xref: /petsc/src/mat/impls/aij/seq/aijfact.c (revision ac450b4331df367ad8a5a6b9d70f54001dfe8d46)
1 #define PETSCMAT_DLL
2 
3 
4 #include "../src/mat/impls/aij/seq/aij.h"
5 #include "petscbt.h"
6 #include "../src/mat/utils/freespace.h"
7 
8 EXTERN_C_BEGIN
9 #undef __FUNCT__
10 #define __FUNCT__ "MatOrdering_Flow_SeqAIJ"
11 /*
12       Computes an ordering to get most of the large numerical values in the lower triangular part of the matrix
13 */
14 PetscErrorCode MatOrdering_Flow_SeqAIJ(Mat mat,const MatOrderingType type,IS *irow,IS *icol)
15 {
16   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)mat->data;
17   PetscErrorCode    ierr;
18   PetscInt          i,j,jj,k, kk,n = mat->rmap->n, current = 0, newcurrent = 0,*order;
19   const PetscInt    *ai = a->i, *aj = a->j;
20   const PetscScalar *aa = a->a;
21   PetscTruth        *done;
22   PetscReal         best,past = 0,future;
23 
24   PetscFunctionBegin;
25   /* pick initial row */
26   best = -1;
27   for (i=0; i<n; i++) {
28     future = 0;
29     for (j=ai[i]; j<ai[i+1]; j++) {
30       if (aj[j] != i) future  += PetscAbsScalar(aa[j]); else past = PetscAbsScalar(aa[j]);
31     }
32     if (!future) future = 1.e-10; /* if there is zero in the upper diagonal part want to rank this row high */
33     if (past/future > best) {
34       best = past/future;
35       current = i;
36     }
37   }
38 
39   ierr = PetscMalloc(n*sizeof(PetscTruth),&done);CHKERRQ(ierr);
40   ierr = PetscMalloc(n*sizeof(PetscInt),&order);CHKERRQ(ierr);
41   ierr = PetscMemzero(done,n*sizeof(PetscTruth));CHKERRQ(ierr);
42   order[0] = current;
43   for (i=0; i<n-1; i++) {
44     done[current] = PETSC_TRUE;
45     best          = -1;
46     /* loop over all neighbors of current pivot */
47     for (j=ai[current]; j<ai[current+1]; j++) {
48       jj = aj[j];
49       if (done[jj]) continue;
50       /* loop over columns of potential next row computing weights for below and above diagonal */
51       past = future = 0.0;
52       for (k=ai[jj]; k<ai[jj+1]; k++) {
53         kk = aj[k];
54         if (done[kk]) past += PetscAbsScalar(aa[k]);
55         else if (kk != jj) future  += PetscAbsScalar(aa[k]);
56       }
57       if (!future) future = 1.e-10; /* if there is zero in the upper diagonal part want to rank this row high */
58       if (past/future > best) {
59         best = past/future;
60         newcurrent = jj;
61       }
62     }
63     if (best == -1) { /* no neighbors to select from so select best of all that remain */
64       best = -1;
65       for (k=0; k<n; k++) {
66         if (done[k]) continue;
67         future = 0;
68         past   = 0;
69         for (j=ai[k]; j<ai[k+1]; j++) {
70           kk = aj[j];
71           if (done[kk]) past += PetscAbsScalar(aa[j]);
72           else if (kk != k) future  += PetscAbsScalar(aa[j]);
73         }
74         if (!future) future = 1.e-10; /* if there is zero in the upper diagonal part want to rank this row high */
75         if (past/future > best) {
76           best = past/future;
77           newcurrent = k;
78         }
79       }
80     }
81     if (current == newcurrent) SETERRQ(PETSC_ERR_PLIB,"newcurrent cannot be current");
82     current = newcurrent;
83     order[i+1] = current;
84   }
85   ierr = ISCreateGeneral(PETSC_COMM_SELF,n,order,irow);CHKERRQ(ierr);
86   *icol = *irow;
87   ierr = PetscObjectReference((PetscObject)*irow);CHKERRQ(ierr);
88   ierr = PetscFree(done);CHKERRQ(ierr);
89   ierr = PetscFree(order);CHKERRQ(ierr);
90   PetscFunctionReturn(0);
91 }
92 EXTERN_C_END
93 
94 EXTERN_C_BEGIN
95 #undef __FUNCT__
96 #define __FUNCT__ "MatGetFactorAvailable_seqaij_petsc"
97 PetscErrorCode MatGetFactorAvailable_seqaij_petsc(Mat A,MatFactorType ftype,PetscTruth *flg)
98 {
99   PetscFunctionBegin;
100   *flg = PETSC_TRUE;
101   PetscFunctionReturn(0);
102 }
103 EXTERN_C_END
104 
105 EXTERN_C_BEGIN
106 #undef __FUNCT__
107 #define __FUNCT__ "MatGetFactor_seqaij_petsc"
108 PetscErrorCode MatGetFactor_seqaij_petsc(Mat A,MatFactorType ftype,Mat *B)
109 {
110   PetscInt           n = A->rmap->n;
111   PetscErrorCode     ierr;
112 
113   PetscFunctionBegin;
114   ierr = MatCreate(((PetscObject)A)->comm,B);CHKERRQ(ierr);
115   ierr = MatSetSizes(*B,n,n,n,n);CHKERRQ(ierr);
116   if (ftype == MAT_FACTOR_LU || ftype == MAT_FACTOR_ILU || ftype == MAT_FACTOR_ILUDT){
117     ierr = MatSetType(*B,MATSEQAIJ);CHKERRQ(ierr);
118     (*B)->ops->ilufactorsymbolic = MatILUFactorSymbolic_SeqAIJ;
119     (*B)->ops->lufactorsymbolic  = MatLUFactorSymbolic_SeqAIJ;
120     (*B)->ops->iludtfactor       = MatILUDTFactor_SeqAIJ;
121   } else if (ftype == MAT_FACTOR_CHOLESKY || ftype == MAT_FACTOR_ICC) {
122     ierr = MatSetType(*B,MATSEQSBAIJ);CHKERRQ(ierr);
123     ierr = MatSeqSBAIJSetPreallocation(*B,1,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
124     (*B)->ops->iccfactorsymbolic      = MatICCFactorSymbolic_SeqAIJ;
125     (*B)->ops->choleskyfactorsymbolic = MatCholeskyFactorSymbolic_SeqAIJ;
126   } else SETERRQ(PETSC_ERR_SUP,"Factor type not supported");
127   (*B)->factor = ftype;
128   PetscFunctionReturn(0);
129 }
130 EXTERN_C_END
131 
132 #undef __FUNCT__
133 #define __FUNCT__ "MatLUFactorSymbolic_SeqAIJ"
134 PetscErrorCode MatLUFactorSymbolic_SeqAIJ(Mat B,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
135 {
136   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
137   IS                 isicol;
138   PetscErrorCode     ierr;
139   const PetscInt     *r,*ic;
140   PetscInt           i,n=A->rmap->n,*ai=a->i,*aj=a->j;
141   PetscInt           *bi,*bj,*ajtmp;
142   PetscInt           *bdiag,row,nnz,nzi,reallocs=0,nzbd,*im;
143   PetscReal          f;
144   PetscInt           nlnk,*lnk,k,**bi_ptr;
145   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
146   PetscBT            lnkbt;
147 
148   PetscFunctionBegin;
149   if (A->rmap->N != A->cmap->N) SETERRQ(PETSC_ERR_ARG_WRONG,"matrix must be square");
150   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
151   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
152   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
153 
154   /* get new row pointers */
155   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
156   bi[0] = 0;
157 
158   /* bdiag is location of diagonal in factor */
159   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);
160   bdiag[0] = 0;
161 
162   /* linked list for storing column indices of the active row */
163   nlnk = n + 1;
164   ierr = PetscLLCreate(n,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
165 
166   ierr = PetscMalloc2(n+1,PetscInt**,&bi_ptr,n+1,PetscInt,&im);CHKERRQ(ierr);
167 
168   /* initial FreeSpace size is f*(ai[n]+1) */
169   f = info->fill;
170   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space);CHKERRQ(ierr);
171   current_space = free_space;
172 
173   for (i=0; i<n; i++) {
174     /* copy previous fill into linked list */
175     nzi = 0;
176     nnz = ai[r[i]+1] - ai[r[i]];
177     if (!nnz) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
178     ajtmp = aj + ai[r[i]];
179     ierr = PetscLLAddPerm(nnz,ajtmp,ic,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
180     nzi += nlnk;
181 
182     /* add pivot rows into linked list */
183     row = lnk[n];
184     while (row < i) {
185       nzbd    = bdiag[row] - bi[row] + 1; /* num of entries in the row with column index <= row */
186       ajtmp   = bi_ptr[row] + nzbd; /* points to the entry next to the diagonal */
187       ierr = PetscLLAddSortedLU(ajtmp,row,nlnk,lnk,lnkbt,i,nzbd,im);CHKERRQ(ierr);
188       nzi += nlnk;
189       row  = lnk[row];
190     }
191     bi[i+1] = bi[i] + nzi;
192     im[i]   = nzi;
193 
194     /* mark bdiag */
195     nzbd = 0;
196     nnz  = nzi;
197     k    = lnk[n];
198     while (nnz-- && k < i){
199       nzbd++;
200       k = lnk[k];
201     }
202     bdiag[i] = bi[i] + nzbd;
203 
204     /* if free space is not available, make more free space */
205     if (current_space->local_remaining<nzi) {
206       nnz = (n - i)*nzi; /* estimated and max additional space needed */
207       ierr = PetscFreeSpaceGet(nnz,&current_space);CHKERRQ(ierr);
208       reallocs++;
209     }
210 
211     /* copy data into free space, then initialize lnk */
212     ierr = PetscLLClean(n,n,nzi,lnk,current_space->array,lnkbt);CHKERRQ(ierr);
213     bi_ptr[i] = current_space->array;
214     current_space->array           += nzi;
215     current_space->local_used      += nzi;
216     current_space->local_remaining -= nzi;
217   }
218 #if defined(PETSC_USE_INFO)
219   if (ai[n] != 0) {
220     PetscReal af = ((PetscReal)bi[n])/((PetscReal)ai[n]);
221     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,f,af);CHKERRQ(ierr);
222     ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
223     ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G);\n",af);CHKERRQ(ierr);
224     ierr = PetscInfo(A,"for best performance.\n");CHKERRQ(ierr);
225   } else {
226     ierr = PetscInfo(A,"Empty matrix\n");CHKERRQ(ierr);
227   }
228 #endif
229 
230   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
231   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
232 
233   /* destroy list of free space and other temporary array(s) */
234   ierr = PetscMalloc((bi[n]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
235   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
236   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
237   ierr = PetscFree2(bi_ptr,im);CHKERRQ(ierr);
238 
239   /* put together the new matrix */
240   ierr = MatSeqAIJSetPreallocation_SeqAIJ(B,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
241   ierr = PetscLogObjectParent(B,isicol);CHKERRQ(ierr);
242   b    = (Mat_SeqAIJ*)(B)->data;
243   b->free_a       = PETSC_TRUE;
244   b->free_ij      = PETSC_TRUE;
245   b->singlemalloc = PETSC_FALSE;
246   ierr          = PetscMalloc((bi[n]+1)*sizeof(PetscScalar),&b->a);CHKERRQ(ierr);
247   b->j          = bj;
248   b->i          = bi;
249   b->diag       = bdiag;
250   b->ilen       = 0;
251   b->imax       = 0;
252   b->row        = isrow;
253   b->col        = iscol;
254   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
255   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
256   b->icol       = isicol;
257   ierr          = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
258 
259   /* In b structure:  Free imax, ilen, old a, old j.  Allocate solve_work, new a, new j */
260   ierr = PetscLogObjectMemory(B,(bi[n]-n)*(sizeof(PetscInt)+sizeof(PetscScalar)));CHKERRQ(ierr);
261   b->maxnz = b->nz = bi[n] ;
262 
263   (B)->factor                = MAT_FACTOR_LU;
264   (B)->info.factor_mallocs   = reallocs;
265   (B)->info.fill_ratio_given = f;
266 
267   if (ai[n]) {
268     (B)->info.fill_ratio_needed = ((PetscReal)bi[n])/((PetscReal)ai[n]);
269   } else {
270     (B)->info.fill_ratio_needed = 0.0;
271   }
272   (B)->ops->lufactornumeric  = MatLUFactorNumeric_SeqAIJ;
273   (B)->ops->solve            = MatSolve_SeqAIJ;
274   (B)->ops->solvetranspose   = MatSolveTranspose_SeqAIJ;
275   /* switch to inodes if appropriate */
276   ierr = MatLUFactorSymbolic_Inode(B,A,isrow,iscol,info);CHKERRQ(ierr);
277   PetscFunctionReturn(0);
278 }
279 
280 /*
281     Trouble in factorization, should we dump the original matrix?
282 */
283 #undef __FUNCT__
284 #define __FUNCT__ "MatFactorDumpMatrix"
285 PetscErrorCode MatFactorDumpMatrix(Mat A)
286 {
287   PetscErrorCode ierr;
288   PetscTruth     flg = PETSC_FALSE;
289 
290   PetscFunctionBegin;
291   ierr = PetscOptionsGetTruth(PETSC_NULL,"-mat_factor_dump_on_error",&flg,PETSC_NULL);CHKERRQ(ierr);
292   if (flg) {
293     PetscViewer viewer;
294     char        filename[PETSC_MAX_PATH_LEN];
295 
296     ierr = PetscSNPrintf(filename,PETSC_MAX_PATH_LEN,"matrix_factor_error.%d",PetscGlobalRank);CHKERRQ(ierr);
297     ierr = PetscViewerBinaryOpen(((PetscObject)A)->comm,filename,FILE_MODE_WRITE,&viewer);CHKERRQ(ierr);
298     ierr = MatView(A,viewer);CHKERRQ(ierr);
299     ierr = PetscViewerDestroy(viewer);CHKERRQ(ierr);
300   }
301   PetscFunctionReturn(0);
302 }
303 
304 extern PetscErrorCode MatSolve_Inode(Mat,Vec,Vec);
305 
306 /* ----------------------------------------------------------- */
307 extern PetscErrorCode MatSolve_SeqAIJ_NaturalOrdering_iludt(Mat,Vec,Vec);
308 extern PetscErrorCode MatSolve_SeqAIJ_iludt(Mat,Vec,Vec);
309 
310 #undef __FUNCT__
311 #define __FUNCT__ "MatLUFactorNumeric_SeqAIJ_newdatastruct"
312 PetscErrorCode MatLUFactorNumeric_SeqAIJ_newdatastruct(Mat B,Mat A,const MatFactorInfo *info)
313 {
314   Mat            C=B;
315   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ *)C->data;
316   IS             isrow = b->row,isicol = b->icol;
317   PetscErrorCode ierr;
318   const PetscInt *r,*ic,*ics;
319   PetscInt       i,j,k,n=A->rmap->n,*ai=a->i,*aj=a->j,*bi=b->i,*bj=b->j;
320   PetscInt       *ajtmp,*bjtmp,nz,nzL,row,*bdiag=b->diag,*pj;
321   MatScalar      *rtmp,*pc,multiplier,*v,*pv,*aa=a->a;
322   PetscReal      shift=info->shiftinblocks;
323   PetscTruth     row_identity, col_identity;
324 
325   PetscFunctionBegin;
326   /* printf("MatLUFactorNumeric_SeqAIJ_newdatastruct is called ...\n"); */
327   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
328   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
329   ierr = PetscMalloc((n+1)*sizeof(MatScalar),&rtmp);CHKERRQ(ierr);
330   ics  = ic;
331 
332   for (i=0; i<n; i++){
333     /* zero rtmp */
334     /* L part */
335     nz    = bi[i+1] - bi[i];
336     bjtmp = bj + bi[i];
337     for  (j=0; j<nz; j++) rtmp[bjtmp[j]] = 0.0;
338 
339     /* U part */
340     nz = bi[2*n-i+1] - bi[2*n-i];
341     bjtmp = bj + bi[2*n-i];
342     for  (j=0; j<nz; j++) rtmp[bjtmp[j]] = 0.0;
343 
344     /* load in initial (unfactored row) */
345     nz    = ai[r[i]+1] - ai[r[i]];
346     ajtmp = aj + ai[r[i]];
347     v     = aa + ai[r[i]];
348     for (j=0; j<nz; j++) {
349       rtmp[ics[ajtmp[j]]] = v[j];
350     }
351     if (rtmp[ics[r[i]]] == 0.0){
352       rtmp[ics[r[i]]] += shift; /* shift the diagonal of the matrix */
353       /* printf("row %d, shift %g\n",i,shift); */
354     }
355 
356     /* elimination */
357     bjtmp = bj + bi[i];
358     row   = *bjtmp++;
359     nzL   = bi[i+1] - bi[i];
360     k   = 0;
361     while  (k < nzL) {
362       pc = rtmp + row;
363       if (*pc != 0.0) {
364         pv         = b->a + bdiag[row];
365         multiplier = *pc * (*pv);
366         *pc        = multiplier;
367         pj         = b->j + bi[2*n-row]; /* begining of U(row,:) */
368         pv         = b->a + bi[2*n-row];
369         nz         = bi[2*n-row+1] - bi[2*n-row] - 1; /* num of entries in U(row,:), excluding diag */
370         for (j=0; j<nz; j++) rtmp[pj[j]] -= multiplier * pv[j];
371         ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr);
372       }
373       row = *bjtmp++; k++;
374     }
375 
376     /* finished row so stick it into b->a */
377     /* L part */
378     pv   = b->a + bi[i] ;
379     pj   = b->j + bi[i] ;
380     nz   = bi[i+1] - bi[i];
381     for (j=0; j<nz; j++) {
382       pv[j] = rtmp[pj[j]];
383     }
384 
385     /* Mark diagonal and invert diagonal for simplier triangular solves */
386     pv  = b->a + bdiag[i];
387     pj  = b->j + bdiag[i];
388     /* if (*pj != i)SETERRQ2(PETSC_ERR_SUP,"row %d != *pj %d",i,*pj) */
389     *pv = 1.0/rtmp[*pj];
390 
391     /* U part */
392     pv = b->a + bi[2*n-i];
393     pj = b->j + bi[2*n-i];
394     nz = bi[2*n-i+1] - bi[2*n-i] - 1;
395     for (j=0; j<nz; j++) pv[j] = rtmp[pj[j]];
396   }
397   ierr = PetscFree(rtmp);CHKERRQ(ierr);
398   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
399   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
400 
401   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
402   ierr = ISIdentity(isicol,&col_identity);CHKERRQ(ierr);
403   if (row_identity && col_identity) {
404     C->ops->solve = MatSolve_SeqAIJ_NaturalOrdering_iludt;
405   } else {
406     C->ops->solve = MatSolve_SeqAIJ_iludt;
407   }
408 
409   C->ops->solveadd           = 0;
410   C->ops->solvetranspose     = 0;
411   C->ops->solvetransposeadd  = 0;
412   C->ops->matsolve           = 0;
413   C->assembled    = PETSC_TRUE;
414   C->preallocated = PETSC_TRUE;
415   ierr = PetscLogFlops(C->cmap->n);CHKERRQ(ierr);
416   PetscFunctionReturn(0);
417 }
418 
419 #undef __FUNCT__
420 #define __FUNCT__ "MatLUFactorNumeric_SeqAIJ"
421 PetscErrorCode MatLUFactorNumeric_SeqAIJ(Mat B,Mat A,const MatFactorInfo *info)
422 {
423   Mat             C=B;
424   Mat_SeqAIJ      *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ *)C->data;
425   IS              isrow = b->row,isicol = b->icol;
426   PetscErrorCode  ierr;
427   const PetscInt   *r,*ic,*ics;
428   PetscInt        nz,row,i,j,n=A->rmap->n,diag;
429   const PetscInt  *ai=a->i,*aj=a->j,*bi=b->i,*bj=b->j;
430   const PetscInt  *ajtmp,*bjtmp,*diag_offset = b->diag,*pj;
431   MatScalar       *pv,*rtmp,*pc,multiplier,d;
432   const MatScalar *v,*aa=a->a;
433   PetscReal       rs=0.0;
434   LUShift_Ctx     sctx;
435   PetscInt        newshift,*ddiag;
436 
437   PetscFunctionBegin;
438   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
439   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
440   ierr = PetscMalloc((n+1)*sizeof(MatScalar),&rtmp);CHKERRQ(ierr);
441   ics  = ic;
442 
443   sctx.shift_top      = 0;
444   sctx.nshift_max     = 0;
445   sctx.shift_lo       = 0;
446   sctx.shift_hi       = 0;
447   sctx.shift_fraction = 0;
448 
449   /* if both shift schemes are chosen by user, only use info->shiftpd */
450   if (info->shiftpd) { /* set sctx.shift_top=max{rs} */
451     ddiag          = a->diag;
452     sctx.shift_top = info->zeropivot;
453     for (i=0; i<n; i++) {
454       /* calculate sum(|aij|)-RealPart(aii), amt of shift needed for this row */
455       d  = (aa)[ddiag[i]];
456       rs = -PetscAbsScalar(d) - PetscRealPart(d);
457       v  = aa+ai[i];
458       nz = ai[i+1] - ai[i];
459       for (j=0; j<nz; j++)
460 	rs += PetscAbsScalar(v[j]);
461       if (rs>sctx.shift_top) sctx.shift_top = rs;
462     }
463     sctx.shift_top   *= 1.1;
464     sctx.nshift_max   = 5;
465     sctx.shift_lo     = 0.;
466     sctx.shift_hi     = 1.;
467   }
468 
469   sctx.shift_amount = 0.0;
470   sctx.nshift       = 0;
471   do {
472     sctx.lushift = PETSC_FALSE;
473     for (i=0; i<n; i++){
474       nz    = bi[i+1] - bi[i];
475       bjtmp = bj + bi[i];
476       for  (j=0; j<nz; j++) rtmp[bjtmp[j]] = 0.0;
477 
478       /* load in initial (unfactored row) */
479       nz    = ai[r[i]+1] - ai[r[i]];
480       ajtmp = aj + ai[r[i]];
481       v     = aa + ai[r[i]];
482       for (j=0; j<nz; j++) {
483         rtmp[ics[ajtmp[j]]] = v[j];
484       }
485       rtmp[ics[r[i]]] += sctx.shift_amount; /* shift the diagonal of the matrix */
486       /* if (sctx.shift_amount > 0.0) printf("row %d, shift %g\n",i,sctx.shift_amount); */
487 
488       row = *bjtmp++;
489       while  (row < i) {
490         pc = rtmp + row;
491         if (*pc != 0.0) {
492           pv         = b->a + diag_offset[row];
493           pj         = b->j + diag_offset[row] + 1;
494           multiplier = *pc / *pv++;
495           *pc        = multiplier;
496           nz         = bi[row+1] - diag_offset[row] - 1;
497           for (j=0; j<nz; j++) rtmp[pj[j]] -= multiplier * pv[j];
498           ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr);
499         }
500         row = *bjtmp++;
501       }
502       /* finished row so stick it into b->a */
503       pv   = b->a + bi[i] ;
504       pj   = b->j + bi[i] ;
505       nz   = bi[i+1] - bi[i];
506       diag = diag_offset[i] - bi[i];
507       rs   = 0.0;
508       for (j=0; j<nz; j++) {
509         pv[j] = rtmp[pj[j]];
510         rs   += PetscAbsScalar(pv[j]);
511       }
512       rs   -= PetscAbsScalar(pv[diag]);
513 
514       /* 9/13/02 Victor Eijkhout suggested scaling zeropivot by rs for matrices with funny scalings */
515       sctx.rs  = rs;
516       sctx.pv  = pv[diag];
517       ierr = MatLUCheckShift_inline(info,sctx,i,newshift);CHKERRQ(ierr);
518       if (newshift == 1) break;
519     }
520 
521     if (info->shiftpd && !sctx.lushift && sctx.shift_fraction>0 && sctx.nshift<sctx.nshift_max) {
522       /*
523        * if no shift in this attempt & shifting & started shifting & can refine,
524        * then try lower shift
525        */
526       sctx.shift_hi       = sctx.shift_fraction;
527       sctx.shift_fraction = (sctx.shift_hi+sctx.shift_lo)/2.;
528       sctx.shift_amount   = sctx.shift_fraction * sctx.shift_top;
529       sctx.lushift        = PETSC_TRUE;
530       sctx.nshift++;
531     }
532   } while (sctx.lushift);
533 
534   /* invert diagonal entries for simplier triangular solves */
535   for (i=0; i<n; i++) {
536     b->a[diag_offset[i]] = 1.0/b->a[diag_offset[i]];
537   }
538   ierr = PetscFree(rtmp);CHKERRQ(ierr);
539   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
540   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
541   if (b->inode.use) {
542     C->ops->solve   = MatSolve_Inode;
543   } else {
544     PetscTruth row_identity, col_identity;
545     ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
546     ierr = ISIdentity(isicol,&col_identity);CHKERRQ(ierr);
547     if (row_identity && col_identity) {
548       C->ops->solve   = MatSolve_SeqAIJ_NaturalOrdering;
549     } else {
550       C->ops->solve   = MatSolve_SeqAIJ;
551     }
552   }
553   C->ops->solveadd           = MatSolveAdd_SeqAIJ;
554   C->ops->solvetranspose     = MatSolveTranspose_SeqAIJ;
555   C->ops->solvetransposeadd  = MatSolveTransposeAdd_SeqAIJ;
556   C->ops->matsolve           = MatMatSolve_SeqAIJ;
557   C->assembled    = PETSC_TRUE;
558   C->preallocated = PETSC_TRUE;
559   ierr = PetscLogFlops(C->cmap->n);CHKERRQ(ierr);
560   if (sctx.nshift){
561      if (info->shiftpd) {
562       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);
563     } else if (info->shiftnz) {
564       ierr = PetscInfo2(A,"number of shift_nz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
565     }
566   }
567   PetscFunctionReturn(0);
568 }
569 
570 /*
571    This routine implements inplace ILU(0) with row or/and column permutations.
572    Input:
573      A - original matrix
574    Output;
575      A - a->i (rowptr) is same as original rowptr, but factored i-the row is stored in rowperm[i]
576          a->j (col index) is permuted by the inverse of colperm, then sorted
577          a->a reordered accordingly with a->j
578          a->diag (ptr to diagonal elements) is updated.
579 */
580 #undef __FUNCT__
581 #define __FUNCT__ "MatLUFactorNumeric_SeqAIJ_InplaceWithPerm"
582 PetscErrorCode MatLUFactorNumeric_SeqAIJ_InplaceWithPerm(Mat B,Mat A,const MatFactorInfo *info)
583 {
584   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data;
585   IS             isrow = a->row,isicol = a->icol;
586   PetscErrorCode ierr;
587   const PetscInt *r,*ic,*ics;
588   PetscInt       i,j,n=A->rmap->n,*ai=a->i,*aj=a->j;
589   PetscInt       *ajtmp,nz,row;
590   PetscInt       *diag = a->diag,nbdiag,*pj;
591   PetscScalar    *rtmp,*pc,multiplier,d;
592   MatScalar      *v,*pv;
593   PetscReal      rs;
594   LUShift_Ctx    sctx;
595   PetscInt       newshift;
596 
597   PetscFunctionBegin;
598   if (A != B) SETERRQ(PETSC_ERR_ARG_INCOMP,"input and output matrix must have same address");
599   ierr  = ISGetIndices(isrow,&r);CHKERRQ(ierr);
600   ierr  = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
601   ierr  = PetscMalloc((n+1)*sizeof(PetscScalar),&rtmp);CHKERRQ(ierr);
602   ierr  = PetscMemzero(rtmp,(n+1)*sizeof(PetscScalar));CHKERRQ(ierr);
603   ics = ic;
604 
605   sctx.shift_top      = 0;
606   sctx.nshift_max     = 0;
607   sctx.shift_lo       = 0;
608   sctx.shift_hi       = 0;
609   sctx.shift_fraction = 0;
610 
611   /* if both shift schemes are chosen by user, only use info->shiftpd */
612   if (info->shiftpd) { /* set sctx.shift_top=max{rs} */
613     sctx.shift_top = 0;
614     for (i=0; i<n; i++) {
615       /* calculate sum(|aij|)-RealPart(aii), amt of shift needed for this row */
616       d  = (a->a)[diag[i]];
617       rs = -PetscAbsScalar(d) - PetscRealPart(d);
618       v  = a->a+ai[i];
619       nz = ai[i+1] - ai[i];
620       for (j=0; j<nz; j++)
621 	rs += PetscAbsScalar(v[j]);
622       if (rs>sctx.shift_top) sctx.shift_top = rs;
623     }
624     if (sctx.shift_top < info->zeropivot) sctx.shift_top = info->zeropivot;
625     sctx.shift_top    *= 1.1;
626     sctx.nshift_max   = 5;
627     sctx.shift_lo     = 0.;
628     sctx.shift_hi     = 1.;
629   }
630 
631   sctx.shift_amount = 0;
632   sctx.nshift       = 0;
633   do {
634     sctx.lushift = PETSC_FALSE;
635     for (i=0; i<n; i++){
636       /* load in initial unfactored row */
637       nz    = ai[r[i]+1] - ai[r[i]];
638       ajtmp = aj + ai[r[i]];
639       v     = a->a + ai[r[i]];
640       /* sort permuted ajtmp and values v accordingly */
641       for (j=0; j<nz; j++) ajtmp[j] = ics[ajtmp[j]];
642       ierr = PetscSortIntWithScalarArray(nz,ajtmp,v);CHKERRQ(ierr);
643 
644       diag[r[i]] = ai[r[i]];
645       for (j=0; j<nz; j++) {
646         rtmp[ajtmp[j]] = v[j];
647         if (ajtmp[j] < i) diag[r[i]]++; /* update a->diag */
648       }
649       rtmp[r[i]] += sctx.shift_amount; /* shift the diagonal of the matrix */
650 
651       row = *ajtmp++;
652       while  (row < i) {
653         pc = rtmp + row;
654         if (*pc != 0.0) {
655           pv         = a->a + diag[r[row]];
656           pj         = aj + diag[r[row]] + 1;
657 
658           multiplier = *pc / *pv++;
659           *pc        = multiplier;
660           nz         = ai[r[row]+1] - diag[r[row]] - 1;
661           for (j=0; j<nz; j++) rtmp[pj[j]] -= multiplier * pv[j];
662           ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr);
663         }
664         row = *ajtmp++;
665       }
666       /* finished row so overwrite it onto a->a */
667       pv   = a->a + ai[r[i]] ;
668       pj   = aj + ai[r[i]] ;
669       nz   = ai[r[i]+1] - ai[r[i]];
670       nbdiag = diag[r[i]] - ai[r[i]]; /* num of entries before the diagonal */
671 
672       rs   = 0.0;
673       for (j=0; j<nz; j++) {
674         pv[j] = rtmp[pj[j]];
675         if (j != nbdiag) rs += PetscAbsScalar(pv[j]);
676       }
677 
678       /* 9/13/02 Victor Eijkhout suggested scaling zeropivot by rs for matrices with funny scalings */
679       sctx.rs  = rs;
680       sctx.pv  = pv[nbdiag];
681       ierr = MatLUCheckShift_inline(info,sctx,i,newshift);CHKERRQ(ierr);
682       if (newshift == 1) break;
683     }
684 
685     if (info->shiftpd && !sctx.lushift && sctx.shift_fraction>0 && sctx.nshift<sctx.nshift_max) {
686       /*
687        * if no shift in this attempt & shifting & started shifting & can refine,
688        * then try lower shift
689        */
690       sctx.shift_hi        = sctx.shift_fraction;
691       sctx.shift_fraction = (sctx.shift_hi+sctx.shift_lo)/2.;
692       sctx.shift_amount    = sctx.shift_fraction * sctx.shift_top;
693       sctx.lushift         = PETSC_TRUE;
694       sctx.nshift++;
695     }
696   } while (sctx.lushift);
697 
698   /* invert diagonal entries for simplier triangular solves */
699   for (i=0; i<n; i++) {
700     a->a[diag[r[i]]] = 1.0/a->a[diag[r[i]]];
701   }
702 
703   ierr = PetscFree(rtmp);CHKERRQ(ierr);
704   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
705   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
706   A->ops->solve             = MatSolve_SeqAIJ_InplaceWithPerm;
707   A->ops->solveadd          = MatSolveAdd_SeqAIJ;
708   A->ops->solvetranspose    = MatSolveTranspose_SeqAIJ;
709   A->ops->solvetransposeadd = MatSolveTransposeAdd_SeqAIJ;
710   A->assembled = PETSC_TRUE;
711   A->preallocated = PETSC_TRUE;
712   ierr = PetscLogFlops(A->cmap->n);CHKERRQ(ierr);
713   if (sctx.nshift){
714     if (info->shiftpd) {
715       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);
716     } else if (info->shiftnz) {
717       ierr = PetscInfo2(A,"number of shift_nz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
718     }
719   }
720   PetscFunctionReturn(0);
721 }
722 
723 /* ----------------------------------------------------------- */
724 #undef __FUNCT__
725 #define __FUNCT__ "MatLUFactor_SeqAIJ"
726 PetscErrorCode MatLUFactor_SeqAIJ(Mat A,IS row,IS col,const MatFactorInfo *info)
727 {
728   PetscErrorCode ierr;
729   Mat            C;
730 
731   PetscFunctionBegin;
732   ierr = MatGetFactor(A,MAT_SOLVER_PETSC,MAT_FACTOR_LU,&C);CHKERRQ(ierr);
733   ierr = MatLUFactorSymbolic(C,A,row,col,info);CHKERRQ(ierr);
734   ierr = MatLUFactorNumeric(C,A,info);CHKERRQ(ierr);
735   A->ops->solve            = C->ops->solve;
736   A->ops->solvetranspose   = C->ops->solvetranspose;
737   ierr = MatHeaderCopy(A,C);CHKERRQ(ierr);
738   ierr = PetscLogObjectParent(A,((Mat_SeqAIJ*)(A->data))->icol);CHKERRQ(ierr);
739   PetscFunctionReturn(0);
740 }
741 /* ----------------------------------------------------------- */
742 
743 
744 #undef __FUNCT__
745 #define __FUNCT__ "MatSolve_SeqAIJ"
746 PetscErrorCode MatSolve_SeqAIJ(Mat A,Vec bb,Vec xx)
747 {
748   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
749   IS                iscol = a->col,isrow = a->row;
750   PetscErrorCode    ierr;
751   PetscInt          i, n = A->rmap->n,*vi,*ai = a->i,*aj = a->j;
752   PetscInt          nz;
753   const PetscInt    *rout,*cout,*r,*c;
754   PetscScalar       *x,*tmp,*tmps,sum;
755   const PetscScalar *b;
756   const MatScalar   *aa = a->a,*v;
757 
758   PetscFunctionBegin;
759   if (!n) PetscFunctionReturn(0);
760 
761   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
762   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
763   tmp  = a->solve_work;
764 
765   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
766   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout + (n-1);
767 
768   /* forward solve the lower triangular */
769   tmp[0] = b[*r++];
770   tmps   = tmp;
771   for (i=1; i<n; i++) {
772     v   = aa + ai[i] ;
773     vi  = aj + ai[i] ;
774     nz  = a->diag[i] - ai[i];
775     sum = b[*r++];
776     PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
777     tmp[i] = sum;
778   }
779 
780   /* backward solve the upper triangular */
781   for (i=n-1; i>=0; i--){
782     v   = aa + a->diag[i] + 1;
783     vi  = aj + a->diag[i] + 1;
784     nz  = ai[i+1] - a->diag[i] - 1;
785     sum = tmp[i];
786     PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
787     x[*c--] = tmp[i] = sum*aa[a->diag[i]];
788   }
789 
790   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
791   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
792   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
793   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
794   ierr = PetscLogFlops(2.0*a->nz - A->cmap->n);CHKERRQ(ierr);
795   PetscFunctionReturn(0);
796 }
797 
798 #undef __FUNCT__
799 #define __FUNCT__ "MatMatSolve_SeqAIJ"
800 PetscErrorCode MatMatSolve_SeqAIJ(Mat A,Mat B,Mat X)
801 {
802   Mat_SeqAIJ      *a = (Mat_SeqAIJ*)A->data;
803   IS              iscol = a->col,isrow = a->row;
804   PetscErrorCode  ierr;
805   PetscInt        i, n = A->rmap->n,*vi,*ai = a->i,*aj = a->j;
806   PetscInt        nz,neq;
807   const PetscInt  *rout,*cout,*r,*c;
808   PetscScalar     *x,*b,*tmp,*tmps,sum;
809   const MatScalar *aa = a->a,*v;
810   PetscTruth      bisdense,xisdense;
811 
812   PetscFunctionBegin;
813   if (!n) PetscFunctionReturn(0);
814 
815   ierr = PetscTypeCompare((PetscObject)B,MATSEQDENSE,&bisdense);CHKERRQ(ierr);
816   if (!bisdense) SETERRQ(PETSC_ERR_ARG_INCOMP,"B matrix must be a SeqDense matrix");
817   ierr = PetscTypeCompare((PetscObject)X,MATSEQDENSE,&xisdense);CHKERRQ(ierr);
818   if (!xisdense) SETERRQ(PETSC_ERR_ARG_INCOMP,"X matrix must be a SeqDense matrix");
819 
820   ierr = MatGetArray(B,&b);CHKERRQ(ierr);
821   ierr = MatGetArray(X,&x);CHKERRQ(ierr);
822 
823   tmp  = a->solve_work;
824   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
825   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
826 
827   for (neq=0; neq<B->cmap->n; neq++){
828     /* forward solve the lower triangular */
829     tmp[0] = b[r[0]];
830     tmps   = tmp;
831     for (i=1; i<n; i++) {
832       v   = aa + ai[i] ;
833       vi  = aj + ai[i] ;
834       nz  = a->diag[i] - ai[i];
835       sum = b[r[i]];
836       PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
837       tmp[i] = sum;
838     }
839     /* backward solve the upper triangular */
840     for (i=n-1; i>=0; i--){
841       v   = aa + a->diag[i] + 1;
842       vi  = aj + a->diag[i] + 1;
843       nz  = ai[i+1] - a->diag[i] - 1;
844       sum = tmp[i];
845       PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
846       x[c[i]] = tmp[i] = sum*aa[a->diag[i]];
847     }
848 
849     b += n;
850     x += n;
851   }
852   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
853   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
854   ierr = MatRestoreArray(B,&b);CHKERRQ(ierr);
855   ierr = MatRestoreArray(X,&x);CHKERRQ(ierr);
856   ierr = PetscLogFlops(B->cmap->n*(2.0*a->nz - n));CHKERRQ(ierr);
857   PetscFunctionReturn(0);
858 }
859 
860 #undef __FUNCT__
861 #define __FUNCT__ "MatSolve_SeqAIJ_InplaceWithPerm"
862 PetscErrorCode MatSolve_SeqAIJ_InplaceWithPerm(Mat A,Vec bb,Vec xx)
863 {
864   Mat_SeqAIJ      *a = (Mat_SeqAIJ*)A->data;
865   IS              iscol = a->col,isrow = a->row;
866   PetscErrorCode  ierr;
867   const PetscInt  *r,*c,*rout,*cout;
868   PetscInt        i, n = A->rmap->n,*vi,*ai = a->i,*aj = a->j;
869   PetscInt        nz,row;
870   PetscScalar     *x,*b,*tmp,*tmps,sum;
871   const MatScalar *aa = a->a,*v;
872 
873   PetscFunctionBegin;
874   if (!n) PetscFunctionReturn(0);
875 
876   ierr = VecGetArray(bb,&b);CHKERRQ(ierr);
877   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
878   tmp  = a->solve_work;
879 
880   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
881   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout + (n-1);
882 
883   /* forward solve the lower triangular */
884   tmp[0] = b[*r++];
885   tmps   = tmp;
886   for (row=1; row<n; row++) {
887     i   = rout[row]; /* permuted row */
888     v   = aa + ai[i] ;
889     vi  = aj + ai[i] ;
890     nz  = a->diag[i] - ai[i];
891     sum = b[*r++];
892     PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
893     tmp[row] = sum;
894   }
895 
896   /* backward solve the upper triangular */
897   for (row=n-1; row>=0; row--){
898     i   = rout[row]; /* permuted row */
899     v   = aa + a->diag[i] + 1;
900     vi  = aj + a->diag[i] + 1;
901     nz  = ai[i+1] - a->diag[i] - 1;
902     sum = tmp[row];
903     PetscSparseDenseMinusDot(sum,tmps,v,vi,nz);
904     x[*c--] = tmp[row] = sum*aa[a->diag[i]];
905   }
906 
907   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
908   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
909   ierr = VecRestoreArray(bb,&b);CHKERRQ(ierr);
910   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
911   ierr = PetscLogFlops(2.0*a->nz - A->cmap->n);CHKERRQ(ierr);
912   PetscFunctionReturn(0);
913 }
914 
915 /* ----------------------------------------------------------- */
916 #include "../src/mat/impls/aij/seq/ftn-kernels/fsolve.h"
917 #undef __FUNCT__
918 #define __FUNCT__ "MatSolve_SeqAIJ_NaturalOrdering"
919 PetscErrorCode MatSolve_SeqAIJ_NaturalOrdering(Mat A,Vec bb,Vec xx)
920 {
921   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
922   PetscErrorCode    ierr;
923   PetscInt          n = A->rmap->n;
924   const PetscInt    *ai = a->i,*aj = a->j,*adiag = a->diag;
925   PetscScalar       *x;
926   const PetscScalar *b;
927   const MatScalar   *aa = a->a;
928 #if !defined(PETSC_USE_FORTRAN_KERNEL_SOLVEAIJ)
929   PetscInt          adiag_i,i,nz,ai_i;
930   const PetscInt    *vi;
931   const MatScalar   *v;
932   PetscScalar       sum;
933 #endif
934 
935   PetscFunctionBegin;
936   if (!n) PetscFunctionReturn(0);
937 
938   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
939   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
940 
941 #if defined(PETSC_USE_FORTRAN_KERNEL_SOLVEAIJ)
942   fortransolveaij_(&n,x,ai,aj,adiag,aa,b);
943 #else
944   /* forward solve the lower triangular */
945   x[0] = b[0];
946   for (i=1; i<n; i++) {
947     ai_i = ai[i];
948     v    = aa + ai_i;
949     vi   = aj + ai_i;
950     nz   = adiag[i] - ai_i;
951     sum  = b[i];
952     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
953     x[i] = sum;
954   }
955 
956   /* backward solve the upper triangular */
957   for (i=n-1; i>=0; i--){
958     adiag_i = adiag[i];
959     v       = aa + adiag_i + 1;
960     vi      = aj + adiag_i + 1;
961     nz      = ai[i+1] - adiag_i - 1;
962     sum     = x[i];
963     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
964     x[i]    = sum*aa[adiag_i];
965   }
966 #endif
967   ierr = PetscLogFlops(2.0*a->nz - A->cmap->n);CHKERRQ(ierr);
968   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
969   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
970   PetscFunctionReturn(0);
971 }
972 
973 #undef __FUNCT__
974 #define __FUNCT__ "MatSolveAdd_SeqAIJ"
975 PetscErrorCode MatSolveAdd_SeqAIJ(Mat A,Vec bb,Vec yy,Vec xx)
976 {
977   Mat_SeqAIJ      *a = (Mat_SeqAIJ*)A->data;
978   IS              iscol = a->col,isrow = a->row;
979   PetscErrorCode  ierr;
980   PetscInt        i, n = A->rmap->n,*vi,*ai = a->i,*aj = a->j;
981   PetscInt        nz;
982   const PetscInt  *rout,*cout,*r,*c;
983   PetscScalar     *x,*b,*tmp,sum;
984   const MatScalar *aa = a->a,*v;
985 
986   PetscFunctionBegin;
987   if (yy != xx) {ierr = VecCopy(yy,xx);CHKERRQ(ierr);}
988 
989   ierr = VecGetArray(bb,&b);CHKERRQ(ierr);
990   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
991   tmp  = a->solve_work;
992 
993   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
994   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout + (n-1);
995 
996   /* forward solve the lower triangular */
997   tmp[0] = b[*r++];
998   for (i=1; i<n; i++) {
999     v   = aa + ai[i] ;
1000     vi  = aj + ai[i] ;
1001     nz  = a->diag[i] - ai[i];
1002     sum = b[*r++];
1003     while (nz--) sum -= *v++ * tmp[*vi++ ];
1004     tmp[i] = sum;
1005   }
1006 
1007   /* backward solve the upper triangular */
1008   for (i=n-1; i>=0; i--){
1009     v   = aa + a->diag[i] + 1;
1010     vi  = aj + a->diag[i] + 1;
1011     nz  = ai[i+1] - a->diag[i] - 1;
1012     sum = tmp[i];
1013     while (nz--) sum -= *v++ * tmp[*vi++ ];
1014     tmp[i] = sum*aa[a->diag[i]];
1015     x[*c--] += tmp[i];
1016   }
1017 
1018   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1019   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1020   ierr = VecRestoreArray(bb,&b);CHKERRQ(ierr);
1021   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1022   ierr = PetscLogFlops(2.0*a->nz);CHKERRQ(ierr);
1023 
1024   PetscFunctionReturn(0);
1025 }
1026 /* -------------------------------------------------------------------*/
1027 #undef __FUNCT__
1028 #define __FUNCT__ "MatSolveTranspose_SeqAIJ"
1029 PetscErrorCode MatSolveTranspose_SeqAIJ(Mat A,Vec bb,Vec xx)
1030 {
1031   Mat_SeqAIJ      *a = (Mat_SeqAIJ*)A->data;
1032   IS              iscol = a->col,isrow = a->row;
1033   PetscErrorCode  ierr;
1034   const PetscInt  *rout,*cout,*r,*c;
1035   PetscInt        i,n = A->rmap->n,*vi,*ai = a->i,*aj = a->j;
1036   PetscInt        nz,*diag = a->diag;
1037   PetscScalar     *x,*b,*tmp,s1;
1038   const MatScalar *aa = a->a,*v;
1039 
1040   PetscFunctionBegin;
1041   ierr = VecGetArray(bb,&b);CHKERRQ(ierr);
1042   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1043   tmp  = a->solve_work;
1044 
1045   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1046   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1047 
1048   /* copy the b into temp work space according to permutation */
1049   for (i=0; i<n; i++) tmp[i] = b[c[i]];
1050 
1051   /* forward solve the U^T */
1052   for (i=0; i<n; i++) {
1053     v   = aa + diag[i] ;
1054     vi  = aj + diag[i] + 1;
1055     nz  = ai[i+1] - diag[i] - 1;
1056     s1  = tmp[i];
1057     s1 *= (*v++);  /* multiply by inverse of diagonal entry */
1058     while (nz--) {
1059       tmp[*vi++ ] -= (*v++)*s1;
1060     }
1061     tmp[i] = s1;
1062   }
1063 
1064   /* backward solve the L^T */
1065   for (i=n-1; i>=0; i--){
1066     v   = aa + diag[i] - 1 ;
1067     vi  = aj + diag[i] - 1 ;
1068     nz  = diag[i] - ai[i];
1069     s1  = tmp[i];
1070     while (nz--) {
1071       tmp[*vi-- ] -= (*v--)*s1;
1072     }
1073   }
1074 
1075   /* copy tmp into x according to permutation */
1076   for (i=0; i<n; i++) x[r[i]] = tmp[i];
1077 
1078   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1079   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1080   ierr = VecRestoreArray(bb,&b);CHKERRQ(ierr);
1081   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1082 
1083   ierr = PetscLogFlops(2.0*a->nz-A->cmap->n);CHKERRQ(ierr);
1084   PetscFunctionReturn(0);
1085 }
1086 
1087 #undef __FUNCT__
1088 #define __FUNCT__ "MatSolveTransposeAdd_SeqAIJ"
1089 PetscErrorCode MatSolveTransposeAdd_SeqAIJ(Mat A,Vec bb,Vec zz,Vec xx)
1090 {
1091   Mat_SeqAIJ      *a = (Mat_SeqAIJ*)A->data;
1092   IS              iscol = a->col,isrow = a->row;
1093   PetscErrorCode  ierr;
1094   const PetscInt  *r,*c,*rout,*cout;
1095   PetscInt        i,n = A->rmap->n,*vi,*ai = a->i,*aj = a->j;
1096   PetscInt        nz,*diag = a->diag;
1097   PetscScalar     *x,*b,*tmp;
1098   const MatScalar *aa = a->a,*v;
1099 
1100   PetscFunctionBegin;
1101   if (zz != xx) {ierr = VecCopy(zz,xx);CHKERRQ(ierr);}
1102 
1103   ierr = VecGetArray(bb,&b);CHKERRQ(ierr);
1104   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
1105   tmp = a->solve_work;
1106 
1107   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
1108   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout;
1109 
1110   /* copy the b into temp work space according to permutation */
1111   for (i=0; i<n; i++) tmp[i] = b[c[i]];
1112 
1113   /* forward solve the U^T */
1114   for (i=0; i<n; i++) {
1115     v   = aa + diag[i] ;
1116     vi  = aj + diag[i] + 1;
1117     nz  = ai[i+1] - diag[i] - 1;
1118     tmp[i] *= *v++;
1119     while (nz--) {
1120       tmp[*vi++ ] -= (*v++)*tmp[i];
1121     }
1122   }
1123 
1124   /* backward solve the L^T */
1125   for (i=n-1; i>=0; i--){
1126     v   = aa + diag[i] - 1 ;
1127     vi  = aj + diag[i] - 1 ;
1128     nz  = diag[i] - ai[i];
1129     while (nz--) {
1130       tmp[*vi-- ] -= (*v--)*tmp[i];
1131     }
1132   }
1133 
1134   /* copy tmp into x according to permutation */
1135   for (i=0; i<n; i++) x[r[i]] += tmp[i];
1136 
1137   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
1138   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
1139   ierr = VecRestoreArray(bb,&b);CHKERRQ(ierr);
1140   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
1141 
1142   ierr = PetscLogFlops(2.0*a->nz);CHKERRQ(ierr);
1143   PetscFunctionReturn(0);
1144 }
1145 /* ----------------------------------------------------------------*/
1146 EXTERN PetscErrorCode Mat_CheckInode(Mat,PetscTruth);
1147 EXTERN PetscErrorCode MatDuplicateNoCreate_SeqAIJ(Mat,Mat,MatDuplicateOption,PetscTruth);
1148 
1149 /*
1150    ilu(0) with natural ordering under new data structure.
1151    Factored arrays bj and ba are stored as
1152      L(0,:), L(1,:), ...,L(n-1,:),  U(n-1,:),...,U(i,:),U(i-1,:),...,U(0,:)
1153 
1154    bi=fact->i is an array of size 2n+2, in which
1155    bi+
1156      bi[i]      ->  1st entry of L(i,:),i=0,...,i-1
1157      bi[n]      ->  points to L(n-1,:)+1
1158      bi[n+1]    ->  1st entry of U(n-1,:)
1159      bi[2n-i]   ->  1st entry of U(i,:)
1160      bi[2n-i+1] ->  end of U(i,:)+1, the 1st entry of U(i-1,:)
1161      bi[2n]     ->  1st entry of U(0,:)
1162      bi[2n+1]   ->  points to U(0,:)+1
1163 
1164    U(i,:) contains diag[i] as its last entry, i.e.,
1165     U(i,:) = (u[i,i+1],...,u[i,n-1],diag[i])
1166 */
1167 #undef __FUNCT__
1168 #define __FUNCT__ "MatILUFactorSymbolic_SeqAIJ_ilu0_newdatastruct"
1169 PetscErrorCode MatILUFactorSymbolic_SeqAIJ_ilu0_newdatastruct(Mat fact,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
1170 {
1171 
1172   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
1173   PetscErrorCode     ierr;
1174   PetscInt           n=A->rmap->n,*ai=a->i,*aj,*adiag=a->diag;
1175   PetscInt           i,j,nz,*bi,*bj,*bdiag;
1176 
1177   PetscFunctionBegin;
1178   ierr = MatDuplicateNoCreate_SeqAIJ(fact,A,MAT_DO_NOT_COPY_VALUES,PETSC_FALSE);CHKERRQ(ierr);
1179   b    = (Mat_SeqAIJ*)(fact)->data;
1180 
1181   /* allocate matrix arrays for new data structure */
1182   ierr = PetscMalloc3(ai[n]+1,PetscScalar,&b->a,ai[n]+1,PetscInt,&b->j,2*n+2,PetscInt,&b->i);CHKERRQ(ierr);
1183   ierr = PetscLogObjectMemory(fact,ai[n]*(sizeof(PetscScalar)+sizeof(PetscInt))+(2*n+2)*sizeof(PetscInt));CHKERRQ(ierr);
1184   b->singlemalloc = PETSC_TRUE;
1185   if (!b->diag){
1186     ierr = PetscMalloc((n+1)*sizeof(PetscInt),&b->diag);CHKERRQ(ierr);
1187   }
1188   bdiag = b->diag;
1189 
1190   if (n > 0) {
1191     ierr = PetscMemzero(b->a,(ai[n])*sizeof(MatScalar));CHKERRQ(ierr);
1192   }
1193 
1194   /* set bi and bj with new data structure */
1195   bi = b->i;
1196   bj = b->j;
1197 
1198   /* L part */
1199   bi[0] = 0;
1200   for (i=0; i<n; i++){
1201     nz = adiag[i] - ai[i];
1202     bi[i+1] = bi[i] + nz;
1203     aj = a->j + ai[i];
1204     for (j=0; j<nz; j++){
1205       *bj = aj[j]; bj++;
1206     }
1207   }
1208 
1209   /* U part */
1210   bi[n+1] = bi[n];
1211   for (i=n-1; i>=0; i--){
1212     nz = ai[i+1] - adiag[i] - 1;
1213     bi[2*n-i+1] = bi[2*n-i] + nz + 1;
1214     aj = a->j + adiag[i] + 1;
1215     for (j=0; j<nz; j++){
1216       *bj = aj[j]; bj++;
1217     }
1218     /* diag[i] */
1219     *bj = i; bj++;
1220     bdiag[i] = bi[2*n-i+1]-1;
1221   }
1222   PetscFunctionReturn(0);
1223 }
1224 
1225 #undef __FUNCT__
1226 #define __FUNCT__ "MatILUFactorSymbolic_SeqAIJ_newdatastruct"
1227 PetscErrorCode MatILUFactorSymbolic_SeqAIJ_newdatastruct(Mat fact,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
1228 {
1229   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
1230   IS                 isicol;
1231   PetscErrorCode     ierr;
1232   const PetscInt     *r,*ic;
1233   PetscInt           n=A->rmap->n,*ai=a->i,*aj=a->j,d;
1234   PetscInt           *bi,*cols,nnz,*cols_lvl;
1235   PetscInt           *bdiag,prow,fm,nzbd,reallocs=0,dcount=0;
1236   PetscInt           i,levels,diagonal_fill;
1237   PetscTruth         col_identity,row_identity;
1238   PetscReal          f;
1239   PetscInt           nlnk,*lnk,*lnk_lvl=PETSC_NULL;
1240   PetscBT            lnkbt;
1241   PetscInt           nzi,*bj,**bj_ptr,**bjlvl_ptr;
1242   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
1243   PetscFreeSpaceList free_space_lvl=PETSC_NULL,current_space_lvl=PETSC_NULL;
1244   PetscTruth         missing;
1245 
1246   PetscFunctionBegin;
1247   //printf("MatILUFactorSymbolic_SeqAIJ_newdatastruct ...\n");
1248   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);
1249   f             = info->fill;
1250   levels        = (PetscInt)info->levels;
1251   diagonal_fill = (PetscInt)info->diagonal_fill;
1252   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
1253 
1254   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
1255   ierr = ISIdentity(iscol,&col_identity);CHKERRQ(ierr);
1256 
1257   if (!levels && row_identity && col_identity) {
1258     /* special case: ilu(0) with natural ordering */
1259     ierr = MatILUFactorSymbolic_SeqAIJ_ilu0_newdatastruct(fact,A,isrow,iscol,info);CHKERRQ(ierr);
1260     (fact)->ops->lufactornumeric =  MatLUFactorNumeric_SeqAIJ_newdatastruct;
1261 
1262     fact->factor = MAT_FACTOR_ILU;
1263     (fact)->info.factor_mallocs    = 0;
1264     (fact)->info.fill_ratio_given  = info->fill;
1265     (fact)->info.fill_ratio_needed = 1.0;
1266     b               = (Mat_SeqAIJ*)(fact)->data;
1267     ierr = MatMissingDiagonal(A,&missing,&d);CHKERRQ(ierr);
1268     if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",d);
1269     b->row              = isrow;
1270     b->col              = iscol;
1271     b->icol             = isicol;
1272     ierr                = PetscMalloc(((fact)->rmap->n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1273     ierr                = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
1274     ierr                = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
1275     /* ierr = MatILUFactorSymbolic_Inode(fact,A,isrow,iscol,info);CHKERRQ(ierr); */
1276     PetscFunctionReturn(0);
1277   }
1278 
1279   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
1280   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
1281 
1282   /* get new row pointers */
1283   ierr = PetscMalloc((2*n+2)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
1284   bi[0] = 0;
1285   /* bdiag is location of diagonal in factor */
1286   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);
1287   bdiag[0]  = 0;
1288 
1289   ierr = PetscMalloc((2*n+1)*sizeof(PetscInt**),&bj_ptr);CHKERRQ(ierr);
1290   bjlvl_ptr = (PetscInt**)(bj_ptr + n);
1291 
1292   /* create a linked list for storing column indices of the active row */
1293   nlnk = n + 1;
1294   ierr = PetscIncompleteLLCreate(n,n,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1295 
1296   /* initial FreeSpace size is f*(ai[n]+1) */
1297   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space);CHKERRQ(ierr);
1298   current_space = free_space;
1299   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space_lvl);CHKERRQ(ierr);
1300   current_space_lvl = free_space_lvl;
1301 
1302   for (i=0; i<n; i++) {
1303     nzi = 0;
1304     /* copy current row into linked list */
1305     nnz  = ai[r[i]+1] - ai[r[i]];
1306     if (!nnz) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
1307     cols = aj + ai[r[i]];
1308     lnk[i] = -1; /* marker to indicate if diagonal exists */
1309     ierr = PetscIncompleteLLInit(nnz,cols,n,ic,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1310     nzi += nlnk;
1311 
1312     /* make sure diagonal entry is included */
1313     if (diagonal_fill && lnk[i] == -1) {
1314       fm = n;
1315       while (lnk[fm] < i) fm = lnk[fm];
1316       lnk[i]     = lnk[fm]; /* insert diagonal into linked list */
1317       lnk[fm]    = i;
1318       lnk_lvl[i] = 0;
1319       nzi++; dcount++;
1320     }
1321 
1322     /* add pivot rows into the active row */
1323     nzbd = 0;
1324     prow = lnk[n];
1325     while (prow < i) {
1326       nnz      = bdiag[prow];
1327       cols     = bj_ptr[prow] + nnz + 1;
1328       cols_lvl = bjlvl_ptr[prow] + nnz + 1;
1329       nnz      = bi[prow+1] - bi[prow] - nnz - 1;
1330       ierr = PetscILULLAddSorted(nnz,cols,levels,cols_lvl,prow,nlnk,lnk,lnk_lvl,lnkbt,prow);CHKERRQ(ierr);
1331       nzi += nlnk;
1332       prow = lnk[prow];
1333       nzbd++;
1334     }
1335     bdiag[i] = nzbd;
1336     bi[i+1]  = bi[i] + nzi;
1337 
1338     /* if free space is not available, make more free space */
1339     if (current_space->local_remaining<nzi) {
1340       nnz = 2*nzi*(n - i); /* estimated and max additional space needed */
1341       ierr = PetscFreeSpaceGet(nnz,&current_space);CHKERRQ(ierr);
1342       ierr = PetscFreeSpaceGet(nnz,&current_space_lvl);CHKERRQ(ierr);
1343       reallocs++;
1344     }
1345 
1346     /* copy data into free_space and free_space_lvl, then initialize lnk */
1347     ierr = PetscIncompleteLLClean(n,n,nzi,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);CHKERRQ(ierr);
1348     bj_ptr[i]    = current_space->array;
1349     bjlvl_ptr[i] = current_space_lvl->array;
1350 
1351     /* make sure the active row i has diagonal entry */
1352     if (*(bj_ptr[i]+bdiag[i]) != i) {
1353       SETERRQ1(PETSC_ERR_MAT_LU_ZRPVT,"Row %D has missing diagonal in factored matrix\n\
1354     try running with -pc_factor_nonzeros_along_diagonal or -pc_factor_diagonal_fill",i);
1355     }
1356 
1357     current_space->array           += nzi;
1358     current_space->local_used      += nzi;
1359     current_space->local_remaining -= nzi;
1360     current_space_lvl->array           += nzi;
1361     current_space_lvl->local_used      += nzi;
1362     current_space_lvl->local_remaining -= nzi;
1363   }
1364 
1365   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
1366   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
1367 
1368   /* destroy list of free space and other temporary arrays */
1369   ierr = PetscMalloc((bi[n]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
1370 
1371   /* copy free_space into bj and free free_space; set bi, bj, bdiag in new datastructure; */
1372   ierr = PetscFreeSpaceContiguous_newdatastruct(&free_space,bj,n,bi,bdiag);CHKERRQ(ierr);
1373 
1374   ierr = PetscIncompleteLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1375   ierr = PetscFreeSpaceDestroy(free_space_lvl);CHKERRQ(ierr);
1376   ierr = PetscFree(bj_ptr);CHKERRQ(ierr);
1377 
1378 #if defined(PETSC_USE_INFO)
1379   {
1380     PetscReal af = ((PetscReal)bi[n])/((PetscReal)ai[n]);
1381     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,f,af);CHKERRQ(ierr);
1382     ierr = PetscInfo1(A,"Run with -[sub_]pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
1383     ierr = PetscInfo1(A,"PCFactorSetFill([sub]pc,%G);\n",af);CHKERRQ(ierr);
1384     ierr = PetscInfo(A,"for best performance.\n");CHKERRQ(ierr);
1385     if (diagonal_fill) {
1386       ierr = PetscInfo1(A,"Detected and replaced %D missing diagonals",dcount);CHKERRQ(ierr);
1387     }
1388   }
1389 #endif
1390 
1391   /* put together the new matrix */
1392   ierr = MatSeqAIJSetPreallocation_SeqAIJ(fact,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
1393   ierr = PetscLogObjectParent(fact,isicol);CHKERRQ(ierr);
1394   b = (Mat_SeqAIJ*)(fact)->data;
1395   b->free_a       = PETSC_TRUE;
1396   b->free_ij      = PETSC_TRUE;
1397   b->singlemalloc = PETSC_FALSE;
1398   ierr = PetscMalloc( (bi[2*n+1] )*sizeof(PetscScalar),&b->a);CHKERRQ(ierr);
1399   b->j          = bj;
1400   b->i          = bi;
1401   b->diag       = bdiag;
1402   b->ilen       = 0;
1403   b->imax       = 0;
1404   b->row        = isrow;
1405   b->col        = iscol;
1406   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
1407   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
1408   b->icol       = isicol;
1409   ierr = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1410   /* In b structure:  Free imax, ilen, old a, old j.
1411      Allocate bdiag, solve_work, new a, new j */
1412   ierr = PetscLogObjectMemory(fact,bi[2*n+1] * (sizeof(PetscInt)+sizeof(PetscScalar)));CHKERRQ(ierr);
1413   b->maxnz = b->nz = bi[2*n+1] ;
1414   (fact)->info.factor_mallocs    = reallocs;
1415   (fact)->info.fill_ratio_given  = f;
1416   (fact)->info.fill_ratio_needed = ((PetscReal)bi[2*n+1])/((PetscReal)ai[n]);
1417   (fact)->ops->lufactornumeric   = MatLUFactorNumeric_SeqAIJ_newdatastruct;
1418   /* ierr = MatILUFactorSymbolic_Inode(fact,A,isrow,iscol,info);CHKERRQ(ierr); */
1419   PetscFunctionReturn(0);
1420 }
1421 
1422 #undef __FUNCT__
1423 #define __FUNCT__ "MatILUFactorSymbolic_SeqAIJ"
1424 PetscErrorCode MatILUFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
1425 {
1426   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
1427   IS                 isicol;
1428   PetscErrorCode     ierr;
1429   const PetscInt     *r,*ic;
1430   PetscInt           n=A->rmap->n,*ai=a->i,*aj=a->j,d;
1431   PetscInt           *bi,*cols,nnz,*cols_lvl;
1432   PetscInt           *bdiag,prow,fm,nzbd,reallocs=0,dcount=0;
1433   PetscInt           i,levels,diagonal_fill;
1434   PetscTruth         col_identity,row_identity;
1435   PetscReal          f;
1436   PetscInt           nlnk,*lnk,*lnk_lvl=PETSC_NULL;
1437   PetscBT            lnkbt;
1438   PetscInt           nzi,*bj,**bj_ptr,**bjlvl_ptr;
1439   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
1440   PetscFreeSpaceList free_space_lvl=PETSC_NULL,current_space_lvl=PETSC_NULL;
1441   PetscTruth         missing;
1442   PetscTruth         newdatastruct=PETSC_FALSE;
1443 
1444   PetscFunctionBegin;
1445   ierr = PetscOptionsGetTruth(PETSC_NULL,"-ilu_new",&newdatastruct,PETSC_NULL);CHKERRQ(ierr);
1446   if (newdatastruct){
1447     ierr = MatILUFactorSymbolic_SeqAIJ_newdatastruct(fact,A,isrow,iscol,info);CHKERRQ(ierr);
1448     PetscFunctionReturn(0);
1449   }
1450 
1451   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);
1452   f             = info->fill;
1453   levels        = (PetscInt)info->levels;
1454   diagonal_fill = (PetscInt)info->diagonal_fill;
1455   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
1456 
1457   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
1458   ierr = ISIdentity(iscol,&col_identity);CHKERRQ(ierr);
1459   if (!levels && row_identity && col_identity) { /* special case: ilu(0) with natural ordering */
1460     ierr = MatDuplicateNoCreate_SeqAIJ(fact,A,MAT_DO_NOT_COPY_VALUES,PETSC_TRUE);CHKERRQ(ierr);
1461     (fact)->ops->lufactornumeric =  MatLUFactorNumeric_SeqAIJ;
1462 
1463     fact->factor = MAT_FACTOR_ILU;
1464     (fact)->info.factor_mallocs    = 0;
1465     (fact)->info.fill_ratio_given  = info->fill;
1466     (fact)->info.fill_ratio_needed = 1.0;
1467     b               = (Mat_SeqAIJ*)(fact)->data;
1468     ierr = MatMissingDiagonal(A,&missing,&d);CHKERRQ(ierr);
1469     if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",d);
1470     b->row              = isrow;
1471     b->col              = iscol;
1472     b->icol             = isicol;
1473     ierr                = PetscMalloc(((fact)->rmap->n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1474     ierr                = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
1475     ierr                = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
1476     ierr = MatILUFactorSymbolic_Inode(fact,A,isrow,iscol,info);CHKERRQ(ierr);
1477     PetscFunctionReturn(0);
1478   }
1479 
1480   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
1481   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
1482 
1483   /* get new row pointers */
1484   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
1485   bi[0] = 0;
1486   /* bdiag is location of diagonal in factor */
1487   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);
1488   bdiag[0]  = 0;
1489 
1490   ierr = PetscMalloc((2*n+1)*sizeof(PetscInt**),&bj_ptr);CHKERRQ(ierr);
1491   bjlvl_ptr = (PetscInt**)(bj_ptr + n);
1492 
1493   /* create a linked list for storing column indices of the active row */
1494   nlnk = n + 1;
1495   ierr = PetscIncompleteLLCreate(n,n,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1496 
1497   /* initial FreeSpace size is f*(ai[n]+1) */
1498   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space);CHKERRQ(ierr);
1499   current_space = free_space;
1500   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space_lvl);CHKERRQ(ierr);
1501   current_space_lvl = free_space_lvl;
1502 
1503   for (i=0; i<n; i++) {
1504     nzi = 0;
1505     /* copy current row into linked list */
1506     nnz  = ai[r[i]+1] - ai[r[i]];
1507     if (!nnz) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
1508     cols = aj + ai[r[i]];
1509     lnk[i] = -1; /* marker to indicate if diagonal exists */
1510     ierr = PetscIncompleteLLInit(nnz,cols,n,ic,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1511     nzi += nlnk;
1512 
1513     /* make sure diagonal entry is included */
1514     if (diagonal_fill && lnk[i] == -1) {
1515       fm = n;
1516       while (lnk[fm] < i) fm = lnk[fm];
1517       lnk[i]     = lnk[fm]; /* insert diagonal into linked list */
1518       lnk[fm]    = i;
1519       lnk_lvl[i] = 0;
1520       nzi++; dcount++;
1521     }
1522 
1523     /* add pivot rows into the active row */
1524     nzbd = 0;
1525     prow = lnk[n];
1526     while (prow < i) {
1527       nnz      = bdiag[prow];
1528       cols     = bj_ptr[prow] + nnz + 1;
1529       cols_lvl = bjlvl_ptr[prow] + nnz + 1;
1530       nnz      = bi[prow+1] - bi[prow] - nnz - 1;
1531       ierr = PetscILULLAddSorted(nnz,cols,levels,cols_lvl,prow,nlnk,lnk,lnk_lvl,lnkbt,prow);CHKERRQ(ierr);
1532       nzi += nlnk;
1533       prow = lnk[prow];
1534       nzbd++;
1535     }
1536     bdiag[i] = nzbd;
1537     bi[i+1]  = bi[i] + nzi;
1538 
1539     /* if free space is not available, make more free space */
1540     if (current_space->local_remaining<nzi) {
1541       nnz = nzi*(n - i); /* estimated and max additional space needed */
1542       ierr = PetscFreeSpaceGet(nnz,&current_space);CHKERRQ(ierr);
1543       ierr = PetscFreeSpaceGet(nnz,&current_space_lvl);CHKERRQ(ierr);
1544       reallocs++;
1545     }
1546 
1547     /* copy data into free_space and free_space_lvl, then initialize lnk */
1548     ierr = PetscIncompleteLLClean(n,n,nzi,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);CHKERRQ(ierr);
1549     bj_ptr[i]    = current_space->array;
1550     bjlvl_ptr[i] = current_space_lvl->array;
1551 
1552     /* make sure the active row i has diagonal entry */
1553     if (*(bj_ptr[i]+bdiag[i]) != i) {
1554       SETERRQ1(PETSC_ERR_MAT_LU_ZRPVT,"Row %D has missing diagonal in factored matrix\n\
1555     try running with -pc_factor_nonzeros_along_diagonal or -pc_factor_diagonal_fill",i);
1556     }
1557 
1558     current_space->array           += nzi;
1559     current_space->local_used      += nzi;
1560     current_space->local_remaining -= nzi;
1561     current_space_lvl->array           += nzi;
1562     current_space_lvl->local_used      += nzi;
1563     current_space_lvl->local_remaining -= nzi;
1564   }
1565 
1566   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
1567   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
1568 
1569   /* destroy list of free space and other temporary arrays */
1570   ierr = PetscMalloc((bi[n]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
1571   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr); /* copy free_space -> bj */
1572   ierr = PetscIncompleteLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1573   ierr = PetscFreeSpaceDestroy(free_space_lvl);CHKERRQ(ierr);
1574   ierr = PetscFree(bj_ptr);CHKERRQ(ierr);
1575 
1576 #if defined(PETSC_USE_INFO)
1577   {
1578     PetscReal af = ((PetscReal)bi[n])/((PetscReal)ai[n]);
1579     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,f,af);CHKERRQ(ierr);
1580     ierr = PetscInfo1(A,"Run with -[sub_]pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
1581     ierr = PetscInfo1(A,"PCFactorSetFill([sub]pc,%G);\n",af);CHKERRQ(ierr);
1582     ierr = PetscInfo(A,"for best performance.\n");CHKERRQ(ierr);
1583     if (diagonal_fill) {
1584       ierr = PetscInfo1(A,"Detected and replaced %D missing diagonals",dcount);CHKERRQ(ierr);
1585     }
1586   }
1587 #endif
1588 
1589   /* put together the new matrix */
1590   ierr = MatSeqAIJSetPreallocation_SeqAIJ(fact,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
1591   ierr = PetscLogObjectParent(fact,isicol);CHKERRQ(ierr);
1592   b = (Mat_SeqAIJ*)(fact)->data;
1593   b->free_a       = PETSC_TRUE;
1594   b->free_ij      = PETSC_TRUE;
1595   b->singlemalloc = PETSC_FALSE;
1596   ierr = PetscMalloc( (bi[n] )*sizeof(PetscScalar),&b->a);CHKERRQ(ierr);
1597   b->j          = bj;
1598   b->i          = bi;
1599   for (i=0; i<n; i++) bdiag[i] += bi[i];
1600   b->diag       = bdiag;
1601   b->ilen       = 0;
1602   b->imax       = 0;
1603   b->row        = isrow;
1604   b->col        = iscol;
1605   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
1606   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
1607   b->icol       = isicol;
1608   ierr = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1609   /* In b structure:  Free imax, ilen, old a, old j.
1610      Allocate bdiag, solve_work, new a, new j */
1611   ierr = PetscLogObjectMemory(fact,(bi[n]-n) * (sizeof(PetscInt)+sizeof(PetscScalar)));CHKERRQ(ierr);
1612   b->maxnz             = b->nz = bi[n] ;
1613   (fact)->info.factor_mallocs    = reallocs;
1614   (fact)->info.fill_ratio_given  = f;
1615   (fact)->info.fill_ratio_needed = ((PetscReal)bi[n])/((PetscReal)ai[n]);
1616   (fact)->ops->lufactornumeric =  MatLUFactorNumeric_SeqAIJ;
1617   ierr = MatILUFactorSymbolic_Inode(fact,A,isrow,iscol,info);CHKERRQ(ierr);
1618   PetscFunctionReturn(0);
1619 }
1620 
1621 #include "../src/mat/impls/sbaij/seq/sbaij.h"
1622 #undef __FUNCT__
1623 #define __FUNCT__ "MatCholeskyFactorNumeric_SeqAIJ"
1624 PetscErrorCode MatCholeskyFactorNumeric_SeqAIJ(Mat B,Mat A,const MatFactorInfo *info)
1625 {
1626   Mat            C = B;
1627   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data;
1628   Mat_SeqSBAIJ   *b=(Mat_SeqSBAIJ*)C->data;
1629   IS             ip=b->row,iip = b->icol;
1630   PetscErrorCode ierr;
1631   const PetscInt *rip,*riip;
1632   PetscInt       i,j,mbs=A->rmap->n,*bi=b->i,*bj=b->j,*bcol;
1633   PetscInt       *ai=a->i,*aj=a->j;
1634   PetscInt       k,jmin,jmax,*jl,*il,col,nexti,ili,nz;
1635   MatScalar      *rtmp,*ba=b->a,*bval,*aa=a->a,dk,uikdi;
1636   PetscReal      zeropivot,rs,shiftnz;
1637   PetscReal      shiftpd;
1638   ChShift_Ctx    sctx;
1639   PetscInt       newshift;
1640   PetscTruth     perm_identity;
1641 
1642   PetscFunctionBegin;
1643 
1644   shiftnz   = info->shiftnz;
1645   shiftpd   = info->shiftpd;
1646   zeropivot = info->zeropivot;
1647 
1648   ierr  = ISGetIndices(ip,&rip);CHKERRQ(ierr);
1649   ierr  = ISGetIndices(iip,&riip);CHKERRQ(ierr);
1650 
1651   /* initialization */
1652   nz   = (2*mbs+1)*sizeof(PetscInt)+mbs*sizeof(MatScalar);
1653   ierr = PetscMalloc(nz,&il);CHKERRQ(ierr);
1654   jl   = il + mbs;
1655   rtmp = (MatScalar*)(jl + mbs);
1656 
1657   sctx.shift_amount = 0;
1658   sctx.nshift       = 0;
1659   do {
1660     sctx.chshift = PETSC_FALSE;
1661     for (i=0; i<mbs; i++) {
1662       rtmp[i] = 0.0; jl[i] = mbs; il[0] = 0;
1663     }
1664 
1665     for (k = 0; k<mbs; k++){
1666       bval = ba + bi[k];
1667       /* initialize k-th row by the perm[k]-th row of A */
1668       jmin = ai[rip[k]]; jmax = ai[rip[k]+1];
1669       for (j = jmin; j < jmax; j++){
1670         col = riip[aj[j]];
1671         if (col >= k){ /* only take upper triangular entry */
1672           rtmp[col] = aa[j];
1673           *bval++  = 0.0; /* for in-place factorization */
1674         }
1675       }
1676       /* shift the diagonal of the matrix */
1677       if (sctx.nshift) rtmp[k] += sctx.shift_amount;
1678 
1679       /* modify k-th row by adding in those rows i with U(i,k)!=0 */
1680       dk = rtmp[k];
1681       i = jl[k]; /* first row to be added to k_th row  */
1682 
1683       while (i < k){
1684         nexti = jl[i]; /* next row to be added to k_th row */
1685 
1686         /* compute multiplier, update diag(k) and U(i,k) */
1687         ili = il[i];  /* index of first nonzero element in U(i,k:bms-1) */
1688         uikdi = - ba[ili]*ba[bi[i]];  /* diagonal(k) */
1689         dk += uikdi*ba[ili];
1690         ba[ili] = uikdi; /* -U(i,k) */
1691 
1692         /* add multiple of row i to k-th row */
1693         jmin = ili + 1; jmax = bi[i+1];
1694         if (jmin < jmax){
1695           for (j=jmin; j<jmax; j++) rtmp[bj[j]] += uikdi*ba[j];
1696           /* update il and jl for row i */
1697           il[i] = jmin;
1698           j = bj[jmin]; jl[i] = jl[j]; jl[j] = i;
1699         }
1700         i = nexti;
1701       }
1702 
1703       /* shift the diagonals when zero pivot is detected */
1704       /* compute rs=sum of abs(off-diagonal) */
1705       rs   = 0.0;
1706       jmin = bi[k]+1;
1707       nz   = bi[k+1] - jmin;
1708       bcol = bj + jmin;
1709       while (nz--){
1710         rs += PetscAbsScalar(rtmp[*bcol]);
1711         bcol++;
1712       }
1713 
1714       sctx.rs = rs;
1715       sctx.pv = dk;
1716       ierr = MatCholeskyCheckShift_inline(info,sctx,k,newshift);CHKERRQ(ierr);
1717 
1718       if (newshift == 1) {
1719         if (!sctx.shift_amount) {
1720           sctx.shift_amount = 1e-5;
1721         }
1722         break;
1723       }
1724 
1725       /* copy data into U(k,:) */
1726       ba[bi[k]] = 1.0/dk; /* U(k,k) */
1727       jmin = bi[k]+1; jmax = bi[k+1];
1728       if (jmin < jmax) {
1729         for (j=jmin; j<jmax; j++){
1730           col = bj[j]; ba[j] = rtmp[col]; rtmp[col] = 0.0;
1731         }
1732         /* add the k-th row into il and jl */
1733         il[k] = jmin;
1734         i = bj[jmin]; jl[k] = jl[i]; jl[i] = k;
1735       }
1736     }
1737   } while (sctx.chshift);
1738   ierr = PetscFree(il);CHKERRQ(ierr);
1739 
1740   ierr = ISRestoreIndices(ip,&rip);CHKERRQ(ierr);
1741   ierr = ISRestoreIndices(iip,&riip);CHKERRQ(ierr);
1742 
1743   ierr = ISIdentity(ip,&perm_identity);CHKERRQ(ierr);
1744   if (perm_identity){
1745     (B)->ops->solve           = MatSolve_SeqSBAIJ_1_NaturalOrdering;
1746     (B)->ops->solvetranspose  = MatSolve_SeqSBAIJ_1_NaturalOrdering;
1747     (B)->ops->forwardsolve    = MatForwardSolve_SeqSBAIJ_1_NaturalOrdering;
1748     (B)->ops->backwardsolve   = MatBackwardSolve_SeqSBAIJ_1_NaturalOrdering;
1749   } else {
1750     (B)->ops->solve           = MatSolve_SeqSBAIJ_1;
1751     (B)->ops->solvetranspose  = MatSolve_SeqSBAIJ_1;
1752     (B)->ops->forwardsolve    = MatForwardSolve_SeqSBAIJ_1;
1753     (B)->ops->backwardsolve   = MatBackwardSolve_SeqSBAIJ_1;
1754   }
1755 
1756   C->assembled    = PETSC_TRUE;
1757   C->preallocated = PETSC_TRUE;
1758   ierr = PetscLogFlops(C->rmap->n);CHKERRQ(ierr);
1759   if (sctx.nshift){
1760     if (shiftnz) {
1761       ierr = PetscInfo2(A,"number of shiftnz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
1762     } else if (shiftpd) {
1763       ierr = PetscInfo2(A,"number of shiftpd tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
1764     }
1765   }
1766   PetscFunctionReturn(0);
1767 }
1768 
1769 #undef __FUNCT__
1770 #define __FUNCT__ "MatICCFactorSymbolic_SeqAIJ"
1771 PetscErrorCode MatICCFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
1772 {
1773   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data;
1774   Mat_SeqSBAIJ       *b;
1775   PetscErrorCode     ierr;
1776   PetscTruth         perm_identity,missing;
1777   PetscInt           reallocs=0,i,*ai=a->i,*aj=a->j,am=A->rmap->n,*ui;
1778   const PetscInt     *rip,*riip;
1779   PetscInt           jmin,jmax,nzk,k,j,*jl,prow,*il,nextprow;
1780   PetscInt           nlnk,*lnk,*lnk_lvl=PETSC_NULL,d;
1781   PetscInt           ncols,ncols_upper,*cols,*ajtmp,*uj,**uj_ptr,**uj_lvl_ptr;
1782   PetscReal          fill=info->fill,levels=info->levels;
1783   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
1784   PetscFreeSpaceList free_space_lvl=PETSC_NULL,current_space_lvl=PETSC_NULL;
1785   PetscBT            lnkbt;
1786   IS                 iperm;
1787 
1788   PetscFunctionBegin;
1789   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);
1790   ierr = MatMissingDiagonal(A,&missing,&d);CHKERRQ(ierr);
1791   if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",d);
1792   ierr = ISIdentity(perm,&perm_identity);CHKERRQ(ierr);
1793   ierr = ISInvertPermutation(perm,PETSC_DECIDE,&iperm);CHKERRQ(ierr);
1794 
1795   ierr = PetscMalloc((am+1)*sizeof(PetscInt),&ui);CHKERRQ(ierr);
1796   ui[0] = 0;
1797 
1798   /* ICC(0) without matrix ordering: simply copies fill pattern */
1799   if (!levels && perm_identity) {
1800 
1801     for (i=0; i<am; i++) {
1802       ui[i+1] = ui[i] + ai[i+1] - a->diag[i];
1803     }
1804     ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
1805     cols = uj;
1806     for (i=0; i<am; i++) {
1807       aj    = a->j + a->diag[i];
1808       ncols = ui[i+1] - ui[i];
1809       for (j=0; j<ncols; j++) *cols++ = *aj++;
1810     }
1811   } else { /* case: levels>0 || (levels=0 && !perm_identity) */
1812     ierr = ISGetIndices(iperm,&riip);CHKERRQ(ierr);
1813     ierr = ISGetIndices(perm,&rip);CHKERRQ(ierr);
1814 
1815     /* initialization */
1816     ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&ajtmp);CHKERRQ(ierr);
1817 
1818     /* jl: linked list for storing indices of the pivot rows
1819        il: il[i] points to the 1st nonzero entry of U(i,k:am-1) */
1820     ierr = PetscMalloc((2*am+1)*sizeof(PetscInt)+2*am*sizeof(PetscInt**),&jl);CHKERRQ(ierr);
1821     il         = jl + am;
1822     uj_ptr     = (PetscInt**)(il + am);
1823     uj_lvl_ptr = (PetscInt**)(uj_ptr + am);
1824     for (i=0; i<am; i++){
1825       jl[i] = am; il[i] = 0;
1826     }
1827 
1828     /* create and initialize a linked list for storing column indices of the active row k */
1829     nlnk = am + 1;
1830     ierr = PetscIncompleteLLCreate(am,am,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1831 
1832     /* initial FreeSpace size is fill*(ai[am]+1) */
1833     ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space);CHKERRQ(ierr);
1834     current_space = free_space;
1835     ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space_lvl);CHKERRQ(ierr);
1836     current_space_lvl = free_space_lvl;
1837 
1838     for (k=0; k<am; k++){  /* for each active row k */
1839       /* initialize lnk by the column indices of row rip[k] of A */
1840       nzk   = 0;
1841       ncols = ai[rip[k]+1] - ai[rip[k]];
1842       if (!ncols) SETERRQ2(PETSC_ERR_MAT_CH_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",rip[k],k);
1843       ncols_upper = 0;
1844       for (j=0; j<ncols; j++){
1845         i = *(aj + ai[rip[k]] + j); /* unpermuted column index */
1846         if (riip[i] >= k){ /* only take upper triangular entry */
1847           ajtmp[ncols_upper] = i;
1848           ncols_upper++;
1849         }
1850       }
1851       ierr = PetscIncompleteLLInit(ncols_upper,ajtmp,am,riip,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1852       nzk += nlnk;
1853 
1854       /* update lnk by computing fill-in for each pivot row to be merged in */
1855       prow = jl[k]; /* 1st pivot row */
1856 
1857       while (prow < k){
1858         nextprow = jl[prow];
1859 
1860         /* merge prow into k-th row */
1861         jmin = il[prow] + 1;  /* index of the 2nd nzero entry in U(prow,k:am-1) */
1862         jmax = ui[prow+1];
1863         ncols = jmax-jmin;
1864         i     = jmin - ui[prow];
1865         cols  = uj_ptr[prow] + i; /* points to the 2nd nzero entry in U(prow,k:am-1) */
1866         uj    = uj_lvl_ptr[prow] + i; /* levels of cols */
1867         j     = *(uj - 1);
1868         ierr = PetscICCLLAddSorted(ncols,cols,levels,uj,am,nlnk,lnk,lnk_lvl,lnkbt,j);CHKERRQ(ierr);
1869         nzk += nlnk;
1870 
1871         /* update il and jl for prow */
1872         if (jmin < jmax){
1873           il[prow] = jmin;
1874           j = *cols; jl[prow] = jl[j]; jl[j] = prow;
1875         }
1876         prow = nextprow;
1877       }
1878 
1879       /* if free space is not available, make more free space */
1880       if (current_space->local_remaining<nzk) {
1881         i = am - k + 1; /* num of unfactored rows */
1882         i = PetscMin(i*nzk, i*(i-1)); /* i*nzk, i*(i-1): estimated and max additional space needed */
1883         ierr = PetscFreeSpaceGet(i,&current_space);CHKERRQ(ierr);
1884         ierr = PetscFreeSpaceGet(i,&current_space_lvl);CHKERRQ(ierr);
1885         reallocs++;
1886       }
1887 
1888       /* copy data into free_space and free_space_lvl, then initialize lnk */
1889       if (nzk == 0) SETERRQ1(PETSC_ERR_ARG_WRONG,"Empty row %D in ICC matrix factor",k);
1890       ierr = PetscIncompleteLLClean(am,am,nzk,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);CHKERRQ(ierr);
1891 
1892       /* add the k-th row into il and jl */
1893       if (nzk > 1){
1894         i = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:am-1) */
1895         jl[k] = jl[i]; jl[i] = k;
1896         il[k] = ui[k] + 1;
1897       }
1898       uj_ptr[k]     = current_space->array;
1899       uj_lvl_ptr[k] = current_space_lvl->array;
1900 
1901       current_space->array           += nzk;
1902       current_space->local_used      += nzk;
1903       current_space->local_remaining -= nzk;
1904 
1905       current_space_lvl->array           += nzk;
1906       current_space_lvl->local_used      += nzk;
1907       current_space_lvl->local_remaining -= nzk;
1908 
1909       ui[k+1] = ui[k] + nzk;
1910     }
1911 
1912 #if defined(PETSC_USE_INFO)
1913     if (ai[am] != 0) {
1914       PetscReal af = (PetscReal)ui[am]/((PetscReal)ai[am]);
1915       ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,fill,af);CHKERRQ(ierr);
1916       ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
1917       ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G) for best performance.\n",af);CHKERRQ(ierr);
1918     } else {
1919       ierr = PetscInfo(A,"Empty matrix.\n");CHKERRQ(ierr);
1920     }
1921 #endif
1922 
1923     ierr = ISRestoreIndices(perm,&rip);CHKERRQ(ierr);
1924     ierr = ISRestoreIndices(iperm,&riip);CHKERRQ(ierr);
1925     ierr = PetscFree(jl);CHKERRQ(ierr);
1926     ierr = PetscFree(ajtmp);CHKERRQ(ierr);
1927 
1928     /* destroy list of free space and other temporary array(s) */
1929     ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
1930     ierr = PetscFreeSpaceContiguous(&free_space,uj);CHKERRQ(ierr);
1931     ierr = PetscIncompleteLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1932     ierr = PetscFreeSpaceDestroy(free_space_lvl);CHKERRQ(ierr);
1933 
1934   } /* end of case: levels>0 || (levels=0 && !perm_identity) */
1935 
1936   /* put together the new matrix in MATSEQSBAIJ format */
1937 
1938   b    = (Mat_SeqSBAIJ*)(fact)->data;
1939   b->singlemalloc = PETSC_FALSE;
1940   ierr = PetscMalloc((ui[am]+1)*sizeof(MatScalar),&b->a);CHKERRQ(ierr);
1941   b->j    = uj;
1942   b->i    = ui;
1943   b->diag = 0;
1944   b->ilen = 0;
1945   b->imax = 0;
1946   b->row  = perm;
1947   b->col  = perm;
1948   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
1949   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
1950   b->icol = iperm;
1951   b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */
1952   ierr    = PetscMalloc((am+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1953   ierr = PetscLogObjectMemory((fact),(ui[am]-am)*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
1954   b->maxnz   = b->nz = ui[am];
1955   b->free_a  = PETSC_TRUE;
1956   b->free_ij = PETSC_TRUE;
1957 
1958   (fact)->info.factor_mallocs    = reallocs;
1959   (fact)->info.fill_ratio_given  = fill;
1960   if (ai[am] != 0) {
1961     (fact)->info.fill_ratio_needed = ((PetscReal)ui[am])/((PetscReal)ai[am]);
1962   } else {
1963     (fact)->info.fill_ratio_needed = 0.0;
1964   }
1965   (fact)->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqAIJ;
1966   PetscFunctionReturn(0);
1967 }
1968 
1969 #undef __FUNCT__
1970 #define __FUNCT__ "MatCholeskyFactorSymbolic_SeqAIJ"
1971 PetscErrorCode MatCholeskyFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
1972 {
1973   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data;
1974   Mat_SeqSBAIJ       *b;
1975   PetscErrorCode     ierr;
1976   PetscTruth         perm_identity;
1977   PetscReal          fill = info->fill;
1978   const PetscInt     *rip,*riip;
1979   PetscInt           i,am=A->rmap->n,*ai=a->i,*aj=a->j,reallocs=0,prow;
1980   PetscInt           *jl,jmin,jmax,nzk,*ui,k,j,*il,nextprow;
1981   PetscInt           nlnk,*lnk,ncols,ncols_upper,*cols,*uj,**ui_ptr,*uj_ptr;
1982   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
1983   PetscBT            lnkbt;
1984   IS                 iperm;
1985 
1986   PetscFunctionBegin;
1987   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);
1988   /* check whether perm is the identity mapping */
1989   ierr = ISIdentity(perm,&perm_identity);CHKERRQ(ierr);
1990   ierr = ISInvertPermutation(perm,PETSC_DECIDE,&iperm);CHKERRQ(ierr);
1991   ierr = ISGetIndices(iperm,&riip);CHKERRQ(ierr);
1992   ierr = ISGetIndices(perm,&rip);CHKERRQ(ierr);
1993 
1994   /* initialization */
1995   ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&ui);CHKERRQ(ierr);
1996   ui[0] = 0;
1997 
1998   /* jl: linked list for storing indices of the pivot rows
1999      il: il[i] points to the 1st nonzero entry of U(i,k:am-1) */
2000   ierr = PetscMalloc((3*am+1)*sizeof(PetscInt)+am*sizeof(PetscInt**),&jl);CHKERRQ(ierr);
2001   il     = jl + am;
2002   cols   = il + am;
2003   ui_ptr = (PetscInt**)(cols + am);
2004   for (i=0; i<am; i++){
2005     jl[i] = am; il[i] = 0;
2006   }
2007 
2008   /* create and initialize a linked list for storing column indices of the active row k */
2009   nlnk = am + 1;
2010   ierr = PetscLLCreate(am,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2011 
2012   /* initial FreeSpace size is fill*(ai[am]+1) */
2013   ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space);CHKERRQ(ierr);
2014   current_space = free_space;
2015 
2016   for (k=0; k<am; k++){  /* for each active row k */
2017     /* initialize lnk by the column indices of row rip[k] of A */
2018     nzk   = 0;
2019     ncols = ai[rip[k]+1] - ai[rip[k]];
2020     if (!ncols) SETERRQ2(PETSC_ERR_MAT_CH_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",rip[k],k);
2021     ncols_upper = 0;
2022     for (j=0; j<ncols; j++){
2023       i = riip[*(aj + ai[rip[k]] + j)];
2024       if (i >= k){ /* only take upper triangular entry */
2025         cols[ncols_upper] = i;
2026         ncols_upper++;
2027       }
2028     }
2029     ierr = PetscLLAdd(ncols_upper,cols,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2030     nzk += nlnk;
2031 
2032     /* update lnk by computing fill-in for each pivot row to be merged in */
2033     prow = jl[k]; /* 1st pivot row */
2034 
2035     while (prow < k){
2036       nextprow = jl[prow];
2037       /* merge prow into k-th row */
2038       jmin = il[prow] + 1;  /* index of the 2nd nzero entry in U(prow,k:am-1) */
2039       jmax = ui[prow+1];
2040       ncols = jmax-jmin;
2041       uj_ptr = ui_ptr[prow] + jmin - ui[prow]; /* points to the 2nd nzero entry in U(prow,k:am-1) */
2042       ierr = PetscLLAddSorted(ncols,uj_ptr,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2043       nzk += nlnk;
2044 
2045       /* update il and jl for prow */
2046       if (jmin < jmax){
2047         il[prow] = jmin;
2048         j = *uj_ptr; jl[prow] = jl[j]; jl[j] = prow;
2049       }
2050       prow = nextprow;
2051     }
2052 
2053     /* if free space is not available, make more free space */
2054     if (current_space->local_remaining<nzk) {
2055       i = am - k + 1; /* num of unfactored rows */
2056       i = PetscMin(i*nzk, i*(i-1)); /* i*nzk, i*(i-1): estimated and max additional space needed */
2057       ierr = PetscFreeSpaceGet(i,&current_space);CHKERRQ(ierr);
2058       reallocs++;
2059     }
2060 
2061     /* copy data into free space, then initialize lnk */
2062     ierr = PetscLLClean(am,am,nzk,lnk,current_space->array,lnkbt);CHKERRQ(ierr);
2063 
2064     /* add the k-th row into il and jl */
2065     if (nzk-1 > 0){
2066       i = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:am-1) */
2067       jl[k] = jl[i]; jl[i] = k;
2068       il[k] = ui[k] + 1;
2069     }
2070     ui_ptr[k] = current_space->array;
2071     current_space->array           += nzk;
2072     current_space->local_used      += nzk;
2073     current_space->local_remaining -= nzk;
2074 
2075     ui[k+1] = ui[k] + nzk;
2076   }
2077 
2078 #if defined(PETSC_USE_INFO)
2079   if (ai[am] != 0) {
2080     PetscReal af = (PetscReal)(ui[am])/((PetscReal)ai[am]);
2081     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,fill,af);CHKERRQ(ierr);
2082     ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
2083     ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G) for best performance.\n",af);CHKERRQ(ierr);
2084   } else {
2085      ierr = PetscInfo(A,"Empty matrix.\n");CHKERRQ(ierr);
2086   }
2087 #endif
2088 
2089   ierr = ISRestoreIndices(perm,&rip);CHKERRQ(ierr);
2090   ierr = ISRestoreIndices(iperm,&riip);CHKERRQ(ierr);
2091   ierr = PetscFree(jl);CHKERRQ(ierr);
2092 
2093   /* destroy list of free space and other temporary array(s) */
2094   ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
2095   ierr = PetscFreeSpaceContiguous(&free_space,uj);CHKERRQ(ierr);
2096   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
2097 
2098   /* put together the new matrix in MATSEQSBAIJ format */
2099 
2100   b = (Mat_SeqSBAIJ*)(fact)->data;
2101   b->singlemalloc = PETSC_FALSE;
2102   b->free_a       = PETSC_TRUE;
2103   b->free_ij      = PETSC_TRUE;
2104   ierr = PetscMalloc((ui[am]+1)*sizeof(MatScalar),&b->a);CHKERRQ(ierr);
2105   b->j    = uj;
2106   b->i    = ui;
2107   b->diag = 0;
2108   b->ilen = 0;
2109   b->imax = 0;
2110   b->row  = perm;
2111   b->col  = perm;
2112   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2113   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2114   b->icol = iperm;
2115   b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */
2116   ierr    = PetscMalloc((am+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
2117   ierr    = PetscLogObjectMemory(fact,(ui[am]-am)*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
2118   b->maxnz = b->nz = ui[am];
2119 
2120   (fact)->info.factor_mallocs    = reallocs;
2121   (fact)->info.fill_ratio_given  = fill;
2122   if (ai[am] != 0) {
2123     (fact)->info.fill_ratio_needed = ((PetscReal)ui[am])/((PetscReal)ai[am]);
2124   } else {
2125     (fact)->info.fill_ratio_needed = 0.0;
2126   }
2127   (fact)->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqAIJ;
2128   PetscFunctionReturn(0);
2129 }
2130 
2131 #undef __FUNCT__
2132 #define __FUNCT__ "MatSolve_SeqAIJ_NaturalOrdering_iludt"
2133 PetscErrorCode MatSolve_SeqAIJ_NaturalOrdering_iludt(Mat A,Vec bb,Vec xx)
2134 {
2135   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
2136   PetscErrorCode    ierr;
2137   PetscInt          n = A->rmap->n;
2138   const PetscInt    *ai = a->i,*aj = a->j,*vi;
2139   PetscScalar       *x,sum;
2140   const PetscScalar *b;
2141   const MatScalar   *aa = a->a,*v;
2142   PetscInt          i,nz;
2143 
2144   PetscFunctionBegin;
2145   if (!n) PetscFunctionReturn(0);
2146 
2147   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
2148   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
2149 
2150   /* forward solve the lower triangular */
2151   x[0] = b[0];
2152   v    = aa;
2153   vi   = aj;
2154   for (i=1; i<n; i++) {
2155     nz  = ai[i+1] - ai[i];
2156     sum = b[i];
2157     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
2158     /*    while (nz--) sum -= *v++ * x[*vi++];*/
2159     v  += nz;
2160     vi += nz;
2161     x[i] = sum;
2162   }
2163 
2164   /* backward solve the upper triangular */
2165   v   = aa + ai[n+1];
2166   vi  = aj + ai[n+1];
2167   for (i=n-1; i>=0; i--){
2168     nz = ai[2*n-i +1] - ai[2*n-i]-1;
2169     sum = x[i];
2170     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
2171     /* while (nz--) sum -= *v++ * x[*vi++]; */
2172     v   += nz;
2173     vi  += nz; vi++;
2174     x[i] = *v++ *sum; /* x[i]=aa[adiag[i]]*sum; v++; */
2175   }
2176 
2177   ierr = PetscLogFlops(2.0*a->nz - A->cmap->n);CHKERRQ(ierr);
2178   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
2179   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
2180   PetscFunctionReturn(0);
2181 }
2182 
2183 #undef __FUNCT__
2184 #define __FUNCT__ "MatSolve_SeqAIJ_iludt"
2185 PetscErrorCode MatSolve_SeqAIJ_iludt(Mat A,Vec bb,Vec xx)
2186 {
2187   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
2188   IS                iscol = a->col,isrow = a->row;
2189   PetscErrorCode    ierr;
2190   PetscInt          i,n=A->rmap->n,*vi,*ai = a->i,*aj = a->j,*adiag=a->diag;
2191   PetscInt          nz;
2192   const PetscInt    *rout,*cout,*r,*c;
2193   PetscScalar       *x,*tmp,*tmps;
2194   const PetscScalar *b;
2195   const MatScalar   *aa = a->a,*v;
2196 
2197   PetscFunctionBegin;
2198   if (!n) PetscFunctionReturn(0);
2199 
2200   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
2201   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
2202   tmp  = a->solve_work;
2203 
2204   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
2205   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout + (n-1);
2206 
2207   /* forward solve the lower triangular */
2208   tmp[0] = b[*r++];
2209   tmps   = tmp;
2210   v      = aa;
2211   vi     = aj;
2212   for (i=1; i<n; i++) {
2213     nz  = ai[i+1] - ai[i];
2214     tmp[i] = b[*r++];
2215     PetscSparseDenseMinusDot(tmp[i],tmps,v,vi,nz);
2216     v += nz; vi += nz;
2217   }
2218 
2219   /* backward solve the upper triangular */
2220   v   = aa + adiag[n] + 1;
2221   vi  = aj + adiag[n] + 1;
2222   for (i=n-1; i>=0; i--){
2223     nz  = adiag[i] - adiag[i+1] - 1;
2224     PetscSparseDenseMinusDot(tmp[i],tmps,v,vi,nz);
2225     x[*c--] = tmp[i] = tmp[i]*aa[adiag[i]];
2226     v += nz+1; vi += nz+1;
2227   }
2228 
2229   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
2230   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
2231   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
2232   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
2233   ierr = PetscLogFlops(2*a->nz - A->cmap->n);CHKERRQ(ierr);
2234   PetscFunctionReturn(0);
2235 }
2236 
2237 #undef __FUNCT__
2238 #define __FUNCT__ "MatILUDTFactor_SeqAIJ"
2239 PetscErrorCode MatILUDTFactor_SeqAIJ(Mat A,IS isrow,IS iscol,const MatFactorInfo *info,Mat *fact)
2240 {
2241   Mat                B = *fact;
2242   Mat_SeqAIJ         *a=(Mat_SeqAIJ*)A->data,*b;
2243   IS                 isicol;
2244   PetscErrorCode     ierr;
2245   const PetscInt     *r,*ic;
2246   PetscInt           i,n=A->rmap->n,*ai=a->i,*aj=a->j,*ajtmp,*adiag;
2247   PetscInt           *bi,*bj,*bdiag,*bdiag_rev;
2248   PetscInt           row,nzi,nzi_bl,nzi_bu,*im,nzi_al,nzi_au;
2249   PetscInt           nlnk,*lnk;
2250   PetscBT            lnkbt;
2251   PetscTruth         row_identity,icol_identity,both_identity;
2252   MatScalar          *aatmp,*pv,*batmp,*ba,*rtmp,*pc,multiplier,*vtmp,diag_tmp;
2253   const PetscInt     *ics;
2254   PetscInt           j,nz,*pj,*bjtmp,k,ncut,*jtmp;
2255   PetscReal          dt=info->dt,dtcol=info->dtcol,shift=info->shiftinblocks;
2256   PetscInt           dtcount=(PetscInt)info->dtcount,nnz_max;
2257   PetscTruth         missing;
2258 
2259   PetscFunctionBegin;
2260 
2261   if (dt      == PETSC_DEFAULT) dt      = 0.005;
2262   if (dtcol   == PETSC_DEFAULT) dtcol   = 0.01; /* XXX unused! */
2263   if (dtcount == PETSC_DEFAULT) dtcount = (PetscInt)(1.5*a->rmax);
2264 
2265   /* ------- symbolic factorization, can be reused ---------*/
2266   ierr = MatMissingDiagonal(A,&missing,&i);CHKERRQ(ierr);
2267   if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",i);
2268   adiag=a->diag;
2269 
2270   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
2271 
2272   /* bdiag is location of diagonal in factor */
2273   ierr = PetscMalloc((2*n+2)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);
2274   bdiag_rev = bdiag + n+1;
2275 
2276   /* allocate row pointers bi */
2277   ierr = PetscMalloc((2*n+2)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
2278 
2279   /* allocate bj and ba; max num of nonzero entries is (ai[n]+2*n*dtcount+2) */
2280   if (dtcount > n-1) dtcount = n-1; /* diagonal is excluded */
2281   nnz_max  = ai[n]+2*n*dtcount+2;
2282 
2283   ierr = PetscMalloc((nnz_max+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
2284   ierr = PetscMalloc((nnz_max+1)*sizeof(MatScalar),&ba);CHKERRQ(ierr);
2285 
2286   /* put together the new matrix */
2287   ierr = MatSeqAIJSetPreallocation_SeqAIJ(B,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
2288   ierr = PetscLogObjectParent(B,isicol);CHKERRQ(ierr);
2289   b    = (Mat_SeqAIJ*)(B)->data;
2290   b->free_a       = PETSC_TRUE;
2291   b->free_ij      = PETSC_TRUE;
2292   b->singlemalloc = PETSC_FALSE;
2293   b->a          = ba;
2294   b->j          = bj;
2295   b->i          = bi;
2296   b->diag       = bdiag;
2297   b->ilen       = 0;
2298   b->imax       = 0;
2299   b->row        = isrow;
2300   b->col        = iscol;
2301   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
2302   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
2303   b->icol       = isicol;
2304   ierr = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
2305 
2306   ierr = PetscLogObjectMemory(B,nnz_max*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
2307   b->maxnz = nnz_max;
2308 
2309   (B)->factor                = MAT_FACTOR_ILUDT;
2310   (B)->info.factor_mallocs   = 0;
2311   (B)->info.fill_ratio_given = ((PetscReal)nnz_max)/((PetscReal)ai[n]);
2312   CHKMEMQ;
2313   /* ------- end of symbolic factorization ---------*/
2314 
2315   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
2316   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
2317   ics  = ic;
2318 
2319   /* linked list for storing column indices of the active row */
2320   nlnk = n + 1;
2321   ierr = PetscLLCreate(n,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2322 
2323   /* im: used by PetscLLAddSortedLU(); jtmp: working array for column indices of active row */
2324   ierr = PetscMalloc((2*n+1)*sizeof(PetscInt),&im);CHKERRQ(ierr);
2325   jtmp = im + n;
2326   /* rtmp, vtmp: working arrays for sparse and contiguous row entries of active row */
2327   ierr = PetscMalloc((2*n+1)*sizeof(MatScalar),&rtmp);CHKERRQ(ierr);
2328   ierr = PetscMemzero(rtmp,(n+1)*sizeof(PetscScalar));CHKERRQ(ierr);
2329   vtmp = rtmp + n;
2330 
2331   bi[0]    = 0;
2332   bdiag[0] = nnz_max-1; /* location of diag[0] in factor B */
2333   bdiag_rev[n] = bdiag[0];
2334   bi[2*n+1] = bdiag[0]+1; /* endof bj and ba array */
2335   for (i=0; i<n; i++) {
2336     /* copy initial fill into linked list */
2337     nzi = 0; /* nonzeros for active row i */
2338     nzi = ai[r[i]+1] - ai[r[i]];
2339     if (!nzi) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
2340     nzi_al = adiag[r[i]] - ai[r[i]];
2341     nzi_au = ai[r[i]+1] - adiag[r[i]] -1;
2342     ajtmp = aj + ai[r[i]];
2343     ierr = PetscLLAddPerm(nzi,ajtmp,ic,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2344 
2345     /* load in initial (unfactored row) */
2346     aatmp = a->a + ai[r[i]];
2347     for (j=0; j<nzi; j++) {
2348       rtmp[ics[*ajtmp++]] = *aatmp++;
2349     }
2350 
2351     /* add pivot rows into linked list */
2352     row = lnk[n];
2353     while (row < i ) {
2354       nzi_bl = bi[row+1] - bi[row] + 1;
2355       bjtmp = bj + bdiag[row+1]+1; /* points to 1st column next to the diagonal in U */
2356       ierr  = PetscLLAddSortedLU(bjtmp,row,nlnk,lnk,lnkbt,i,nzi_bl,im);CHKERRQ(ierr);
2357       nzi  += nlnk;
2358       row   = lnk[row];
2359     }
2360 
2361     /* copy data from lnk into jtmp, then initialize lnk */
2362     ierr = PetscLLClean(n,n,nzi,lnk,jtmp,lnkbt);CHKERRQ(ierr);
2363 
2364     /* numerical factorization */
2365     bjtmp = jtmp;
2366     row   = *bjtmp++; /* 1st pivot row */
2367     while  ( row < i ) {
2368       pc         = rtmp + row;
2369       pv         = ba + bdiag[row]; /* 1./(diag of the pivot row) */
2370       multiplier = (*pc) * (*pv);
2371       *pc        = multiplier;
2372       if (PetscAbsScalar(*pc) > dt){ /* apply tolerance dropping rule */
2373         pj         = bj + bdiag[row+1] + 1; /* point to 1st entry of U(row,:) */
2374         pv         = ba + bdiag[row+1] + 1;
2375         /* if (multiplier < -1.0 or multiplier >1.0) printf("row/prow %d, %d, multiplier %g\n",i,row,multiplier); */
2376         nz         = bdiag[row] - bdiag[row+1] - 1; /* num of entries in U(row,:), excluding diagonal */
2377         for (j=0; j<nz; j++) rtmp[*pj++] -= multiplier * (*pv++);
2378         ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr);
2379       }
2380       row = *bjtmp++;
2381     }
2382 
2383     /* copy sparse rtmp into contiguous vtmp; separate L and U part */
2384     diag_tmp = rtmp[i];  /* save diagonal value - may not needed?? */
2385     nzi_bl = 0; j = 0;
2386     while (jtmp[j] < i){ /* Note: jtmp is sorted */
2387       vtmp[j] = rtmp[jtmp[j]]; rtmp[jtmp[j]]=0.0;
2388       nzi_bl++; j++;
2389     }
2390     nzi_bu = nzi - nzi_bl -1;
2391     while (j < nzi){
2392       vtmp[j] = rtmp[jtmp[j]]; rtmp[jtmp[j]]=0.0;
2393       j++;
2394     }
2395 
2396     bjtmp = bj + bi[i];
2397     batmp = ba + bi[i];
2398     /* apply level dropping rule to L part */
2399     ncut = nzi_al + dtcount;
2400     if (ncut < nzi_bl){
2401       ierr = PetscSortSplit(ncut,nzi_bl,vtmp,jtmp);CHKERRQ(ierr);
2402       ierr = PetscSortIntWithScalarArray(ncut,jtmp,vtmp);CHKERRQ(ierr);
2403     } else {
2404       ncut = nzi_bl;
2405     }
2406     for (j=0; j<ncut; j++){
2407       bjtmp[j] = jtmp[j];
2408       batmp[j] = vtmp[j];
2409       /* printf(" (%d,%g),",bjtmp[j],batmp[j]); */
2410     }
2411     bi[i+1] = bi[i] + ncut;
2412     nzi = ncut + 1;
2413 
2414     /* apply level dropping rule to U part */
2415     ncut = nzi_au + dtcount;
2416     if (ncut < nzi_bu){
2417       ierr = PetscSortSplit(ncut,nzi_bu,vtmp+nzi_bl+1,jtmp+nzi_bl+1);CHKERRQ(ierr);
2418       ierr = PetscSortIntWithScalarArray(ncut,jtmp+nzi_bl+1,vtmp+nzi_bl+1);CHKERRQ(ierr);
2419     } else {
2420       ncut = nzi_bu;
2421     }
2422     nzi += ncut;
2423 
2424     /* mark bdiagonal */
2425     bdiag[i+1]       = bdiag[i] - (ncut + 1);
2426     bdiag_rev[n-i-1] = bdiag[i+1];
2427     bi[2*n - i]      = bi[2*n - i +1] - (ncut + 1);
2428     bjtmp = bj + bdiag[i];
2429     batmp = ba + bdiag[i];
2430     *bjtmp = i;
2431     *batmp = diag_tmp; /* rtmp[i]; */
2432     if (*batmp == 0.0) {
2433       *batmp = dt+shift;
2434       /* printf(" row %d add shift %g\n",i,shift); */
2435     }
2436     *batmp = 1.0/(*batmp); /* invert diagonal entries for simplier triangular solves */
2437     /* printf(" (%d,%g),",*bjtmp,*batmp); */
2438 
2439     bjtmp = bj + bdiag[i+1]+1;
2440     batmp = ba + bdiag[i+1]+1;
2441     for (k=0; k<ncut; k++){
2442       bjtmp[k] = jtmp[nzi_bl+1+k];
2443       batmp[k] = vtmp[nzi_bl+1+k];
2444       /* printf(" (%d,%g),",bjtmp[k],batmp[k]); */
2445     }
2446     /* printf("\n"); */
2447 
2448     im[i]   = nzi; /* used by PetscLLAddSortedLU() */
2449     /*
2450     printf("row %d: bi %d, bdiag %d\n",i,bi[i],bdiag[i]);
2451     printf(" ----------------------------\n");
2452     */
2453   } /* for (i=0; i<n; i++) */
2454   /* printf("end of L %d, beginning of U %d\n",bi[n],bdiag[n]); */
2455   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]);
2456 
2457   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
2458   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
2459 
2460   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
2461   ierr = PetscFree(im);CHKERRQ(ierr);
2462   ierr = PetscFree(rtmp);CHKERRQ(ierr);
2463 
2464   ierr = PetscLogFlops(B->cmap->n);CHKERRQ(ierr);
2465   b->maxnz = b->nz = bi[n] + bdiag[0] - bdiag[n];
2466 
2467   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
2468   ierr = ISIdentity(isicol,&icol_identity);CHKERRQ(ierr);
2469   both_identity = (PetscTruth) (row_identity && icol_identity);
2470   if (row_identity && icol_identity) {
2471     B->ops->solve = MatSolve_SeqAIJ_NaturalOrdering_iludt;
2472   } else {
2473     B->ops->solve = MatSolve_SeqAIJ_iludt;
2474   }
2475 
2476   B->ops->lufactorsymbolic  = MatILUDTFactorSymbolic_SeqAIJ;
2477   B->ops->lufactornumeric   = MatILUDTFactorNumeric_SeqAIJ;
2478   B->ops->solveadd          = 0;
2479   B->ops->solvetranspose    = 0;
2480   B->ops->solvetransposeadd = 0;
2481   B->ops->matsolve          = 0;
2482   B->assembled              = PETSC_TRUE;
2483   B->preallocated           = PETSC_TRUE;
2484   PetscFunctionReturn(0);
2485 }
2486 
2487 /* a wraper of MatILUDTFactor_SeqAIJ() */
2488 #undef __FUNCT__
2489 #define __FUNCT__ "MatILUDTFactorSymbolic_SeqAIJ"
2490 PetscErrorCode PETSCMAT_DLLEXPORT MatILUDTFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS row,IS col,const MatFactorInfo *info)
2491 {
2492   PetscErrorCode     ierr;
2493 
2494   PetscFunctionBegin;
2495   ierr = MatILUDTFactor_SeqAIJ(A,row,col,info,&fact);CHKERRQ(ierr);
2496 
2497   fact->ops->lufactornumeric = MatILUDTFactorNumeric_SeqAIJ;
2498   PetscFunctionReturn(0);
2499 }
2500 
2501 /*
2502    same as MatLUFactorNumeric_SeqAIJ(), except using contiguous array matrix factors
2503    - intend to replace existing MatLUFactorNumeric_SeqAIJ()
2504 */
2505 #undef __FUNCT__
2506 #define __FUNCT__ "MatILUDTFactorNumeric_SeqAIJ"
2507 PetscErrorCode PETSCMAT_DLLEXPORT MatILUDTFactorNumeric_SeqAIJ(Mat fact,Mat A,const MatFactorInfo *info)
2508 {
2509   Mat            C=fact;
2510   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ *)C->data;
2511   IS             isrow = b->row,isicol = b->icol;
2512   PetscErrorCode ierr;
2513   const PetscInt *r,*ic,*ics;
2514   PetscInt       i,j,k,n=A->rmap->n,*ai=a->i,*aj=a->j,*bi=b->i,*bj=b->j;
2515   PetscInt       *ajtmp,*bjtmp,nz,nzl,nzu,row,*bdiag = b->diag,*pj;
2516   MatScalar      *rtmp,*pc,multiplier,*v,*pv,*aa=a->a;
2517   PetscReal      dt=info->dt,shift=info->shiftinblocks;
2518   PetscTruth     row_identity, col_identity;
2519 
2520   PetscFunctionBegin;
2521   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
2522   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
2523   ierr = PetscMalloc((n+1)*sizeof(MatScalar),&rtmp);CHKERRQ(ierr);
2524   ics  = ic;
2525 
2526   for (i=0; i<n; i++){
2527     /* initialize rtmp array */
2528     nzl   = bi[i+1] - bi[i];       /* num of nozeros in L(i,:) */
2529     bjtmp = bj + bi[i];
2530     for  (j=0; j<nzl; j++) rtmp[*bjtmp++] = 0.0;
2531     rtmp[i] = 0.0;
2532     nzu   = bdiag[i] - bdiag[i+1]; /* num of nozeros in U(i,:) */
2533     bjtmp = bj + bdiag[i+1] + 1;
2534     for  (j=0; j<nzu; j++) rtmp[*bjtmp++] = 0.0;
2535 
2536     /* load in initial unfactored row of A */
2537     /* printf("row %d\n",i); */
2538     nz    = ai[r[i]+1] - ai[r[i]];
2539     ajtmp = aj + ai[r[i]];
2540     v     = aa + ai[r[i]];
2541     for (j=0; j<nz; j++) {
2542       rtmp[ics[*ajtmp++]] = v[j];
2543       /* printf(" (%d,%g),",ics[ajtmp[j]],rtmp[ics[ajtmp[j]]]); */
2544     }
2545     /* printf("\n"); */
2546 
2547     /* numerical factorization */
2548     bjtmp = bj + bi[i]; /* point to 1st entry of L(i,:) */
2549     nzl   = bi[i+1] - bi[i]; /* num of entries in L(i,:) */
2550     k = 0;
2551     while (k < nzl){
2552       row   = *bjtmp++;
2553       /* printf("  prow %d\n",row); */
2554       pc         = rtmp + row;
2555       pv         = b->a + bdiag[row]; /* 1./(diag of the pivot row) */
2556       multiplier = (*pc) * (*pv);
2557       *pc        = multiplier;
2558       if (PetscAbsScalar(multiplier) > dt){
2559         pj         = bj + bdiag[row+1] + 1; /* point to 1st entry of U(row,:) */
2560         pv         = b->a + bdiag[row+1] + 1;
2561         nz         = bdiag[row] - bdiag[row+1] - 1; /* num of entries in U(row,:), excluding diagonal */
2562         for (j=0; j<nz; j++) rtmp[*pj++] -= multiplier * (*pv++);
2563         /* ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr); */
2564       }
2565       k++;
2566     }
2567 
2568     /* finished row so stick it into b->a */
2569     /* L-part */
2570     pv = b->a + bi[i] ;
2571     pj = bj + bi[i] ;
2572     nzl = bi[i+1] - bi[i];
2573     for (j=0; j<nzl; j++) {
2574       pv[j] = rtmp[pj[j]];
2575       /* printf(" (%d,%g),",pj[j],pv[j]); */
2576     }
2577 
2578     /* diagonal: invert diagonal entries for simplier triangular solves */
2579     if (rtmp[i] == 0.0) rtmp[i] = dt+shift;
2580     b->a[bdiag[i]] = 1.0/rtmp[i];
2581     /* printf(" (%d,%g),",i,b->a[bdiag[i]]); */
2582 
2583     /* U-part */
2584     pv = b->a + bdiag[i+1] + 1;
2585     pj = bj + bdiag[i+1] + 1;
2586     nzu = bdiag[i] - bdiag[i+1] - 1;
2587     for (j=0; j<nzu; j++) {
2588       pv[j] = rtmp[pj[j]];
2589       /* printf(" (%d,%g),",pj[j],pv[j]); */
2590     }
2591     /* printf("\n"); */
2592   }
2593 
2594   ierr = PetscFree(rtmp);CHKERRQ(ierr);
2595   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
2596   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
2597 
2598   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
2599   ierr = ISIdentity(isicol,&col_identity);CHKERRQ(ierr);
2600   if (row_identity && col_identity) {
2601     C->ops->solve   = MatSolve_SeqAIJ_NaturalOrdering_iludt;
2602   } else {
2603     C->ops->solve   = MatSolve_SeqAIJ_iludt;
2604   }
2605   C->ops->solveadd           = 0;
2606   C->ops->solvetranspose     = 0;
2607   C->ops->solvetransposeadd  = 0;
2608   C->ops->matsolve           = 0;
2609   C->assembled    = PETSC_TRUE;
2610   C->preallocated = PETSC_TRUE;
2611   ierr = PetscLogFlops(C->cmap->n);CHKERRQ(ierr);
2612   PetscFunctionReturn(0);
2613 }
2614