xref: /petsc/src/mat/impls/aij/seq/matmatmult.c (revision 5a856986583887c326abe5dfd149e8184a29cd80)
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 
25    ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
26    ierr = MatMatMultNumeric_SeqAIJ_SeqAIJ(A,B,*C);CHKERRQ(ierr);
27    ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
28    PetscFunctionReturn(0);
29  }
30 
31  PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ(Mat A,Mat B,Mat C)
32  {
33    PetscErrorCode ierr;
34 
35    PetscFunctionBegin;
36    if (C->ops->matmultnumeric) {
37      ierr = (*C->ops->matmultnumeric)(A,B,C);CHKERRQ(ierr);
38    } else {
39      ierr = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted(A,B,C);CHKERRQ(ierr);
40    }
41    PetscFunctionReturn(0);
42  }
43 
44  PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ(Mat A,Mat B,PetscReal fill,Mat *C)
45  {
46    PetscErrorCode ierr;
47  #if !defined(PETSC_HAVE_HYPRE)
48    const char     *algTypes[8] = {"sorted","scalable","scalable_fast","heap","btheap","llcondensed","combined","rowmerge"};
49    PetscInt       nalg = 8;
50  #else
51    const char     *algTypes[9] = {"sorted","scalable","scalable_fast","heap","btheap","llcondensed","combined","rowmerge","hypre"};
52    PetscInt       nalg = 9;
53  #endif
54    PetscInt       alg = 0; /* set default algorithm */
55 
56    PetscFunctionBegin;
57    ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)A),((PetscObject)A)->prefix,"MatMatMult","Mat");CHKERRQ(ierr);
58    ierr = PetscOptionsEList("-matmatmult_via","Algorithmic approach","MatMatMult",algTypes,nalg,algTypes[0],&alg,NULL);CHKERRQ(ierr);
59    ierr = PetscOptionsEnd();CHKERRQ(ierr);
60    switch (alg) {
61    case 1:
62      ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable(A,B,fill,C);CHKERRQ(ierr);
63      break;
64    case 2:
65      ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable_fast(A,B,fill,C);CHKERRQ(ierr);
66      break;
67    case 3:
68      ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Heap(A,B,fill,C);CHKERRQ(ierr);
69      break;
70    case 4:
71      ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_BTHeap(A,B,fill,C);CHKERRQ(ierr);
72      break;
73    case 5:
74      ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_LLCondensed(A,B,fill,C);CHKERRQ(ierr);
75      break;
76    case 6:
77      ierr = MatMatMult_SeqAIJ_SeqAIJ_Combined(A,B,fill,C);CHKERRQ(ierr);
78      break;
79    case 7:
80      ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_RowMerge(A,B,fill,C);CHKERRQ(ierr);
81      break;
82  #if defined(PETSC_HAVE_HYPRE)
83    case 8:
84      ierr = MatMatMultSymbolic_AIJ_AIJ_wHYPRE(A,B,fill,C);CHKERRQ(ierr);
85      break;
86  #endif
87    default:
88      ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Sorted(A,B,fill,C);CHKERRQ(ierr);
89      break;
90    }
91    PetscFunctionReturn(0);
92  }
93 
94  PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_LLCondensed(Mat A,Mat B,PetscReal fill,Mat *C)
95  {
96    PetscErrorCode     ierr;
97    Mat_SeqAIJ         *a =(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
98    PetscInt           *ai=a->i,*bi=b->i,*ci,*cj;
99    PetscInt           am =A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
100    PetscReal          afill;
101    PetscInt           i,j,anzi,brow,bnzj,cnzi,*bj,*aj,*lnk,ndouble=0,Crmax;
102    PetscTable         ta;
103    PetscBT            lnkbt;
104    PetscFreeSpaceList free_space=NULL,current_space=NULL;
105 
106    PetscFunctionBegin;
107    /* Get ci and cj */
108    /*---------------*/
109    /* Allocate ci array, arrays for fill computation and */
110    /* free space for accumulating nonzero column info */
111    ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
112    ci[0] = 0;
113 
114    /* create and initialize a linked list */
115    ierr = PetscTableCreate(bn,bn,&ta);CHKERRQ(ierr);
116    MatRowMergeMax_SeqAIJ(b,bm,ta);
117    ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
118    ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
119 
120    ierr = PetscLLCondensedCreate(Crmax,bn,&lnk,&lnkbt);CHKERRQ(ierr);
121 
122    /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
123    ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
124 
125    current_space = free_space;
126 
127    /* Determine ci and cj */
128    for (i=0; i<am; i++) {
129      anzi = ai[i+1] - ai[i];
130      aj   = a->j + ai[i];
131      for (j=0; j<anzi; j++) {
132        brow = aj[j];
133        bnzj = bi[brow+1] - bi[brow];
134        bj   = b->j + bi[brow];
135        /* add non-zero cols of B into the sorted linked list lnk */
136        ierr = PetscLLCondensedAddSorted(bnzj,bj,lnk,lnkbt);CHKERRQ(ierr);
137      }
138      cnzi = lnk[0];
139 
140      /* If free space is not available, make more free space */
141      /* Double the amount of total space in the list */
142      if (current_space->local_remaining<cnzi) {
143        ierr = PetscFreeSpaceGet(PetscIntSumTruncate(cnzi,current_space->total_array_size),&current_space);CHKERRQ(ierr);
144        ndouble++;
145      }
146 
147      /* Copy data into free space, then initialize lnk */
148      ierr = PetscLLCondensedClean(bn,cnzi,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
149 
150      current_space->array           += cnzi;
151      current_space->local_used      += cnzi;
152      current_space->local_remaining -= cnzi;
153 
154      ci[i+1] = ci[i] + cnzi;
155    }
156 
157    /* Column indices are in the list of free space */
158    /* Allocate space for cj, initialize cj, and */
159    /* destroy list of free space and other temporary array(s) */
160    ierr = PetscMalloc1(ci[am]+1,&cj);CHKERRQ(ierr);
161    ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
162    ierr = PetscLLCondensedDestroy(lnk,lnkbt);CHKERRQ(ierr);
163 
164    /* put together the new symbolic matrix */
165    ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
166    ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
167    ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
168 
169   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
170   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
171   c                         = (Mat_SeqAIJ*)((*C)->data);
172   c->free_a                 = PETSC_FALSE;
173   c->free_ij                = PETSC_TRUE;
174   c->nonew                  = 0;
175   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted; /* fast, needs non-scalable O(bn) array 'abdense' */
176 
177   /* set MatInfo */
178   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
179   if (afill < 1.0) afill = 1.0;
180   c->maxnz                     = ci[am];
181   c->nz                        = ci[am];
182   (*C)->info.mallocs           = ndouble;
183   (*C)->info.fill_ratio_given  = fill;
184   (*C)->info.fill_ratio_needed = afill;
185 
186 #if defined(PETSC_USE_INFO)
187   if (ci[am]) {
188     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
189     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
190   } else {
191     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
192   }
193 #endif
194   PetscFunctionReturn(0);
195 }
196 
197 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted(Mat A,Mat B,Mat C)
198 {
199   PetscErrorCode ierr;
200   PetscLogDouble flops=0.0;
201   Mat_SeqAIJ     *a   = (Mat_SeqAIJ*)A->data;
202   Mat_SeqAIJ     *b   = (Mat_SeqAIJ*)B->data;
203   Mat_SeqAIJ     *c   = (Mat_SeqAIJ*)C->data;
204   PetscInt       *ai  =a->i,*aj=a->j,*bi=b->i,*bj=b->j,*bjj,*ci=c->i,*cj=c->j;
205   PetscInt       am   =A->rmap->n,cm=C->rmap->n;
206   PetscInt       i,j,k,anzi,bnzi,cnzi,brow;
207   PetscScalar    *aa=a->a,*ba=b->a,*baj,*ca,valtmp;
208   PetscScalar    *ab_dense;
209 
210   PetscFunctionBegin;
211   if (!c->a) { /* first call of MatMatMultNumeric_SeqAIJ_SeqAIJ, allocate ca and matmult_abdense */
212     ierr      = PetscMalloc1(ci[cm]+1,&ca);CHKERRQ(ierr);
213     c->a      = ca;
214     c->free_a = PETSC_TRUE;
215   } else {
216     ca        = c->a;
217   }
218   if (!c->matmult_abdense) {
219     ierr = PetscCalloc1(B->cmap->N,&ab_dense);CHKERRQ(ierr);
220     c->matmult_abdense = ab_dense;
221   } else {
222     ab_dense = c->matmult_abdense;
223   }
224 
225   /* clean old values in C */
226   ierr = PetscArrayzero(ca,ci[cm]);CHKERRQ(ierr);
227   /* Traverse A row-wise. */
228   /* Build the ith row in C by summing over nonzero columns in A, */
229   /* the rows of B corresponding to nonzeros of A. */
230   for (i=0; i<am; i++) {
231     anzi = ai[i+1] - ai[i];
232     for (j=0; j<anzi; j++) {
233       brow = aj[j];
234       bnzi = bi[brow+1] - bi[brow];
235       bjj  = bj + bi[brow];
236       baj  = ba + bi[brow];
237       /* perform dense axpy */
238       valtmp = aa[j];
239       for (k=0; k<bnzi; k++) {
240         ab_dense[bjj[k]] += valtmp*baj[k];
241       }
242       flops += 2*bnzi;
243     }
244     aj += anzi; aa += anzi;
245 
246     cnzi = ci[i+1] - ci[i];
247     for (k=0; k<cnzi; k++) {
248       ca[k]          += ab_dense[cj[k]];
249       ab_dense[cj[k]] = 0.0; /* zero ab_dense */
250     }
251     flops += cnzi;
252     cj    += cnzi; ca += cnzi;
253   }
254   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
255   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
256   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
257   PetscFunctionReturn(0);
258 }
259 
260 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable(Mat A,Mat B,Mat C)
261 {
262   PetscErrorCode ierr;
263   PetscLogDouble flops=0.0;
264   Mat_SeqAIJ     *a   = (Mat_SeqAIJ*)A->data;
265   Mat_SeqAIJ     *b   = (Mat_SeqAIJ*)B->data;
266   Mat_SeqAIJ     *c   = (Mat_SeqAIJ*)C->data;
267   PetscInt       *ai  = a->i,*aj=a->j,*bi=b->i,*bj=b->j,*bjj,*ci=c->i,*cj=c->j;
268   PetscInt       am   = A->rmap->N,cm=C->rmap->N;
269   PetscInt       i,j,k,anzi,bnzi,cnzi,brow;
270   PetscScalar    *aa=a->a,*ba=b->a,*baj,*ca=c->a,valtmp;
271   PetscInt       nextb;
272 
273   PetscFunctionBegin;
274   if (!ca) { /* first call of MatMatMultNumeric_SeqAIJ_SeqAIJ, allocate ca and matmult_abdense */
275     ierr      = PetscMalloc1(ci[cm]+1,&ca);CHKERRQ(ierr);
276     c->a      = ca;
277     c->free_a = PETSC_TRUE;
278   }
279 
280   /* clean old values in C */
281   ierr = PetscArrayzero(ca,ci[cm]);CHKERRQ(ierr);
282   /* Traverse A row-wise. */
283   /* Build the ith row in C by summing over nonzero columns in A, */
284   /* the rows of B corresponding to nonzeros of A. */
285   for (i=0; i<am; i++) {
286     anzi = ai[i+1] - ai[i];
287     cnzi = ci[i+1] - ci[i];
288     for (j=0; j<anzi; j++) {
289       brow = aj[j];
290       bnzi = bi[brow+1] - bi[brow];
291       bjj  = bj + bi[brow];
292       baj  = ba + bi[brow];
293       /* perform sparse axpy */
294       valtmp = aa[j];
295       nextb  = 0;
296       for (k=0; nextb<bnzi; k++) {
297         if (cj[k] == bjj[nextb]) { /* ccol == bcol */
298           ca[k] += valtmp*baj[nextb++];
299         }
300       }
301       flops += 2*bnzi;
302     }
303     aj += anzi; aa += anzi;
304     cj += cnzi; ca += cnzi;
305   }
306 
307   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
308   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
309   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
310   PetscFunctionReturn(0);
311 }
312 
313 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable_fast(Mat A,Mat B,PetscReal fill,Mat *C)
314 {
315   PetscErrorCode     ierr;
316   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
317   PetscInt           *ai = a->i,*bi=b->i,*ci,*cj;
318   PetscInt           am  = A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
319   MatScalar          *ca;
320   PetscReal          afill;
321   PetscInt           i,j,anzi,brow,bnzj,cnzi,*bj,*aj,*lnk,ndouble=0,Crmax;
322   PetscTable         ta;
323   PetscFreeSpaceList free_space=NULL,current_space=NULL;
324 
325   PetscFunctionBegin;
326   /* Get ci and cj - same as MatMatMultSymbolic_SeqAIJ_SeqAIJ except using PetscLLxxx_fast() */
327   /*-----------------------------------------------------------------------------------------*/
328   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
329   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
330   ci[0] = 0;
331 
332   /* create and initialize a linked list */
333   ierr = PetscTableCreate(bn,bn,&ta);CHKERRQ(ierr);
334   MatRowMergeMax_SeqAIJ(b,bm,ta);
335   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
336   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
337 
338   ierr = PetscLLCondensedCreate_fast(Crmax,&lnk);CHKERRQ(ierr);
339 
340   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
341   ierr          = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
342   current_space = free_space;
343 
344   /* Determine ci and cj */
345   for (i=0; i<am; i++) {
346     anzi = ai[i+1] - ai[i];
347     aj   = a->j + ai[i];
348     for (j=0; j<anzi; j++) {
349       brow = aj[j];
350       bnzj = bi[brow+1] - bi[brow];
351       bj   = b->j + bi[brow];
352       /* add non-zero cols of B into the sorted linked list lnk */
353       ierr = PetscLLCondensedAddSorted_fast(bnzj,bj,lnk);CHKERRQ(ierr);
354     }
355     cnzi = lnk[1];
356 
357     /* If free space is not available, make more free space */
358     /* Double the amount of total space in the list */
359     if (current_space->local_remaining<cnzi) {
360       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(cnzi,current_space->total_array_size),&current_space);CHKERRQ(ierr);
361       ndouble++;
362     }
363 
364     /* Copy data into free space, then initialize lnk */
365     ierr = PetscLLCondensedClean_fast(cnzi,current_space->array,lnk);CHKERRQ(ierr);
366 
367     current_space->array           += cnzi;
368     current_space->local_used      += cnzi;
369     current_space->local_remaining -= cnzi;
370 
371     ci[i+1] = ci[i] + cnzi;
372   }
373 
374   /* Column indices are in the list of free space */
375   /* Allocate space for cj, initialize cj, and */
376   /* destroy list of free space and other temporary array(s) */
377   ierr = PetscMalloc1(ci[am]+1,&cj);CHKERRQ(ierr);
378   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
379   ierr = PetscLLCondensedDestroy_fast(lnk);CHKERRQ(ierr);
380 
381   /* Allocate space for ca */
382   ierr = PetscCalloc1(ci[am]+1,&ca);CHKERRQ(ierr);
383 
384   /* put together the new symbolic matrix */
385   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,ca,C);CHKERRQ(ierr);
386   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
387   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
388 
389   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
390   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
391   c          = (Mat_SeqAIJ*)((*C)->data);
392   c->free_a  = PETSC_TRUE;
393   c->free_ij = PETSC_TRUE;
394   c->nonew   = 0;
395 
396   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable; /* slower, less memory */
397 
398   /* set MatInfo */
399   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
400   if (afill < 1.0) afill = 1.0;
401   c->maxnz                     = ci[am];
402   c->nz                        = ci[am];
403   (*C)->info.mallocs           = ndouble;
404   (*C)->info.fill_ratio_given  = fill;
405   (*C)->info.fill_ratio_needed = afill;
406 
407 #if defined(PETSC_USE_INFO)
408   if (ci[am]) {
409     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
410     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
411   } else {
412     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
413   }
414 #endif
415   PetscFunctionReturn(0);
416 }
417 
418 
419 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable(Mat A,Mat B,PetscReal fill,Mat *C)
420 {
421   PetscErrorCode     ierr;
422   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
423   PetscInt           *ai = a->i,*bi=b->i,*ci,*cj;
424   PetscInt           am  = A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
425   MatScalar          *ca;
426   PetscReal          afill;
427   PetscInt           i,j,anzi,brow,bnzj,cnzi,*bj,*aj,*lnk,ndouble=0,Crmax;
428   PetscTable         ta;
429   PetscFreeSpaceList free_space=NULL,current_space=NULL;
430 
431   PetscFunctionBegin;
432   /* Get ci and cj - same as MatMatMultSymbolic_SeqAIJ_SeqAIJ except using PetscLLxxx_Scalalbe() */
433   /*---------------------------------------------------------------------------------------------*/
434   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
435   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
436   ci[0] = 0;
437 
438   /* create and initialize a linked list */
439   ierr = PetscTableCreate(bn,bn,&ta);CHKERRQ(ierr);
440   MatRowMergeMax_SeqAIJ(b,bm,ta);
441   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
442   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
443   ierr = PetscLLCondensedCreate_Scalable(Crmax,&lnk);CHKERRQ(ierr);
444 
445   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
446   ierr          = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
447   current_space = free_space;
448 
449   /* Determine ci and cj */
450   for (i=0; i<am; i++) {
451     anzi = ai[i+1] - ai[i];
452     aj   = a->j + ai[i];
453     for (j=0; j<anzi; j++) {
454       brow = aj[j];
455       bnzj = bi[brow+1] - bi[brow];
456       bj   = b->j + bi[brow];
457       /* add non-zero cols of B into the sorted linked list lnk */
458       ierr = PetscLLCondensedAddSorted_Scalable(bnzj,bj,lnk);CHKERRQ(ierr);
459     }
460     cnzi = lnk[0];
461 
462     /* If free space is not available, make more free space */
463     /* Double the amount of total space in the list */
464     if (current_space->local_remaining<cnzi) {
465       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(cnzi,current_space->total_array_size),&current_space);CHKERRQ(ierr);
466       ndouble++;
467     }
468 
469     /* Copy data into free space, then initialize lnk */
470     ierr = PetscLLCondensedClean_Scalable(cnzi,current_space->array,lnk);CHKERRQ(ierr);
471 
472     current_space->array           += cnzi;
473     current_space->local_used      += cnzi;
474     current_space->local_remaining -= cnzi;
475 
476     ci[i+1] = ci[i] + cnzi;
477   }
478 
479   /* Column indices are in the list of free space */
480   /* Allocate space for cj, initialize cj, and */
481   /* destroy list of free space and other temporary array(s) */
482   ierr = PetscMalloc1(ci[am]+1,&cj);CHKERRQ(ierr);
483   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
484   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
485 
486   /* Allocate space for ca */
487   /*-----------------------*/
488   ierr = PetscCalloc1(ci[am]+1,&ca);CHKERRQ(ierr);
489 
490   /* put together the new symbolic matrix */
491   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,ca,C);CHKERRQ(ierr);
492   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
493   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
494 
495   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
496   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
497   c          = (Mat_SeqAIJ*)((*C)->data);
498   c->free_a  = PETSC_TRUE;
499   c->free_ij = PETSC_TRUE;
500   c->nonew   = 0;
501 
502   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable; /* slower, less memory */
503 
504   /* set MatInfo */
505   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
506   if (afill < 1.0) afill = 1.0;
507   c->maxnz                     = ci[am];
508   c->nz                        = ci[am];
509   (*C)->info.mallocs           = ndouble;
510   (*C)->info.fill_ratio_given  = fill;
511   (*C)->info.fill_ratio_needed = afill;
512 
513 #if defined(PETSC_USE_INFO)
514   if (ci[am]) {
515     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
516     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
517   } else {
518     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
519   }
520 #endif
521   PetscFunctionReturn(0);
522 }
523 
524 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Heap(Mat A,Mat B,PetscReal fill,Mat *C)
525 {
526   PetscErrorCode     ierr;
527   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
528   const PetscInt     *ai=a->i,*bi=b->i,*aj=a->j,*bj=b->j;
529   PetscInt           *ci,*cj,*bb;
530   PetscInt           am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
531   PetscReal          afill;
532   PetscInt           i,j,col,ndouble = 0;
533   PetscFreeSpaceList free_space=NULL,current_space=NULL;
534   PetscHeap          h;
535 
536   PetscFunctionBegin;
537   /* Get ci and cj - by merging sorted rows using a heap */
538   /*---------------------------------------------------------------------------------------------*/
539   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
540   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
541   ci[0] = 0;
542 
543   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
544   ierr          = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
545   current_space = free_space;
546 
547   ierr = PetscHeapCreate(a->rmax,&h);CHKERRQ(ierr);
548   ierr = PetscMalloc1(a->rmax,&bb);CHKERRQ(ierr);
549 
550   /* Determine ci and cj */
551   for (i=0; i<am; i++) {
552     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 */
553     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
554     ci[i+1] = ci[i];
555     /* Populate the min heap */
556     for (j=0; j<anzi; j++) {
557       bb[j] = bi[acol[j]];         /* bb points at the start of the row */
558       if (bb[j] < bi[acol[j]+1]) { /* Add if row is nonempty */
559         ierr = PetscHeapAdd(h,j,bj[bb[j]++]);CHKERRQ(ierr);
560       }
561     }
562     /* Pick off the min element, adding it to free space */
563     ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
564     while (j >= 0) {
565       if (current_space->local_remaining < 1) { /* double the size, but don't exceed 16 MiB */
566         ierr = PetscFreeSpaceGet(PetscMin(PetscIntMultTruncate(2,current_space->total_array_size),16 << 20),&current_space);CHKERRQ(ierr);
567         ndouble++;
568       }
569       *(current_space->array++) = col;
570       current_space->local_used++;
571       current_space->local_remaining--;
572       ci[i+1]++;
573 
574       /* stash if anything else remains in this row of B */
575       if (bb[j] < bi[acol[j]+1]) {ierr = PetscHeapStash(h,j,bj[bb[j]++]);CHKERRQ(ierr);}
576       while (1) {               /* pop and stash any other rows of B that also had an entry in this column */
577         PetscInt j2,col2;
578         ierr = PetscHeapPeek(h,&j2,&col2);CHKERRQ(ierr);
579         if (col2 != col) break;
580         ierr = PetscHeapPop(h,&j2,&col2);CHKERRQ(ierr);
581         if (bb[j2] < bi[acol[j2]+1]) {ierr = PetscHeapStash(h,j2,bj[bb[j2]++]);CHKERRQ(ierr);}
582       }
583       /* Put any stashed elements back into the min heap */
584       ierr = PetscHeapUnstash(h);CHKERRQ(ierr);
585       ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
586     }
587   }
588   ierr = PetscFree(bb);CHKERRQ(ierr);
589   ierr = PetscHeapDestroy(&h);CHKERRQ(ierr);
590 
591   /* Column indices are in the list of free space */
592   /* Allocate space for cj, initialize cj, and */
593   /* destroy list of free space and other temporary array(s) */
594   ierr = PetscMalloc1(ci[am],&cj);CHKERRQ(ierr);
595   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
596 
597   /* put together the new symbolic matrix */
598   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
599   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
600   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
601 
602   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
603   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
604   c          = (Mat_SeqAIJ*)((*C)->data);
605   c->free_a  = PETSC_TRUE;
606   c->free_ij = PETSC_TRUE;
607   c->nonew   = 0;
608 
609   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;
610 
611   /* set MatInfo */
612   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
613   if (afill < 1.0) afill = 1.0;
614   c->maxnz                     = ci[am];
615   c->nz                        = ci[am];
616   (*C)->info.mallocs           = ndouble;
617   (*C)->info.fill_ratio_given  = fill;
618   (*C)->info.fill_ratio_needed = afill;
619 
620 #if defined(PETSC_USE_INFO)
621   if (ci[am]) {
622     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
623     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
624   } else {
625     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
626   }
627 #endif
628   PetscFunctionReturn(0);
629 }
630 
631 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_BTHeap(Mat A,Mat B,PetscReal fill,Mat *C)
632 {
633   PetscErrorCode     ierr;
634   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
635   const PetscInt     *ai = a->i,*bi=b->i,*aj=a->j,*bj=b->j;
636   PetscInt           *ci,*cj,*bb;
637   PetscInt           am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
638   PetscReal          afill;
639   PetscInt           i,j,col,ndouble = 0;
640   PetscFreeSpaceList free_space=NULL,current_space=NULL;
641   PetscHeap          h;
642   PetscBT            bt;
643 
644   PetscFunctionBegin;
645   /* Get ci and cj - using a heap for the sorted rows, but use BT so that each index is only added once */
646   /*---------------------------------------------------------------------------------------------*/
647   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
648   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
649   ci[0] = 0;
650 
651   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
652   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
653 
654   current_space = free_space;
655 
656   ierr = PetscHeapCreate(a->rmax,&h);CHKERRQ(ierr);
657   ierr = PetscMalloc1(a->rmax,&bb);CHKERRQ(ierr);
658   ierr = PetscBTCreate(bn,&bt);CHKERRQ(ierr);
659 
660   /* Determine ci and cj */
661   for (i=0; i<am; i++) {
662     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 */
663     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
664     const PetscInt *fptr = current_space->array; /* Save beginning of the row so we can clear the BT later */
665     ci[i+1] = ci[i];
666     /* Populate the min heap */
667     for (j=0; j<anzi; j++) {
668       PetscInt brow = acol[j];
669       for (bb[j] = bi[brow]; bb[j] < bi[brow+1]; bb[j]++) {
670         PetscInt bcol = bj[bb[j]];
671         if (!PetscBTLookupSet(bt,bcol)) { /* new entry */
672           ierr = PetscHeapAdd(h,j,bcol);CHKERRQ(ierr);
673           bb[j]++;
674           break;
675         }
676       }
677     }
678     /* Pick off the min element, adding it to free space */
679     ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
680     while (j >= 0) {
681       if (current_space->local_remaining < 1) { /* double the size, but don't exceed 16 MiB */
682         fptr = NULL;                      /* need PetscBTMemzero */
683         ierr = PetscFreeSpaceGet(PetscMin(PetscIntMultTruncate(2,current_space->total_array_size),16 << 20),&current_space);CHKERRQ(ierr);
684         ndouble++;
685       }
686       *(current_space->array++) = col;
687       current_space->local_used++;
688       current_space->local_remaining--;
689       ci[i+1]++;
690 
691       /* stash if anything else remains in this row of B */
692       for (; bb[j] < bi[acol[j]+1]; bb[j]++) {
693         PetscInt bcol = bj[bb[j]];
694         if (!PetscBTLookupSet(bt,bcol)) { /* new entry */
695           ierr = PetscHeapAdd(h,j,bcol);CHKERRQ(ierr);
696           bb[j]++;
697           break;
698         }
699       }
700       ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
701     }
702     if (fptr) {                 /* Clear the bits for this row */
703       for (; fptr<current_space->array; fptr++) {ierr = PetscBTClear(bt,*fptr);CHKERRQ(ierr);}
704     } else {                    /* We reallocated so we don't remember (easily) how to clear only the bits we changed */
705       ierr = PetscBTMemzero(bn,bt);CHKERRQ(ierr);
706     }
707   }
708   ierr = PetscFree(bb);CHKERRQ(ierr);
709   ierr = PetscHeapDestroy(&h);CHKERRQ(ierr);
710   ierr = PetscBTDestroy(&bt);CHKERRQ(ierr);
711 
712   /* Column indices are in the list of free space */
713   /* Allocate space for cj, initialize cj, and */
714   /* destroy list of free space and other temporary array(s) */
715   ierr = PetscMalloc1(ci[am],&cj);CHKERRQ(ierr);
716   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
717 
718   /* put together the new symbolic matrix */
719   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
720   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
721   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
722 
723   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
724   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
725   c          = (Mat_SeqAIJ*)((*C)->data);
726   c->free_a  = PETSC_TRUE;
727   c->free_ij = PETSC_TRUE;
728   c->nonew   = 0;
729 
730   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;
731 
732   /* set MatInfo */
733   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
734   if (afill < 1.0) afill = 1.0;
735   c->maxnz                     = ci[am];
736   c->nz                        = ci[am];
737   (*C)->info.mallocs           = ndouble;
738   (*C)->info.fill_ratio_given  = fill;
739   (*C)->info.fill_ratio_needed = afill;
740 
741 #if defined(PETSC_USE_INFO)
742   if (ci[am]) {
743     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
744     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
745   } else {
746     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
747   }
748 #endif
749   PetscFunctionReturn(0);
750 }
751 
752 
753 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_RowMerge(Mat A,Mat B,PetscReal fill,Mat *C)
754 {
755   PetscErrorCode     ierr;
756   Mat_SeqAIJ         *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
757   const PetscInt     *ai=a->i,*bi=b->i,*aj=a->j,*bj=b->j,*inputi,*inputj,*inputcol,*inputcol_L1;
758   PetscInt           *ci,*cj,*outputj,worki_L1[9],worki_L2[9];
759   PetscInt           c_maxmem,a_maxrownnz=0,a_rownnz;
760   const PetscInt     workcol[8]={0,1,2,3,4,5,6,7};
761   const PetscInt     am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
762   const PetscInt     *brow_ptr[8],*brow_end[8];
763   PetscInt           window[8];
764   PetscInt           window_min,old_window_min,ci_nnz,outputi_nnz=0,L1_nrows,L2_nrows;
765   PetscInt           i,k,ndouble=0,L1_rowsleft,rowsleft;
766   PetscReal          afill;
767   PetscInt           *workj_L1,*workj_L2,*workj_L3;
768   PetscInt           L1_nnz,L2_nnz;
769 
770   /* Step 1: Get upper bound on memory required for allocation.
771              Because of the way virtual memory works,
772              only the memory pages that are actually needed will be physically allocated. */
773   PetscFunctionBegin;
774   ierr = PetscMalloc1(am+1,&ci);CHKERRQ(ierr);
775   for (i=0; i<am; i++) {
776     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 */
777     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
778     a_rownnz = 0;
779     for (k=0; k<anzi; ++k) {
780       a_rownnz += bi[acol[k]+1] - bi[acol[k]];
781       if (a_rownnz > bn) {
782         a_rownnz = bn;
783         break;
784       }
785     }
786     a_maxrownnz = PetscMax(a_maxrownnz, a_rownnz);
787   }
788   /* temporary work areas for merging rows */
789   ierr = PetscMalloc1(a_maxrownnz*8,&workj_L1);CHKERRQ(ierr);
790   ierr = PetscMalloc1(a_maxrownnz*8,&workj_L2);CHKERRQ(ierr);
791   ierr = PetscMalloc1(a_maxrownnz,&workj_L3);CHKERRQ(ierr);
792 
793   /* This should be enough for almost all matrices. If not, memory is reallocated later. */
794   c_maxmem = 8*(ai[am]+bi[bm]);
795   /* Step 2: Populate pattern for C */
796   ierr  = PetscMalloc1(c_maxmem,&cj);CHKERRQ(ierr);
797 
798   ci_nnz       = 0;
799   ci[0]        = 0;
800   worki_L1[0]  = 0;
801   worki_L2[0]  = 0;
802   for (i=0; i<am; i++) {
803     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 */
804     const PetscInt *acol = aj + ai[i];      /* column indices of nonzero entries in this row */
805     rowsleft             = anzi;
806     inputcol_L1          = acol;
807     L2_nnz               = 0;
808     L2_nrows             = 1;  /* Number of rows to be merged on Level 3. output of L3 already exists -> initial value 1   */
809     worki_L2[1]          = 0;
810     outputi_nnz          = 0;
811 
812     /* If the number of indices in C so far + the max number of columns in the next row > c_maxmem  -> allocate more memory */
813     while (ci_nnz+a_maxrownnz > c_maxmem) {
814       c_maxmem *= 2;
815       ndouble++;
816       ierr = PetscRealloc(sizeof(PetscInt)*c_maxmem,&cj);CHKERRQ(ierr);
817     }
818 
819     while (rowsleft) {
820       L1_rowsleft = PetscMin(64, rowsleft); /* In the inner loop max 64 rows of B can be merged */
821       L1_nrows    = 0;
822       L1_nnz      = 0;
823       inputcol    = inputcol_L1;
824       inputi      = bi;
825       inputj      = bj;
826 
827       /* The following macro is used to specialize for small rows in A.
828          This helps with compiler unrolling, improving performance substantially.
829           Input:  inputj   inputi  inputcol  bn
830           Output: outputj  outputi_nnz                       */
831        #define MatMatMultSymbolic_RowMergeMacro(ANNZ)                        \
832          window_min  = bn;                                                   \
833          outputi_nnz = 0;                                                    \
834          for (k=0; k<ANNZ; ++k) {                                            \
835            brow_ptr[k] = inputj + inputi[inputcol[k]];                       \
836            brow_end[k] = inputj + inputi[inputcol[k]+1];                     \
837            window[k]   = (brow_ptr[k] != brow_end[k]) ? *brow_ptr[k] : bn;   \
838            window_min  = PetscMin(window[k], window_min);                    \
839          }                                                                   \
840          while (window_min < bn) {                                           \
841            outputj[outputi_nnz++] = window_min;                              \
842            /* advance front and compute new minimum */                       \
843            old_window_min = window_min;                                      \
844            window_min = bn;                                                  \
845            for (k=0; k<ANNZ; ++k) {                                          \
846              if (window[k] == old_window_min) {                              \
847                brow_ptr[k]++;                                                \
848                window[k] = (brow_ptr[k] != brow_end[k]) ? *brow_ptr[k] : bn; \
849              }                                                               \
850              window_min = PetscMin(window[k], window_min);                   \
851            }                                                                 \
852          }
853 
854       /************** L E V E L  1 ***************/
855       /* Merge up to 8 rows of B to L1 work array*/
856       while (L1_rowsleft) {
857         outputi_nnz = 0;
858         if (anzi > 8)  outputj = workj_L1 + L1_nnz;     /* Level 1 rowmerge*/
859         else           outputj = cj + ci_nnz;           /* Merge directly to C */
860 
861         switch (L1_rowsleft) {
862         case 1:  brow_ptr[0] = inputj + inputi[inputcol[0]];
863                  brow_end[0] = inputj + inputi[inputcol[0]+1];
864                  for (; brow_ptr[0] != brow_end[0]; ++brow_ptr[0]) outputj[outputi_nnz++] = *brow_ptr[0]; /* copy row in b over */
865                  inputcol    += L1_rowsleft;
866                  rowsleft    -= L1_rowsleft;
867                  L1_rowsleft  = 0;
868                  break;
869         case 2:  MatMatMultSymbolic_RowMergeMacro(2);
870                  inputcol    += L1_rowsleft;
871                  rowsleft    -= L1_rowsleft;
872                  L1_rowsleft  = 0;
873                  break;
874         case 3: MatMatMultSymbolic_RowMergeMacro(3);
875                  inputcol    += L1_rowsleft;
876                  rowsleft    -= L1_rowsleft;
877                  L1_rowsleft  = 0;
878                  break;
879         case 4:  MatMatMultSymbolic_RowMergeMacro(4);
880                  inputcol    += L1_rowsleft;
881                  rowsleft    -= L1_rowsleft;
882                  L1_rowsleft  = 0;
883                  break;
884         case 5:  MatMatMultSymbolic_RowMergeMacro(5);
885                  inputcol    += L1_rowsleft;
886                  rowsleft    -= L1_rowsleft;
887                  L1_rowsleft  = 0;
888                  break;
889         case 6:  MatMatMultSymbolic_RowMergeMacro(6);
890                  inputcol    += L1_rowsleft;
891                  rowsleft    -= L1_rowsleft;
892                  L1_rowsleft  = 0;
893                  break;
894         case 7:  MatMatMultSymbolic_RowMergeMacro(7);
895                  inputcol    += L1_rowsleft;
896                  rowsleft    -= L1_rowsleft;
897                  L1_rowsleft  = 0;
898                  break;
899         default: MatMatMultSymbolic_RowMergeMacro(8);
900                  inputcol    += 8;
901                  rowsleft    -= 8;
902                  L1_rowsleft -= 8;
903                  break;
904         }
905         inputcol_L1           = inputcol;
906         L1_nnz               += outputi_nnz;
907         worki_L1[++L1_nrows]  = L1_nnz;
908       }
909 
910       /********************** L E V E L  2 ************************/
911       /* Merge from L1 work array to either C or to L2 work array */
912       if (anzi > 8) {
913         inputi      = worki_L1;
914         inputj      = workj_L1;
915         inputcol    = workcol;
916         outputi_nnz = 0;
917 
918         if (anzi <= 64) outputj = cj + ci_nnz;        /* Merge from L1 work array to C */
919         else            outputj = workj_L2 + L2_nnz;  /* Merge from L1 work array to L2 work array */
920 
921         switch (L1_nrows) {
922         case 1:  brow_ptr[0] = inputj + inputi[inputcol[0]];
923                  brow_end[0] = inputj + inputi[inputcol[0]+1];
924                  for (; brow_ptr[0] != brow_end[0]; ++brow_ptr[0]) outputj[outputi_nnz++] = *brow_ptr[0]; /* copy row in b over */
925                  break;
926         case 2:  MatMatMultSymbolic_RowMergeMacro(2); break;
927         case 3:  MatMatMultSymbolic_RowMergeMacro(3); break;
928         case 4:  MatMatMultSymbolic_RowMergeMacro(4); break;
929         case 5:  MatMatMultSymbolic_RowMergeMacro(5); break;
930         case 6:  MatMatMultSymbolic_RowMergeMacro(6); break;
931         case 7:  MatMatMultSymbolic_RowMergeMacro(7); break;
932         case 8:  MatMatMultSymbolic_RowMergeMacro(8); break;
933         default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"MatMatMult logic error: Not merging 1-8 rows from L1 work array!");
934         }
935         L2_nnz               += outputi_nnz;
936         worki_L2[++L2_nrows]  = L2_nnz;
937 
938         /************************ L E V E L  3 **********************/
939         /* Merge from L2 work array to either C or to L2 work array */
940         if (anzi > 64 && (L2_nrows == 8 || rowsleft == 0)) {
941           inputi      = worki_L2;
942           inputj      = workj_L2;
943           inputcol    = workcol;
944           outputi_nnz = 0;
945           if (rowsleft) outputj = workj_L3;
946           else          outputj = cj + ci_nnz;
947           switch (L2_nrows) {
948           case 1:  brow_ptr[0] = inputj + inputi[inputcol[0]];
949                    brow_end[0] = inputj + inputi[inputcol[0]+1];
950                    for (; brow_ptr[0] != brow_end[0]; ++brow_ptr[0]) outputj[outputi_nnz++] = *brow_ptr[0]; /* copy row in b over */
951                    break;
952           case 2:  MatMatMultSymbolic_RowMergeMacro(2); break;
953           case 3:  MatMatMultSymbolic_RowMergeMacro(3); break;
954           case 4:  MatMatMultSymbolic_RowMergeMacro(4); break;
955           case 5:  MatMatMultSymbolic_RowMergeMacro(5); break;
956           case 6:  MatMatMultSymbolic_RowMergeMacro(6); break;
957           case 7:  MatMatMultSymbolic_RowMergeMacro(7); break;
958           case 8:  MatMatMultSymbolic_RowMergeMacro(8); break;
959           default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"MatMatMult logic error: Not merging 1-8 rows from L2 work array!");
960           }
961           L2_nrows    = 1;
962           L2_nnz      = outputi_nnz;
963           worki_L2[1] = outputi_nnz;
964           /* Copy to workj_L2 */
965           if (rowsleft) {
966             for (k=0; k<outputi_nnz; ++k)  workj_L2[k] = outputj[k];
967           }
968         }
969       }
970     }  /* while (rowsleft) */
971 #undef MatMatMultSymbolic_RowMergeMacro
972 
973     /* terminate current row */
974     ci_nnz += outputi_nnz;
975     ci[i+1] = ci_nnz;
976   }
977 
978   /* Step 3: Create the new symbolic matrix */
979   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
980   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
981   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
982 
983   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
984   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
985   c          = (Mat_SeqAIJ*)((*C)->data);
986   c->free_a  = PETSC_TRUE;
987   c->free_ij = PETSC_TRUE;
988   c->nonew   = 0;
989 
990   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;
991 
992   /* set MatInfo */
993   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
994   if (afill < 1.0) afill = 1.0;
995   c->maxnz                     = ci[am];
996   c->nz                        = ci[am];
997   (*C)->info.mallocs           = ndouble;
998   (*C)->info.fill_ratio_given  = fill;
999   (*C)->info.fill_ratio_needed = afill;
1000 
1001 #if defined(PETSC_USE_INFO)
1002   if (ci[am]) {
1003     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
1004     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
1005   } else {
1006     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
1007   }
1008 #endif
1009 
1010   /* Step 4: Free temporary work areas */
1011   ierr = PetscFree(workj_L1);CHKERRQ(ierr);
1012   ierr = PetscFree(workj_L2);CHKERRQ(ierr);
1013   ierr = PetscFree(workj_L3);CHKERRQ(ierr);
1014   PetscFunctionReturn(0);
1015 }
1016 
1017 /* concatenate unique entries and then sort */
1018 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Sorted(Mat A,Mat B,PetscReal fill,Mat *C)
1019 {
1020   PetscErrorCode     ierr;
1021   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
1022   const PetscInt     *ai = a->i,*bi=b->i,*aj=a->j,*bj=b->j;
1023   PetscInt           *ci,*cj;
1024   PetscInt           am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
1025   PetscReal          afill;
1026   PetscInt           i,j,ndouble = 0;
1027   PetscSegBuffer     seg,segrow;
1028   char               *seen;
1029 
1030   PetscFunctionBegin;
1031   ierr  = PetscMalloc1(am+1,&ci);CHKERRQ(ierr);
1032   ci[0] = 0;
1033 
1034   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
1035   ierr = PetscSegBufferCreate(sizeof(PetscInt),(PetscInt)(fill*(ai[am]+bi[bm])),&seg);CHKERRQ(ierr);
1036   ierr = PetscSegBufferCreate(sizeof(PetscInt),100,&segrow);CHKERRQ(ierr);
1037   ierr = PetscCalloc1(bn,&seen);CHKERRQ(ierr);
1038 
1039   /* Determine ci and cj */
1040   for (i=0; i<am; i++) {
1041     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 */
1042     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
1043     PetscInt packlen = 0,*PETSC_RESTRICT crow;
1044     /* Pack segrow */
1045     for (j=0; j<anzi; j++) {
1046       PetscInt brow = acol[j],bjstart = bi[brow],bjend = bi[brow+1],k;
1047       for (k=bjstart; k<bjend; k++) {
1048         PetscInt bcol = bj[k];
1049         if (!seen[bcol]) { /* new entry */
1050           PetscInt *PETSC_RESTRICT slot;
1051           ierr = PetscSegBufferGetInts(segrow,1,&slot);CHKERRQ(ierr);
1052           *slot = bcol;
1053           seen[bcol] = 1;
1054           packlen++;
1055         }
1056       }
1057     }
1058     ierr = PetscSegBufferGetInts(seg,packlen,&crow);CHKERRQ(ierr);
1059     ierr = PetscSegBufferExtractTo(segrow,crow);CHKERRQ(ierr);
1060     ierr = PetscSortInt(packlen,crow);CHKERRQ(ierr);
1061     ci[i+1] = ci[i] + packlen;
1062     for (j=0; j<packlen; j++) seen[crow[j]] = 0;
1063   }
1064   ierr = PetscSegBufferDestroy(&segrow);CHKERRQ(ierr);
1065   ierr = PetscFree(seen);CHKERRQ(ierr);
1066 
1067   /* Column indices are in the segmented buffer */
1068   ierr = PetscSegBufferExtractAlloc(seg,&cj);CHKERRQ(ierr);
1069   ierr = PetscSegBufferDestroy(&seg);CHKERRQ(ierr);
1070 
1071   /* put together the new symbolic matrix */
1072   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
1073   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
1074   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
1075 
1076   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
1077   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
1078   c          = (Mat_SeqAIJ*)((*C)->data);
1079   c->free_a  = PETSC_TRUE;
1080   c->free_ij = PETSC_TRUE;
1081   c->nonew   = 0;
1082 
1083   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;
1084 
1085   /* set MatInfo */
1086   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
1087   if (afill < 1.0) afill = 1.0;
1088   c->maxnz                     = ci[am];
1089   c->nz                        = ci[am];
1090   (*C)->info.mallocs           = ndouble;
1091   (*C)->info.fill_ratio_given  = fill;
1092   (*C)->info.fill_ratio_needed = afill;
1093 
1094 #if defined(PETSC_USE_INFO)
1095   if (ci[am]) {
1096     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
1097     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
1098   } else {
1099     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
1100   }
1101 #endif
1102   PetscFunctionReturn(0);
1103 }
1104 
1105 /* This routine is not used. Should be removed! */
1106 PetscErrorCode MatMatTransposeMult_SeqAIJ_SeqAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
1107 {
1108   PetscErrorCode ierr;
1109 
1110   PetscFunctionBegin;
1111   if (scall == MAT_INITIAL_MATRIX) {
1112     ierr = PetscLogEventBegin(MAT_MatTransposeMultSymbolic,A,B,0,0);CHKERRQ(ierr);
1113     ierr = MatMatTransposeMultSymbolic_SeqAIJ_SeqAIJ(A,B,fill,C);CHKERRQ(ierr);
1114     ierr = PetscLogEventEnd(MAT_MatTransposeMultSymbolic,A,B,0,0);CHKERRQ(ierr);
1115   }
1116   ierr = PetscLogEventBegin(MAT_MatTransposeMultNumeric,A,B,0,0);CHKERRQ(ierr);
1117   ierr = MatMatTransposeMultNumeric_SeqAIJ_SeqAIJ(A,B,*C);CHKERRQ(ierr);
1118   ierr = PetscLogEventEnd(MAT_MatTransposeMultNumeric,A,B,0,0);CHKERRQ(ierr);
1119   PetscFunctionReturn(0);
1120 }
1121 
1122 PetscErrorCode MatDestroy_SeqAIJ_MatMatMultTrans(Mat A)
1123 {
1124   PetscErrorCode      ierr;
1125   Mat_SeqAIJ          *a=(Mat_SeqAIJ*)A->data;
1126   Mat_MatMatTransMult *abt=a->abt;
1127 
1128   PetscFunctionBegin;
1129   ierr = (abt->destroy)(A);CHKERRQ(ierr);
1130   ierr = MatTransposeColoringDestroy(&abt->matcoloring);CHKERRQ(ierr);
1131   ierr = MatDestroy(&abt->Bt_den);CHKERRQ(ierr);
1132   ierr = MatDestroy(&abt->ABt_den);CHKERRQ(ierr);
1133   ierr = PetscFree(abt);CHKERRQ(ierr);
1134   PetscFunctionReturn(0);
1135 }
1136 
1137 PetscErrorCode MatMatTransposeMultSymbolic_SeqAIJ_SeqAIJ(Mat A,Mat B,PetscReal fill,Mat *C)
1138 {
1139   PetscErrorCode      ierr;
1140   Mat                 Bt;
1141   PetscInt            *bti,*btj;
1142   Mat_MatMatTransMult *abt;
1143   Mat_SeqAIJ          *c;
1144 
1145   PetscFunctionBegin;
1146   /* create symbolic Bt */
1147   ierr = MatGetSymbolicTranspose_SeqAIJ(B,&bti,&btj);CHKERRQ(ierr);
1148   ierr = MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,B->cmap->n,B->rmap->n,bti,btj,NULL,&Bt);CHKERRQ(ierr);
1149   ierr = MatSetBlockSizes(Bt,PetscAbs(A->cmap->bs),PetscAbs(B->cmap->bs));CHKERRQ(ierr);
1150   ierr = MatSetType(Bt,((PetscObject)A)->type_name);CHKERRQ(ierr);
1151 
1152   /* get symbolic C=A*Bt */
1153   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(A,Bt,fill,C);CHKERRQ(ierr);
1154 
1155   /* create a supporting struct for reuse intermidiate dense matrices with matcoloring */
1156   ierr   = PetscNew(&abt);CHKERRQ(ierr);
1157   c      = (Mat_SeqAIJ*)(*C)->data;
1158   c->abt = abt;
1159 
1160   abt->usecoloring = PETSC_FALSE;
1161   abt->destroy     = (*C)->ops->destroy;
1162   (*C)->ops->destroy     = MatDestroy_SeqAIJ_MatMatMultTrans;
1163 
1164   ierr = PetscOptionsGetBool(((PetscObject)A)->options,NULL,"-matmattransmult_color",&abt->usecoloring,NULL);CHKERRQ(ierr);
1165   if (abt->usecoloring) {
1166     /* Create MatTransposeColoring from symbolic C=A*B^T */
1167     MatTransposeColoring matcoloring;
1168     MatColoring          coloring;
1169     ISColoring           iscoloring;
1170     Mat                  Bt_dense,C_dense;
1171     Mat_SeqAIJ           *c=(Mat_SeqAIJ*)(*C)->data;
1172     /* inode causes memory problem, don't know why */
1173     if (c->inode.use) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"MAT_USE_INODES is not supported. Use '-mat_no_inode'");
1174 
1175     ierr = MatColoringCreate(*C,&coloring);CHKERRQ(ierr);
1176     ierr = MatColoringSetDistance(coloring,2);CHKERRQ(ierr);
1177     ierr = MatColoringSetType(coloring,MATCOLORINGSL);CHKERRQ(ierr);
1178     ierr = MatColoringSetFromOptions(coloring);CHKERRQ(ierr);
1179     ierr = MatColoringApply(coloring,&iscoloring);CHKERRQ(ierr);
1180     ierr = MatColoringDestroy(&coloring);CHKERRQ(ierr);
1181     ierr = MatTransposeColoringCreate(*C,iscoloring,&matcoloring);CHKERRQ(ierr);
1182 
1183     abt->matcoloring = matcoloring;
1184 
1185     ierr = ISColoringDestroy(&iscoloring);CHKERRQ(ierr);
1186 
1187     /* Create Bt_dense and C_dense = A*Bt_dense */
1188     ierr = MatCreate(PETSC_COMM_SELF,&Bt_dense);CHKERRQ(ierr);
1189     ierr = MatSetSizes(Bt_dense,A->cmap->n,matcoloring->ncolors,A->cmap->n,matcoloring->ncolors);CHKERRQ(ierr);
1190     ierr = MatSetType(Bt_dense,MATSEQDENSE);CHKERRQ(ierr);
1191     ierr = MatSeqDenseSetPreallocation(Bt_dense,NULL);CHKERRQ(ierr);
1192 
1193     Bt_dense->assembled = PETSC_TRUE;
1194     abt->Bt_den   = Bt_dense;
1195 
1196     ierr = MatCreate(PETSC_COMM_SELF,&C_dense);CHKERRQ(ierr);
1197     ierr = MatSetSizes(C_dense,A->rmap->n,matcoloring->ncolors,A->rmap->n,matcoloring->ncolors);CHKERRQ(ierr);
1198     ierr = MatSetType(C_dense,MATSEQDENSE);CHKERRQ(ierr);
1199     ierr = MatSeqDenseSetPreallocation(C_dense,NULL);CHKERRQ(ierr);
1200 
1201     Bt_dense->assembled = PETSC_TRUE;
1202     abt->ABt_den  = C_dense;
1203 
1204 #if defined(PETSC_USE_INFO)
1205     {
1206       Mat_SeqAIJ *c = (Mat_SeqAIJ*)(*C)->data;
1207       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);
1208     }
1209 #endif
1210   }
1211   /* clean up */
1212   ierr = MatDestroy(&Bt);CHKERRQ(ierr);
1213   ierr = MatRestoreSymbolicTranspose_SeqAIJ(B,&bti,&btj);CHKERRQ(ierr);
1214   PetscFunctionReturn(0);
1215 }
1216 
1217 PetscErrorCode MatMatTransposeMultNumeric_SeqAIJ_SeqAIJ(Mat A,Mat B,Mat C)
1218 {
1219   PetscErrorCode      ierr;
1220   Mat_SeqAIJ          *a   =(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c=(Mat_SeqAIJ*)C->data;
1221   PetscInt            *ai  =a->i,*aj=a->j,*bi=b->i,*bj=b->j,anzi,bnzj,nexta,nextb,*acol,*bcol,brow;
1222   PetscInt            cm   =C->rmap->n,*ci=c->i,*cj=c->j,i,j,cnzi,*ccol;
1223   PetscLogDouble      flops=0.0;
1224   MatScalar           *aa  =a->a,*aval,*ba=b->a,*bval,*ca,*cval;
1225   Mat_MatMatTransMult *abt = c->abt;
1226 
1227   PetscFunctionBegin;
1228   /* clear old values in C */
1229   if (!c->a) {
1230     ierr      = PetscCalloc1(ci[cm]+1,&ca);CHKERRQ(ierr);
1231     c->a      = ca;
1232     c->free_a = PETSC_TRUE;
1233   } else {
1234     ca =  c->a;
1235     ierr = PetscArrayzero(ca,ci[cm]+1);CHKERRQ(ierr);
1236   }
1237 
1238   if (abt->usecoloring) {
1239     MatTransposeColoring matcoloring = abt->matcoloring;
1240     Mat                  Bt_dense,C_dense = abt->ABt_den;
1241 
1242     /* Get Bt_dense by Apply MatTransposeColoring to B */
1243     Bt_dense = abt->Bt_den;
1244     ierr = MatTransColoringApplySpToDen(matcoloring,B,Bt_dense);CHKERRQ(ierr);
1245 
1246     /* C_dense = A*Bt_dense */
1247     ierr = MatMatMultNumeric_SeqAIJ_SeqDense(A,Bt_dense,C_dense);CHKERRQ(ierr);
1248 
1249     /* Recover C from C_dense */
1250     ierr = MatTransColoringApplyDenToSp(matcoloring,C_dense,C);CHKERRQ(ierr);
1251     PetscFunctionReturn(0);
1252   }
1253 
1254   for (i=0; i<cm; i++) {
1255     anzi = ai[i+1] - ai[i];
1256     acol = aj + ai[i];
1257     aval = aa + ai[i];
1258     cnzi = ci[i+1] - ci[i];
1259     ccol = cj + ci[i];
1260     cval = ca + ci[i];
1261     for (j=0; j<cnzi; j++) {
1262       brow = ccol[j];
1263       bnzj = bi[brow+1] - bi[brow];
1264       bcol = bj + bi[brow];
1265       bval = ba + bi[brow];
1266 
1267       /* perform sparse inner-product c(i,j)=A[i,:]*B[j,:]^T */
1268       nexta = 0; nextb = 0;
1269       while (nexta<anzi && nextb<bnzj) {
1270         while (nexta < anzi && acol[nexta] < bcol[nextb]) nexta++;
1271         if (nexta == anzi) break;
1272         while (nextb < bnzj && acol[nexta] > bcol[nextb]) nextb++;
1273         if (nextb == bnzj) break;
1274         if (acol[nexta] == bcol[nextb]) {
1275           cval[j] += aval[nexta]*bval[nextb];
1276           nexta++; nextb++;
1277           flops += 2;
1278         }
1279       }
1280     }
1281   }
1282   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1283   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1284   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
1285   PetscFunctionReturn(0);
1286 }
1287 
1288 PetscErrorCode MatDestroy_SeqAIJ_MatTransMatMult(Mat A)
1289 {
1290   PetscErrorCode      ierr;
1291   Mat_SeqAIJ          *a = (Mat_SeqAIJ*)A->data;
1292   Mat_MatTransMatMult *atb = a->atb;
1293 
1294   PetscFunctionBegin;
1295   if (atb) {
1296     ierr = MatDestroy(&atb->At);CHKERRQ(ierr);
1297     ierr = (*atb->destroy)(A);CHKERRQ(ierr);
1298   }
1299   ierr = PetscFree(atb);CHKERRQ(ierr);
1300   PetscFunctionReturn(0);
1301 }
1302 
1303 PetscErrorCode MatTransposeMatMult_SeqAIJ_SeqAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
1304 {
1305   PetscErrorCode      ierr;
1306   const char          *algTypes[2] = {"matmatmult","outerproduct"};
1307   PetscInt            alg=0; /* set default algorithm */
1308   Mat                 At;
1309   Mat_MatTransMatMult *atb;
1310   Mat_SeqAIJ          *c;
1311 
1312   PetscFunctionBegin;
1313   if (scall == MAT_INITIAL_MATRIX) {
1314     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)A),((PetscObject)A)->prefix,"MatTransposeMatMult","Mat");CHKERRQ(ierr);
1315     ierr = PetscOptionsEList("-mattransposematmult_via","Algorithmic approach","MatTransposeMatMult",algTypes,2,algTypes[0],&alg,NULL);CHKERRQ(ierr);
1316     ierr = PetscOptionsEnd();CHKERRQ(ierr);
1317 
1318     switch (alg) {
1319     case 1:
1320       ierr = MatTransposeMatMultSymbolic_SeqAIJ_SeqAIJ(A,B,fill,C);CHKERRQ(ierr);
1321       break;
1322     default:
1323       ierr = PetscNew(&atb);CHKERRQ(ierr);
1324       ierr = MatTranspose_SeqAIJ(A,MAT_INITIAL_MATRIX,&At);CHKERRQ(ierr);
1325       ierr = MatMatMult_SeqAIJ_SeqAIJ(At,B,MAT_INITIAL_MATRIX,fill,C);CHKERRQ(ierr);
1326 
1327       c                  = (Mat_SeqAIJ*)(*C)->data;
1328       c->atb             = atb;
1329       atb->At            = At;
1330       atb->destroy       = (*C)->ops->destroy;
1331       (*C)->ops->destroy = MatDestroy_SeqAIJ_MatTransMatMult;
1332 
1333       break;
1334     }
1335   }
1336   if (alg) {
1337     ierr = (*(*C)->ops->mattransposemultnumeric)(A,B,*C);CHKERRQ(ierr);
1338   } else if (!alg && scall == MAT_REUSE_MATRIX) {
1339     c   = (Mat_SeqAIJ*)(*C)->data;
1340     atb = c->atb;
1341     At  = atb->At;
1342     ierr = MatTranspose_SeqAIJ(A,MAT_REUSE_MATRIX,&At);CHKERRQ(ierr);
1343     ierr = MatMatMult_SeqAIJ_SeqAIJ(At,B,MAT_REUSE_MATRIX,fill,C);CHKERRQ(ierr);
1344   }
1345   PetscFunctionReturn(0);
1346 }
1347 
1348 PetscErrorCode MatTransposeMatMultSymbolic_SeqAIJ_SeqAIJ(Mat A,Mat B,PetscReal fill,Mat *C)
1349 {
1350   PetscErrorCode ierr;
1351   Mat            At;
1352   PetscInt       *ati,*atj;
1353 
1354   PetscFunctionBegin;
1355   /* create symbolic At */
1356   ierr = MatGetSymbolicTranspose_SeqAIJ(A,&ati,&atj);CHKERRQ(ierr);
1357   ierr = MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,A->cmap->n,A->rmap->n,ati,atj,NULL,&At);CHKERRQ(ierr);
1358   ierr = MatSetBlockSizes(At,PetscAbs(A->cmap->bs),PetscAbs(B->cmap->bs));CHKERRQ(ierr);
1359   ierr = MatSetType(At,((PetscObject)A)->type_name);CHKERRQ(ierr);
1360 
1361   /* get symbolic C=At*B */
1362   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(At,B,fill,C);CHKERRQ(ierr);
1363 
1364   /* clean up */
1365   ierr = MatDestroy(&At);CHKERRQ(ierr);
1366   ierr = MatRestoreSymbolicTranspose_SeqAIJ(A,&ati,&atj);CHKERRQ(ierr);
1367 
1368   (*C)->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_SeqAIJ_SeqAIJ;
1369   PetscFunctionReturn(0);
1370 }
1371 
1372 PetscErrorCode MatTransposeMatMultNumeric_SeqAIJ_SeqAIJ(Mat A,Mat B,Mat C)
1373 {
1374   PetscErrorCode ierr;
1375   Mat_SeqAIJ     *a   =(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c=(Mat_SeqAIJ*)C->data;
1376   PetscInt       am   =A->rmap->n,anzi,*ai=a->i,*aj=a->j,*bi=b->i,*bj,bnzi,nextb;
1377   PetscInt       cm   =C->rmap->n,*ci=c->i,*cj=c->j,crow,*cjj,i,j,k;
1378   PetscLogDouble flops=0.0;
1379   MatScalar      *aa  =a->a,*ba,*ca,*caj;
1380 
1381   PetscFunctionBegin;
1382   if (!c->a) {
1383     ierr = PetscCalloc1(ci[cm]+1,&ca);CHKERRQ(ierr);
1384 
1385     c->a      = ca;
1386     c->free_a = PETSC_TRUE;
1387   } else {
1388     ca   = c->a;
1389     ierr = PetscArrayzero(ca,ci[cm]);CHKERRQ(ierr);
1390   }
1391 
1392   /* compute A^T*B using outer product (A^T)[:,i]*B[i,:] */
1393   for (i=0; i<am; i++) {
1394     bj   = b->j + bi[i];
1395     ba   = b->a + bi[i];
1396     bnzi = bi[i+1] - bi[i];
1397     anzi = ai[i+1] - ai[i];
1398     for (j=0; j<anzi; j++) {
1399       nextb = 0;
1400       crow  = *aj++;
1401       cjj   = cj + ci[crow];
1402       caj   = ca + ci[crow];
1403       /* perform sparse axpy operation.  Note cjj includes bj. */
1404       for (k=0; nextb<bnzi; k++) {
1405         if (cjj[k] == *(bj+nextb)) { /* ccol == bcol */
1406           caj[k] += (*aa)*(*(ba+nextb));
1407           nextb++;
1408         }
1409       }
1410       flops += 2*bnzi;
1411       aa++;
1412     }
1413   }
1414 
1415   /* Assemble the final matrix and clean up */
1416   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1417   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1418   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
1419   PetscFunctionReturn(0);
1420 }
1421 
1422 PETSC_INTERN PetscErrorCode MatMatMult_SeqAIJ_SeqDense(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
1423 {
1424   PetscErrorCode ierr;
1425 
1426   PetscFunctionBegin;
1427   if (scall == MAT_INITIAL_MATRIX) {
1428     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
1429     ierr = MatMatMultSymbolic_SeqAIJ_SeqDense(A,B,fill,C);CHKERRQ(ierr);
1430     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
1431   }
1432   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
1433   ierr = MatMatMultNumeric_SeqAIJ_SeqDense(A,B,*C);CHKERRQ(ierr);
1434   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
1435   PetscFunctionReturn(0);
1436 }
1437 
1438 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqDense(Mat A,Mat B,PetscReal fill,Mat *C)
1439 {
1440   PetscErrorCode ierr;
1441 
1442   PetscFunctionBegin;
1443   ierr = MatMatMultSymbolic_SeqDense_SeqDense(A,B,0.0,C);CHKERRQ(ierr);
1444 
1445   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqDense;
1446   PetscFunctionReturn(0);
1447 }
1448 
1449 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqDense(Mat A,Mat B,Mat C)
1450 {
1451   Mat_SeqAIJ        *a=(Mat_SeqAIJ*)A->data;
1452   Mat_SeqDense      *bd = (Mat_SeqDense*)B->data;
1453   PetscErrorCode    ierr;
1454   PetscScalar       *c,*b,r1,r2,r3,r4,*c1,*c2,*c3,*c4,aatmp;
1455   const PetscScalar *aa,*b1,*b2,*b3,*b4;
1456   const PetscInt    *aj;
1457   PetscInt          cm=C->rmap->n,cn=B->cmap->n,bm=bd->lda,am=A->rmap->n;
1458   PetscInt          am4=4*am,bm4=4*bm,col,i,j,n,ajtmp;
1459 
1460   PetscFunctionBegin;
1461   if (!cm || !cn) PetscFunctionReturn(0);
1462   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);
1463   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);
1464   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);
1465   b = bd->v;
1466   ierr = MatDenseGetArray(C,&c);CHKERRQ(ierr);
1467   b1 = b; b2 = b1 + bm; b3 = b2 + bm; b4 = b3 + bm;
1468   c1 = c; c2 = c1 + am; c3 = c2 + am; c4 = c3 + am;
1469   for (col=0; col<cn-4; col += 4) {  /* over columns of C */
1470     for (i=0; i<am; i++) {        /* over rows of C in those columns */
1471       r1 = r2 = r3 = r4 = 0.0;
1472       n  = a->i[i+1] - a->i[i];
1473       aj = a->j + a->i[i];
1474       aa = a->a + a->i[i];
1475       for (j=0; j<n; j++) {
1476         aatmp = aa[j]; ajtmp = aj[j];
1477         r1 += aatmp*b1[ajtmp];
1478         r2 += aatmp*b2[ajtmp];
1479         r3 += aatmp*b3[ajtmp];
1480         r4 += aatmp*b4[ajtmp];
1481       }
1482       c1[i] = r1;
1483       c2[i] = r2;
1484       c3[i] = r3;
1485       c4[i] = r4;
1486     }
1487     b1 += bm4; b2 += bm4; b3 += bm4; b4 += bm4;
1488     c1 += am4; c2 += am4; c3 += am4; c4 += am4;
1489   }
1490   for (; col<cn; col++) {   /* over extra columns of C */
1491     for (i=0; i<am; i++) {  /* over rows of C in those columns */
1492       r1 = 0.0;
1493       n  = a->i[i+1] - a->i[i];
1494       aj = a->j + a->i[i];
1495       aa = a->a + a->i[i];
1496       for (j=0; j<n; j++) {
1497         r1 += aa[j]*b1[aj[j]];
1498       }
1499       c1[i] = r1;
1500     }
1501     b1 += bm;
1502     c1 += am;
1503   }
1504   ierr = PetscLogFlops(cn*(2.0*a->nz));CHKERRQ(ierr);
1505   ierr = MatDenseRestoreArray(C,&c);CHKERRQ(ierr);
1506   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1507   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1508   PetscFunctionReturn(0);
1509 }
1510 
1511 /*
1512    Note very similar to MatMult_SeqAIJ(), should generate both codes from same base
1513 */
1514 PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat A,Mat B,Mat C)
1515 {
1516   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data;
1517   Mat_SeqDense   *bd = (Mat_SeqDense*)B->data;
1518   PetscErrorCode ierr;
1519   PetscScalar    *b,*c,r1,r2,r3,r4,*b1,*b2,*b3,*b4;
1520   MatScalar      *aa;
1521   PetscInt       cm  = C->rmap->n, cn=B->cmap->n, bm=bd->lda, col, i,j,n,*aj, am = A->rmap->n,*ii,arm;
1522   PetscInt       am2 = 2*am, am3 = 3*am,  bm4 = 4*bm,colam,*ridx;
1523 
1524   PetscFunctionBegin;
1525   if (!cm || !cn) PetscFunctionReturn(0);
1526   b = bd->v;
1527   ierr = MatDenseGetArray(C,&c);CHKERRQ(ierr);
1528   b1   = b; b2 = b1 + bm; b3 = b2 + bm; b4 = b3 + bm;
1529 
1530   if (a->compressedrow.use) { /* use compressed row format */
1531     for (col=0; col<cn-4; col += 4) {  /* over columns of C */
1532       colam = col*am;
1533       arm   = a->compressedrow.nrows;
1534       ii    = a->compressedrow.i;
1535       ridx  = a->compressedrow.rindex;
1536       for (i=0; i<arm; i++) {        /* over rows of C in those columns */
1537         r1 = r2 = r3 = r4 = 0.0;
1538         n  = ii[i+1] - ii[i];
1539         aj = a->j + ii[i];
1540         aa = a->a + ii[i];
1541         for (j=0; j<n; j++) {
1542           r1 += (*aa)*b1[*aj];
1543           r2 += (*aa)*b2[*aj];
1544           r3 += (*aa)*b3[*aj];
1545           r4 += (*aa++)*b4[*aj++];
1546         }
1547         c[colam       + ridx[i]] += r1;
1548         c[colam + am  + ridx[i]] += r2;
1549         c[colam + am2 + ridx[i]] += r3;
1550         c[colam + am3 + ridx[i]] += r4;
1551       }
1552       b1 += bm4;
1553       b2 += bm4;
1554       b3 += bm4;
1555       b4 += bm4;
1556     }
1557     for (; col<cn; col++) {     /* over extra columns of C */
1558       colam = col*am;
1559       arm   = a->compressedrow.nrows;
1560       ii    = a->compressedrow.i;
1561       ridx  = a->compressedrow.rindex;
1562       for (i=0; i<arm; i++) {  /* over rows of C in those columns */
1563         r1 = 0.0;
1564         n  = ii[i+1] - ii[i];
1565         aj = a->j + ii[i];
1566         aa = a->a + ii[i];
1567 
1568         for (j=0; j<n; j++) {
1569           r1 += (*aa++)*b1[*aj++];
1570         }
1571         c[colam + ridx[i]] += r1;
1572       }
1573       b1 += bm;
1574     }
1575   } else {
1576     for (col=0; col<cn-4; col += 4) {  /* over columns of C */
1577       colam = col*am;
1578       for (i=0; i<am; i++) {        /* over rows of C in those columns */
1579         r1 = r2 = r3 = r4 = 0.0;
1580         n  = a->i[i+1] - a->i[i];
1581         aj = a->j + a->i[i];
1582         aa = a->a + a->i[i];
1583         for (j=0; j<n; j++) {
1584           r1 += (*aa)*b1[*aj];
1585           r2 += (*aa)*b2[*aj];
1586           r3 += (*aa)*b3[*aj];
1587           r4 += (*aa++)*b4[*aj++];
1588         }
1589         c[colam + i]       += r1;
1590         c[colam + am + i]  += r2;
1591         c[colam + am2 + i] += r3;
1592         c[colam + am3 + i] += r4;
1593       }
1594       b1 += bm4;
1595       b2 += bm4;
1596       b3 += bm4;
1597       b4 += bm4;
1598     }
1599     for (; col<cn; col++) {     /* over extra columns of C */
1600       colam = col*am;
1601       for (i=0; i<am; i++) {  /* over rows of C in those columns */
1602         r1 = 0.0;
1603         n  = a->i[i+1] - a->i[i];
1604         aj = a->j + a->i[i];
1605         aa = a->a + a->i[i];
1606 
1607         for (j=0; j<n; j++) {
1608           r1 += (*aa++)*b1[*aj++];
1609         }
1610         c[colam + i] += r1;
1611       }
1612       b1 += bm;
1613     }
1614   }
1615   ierr = PetscLogFlops(cn*2.0*a->nz);CHKERRQ(ierr);
1616   ierr = MatDenseRestoreArray(C,&c);CHKERRQ(ierr);
1617   PetscFunctionReturn(0);
1618 }
1619 
1620 PetscErrorCode  MatTransColoringApplySpToDen_SeqAIJ(MatTransposeColoring coloring,Mat B,Mat Btdense)
1621 {
1622   PetscErrorCode ierr;
1623   Mat_SeqAIJ     *b       = (Mat_SeqAIJ*)B->data;
1624   Mat_SeqDense   *btdense = (Mat_SeqDense*)Btdense->data;
1625   PetscInt       *bi      = b->i,*bj=b->j;
1626   PetscInt       m        = Btdense->rmap->n,n=Btdense->cmap->n,j,k,l,col,anz,*btcol,brow,ncolumns;
1627   MatScalar      *btval,*btval_den,*ba=b->a;
1628   PetscInt       *columns=coloring->columns,*colorforcol=coloring->colorforcol,ncolors=coloring->ncolors;
1629 
1630   PetscFunctionBegin;
1631   btval_den=btdense->v;
1632   ierr     = PetscArrayzero(btval_den,m*n);CHKERRQ(ierr);
1633   for (k=0; k<ncolors; k++) {
1634     ncolumns = coloring->ncolumns[k];
1635     for (l=0; l<ncolumns; l++) { /* insert a row of B to a column of Btdense */
1636       col   = *(columns + colorforcol[k] + l);
1637       btcol = bj + bi[col];
1638       btval = ba + bi[col];
1639       anz   = bi[col+1] - bi[col];
1640       for (j=0; j<anz; j++) {
1641         brow            = btcol[j];
1642         btval_den[brow] = btval[j];
1643       }
1644     }
1645     btval_den += m;
1646   }
1647   PetscFunctionReturn(0);
1648 }
1649 
1650 PetscErrorCode MatTransColoringApplyDenToSp_SeqAIJ(MatTransposeColoring matcoloring,Mat Cden,Mat Csp)
1651 {
1652   PetscErrorCode    ierr;
1653   Mat_SeqAIJ        *csp = (Mat_SeqAIJ*)Csp->data;
1654   const PetscScalar *ca_den,*ca_den_ptr;
1655   PetscScalar       *ca=csp->a;
1656   PetscInt          k,l,m=Cden->rmap->n,ncolors=matcoloring->ncolors;
1657   PetscInt          brows=matcoloring->brows,*den2sp=matcoloring->den2sp;
1658   PetscInt          nrows,*row,*idx;
1659   PetscInt          *rows=matcoloring->rows,*colorforrow=matcoloring->colorforrow;
1660 
1661   PetscFunctionBegin;
1662   ierr   = MatDenseGetArrayRead(Cden,&ca_den);CHKERRQ(ierr);
1663 
1664   if (brows > 0) {
1665     PetscInt *lstart,row_end,row_start;
1666     lstart = matcoloring->lstart;
1667     ierr = PetscArrayzero(lstart,ncolors);CHKERRQ(ierr);
1668 
1669     row_end = brows;
1670     if (row_end > m) row_end = m;
1671     for (row_start=0; row_start<m; row_start+=brows) { /* loop over row blocks of Csp */
1672       ca_den_ptr = ca_den;
1673       for (k=0; k<ncolors; k++) { /* loop over colors (columns of Cden) */
1674         nrows = matcoloring->nrows[k];
1675         row   = rows  + colorforrow[k];
1676         idx   = den2sp + colorforrow[k];
1677         for (l=lstart[k]; l<nrows; l++) {
1678           if (row[l] >= row_end) {
1679             lstart[k] = l;
1680             break;
1681           } else {
1682             ca[idx[l]] = ca_den_ptr[row[l]];
1683           }
1684         }
1685         ca_den_ptr += m;
1686       }
1687       row_end += brows;
1688       if (row_end > m) row_end = m;
1689     }
1690   } else { /* non-blocked impl: loop over columns of Csp - slow if Csp is large */
1691     ca_den_ptr = ca_den;
1692     for (k=0; k<ncolors; k++) {
1693       nrows = matcoloring->nrows[k];
1694       row   = rows  + colorforrow[k];
1695       idx   = den2sp + colorforrow[k];
1696       for (l=0; l<nrows; l++) {
1697         ca[idx[l]] = ca_den_ptr[row[l]];
1698       }
1699       ca_den_ptr += m;
1700     }
1701   }
1702 
1703   ierr = MatDenseRestoreArrayRead(Cden,&ca_den);CHKERRQ(ierr);
1704 #if defined(PETSC_USE_INFO)
1705   if (matcoloring->brows > 0) {
1706     ierr = PetscInfo1(Csp,"Loop over %D row blocks for den2sp\n",brows);CHKERRQ(ierr);
1707   } else {
1708     ierr = PetscInfo(Csp,"Loop over colors/columns of Cden, inefficient for large sparse matrix product \n");CHKERRQ(ierr);
1709   }
1710 #endif
1711   PetscFunctionReturn(0);
1712 }
1713 
1714 PetscErrorCode MatTransposeColoringCreate_SeqAIJ(Mat mat,ISColoring iscoloring,MatTransposeColoring c)
1715 {
1716   PetscErrorCode ierr;
1717   PetscInt       i,n,nrows,Nbs,j,k,m,ncols,col,cm;
1718   const PetscInt *is,*ci,*cj,*row_idx;
1719   PetscInt       nis = iscoloring->n,*rowhit,bs = 1;
1720   IS             *isa;
1721   Mat_SeqAIJ     *csp = (Mat_SeqAIJ*)mat->data;
1722   PetscInt       *colorforrow,*rows,*rows_i,*idxhit,*spidx,*den2sp,*den2sp_i;
1723   PetscInt       *colorforcol,*columns,*columns_i,brows;
1724   PetscBool      flg;
1725 
1726   PetscFunctionBegin;
1727   ierr = ISColoringGetIS(iscoloring,PETSC_USE_POINTER,PETSC_IGNORE,&isa);CHKERRQ(ierr);
1728 
1729   /* bs >1 is not being tested yet! */
1730   Nbs       = mat->cmap->N/bs;
1731   c->M      = mat->rmap->N/bs;  /* set total rows, columns and local rows */
1732   c->N      = Nbs;
1733   c->m      = c->M;
1734   c->rstart = 0;
1735   c->brows  = 100;
1736 
1737   c->ncolors = nis;
1738   ierr = PetscMalloc3(nis,&c->ncolumns,nis,&c->nrows,nis+1,&colorforrow);CHKERRQ(ierr);
1739   ierr = PetscMalloc1(csp->nz+1,&rows);CHKERRQ(ierr);
1740   ierr = PetscMalloc1(csp->nz+1,&den2sp);CHKERRQ(ierr);
1741 
1742   brows = c->brows;
1743   ierr = PetscOptionsGetInt(NULL,NULL,"-matden2sp_brows",&brows,&flg);CHKERRQ(ierr);
1744   if (flg) c->brows = brows;
1745   if (brows > 0) {
1746     ierr = PetscMalloc1(nis+1,&c->lstart);CHKERRQ(ierr);
1747   }
1748 
1749   colorforrow[0] = 0;
1750   rows_i         = rows;
1751   den2sp_i       = den2sp;
1752 
1753   ierr = PetscMalloc1(nis+1,&colorforcol);CHKERRQ(ierr);
1754   ierr = PetscMalloc1(Nbs+1,&columns);CHKERRQ(ierr);
1755 
1756   colorforcol[0] = 0;
1757   columns_i      = columns;
1758 
1759   /* get column-wise storage of mat */
1760   ierr = MatGetColumnIJ_SeqAIJ_Color(mat,0,PETSC_FALSE,PETSC_FALSE,&ncols,&ci,&cj,&spidx,NULL);CHKERRQ(ierr);
1761 
1762   cm   = c->m;
1763   ierr = PetscMalloc1(cm+1,&rowhit);CHKERRQ(ierr);
1764   ierr = PetscMalloc1(cm+1,&idxhit);CHKERRQ(ierr);
1765   for (i=0; i<nis; i++) { /* loop over color */
1766     ierr = ISGetLocalSize(isa[i],&n);CHKERRQ(ierr);
1767     ierr = ISGetIndices(isa[i],&is);CHKERRQ(ierr);
1768 
1769     c->ncolumns[i] = n;
1770     if (n) {
1771       ierr = PetscArraycpy(columns_i,is,n);CHKERRQ(ierr);
1772     }
1773     colorforcol[i+1] = colorforcol[i] + n;
1774     columns_i       += n;
1775 
1776     /* fast, crude version requires O(N*N) work */
1777     ierr = PetscArrayzero(rowhit,cm);CHKERRQ(ierr);
1778 
1779     for (j=0; j<n; j++) { /* loop over columns*/
1780       col     = is[j];
1781       row_idx = cj + ci[col];
1782       m       = ci[col+1] - ci[col];
1783       for (k=0; k<m; k++) { /* loop over columns marking them in rowhit */
1784         idxhit[*row_idx]   = spidx[ci[col] + k];
1785         rowhit[*row_idx++] = col + 1;
1786       }
1787     }
1788     /* count the number of hits */
1789     nrows = 0;
1790     for (j=0; j<cm; j++) {
1791       if (rowhit[j]) nrows++;
1792     }
1793     c->nrows[i]      = nrows;
1794     colorforrow[i+1] = colorforrow[i] + nrows;
1795 
1796     nrows = 0;
1797     for (j=0; j<cm; j++) { /* loop over rows */
1798       if (rowhit[j]) {
1799         rows_i[nrows]   = j;
1800         den2sp_i[nrows] = idxhit[j];
1801         nrows++;
1802       }
1803     }
1804     den2sp_i += nrows;
1805 
1806     ierr    = ISRestoreIndices(isa[i],&is);CHKERRQ(ierr);
1807     rows_i += nrows;
1808   }
1809   ierr = MatRestoreColumnIJ_SeqAIJ_Color(mat,0,PETSC_FALSE,PETSC_FALSE,&ncols,&ci,&cj,&spidx,NULL);CHKERRQ(ierr);
1810   ierr = PetscFree(rowhit);CHKERRQ(ierr);
1811   ierr = ISColoringRestoreIS(iscoloring,PETSC_USE_POINTER,&isa);CHKERRQ(ierr);
1812   if (csp->nz != colorforrow[nis]) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_PLIB,"csp->nz %d != colorforrow[nis] %d",csp->nz,colorforrow[nis]);
1813 
1814   c->colorforrow = colorforrow;
1815   c->rows        = rows;
1816   c->den2sp      = den2sp;
1817   c->colorforcol = colorforcol;
1818   c->columns     = columns;
1819 
1820   ierr = PetscFree(idxhit);CHKERRQ(ierr);
1821   PetscFunctionReturn(0);
1822 }
1823 
1824 /* The combine method has done the symbolic and numeric in the first phase, and so we just return here */
1825 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ_Combined(Mat A,Mat B,Mat C)
1826 {
1827   PetscFunctionBegin;
1828   /* Empty function */
1829   PetscFunctionReturn(0);
1830 }
1831 
1832 /* This algorithm combines the symbolic and numeric phase of matrix-matrix multiplication. */
1833 PetscErrorCode MatMatMult_SeqAIJ_SeqAIJ_Combined(Mat A,Mat B,PetscReal fill,Mat *C)
1834 {
1835   PetscErrorCode     ierr;
1836   PetscLogDouble     flops=0.0;
1837   Mat_SeqAIJ         *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
1838   const PetscInt     *ai=a->i,*bi=b->i;
1839   PetscInt           *ci,*cj,*cj_i;
1840   PetscScalar        *ca,*ca_i;
1841   PetscInt           b_maxmemrow,c_maxmem,a_col;
1842   PetscInt           am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
1843   PetscInt           i,k,ndouble=0;
1844   PetscReal          afill;
1845   PetscScalar        *c_row_val_dense;
1846   PetscBool          *c_row_idx_flags;
1847   PetscInt           *aj_i=a->j;
1848   PetscScalar        *aa_i=a->a;
1849 
1850   PetscFunctionBegin;
1851 
1852   /* Step 1: Determine upper bounds on memory for C and allocate memory */
1853   /* This should be enough for almost all matrices. If still more memory is needed, it is reallocated later. */
1854   c_maxmem    = 8*(ai[am]+bi[bm]);
1855   b_maxmemrow = PetscMin(bi[bm],bn);
1856   ierr  = PetscMalloc1(am+1,&ci);CHKERRQ(ierr);
1857   ierr  = PetscCalloc1(bn,&c_row_val_dense);CHKERRQ(ierr);
1858   ierr  = PetscCalloc1(bn,&c_row_idx_flags);CHKERRQ(ierr);
1859   ierr  = PetscMalloc1(c_maxmem,&cj);CHKERRQ(ierr);
1860   ierr  = PetscMalloc1(c_maxmem,&ca);CHKERRQ(ierr);
1861   ca_i  = ca;
1862   cj_i  = cj;
1863   ci[0] = 0;
1864   for (i=0; i<am; i++) {
1865     /* Step 2: Initialize the dense row vector for C  */
1866     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 */
1867     PetscInt       cnzi = 0;
1868     PetscInt       *bj_i;
1869     PetscScalar    *ba_i;
1870     /* If the number of indices in C so far + the max number of columns in the next row > c_maxmem  -> allocate more memory
1871        Usually, there is enough memory in the first place, so this is not executed. */
1872     while (ci[i] + b_maxmemrow > c_maxmem) {
1873       c_maxmem *= 2;
1874       ndouble++;
1875       ierr = PetscRealloc(sizeof(PetscInt)*c_maxmem,&cj);CHKERRQ(ierr);
1876       ierr = PetscRealloc(sizeof(PetscScalar)*c_maxmem,&ca);CHKERRQ(ierr);
1877     }
1878 
1879     /* Step 3: Do the numerical calculations */
1880     for (a_col=0; a_col<anzi; a_col++) {          /* iterate over all non zero values in a row of A */
1881       PetscInt       a_col_index = aj_i[a_col];
1882       const PetscInt bnzi        = bi[a_col_index+1] - bi[a_col_index];
1883       flops += 2*bnzi;
1884       bj_i   = b->j + bi[a_col_index];   /* points to the current row in bj */
1885       ba_i   = b->a + bi[a_col_index];   /* points to the current row in ba */
1886       for (k=0; k<bnzi; ++k) { /* iterate over all non zeros of this row in B */
1887         if (c_row_idx_flags[bj_i[k]] == PETSC_FALSE) {
1888           cj_i[cnzi++]             = bj_i[k];
1889           c_row_idx_flags[bj_i[k]] = PETSC_TRUE;
1890         }
1891         c_row_val_dense[bj_i[k]] += aa_i[a_col] * ba_i[k];
1892       }
1893     }
1894 
1895     /* Sort array */
1896     ierr = PetscSortInt(cnzi,cj_i);CHKERRQ(ierr);
1897     /* Step 4 */
1898     for (k=0; k<cnzi; k++) {
1899       ca_i[k]                  = c_row_val_dense[cj_i[k]];
1900       c_row_val_dense[cj_i[k]] = 0.;
1901       c_row_idx_flags[cj_i[k]] = PETSC_FALSE;
1902     }
1903     /* terminate current row */
1904     aa_i   += anzi;
1905     aj_i   += anzi;
1906     ca_i   += cnzi;
1907     cj_i   += cnzi;
1908     ci[i+1] = ci[i] + cnzi;
1909     flops  += cnzi;
1910   }
1911 
1912   /* Step 5 */
1913   /* Create the new matrix */
1914   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
1915   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
1916   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
1917 
1918   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
1919   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
1920   c          = (Mat_SeqAIJ*)((*C)->data);
1921   c->a       = ca;
1922   c->free_a  = PETSC_TRUE;
1923   c->free_ij = PETSC_TRUE;
1924   c->nonew   = 0;
1925 
1926   /* set MatInfo */
1927   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
1928   if (afill < 1.0) afill = 1.0;
1929   c->maxnz                     = ci[am];
1930   c->nz                        = ci[am];
1931   (*C)->info.mallocs           = ndouble;
1932   (*C)->info.fill_ratio_given  = fill;
1933   (*C)->info.fill_ratio_needed = afill;
1934   (*C)->ops->matmultnumeric    = MatMatMultNumeric_SeqAIJ_SeqAIJ_Combined;
1935 
1936   ierr = PetscFree(c_row_val_dense);CHKERRQ(ierr);
1937   ierr = PetscFree(c_row_idx_flags);CHKERRQ(ierr);
1938 
1939   ierr = MatAssemblyBegin(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1940   ierr = MatAssemblyEnd(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1941   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
1942   PetscFunctionReturn(0);
1943 }
1944