bdiag[0]+1; /* endof bj and ba array */
3303   for (i=0; i<n; i++) {
3304     /* copy initial fill into linked list */
3305     nzi = 0; /* nonzeros for active row i */
3306     nzi = ai[r[i]+1] - ai[r[i]];
3307     if (!nzi) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
3308     nzi_al = adiag[r[i]] - ai[r[i]];
3309     nzi_au = ai[r[i]+1] - adiag[r[i]] -1;
3310     ajtmp = aj + ai[r[i]];
3311     ierr = PetscLLAddPerm(nzi,ajtmp,ic,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
3312 
3313     /* load in initial (unfactored row) */
3314     aatmp = a->a + ai[r[i]];
3315     for (j=0; j<nzi; j++) {
3316       rtmp[ics[*ajtmp++]] = *aatmp++;
3317     }
3318 
3319     /* add pivot rows into linked list */
3320     row = lnk[n];
3321     while (row < i ) {
3322       nzi_bl = bi[row+1] - bi[row] + 1;
3323       bjtmp = bj + bdiag[row+1]+1; /* points to 1st column next to the diagonal in U */
3324       ierr  = PetscLLAddSortedLU(bjtmp,row,nlnk,lnk,lnkbt,i,nzi_bl,im);CHKERRQ(ierr);
3325       nzi  += nlnk;
3326       row   = lnk[row];
3327     }
3328 
3329     /* copy data from lnk into jtmp, then initialize lnk */
3330     ierr = PetscLLClean(n,n,nzi,lnk,jtmp,lnkbt);CHKERRQ(ierr);
3331 
3332     /* numerical factorization */
3333     bjtmp = jtmp;
3334     row   = *bjtmp++; /* 1st pivot row */
3335     while  ( row < i ) {
3336       pc         = rtmp + row;
3337       pv         = ba + bdiag[row]; /* 1./(diag of the pivot row) */
3338       multiplier = (*pc) * (*pv);
3339       *pc        = multiplier;
3340       if (PetscAbsScalar(*pc) > dt){ /* apply tolerance dropping rule */
3341         pj         = bj + bdiag[row+1] + 1; /* point to 1st entry of U(row,:) */
3342         pv         = ba + bdiag[row+1] + 1;
3343         /* if (multiplier < -1.0 or multiplier >1.0) printf("row/prow %d, %d, multiplier %g\n",i,row,multiplier); */
3344         nz         = bdiag[row] - bdiag[row+1] - 1; /* num of entries in U(row,:), excluding diagonal */
3345         for (j=0; j<nz; j++) rtmp[*pj++] -= multiplier * (*pv++);
3346         ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr);
3347       }
3348       row = *bjtmp++;
3349     }
3350 
3351     /* copy sparse rtmp into contiguous vtmp; separate L and U part */
3352     diag_tmp = rtmp[i];  /* save diagonal value - may not needed?? */
3353     nzi_bl = 0; j = 0;
3354     while (jtmp[j] < i){ /* Note: jtmp is sorted */
3355       vtmp[j] = rtmp[jtmp[j]]; rtmp[jtmp[j]]=0.0;
3356       nzi_bl++; j++;
3357     }
3358     nzi_bu = nzi - nzi_bl -1;
3359     while (j < nzi){
3360       vtmp[j] = rtmp[jtmp[j]]; rtmp[jtmp[j]]=0.0;
3361       j++;
3362     }
3363 
3364     bjtmp = bj + bi[i];
3365     batmp = ba + bi[i];
3366     /* apply level dropping rule to L part */
3367     ncut = nzi_al + dtcount;
3368     if (ncut < nzi_bl){
3369       ierr = PetscSortSplit(ncut,nzi_bl,vtmp,jtmp);CHKERRQ(ierr);
3370       ierr = PetscSortIntWithScalarArray(ncut,jtmp,vtmp);CHKERRQ(ierr);
3371     } else {
3372       ncut = nzi_bl;
3373     }
3374     for (j=0; j<ncut; j++){
3375       bjtmp[j] = jtmp[j];
3376       batmp[j] = vtmp[j];
3377       /* printf(" (%d,%g),",bjtmp[j],batmp[j]); */
3378     }
3379     bi[i+1] = bi[i] + ncut;
3380     nzi = ncut + 1;
3381 
3382     /* apply level dropping rule to U part */
3383     ncut = nzi_au + dtcount;
3384     if (ncut < nzi_bu){
3385       ierr = PetscSortSplit(ncut,nzi_bu,vtmp+nzi_bl+1,jtmp+nzi_bl+1);CHKERRQ(ierr);
3386       ierr = PetscSortIntWithScalarArray(ncut,jtmp+nzi_bl+1,vtmp+nzi_bl+1);CHKERRQ(ierr);
3387     } else {
3388       ncut = nzi_bu;
3389     }
3390     nzi += ncut;
3391 
3392     /* mark bdiagonal */
3393     bdiag[i+1]       = bdiag[i] - (ncut + 1);
3394     bdiag_rev[n-i-1] = bdiag[i+1];
3395     bi[2*n - i]      = bi[2*n - i +1] - (ncut + 1);
3396     bjtmp = bj + bdiag[i];
3397     batmp = ba + bdiag[i];
3398     *bjtmp = i;
3399     *batmp = diag_tmp; /* rtmp[i]; */
3400     if (*batmp == 0.0) {
3401       *batmp = dt+shift;
3402       /* printf(" row %d add shift %g\n",i,shift); */
3403     }
3404     *batmp = 1.0/(*batmp); /* invert diagonal entries for simplier triangular solves */
3405     /* printf(" (%d,%g),",*bjtmp,*batmp); */
3406 
3407     bjtmp = bj + bdiag[i+1]+1;
3408     batmp = ba + bdiag[i+1]+1;
3409     for (k=0; k<ncut; k++){
3410       bjtmp[k] = jtmp[nzi_bl+1+k];
3411       batmp[k] = vtmp[nzi_bl+1+k];
3412       /* printf(" (%d,%g),",bjtmp[k],batmp[k]); */
3413     }
3414     /* printf("\n"); */
3415 
3416     im[i]   = nzi; /* used by PetscLLAddSortedLU() */
3417     /*
3418     printf("row %d: bi %d, bdiag %d\n",i,bi[i],bdiag[i]);
3419     printf(" ----------------------------\n");
3420     */
3421   } /* for (i=0; i<n; i++) */
3422   /* printf("end of L %d, beginning of U %d\n",bi[n],bdiag[n]); */
3423   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]);
3424 
3425   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
3426   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
3427 
3428   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
3429   ierr = PetscFree2(im,jtmp);CHKERRQ(ierr);
3430   ierr = PetscFree2(rtmp,vtmp);CHKERRQ(ierr);
3431   ierr = PetscFree(bdiag_rev);CHKERRQ(ierr);
3432 
3433   ierr = PetscLogFlops(B->cmap->n);CHKERRQ(ierr);
3434   b->maxnz = b->nz = bi[n] + bdiag[0] - bdiag[n];
3435 
3436   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
3437   ierr = ISIdentity(isicol,&icol_identity);CHKERRQ(ierr);
3438   both_identity = (PetscTruth) (row_identity && icol_identity);
3439   if (row_identity && icol_identity) {
3440     B->ops->solve = MatSolve_SeqAIJ_NaturalOrdering;
3441   } else {
3442     B->ops->solve = MatSolve_SeqAIJ;
3443   }
3444 
3445   B->ops->solveadd          = 0;
3446   B->ops->solvetranspose    = 0;
3447   B->ops->solvetransposeadd = 0;
3448   B->ops->matsolve          = 0;
3449   B->assembled              = PETSC_TRUE;
3450   B->preallocated           = PETSC_TRUE;
3451   PetscFunctionReturn(0);
3452 }
3453 
3454 /* a wraper of MatILUDTFactor_SeqAIJ() */
3455 #undef __FUNCT__
3456 #define __FUNCT__ "MatILUDTFactorSymbolic_SeqAIJ"
3457 /*
3458     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
3459 */
3460 
3461 PetscErrorCode PETSCMAT_DLLEXPORT MatILUDTFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS row,IS col,const MatFactorInfo *info)
3462 {
3463   PetscErrorCode     ierr;
3464 
3465   PetscFunctionBegin;
3466   ierr = MatILUDTFactor_SeqAIJ(A,row,col,info,&fact);CHKERRQ(ierr);
3467   PetscFunctionReturn(0);
3468 }
3469 
3470 /*
3471    same as MatLUFactorNumeric_SeqAIJ(), except using contiguous array matrix factors
3472    - intend to replace existing MatLUFactorNumeric_SeqAIJ()
3473 */
3474 #undef __FUNCT__
3475 #define __FUNCT__ "MatILUDTFactorNumeric_SeqAIJ"
3476 /*
3477     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
3478 */
3479 
3480 PetscErrorCode PETSCMAT_DLLEXPORT MatILUDTFactorNumeric_SeqAIJ(Mat fact,Mat A,const MatFactorInfo *info)
3481 {
3482   Mat            C=fact;
3483   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ *)C->data;
3484   IS             isrow = b->row,isicol = b->icol;
3485   PetscErrorCode ierr;
3486   const PetscInt *r,*ic,*ics;
3487   PetscInt       i,j,k,n=A->rmap->n,*ai=a->i,*aj=a->j,*bi=b->i,*bj=b->j;
3488   PetscInt       *ajtmp,*bjtmp,nz,nzl,nzu,row,*bdiag = b->diag,*pj;
3489   MatScalar      *rtmp,*pc,multiplier,*v,*pv,*aa=a->a;
3490   PetscReal      dt=info->dt,shift=info->shiftamount;
3491   PetscTruth     row_identity, col_identity;
3492 
3493   PetscFunctionBegin;
3494   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
3495   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
3496   ierr = PetscMalloc((n+1)*sizeof(MatScalar),&rtmp);CHKERRQ(ierr);
3497   ics  = ic;
3498 
3499   for (i=0; i<n; i++){
3500     /* initialize rtmp array */
3501     nzl   = bi[i+1] - bi[i];       /* num of nozeros in L(i,:) */
3502     bjtmp = bj + bi[i];
3503     for  (j=0; j<nzl; j++) rtmp[*bjtmp++] = 0.0;
3504     rtmp[i] = 0.0;
3505     nzu   = bdiag[i] - bdiag[i+1]; /* num of nozeros in U(i,:) */
3506     bjtmp = bj + bdiag[i+1] + 1;
3507     for  (j=0; j<nzu; j++) rtmp[*bjtmp++] = 0.0;
3508 
3509     /* load in initial unfactored row of A */
3510     /* printf("row %d\n",i); */
3511     nz    = ai[r[i]+1] - ai[r[i]];
3512     ajtmp = aj + ai[r[i]];
3513     v     = aa + ai[r[i]];
3514     for (j=0; j<nz; j++) {
3515       rtmp[ics[*ajtmp++]] = v[j];
3516       /* printf(" (%d,%g),",ics[ajtmp[j]],rtmp[ics[ajtmp[j]]]); */
3517     }
3518     /* printf("\n"); */
3519 
3520     /* numerical factorization */
3521     bjtmp = bj + bi[i]; /* point to 1st entry of L(i,:) */
3522     nzl   = bi[i+1] - bi[i]; /* num of entries in L(i,:) */
3523     k = 0;
3524     while (k < nzl){
3525       row   = *bjtmp++;
3526       /* printf("  prow %d\n",row); */
3527       pc         = rtmp + row;
3528       pv         = b->a + bdiag[row]; /* 1./(diag of the pivot row) */
3529       multiplier = (*pc) * (*pv);
3530       *pc        = multiplier;
3531       if (PetscAbsScalar(multiplier) > dt){
3532         pj         = bj + bdiag[row+1] + 1; /* point to 1st entry of U(row,:) */
3533         pv         = b->a + bdiag[row+1] + 1;
3534         nz         = bdiag[row] - bdiag[row+1] - 1; /* num of entries in U(row,:), excluding diagonal */
3535         for (j=0; j<nz; j++) rtmp[*pj++] -= multiplier * (*pv++);
3536         /* ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr); */
3537       }
3538       k++;
3539     }
3540 
3541     /* finished row so stick it into b->a */
3542     /* L-part */
3543     pv = b->a + bi[i] ;
3544     pj = bj + bi[i] ;
3545     nzl = bi[i+1] - bi[i];
3546     for (j=0; j<nzl; j++) {
3547       pv[j] = rtmp[pj[j]];
3548       /* printf(" (%d,%g),",pj[j],pv[j]); */
3549     }
3550 
3551     /* diagonal: invert diagonal entries for simplier triangular solves */
3552     if (rtmp[i] == 0.0) rtmp[i] = dt+shift;
3553     b->a[bdiag[i]] = 1.0/rtmp[i];
3554     /* printf(" (%d,%g),",i,b->a[bdiag[i]]); */
3555 
3556     /* U-part */
3557     pv = b->a + bdiag[i+1] + 1;
3558     pj = bj + bdiag[i+1] + 1;
3559     nzu = bdiag[i] - bdiag[i+1] - 1;
3560     for (j=0; j<nzu; j++) {
3561       pv[j] = rtmp[pj[j]];
3562       /* printf(" (%d,%g),",pj[j],pv[j]); */
3563     }
3564     /* printf("\n"); */
3565   }
3566 
3567   ierr = PetscFree(rtmp);CHKERRQ(ierr);
3568   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
3569   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
3570 
3571   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
3572   ierr = ISIdentity(isicol,&col_identity);CHKERRQ(ierr);
3573   if (row_identity && col_identity) {
3574     C->ops->solve   = MatSolve_SeqAIJ_NaturalOrdering;
3575   } else {
3576     C->ops->solve   = MatSolve_SeqAIJ;
3577   }
3578   C->ops->solveadd           = 0;
3579   C->ops->solvetranspose     = 0;
3580   C->ops->solvetransposeadd  = 0;
3581   C->ops->matsolve           = 0;
3582   C->assembled    = PETSC_TRUE;
3583   C->preallocated = PETSC_TRUE;
3584   ierr = PetscLogFlops(C->cmap->n);CHKERRQ(ierr);
3585   PetscFunctionReturn(0);
3586 }
3587