xref: /petsc/src/mat/impls/aij/seq/matmatmult.c (revision 38cfc46e4752cf65941cec0be9bac4d1d6832cc8)
1 
2 /*
3   Defines matrix-matrix product routines for pairs of SeqAIJ matrices
4           C = A * B
5 */
6 
7 #include <../src/mat/impls/aij/seq/aij.h> /*I "petscmat.h" I*/
8 #include <../src/mat/utils/freespace.h>
9 #include <petscbt.h>
10 #include <petsc/private/isimpl.h>
11 #include <../src/mat/impls/dense/seq/dense.h>
12 
13 
14 PETSC_INTERN PetscErrorCode MatMatMult_SeqAIJ_SeqAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
15 {
16   PetscErrorCode ierr;
17 
18   PetscFunctionBegin;
19   if (scall == MAT_INITIAL_MATRIX) {
20     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
21     ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(A,B,fill,C);CHKERRQ(ierr);
22     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
23   }
24   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
25   ierr = MatMatMultNumeric_SeqAIJ_SeqAIJ(A,B,*C);CHKERRQ(ierr);
26   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
27   PetscFunctionReturn(0);
28 }
29 
30 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ(Mat A,Mat B,Mat C)
31 {
32   PetscErrorCode ierr;
33 
34   PetscFunctionBegin;
35   if (C->ops->matmultnumeric) {
36     ierr = (*C->ops->matmultnumeric)(A,B,C);CHKERRQ(ierr);
37   } else {
38     ierr = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted(A,B,C);CHKERRQ(ierr);
39   }
40   PetscFunctionReturn(0);
41 }
42 
43 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ(Mat A,Mat B,PetscReal fill,Mat *C)
44 {
45   PetscErrorCode ierr;
46 #if !defined(PETSC_HAVE_HYPRE)
47   const char     *algTypes[8] = {"sorted","scalable","scalable_fast","heap","btheap","llcondensed","combined","rowmerge"};
48   PetscInt       nalg = 8;
49 #else
50   const char     *algTypes[9] = {"sorted","scalable","scalable_fast","heap","btheap","llcondensed","combined","rowmerge","hypre"};
51   PetscInt       nalg = 9;
52 #endif
53   PetscInt       alg = 0; /* set default algorithm */
54 
55   PetscFunctionBegin;
56   ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)A),((PetscObject)A)->prefix,"MatMatMult","Mat");CHKERRQ(ierr);
57   ierr = PetscOptionsEList("-matmatmult_via","Algorithmic approach","MatMatMult",algTypes,nalg,algTypes[0],&alg,NULL);CHKERRQ(ierr);
58   ierr = PetscOptionsEnd();CHKERRQ(ierr);
59   switch (alg) {
60   case 1:
61     ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable(A,B,fill,C);CHKERRQ(ierr);
62     break;
63   case 2:
64     ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable_fast(A,B,fill,C);CHKERRQ(ierr);
65     break;
66   case 3:
67     ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Heap(A,B,fill,C);CHKERRQ(ierr);
68     break;
69   case 4:
70     ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_BTHeap(A,B,fill,C);CHKERRQ(ierr);
71     break;
72   case 5:
73     ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_LLCondensed(A,B,fill,C);CHKERRQ(ierr);
74     break;
75   case 6:
76     ierr = MatMatMult_SeqAIJ_SeqAIJ_Combined(A,B,fill,C);CHKERRQ(ierr);
77     break;
78   case 7:
79     ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_RowMerge(A,B,fill,C);CHKERRQ(ierr);
80     break;
81 #if defined(PETSC_HAVE_HYPRE)
82   case 8:
83     ierr = MatMatMultSymbolic_AIJ_AIJ_wHYPRE(A,B,fill,C);CHKERRQ(ierr);
84     break;
85 #endif
86   default:
87     ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Sorted(A,B,fill,C);CHKERRQ(ierr);
88     break;
89   }
90   PetscFunctionReturn(0);
91 }
92 
93 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_LLCondensed(Mat A,Mat B,PetscReal fill,Mat *C)
94 {
95   PetscErrorCode     ierr;
96   Mat_SeqAIJ         *a =(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
97   PetscInt           *ai=a->i,*bi=b->i,*ci,*cj;
98   PetscInt           am =A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
99   PetscReal          afill;
100   PetscInt           i,j,anzi,brow,bnzj,cnzi,*bj,*aj,*lnk,ndouble=0,Crmax;
101   PetscTable         ta;
102   PetscBT            lnkbt;
103   PetscFreeSpaceList free_space=NULL,current_space=NULL;
104 
105   PetscFunctionBegin;
106   /* Get ci and cj */
107   /*---------------*/
108   /* Allocate ci array, arrays for fill computation and */
109   /* free space for accumulating nonzero column info */
110   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
111   ci[0] = 0;
112 
113   /* create and initialize a linked list */
114   ierr = PetscTableCreate(bn,bn,&ta);CHKERRQ(ierr);
115   MatRowMergeMax_SeqAIJ(b,bm,ta);
116   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
117   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
118 
119   ierr = PetscLLCondensedCreate(Crmax,bn,&lnk,&lnkbt);CHKERRQ(ierr);
120 
121   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
122   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
123 
124   current_space = free_space;
125 
126   /* Determine ci and cj */
127   for (i=0; i<am; i++) {
128     anzi = ai[i+1] - ai[i];
129     aj   = a->j + ai[i];
130     for (j=0; j<anzi; j++) {
131       brow = aj[j];
132       bnzj = bi[brow+1] - bi[brow];
133       bj   = b->j + bi[brow];
134       /* add non-zero cols of B into the sorted linked list lnk */
135       ierr = PetscLLCondensedAddSorted(bnzj,bj,lnk,lnkbt);CHKERRQ(ierr);
136     }
137     cnzi = lnk[0];
138 
139     /* If free space is not available, make more free space */
140     /* Double the amount of total space in the list */
141     if (current_space->local_remaining<cnzi) {
142       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(cnzi,current_space->total_array_size),&current_space);CHKERRQ(ierr);
143       ndouble++;
144     }
145 
146     /* Copy data into free space, then initialize lnk */
147     ierr = PetscLLCondensedClean(bn,cnzi,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
148 
149     current_space->array           += cnzi;
150     current_space->local_used      += cnzi;
151     current_space->local_remaining -= cnzi;
152 
153     ci[i+1] = ci[i] + cnzi;
154   }
155 
156   /* Column indices are in the list of free space */
157   /* Allocate space for cj, initialize cj, and */
158   /* destroy list of free space and other temporary array(s) */
159   ierr = PetscMalloc1(ci[am]+1,&cj);CHKERRQ(ierr);
160   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
161   ierr = PetscLLCondensedDestroy(lnk,lnkbt);CHKERRQ(ierr);
162 
163   /* put together the new symbolic matrix */
164   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
165   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
166   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
167 
168   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
169   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
170   c                         = (Mat_SeqAIJ*)((*C)->data);
171   c->free_a                 = PETSC_FALSE;
172   c->free_ij                = PETSC_TRUE;
173   c->nonew                  = 0;
174   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted; /* fast, needs non-scalable O(bn) array 'abdense' */
175 
176   /* set MatInfo */
177   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
178   if (afill < 1.0) afill = 1.0;
179   c->maxnz                     = ci[am];
180   c->nz                        = ci[am];
181   (*C)->info.mallocs           = ndouble;
182   (*C)->info.fill_ratio_given  = fill;
183   (*C)->info.fill_ratio_needed = afill;
184 
185 #if defined(PETSC_USE_INFO)
186   if (ci[am]) {
187     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
188     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
189   } else {
190     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
191   }
192 #endif
193   PetscFunctionReturn(0);
194 }
195 
196 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted(Mat A,Mat B,Mat C)
197 {
198   PetscErrorCode ierr;
199   PetscLogDouble flops=0.0;
200   Mat_SeqAIJ     *a   = (Mat_SeqAIJ*)A->data;
201   Mat_SeqAIJ     *b   = (Mat_SeqAIJ*)B->data;
202   Mat_SeqAIJ     *c   = (Mat_SeqAIJ*)C->data;
203   PetscInt       *ai  =a->i,*aj=a->j,*bi=b->i,*bj=b->j,*bjj,*ci=c->i,*cj=c->j;
204   PetscInt       am   =A->rmap->n,cm=C->rmap->n;
205   PetscInt       i,j,k,anzi,bnzi,cnzi,brow;
206   PetscScalar    *aa=a->a,*ba=b->a,*baj,*ca,valtmp;
207   PetscScalar    *ab_dense;
208 
209   PetscFunctionBegin;
210   if (!c->a) { /* first call of MatMatMultNumeric_SeqAIJ_SeqAIJ, allocate ca and matmult_abdense */
211     ierr      = PetscMalloc1(ci[cm]+1,&ca);CHKERRQ(ierr);
212     c->a      = ca;
213     c->free_a = PETSC_TRUE;
214   } else {
215     ca        = c->a;
216   }
217   if (!c->matmult_abdense) {
218     ierr = PetscCalloc1(B->cmap->N,&ab_dense);CHKERRQ(ierr);
219     c->matmult_abdense = ab_dense;
220   } else {
221     ab_dense = c->matmult_abdense;
222   }
223 
224   /* clean old values in C */
225   ierr = PetscArrayzero(ca,ci[cm]);CHKERRQ(ierr);
226   /* Traverse A row-wise. */
227   /* Build the ith row in C by summing over nonzero columns in A, */
228   /* the rows of B corresponding to nonzeros of A. */
229   for (i=0; i<am; i++) {
230     anzi = ai[i+1] - ai[i];
231     for (j=0; j<anzi; j++) {
232       brow = aj[j];
233       bnzi = bi[brow+1] - bi[brow];
234       bjj  = bj + bi[brow];
235       baj  = ba + bi[brow];
236       /* perform dense axpy */
237       valtmp = aa[j];
238       for (k=0; k<bnzi; k++) {
239         ab_dense[bjj[k]] += valtmp*baj[k];
240       }
241       flops += 2*bnzi;
242     }
243     aj += anzi; aa += anzi;
244 
245     cnzi = ci[i+1] - ci[i];
246     for (k=0; k<cnzi; k++) {
247       ca[k]          += ab_dense[cj[k]];
248       ab_dense[cj[k]] = 0.0; /* zero ab_dense */
249     }
250     flops += cnzi;
251     cj    += cnzi; ca += cnzi;
252   }
253   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
254   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
255   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
256   PetscFunctionReturn(0);
257 }
258 
259 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable(Mat A,Mat B,Mat C)
260 {
261   PetscErrorCode ierr;
262   PetscLogDouble flops=0.0;
263   Mat_SeqAIJ     *a   = (Mat_SeqAIJ*)A->data;
264   Mat_SeqAIJ     *b   = (Mat_SeqAIJ*)B->data;
265   Mat_SeqAIJ     *c   = (Mat_SeqAIJ*)C->data;
266   PetscInt       *ai  = a->i,*aj=a->j,*bi=b->i,*bj=b->j,*bjj,*ci=c->i,*cj=c->j;
267   PetscInt       am   = A->rmap->N,cm=C->rmap->N;
268   PetscInt       i,j,k,anzi,bnzi,cnzi,brow;
269   PetscScalar    *aa=a->a,*ba=b->a,*baj,*ca=c->a,valtmp;
270   PetscInt       nextb;
271 
272   PetscFunctionBegin;
273   if (!ca) { /* first call of MatMatMultNumeric_SeqAIJ_SeqAIJ, allocate ca and matmult_abdense */
274     ierr      = PetscMalloc1(ci[cm]+1,&ca);CHKERRQ(ierr);
275     c->a      = ca;
276     c->free_a = PETSC_TRUE;
277   }
278 
279   /* clean old values in C */
280   ierr = PetscArrayzero(ca,ci[cm]);CHKERRQ(ierr);
281   /* Traverse A row-wise. */
282   /* Build the ith row in C by summing over nonzero columns in A, */
283   /* the rows of B corresponding to nonzeros of A. */
284   for (i=0; i<am; i++) {
285     anzi = ai[i+1] - ai[i];
286     cnzi = ci[i+1] - ci[i];
287     for (j=0; j<anzi; j++) {
288       brow = aj[j];
289       bnzi = bi[brow+1] - bi[brow];
290       bjj  = bj + bi[brow];
291       baj  = ba + bi[brow];
292       /* perform sparse axpy */
293       valtmp = aa[j];
294       nextb  = 0;
295       for (k=0; nextb<bnzi; k++) {
296         if (cj[k] == bjj[nextb]) { /* ccol == bcol */
297           ca[k] += valtmp*baj[nextb++];
298         }
299       }
300       flops += 2*bnzi;
301     }
302     aj += anzi; aa += anzi;
303     cj += cnzi; ca += cnzi;
304   }
305 
306   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
307   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
308   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
309   PetscFunctionReturn(0);
310 }
311 
312 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable_fast(Mat A,Mat B,PetscReal fill,Mat *C)
313 {
314   PetscErrorCode     ierr;
315   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
316   PetscInt           *ai = a->i,*bi=b->i,*ci,*cj;
317   PetscInt           am  = A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
318   MatScalar          *ca;
319   PetscReal          afill;
320   PetscInt           i,j,anzi,brow,bnzj,cnzi,*bj,*aj,*lnk,ndouble=0,Crmax;
321   PetscTable         ta;
322   PetscFreeSpaceList free_space=NULL,current_space=NULL;
323 
324   PetscFunctionBegin;
325   /* Get ci and cj - same as MatMatMultSymbolic_SeqAIJ_SeqAIJ except using PetscLLxxx_fast() */
326   /*-----------------------------------------------------------------------------------------*/
327   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
328   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
329   ci[0] = 0;
330 
331   /* create and initialize a linked list */
332   ierr = PetscTableCreate(bn,bn,&ta);CHKERRQ(ierr);
333   MatRowMergeMax_SeqAIJ(b,bm,ta);
334   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
335   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
336 
337   ierr = PetscLLCondensedCreate_fast(Crmax,&lnk);CHKERRQ(ierr);
338 
339   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
340   ierr          = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
341   current_space = free_space;
342 
343   /* Determine ci and cj */
344   for (i=0; i<am; i++) {
345     anzi = ai[i+1] - ai[i];
346     aj   = a->j + ai[i];
347     for (j=0; j<anzi; j++) {
348       brow = aj[j];
349       bnzj = bi[brow+1] - bi[brow];
350       bj   = b->j + bi[brow];
351       /* add non-zero cols of B into the sorted linked list lnk */
352       ierr = PetscLLCondensedAddSorted_fast(bnzj,bj,lnk);CHKERRQ(ierr);
353     }
354     cnzi = lnk[1];
355 
356     /* If free space is not available, make more free space */
357     /* Double the amount of total space in the list */
358     if (current_space->local_remaining<cnzi) {
359       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(cnzi,current_space->total_array_size),&current_space);CHKERRQ(ierr);
360       ndouble++;
361     }
362 
363     /* Copy data into free space, then initialize lnk */
364     ierr = PetscLLCondensedClean_fast(cnzi,current_space->array,lnk);CHKERRQ(ierr);
365 
366     current_space->array           += cnzi;
367     current_space->local_used      += cnzi;
368     current_space->local_remaining -= cnzi;
369 
370     ci[i+1] = ci[i] + cnzi;
371   }
372 
373   /* Column indices are in the list of free space */
374   /* Allocate space for cj, initialize cj, and */
375   /* destroy list of free space and other temporary array(s) */
376   ierr = PetscMalloc1(ci[am]+1,&cj);CHKERRQ(ierr);
377   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
378   ierr = PetscLLCondensedDestroy_fast(lnk);CHKERRQ(ierr);
379 
380   /* Allocate space for ca */
381   ierr = PetscCalloc1(ci[am]+1,&ca);CHKERRQ(ierr);
382 
383   /* put together the new symbolic matrix */
384   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,ca,C);CHKERRQ(ierr);
385   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
386   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
387 
388   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
389   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
390   c          = (Mat_SeqAIJ*)((*C)->data);
391   c->free_a  = PETSC_TRUE;
392   c->free_ij = PETSC_TRUE;
393   c->nonew   = 0;
394 
395   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable; /* slower, less memory */
396 
397   /* set MatInfo */
398   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
399   if (afill < 1.0) afill = 1.0;
400   c->maxnz                     = ci[am];
401   c->nz                        = ci[am];
402   (*C)->info.mallocs           = ndouble;
403   (*C)->info.fill_ratio_given  = fill;
404   (*C)->info.fill_ratio_needed = afill;
405 
406 #if defined(PETSC_USE_INFO)
407   if (ci[am]) {
408     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
409     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
410   } else {
411     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
412   }
413 #endif
414   PetscFunctionReturn(0);
415 }
416 
417 
418 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable(Mat A,Mat B,PetscReal fill,Mat *C)
419 {
420   PetscErrorCode     ierr;
421   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
422   PetscInt           *ai = a->i,*bi=b->i,*ci,*cj;
423   PetscInt           am  = A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
424   MatScalar          *ca;
425   PetscReal          afill;
426   PetscInt           i,j,anzi,brow,bnzj,cnzi,*bj,*aj,*lnk,ndouble=0,Crmax;
427   PetscTable         ta;
428   PetscFreeSpaceList free_space=NULL,current_space=NULL;
429 
430   PetscFunctionBegin;
431   /* Get ci and cj - same as MatMatMultSymbolic_SeqAIJ_SeqAIJ except using PetscLLxxx_Scalalbe() */
432   /*---------------------------------------------------------------------------------------------*/
433   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
434   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
435   ci[0] = 0;
436 
437   /* create and initialize a linked list */
438   ierr = PetscTableCreate(bn,bn,&ta);CHKERRQ(ierr);
439   MatRowMergeMax_SeqAIJ(b,bm,ta);
440   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
441   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
442   ierr = PetscLLCondensedCreate_Scalable(Crmax,&lnk);CHKERRQ(ierr);
443 
444   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
445   ierr          = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
446   current_space = free_space;
447 
448   /* Determine ci and cj */
449   for (i=0; i<am; i++) {
450     anzi = ai[i+1] - ai[i];
451     aj   = a->j + ai[i];
452     for (j=0; j<anzi; j++) {
453       brow = aj[j];
454       bnzj = bi[brow+1] - bi[brow];
455       bj   = b->j + bi[brow];
456       /* add non-zero cols of B into the sorted linked list lnk */
457       ierr = PetscLLCondensedAddSorted_Scalable(bnzj,bj,lnk);CHKERRQ(ierr);
458     }
459     cnzi = lnk[0];
460 
461     /* If free space is not available, make more free space */
462     /* Double the amount of total space in the list */
463     if (current_space->local_remaining<cnzi) {
464       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(cnzi,current_space->total_array_size),&current_space);CHKERRQ(ierr);
465       ndouble++;
466     }
467 
468     /* Copy data into free space, then initialize lnk */
469     ierr = PetscLLCondensedClean_Scalable(cnzi,current_space->array,lnk);CHKERRQ(ierr);
470 
471     current_space->array           += cnzi;
472     current_space->local_used      += cnzi;
473     current_space->local_remaining -= cnzi;
474 
475     ci[i+1] = ci[i] + cnzi;
476   }
477 
478   /* Column indices are in the list of free space */
479   /* Allocate space for cj, initialize cj, and */
480   /* destroy list of free space and other temporary array(s) */
481   ierr = PetscMalloc1(ci[am]+1,&cj);CHKERRQ(ierr);
482   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
483   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
484 
485   /* Allocate space for ca */
486   /*-----------------------*/
487   ierr = PetscCalloc1(ci[am]+1,&ca);CHKERRQ(ierr);
488 
489   /* put together the new symbolic matrix */
490   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,ca,C);CHKERRQ(ierr);
491   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
492   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
493 
494   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
495   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
496   c          = (Mat_SeqAIJ*)((*C)->data);
497   c->free_a  = PETSC_TRUE;
498   c->free_ij = PETSC_TRUE;
499   c->nonew   = 0;
500 
501   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable; /* slower, less memory */
502 
503   /* set MatInfo */
504   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
505   if (afill < 1.0) afill = 1.0;
506   c->maxnz                     = ci[am];
507   c->nz                        = ci[am];
508   (*C)->info.mallocs           = ndouble;
509   (*C)->info.fill_ratio_given  = fill;
510   (*C)->info.fill_ratio_needed = afill;
511 
512 #if defined(PETSC_USE_INFO)
513   if (ci[am]) {
514     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
515     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
516   } else {
517     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
518   }
519 #endif
520   PetscFunctionReturn(0);
521 }
522 
523 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Heap(Mat A,Mat B,PetscReal fill,Mat *C)
524 {
525   PetscErrorCode     ierr;
526   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
527   const PetscInt     *ai=a->i,*bi=b->i,*aj=a->j,*bj=b->j;
528   PetscInt           *ci,*cj,*bb;
529   PetscInt           am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
530   PetscReal          afill;
531   PetscInt           i,j,col,ndouble = 0;
532   PetscFreeSpaceList free_space=NULL,current_space=NULL;
533   PetscHeap          h;
534 
535   PetscFunctionBegin;
536   /* Get ci and cj - by merging sorted rows using a heap */
537   /*---------------------------------------------------------------------------------------------*/
538   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
539   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
540   ci[0] = 0;
541 
542   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
543   ierr          = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
544   current_space = free_space;
545 
546   ierr = PetscHeapCreate(a->rmax,&h);CHKERRQ(ierr);
547   ierr = PetscMalloc1(a->rmax,&bb);CHKERRQ(ierr);
548 
549   /* Determine ci and cj */
550   for (i=0; i<am; i++) {
551     const PetscInt anzi  = ai[i+1] - ai[i]; /* number of nonzeros in this row of A, this is the number of rows of B that we merge */
552     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
553     ci[i+1] = ci[i];
554     /* Populate the min heap */
555     for (j=0; j<anzi; j++) {
556       bb[j] = bi[acol[j]];         /* bb points at the start of the row */
557       if (bb[j] < bi[acol[j]+1]) { /* Add if row is nonempty */
558         ierr = PetscHeapAdd(h,j,bj[bb[j]++]);CHKERRQ(ierr);
559       }
560     }
561     /* Pick off the min element, adding it to free space */
562     ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
563     while (j >= 0) {
564       if (current_space->local_remaining < 1) { /* double the size, but don't exceed 16 MiB */
565         ierr = PetscFreeSpaceGet(PetscMin(PetscIntMultTruncate(2,current_space->total_array_size),16 << 20),&current_space);CHKERRQ(ierr);
566         ndouble++;
567       }
568       *(current_space->array++) = col;
569       current_space->local_used++;
570       current_space->local_remaining--;
571       ci[i+1]++;
572 
573       /* stash if anything else remains in this row of B */
574       if (bb[j] < bi[acol[j]+1]) {ierr = PetscHeapStash(h,j,bj[bb[j]++]);CHKERRQ(ierr);}
575       while (1) {               /* pop and stash any other rows of B that also had an entry in this column */
576         PetscInt j2,col2;
577         ierr = PetscHeapPeek(h,&j2,&col2);CHKERRQ(ierr);
578         if (col2 != col) break;
579         ierr = PetscHeapPop(h,&j2,&col2);CHKERRQ(ierr);
580         if (bb[j2] < bi[acol[j2]+1]) {ierr = PetscHeapStash(h,j2,bj[bb[j2]++]);CHKERRQ(ierr);}
581       }
582       /* Put any stashed elements back into the min heap */
583       ierr = PetscHeapUnstash(h);CHKERRQ(ierr);
584       ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
585     }
586   }
587   ierr = PetscFree(bb);CHKERRQ(ierr);
588   ierr = PetscHeapDestroy(&h);CHKERRQ(ierr);
589 
590   /* Column indices are in the list of free space */
591   /* Allocate space for cj, initialize cj, and */
592   /* destroy list of free space and other temporary array(s) */
593   ierr = PetscMalloc1(ci[am],&cj);CHKERRQ(ierr);
594   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
595 
596   /* put together the new symbolic matrix */
597   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
598   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
599   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
600 
601   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
602   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
603   c          = (Mat_SeqAIJ*)((*C)->data);
604   c->free_a  = PETSC_TRUE;
605   c->free_ij = PETSC_TRUE;
606   c->nonew   = 0;
607 
608   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;
609 
610   /* set MatInfo */
611   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
612   if (afill < 1.0) afill = 1.0;
613   c->maxnz                     = ci[am];
614   c->nz                        = ci[am];
615   (*C)->info.mallocs           = ndouble;
616   (*C)->info.fill_ratio_given  = fill;
617   (*C)->info.fill_ratio_needed = afill;
618 
619 #if defined(PETSC_USE_INFO)
620   if (ci[am]) {
621     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
622     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
623   } else {
624     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
625   }
626 #endif
627   PetscFunctionReturn(0);
628 }
629 
630 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_BTHeap(Mat A,Mat B,PetscReal fill,Mat *C)
631 {
632   PetscErrorCode     ierr;
633   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
634   const PetscInt     *ai = a->i,*bi=b->i,*aj=a->j,*bj=b->j;
635   PetscInt           *ci,*cj,*bb;
636   PetscInt           am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
637   PetscReal          afill;
638   PetscInt           i,j,col,ndouble = 0;
639   PetscFreeSpaceList free_space=NULL,current_space=NULL;
640   PetscHeap          h;
641   PetscBT            bt;
642 
643   PetscFunctionBegin;
644   /* Get ci and cj - using a heap for the sorted rows, but use BT so that each index is only added once */
645   /*---------------------------------------------------------------------------------------------*/
646   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
647   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
648   ci[0] = 0;
649 
650   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
651   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
652 
653   current_space = free_space;
654 
655   ierr = PetscHeapCreate(a->rmax,&h);CHKERRQ(ierr);
656   ierr = PetscMalloc1(a->rmax,&bb);CHKERRQ(ierr);
657   ierr = PetscBTCreate(bn,&bt);CHKERRQ(ierr);
658 
659   /* Determine ci and cj */
660   for (i=0; i<am; i++) {
661     const PetscInt anzi  = ai[i+1] - ai[i]; /* number of nonzeros in this row of A, this is the number of rows of B that we merge */
662     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
663     const PetscInt *fptr = current_space->array; /* Save beginning of the row so we can clear the BT later */
664     ci[i+1] = ci[i];
665     /* Populate the min heap */
666     for (j=0; j<anzi; j++) {
667       PetscInt brow = acol[j];
668       for (bb[j] = bi[brow]; bb[j] < bi[brow+1]; bb[j]++) {
669         PetscInt bcol = bj[bb[j]];
670         if (!PetscBTLookupSet(bt,bcol)) { /* new entry */
671           ierr = PetscHeapAdd(h,j,bcol);CHKERRQ(ierr);
672           bb[j]++;
673           break;
674         }
675       }
676     }
677     /* Pick off the min element, adding it to free space */
678     ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
679     while (j >= 0) {
680       if (current_space->local_remaining < 1) { /* double the size, but don't exceed 16 MiB */
681         fptr = NULL;                      /* need PetscBTMemzero */
682         ierr = PetscFreeSpaceGet(PetscMin(PetscIntMultTruncate(2,current_space->total_array_size),16 << 20),&current_space);CHKERRQ(ierr);
683         ndouble++;
684       }
685       *(current_space->array++) = col;
686       current_space->local_used++;
687       current_space->local_remaining--;
688       ci[i+1]++;
689 
690       /* stash if anything else remains in this row of B */
691       for (; bb[j] < bi[acol[j]+1]; bb[j]++) {
692         PetscInt bcol = bj[bb[j]];
693         if (!PetscBTLookupSet(bt,bcol)) { /* new entry */
694           ierr = PetscHeapAdd(h,j,bcol);CHKERRQ(ierr);
695           bb[j]++;
696           break;
697         }
698       }
699       ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
700     }
701     if (fptr) {                 /* Clear the bits for this row */
702       for (; fptr<current_space->array; fptr++) {ierr = PetscBTClear(bt,*fptr);CHKERRQ(ierr);}
703     } else {                    /* We reallocated so we don't remember (easily) how to clear only the bits we changed */
704       ierr = PetscBTMemzero(bn,bt);CHKERRQ(ierr);
705     }
706   }
707   ierr = PetscFree(bb);CHKERRQ(ierr);
708   ierr = PetscHeapDestroy(&h);CHKERRQ(ierr);
709   ierr = PetscBTDestroy(&bt);CHKERRQ(ierr);
710 
711   /* Column indices are in the list of free space */
712   /* Allocate space for cj, initialize cj, and */
713   /* destroy list of free space and other temporary array(s) */
714   ierr = PetscMalloc1(ci[am],&cj);CHKERRQ(ierr);
715   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
716 
717   /* put together the new symbolic matrix */
718   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
719   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
720   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
721 
722   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
723   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
724   c          = (Mat_SeqAIJ*)((*C)->data);
725   c->free_a  = PETSC_TRUE;
726   c->free_ij = PETSC_TRUE;
727   c->nonew   = 0;
728 
729   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;
730 
731   /* set MatInfo */
732   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
733   if (afill < 1.0) afill = 1.0;
734   c->maxnz                     = ci[am];
735   c->nz                        = ci[am];
736   (*C)->info.mallocs           = ndouble;
737   (*C)->info.fill_ratio_given  = fill;
738   (*C)->info.fill_ratio_needed = afill;
739 
740 #if defined(PETSC_USE_INFO)
741   if (ci[am]) {
742     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
743     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
744   } else {
745     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
746   }
747 #endif
748   PetscFunctionReturn(0);
749 }
750 
751 
752 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_RowMerge(Mat A,Mat B,PetscReal fill,Mat *C)
753 {
754   PetscErrorCode     ierr;
755   Mat_SeqAIJ         *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
756   const PetscInt     *ai=a->i,*bi=b->i,*aj=a->j,*bj=b->j,*inputi,*inputj,*inputcol,*inputcol_L1;
757   PetscInt           *ci,*cj,*outputj,worki_L1[9],worki_L2[9];
758   PetscInt           c_maxmem,a_maxrownnz=0,a_rownnz;
759   const PetscInt     workcol[8]={0,1,2,3,4,5,6,7};
760   const PetscInt     am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
761   const PetscInt     *brow_ptr[8],*brow_end[8];
762   PetscInt           window[8];
763   PetscInt           window_min,old_window_min,ci_nnz,outputi_nnz=0,L1_nrows,L2_nrows;
764   PetscInt           i,k,ndouble=0,L1_rowsleft,rowsleft;
765   PetscReal          afill;
766   PetscInt           *workj_L1,*workj_L2,*workj_L3;
767   PetscInt           L1_nnz,L2_nnz;
768 
769   /* Step 1: Get upper bound on memory required for allocation.
770              Because of the way virtual memory works,
771              only the memory pages that are actually needed will be physically allocated. */
772   PetscFunctionBegin;
773   ierr = PetscMalloc1(am+1,&ci);CHKERRQ(ierr);
774   for (i=0; i<am; i++) {
775     const PetscInt anzi  = ai[i+1] - ai[i]; /* number of nonzeros in this row of A, this is the number of rows of B that we merge */
776     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
777     a_rownnz = 0;
778     for (k=0; k<anzi; ++k) {
779       a_rownnz += bi[acol[k]+1] - bi[acol[k]];
780       if (a_rownnz > bn) {
781         a_rownnz = bn;
782         break;
783       }
784     }
785     a_maxrownnz = PetscMax(a_maxrownnz, a_rownnz);
786   }
787   /* temporary work areas for merging rows */
788   ierr = PetscMalloc1(a_maxrownnz*8,&workj_L1);CHKERRQ(ierr);
789   ierr = PetscMalloc1(a_maxrownnz*8,&workj_L2);CHKERRQ(ierr);
790   ierr = PetscMalloc1(a_maxrownnz,&workj_L3);CHKERRQ(ierr);
791 
792   /* This should be enough for almost all matrices. If not, memory is reallocated later. */
793   c_maxmem = 8*(ai[am]+bi[bm]);
794   /* Step 2: Populate pattern for C */
795   ierr  = PetscMalloc1(c_maxmem,&cj);CHKERRQ(ierr);
796 
797   ci_nnz       = 0;
798   ci[0]        = 0;
799   worki_L1[0]  = 0;
800   worki_L2[0]  = 0;
801   for (i=0; i<am; i++) {
802     const PetscInt anzi  = ai[i+1] - ai[i]; /* number of nonzeros in this row of A, this is the number of rows of B that we merge */
803     const PetscInt *acol = aj + ai[i];      /* column indices of nonzero entries in this row */
804     rowsleft             = anzi;
805     inputcol_L1          = acol;
806     L2_nnz               = 0;
807     L2_nrows             = 1;  /* Number of rows to be merged on Level 3. output of L3 already exists -> initial value 1   */
808     worki_L2[1]          = 0;
809     outputi_nnz          = 0;
810 
811     /* If the number of indices in C so far + the max number of columns in the next row > c_maxmem  -> allocate more memory */
812     while (ci_nnz+a_maxrownnz > c_maxmem) {
813       c_maxmem *= 2;
814       ndouble++;
815       ierr = PetscRealloc(sizeof(PetscInt)*c_maxmem,&cj);CHKERRQ(ierr);
816     }
817 
818     while (rowsleft) {
819       L1_rowsleft = PetscMin(64, rowsleft); /* In the inner loop max 64 rows of B can be merged */
820       L1_nrows    = 0;
821       L1_nnz      = 0;
822       inputcol    = inputcol_L1;
823       inputi      = bi;
824       inputj      = bj;
825 
826       /* The following macro is used to specialize for small rows in A.
827          This helps with compiler unrolling, improving performance substantially.
828           Input:  inputj   inputi  inputcol  bn
829           Output: outputj  outputi_nnz                       */
830        #define MatMatMultSymbolic_RowMergeMacro(ANNZ)                        \
831          window_min  = bn;                                                   \
832          outputi_nnz = 0;                                                    \
833          for (k=0; k<ANNZ; ++k) {                                            \
834            brow_ptr[k] = inputj + inputi[inputcol[k]];                       \
835            brow_end[k] = inputj + inputi[inputcol[k]+1];                     \
836            window[k]   = (brow_ptr[k] != brow_end[k]) ? *brow_ptr[k] : bn;   \
837            window_min  = PetscMin(window[k], window_min);                    \
838          }                                                                   \
839          while (window_min < bn) {                                           \
840            outputj[outputi_nnz++] = window_min;                              \
841            /* advance front and compute new minimum */                       \
842            old_window_min = window_min;                                      \
843            window_min = bn;                                                  \
844            for (k=0; k<ANNZ; ++k) {                                          \
845              if (window[k] == old_window_min) {                              \
846                brow_ptr[k]++;                                                \
847                window[k] = (brow_ptr[k] != brow_end[k]) ? *brow_ptr[k] : bn; \
848              }                                                               \
849              window_min = PetscMin(window[k], window_min);                   \
850            }                                                                 \
851          }
852 
853       /************** L E V E L  1 ***************/
854       /* Merge up to 8 rows of B to L1 work array*/
855       while (L1_rowsleft) {
856         outputi_nnz = 0;
857         if (anzi > 8)  outputj = workj_L1 + L1_nnz;     /* Level 1 rowmerge*/
858         else           outputj = cj + ci_nnz;           /* Merge directly to C */
859 
860         switch (L1_rowsleft) {
861         case 1:  brow_ptr[0] = inputj + inputi[inputcol[0]];
862                  brow_end[0] = inputj + inputi[inputcol[0]+1];
863                  for (; brow_ptr[0] != brow_end[0]; ++brow_ptr[0]) outputj[outputi_nnz++] = *brow_ptr[0]; /* copy row in b over */
864                  inputcol    += L1_rowsleft;
865                  rowsleft    -= L1_rowsleft;
866                  L1_rowsleft  = 0;
867                  break;
868         case 2:  MatMatMultSymbolic_RowMergeMacro(2);
869                  inputcol    += L1_rowsleft;
870                  rowsleft    -= L1_rowsleft;
871                  L1_rowsleft  = 0;
872                  break;
873         case 3: MatMatMultSymbolic_RowMergeMacro(3);
874                  inputcol    += L1_rowsleft;
875                  rowsleft    -= L1_rowsleft;
876                  L1_rowsleft  = 0;
877                  break;
878         case 4:  MatMatMultSymbolic_RowMergeMacro(4);
879                  inputcol    += L1_rowsleft;
880                  rowsleft    -= L1_rowsleft;
881                  L1_rowsleft  = 0;
882                  break;
883         case 5:  MatMatMultSymbolic_RowMergeMacro(5);
884                  inputcol    += L1_rowsleft;
885                  rowsleft    -= L1_rowsleft;
886                  L1_rowsleft  = 0;
887                  break;
888         case 6:  MatMatMultSymbolic_RowMergeMacro(6);
889                  inputcol    += L1_rowsleft;
890                  rowsleft    -= L1_rowsleft;
891                  L1_rowsleft  = 0;
892                  break;
893         case 7:  MatMatMultSymbolic_RowMergeMacro(7);
894                  inputcol    += L1_rowsleft;
895                  rowsleft    -= L1_rowsleft;
896                  L1_rowsleft  = 0;
897                  break;
898         default: MatMatMultSymbolic_RowMergeMacro(8);
899                  inputcol    += 8;
900                  rowsleft    -= 8;
901                  L1_rowsleft -= 8;
902                  break;
903         }
904         inputcol_L1           = inputcol;
905         L1_nnz               += outputi_nnz;
906         worki_L1[++L1_nrows]  = L1_nnz;
907       }
908 
909       /********************** L E V E L  2 ************************/
910       /* Merge from L1 work array to either C or to L2 work array */
911       if (anzi > 8) {
912         inputi      = worki_L1;
913         inputj      = workj_L1;
914         inputcol    = workcol;
915         outputi_nnz = 0;
916 
917         if (anzi <= 64) outputj = cj + ci_nnz;        /* Merge from L1 work array to C */
918         else            outputj = workj_L2 + L2_nnz;  /* Merge from L1 work array to L2 work array */
919 
920         switch (L1_nrows) {
921         case 1:  brow_ptr[0] = inputj + inputi[inputcol[0]];
922                  brow_end[0] = inputj + inputi[inputcol[0]+1];
923                  for (; brow_ptr[0] != brow_end[0]; ++brow_ptr[0]) outputj[outputi_nnz++] = *brow_ptr[0]; /* copy row in b over */
924                  break;
925         case 2:  MatMatMultSymbolic_RowMergeMacro(2); break;
926         case 3:  MatMatMultSymbolic_RowMergeMacro(3); break;
927         case 4:  MatMatMultSymbolic_RowMergeMacro(4); break;
928         case 5:  MatMatMultSymbolic_RowMergeMacro(5); break;
929         case 6:  MatMatMultSymbolic_RowMergeMacro(6); break;
930         case 7:  MatMatMultSymbolic_RowMergeMacro(7); break;
931         case 8:  MatMatMultSymbolic_RowMergeMacro(8); break;
932         default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"MatMatMult logic error: Not merging 1-8 rows from L1 work array!");
933         }
934         L2_nnz               += outputi_nnz;
935         worki_L2[++L2_nrows]  = L2_nnz;
936 
937         /************************ L E V E L  3 **********************/
938         /* Merge from L2 work array to either C or to L2 work array */
939         if (anzi > 64 && (L2_nrows == 8 || rowsleft == 0)) {
940           inputi      = worki_L2;
941           inputj      = workj_L2;
942           inputcol    = workcol;
943           outputi_nnz = 0;
944           if (rowsleft) outputj = workj_L3;
945           else          outputj = cj + ci_nnz;
946           switch (L2_nrows) {
947           case 1:  brow_ptr[0] = inputj + inputi[inputcol[0]];
948                    brow_end[0] = inputj + inputi[inputcol[0]+1];
949                    for (; brow_ptr[0] != brow_end[0]; ++brow_ptr[0]) outputj[outputi_nnz++] = *brow_ptr[0]; /* copy row in b over */
950                    break;
951           case 2:  MatMatMultSymbolic_RowMergeMacro(2); break;
952           case 3:  MatMatMultSymbolic_RowMergeMacro(3); break;
953           case 4:  MatMatMultSymbolic_RowMergeMacro(4); break;
954           case 5:  MatMatMultSymbolic_RowMergeMacro(5); break;
955           case 6:  MatMatMultSymbolic_RowMergeMacro(6); break;
956           case 7:  MatMatMultSymbolic_RowMergeMacro(7); break;
957           case 8:  MatMatMultSymbolic_RowMergeMacro(8); break;
958           default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"MatMatMult logic error: Not merging 1-8 rows from L2 work array!");
959           }
960           L2_nrows    = 1;
961           L2_nnz      = outputi_nnz;
962           worki_L2[1] = outputi_nnz;
963           /* Copy to workj_L2 */
964           if (rowsleft) {
965             for (k=0; k<outputi_nnz; ++k)  workj_L2[k] = outputj[k];
966           }
967         }
968       }
969     }  /* while (rowsleft) */
970 #undef MatMatMultSymbolic_RowMergeMacro
971 
972     /* terminate current row */
973     ci_nnz += outputi_nnz;
974     ci[i+1] = ci_nnz;
975   }
976 
977   /* Step 3: Create the new symbolic matrix */
978   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
979   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
980   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
981 
982   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
983   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
984   c          = (Mat_SeqAIJ*)((*C)->data);
985   c->free_a  = PETSC_TRUE;
986   c->free_ij = PETSC_TRUE;
987   c->nonew   = 0;
988 
989   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;
990 
991   /* set MatInfo */
992   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
993   if (afill < 1.0) afill = 1.0;
994   c->maxnz                     = ci[am];
995   c->nz                        = ci[am];
996   (*C)->info.mallocs           = ndouble;
997   (*C)->info.fill_ratio_given  = fill;
998   (*C)->info.fill_ratio_needed = afill;
999 
1000 #if defined(PETSC_USE_INFO)
1001   if (ci[am]) {
1002     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
1003     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
1004   } else {
1005     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
1006   }
1007 #endif
1008 
1009   /* Step 4: Free temporary work areas */
1010   ierr = PetscFree(workj_L1);CHKERRQ(ierr);
1011   ierr = PetscFree(workj_L2);CHKERRQ(ierr);
1012   ierr = PetscFree(workj_L3);CHKERRQ(ierr);
1013   PetscFunctionReturn(0);
1014 }
1015 
1016 /* concatenate unique entries and then sort */
1017 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Sorted(Mat A,Mat B,PetscReal fill,Mat *C)
1018 {
1019   PetscErrorCode     ierr;
1020   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
1021   const PetscInt     *ai = a->i,*bi=b->i,*aj=a->j,*bj=b->j;
1022   PetscInt           *ci,*cj;
1023   PetscInt           am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
1024   PetscReal          afill;
1025   PetscInt           i,j,ndouble = 0;
1026   PetscSegBuffer     seg,segrow;
1027   char               *seen;
1028 
1029   PetscFunctionBegin;
1030   ierr  = PetscMalloc1(am+1,&ci);CHKERRQ(ierr);
1031   ci[0] = 0;
1032 
1033   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
1034   ierr = PetscSegBufferCreate(sizeof(PetscInt),(PetscInt)(fill*(ai[am]+bi[bm])),&seg);CHKERRQ(ierr);
1035   ierr = PetscSegBufferCreate(sizeof(PetscInt),100,&segrow);CHKERRQ(ierr);
1036   ierr = PetscCalloc1(bn,&seen);CHKERRQ(ierr);
1037 
1038   /* Determine ci and cj */
1039   for (i=0; i<am; i++) {
1040     const PetscInt anzi  = ai[i+1] - ai[i]; /* number of nonzeros in this row of A, this is the number of rows of B that we merge */
1041     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
1042     PetscInt packlen = 0,*PETSC_RESTRICT crow;
1043     /* Pack segrow */
1044     for (j=0; j<anzi; j++) {
1045       PetscInt brow = acol[j],bjstart = bi[brow],bjend = bi[brow+1],k;
1046       for (k=bjstart; k<bjend; k++) {
1047         PetscInt bcol = bj[k];
1048         if (!seen[bcol]) { /* new entry */
1049           PetscInt *PETSC_RESTRICT slot;
1050           ierr = PetscSegBufferGetInts(segrow,1,&slot);CHKERRQ(ierr);
1051           *slot = bcol;
1052           seen[bcol] = 1;
1053           packlen++;
1054         }
1055       }
1056     }
1057     ierr = PetscSegBufferGetInts(seg,packlen,&crow);CHKERRQ(ierr);
1058     ierr = PetscSegBufferExtractTo(segrow,crow);CHKERRQ(ierr);
1059     ierr = PetscSortInt(packlen,crow);CHKERRQ(ierr);
1060     ci[i+1] = ci[i] + packlen;
1061     for (j=0; j<packlen; j++) seen[crow[j]] = 0;
1062   }
1063   ierr = PetscSegBufferDestroy(&segrow);CHKERRQ(ierr);
1064   ierr = PetscFree(seen);CHKERRQ(ierr);
1065 
1066   /* Column indices are in the segmented buffer */
1067   ierr = PetscSegBufferExtractAlloc(seg,&cj);CHKERRQ(ierr);
1068   ierr = PetscSegBufferDestroy(&seg);CHKERRQ(ierr);
1069 
1070   /* put together the new symbolic matrix */
1071   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
1072   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
1073   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
1074 
1075   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
1076   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
1077   c          = (Mat_SeqAIJ*)((*C)->data);
1078   c->free_a  = PETSC_TRUE;
1079   c->free_ij = PETSC_TRUE;
1080   c->nonew   = 0;
1081 
1082   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;
1083 
1084   /* set MatInfo */
1085   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
1086   if (afill < 1.0) afill = 1.0;
1087   c->maxnz                     = ci[am];
1088   c->nz                        = ci[am];
1089   (*C)->info.mallocs           = ndouble;
1090   (*C)->info.fill_ratio_given  = fill;
1091   (*C)->info.fill_ratio_needed = afill;
1092 
1093 #if defined(PETSC_USE_INFO)
1094   if (ci[am]) {
1095     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
1096     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
1097   } else {
1098     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
1099   }
1100 #endif
1101   PetscFunctionReturn(0);
1102 }
1103 
1104 /* This routine is not used. Should be removed! */
1105 PetscErrorCode MatMatTransposeMult_SeqAIJ_SeqAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
1106 {
1107   PetscErrorCode ierr;
1108 
1109   PetscFunctionBegin;
1110   if (scall == MAT_INITIAL_MATRIX) {
1111     ierr = PetscLogEventBegin(MAT_MatTransposeMultSymbolic,A,B,0,0);CHKERRQ(ierr);
1112     ierr = MatMatTransposeMultSymbolic_SeqAIJ_SeqAIJ(A,B,fill,C);CHKERRQ(ierr);
1113     ierr = PetscLogEventEnd(MAT_MatTransposeMultSymbolic,A,B,0,0);CHKERRQ(ierr);
1114   }
1115   ierr = PetscLogEventBegin(MAT_MatTransposeMultNumeric,A,B,0,0);CHKERRQ(ierr);
1116   ierr = MatMatTransposeMultNumeric_SeqAIJ_SeqAIJ(A,B,*C);CHKERRQ(ierr);
1117   ierr = PetscLogEventEnd(MAT_MatTransposeMultNumeric,A,B,0,0);CHKERRQ(ierr);
1118   PetscFunctionReturn(0);
1119 }
1120 
1121 PetscErrorCode MatDestroy_SeqAIJ_MatMatMultTrans(Mat A)
1122 {
1123   PetscErrorCode      ierr;
1124   Mat_SeqAIJ          *a=(Mat_SeqAIJ*)A->data;
1125   Mat_MatMatTransMult *abt=a->abt;
1126 
1127   PetscFunctionBegin;
1128   ierr = (abt->destroy)(A);CHKERRQ(ierr);
1129   ierr = MatTransposeColoringDestroy(&abt->matcoloring);CHKERRQ(ierr);
1130   ierr = MatDestroy(&abt->Bt_den);CHKERRQ(ierr);
1131   ierr = MatDestroy(&abt->ABt_den);CHKERRQ(ierr);
1132   ierr = PetscFree(abt);CHKERRQ(ierr);
1133   PetscFunctionReturn(0);
1134 }
1135 
1136 PetscErrorCode MatMatTransposeMultSymbolic_SeqAIJ_SeqAIJ(Mat A,Mat B,PetscReal fill,Mat *C)
1137 {
1138   PetscErrorCode      ierr;
1139   Mat                 Bt;
1140   PetscInt            *bti,*btj;
1141   Mat_MatMatTransMult *abt;
1142   Mat_SeqAIJ          *c;
1143 
1144   PetscFunctionBegin;
1145   /* create symbolic Bt */
1146   ierr = MatGetSymbolicTranspose_SeqAIJ(B,&bti,&btj);CHKERRQ(ierr);
1147   ierr = MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,B->cmap->n,B->rmap->n,bti,btj,NULL,&Bt);CHKERRQ(ierr);
1148   ierr = MatSetBlockSizes(Bt,PetscAbs(A->cmap->bs),PetscAbs(B->cmap->bs));CHKERRQ(ierr);
1149   ierr = MatSetType(Bt,((PetscObject)A)->type_name);CHKERRQ(ierr);
1150 
1151   /* get symbolic C=A*Bt */
1152   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(A,Bt,fill,C);CHKERRQ(ierr);
1153 
1154   /* create a supporting struct for reuse intermidiate dense matrices with matcoloring */
1155   ierr   = PetscNew(&abt);CHKERRQ(ierr);
1156   c      = (Mat_SeqAIJ*)(*C)->data;
1157   c->abt = abt;
1158 
1159   abt->usecoloring = PETSC_FALSE;
1160   abt->destroy     = (*C)->ops->destroy;
1161   (*C)->ops->destroy     = MatDestroy_SeqAIJ_MatMatMultTrans;
1162 
1163   ierr = PetscOptionsGetBool(((PetscObject)A)->options,NULL,"-matmattransmult_color",&abt->usecoloring,NULL);CHKERRQ(ierr);
1164   if (abt->usecoloring) {
1165     /* Create MatTransposeColoring from symbolic C=A*B^T */
1166     MatTransposeColoring matcoloring;
1167     MatColoring          coloring;
1168     ISColoring           iscoloring;
1169     Mat                  Bt_dense,C_dense;
1170     Mat_SeqAIJ           *c=(Mat_SeqAIJ*)(*C)->data;
1171     /* inode causes memory problem, don't know why */
1172     if (c->inode.use) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"MAT_USE_INODES is not supported. Use '-mat_no_inode'");
1173 
1174     ierr = MatColoringCreate(*C,&coloring);CHKERRQ(ierr);
1175     ierr = MatColoringSetDistance(coloring,2);CHKERRQ(ierr);
1176     ierr = MatColoringSetType(coloring,MATCOLORINGSL);CHKERRQ(ierr);
1177     ierr = MatColoringSetFromOptions(coloring);CHKERRQ(ierr);
1178     ierr = MatColoringApply(coloring,&iscoloring);CHKERRQ(ierr);
1179     ierr = MatColoringDestroy(&coloring);CHKERRQ(ierr);
1180     ierr = MatTransposeColoringCreate(*C,iscoloring,&matcoloring);CHKERRQ(ierr);
1181 
1182     abt->matcoloring = matcoloring;
1183 
1184     ierr = ISColoringDestroy(&iscoloring);CHKERRQ(ierr);
1185 
1186     /* Create Bt_dense and C_dense = A*Bt_dense */
1187     ierr = MatCreate(PETSC_COMM_SELF,&Bt_dense);CHKERRQ(ierr);
1188     ierr = MatSetSizes(Bt_dense,A->cmap->n,matcoloring->ncolors,A->cmap->n,matcoloring->ncolors);CHKERRQ(ierr);
1189     ierr = MatSetType(Bt_dense,MATSEQDENSE);CHKERRQ(ierr);
1190     ierr = MatSeqDenseSetPreallocation(Bt_dense,NULL);CHKERRQ(ierr);
1191 
1192     Bt_dense->assembled = PETSC_TRUE;
1193     abt->Bt_den   = Bt_dense;
1194 
1195     ierr = MatCreate(PETSC_COMM_SELF,&C_dense);CHKERRQ(ierr);
1196     ierr = MatSetSizes(C_dense,A->rmap->n,matcoloring->ncolors,A->rmap->n,matcoloring->ncolors);CHKERRQ(ierr);
1197     ierr = MatSetType(C_dense,MATSEQDENSE);CHKERRQ(ierr);
1198     ierr = MatSeqDenseSetPreallocation(C_dense,NULL);CHKERRQ(ierr);
1199 
1200     Bt_dense->assembled = PETSC_TRUE;
1201     abt->ABt_den  = C_dense;
1202 
1203 #if defined(PETSC_USE_INFO)
1204     {
1205       Mat_SeqAIJ *c = (Mat_SeqAIJ*)(*C)->data;
1206       ierr = PetscInfo7(*C,"Use coloring of C=A*B^T; B^T: %D %D, Bt_dense: %D,%D; Cnz %D / (cm*ncolors %D) = %g\n",B->cmap->n,B->rmap->n,Bt_dense->rmap->n,Bt_dense->cmap->n,c->nz,A->rmap->n*matcoloring->ncolors,(PetscReal)(c->nz)/(A->rmap->n*matcoloring->ncolors));CHKERRQ(ierr);
1207     }
1208 #endif
1209   }
1210   /* clean up */
1211   ierr = MatDestroy(&Bt);CHKERRQ(ierr);
1212   ierr = MatRestoreSymbolicTranspose_SeqAIJ(B,&bti,&btj);CHKERRQ(ierr);
1213   PetscFunctionReturn(0);
1214 }
1215 
1216 PetscErrorCode MatMatTransposeMultNumeric_SeqAIJ_SeqAIJ(Mat A,Mat B,Mat C)
1217 {
1218   PetscErrorCode      ierr;
1219   Mat_SeqAIJ          *a   =(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c=(Mat_SeqAIJ*)C->data;
1220   PetscInt            *ai  =a->i,*aj=a->j,*bi=b->i,*bj=b->j,anzi,bnzj,nexta,nextb,*acol,*bcol,brow;
1221   PetscInt            cm   =C->rmap->n,*ci=c->i,*cj=c->j,i,j,cnzi,*ccol;
1222   PetscLogDouble      flops=0.0;
1223   MatScalar           *aa  =a->a,*aval,*ba=b->a,*bval,*ca,*cval;
1224   Mat_MatMatTransMult *abt = c->abt;
1225 
1226   PetscFunctionBegin;
1227   /* clear old values in C */
1228   if (!c->a) {
1229     ierr      = PetscCalloc1(ci[cm]+1,&ca);CHKERRQ(ierr);
1230     c->a      = ca;
1231     c->free_a = PETSC_TRUE;
1232   } else {
1233     ca =  c->a;
1234     ierr = PetscArrayzero(ca,ci[cm]+1);CHKERRQ(ierr);
1235   }
1236 
1237   if (abt->usecoloring) {
1238     MatTransposeColoring matcoloring = abt->matcoloring;
1239     Mat                  Bt_dense,C_dense = abt->ABt_den;
1240 
1241     /* Get Bt_dense by Apply MatTransposeColoring to B */
1242     Bt_dense = abt->Bt_den;
1243     ierr = MatTransColoringApplySpToDen(matcoloring,B,Bt_dense);CHKERRQ(ierr);
1244 
1245     /* C_dense = A*Bt_dense */
1246     ierr = MatMatMultNumeric_SeqAIJ_SeqDense(A,Bt_dense,C_dense);CHKERRQ(ierr);
1247 
1248     /* Recover C from C_dense */
1249     ierr = MatTransColoringApplyDenToSp(matcoloring,C_dense,C);CHKERRQ(ierr);
1250     PetscFunctionReturn(0);
1251   }
1252 
1253   for (i=0; i<cm; i++) {
1254     anzi = ai[i+1] - ai[i];
1255     acol = aj + ai[i];
1256     aval = aa + ai[i];
1257     cnzi = ci[i+1] - ci[i];
1258     ccol = cj + ci[i];
1259     cval = ca + ci[i];
1260     for (j=0; j<cnzi; j++) {
1261       brow = ccol[j];
1262       bnzj = bi[brow+1] - bi[brow];
1263       bcol = bj + bi[brow];
1264       bval = ba + bi[brow];
1265 
1266       /* perform sparse inner-product c(i,j)=A[i,:]*B[j,:]^T */
1267       nexta = 0; nextb = 0;
1268       while (nexta<anzi && nextb<bnzj) {
1269         while (nexta < anzi && acol[nexta] < bcol[nextb]) nexta++;
1270         if (nexta == anzi) break;
1271         while (nextb < bnzj && acol[nexta] > bcol[nextb]) nextb++;
1272         if (nextb == bnzj) break;
1273         if (acol[nexta] == bcol[nextb]) {
1274           cval[j] += aval[nexta]*bval[nextb];
1275           nexta++; nextb++;
1276           flops += 2;
1277         }
1278       }
1279     }
1280   }
1281   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1282   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1283   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
1284   PetscFunctionReturn(0);
1285 }
1286 
1287 PetscErrorCode MatDestroy_SeqAIJ_MatTransMatMult(Mat A)
1288 {
1289   PetscErrorCode      ierr;
1290   Mat_SeqAIJ          *a = (Mat_SeqAIJ*)A->data;
1291   Mat_MatTransMatMult *atb = a->atb;
1292 
1293   PetscFunctionBegin;
1294   if (atb) {
1295     ierr = MatDestroy(&atb->At);CHKERRQ(ierr);
1296     ierr = (*atb->destroy)(A);CHKERRQ(ierr);
1297   }
1298   ierr = PetscFree(atb);CHKERRQ(ierr);
1299   PetscFunctionReturn(0);
1300 }
1301 
1302 PetscErrorCode MatTransposeMatMult_SeqAIJ_SeqAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
1303 {
1304   PetscErrorCode      ierr;
1305   const char          *algTypes[2] = {"matmatmult","outerproduct"};
1306   PetscInt            alg=0; /* set default algorithm */
1307   Mat                 At;
1308   Mat_MatTransMatMult *atb;
1309   Mat_SeqAIJ          *c;
1310 
1311   PetscFunctionBegin;
1312   if (scall == MAT_INITIAL_MATRIX) {
1313     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)A),((PetscObject)A)->prefix,"MatTransposeMatMult","Mat");CHKERRQ(ierr);
1314     ierr = PetscOptionsEList("-mattransposematmult_via","Algorithmic approach","MatTransposeMatMult",algTypes,2,algTypes[0],&alg,NULL);CHKERRQ(ierr);
1315     ierr = PetscOptionsEnd();CHKERRQ(ierr);
1316 
1317     switch (alg) {
1318     case 1:
1319       ierr = MatTransposeMatMultSymbolic_SeqAIJ_SeqAIJ(A,B,fill,C);CHKERRQ(ierr);
1320       break;
1321     default:
1322       ierr = PetscNew(&atb);CHKERRQ(ierr);
1323       ierr = MatTranspose_SeqAIJ(A,MAT_INITIAL_MATRIX,&At);CHKERRQ(ierr);
1324       ierr = MatMatMult_SeqAIJ_SeqAIJ(At,B,MAT_INITIAL_MATRIX,fill,C);CHKERRQ(ierr);
1325 
1326       c                  = (Mat_SeqAIJ*)(*C)->data;
1327       c->atb             = atb;
1328       atb->At            = At;
1329       atb->destroy       = (*C)->ops->destroy;
1330       (*C)->ops->destroy = MatDestroy_SeqAIJ_MatTransMatMult;
1331 
1332       break;
1333     }
1334   }
1335   if (alg) {
1336     ierr = (*(*C)->ops->mattransposemultnumeric)(A,B,*C);CHKERRQ(ierr);
1337   } else if (!alg && scall == MAT_REUSE_MATRIX) {
1338     c   = (Mat_SeqAIJ*)(*C)->data;
1339     atb = c->atb;
1340     At  = atb->At;
1341     ierr = MatTranspose_SeqAIJ(A,MAT_REUSE_MATRIX,&At);CHKERRQ(ierr);
1342     ierr = MatMatMult_SeqAIJ_SeqAIJ(At,B,MAT_REUSE_MATRIX,fill,C);CHKERRQ(ierr);
1343   }
1344   PetscFunctionReturn(0);
1345 }
1346 
1347 PetscErrorCode MatTransposeMatMultSymbolic_SeqAIJ_SeqAIJ(Mat A,Mat B,PetscReal fill,Mat *C)
1348 {
1349   PetscErrorCode ierr;
1350   Mat            At;
1351   PetscInt       *ati,*atj;
1352 
1353   PetscFunctionBegin;
1354   /* create symbolic At */
1355   ierr = MatGetSymbolicTranspose_SeqAIJ(A,&ati,&atj);CHKERRQ(ierr);
1356   ierr = MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,A->cmap->n,A->rmap->n,ati,atj,NULL,&At);CHKERRQ(ierr);
1357   ierr = MatSetBlockSizes(At,PetscAbs(A->cmap->bs),PetscAbs(B->cmap->bs));CHKERRQ(ierr);
1358   ierr = MatSetType(At,((PetscObject)A)->type_name);CHKERRQ(ierr);
1359 
1360   /* get symbolic C=At*B */
1361   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(At,B,fill,C);CHKERRQ(ierr);
1362 
1363   /* clean up */
1364   ierr = MatDestroy(&At);CHKERRQ(ierr);
1365   ierr = MatRestoreSymbolicTranspose_SeqAIJ(A,&ati,&atj);CHKERRQ(ierr);
1366 
1367   (*C)->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_SeqAIJ_SeqAIJ;
1368   PetscFunctionReturn(0);
1369 }
1370 
1371 PetscErrorCode MatTransposeMatMultNumeric_SeqAIJ_SeqAIJ(Mat A,Mat B,Mat C)
1372 {
1373   PetscErrorCode ierr;
1374   Mat_SeqAIJ     *a   =(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c=(Mat_SeqAIJ*)C->data;
1375   PetscInt       am   =A->rmap->n,anzi,*ai=a->i,*aj=a->j,*bi=b->i,*bj,bnzi,nextb;
1376   PetscInt       cm   =C->rmap->n,*ci=c->i,*cj=c->j,crow,*cjj,i,j,k;
1377   PetscLogDouble flops=0.0;
1378   MatScalar      *aa  =a->a,*ba,*ca,*caj;
1379 
1380   PetscFunctionBegin;
1381   if (!c->a) {
1382     ierr = PetscCalloc1(ci[cm]+1,&ca);CHKERRQ(ierr);
1383 
1384     c->a      = ca;
1385     c->free_a = PETSC_TRUE;
1386   } else {
1387     ca   = c->a;
1388     ierr = PetscArrayzero(ca,ci[cm]);CHKERRQ(ierr);
1389   }
1390 
1391   /* compute A^T*B using outer product (A^T)[:,i]*B[i,:] */
1392   for (i=0; i<am; i++) {
1393     bj   = b->j + bi[i];
1394     ba   = b->a + bi[i];
1395     bnzi = bi[i+1] - bi[i];
1396     anzi = ai[i+1] - ai[i];
1397     for (j=0; j<anzi; j++) {
1398       nextb = 0;
1399       crow  = *aj++;
1400       cjj   = cj + ci[crow];
1401       caj   = ca + ci[crow];
1402       /* perform sparse axpy operation.  Note cjj includes bj. */
1403       for (k=0; nextb<bnzi; k++) {
1404         if (cjj[k] == *(bj+nextb)) { /* ccol == bcol */
1405           caj[k] += (*aa)*(*(ba+nextb));
1406           nextb++;
1407         }
1408       }
1409       flops += 2*bnzi;
1410       aa++;
1411     }
1412   }
1413 
1414   /* Assemble the final matrix and clean up */
1415   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1416   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1417   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
1418   PetscFunctionReturn(0);
1419 }
1420 
1421 PETSC_INTERN PetscErrorCode MatMatMult_SeqAIJ_SeqDense(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
1422 {
1423   PetscErrorCode ierr;
1424 
1425   PetscFunctionBegin;
1426   if (scall == MAT_INITIAL_MATRIX) {
1427     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
1428     ierr = MatMatMultSymbolic_SeqAIJ_SeqDense(A,B,fill,C);CHKERRQ(ierr);
1429     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
1430   }
1431   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
1432   ierr = MatMatMultNumeric_SeqAIJ_SeqDense(A,B,*C);CHKERRQ(ierr);
1433   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
1434   PetscFunctionReturn(0);
1435 }
1436 
1437 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqDense(Mat A,Mat B,PetscReal fill,Mat *C)
1438 {
1439   PetscErrorCode ierr;
1440 
1441   PetscFunctionBegin;
1442   ierr = MatMatMultSymbolic_SeqDense_SeqDense(A,B,0.0,C);CHKERRQ(ierr);
1443 
1444   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqDense;
1445   PetscFunctionReturn(0);
1446 }
1447 
1448 PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat A,Mat B,Mat C)
1449 {
1450   Mat_SeqAIJ        *a=(Mat_SeqAIJ*)A->data;
1451   Mat_SeqDense      *bd = (Mat_SeqDense*)B->data;
1452   PetscErrorCode    ierr;
1453   PetscScalar       *c,*b,r1,r2,r3,r4,*c1,*c2,*c3,*c4,aatmp;
1454   const PetscScalar *aa,*b1,*b2,*b3,*b4;
1455   const PetscInt    *aj;
1456   PetscInt          cm=C->rmap->n,cn=B->cmap->n,bm=bd->lda,am=A->rmap->n;
1457   PetscInt          am4=4*am,bm4=4*bm,col,i,j,n,ajtmp;
1458 
1459   PetscFunctionBegin;
1460   if (!cm || !cn) PetscFunctionReturn(0);
1461 
1462   b = bd->v;
1463   ierr = MatDenseGetArray(C,&c);CHKERRQ(ierr);
1464   b1 = b; b2 = b1 + bm; b3 = b2 + bm; b4 = b3 + bm;
1465   c1 = c; c2 = c1 + am; c3 = c2 + am; c4 = c3 + am;
1466   for (col=0; col<cn-4; col += 4) {  /* over columns of C */
1467     for (i=0; i<am; i++) {        /* over rows of C in those columns */
1468       r1 = r2 = r3 = r4 = 0.0;
1469       n  = a->i[i+1] - a->i[i];
1470       aj = a->j + a->i[i];
1471       aa = a->a + a->i[i];
1472       for (j=0; j<n; j++) {
1473         aatmp = aa[j]; ajtmp = aj[j];
1474         r1 += aatmp*b1[ajtmp];
1475         r2 += aatmp*b2[ajtmp];
1476         r3 += aatmp*b3[ajtmp];
1477         r4 += aatmp*b4[ajtmp];
1478       }
1479       c1[i] += r1;
1480       c2[i] += r2;
1481       c3[i] += r3;
1482       c4[i] += r4;
1483     }
1484     b1 += bm4; b2 += bm4; b3 += bm4; b4 += bm4;
1485     c1 += am4; c2 += am4; c3 += am4; c4 += am4;
1486   }
1487   for (; col<cn; col++) {   /* over extra columns of C */
1488     for (i=0; i<am; i++) {  /* over rows of C in those columns */
1489       r1 = 0.0;
1490       n  = a->i[i+1] - a->i[i];
1491       aj = a->j + a->i[i];
1492       aa = a->a + a->i[i];
1493       for (j=0; j<n; j++) {
1494         r1 += aa[j]*b1[aj[j]];
1495       }
1496       c1[i] += r1;
1497     }
1498     b1 += bm;
1499     c1 += am;
1500   }
1501   ierr = PetscLogFlops(cn*(2.0*a->nz));CHKERRQ(ierr);
1502   ierr = MatDenseRestoreArray(C,&c);CHKERRQ(ierr);
1503   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1504   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1505   PetscFunctionReturn(0);
1506 }
1507 
1508 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqDense(Mat A,Mat B,Mat C)
1509 {
1510   PetscErrorCode ierr;
1511 
1512   PetscFunctionBegin;
1513   if (B->rmap->n != A->cmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Number columns in A %D not equal rows in B %D\n",A->cmap->n,B->rmap->n);
1514   if (A->rmap->n != C->rmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Number rows in C %D not equal rows in A %D\n",C->rmap->n,A->rmap->n);
1515   if (B->cmap->n != C->cmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Number columns in B %D not equal columns in C %D\n",B->cmap->n,C->cmap->n);
1516 
1517   ierr = MatZeroEntries(C);CHKERRQ(ierr);
1518   ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(A,B,C);CHKERRQ(ierr);
1519   PetscFunctionReturn(0);
1520 }
1521 
1522 PetscErrorCode  MatTransColoringApplySpToDen_SeqAIJ(MatTransposeColoring coloring,Mat B,Mat Btdense)
1523 {
1524   PetscErrorCode ierr;
1525   Mat_SeqAIJ     *b       = (Mat_SeqAIJ*)B->data;
1526   Mat_SeqDense   *btdense = (Mat_SeqDense*)Btdense->data;
1527   PetscInt       *bi      = b->i,*bj=b->j;
1528   PetscInt       m        = Btdense->rmap->n,n=Btdense->cmap->n,j,k,l,col,anz,*btcol,brow,ncolumns;
1529   MatScalar      *btval,*btval_den,*ba=b->a;
1530   PetscInt       *columns=coloring->columns,*colorforcol=coloring->colorforcol,ncolors=coloring->ncolors;
1531 
1532   PetscFunctionBegin;
1533   btval_den=btdense->v;
1534   ierr     = PetscArrayzero(btval_den,m*n);CHKERRQ(ierr);
1535   for (k=0; k<ncolors; k++) {
1536     ncolumns = coloring->ncolumns[k];
1537     for (l=0; l<ncolumns; l++) { /* insert a row of B to a column of Btdense */
1538       col   = *(columns + colorforcol[k] + l);
1539       btcol = bj + bi[col];
1540       btval = ba + bi[col];
1541       anz   = bi[col+1] - bi[col];
1542       for (j=0; j<anz; j++) {
1543         brow            = btcol[j];
1544         btval_den[brow] = btval[j];
1545       }
1546     }
1547     btval_den += m;
1548   }
1549   PetscFunctionReturn(0);
1550 }
1551 
1552 PetscErrorCode MatTransColoringApplyDenToSp_SeqAIJ(MatTransposeColoring matcoloring,Mat Cden,Mat Csp)
1553 {
1554   PetscErrorCode    ierr;
1555   Mat_SeqAIJ        *csp = (Mat_SeqAIJ*)Csp->data;
1556   const PetscScalar *ca_den,*ca_den_ptr;
1557   PetscScalar       *ca=csp->a;
1558   PetscInt          k,l,m=Cden->rmap->n,ncolors=matcoloring->ncolors;
1559   PetscInt          brows=matcoloring->brows,*den2sp=matcoloring->den2sp;
1560   PetscInt          nrows,*row,*idx;
1561   PetscInt          *rows=matcoloring->rows,*colorforrow=matcoloring->colorforrow;
1562 
1563   PetscFunctionBegin;
1564   ierr   = MatDenseGetArrayRead(Cden,&ca_den);CHKERRQ(ierr);
1565 
1566   if (brows > 0) {
1567     PetscInt *lstart,row_end,row_start;
1568     lstart = matcoloring->lstart;
1569     ierr = PetscArrayzero(lstart,ncolors);CHKERRQ(ierr);
1570 
1571     row_end = brows;
1572     if (row_end > m) row_end = m;
1573     for (row_start=0; row_start<m; row_start+=brows) { /* loop over row blocks of Csp */
1574       ca_den_ptr = ca_den;
1575       for (k=0; k<ncolors; k++) { /* loop over colors (columns of Cden) */
1576         nrows = matcoloring->nrows[k];
1577         row   = rows  + colorforrow[k];
1578         idx   = den2sp + colorforrow[k];
1579         for (l=lstart[k]; l<nrows; l++) {
1580           if (row[l] >= row_end) {
1581             lstart[k] = l;
1582             break;
1583           } else {
1584             ca[idx[l]] = ca_den_ptr[row[l]];
1585           }
1586         }
1587         ca_den_ptr += m;
1588       }
1589       row_end += brows;
1590       if (row_end > m) row_end = m;
1591     }
1592   } else { /* non-blocked impl: loop over columns of Csp - slow if Csp is large */
1593     ca_den_ptr = ca_den;
1594     for (k=0; k<ncolors; k++) {
1595       nrows = matcoloring->nrows[k];
1596       row   = rows  + colorforrow[k];
1597       idx   = den2sp + colorforrow[k];
1598       for (l=0; l<nrows; l++) {
1599         ca[idx[l]] = ca_den_ptr[row[l]];
1600       }
1601       ca_den_ptr += m;
1602     }
1603   }
1604 
1605   ierr = MatDenseRestoreArrayRead(Cden,&ca_den);CHKERRQ(ierr);
1606 #if defined(PETSC_USE_INFO)
1607   if (matcoloring->brows > 0) {
1608     ierr = PetscInfo1(Csp,"Loop over %D row blocks for den2sp\n",brows);CHKERRQ(ierr);
1609   } else {
1610     ierr = PetscInfo(Csp,"Loop over colors/columns of Cden, inefficient for large sparse matrix product \n");CHKERRQ(ierr);
1611   }
1612 #endif
1613   PetscFunctionReturn(0);
1614 }
1615 
1616 PetscErrorCode MatTransposeColoringCreate_SeqAIJ(Mat mat,ISColoring iscoloring,MatTransposeColoring c)
1617 {
1618   PetscErrorCode ierr;
1619   PetscInt       i,n,nrows,Nbs,j,k,m,ncols,col,cm;
1620   const PetscInt *is,*ci,*cj,*row_idx;
1621   PetscInt       nis = iscoloring->n,*rowhit,bs = 1;
1622   IS             *isa;
1623   Mat_SeqAIJ     *csp = (Mat_SeqAIJ*)mat->data;
1624   PetscInt       *colorforrow,*rows,*rows_i,*idxhit,*spidx,*den2sp,*den2sp_i;
1625   PetscInt       *colorforcol,*columns,*columns_i,brows;
1626   PetscBool      flg;
1627 
1628   PetscFunctionBegin;
1629   ierr = ISColoringGetIS(iscoloring,PETSC_USE_POINTER,PETSC_IGNORE,&isa);CHKERRQ(ierr);
1630 
1631   /* bs >1 is not being tested yet! */
1632   Nbs       = mat->cmap->N/bs;
1633   c->M      = mat->rmap->N/bs;  /* set total rows, columns and local rows */
1634   c->N      = Nbs;
1635   c->m      = c->M;
1636   c->rstart = 0;
1637   c->brows  = 100;
1638 
1639   c->ncolors = nis;
1640   ierr = PetscMalloc3(nis,&c->ncolumns,nis,&c->nrows,nis+1,&colorforrow);CHKERRQ(ierr);
1641   ierr = PetscMalloc1(csp->nz+1,&rows);CHKERRQ(ierr);
1642   ierr = PetscMalloc1(csp->nz+1,&den2sp);CHKERRQ(ierr);
1643 
1644   brows = c->brows;
1645   ierr = PetscOptionsGetInt(NULL,NULL,"-matden2sp_brows",&brows,&flg);CHKERRQ(ierr);
1646   if (flg) c->brows = brows;
1647   if (brows > 0) {
1648     ierr = PetscMalloc1(nis+1,&c->lstart);CHKERRQ(ierr);
1649   }
1650 
1651   colorforrow[0] = 0;
1652   rows_i         = rows;
1653   den2sp_i       = den2sp;
1654 
1655   ierr = PetscMalloc1(nis+1,&colorforcol);CHKERRQ(ierr);
1656   ierr = PetscMalloc1(Nbs+1,&columns);CHKERRQ(ierr);
1657 
1658   colorforcol[0] = 0;
1659   columns_i      = columns;
1660 
1661   /* get column-wise storage of mat */
1662   ierr = MatGetColumnIJ_SeqAIJ_Color(mat,0,PETSC_FALSE,PETSC_FALSE,&ncols,&ci,&cj,&spidx,NULL);CHKERRQ(ierr);
1663 
1664   cm   = c->m;
1665   ierr = PetscMalloc1(cm+1,&rowhit);CHKERRQ(ierr);
1666   ierr = PetscMalloc1(cm+1,&idxhit);CHKERRQ(ierr);
1667   for (i=0; i<nis; i++) { /* loop over color */
1668     ierr = ISGetLocalSize(isa[i],&n);CHKERRQ(ierr);
1669     ierr = ISGetIndices(isa[i],&is);CHKERRQ(ierr);
1670 
1671     c->ncolumns[i] = n;
1672     if (n) {
1673       ierr = PetscArraycpy(columns_i,is,n);CHKERRQ(ierr);
1674     }
1675     colorforcol[i+1] = colorforcol[i] + n;
1676     columns_i       += n;
1677 
1678     /* fast, crude version requires O(N*N) work */
1679     ierr = PetscArrayzero(rowhit,cm);CHKERRQ(ierr);
1680 
1681     for (j=0; j<n; j++) { /* loop over columns*/
1682       col     = is[j];
1683       row_idx = cj + ci[col];
1684       m       = ci[col+1] - ci[col];
1685       for (k=0; k<m; k++) { /* loop over columns marking them in rowhit */
1686         idxhit[*row_idx]   = spidx[ci[col] + k];
1687         rowhit[*row_idx++] = col + 1;
1688       }
1689     }
1690     /* count the number of hits */
1691     nrows = 0;
1692     for (j=0; j<cm; j++) {
1693       if (rowhit[j]) nrows++;
1694     }
1695     c->nrows[i]      = nrows;
1696     colorforrow[i+1] = colorforrow[i] + nrows;
1697 
1698     nrows = 0;
1699     for (j=0; j<cm; j++) { /* loop over rows */
1700       if (rowhit[j]) {
1701         rows_i[nrows]   = j;
1702         den2sp_i[nrows] = idxhit[j];
1703         nrows++;
1704       }
1705     }
1706     den2sp_i += nrows;
1707 
1708     ierr    = ISRestoreIndices(isa[i],&is);CHKERRQ(ierr);
1709     rows_i += nrows;
1710   }
1711   ierr = MatRestoreColumnIJ_SeqAIJ_Color(mat,0,PETSC_FALSE,PETSC_FALSE,&ncols,&ci,&cj,&spidx,NULL);CHKERRQ(ierr);
1712   ierr = PetscFree(rowhit);CHKERRQ(ierr);
1713   ierr = ISColoringRestoreIS(iscoloring,PETSC_USE_POINTER,&isa);CHKERRQ(ierr);
1714   if (csp->nz != colorforrow[nis]) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_PLIB,"csp->nz %d != colorforrow[nis] %d",csp->nz,colorforrow[nis]);
1715 
1716   c->colorforrow = colorforrow;
1717   c->rows        = rows;
1718   c->den2sp      = den2sp;
1719   c->colorforcol = colorforcol;
1720   c->columns     = columns;
1721 
1722   ierr = PetscFree(idxhit);CHKERRQ(ierr);
1723   PetscFunctionReturn(0);
1724 }
1725 
1726 /* The combine method has done the symbolic and numeric in the first phase, and so we just return here */
1727 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ_Combined(Mat A,Mat B,Mat C)
1728 {
1729   PetscFunctionBegin;
1730   /* Empty function */
1731   PetscFunctionReturn(0);
1732 }
1733 
1734 /* This algorithm combines the symbolic and numeric phase of matrix-matrix multiplication. */
1735 PetscErrorCode MatMatMult_SeqAIJ_SeqAIJ_Combined(Mat A,Mat B,PetscReal fill,Mat *C)
1736 {
1737   PetscErrorCode     ierr;
1738   PetscLogDouble     flops=0.0;
1739   Mat_SeqAIJ         *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
1740   const PetscInt     *ai=a->i,*bi=b->i;
1741   PetscInt           *ci,*cj,*cj_i;
1742   PetscScalar        *ca,*ca_i;
1743   PetscInt           b_maxmemrow,c_maxmem,a_col;
1744   PetscInt           am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
1745   PetscInt           i,k,ndouble=0;
1746   PetscReal          afill;
1747   PetscScalar        *c_row_val_dense;
1748   PetscBool          *c_row_idx_flags;
1749   PetscInt           *aj_i=a->j;
1750   PetscScalar        *aa_i=a->a;
1751 
1752   PetscFunctionBegin;
1753 
1754   /* Step 1: Determine upper bounds on memory for C and allocate memory */
1755   /* This should be enough for almost all matrices. If still more memory is needed, it is reallocated later. */
1756   c_maxmem    = 8*(ai[am]+bi[bm]);
1757   b_maxmemrow = PetscMin(bi[bm],bn);
1758   ierr  = PetscMalloc1(am+1,&ci);CHKERRQ(ierr);
1759   ierr  = PetscCalloc1(bn,&c_row_val_dense);CHKERRQ(ierr);
1760   ierr  = PetscCalloc1(bn,&c_row_idx_flags);CHKERRQ(ierr);
1761   ierr  = PetscMalloc1(c_maxmem,&cj);CHKERRQ(ierr);
1762   ierr  = PetscMalloc1(c_maxmem,&ca);CHKERRQ(ierr);
1763   ca_i  = ca;
1764   cj_i  = cj;
1765   ci[0] = 0;
1766   for (i=0; i<am; i++) {
1767     /* Step 2: Initialize the dense row vector for C  */
1768     const PetscInt anzi = ai[i+1] - ai[i]; /* number of nonzeros in this row of A, this is the number of rows of B that we merge */
1769     PetscInt       cnzi = 0;
1770     PetscInt       *bj_i;
1771     PetscScalar    *ba_i;
1772     /* If the number of indices in C so far + the max number of columns in the next row > c_maxmem  -> allocate more memory
1773        Usually, there is enough memory in the first place, so this is not executed. */
1774     while (ci[i] + b_maxmemrow > c_maxmem) {
1775       c_maxmem *= 2;
1776       ndouble++;
1777       ierr = PetscRealloc(sizeof(PetscInt)*c_maxmem,&cj);CHKERRQ(ierr);
1778       ierr = PetscRealloc(sizeof(PetscScalar)*c_maxmem,&ca);CHKERRQ(ierr);
1779     }
1780 
1781     /* Step 3: Do the numerical calculations */
1782     for (a_col=0; a_col<anzi; a_col++) {          /* iterate over all non zero values in a row of A */
1783       PetscInt       a_col_index = aj_i[a_col];
1784       const PetscInt bnzi        = bi[a_col_index+1] - bi[a_col_index];
1785       flops += 2*bnzi;
1786       bj_i   = b->j + bi[a_col_index];   /* points to the current row in bj */
1787       ba_i   = b->a + bi[a_col_index];   /* points to the current row in ba */
1788       for (k=0; k<bnzi; ++k) { /* iterate over all non zeros of this row in B */
1789         if (c_row_idx_flags[bj_i[k]] == PETSC_FALSE) {
1790           cj_i[cnzi++]             = bj_i[k];
1791           c_row_idx_flags[bj_i[k]] = PETSC_TRUE;
1792         }
1793         c_row_val_dense[bj_i[k]] += aa_i[a_col] * ba_i[k];
1794       }
1795     }
1796 
1797     /* Sort array */
1798     ierr = PetscSortInt(cnzi,cj_i);CHKERRQ(ierr);
1799     /* Step 4 */
1800     for (k=0; k<cnzi; k++) {
1801       ca_i[k]                  = c_row_val_dense[cj_i[k]];
1802       c_row_val_dense[cj_i[k]] = 0.;
1803       c_row_idx_flags[cj_i[k]] = PETSC_FALSE;
1804     }
1805     /* terminate current row */
1806     aa_i   += anzi;
1807     aj_i   += anzi;
1808     ca_i   += cnzi;
1809     cj_i   += cnzi;
1810     ci[i+1] = ci[i] + cnzi;
1811     flops  += cnzi;
1812   }
1813 
1814   /* Step 5 */
1815   /* Create the new matrix */
1816   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
1817   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
1818   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
1819 
1820   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
1821   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
1822   c          = (Mat_SeqAIJ*)((*C)->data);
1823   c->a       = ca;
1824   c->free_a  = PETSC_TRUE;
1825   c->free_ij = PETSC_TRUE;
1826   c->nonew   = 0;
1827 
1828   /* set MatInfo */
1829   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
1830   if (afill < 1.0) afill = 1.0;
1831   c->maxnz                     = ci[am];
1832   c->nz                        = ci[am];
1833   (*C)->info.mallocs           = ndouble;
1834   (*C)->info.fill_ratio_given  = fill;
1835   (*C)->info.fill_ratio_needed = afill;
1836   (*C)->ops->matmultnumeric    = MatMatMultNumeric_SeqAIJ_SeqAIJ_Combined;
1837 
1838   ierr = PetscFree(c_row_val_dense);CHKERRQ(ierr);
1839   ierr = PetscFree(c_row_idx_flags);CHKERRQ(ierr);
1840 
1841   ierr = MatAssemblyBegin(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1842   ierr = MatAssemblyEnd(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1843   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
1844   PetscFunctionReturn(0);
1845 }
1846