xref: /petsc/src/mat/impls/aij/seq/aijfact.c (revision 1999ca518cb2b04afbdf9d234e189ae23820116a)
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__ "PetscFreeSpaceContiguous_newdatastruct"
1227 PetscErrorCode PetscFreeSpaceContiguous_newdatastruct(PetscFreeSpaceList *head,PetscInt *space,PetscInt n,PetscInt *bi,PetscInt *bdiag)
1228 {
1229   PetscFreeSpaceList a;
1230   PetscErrorCode     ierr;
1231   PetscInt           row,nnz,*bj,*array,total;
1232   PetscInt           nnzL,nnzU;
1233 
1234   PetscFunctionBegin;
1235   bi[2*n+1] = bi[n];
1236   row   = 1;
1237   total = 0;
1238   nnzL  = bdiag[0];
1239   while ((*head)!=NULL) {
1240     total += (*head)->local_used;
1241     array  = (*head)->array_head;
1242 
1243     while (bi[row] <= total && row <=n){
1244       /* copy array entries into bj for row-1 */
1245       nnz  = bi[row] - bi[row-1];
1246       /* set bi[row-1] for new datastruct */
1247       if (row -1 <= 1 ){
1248         bi[row -1] = 0;
1249       } else {
1250         bi[row-1] = bi[row-2] + nnzL; /* nnzL of previous row */
1251       }
1252 
1253       /* L part */
1254       nnzL = bdiag[row-1];
1255       bj   = space+bi[row-1];
1256       ierr = PetscMemcpy(bj,array,nnzL*sizeof(PetscInt));CHKERRQ(ierr);
1257 
1258       /* diagonal entry */
1259       bdiag[row-1]        = bi[2*n-(row-1)+1]-1;
1260       space[bdiag[row-1]] = row-1;
1261 
1262       /* U part */
1263       nnzU = nnz - nnzL;
1264       bi[2*n-(row-1)] = bi[2*n-(row-1)+1] - nnzU;
1265       nnzU --; /* exclude diagonal */
1266       bj  = space + bi[2*n-(row-1)];
1267       ierr = PetscMemcpy(bj,array+nnzL+1,nnzU*sizeof(PetscInt));CHKERRQ(ierr);
1268 
1269       array += nnz;
1270       row++;
1271     }
1272 
1273     a     =  (*head)->more_space;
1274     ierr  =  PetscFree((*head)->array_head);CHKERRQ(ierr);
1275     ierr  =  PetscFree(*head);CHKERRQ(ierr);
1276     *head =  a;
1277   }
1278   bi[n] = bi[n-1] + nnzL;
1279   if (bi[n] != bi[n+1]) SETERRQ2(1,"bi[n] %d != bi[n+1] %d",bi[n],bi[n+1]);
1280   PetscFunctionReturn(0);
1281 }
1282 
1283 #undef __FUNCT__
1284 #define __FUNCT__ "MatILUFactorSymbolic_SeqAIJ_newdatastruct"
1285 PetscErrorCode MatILUFactorSymbolic_SeqAIJ_newdatastruct(Mat fact,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
1286 {
1287   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
1288   IS                 isicol;
1289   PetscErrorCode     ierr;
1290   const PetscInt     *r,*ic;
1291   PetscInt           n=A->rmap->n,*ai=a->i,*aj=a->j,d;
1292   PetscInt           *bi,*cols,nnz,*cols_lvl;
1293   PetscInt           *bdiag,prow,fm,nzbd,reallocs=0,dcount=0;
1294   PetscInt           i,levels,diagonal_fill;
1295   PetscTruth         col_identity,row_identity;
1296   PetscReal          f;
1297   PetscInt           nlnk,*lnk,*lnk_lvl=PETSC_NULL;
1298   PetscBT            lnkbt;
1299   PetscInt           nzi,*bj,**bj_ptr,**bjlvl_ptr;
1300   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
1301   PetscFreeSpaceList free_space_lvl=PETSC_NULL,current_space_lvl=PETSC_NULL;
1302   PetscTruth         missing;
1303 
1304   PetscFunctionBegin;
1305   //printf("MatILUFactorSymbolic_SeqAIJ_newdatastruct ...\n");
1306   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);
1307   f             = info->fill;
1308   levels        = (PetscInt)info->levels;
1309   diagonal_fill = (PetscInt)info->diagonal_fill;
1310   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
1311 
1312   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
1313   ierr = ISIdentity(iscol,&col_identity);CHKERRQ(ierr);
1314 
1315   if (!levels && row_identity && col_identity) {
1316     /* special case: ilu(0) with natural ordering */
1317     ierr = MatILUFactorSymbolic_SeqAIJ_ilu0_newdatastruct(fact,A,isrow,iscol,info);CHKERRQ(ierr);
1318     (fact)->ops->lufactornumeric =  MatLUFactorNumeric_SeqAIJ_newdatastruct;
1319 
1320     fact->factor = MAT_FACTOR_ILU;
1321     (fact)->info.factor_mallocs    = 0;
1322     (fact)->info.fill_ratio_given  = info->fill;
1323     (fact)->info.fill_ratio_needed = 1.0;
1324     b               = (Mat_SeqAIJ*)(fact)->data;
1325     ierr = MatMissingDiagonal(A,&missing,&d);CHKERRQ(ierr);
1326     if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",d);
1327     b->row              = isrow;
1328     b->col              = iscol;
1329     b->icol             = isicol;
1330     ierr                = PetscMalloc(((fact)->rmap->n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1331     ierr                = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
1332     ierr                = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
1333     /* ierr = MatILUFactorSymbolic_Inode(fact,A,isrow,iscol,info);CHKERRQ(ierr); */
1334     PetscFunctionReturn(0);
1335   }
1336 
1337   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
1338   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
1339 
1340   /* get new row pointers */
1341   ierr = PetscMalloc((2*n+2)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
1342   bi[0] = 0;
1343   /* bdiag is location of diagonal in factor */
1344   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);
1345   bdiag[0]  = 0;
1346 
1347   ierr = PetscMalloc((2*n+1)*sizeof(PetscInt**),&bj_ptr);CHKERRQ(ierr);
1348   bjlvl_ptr = (PetscInt**)(bj_ptr + n);
1349 
1350   /* create a linked list for storing column indices of the active row */
1351   nlnk = n + 1;
1352   ierr = PetscIncompleteLLCreate(n,n,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1353 
1354   /* initial FreeSpace size is f*(ai[n]+1) */
1355   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space);CHKERRQ(ierr);
1356   current_space = free_space;
1357   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space_lvl);CHKERRQ(ierr);
1358   current_space_lvl = free_space_lvl;
1359 
1360   for (i=0; i<n; i++) {
1361     nzi = 0;
1362     /* copy current row into linked list */
1363     nnz  = ai[r[i]+1] - ai[r[i]];
1364     if (!nnz) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
1365     cols = aj + ai[r[i]];
1366     lnk[i] = -1; /* marker to indicate if diagonal exists */
1367     ierr = PetscIncompleteLLInit(nnz,cols,n,ic,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1368     nzi += nlnk;
1369 
1370     /* make sure diagonal entry is included */
1371     if (diagonal_fill && lnk[i] == -1) {
1372       fm = n;
1373       while (lnk[fm] < i) fm = lnk[fm];
1374       lnk[i]     = lnk[fm]; /* insert diagonal into linked list */
1375       lnk[fm]    = i;
1376       lnk_lvl[i] = 0;
1377       nzi++; dcount++;
1378     }
1379 
1380     /* add pivot rows into the active row */
1381     nzbd = 0;
1382     prow = lnk[n];
1383     while (prow < i) {
1384       nnz      = bdiag[prow];
1385       cols     = bj_ptr[prow] + nnz + 1;
1386       cols_lvl = bjlvl_ptr[prow] + nnz + 1;
1387       nnz      = bi[prow+1] - bi[prow] - nnz - 1;
1388       ierr = PetscILULLAddSorted(nnz,cols,levels,cols_lvl,prow,nlnk,lnk,lnk_lvl,lnkbt,prow);CHKERRQ(ierr);
1389       nzi += nlnk;
1390       prow = lnk[prow];
1391       nzbd++;
1392     }
1393     bdiag[i] = nzbd;
1394     bi[i+1]  = bi[i] + nzi;
1395 
1396     /* if free space is not available, make more free space */
1397     if (current_space->local_remaining<nzi) {
1398       nnz = 2*nzi*(n - i); /* estimated and max additional space needed */
1399       ierr = PetscFreeSpaceGet(nnz,&current_space);CHKERRQ(ierr);
1400       ierr = PetscFreeSpaceGet(nnz,&current_space_lvl);CHKERRQ(ierr);
1401       reallocs++;
1402     }
1403 
1404     /* copy data into free_space and free_space_lvl, then initialize lnk */
1405     ierr = PetscIncompleteLLClean(n,n,nzi,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);CHKERRQ(ierr);
1406     bj_ptr[i]    = current_space->array;
1407     bjlvl_ptr[i] = current_space_lvl->array;
1408 
1409     /* make sure the active row i has diagonal entry */
1410     if (*(bj_ptr[i]+bdiag[i]) != i) {
1411       SETERRQ1(PETSC_ERR_MAT_LU_ZRPVT,"Row %D has missing diagonal in factored matrix\n\
1412     try running with -pc_factor_nonzeros_along_diagonal or -pc_factor_diagonal_fill",i);
1413     }
1414 
1415     current_space->array           += nzi;
1416     current_space->local_used      += nzi;
1417     current_space->local_remaining -= nzi;
1418     current_space_lvl->array           += nzi;
1419     current_space_lvl->local_used      += nzi;
1420     current_space_lvl->local_remaining -= nzi;
1421   }
1422 
1423   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
1424   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
1425 
1426   /* destroy list of free space and other temporary arrays */
1427   ierr = PetscMalloc((bi[n]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
1428 
1429   /* copy free_space into bj and free free_space; set bi, bj, bdiag in new datastructure; */
1430   ierr = PetscFreeSpaceContiguous_newdatastruct(&free_space,bj,n,bi,bdiag);CHKERRQ(ierr);
1431 
1432   ierr = PetscIncompleteLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1433   ierr = PetscFreeSpaceDestroy(free_space_lvl);CHKERRQ(ierr);
1434   ierr = PetscFree(bj_ptr);CHKERRQ(ierr);
1435 
1436 #if defined(PETSC_USE_INFO)
1437   {
1438     PetscReal af = ((PetscReal)bi[n])/((PetscReal)ai[n]);
1439     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,f,af);CHKERRQ(ierr);
1440     ierr = PetscInfo1(A,"Run with -[sub_]pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
1441     ierr = PetscInfo1(A,"PCFactorSetFill([sub]pc,%G);\n",af);CHKERRQ(ierr);
1442     ierr = PetscInfo(A,"for best performance.\n");CHKERRQ(ierr);
1443     if (diagonal_fill) {
1444       ierr = PetscInfo1(A,"Detected and replaced %D missing diagonals",dcount);CHKERRQ(ierr);
1445     }
1446   }
1447 #endif
1448 
1449   /* put together the new matrix */
1450   ierr = MatSeqAIJSetPreallocation_SeqAIJ(fact,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
1451   ierr = PetscLogObjectParent(fact,isicol);CHKERRQ(ierr);
1452   b = (Mat_SeqAIJ*)(fact)->data;
1453   b->free_a       = PETSC_TRUE;
1454   b->free_ij      = PETSC_TRUE;
1455   b->singlemalloc = PETSC_FALSE;
1456   ierr = PetscMalloc( (bi[2*n+1] )*sizeof(PetscScalar),&b->a);CHKERRQ(ierr);
1457   b->j          = bj;
1458   b->i          = bi;
1459   b->diag       = bdiag;
1460   b->ilen       = 0;
1461   b->imax       = 0;
1462   b->row        = isrow;
1463   b->col        = iscol;
1464   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
1465   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
1466   b->icol       = isicol;
1467   ierr = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1468   /* In b structure:  Free imax, ilen, old a, old j.
1469      Allocate bdiag, solve_work, new a, new j */
1470   ierr = PetscLogObjectMemory(fact,bi[2*n+1] * (sizeof(PetscInt)+sizeof(PetscScalar)));CHKERRQ(ierr);
1471   b->maxnz = b->nz = bi[2*n+1] ;
1472   (fact)->info.factor_mallocs    = reallocs;
1473   (fact)->info.fill_ratio_given  = f;
1474   (fact)->info.fill_ratio_needed = ((PetscReal)bi[2*n+1])/((PetscReal)ai[n]);
1475   (fact)->ops->lufactornumeric   = MatLUFactorNumeric_SeqAIJ_newdatastruct;
1476   /* ierr = MatILUFactorSymbolic_Inode(fact,A,isrow,iscol,info);CHKERRQ(ierr); */
1477   PetscFunctionReturn(0);
1478 }
1479 
1480 #undef __FUNCT__
1481 #define __FUNCT__ "MatILUFactorSymbolic_SeqAIJ"
1482 PetscErrorCode MatILUFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS isrow,IS iscol,const MatFactorInfo *info)
1483 {
1484   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b;
1485   IS                 isicol;
1486   PetscErrorCode     ierr;
1487   const PetscInt     *r,*ic;
1488   PetscInt           n=A->rmap->n,*ai=a->i,*aj=a->j,d;
1489   PetscInt           *bi,*cols,nnz,*cols_lvl;
1490   PetscInt           *bdiag,prow,fm,nzbd,reallocs=0,dcount=0;
1491   PetscInt           i,levels,diagonal_fill;
1492   PetscTruth         col_identity,row_identity;
1493   PetscReal          f;
1494   PetscInt           nlnk,*lnk,*lnk_lvl=PETSC_NULL;
1495   PetscBT            lnkbt;
1496   PetscInt           nzi,*bj,**bj_ptr,**bjlvl_ptr;
1497   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
1498   PetscFreeSpaceList free_space_lvl=PETSC_NULL,current_space_lvl=PETSC_NULL;
1499   PetscTruth         missing;
1500   PetscTruth         newdatastruct=PETSC_FALSE;
1501 
1502   PetscFunctionBegin;
1503   ierr = PetscOptionsGetTruth(PETSC_NULL,"-ilu_new",&newdatastruct,PETSC_NULL);CHKERRQ(ierr);
1504   if (newdatastruct){
1505     ierr = MatILUFactorSymbolic_SeqAIJ_newdatastruct(fact,A,isrow,iscol,info);CHKERRQ(ierr);
1506     PetscFunctionReturn(0);
1507   }
1508 
1509   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);
1510   f             = info->fill;
1511   levels        = (PetscInt)info->levels;
1512   diagonal_fill = (PetscInt)info->diagonal_fill;
1513   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
1514 
1515   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
1516   ierr = ISIdentity(iscol,&col_identity);CHKERRQ(ierr);
1517   if (!levels && row_identity && col_identity) { /* special case: ilu(0) with natural ordering */
1518     ierr = MatDuplicateNoCreate_SeqAIJ(fact,A,MAT_DO_NOT_COPY_VALUES,PETSC_TRUE);CHKERRQ(ierr);
1519     (fact)->ops->lufactornumeric =  MatLUFactorNumeric_SeqAIJ;
1520 
1521     fact->factor = MAT_FACTOR_ILU;
1522     (fact)->info.factor_mallocs    = 0;
1523     (fact)->info.fill_ratio_given  = info->fill;
1524     (fact)->info.fill_ratio_needed = 1.0;
1525     b               = (Mat_SeqAIJ*)(fact)->data;
1526     ierr = MatMissingDiagonal(A,&missing,&d);CHKERRQ(ierr);
1527     if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",d);
1528     b->row              = isrow;
1529     b->col              = iscol;
1530     b->icol             = isicol;
1531     ierr                = PetscMalloc(((fact)->rmap->n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1532     ierr                = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
1533     ierr                = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
1534     ierr = MatILUFactorSymbolic_Inode(fact,A,isrow,iscol,info);CHKERRQ(ierr);
1535     PetscFunctionReturn(0);
1536   }
1537 
1538   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
1539   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
1540 
1541   /* get new row pointers */
1542   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
1543   bi[0] = 0;
1544   /* bdiag is location of diagonal in factor */
1545   ierr = PetscMalloc((n+1)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);
1546   bdiag[0]  = 0;
1547 
1548   ierr = PetscMalloc((2*n+1)*sizeof(PetscInt**),&bj_ptr);CHKERRQ(ierr);
1549   bjlvl_ptr = (PetscInt**)(bj_ptr + n);
1550 
1551   /* create a linked list for storing column indices of the active row */
1552   nlnk = n + 1;
1553   ierr = PetscIncompleteLLCreate(n,n,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1554 
1555   /* initial FreeSpace size is f*(ai[n]+1) */
1556   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space);CHKERRQ(ierr);
1557   current_space = free_space;
1558   ierr = PetscFreeSpaceGet((PetscInt)(f*(ai[n]+1)),&free_space_lvl);CHKERRQ(ierr);
1559   current_space_lvl = free_space_lvl;
1560 
1561   for (i=0; i<n; i++) {
1562     nzi = 0;
1563     /* copy current row into linked list */
1564     nnz  = ai[r[i]+1] - ai[r[i]];
1565     if (!nnz) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
1566     cols = aj + ai[r[i]];
1567     lnk[i] = -1; /* marker to indicate if diagonal exists */
1568     ierr = PetscIncompleteLLInit(nnz,cols,n,ic,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1569     nzi += nlnk;
1570 
1571     /* make sure diagonal entry is included */
1572     if (diagonal_fill && lnk[i] == -1) {
1573       fm = n;
1574       while (lnk[fm] < i) fm = lnk[fm];
1575       lnk[i]     = lnk[fm]; /* insert diagonal into linked list */
1576       lnk[fm]    = i;
1577       lnk_lvl[i] = 0;
1578       nzi++; dcount++;
1579     }
1580 
1581     /* add pivot rows into the active row */
1582     nzbd = 0;
1583     prow = lnk[n];
1584     while (prow < i) {
1585       nnz      = bdiag[prow];
1586       cols     = bj_ptr[prow] + nnz + 1;
1587       cols_lvl = bjlvl_ptr[prow] + nnz + 1;
1588       nnz      = bi[prow+1] - bi[prow] - nnz - 1;
1589       ierr = PetscILULLAddSorted(nnz,cols,levels,cols_lvl,prow,nlnk,lnk,lnk_lvl,lnkbt,prow);CHKERRQ(ierr);
1590       nzi += nlnk;
1591       prow = lnk[prow];
1592       nzbd++;
1593     }
1594     bdiag[i] = nzbd;
1595     bi[i+1]  = bi[i] + nzi;
1596 
1597     /* if free space is not available, make more free space */
1598     if (current_space->local_remaining<nzi) {
1599       nnz = nzi*(n - i); /* estimated and max additional space needed */
1600       ierr = PetscFreeSpaceGet(nnz,&current_space);CHKERRQ(ierr);
1601       ierr = PetscFreeSpaceGet(nnz,&current_space_lvl);CHKERRQ(ierr);
1602       reallocs++;
1603     }
1604 
1605     /* copy data into free_space and free_space_lvl, then initialize lnk */
1606     ierr = PetscIncompleteLLClean(n,n,nzi,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);CHKERRQ(ierr);
1607     bj_ptr[i]    = current_space->array;
1608     bjlvl_ptr[i] = current_space_lvl->array;
1609 
1610     /* make sure the active row i has diagonal entry */
1611     if (*(bj_ptr[i]+bdiag[i]) != i) {
1612       SETERRQ1(PETSC_ERR_MAT_LU_ZRPVT,"Row %D has missing diagonal in factored matrix\n\
1613     try running with -pc_factor_nonzeros_along_diagonal or -pc_factor_diagonal_fill",i);
1614     }
1615 
1616     current_space->array           += nzi;
1617     current_space->local_used      += nzi;
1618     current_space->local_remaining -= nzi;
1619     current_space_lvl->array           += nzi;
1620     current_space_lvl->local_used      += nzi;
1621     current_space_lvl->local_remaining -= nzi;
1622   }
1623 
1624   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
1625   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
1626 
1627   /* destroy list of free space and other temporary arrays */
1628   ierr = PetscMalloc((bi[n]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
1629   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr); /* copy free_space -> bj */
1630   ierr = PetscIncompleteLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1631   ierr = PetscFreeSpaceDestroy(free_space_lvl);CHKERRQ(ierr);
1632   ierr = PetscFree(bj_ptr);CHKERRQ(ierr);
1633 
1634 #if defined(PETSC_USE_INFO)
1635   {
1636     PetscReal af = ((PetscReal)bi[n])/((PetscReal)ai[n]);
1637     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,f,af);CHKERRQ(ierr);
1638     ierr = PetscInfo1(A,"Run with -[sub_]pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
1639     ierr = PetscInfo1(A,"PCFactorSetFill([sub]pc,%G);\n",af);CHKERRQ(ierr);
1640     ierr = PetscInfo(A,"for best performance.\n");CHKERRQ(ierr);
1641     if (diagonal_fill) {
1642       ierr = PetscInfo1(A,"Detected and replaced %D missing diagonals",dcount);CHKERRQ(ierr);
1643     }
1644   }
1645 #endif
1646 
1647   /* put together the new matrix */
1648   ierr = MatSeqAIJSetPreallocation_SeqAIJ(fact,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
1649   ierr = PetscLogObjectParent(fact,isicol);CHKERRQ(ierr);
1650   b = (Mat_SeqAIJ*)(fact)->data;
1651   b->free_a       = PETSC_TRUE;
1652   b->free_ij      = PETSC_TRUE;
1653   b->singlemalloc = PETSC_FALSE;
1654   ierr = PetscMalloc( (bi[n] )*sizeof(PetscScalar),&b->a);CHKERRQ(ierr);
1655   b->j          = bj;
1656   b->i          = bi;
1657   for (i=0; i<n; i++) bdiag[i] += bi[i];
1658   b->diag       = bdiag;
1659   b->ilen       = 0;
1660   b->imax       = 0;
1661   b->row        = isrow;
1662   b->col        = iscol;
1663   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
1664   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
1665   b->icol       = isicol;
1666   ierr = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
1667   /* In b structure:  Free imax, ilen, old a, old j.
1668      Allocate bdiag, solve_work, new a, new j */
1669   ierr = PetscLogObjectMemory(fact,(bi[n]-n) * (sizeof(PetscInt)+sizeof(PetscScalar)));CHKERRQ(ierr);
1670   b->maxnz             = b->nz = bi[n] ;
1671   (fact)->info.factor_mallocs    = reallocs;
1672   (fact)->info.fill_ratio_given  = f;
1673   (fact)->info.fill_ratio_needed = ((PetscReal)bi[n])/((PetscReal)ai[n]);
1674   (fact)->ops->lufactornumeric =  MatLUFactorNumeric_SeqAIJ;
1675   ierr = MatILUFactorSymbolic_Inode(fact,A,isrow,iscol,info);CHKERRQ(ierr);
1676   PetscFunctionReturn(0);
1677 }
1678 
1679 #include "../src/mat/impls/sbaij/seq/sbaij.h"
1680 #undef __FUNCT__
1681 #define __FUNCT__ "MatCholeskyFactorNumeric_SeqAIJ"
1682 PetscErrorCode MatCholeskyFactorNumeric_SeqAIJ(Mat B,Mat A,const MatFactorInfo *info)
1683 {
1684   Mat            C = B;
1685   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data;
1686   Mat_SeqSBAIJ   *b=(Mat_SeqSBAIJ*)C->data;
1687   IS             ip=b->row,iip = b->icol;
1688   PetscErrorCode ierr;
1689   const PetscInt *rip,*riip;
1690   PetscInt       i,j,mbs=A->rmap->n,*bi=b->i,*bj=b->j,*bcol;
1691   PetscInt       *ai=a->i,*aj=a->j;
1692   PetscInt       k,jmin,jmax,*jl,*il,col,nexti,ili,nz;
1693   MatScalar      *rtmp,*ba=b->a,*bval,*aa=a->a,dk,uikdi;
1694   PetscReal      zeropivot,rs,shiftnz;
1695   PetscReal      shiftpd;
1696   ChShift_Ctx    sctx;
1697   PetscInt       newshift;
1698   PetscTruth     perm_identity;
1699 
1700   PetscFunctionBegin;
1701 
1702   shiftnz   = info->shiftnz;
1703   shiftpd   = info->shiftpd;
1704   zeropivot = info->zeropivot;
1705 
1706   ierr  = ISGetIndices(ip,&rip);CHKERRQ(ierr);
1707   ierr  = ISGetIndices(iip,&riip);CHKERRQ(ierr);
1708 
1709   /* initialization */
1710   nz   = (2*mbs+1)*sizeof(PetscInt)+mbs*sizeof(MatScalar);
1711   ierr = PetscMalloc(nz,&il);CHKERRQ(ierr);
1712   jl   = il + mbs;
1713   rtmp = (MatScalar*)(jl + mbs);
1714 
1715   sctx.shift_amount = 0;
1716   sctx.nshift       = 0;
1717   do {
1718     sctx.chshift = PETSC_FALSE;
1719     for (i=0; i<mbs; i++) {
1720       rtmp[i] = 0.0; jl[i] = mbs; il[0] = 0;
1721     }
1722 
1723     for (k = 0; k<mbs; k++){
1724       bval = ba + bi[k];
1725       /* initialize k-th row by the perm[k]-th row of A */
1726       jmin = ai[rip[k]]; jmax = ai[rip[k]+1];
1727       for (j = jmin; j < jmax; j++){
1728         col = riip[aj[j]];
1729         if (col >= k){ /* only take upper triangular entry */
1730           rtmp[col] = aa[j];
1731           *bval++  = 0.0; /* for in-place factorization */
1732         }
1733       }
1734       /* shift the diagonal of the matrix */
1735       if (sctx.nshift) rtmp[k] += sctx.shift_amount;
1736 
1737       /* modify k-th row by adding in those rows i with U(i,k)!=0 */
1738       dk = rtmp[k];
1739       i = jl[k]; /* first row to be added to k_th row  */
1740 
1741       while (i < k){
1742         nexti = jl[i]; /* next row to be added to k_th row */
1743 
1744         /* compute multiplier, update diag(k) and U(i,k) */
1745         ili = il[i];  /* index of first nonzero element in U(i,k:bms-1) */
1746         uikdi = - ba[ili]*ba[bi[i]];  /* diagonal(k) */
1747         dk += uikdi*ba[ili];
1748         ba[ili] = uikdi; /* -U(i,k) */
1749 
1750         /* add multiple of row i to k-th row */
1751         jmin = ili + 1; jmax = bi[i+1];
1752         if (jmin < jmax){
1753           for (j=jmin; j<jmax; j++) rtmp[bj[j]] += uikdi*ba[j];
1754           /* update il and jl for row i */
1755           il[i] = jmin;
1756           j = bj[jmin]; jl[i] = jl[j]; jl[j] = i;
1757         }
1758         i = nexti;
1759       }
1760 
1761       /* shift the diagonals when zero pivot is detected */
1762       /* compute rs=sum of abs(off-diagonal) */
1763       rs   = 0.0;
1764       jmin = bi[k]+1;
1765       nz   = bi[k+1] - jmin;
1766       bcol = bj + jmin;
1767       while (nz--){
1768         rs += PetscAbsScalar(rtmp[*bcol]);
1769         bcol++;
1770       }
1771 
1772       sctx.rs = rs;
1773       sctx.pv = dk;
1774       ierr = MatCholeskyCheckShift_inline(info,sctx,k,newshift);CHKERRQ(ierr);
1775 
1776       if (newshift == 1) {
1777         if (!sctx.shift_amount) {
1778           sctx.shift_amount = 1e-5;
1779         }
1780         break;
1781       }
1782 
1783       /* copy data into U(k,:) */
1784       ba[bi[k]] = 1.0/dk; /* U(k,k) */
1785       jmin = bi[k]+1; jmax = bi[k+1];
1786       if (jmin < jmax) {
1787         for (j=jmin; j<jmax; j++){
1788           col = bj[j]; ba[j] = rtmp[col]; rtmp[col] = 0.0;
1789         }
1790         /* add the k-th row into il and jl */
1791         il[k] = jmin;
1792         i = bj[jmin]; jl[k] = jl[i]; jl[i] = k;
1793       }
1794     }
1795   } while (sctx.chshift);
1796   ierr = PetscFree(il);CHKERRQ(ierr);
1797 
1798   ierr = ISRestoreIndices(ip,&rip);CHKERRQ(ierr);
1799   ierr = ISRestoreIndices(iip,&riip);CHKERRQ(ierr);
1800 
1801   ierr = ISIdentity(ip,&perm_identity);CHKERRQ(ierr);
1802   if (perm_identity){
1803     (B)->ops->solve           = MatSolve_SeqSBAIJ_1_NaturalOrdering;
1804     (B)->ops->solvetranspose  = MatSolve_SeqSBAIJ_1_NaturalOrdering;
1805     (B)->ops->forwardsolve    = MatForwardSolve_SeqSBAIJ_1_NaturalOrdering;
1806     (B)->ops->backwardsolve   = MatBackwardSolve_SeqSBAIJ_1_NaturalOrdering;
1807   } else {
1808     (B)->ops->solve           = MatSolve_SeqSBAIJ_1;
1809     (B)->ops->solvetranspose  = MatSolve_SeqSBAIJ_1;
1810     (B)->ops->forwardsolve    = MatForwardSolve_SeqSBAIJ_1;
1811     (B)->ops->backwardsolve   = MatBackwardSolve_SeqSBAIJ_1;
1812   }
1813 
1814   C->assembled    = PETSC_TRUE;
1815   C->preallocated = PETSC_TRUE;
1816   ierr = PetscLogFlops(C->rmap->n);CHKERRQ(ierr);
1817   if (sctx.nshift){
1818     if (shiftnz) {
1819       ierr = PetscInfo2(A,"number of shiftnz tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
1820     } else if (shiftpd) {
1821       ierr = PetscInfo2(A,"number of shiftpd tries %D, shift_amount %G\n",sctx.nshift,sctx.shift_amount);CHKERRQ(ierr);
1822     }
1823   }
1824   PetscFunctionReturn(0);
1825 }
1826 
1827 #undef __FUNCT__
1828 #define __FUNCT__ "MatICCFactorSymbolic_SeqAIJ"
1829 PetscErrorCode MatICCFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
1830 {
1831   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data;
1832   Mat_SeqSBAIJ       *b;
1833   PetscErrorCode     ierr;
1834   PetscTruth         perm_identity,missing;
1835   PetscInt           reallocs=0,i,*ai=a->i,*aj=a->j,am=A->rmap->n,*ui;
1836   const PetscInt     *rip,*riip;
1837   PetscInt           jmin,jmax,nzk,k,j,*jl,prow,*il,nextprow;
1838   PetscInt           nlnk,*lnk,*lnk_lvl=PETSC_NULL,d;
1839   PetscInt           ncols,ncols_upper,*cols,*ajtmp,*uj,**uj_ptr,**uj_lvl_ptr;
1840   PetscReal          fill=info->fill,levels=info->levels;
1841   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
1842   PetscFreeSpaceList free_space_lvl=PETSC_NULL,current_space_lvl=PETSC_NULL;
1843   PetscBT            lnkbt;
1844   IS                 iperm;
1845 
1846   PetscFunctionBegin;
1847   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);
1848   ierr = MatMissingDiagonal(A,&missing,&d);CHKERRQ(ierr);
1849   if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",d);
1850   ierr = ISIdentity(perm,&perm_identity);CHKERRQ(ierr);
1851   ierr = ISInvertPermutation(perm,PETSC_DECIDE,&iperm);CHKERRQ(ierr);
1852 
1853   ierr = PetscMalloc((am+1)*sizeof(PetscInt),&ui);CHKERRQ(ierr);
1854   ui[0] = 0;
1855 
1856   /* ICC(0) without matrix ordering: simply copies fill pattern */
1857   if (!levels && perm_identity) {
1858 
1859     for (i=0; i<am; i++) {
1860       ui[i+1] = ui[i] + ai[i+1] - a->diag[i];
1861     }
1862     ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
1863     cols = uj;
1864     for (i=0; i<am; i++) {
1865       aj    = a->j + a->diag[i];
1866       ncols = ui[i+1] - ui[i];
1867       for (j=0; j<ncols; j++) *cols++ = *aj++;
1868     }
1869   } else { /* case: levels>0 || (levels=0 && !perm_identity) */
1870     ierr = ISGetIndices(iperm,&riip);CHKERRQ(ierr);
1871     ierr = ISGetIndices(perm,&rip);CHKERRQ(ierr);
1872 
1873     /* initialization */
1874     ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&ajtmp);CHKERRQ(ierr);
1875 
1876     /* jl: linked list for storing indices of the pivot rows
1877        il: il[i] points to the 1st nonzero entry of U(i,k:am-1) */
1878     ierr = PetscMalloc((2*am+1)*sizeof(PetscInt)+2*am*sizeof(PetscInt**),&jl);CHKERRQ(ierr);
1879     il         = jl + am;
1880     uj_ptr     = (PetscInt**)(il + am);
1881     uj_lvl_ptr = (PetscInt**)(uj_ptr + am);
1882     for (i=0; i<am; i++){
1883       jl[i] = am; il[i] = 0;
1884     }
1885 
1886     /* create and initialize a linked list for storing column indices of the active row k */
1887     nlnk = am + 1;
1888     ierr = PetscIncompleteLLCreate(am,am,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1889 
1890     /* initial FreeSpace size is fill*(ai[am]+1) */
1891     ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space);CHKERRQ(ierr);
1892     current_space = free_space;
1893     ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space_lvl);CHKERRQ(ierr);
1894     current_space_lvl = free_space_lvl;
1895 
1896     for (k=0; k<am; k++){  /* for each active row k */
1897       /* initialize lnk by the column indices of row rip[k] of A */
1898       nzk   = 0;
1899       ncols = ai[rip[k]+1] - ai[rip[k]];
1900       if (!ncols) SETERRQ2(PETSC_ERR_MAT_CH_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",rip[k],k);
1901       ncols_upper = 0;
1902       for (j=0; j<ncols; j++){
1903         i = *(aj + ai[rip[k]] + j); /* unpermuted column index */
1904         if (riip[i] >= k){ /* only take upper triangular entry */
1905           ajtmp[ncols_upper] = i;
1906           ncols_upper++;
1907         }
1908       }
1909       ierr = PetscIncompleteLLInit(ncols_upper,ajtmp,am,riip,nlnk,lnk,lnk_lvl,lnkbt);CHKERRQ(ierr);
1910       nzk += nlnk;
1911 
1912       /* update lnk by computing fill-in for each pivot row to be merged in */
1913       prow = jl[k]; /* 1st pivot row */
1914 
1915       while (prow < k){
1916         nextprow = jl[prow];
1917 
1918         /* merge prow into k-th row */
1919         jmin = il[prow] + 1;  /* index of the 2nd nzero entry in U(prow,k:am-1) */
1920         jmax = ui[prow+1];
1921         ncols = jmax-jmin;
1922         i     = jmin - ui[prow];
1923         cols  = uj_ptr[prow] + i; /* points to the 2nd nzero entry in U(prow,k:am-1) */
1924         uj    = uj_lvl_ptr[prow] + i; /* levels of cols */
1925         j     = *(uj - 1);
1926         ierr = PetscICCLLAddSorted(ncols,cols,levels,uj,am,nlnk,lnk,lnk_lvl,lnkbt,j);CHKERRQ(ierr);
1927         nzk += nlnk;
1928 
1929         /* update il and jl for prow */
1930         if (jmin < jmax){
1931           il[prow] = jmin;
1932           j = *cols; jl[prow] = jl[j]; jl[j] = prow;
1933         }
1934         prow = nextprow;
1935       }
1936 
1937       /* if free space is not available, make more free space */
1938       if (current_space->local_remaining<nzk) {
1939         i = am - k + 1; /* num of unfactored rows */
1940         i = PetscMin(i*nzk, i*(i-1)); /* i*nzk, i*(i-1): estimated and max additional space needed */
1941         ierr = PetscFreeSpaceGet(i,&current_space);CHKERRQ(ierr);
1942         ierr = PetscFreeSpaceGet(i,&current_space_lvl);CHKERRQ(ierr);
1943         reallocs++;
1944       }
1945 
1946       /* copy data into free_space and free_space_lvl, then initialize lnk */
1947       if (nzk == 0) SETERRQ1(PETSC_ERR_ARG_WRONG,"Empty row %D in ICC matrix factor",k);
1948       ierr = PetscIncompleteLLClean(am,am,nzk,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);CHKERRQ(ierr);
1949 
1950       /* add the k-th row into il and jl */
1951       if (nzk > 1){
1952         i = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:am-1) */
1953         jl[k] = jl[i]; jl[i] = k;
1954         il[k] = ui[k] + 1;
1955       }
1956       uj_ptr[k]     = current_space->array;
1957       uj_lvl_ptr[k] = current_space_lvl->array;
1958 
1959       current_space->array           += nzk;
1960       current_space->local_used      += nzk;
1961       current_space->local_remaining -= nzk;
1962 
1963       current_space_lvl->array           += nzk;
1964       current_space_lvl->local_used      += nzk;
1965       current_space_lvl->local_remaining -= nzk;
1966 
1967       ui[k+1] = ui[k] + nzk;
1968     }
1969 
1970 #if defined(PETSC_USE_INFO)
1971     if (ai[am] != 0) {
1972       PetscReal af = (PetscReal)ui[am]/((PetscReal)ai[am]);
1973       ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,fill,af);CHKERRQ(ierr);
1974       ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
1975       ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G) for best performance.\n",af);CHKERRQ(ierr);
1976     } else {
1977       ierr = PetscInfo(A,"Empty matrix.\n");CHKERRQ(ierr);
1978     }
1979 #endif
1980 
1981     ierr = ISRestoreIndices(perm,&rip);CHKERRQ(ierr);
1982     ierr = ISRestoreIndices(iperm,&riip);CHKERRQ(ierr);
1983     ierr = PetscFree(jl);CHKERRQ(ierr);
1984     ierr = PetscFree(ajtmp);CHKERRQ(ierr);
1985 
1986     /* destroy list of free space and other temporary array(s) */
1987     ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
1988     ierr = PetscFreeSpaceContiguous(&free_space,uj);CHKERRQ(ierr);
1989     ierr = PetscIncompleteLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1990     ierr = PetscFreeSpaceDestroy(free_space_lvl);CHKERRQ(ierr);
1991 
1992   } /* end of case: levels>0 || (levels=0 && !perm_identity) */
1993 
1994   /* put together the new matrix in MATSEQSBAIJ format */
1995 
1996   b    = (Mat_SeqSBAIJ*)(fact)->data;
1997   b->singlemalloc = PETSC_FALSE;
1998   ierr = PetscMalloc((ui[am]+1)*sizeof(MatScalar),&b->a);CHKERRQ(ierr);
1999   b->j    = uj;
2000   b->i    = ui;
2001   b->diag = 0;
2002   b->ilen = 0;
2003   b->imax = 0;
2004   b->row  = perm;
2005   b->col  = perm;
2006   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2007   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2008   b->icol = iperm;
2009   b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */
2010   ierr    = PetscMalloc((am+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
2011   ierr = PetscLogObjectMemory((fact),(ui[am]-am)*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
2012   b->maxnz   = b->nz = ui[am];
2013   b->free_a  = PETSC_TRUE;
2014   b->free_ij = PETSC_TRUE;
2015 
2016   (fact)->info.factor_mallocs    = reallocs;
2017   (fact)->info.fill_ratio_given  = fill;
2018   if (ai[am] != 0) {
2019     (fact)->info.fill_ratio_needed = ((PetscReal)ui[am])/((PetscReal)ai[am]);
2020   } else {
2021     (fact)->info.fill_ratio_needed = 0.0;
2022   }
2023   (fact)->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqAIJ;
2024   PetscFunctionReturn(0);
2025 }
2026 
2027 #undef __FUNCT__
2028 #define __FUNCT__ "MatCholeskyFactorSymbolic_SeqAIJ"
2029 PetscErrorCode MatCholeskyFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
2030 {
2031   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data;
2032   Mat_SeqSBAIJ       *b;
2033   PetscErrorCode     ierr;
2034   PetscTruth         perm_identity;
2035   PetscReal          fill = info->fill;
2036   const PetscInt     *rip,*riip;
2037   PetscInt           i,am=A->rmap->n,*ai=a->i,*aj=a->j,reallocs=0,prow;
2038   PetscInt           *jl,jmin,jmax,nzk,*ui,k,j,*il,nextprow;
2039   PetscInt           nlnk,*lnk,ncols,ncols_upper,*cols,*uj,**ui_ptr,*uj_ptr;
2040   PetscFreeSpaceList free_space=PETSC_NULL,current_space=PETSC_NULL;
2041   PetscBT            lnkbt;
2042   IS                 iperm;
2043 
2044   PetscFunctionBegin;
2045   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);
2046   /* check whether perm is the identity mapping */
2047   ierr = ISIdentity(perm,&perm_identity);CHKERRQ(ierr);
2048   ierr = ISInvertPermutation(perm,PETSC_DECIDE,&iperm);CHKERRQ(ierr);
2049   ierr = ISGetIndices(iperm,&riip);CHKERRQ(ierr);
2050   ierr = ISGetIndices(perm,&rip);CHKERRQ(ierr);
2051 
2052   /* initialization */
2053   ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&ui);CHKERRQ(ierr);
2054   ui[0] = 0;
2055 
2056   /* jl: linked list for storing indices of the pivot rows
2057      il: il[i] points to the 1st nonzero entry of U(i,k:am-1) */
2058   ierr = PetscMalloc((3*am+1)*sizeof(PetscInt)+am*sizeof(PetscInt**),&jl);CHKERRQ(ierr);
2059   il     = jl + am;
2060   cols   = il + am;
2061   ui_ptr = (PetscInt**)(cols + am);
2062   for (i=0; i<am; i++){
2063     jl[i] = am; il[i] = 0;
2064   }
2065 
2066   /* create and initialize a linked list for storing column indices of the active row k */
2067   nlnk = am + 1;
2068   ierr = PetscLLCreate(am,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2069 
2070   /* initial FreeSpace size is fill*(ai[am]+1) */
2071   ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space);CHKERRQ(ierr);
2072   current_space = free_space;
2073 
2074   for (k=0; k<am; k++){  /* for each active row k */
2075     /* initialize lnk by the column indices of row rip[k] of A */
2076     nzk   = 0;
2077     ncols = ai[rip[k]+1] - ai[rip[k]];
2078     if (!ncols) SETERRQ2(PETSC_ERR_MAT_CH_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",rip[k],k);
2079     ncols_upper = 0;
2080     for (j=0; j<ncols; j++){
2081       i = riip[*(aj + ai[rip[k]] + j)];
2082       if (i >= k){ /* only take upper triangular entry */
2083         cols[ncols_upper] = i;
2084         ncols_upper++;
2085       }
2086     }
2087     ierr = PetscLLAdd(ncols_upper,cols,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2088     nzk += nlnk;
2089 
2090     /* update lnk by computing fill-in for each pivot row to be merged in */
2091     prow = jl[k]; /* 1st pivot row */
2092 
2093     while (prow < k){
2094       nextprow = jl[prow];
2095       /* merge prow into k-th row */
2096       jmin = il[prow] + 1;  /* index of the 2nd nzero entry in U(prow,k:am-1) */
2097       jmax = ui[prow+1];
2098       ncols = jmax-jmin;
2099       uj_ptr = ui_ptr[prow] + jmin - ui[prow]; /* points to the 2nd nzero entry in U(prow,k:am-1) */
2100       ierr = PetscLLAddSorted(ncols,uj_ptr,am,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2101       nzk += nlnk;
2102 
2103       /* update il and jl for prow */
2104       if (jmin < jmax){
2105         il[prow] = jmin;
2106         j = *uj_ptr; jl[prow] = jl[j]; jl[j] = prow;
2107       }
2108       prow = nextprow;
2109     }
2110 
2111     /* if free space is not available, make more free space */
2112     if (current_space->local_remaining<nzk) {
2113       i = am - k + 1; /* num of unfactored rows */
2114       i = PetscMin(i*nzk, i*(i-1)); /* i*nzk, i*(i-1): estimated and max additional space needed */
2115       ierr = PetscFreeSpaceGet(i,&current_space);CHKERRQ(ierr);
2116       reallocs++;
2117     }
2118 
2119     /* copy data into free space, then initialize lnk */
2120     ierr = PetscLLClean(am,am,nzk,lnk,current_space->array,lnkbt);CHKERRQ(ierr);
2121 
2122     /* add the k-th row into il and jl */
2123     if (nzk-1 > 0){
2124       i = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:am-1) */
2125       jl[k] = jl[i]; jl[i] = k;
2126       il[k] = ui[k] + 1;
2127     }
2128     ui_ptr[k] = current_space->array;
2129     current_space->array           += nzk;
2130     current_space->local_used      += nzk;
2131     current_space->local_remaining -= nzk;
2132 
2133     ui[k+1] = ui[k] + nzk;
2134   }
2135 
2136 #if defined(PETSC_USE_INFO)
2137   if (ai[am] != 0) {
2138     PetscReal af = (PetscReal)(ui[am])/((PetscReal)ai[am]);
2139     ierr = PetscInfo3(A,"Reallocs %D Fill ratio:given %G needed %G\n",reallocs,fill,af);CHKERRQ(ierr);
2140     ierr = PetscInfo1(A,"Run with -pc_factor_fill %G or use \n",af);CHKERRQ(ierr);
2141     ierr = PetscInfo1(A,"PCFactorSetFill(pc,%G) for best performance.\n",af);CHKERRQ(ierr);
2142   } else {
2143      ierr = PetscInfo(A,"Empty matrix.\n");CHKERRQ(ierr);
2144   }
2145 #endif
2146 
2147   ierr = ISRestoreIndices(perm,&rip);CHKERRQ(ierr);
2148   ierr = ISRestoreIndices(iperm,&riip);CHKERRQ(ierr);
2149   ierr = PetscFree(jl);CHKERRQ(ierr);
2150 
2151   /* destroy list of free space and other temporary array(s) */
2152   ierr = PetscMalloc((ui[am]+1)*sizeof(PetscInt),&uj);CHKERRQ(ierr);
2153   ierr = PetscFreeSpaceContiguous(&free_space,uj);CHKERRQ(ierr);
2154   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
2155 
2156   /* put together the new matrix in MATSEQSBAIJ format */
2157 
2158   b = (Mat_SeqSBAIJ*)(fact)->data;
2159   b->singlemalloc = PETSC_FALSE;
2160   b->free_a       = PETSC_TRUE;
2161   b->free_ij      = PETSC_TRUE;
2162   ierr = PetscMalloc((ui[am]+1)*sizeof(MatScalar),&b->a);CHKERRQ(ierr);
2163   b->j    = uj;
2164   b->i    = ui;
2165   b->diag = 0;
2166   b->ilen = 0;
2167   b->imax = 0;
2168   b->row  = perm;
2169   b->col  = perm;
2170   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2171   ierr    = PetscObjectReference((PetscObject)perm);CHKERRQ(ierr);
2172   b->icol = iperm;
2173   b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */
2174   ierr    = PetscMalloc((am+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
2175   ierr    = PetscLogObjectMemory(fact,(ui[am]-am)*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
2176   b->maxnz = b->nz = ui[am];
2177 
2178   (fact)->info.factor_mallocs    = reallocs;
2179   (fact)->info.fill_ratio_given  = fill;
2180   if (ai[am] != 0) {
2181     (fact)->info.fill_ratio_needed = ((PetscReal)ui[am])/((PetscReal)ai[am]);
2182   } else {
2183     (fact)->info.fill_ratio_needed = 0.0;
2184   }
2185   (fact)->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqAIJ;
2186   PetscFunctionReturn(0);
2187 }
2188 
2189 #undef __FUNCT__
2190 #define __FUNCT__ "MatSolve_SeqAIJ_NaturalOrdering_iludt"
2191 PetscErrorCode MatSolve_SeqAIJ_NaturalOrdering_iludt(Mat A,Vec bb,Vec xx)
2192 {
2193   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
2194   PetscErrorCode    ierr;
2195   PetscInt          n = A->rmap->n;
2196   const PetscInt    *ai = a->i,*aj = a->j,*vi;
2197   PetscScalar       *x,sum;
2198   const PetscScalar *b;
2199   const MatScalar   *aa = a->a,*v;
2200   PetscInt          i,nz;
2201 
2202   PetscFunctionBegin;
2203   if (!n) PetscFunctionReturn(0);
2204 
2205   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
2206   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
2207 
2208   /* forward solve the lower triangular */
2209   x[0] = b[0];
2210   v    = aa;
2211   vi   = aj;
2212   for (i=1; i<n; i++) {
2213     nz  = ai[i+1] - ai[i];
2214     sum = b[i];
2215     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
2216     /*    while (nz--) sum -= *v++ * x[*vi++];*/
2217     v  += nz;
2218     vi += nz;
2219     x[i] = sum;
2220   }
2221 
2222   /* backward solve the upper triangular */
2223   v   = aa + ai[n+1];
2224   vi  = aj + ai[n+1];
2225   for (i=n-1; i>=0; i--){
2226     nz = ai[2*n-i +1] - ai[2*n-i]-1;
2227     sum = x[i];
2228     PetscSparseDenseMinusDot(sum,x,v,vi,nz);
2229     /* while (nz--) sum -= *v++ * x[*vi++]; */
2230     v   += nz;
2231     vi  += nz; vi++;
2232     x[i] = *v++ *sum; /* x[i]=aa[adiag[i]]*sum; v++; */
2233   }
2234 
2235   ierr = PetscLogFlops(2.0*a->nz - A->cmap->n);CHKERRQ(ierr);
2236   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
2237   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
2238   PetscFunctionReturn(0);
2239 }
2240 
2241 #undef __FUNCT__
2242 #define __FUNCT__ "MatSolve_SeqAIJ_iludt"
2243 PetscErrorCode MatSolve_SeqAIJ_iludt(Mat A,Vec bb,Vec xx)
2244 {
2245   Mat_SeqAIJ        *a = (Mat_SeqAIJ*)A->data;
2246   IS                iscol = a->col,isrow = a->row;
2247   PetscErrorCode    ierr;
2248   PetscInt          i,n=A->rmap->n,*vi,*ai = a->i,*aj = a->j,*adiag=a->diag;
2249   PetscInt          nz;
2250   const PetscInt    *rout,*cout,*r,*c;
2251   PetscScalar       *x,*tmp,*tmps;
2252   const PetscScalar *b;
2253   const MatScalar   *aa = a->a,*v;
2254 
2255   PetscFunctionBegin;
2256   if (!n) PetscFunctionReturn(0);
2257 
2258   ierr = VecGetArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
2259   ierr = VecGetArray(xx,&x);CHKERRQ(ierr);
2260   tmp  = a->solve_work;
2261 
2262   ierr = ISGetIndices(isrow,&rout);CHKERRQ(ierr); r = rout;
2263   ierr = ISGetIndices(iscol,&cout);CHKERRQ(ierr); c = cout + (n-1);
2264 
2265   /* forward solve the lower triangular */
2266   tmp[0] = b[*r++];
2267   tmps   = tmp;
2268   v      = aa;
2269   vi     = aj;
2270   for (i=1; i<n; i++) {
2271     nz  = ai[i+1] - ai[i];
2272     tmp[i] = b[*r++];
2273     PetscSparseDenseMinusDot(tmp[i],tmps,v,vi,nz);
2274     v += nz; vi += nz;
2275   }
2276 
2277   /* backward solve the upper triangular */
2278   v   = aa + adiag[n] + 1;
2279   vi  = aj + adiag[n] + 1;
2280   for (i=n-1; i>=0; i--){
2281     nz  = adiag[i] - adiag[i+1] - 1;
2282     PetscSparseDenseMinusDot(tmp[i],tmps,v,vi,nz);
2283     x[*c--] = tmp[i] = tmp[i]*aa[adiag[i]];
2284     v += nz+1; vi += nz+1;
2285   }
2286 
2287   ierr = ISRestoreIndices(isrow,&rout);CHKERRQ(ierr);
2288   ierr = ISRestoreIndices(iscol,&cout);CHKERRQ(ierr);
2289   ierr = VecRestoreArray(bb,(PetscScalar**)&b);CHKERRQ(ierr);
2290   ierr = VecRestoreArray(xx,&x);CHKERRQ(ierr);
2291   ierr = PetscLogFlops(2*a->nz - A->cmap->n);CHKERRQ(ierr);
2292   PetscFunctionReturn(0);
2293 }
2294 
2295 #undef __FUNCT__
2296 #define __FUNCT__ "MatILUDTFactor_SeqAIJ"
2297 PetscErrorCode MatILUDTFactor_SeqAIJ(Mat A,IS isrow,IS iscol,const MatFactorInfo *info,Mat *fact)
2298 {
2299   Mat                B = *fact;
2300   Mat_SeqAIJ         *a=(Mat_SeqAIJ*)A->data,*b;
2301   IS                 isicol;
2302   PetscErrorCode     ierr;
2303   const PetscInt     *r,*ic;
2304   PetscInt           i,n=A->rmap->n,*ai=a->i,*aj=a->j,*ajtmp,*adiag;
2305   PetscInt           *bi,*bj,*bdiag,*bdiag_rev;
2306   PetscInt           row,nzi,nzi_bl,nzi_bu,*im,dtcount,nzi_al,nzi_au;
2307   PetscInt           nlnk,*lnk;
2308   PetscBT            lnkbt;
2309   PetscTruth         row_identity,icol_identity,both_identity;
2310   MatScalar          *aatmp,*pv,*batmp,*ba,*rtmp,*pc,multiplier,*vtmp,diag_tmp;
2311   const PetscInt     *ics;
2312   PetscInt           j,nz,*pj,*bjtmp,k,ncut,*jtmp;
2313   PetscReal          dt=info->dt,shift=info->shiftinblocks;
2314   PetscInt           nnz_max;
2315   PetscTruth         missing;
2316 
2317   PetscFunctionBegin;
2318   /* ------- symbolic factorization, can be reused ---------*/
2319   ierr = MatMissingDiagonal(A,&missing,&i);CHKERRQ(ierr);
2320   if (missing) SETERRQ1(PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",i);
2321   adiag=a->diag;
2322 
2323   ierr = ISInvertPermutation(iscol,PETSC_DECIDE,&isicol);CHKERRQ(ierr);
2324 
2325   /* bdiag is location of diagonal in factor */
2326   ierr = PetscMalloc((2*n+2)*sizeof(PetscInt),&bdiag);CHKERRQ(ierr);
2327   bdiag_rev = bdiag + n+1;
2328 
2329   /* allocate row pointers bi */
2330   ierr = PetscMalloc((2*n+2)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
2331 
2332   /* allocate bj and ba; max num of nonzero entries is (ai[n]+2*n*dtcount+2) */
2333   dtcount = (PetscInt)info->dtcount;
2334   if (dtcount > n-1) dtcount = n-1; /* diagonal is excluded */
2335   nnz_max  = ai[n]+2*n*dtcount+2;
2336 
2337   ierr = PetscMalloc((nnz_max+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
2338   ierr = PetscMalloc((nnz_max+1)*sizeof(MatScalar),&ba);CHKERRQ(ierr);
2339 
2340   /* put together the new matrix */
2341   ierr = MatSeqAIJSetPreallocation_SeqAIJ(B,MAT_SKIP_ALLOCATION,PETSC_NULL);CHKERRQ(ierr);
2342   ierr = PetscLogObjectParent(B,isicol);CHKERRQ(ierr);
2343   b    = (Mat_SeqAIJ*)(B)->data;
2344   b->free_a       = PETSC_TRUE;
2345   b->free_ij      = PETSC_TRUE;
2346   b->singlemalloc = PETSC_FALSE;
2347   b->a          = ba;
2348   b->j          = bj;
2349   b->i          = bi;
2350   b->diag       = bdiag;
2351   b->ilen       = 0;
2352   b->imax       = 0;
2353   b->row        = isrow;
2354   b->col        = iscol;
2355   ierr          = PetscObjectReference((PetscObject)isrow);CHKERRQ(ierr);
2356   ierr          = PetscObjectReference((PetscObject)iscol);CHKERRQ(ierr);
2357   b->icol       = isicol;
2358   ierr = PetscMalloc((n+1)*sizeof(PetscScalar),&b->solve_work);CHKERRQ(ierr);
2359 
2360   ierr = PetscLogObjectMemory(B,nnz_max*(sizeof(PetscInt)+sizeof(MatScalar)));CHKERRQ(ierr);
2361   b->maxnz = nnz_max;
2362 
2363   (B)->factor                = MAT_FACTOR_ILUDT;
2364   (B)->info.factor_mallocs   = 0;
2365   (B)->info.fill_ratio_given = ((PetscReal)nnz_max)/((PetscReal)ai[n]);
2366   CHKMEMQ;
2367   /* ------- end of symbolic factorization ---------*/
2368 
2369   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
2370   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
2371   ics  = ic;
2372 
2373   /* linked list for storing column indices of the active row */
2374   nlnk = n + 1;
2375   ierr = PetscLLCreate(n,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2376 
2377   /* im: used by PetscLLAddSortedLU(); jtmp: working array for column indices of active row */
2378   ierr = PetscMalloc((2*n+1)*sizeof(PetscInt),&im);CHKERRQ(ierr);
2379   jtmp = im + n;
2380   /* rtmp, vtmp: working arrays for sparse and contiguous row entries of active row */
2381   ierr = PetscMalloc((2*n+1)*sizeof(MatScalar),&rtmp);CHKERRQ(ierr);
2382   ierr = PetscMemzero(rtmp,(n+1)*sizeof(PetscScalar));CHKERRQ(ierr);
2383   vtmp = rtmp + n;
2384 
2385   bi[0]    = 0;
2386   bdiag[0] = nnz_max-1; /* location of diag[0] in factor B */
2387   bdiag_rev[n] = bdiag[0];
2388   bi[2*n+1] = bdiag[0]+1; /* endof bj and ba array */
2389   for (i=0; i<n; i++) {
2390     /* copy initial fill into linked list */
2391     nzi = 0; /* nonzeros for active row i */
2392     nzi = ai[r[i]+1] - ai[r[i]];
2393     if (!nzi) SETERRQ2(PETSC_ERR_MAT_LU_ZRPVT,"Empty row in matrix: row in original ordering %D in permuted ordering %D",r[i],i);
2394     nzi_al = adiag[r[i]] - ai[r[i]];
2395     nzi_au = ai[r[i]+1] - adiag[r[i]] -1;
2396     ajtmp = aj + ai[r[i]];
2397     ierr = PetscLLAddPerm(nzi,ajtmp,ic,n,nlnk,lnk,lnkbt);CHKERRQ(ierr);
2398 
2399     /* load in initial (unfactored row) */
2400     aatmp = a->a + ai[r[i]];
2401     for (j=0; j<nzi; j++) {
2402       rtmp[ics[*ajtmp++]] = *aatmp++;
2403     }
2404 
2405     /* add pivot rows into linked list */
2406     row = lnk[n];
2407     while (row < i ) {
2408       nzi_bl = bi[row+1] - bi[row] + 1;
2409       bjtmp = bj + bdiag[row+1]+1; /* points to 1st column next to the diagonal in U */
2410       ierr  = PetscLLAddSortedLU(bjtmp,row,nlnk,lnk,lnkbt,i,nzi_bl,im);CHKERRQ(ierr);
2411       nzi  += nlnk;
2412       row   = lnk[row];
2413     }
2414 
2415     /* copy data from lnk into jtmp, then initialize lnk */
2416     ierr = PetscLLClean(n,n,nzi,lnk,jtmp,lnkbt);CHKERRQ(ierr);
2417 
2418     /* numerical factorization */
2419     bjtmp = jtmp;
2420     row   = *bjtmp++; /* 1st pivot row */
2421     while  ( row < i ) {
2422       pc         = rtmp + row;
2423       pv         = ba + bdiag[row]; /* 1./(diag of the pivot row) */
2424       multiplier = (*pc) * (*pv);
2425       *pc        = multiplier;
2426       if (PetscAbsScalar(*pc) > dt){ /* apply tolerance dropping rule */
2427         pj         = bj + bdiag[row+1] + 1; /* point to 1st entry of U(row,:) */
2428         pv         = ba + bdiag[row+1] + 1;
2429         /* if (multiplier < -1.0 or multiplier >1.0) printf("row/prow %d, %d, multiplier %g\n",i,row,multiplier); */
2430         nz         = bdiag[row] - bdiag[row+1] - 1; /* num of entries in U(row,:), excluding diagonal */
2431         for (j=0; j<nz; j++) rtmp[*pj++] -= multiplier * (*pv++);
2432         ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr);
2433       }
2434       row = *bjtmp++;
2435     }
2436 
2437     /* copy sparse rtmp into contiguous vtmp; separate L and U part */
2438     diag_tmp = rtmp[i];  /* save diagonal value - may not needed?? */
2439     nzi_bl = 0; j = 0;
2440     while (jtmp[j] < i){ /* Note: jtmp is sorted */
2441       vtmp[j] = rtmp[jtmp[j]]; rtmp[jtmp[j]]=0.0;
2442       nzi_bl++; j++;
2443     }
2444     nzi_bu = nzi - nzi_bl -1;
2445     while (j < nzi){
2446       vtmp[j] = rtmp[jtmp[j]]; rtmp[jtmp[j]]=0.0;
2447       j++;
2448     }
2449 
2450     bjtmp = bj + bi[i];
2451     batmp = ba + bi[i];
2452     /* apply level dropping rule to L part */
2453     ncut = nzi_al + dtcount;
2454     if (ncut < nzi_bl){
2455       ierr = PetscSortSplit(ncut,nzi_bl,vtmp,jtmp);CHKERRQ(ierr);
2456       ierr = PetscSortIntWithScalarArray(ncut,jtmp,vtmp);CHKERRQ(ierr);
2457     } else {
2458       ncut = nzi_bl;
2459     }
2460     for (j=0; j<ncut; j++){
2461       bjtmp[j] = jtmp[j];
2462       batmp[j] = vtmp[j];
2463       /* printf(" (%d,%g),",bjtmp[j],batmp[j]); */
2464     }
2465     bi[i+1] = bi[i] + ncut;
2466     nzi = ncut + 1;
2467 
2468     /* apply level dropping rule to U part */
2469     ncut = nzi_au + dtcount;
2470     if (ncut < nzi_bu){
2471       ierr = PetscSortSplit(ncut,nzi_bu,vtmp+nzi_bl+1,jtmp+nzi_bl+1);CHKERRQ(ierr);
2472       ierr = PetscSortIntWithScalarArray(ncut,jtmp+nzi_bl+1,vtmp+nzi_bl+1);CHKERRQ(ierr);
2473     } else {
2474       ncut = nzi_bu;
2475     }
2476     nzi += ncut;
2477 
2478     /* mark bdiagonal */
2479     bdiag[i+1]       = bdiag[i] - (ncut + 1);
2480     bdiag_rev[n-i-1] = bdiag[i+1];
2481     bi[2*n - i]      = bi[2*n - i +1] - (ncut + 1);
2482     bjtmp = bj + bdiag[i];
2483     batmp = ba + bdiag[i];
2484     *bjtmp = i;
2485     *batmp = diag_tmp; /* rtmp[i]; */
2486     if (*batmp == 0.0) {
2487       *batmp = dt+shift;
2488       /* printf(" row %d add shift %g\n",i,shift); */
2489     }
2490     *batmp = 1.0/(*batmp); /* invert diagonal entries for simplier triangular solves */
2491     /* printf(" (%d,%g),",*bjtmp,*batmp); */
2492 
2493     bjtmp = bj + bdiag[i+1]+1;
2494     batmp = ba + bdiag[i+1]+1;
2495     for (k=0; k<ncut; k++){
2496       bjtmp[k] = jtmp[nzi_bl+1+k];
2497       batmp[k] = vtmp[nzi_bl+1+k];
2498       /* printf(" (%d,%g),",bjtmp[k],batmp[k]); */
2499     }
2500     /* printf("\n"); */
2501 
2502     im[i]   = nzi; /* used by PetscLLAddSortedLU() */
2503     /*
2504     printf("row %d: bi %d, bdiag %d\n",i,bi[i],bdiag[i]);
2505     printf(" ----------------------------\n");
2506     */
2507   } /* for (i=0; i<n; i++) */
2508   /* printf("end of L %d, beginning of U %d\n",bi[n],bdiag[n]); */
2509   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]);
2510 
2511   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
2512   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
2513 
2514   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
2515   ierr = PetscFree(im);CHKERRQ(ierr);
2516   ierr = PetscFree(rtmp);CHKERRQ(ierr);
2517 
2518   ierr = PetscLogFlops(B->cmap->n);CHKERRQ(ierr);
2519   b->maxnz = b->nz = bi[n] + bdiag[0] - bdiag[n];
2520 
2521   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
2522   ierr = ISIdentity(isicol,&icol_identity);CHKERRQ(ierr);
2523   both_identity = (PetscTruth) (row_identity && icol_identity);
2524   if (row_identity && icol_identity) {
2525     B->ops->solve = MatSolve_SeqAIJ_NaturalOrdering_iludt;
2526   } else {
2527     B->ops->solve = MatSolve_SeqAIJ_iludt;
2528   }
2529 
2530   B->ops->lufactorsymbolic  = MatILUDTFactorSymbolic_SeqAIJ;
2531   B->ops->lufactornumeric   = MatILUDTFactorNumeric_SeqAIJ;
2532   B->ops->solveadd          = 0;
2533   B->ops->solvetranspose    = 0;
2534   B->ops->solvetransposeadd = 0;
2535   B->ops->matsolve          = 0;
2536   B->assembled              = PETSC_TRUE;
2537   B->preallocated           = PETSC_TRUE;
2538   PetscFunctionReturn(0);
2539 }
2540 
2541 /* a wraper of MatILUDTFactor_SeqAIJ() */
2542 #undef __FUNCT__
2543 #define __FUNCT__ "MatILUDTFactorSymbolic_SeqAIJ"
2544 PetscErrorCode PETSCMAT_DLLEXPORT MatILUDTFactorSymbolic_SeqAIJ(Mat fact,Mat A,IS row,IS col,const MatFactorInfo *info)
2545 {
2546   PetscErrorCode     ierr;
2547 
2548   PetscFunctionBegin;
2549   ierr = MatILUDTFactor_SeqAIJ(A,row,col,info,&fact);CHKERRQ(ierr);
2550 
2551   fact->ops->lufactornumeric = MatILUDTFactorNumeric_SeqAIJ;
2552   PetscFunctionReturn(0);
2553 }
2554 
2555 /*
2556    same as MatLUFactorNumeric_SeqAIJ(), except using contiguous array matrix factors
2557    - intend to replace existing MatLUFactorNumeric_SeqAIJ()
2558 */
2559 #undef __FUNCT__
2560 #define __FUNCT__ "MatILUDTFactorNumeric_SeqAIJ"
2561 PetscErrorCode PETSCMAT_DLLEXPORT MatILUDTFactorNumeric_SeqAIJ(Mat fact,Mat A,const MatFactorInfo *info)
2562 {
2563   Mat            C=fact;
2564   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ *)C->data;
2565   IS             isrow = b->row,isicol = b->icol;
2566   PetscErrorCode ierr;
2567   const PetscInt *r,*ic,*ics;
2568   PetscInt       i,j,k,n=A->rmap->n,*ai=a->i,*aj=a->j,*bi=b->i,*bj=b->j;
2569   PetscInt       *ajtmp,*bjtmp,nz,nzl,nzu,row,*bdiag = b->diag,*pj;
2570   MatScalar      *rtmp,*pc,multiplier,*v,*pv,*aa=a->a;
2571   PetscReal      dt=info->dt,shift=info->shiftinblocks;
2572   PetscTruth     row_identity, col_identity;
2573 
2574   PetscFunctionBegin;
2575   ierr = ISGetIndices(isrow,&r);CHKERRQ(ierr);
2576   ierr = ISGetIndices(isicol,&ic);CHKERRQ(ierr);
2577   ierr = PetscMalloc((n+1)*sizeof(MatScalar),&rtmp);CHKERRQ(ierr);
2578   ics  = ic;
2579 
2580   for (i=0; i<n; i++){
2581     /* initialize rtmp array */
2582     nzl   = bi[i+1] - bi[i];       /* num of nozeros in L(i,:) */
2583     bjtmp = bj + bi[i];
2584     for  (j=0; j<nzl; j++) rtmp[*bjtmp++] = 0.0;
2585     rtmp[i] = 0.0;
2586     nzu   = bdiag[i] - bdiag[i+1]; /* num of nozeros in U(i,:) */
2587     bjtmp = bj + bdiag[i+1] + 1;
2588     for  (j=0; j<nzu; j++) rtmp[*bjtmp++] = 0.0;
2589 
2590     /* load in initial unfactored row of A */
2591     /* printf("row %d\n",i); */
2592     nz    = ai[r[i]+1] - ai[r[i]];
2593     ajtmp = aj + ai[r[i]];
2594     v     = aa + ai[r[i]];
2595     for (j=0; j<nz; j++) {
2596       rtmp[ics[*ajtmp++]] = v[j];
2597       /* printf(" (%d,%g),",ics[ajtmp[j]],rtmp[ics[ajtmp[j]]]); */
2598     }
2599     /* printf("\n"); */
2600 
2601     /* numerical factorization */
2602     bjtmp = bj + bi[i]; /* point to 1st entry of L(i,:) */
2603     nzl   = bi[i+1] - bi[i]; /* num of entries in L(i,:) */
2604     k = 0;
2605     while (k < nzl){
2606       row   = *bjtmp++;
2607       /* printf("  prow %d\n",row); */
2608       pc         = rtmp + row;
2609       pv         = b->a + bdiag[row]; /* 1./(diag of the pivot row) */
2610       multiplier = (*pc) * (*pv);
2611       *pc        = multiplier;
2612       if (PetscAbsScalar(multiplier) > dt){
2613         pj         = bj + bdiag[row+1] + 1; /* point to 1st entry of U(row,:) */
2614         pv         = b->a + bdiag[row+1] + 1;
2615         nz         = bdiag[row] - bdiag[row+1] - 1; /* num of entries in U(row,:), excluding diagonal */
2616         for (j=0; j<nz; j++) rtmp[*pj++] -= multiplier * (*pv++);
2617         /* ierr = PetscLogFlops(2.0*nz);CHKERRQ(ierr); */
2618       }
2619       k++;
2620     }
2621 
2622     /* finished row so stick it into b->a */
2623     /* L-part */
2624     pv = b->a + bi[i] ;
2625     pj = bj + bi[i] ;
2626     nzl = bi[i+1] - bi[i];
2627     for (j=0; j<nzl; j++) {
2628       pv[j] = rtmp[pj[j]];
2629       /* printf(" (%d,%g),",pj[j],pv[j]); */
2630     }
2631 
2632     /* diagonal: invert diagonal entries for simplier triangular solves */
2633     if (rtmp[i] == 0.0) rtmp[i] = dt+shift;
2634     b->a[bdiag[i]] = 1.0/rtmp[i];
2635     /* printf(" (%d,%g),",i,b->a[bdiag[i]]); */
2636 
2637     /* U-part */
2638     pv = b->a + bdiag[i+1] + 1;
2639     pj = bj + bdiag[i+1] + 1;
2640     nzu = bdiag[i] - bdiag[i+1] - 1;
2641     for (j=0; j<nzu; j++) {
2642       pv[j] = rtmp[pj[j]];
2643       /* printf(" (%d,%g),",pj[j],pv[j]); */
2644     }
2645     /* printf("\n"); */
2646   }
2647 
2648   ierr = PetscFree(rtmp);CHKERRQ(ierr);
2649   ierr = ISRestoreIndices(isicol,&ic);CHKERRQ(ierr);
2650   ierr = ISRestoreIndices(isrow,&r);CHKERRQ(ierr);
2651 
2652   ierr = ISIdentity(isrow,&row_identity);CHKERRQ(ierr);
2653   ierr = ISIdentity(isicol,&col_identity);CHKERRQ(ierr);
2654   if (row_identity && col_identity) {
2655     C->ops->solve   = MatSolve_SeqAIJ_NaturalOrdering_iludt;
2656   } else {
2657     C->ops->solve   = MatSolve_SeqAIJ_iludt;
2658   }
2659   C->ops->solveadd           = 0;
2660   C->ops->solvetranspose     = 0;
2661   C->ops->solvetransposeadd  = 0;
2662   C->ops->matsolve           = 0;
2663   C->assembled    = PETSC_TRUE;
2664   C->preallocated = PETSC_TRUE;
2665   ierr = PetscLogFlops(C->cmap->n);CHKERRQ(ierr);
2666   PetscFunctionReturn(0);
2667 }
2668