xref: /petsc/src/mat/impls/aij/seq/matmatmult.c (revision a685ae26bbf9fb229cf02e15e73d16b8a9cd69e6)
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  static PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_LLCondensed(Mat,Mat,PetscReal,Mat*);
14 
15  #if defined(PETSC_HAVE_HYPRE)
16  PETSC_INTERN PetscErrorCode MatMatMultSymbolic_AIJ_AIJ_wHYPRE(Mat,Mat,PetscReal,Mat*);
17  #endif
18 
19  PETSC_INTERN PetscErrorCode MatMatMult_SeqAIJ_SeqAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
20  {
21    PetscErrorCode ierr;
22  #if !defined(PETSC_HAVE_HYPRE)
23    const char     *algTypes[8] = {"sorted","scalable","scalable_fast","heap","btheap","llcondensed","combined","rowmerge"};
24    PetscInt       nalg = 8;
25  #else
26    const char     *algTypes[9] = {"sorted","scalable","scalable_fast","heap","btheap","llcondensed","combined","rowmerge","hypre"};
27    PetscInt       nalg = 9;
28  #endif
29    PetscInt       alg = 0; /* set default algorithm */
30    PetscBool      combined = PETSC_FALSE;  /* Indicates whether the symbolic stage already computed the numerical values. */
31 
32    PetscFunctionBegin;
33    if (scall == MAT_INITIAL_MATRIX) {
34      ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)A),((PetscObject)A)->prefix,"MatMatMult","Mat");CHKERRQ(ierr);
35      ierr = PetscOptionsEList("-matmatmult_via","Algorithmic approach","MatMatMult",algTypes,nalg,algTypes[0],&alg,NULL);CHKERRQ(ierr);
36      ierr = PetscOptionsEnd();CHKERRQ(ierr);
37      ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
38      switch (alg) {
39      case 1:
40        ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable(A,B,fill,C);CHKERRQ(ierr);
41        break;
42      case 2:
43        ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable_fast(A,B,fill,C);CHKERRQ(ierr);
44        break;
45      case 3:
46        ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Heap(A,B,fill,C);CHKERRQ(ierr);
47        break;
48      case 4:
49        ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_BTHeap(A,B,fill,C);CHKERRQ(ierr);
50        break;
51      case 5:
52        ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_LLCondensed(A,B,fill,C);CHKERRQ(ierr);
53        break;
54      case 6:
55        ierr = MatMatMult_SeqAIJ_SeqAIJ_Combined(A,B,fill,C);CHKERRQ(ierr);
56        combined = PETSC_TRUE;
57        break;
58     case 7:
59        ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_RowMerge(A,B,fill,C);CHKERRQ(ierr);
60        break;
61  #if defined(PETSC_HAVE_HYPRE)
62      case 8:
63        ierr = MatMatMultSymbolic_AIJ_AIJ_wHYPRE(A,B,fill,C);CHKERRQ(ierr);
64        break;
65  #endif
66      default:
67        ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(A,B,fill,C);CHKERRQ(ierr);
68       break;
69      }
70      ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
71    }
72 
73    ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
74    if (!combined) {
75      ierr = (*(*C)->ops->matmultnumeric)(A,B,*C);CHKERRQ(ierr);
76    }
77    ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
78    PetscFunctionReturn(0);
79  }
80 
81  static PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_LLCondensed(Mat A,Mat B,PetscReal fill,Mat *C)
82  {
83    PetscErrorCode     ierr;
84    Mat_SeqAIJ         *a =(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
85    PetscInt           *ai=a->i,*bi=b->i,*ci,*cj;
86    PetscInt           am =A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
87    PetscReal          afill;
88    PetscInt           i,j,anzi,brow,bnzj,cnzi,*bj,*aj,*lnk,ndouble=0,Crmax;
89    PetscTable         ta;
90    PetscBT            lnkbt;
91    PetscFreeSpaceList free_space=NULL,current_space=NULL;
92 
93    PetscFunctionBegin;
94    /* Get ci and cj */
95    /*---------------*/
96    /* Allocate ci array, arrays for fill computation and */
97    /* free space for accumulating nonzero column info */
98    ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
99    ci[0] = 0;
100 
101    /* create and initialize a linked list */
102    ierr = PetscTableCreate(bn,bn,&ta);CHKERRQ(ierr);
103    MatRowMergeMax_SeqAIJ(b,bm,ta);
104    ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
105    ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
106 
107    ierr = PetscLLCondensedCreate(Crmax,bn,&lnk,&lnkbt);CHKERRQ(ierr);
108 
109    /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
110    ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
111 
112    current_space = free_space;
113 
114    /* Determine ci and cj */
115    for (i=0; i<am; i++) {
116      anzi = ai[i+1] - ai[i];
117      aj   = a->j + ai[i];
118      for (j=0; j<anzi; j++) {
119        brow = aj[j];
120        bnzj = bi[brow+1] - bi[brow];
121        bj   = b->j + bi[brow];
122        /* add non-zero cols of B into the sorted linked list lnk */
123        ierr = PetscLLCondensedAddSorted(bnzj,bj,lnk,lnkbt);CHKERRQ(ierr);
124      }
125      cnzi = lnk[0];
126 
127      /* If free space is not available, make more free space */
128      /* Double the amount of total space in the list */
129      if (current_space->local_remaining<cnzi) {
130        ierr = PetscFreeSpaceGet(PetscIntSumTruncate(cnzi,current_space->total_array_size),&current_space);CHKERRQ(ierr);
131        ndouble++;
132      }
133 
134      /* Copy data into free space, then initialize lnk */
135      ierr = PetscLLCondensedClean(bn,cnzi,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
136 
137      current_space->array           += cnzi;
138      current_space->local_used      += cnzi;
139      current_space->local_remaining -= cnzi;
140 
141      ci[i+1] = ci[i] + cnzi;
142    }
143 
144    /* Column indices are in the list of free space */
145    /* Allocate space for cj, initialize cj, and */
146    /* destroy list of free space and other temporary array(s) */
147    ierr = PetscMalloc1(ci[am]+1,&cj);CHKERRQ(ierr);
148    ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
149    ierr = PetscLLCondensedDestroy(lnk,lnkbt);CHKERRQ(ierr);
150 
151    /* put together the new symbolic matrix */
152    ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
153    ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
154    ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
155 
156   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
157   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
158   c                         = (Mat_SeqAIJ*)((*C)->data);
159   c->free_a                 = PETSC_FALSE;
160   c->free_ij                = PETSC_TRUE;
161   c->nonew                  = 0;
162   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ; /* fast, needs non-scalable O(bn) array 'abdense' */
163 
164   /* set MatInfo */
165   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
166   if (afill < 1.0) afill = 1.0;
167   c->maxnz                     = ci[am];
168   c->nz                        = ci[am];
169   (*C)->info.mallocs           = ndouble;
170   (*C)->info.fill_ratio_given  = fill;
171   (*C)->info.fill_ratio_needed = afill;
172 
173 #if defined(PETSC_USE_INFO)
174   if (ci[am]) {
175     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
176     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
177   } else {
178     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
179   }
180 #endif
181   PetscFunctionReturn(0);
182 }
183 
184 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ(Mat A,Mat B,Mat C)
185 {
186   PetscErrorCode ierr;
187   PetscLogDouble flops=0.0;
188   Mat_SeqAIJ     *a   = (Mat_SeqAIJ*)A->data;
189   Mat_SeqAIJ     *b   = (Mat_SeqAIJ*)B->data;
190   Mat_SeqAIJ     *c   = (Mat_SeqAIJ*)C->data;
191   PetscInt       *ai  =a->i,*aj=a->j,*bi=b->i,*bj=b->j,*bjj,*ci=c->i,*cj=c->j;
192   PetscInt       am   =A->rmap->n,cm=C->rmap->n;
193   PetscInt       i,j,k,anzi,bnzi,cnzi,brow;
194   PetscScalar    *aa=a->a,*ba=b->a,*baj,*ca,valtmp;
195   PetscScalar    *ab_dense;
196 
197   PetscFunctionBegin;
198   if (!c->a) { /* first call of MatMatMultNumeric_SeqAIJ_SeqAIJ, allocate ca and matmult_abdense */
199     ierr      = PetscMalloc1(ci[cm]+1,&ca);CHKERRQ(ierr);
200     c->a      = ca;
201     c->free_a = PETSC_TRUE;
202   } else {
203     ca        = c->a;
204   }
205   if (!c->matmult_abdense) {
206     ierr = PetscCalloc1(B->cmap->N,&ab_dense);CHKERRQ(ierr);
207     c->matmult_abdense = ab_dense;
208   } else {
209     ab_dense = c->matmult_abdense;
210   }
211 
212   /* clean old values in C */
213   ierr = PetscMemzero(ca,ci[cm]*sizeof(MatScalar));CHKERRQ(ierr);
214   /* Traverse A row-wise. */
215   /* Build the ith row in C by summing over nonzero columns in A, */
216   /* the rows of B corresponding to nonzeros of A. */
217   for (i=0; i<am; i++) {
218     anzi = ai[i+1] - ai[i];
219     for (j=0; j<anzi; j++) {
220       brow = aj[j];
221       bnzi = bi[brow+1] - bi[brow];
222       bjj  = bj + bi[brow];
223       baj  = ba + bi[brow];
224       /* perform dense axpy */
225       valtmp = aa[j];
226       for (k=0; k<bnzi; k++) {
227         ab_dense[bjj[k]] += valtmp*baj[k];
228       }
229       flops += 2*bnzi;
230     }
231     aj += anzi; aa += anzi;
232 
233     cnzi = ci[i+1] - ci[i];
234     for (k=0; k<cnzi; k++) {
235       ca[k]          += ab_dense[cj[k]];
236       ab_dense[cj[k]] = 0.0; /* zero ab_dense */
237     }
238     flops += cnzi;
239     cj    += cnzi; ca += cnzi;
240   }
241   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
242   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
243   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
244   PetscFunctionReturn(0);
245 }
246 
247 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable(Mat A,Mat B,Mat C)
248 {
249   PetscErrorCode ierr;
250   PetscLogDouble flops=0.0;
251   Mat_SeqAIJ     *a   = (Mat_SeqAIJ*)A->data;
252   Mat_SeqAIJ     *b   = (Mat_SeqAIJ*)B->data;
253   Mat_SeqAIJ     *c   = (Mat_SeqAIJ*)C->data;
254   PetscInt       *ai  = a->i,*aj=a->j,*bi=b->i,*bj=b->j,*bjj,*ci=c->i,*cj=c->j;
255   PetscInt       am   = A->rmap->N,cm=C->rmap->N;
256   PetscInt       i,j,k,anzi,bnzi,cnzi,brow;
257   PetscScalar    *aa=a->a,*ba=b->a,*baj,*ca=c->a,valtmp;
258   PetscInt       nextb;
259 
260   PetscFunctionBegin;
261   if (!ca) { /* first call of MatMatMultNumeric_SeqAIJ_SeqAIJ, allocate ca and matmult_abdense */
262     ierr      = PetscMalloc1(ci[cm]+1,&ca);CHKERRQ(ierr);
263     c->a      = ca;
264     c->free_a = PETSC_TRUE;
265   }
266 
267   /* clean old values in C */
268   ierr = PetscMemzero(ca,ci[cm]*sizeof(MatScalar));CHKERRQ(ierr);
269   /* Traverse A row-wise. */
270   /* Build the ith row in C by summing over nonzero columns in A, */
271   /* the rows of B corresponding to nonzeros of A. */
272   for (i=0; i<am; i++) {
273     anzi = ai[i+1] - ai[i];
274     cnzi = ci[i+1] - ci[i];
275     for (j=0; j<anzi; j++) {
276       brow = aj[j];
277       bnzi = bi[brow+1] - bi[brow];
278       bjj  = bj + bi[brow];
279       baj  = ba + bi[brow];
280       /* perform sparse axpy */
281       valtmp = aa[j];
282       nextb  = 0;
283       for (k=0; nextb<bnzi; k++) {
284         if (cj[k] == bjj[nextb]) { /* ccol == bcol */
285           ca[k] += valtmp*baj[nextb++];
286         }
287       }
288       flops += 2*bnzi;
289     }
290     aj += anzi; aa += anzi;
291     cj += cnzi; ca += cnzi;
292   }
293 
294   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
295   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
296   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
297   PetscFunctionReturn(0);
298 }
299 
300 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable_fast(Mat A,Mat B,PetscReal fill,Mat *C)
301 {
302   PetscErrorCode     ierr;
303   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
304   PetscInt           *ai = a->i,*bi=b->i,*ci,*cj;
305   PetscInt           am  = A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
306   MatScalar          *ca;
307   PetscReal          afill;
308   PetscInt           i,j,anzi,brow,bnzj,cnzi,*bj,*aj,*lnk,ndouble=0,Crmax;
309   PetscTable         ta;
310   PetscFreeSpaceList free_space=NULL,current_space=NULL;
311 
312   PetscFunctionBegin;
313   /* Get ci and cj - same as MatMatMultSymbolic_SeqAIJ_SeqAIJ except using PetscLLxxx_fast() */
314   /*-----------------------------------------------------------------------------------------*/
315   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
316   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
317   ci[0] = 0;
318 
319   /* create and initialize a linked list */
320   ierr = PetscTableCreate(bn,bn,&ta);CHKERRQ(ierr);
321   MatRowMergeMax_SeqAIJ(b,bm,ta);
322   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
323   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
324 
325   ierr = PetscLLCondensedCreate_fast(Crmax,&lnk);CHKERRQ(ierr);
326 
327   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
328   ierr          = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
329   current_space = free_space;
330 
331   /* Determine ci and cj */
332   for (i=0; i<am; i++) {
333     anzi = ai[i+1] - ai[i];
334     aj   = a->j + ai[i];
335     for (j=0; j<anzi; j++) {
336       brow = aj[j];
337       bnzj = bi[brow+1] - bi[brow];
338       bj   = b->j + bi[brow];
339       /* add non-zero cols of B into the sorted linked list lnk */
340       ierr = PetscLLCondensedAddSorted_fast(bnzj,bj,lnk);CHKERRQ(ierr);
341     }
342     cnzi = lnk[1];
343 
344     /* If free space is not available, make more free space */
345     /* Double the amount of total space in the list */
346     if (current_space->local_remaining<cnzi) {
347       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(cnzi,current_space->total_array_size),&current_space);CHKERRQ(ierr);
348       ndouble++;
349     }
350 
351     /* Copy data into free space, then initialize lnk */
352     ierr = PetscLLCondensedClean_fast(cnzi,current_space->array,lnk);CHKERRQ(ierr);
353 
354     current_space->array           += cnzi;
355     current_space->local_used      += cnzi;
356     current_space->local_remaining -= cnzi;
357 
358     ci[i+1] = ci[i] + cnzi;
359   }
360 
361   /* Column indices are in the list of free space */
362   /* Allocate space for cj, initialize cj, and */
363   /* destroy list of free space and other temporary array(s) */
364   ierr = PetscMalloc1(ci[am]+1,&cj);CHKERRQ(ierr);
365   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
366   ierr = PetscLLCondensedDestroy_fast(lnk);CHKERRQ(ierr);
367 
368   /* Allocate space for ca */
369   ierr = PetscMalloc1(ci[am]+1,&ca);CHKERRQ(ierr);
370   ierr = PetscMemzero(ca,(ci[am]+1)*sizeof(MatScalar));CHKERRQ(ierr);
371 
372   /* put together the new symbolic matrix */
373   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,ca,C);CHKERRQ(ierr);
374   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
375   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
376 
377   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
378   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
379   c          = (Mat_SeqAIJ*)((*C)->data);
380   c->free_a  = PETSC_TRUE;
381   c->free_ij = PETSC_TRUE;
382   c->nonew   = 0;
383 
384   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable; /* slower, less memory */
385 
386   /* set MatInfo */
387   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
388   if (afill < 1.0) afill = 1.0;
389   c->maxnz                     = ci[am];
390   c->nz                        = ci[am];
391   (*C)->info.mallocs           = ndouble;
392   (*C)->info.fill_ratio_given  = fill;
393   (*C)->info.fill_ratio_needed = afill;
394 
395 #if defined(PETSC_USE_INFO)
396   if (ci[am]) {
397     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
398     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
399   } else {
400     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
401   }
402 #endif
403   PetscFunctionReturn(0);
404 }
405 
406 
407 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable(Mat A,Mat B,PetscReal fill,Mat *C)
408 {
409   PetscErrorCode     ierr;
410   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
411   PetscInt           *ai = a->i,*bi=b->i,*ci,*cj;
412   PetscInt           am  = A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
413   MatScalar          *ca;
414   PetscReal          afill;
415   PetscInt           i,j,anzi,brow,bnzj,cnzi,*bj,*aj,*lnk,ndouble=0,Crmax;
416   PetscTable         ta;
417   PetscFreeSpaceList free_space=NULL,current_space=NULL;
418 
419   PetscFunctionBegin;
420   /* Get ci and cj - same as MatMatMultSymbolic_SeqAIJ_SeqAIJ except using PetscLLxxx_Scalalbe() */
421   /*---------------------------------------------------------------------------------------------*/
422   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
423   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
424   ci[0] = 0;
425 
426   /* create and initialize a linked list */
427   ierr = PetscTableCreate(bn,bn,&ta);CHKERRQ(ierr);
428   MatRowMergeMax_SeqAIJ(b,bm,ta);
429   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
430   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
431   ierr = PetscLLCondensedCreate_Scalable(Crmax,&lnk);CHKERRQ(ierr);
432 
433   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
434   ierr          = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
435   current_space = free_space;
436 
437   /* Determine ci and cj */
438   for (i=0; i<am; i++) {
439     anzi = ai[i+1] - ai[i];
440     aj   = a->j + ai[i];
441     for (j=0; j<anzi; j++) {
442       brow = aj[j];
443       bnzj = bi[brow+1] - bi[brow];
444       bj   = b->j + bi[brow];
445       /* add non-zero cols of B into the sorted linked list lnk */
446       ierr = PetscLLCondensedAddSorted_Scalable(bnzj,bj,lnk);CHKERRQ(ierr);
447     }
448     cnzi = lnk[0];
449 
450     /* If free space is not available, make more free space */
451     /* Double the amount of total space in the list */
452     if (current_space->local_remaining<cnzi) {
453       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(cnzi,current_space->total_array_size),&current_space);CHKERRQ(ierr);
454       ndouble++;
455     }
456 
457     /* Copy data into free space, then initialize lnk */
458     ierr = PetscLLCondensedClean_Scalable(cnzi,current_space->array,lnk);CHKERRQ(ierr);
459 
460     current_space->array           += cnzi;
461     current_space->local_used      += cnzi;
462     current_space->local_remaining -= cnzi;
463 
464     ci[i+1] = ci[i] + cnzi;
465   }
466 
467   /* Column indices are in the list of free space */
468   /* Allocate space for cj, initialize cj, and */
469   /* destroy list of free space and other temporary array(s) */
470   ierr = PetscMalloc1(ci[am]+1,&cj);CHKERRQ(ierr);
471   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
472   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
473 
474   /* Allocate space for ca */
475   /*-----------------------*/
476   ierr = PetscMalloc1(ci[am]+1,&ca);CHKERRQ(ierr);
477   ierr = PetscMemzero(ca,(ci[am]+1)*sizeof(MatScalar));CHKERRQ(ierr);
478 
479   /* put together the new symbolic matrix */
480   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,ca,C);CHKERRQ(ierr);
481   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
482   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
483 
484   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
485   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
486   c          = (Mat_SeqAIJ*)((*C)->data);
487   c->free_a  = PETSC_TRUE;
488   c->free_ij = PETSC_TRUE;
489   c->nonew   = 0;
490 
491   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable; /* slower, less memory */
492 
493   /* set MatInfo */
494   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
495   if (afill < 1.0) afill = 1.0;
496   c->maxnz                     = ci[am];
497   c->nz                        = ci[am];
498   (*C)->info.mallocs           = ndouble;
499   (*C)->info.fill_ratio_given  = fill;
500   (*C)->info.fill_ratio_needed = afill;
501 
502 #if defined(PETSC_USE_INFO)
503   if (ci[am]) {
504     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
505     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
506   } else {
507     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
508   }
509 #endif
510   PetscFunctionReturn(0);
511 }
512 
513 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Heap(Mat A,Mat B,PetscReal fill,Mat *C)
514 {
515   PetscErrorCode     ierr;
516   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
517   const PetscInt     *ai=a->i,*bi=b->i,*aj=a->j,*bj=b->j;
518   PetscInt           *ci,*cj,*bb;
519   PetscInt           am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
520   PetscReal          afill;
521   PetscInt           i,j,col,ndouble = 0;
522   PetscFreeSpaceList free_space=NULL,current_space=NULL;
523   PetscHeap          h;
524 
525   PetscFunctionBegin;
526   /* Get ci and cj - by merging sorted rows using a heap */
527   /*---------------------------------------------------------------------------------------------*/
528   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
529   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
530   ci[0] = 0;
531 
532   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
533   ierr          = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
534   current_space = free_space;
535 
536   ierr = PetscHeapCreate(a->rmax,&h);CHKERRQ(ierr);
537   ierr = PetscMalloc1(a->rmax,&bb);CHKERRQ(ierr);
538 
539   /* Determine ci and cj */
540   for (i=0; i<am; i++) {
541     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 */
542     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
543     ci[i+1] = ci[i];
544     /* Populate the min heap */
545     for (j=0; j<anzi; j++) {
546       bb[j] = bi[acol[j]];         /* bb points at the start of the row */
547       if (bb[j] < bi[acol[j]+1]) { /* Add if row is nonempty */
548         ierr = PetscHeapAdd(h,j,bj[bb[j]++]);CHKERRQ(ierr);
549       }
550     }
551     /* Pick off the min element, adding it to free space */
552     ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
553     while (j >= 0) {
554       if (current_space->local_remaining < 1) { /* double the size, but don't exceed 16 MiB */
555         ierr = PetscFreeSpaceGet(PetscMin(PetscIntMultTruncate(2,current_space->total_array_size),16 << 20),&current_space);CHKERRQ(ierr);
556         ndouble++;
557       }
558       *(current_space->array++) = col;
559       current_space->local_used++;
560       current_space->local_remaining--;
561       ci[i+1]++;
562 
563       /* stash if anything else remains in this row of B */
564       if (bb[j] < bi[acol[j]+1]) {ierr = PetscHeapStash(h,j,bj[bb[j]++]);CHKERRQ(ierr);}
565       while (1) {               /* pop and stash any other rows of B that also had an entry in this column */
566         PetscInt j2,col2;
567         ierr = PetscHeapPeek(h,&j2,&col2);CHKERRQ(ierr);
568         if (col2 != col) break;
569         ierr = PetscHeapPop(h,&j2,&col2);CHKERRQ(ierr);
570         if (bb[j2] < bi[acol[j2]+1]) {ierr = PetscHeapStash(h,j2,bj[bb[j2]++]);CHKERRQ(ierr);}
571       }
572       /* Put any stashed elements back into the min heap */
573       ierr = PetscHeapUnstash(h);CHKERRQ(ierr);
574       ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
575     }
576   }
577   ierr = PetscFree(bb);CHKERRQ(ierr);
578   ierr = PetscHeapDestroy(&h);CHKERRQ(ierr);
579 
580   /* Column indices are in the list of free space */
581   /* Allocate space for cj, initialize cj, and */
582   /* destroy list of free space and other temporary array(s) */
583   ierr = PetscMalloc1(ci[am],&cj);CHKERRQ(ierr);
584   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
585 
586   /* put together the new symbolic matrix */
587   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
588   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
589   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
590 
591   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
592   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
593   c          = (Mat_SeqAIJ*)((*C)->data);
594   c->free_a  = PETSC_TRUE;
595   c->free_ij = PETSC_TRUE;
596   c->nonew   = 0;
597 
598   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ;
599 
600   /* set MatInfo */
601   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
602   if (afill < 1.0) afill = 1.0;
603   c->maxnz                     = ci[am];
604   c->nz                        = ci[am];
605   (*C)->info.mallocs           = ndouble;
606   (*C)->info.fill_ratio_given  = fill;
607   (*C)->info.fill_ratio_needed = afill;
608 
609 #if defined(PETSC_USE_INFO)
610   if (ci[am]) {
611     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
612     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
613   } else {
614     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
615   }
616 #endif
617   PetscFunctionReturn(0);
618 }
619 
620 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_BTHeap(Mat A,Mat B,PetscReal fill,Mat *C)
621 {
622   PetscErrorCode     ierr;
623   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
624   const PetscInt     *ai = a->i,*bi=b->i,*aj=a->j,*bj=b->j;
625   PetscInt           *ci,*cj,*bb;
626   PetscInt           am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
627   PetscReal          afill;
628   PetscInt           i,j,col,ndouble = 0;
629   PetscFreeSpaceList free_space=NULL,current_space=NULL;
630   PetscHeap          h;
631   PetscBT            bt;
632 
633   PetscFunctionBegin;
634   /* Get ci and cj - using a heap for the sorted rows, but use BT so that each index is only added once */
635   /*---------------------------------------------------------------------------------------------*/
636   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
637   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
638   ci[0] = 0;
639 
640   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
641   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
642 
643   current_space = free_space;
644 
645   ierr = PetscHeapCreate(a->rmax,&h);CHKERRQ(ierr);
646   ierr = PetscMalloc1(a->rmax,&bb);CHKERRQ(ierr);
647   ierr = PetscBTCreate(bn,&bt);CHKERRQ(ierr);
648 
649   /* Determine ci and cj */
650   for (i=0; i<am; i++) {
651     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 */
652     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
653     const PetscInt *fptr = current_space->array; /* Save beginning of the row so we can clear the BT later */
654     ci[i+1] = ci[i];
655     /* Populate the min heap */
656     for (j=0; j<anzi; j++) {
657       PetscInt brow = acol[j];
658       for (bb[j] = bi[brow]; bb[j] < bi[brow+1]; bb[j]++) {
659         PetscInt bcol = bj[bb[j]];
660         if (!PetscBTLookupSet(bt,bcol)) { /* new entry */
661           ierr = PetscHeapAdd(h,j,bcol);CHKERRQ(ierr);
662           bb[j]++;
663           break;
664         }
665       }
666     }
667     /* Pick off the min element, adding it to free space */
668     ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
669     while (j >= 0) {
670       if (current_space->local_remaining < 1) { /* double the size, but don't exceed 16 MiB */
671         fptr = NULL;                      /* need PetscBTMemzero */
672         ierr = PetscFreeSpaceGet(PetscMin(PetscIntMultTruncate(2,current_space->total_array_size),16 << 20),&current_space);CHKERRQ(ierr);
673         ndouble++;
674       }
675       *(current_space->array++) = col;
676       current_space->local_used++;
677       current_space->local_remaining--;
678       ci[i+1]++;
679 
680       /* stash if anything else remains in this row of B */
681       for (; bb[j] < bi[acol[j]+1]; bb[j]++) {
682         PetscInt bcol = bj[bb[j]];
683         if (!PetscBTLookupSet(bt,bcol)) { /* new entry */
684           ierr = PetscHeapAdd(h,j,bcol);CHKERRQ(ierr);
685           bb[j]++;
686           break;
687         }
688       }
689       ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
690     }
691     if (fptr) {                 /* Clear the bits for this row */
692       for (; fptr<current_space->array; fptr++) {ierr = PetscBTClear(bt,*fptr);CHKERRQ(ierr);}
693     } else {                    /* We reallocated so we don't remember (easily) how to clear only the bits we changed */
694       ierr = PetscBTMemzero(bn,bt);CHKERRQ(ierr);
695     }
696   }
697   ierr = PetscFree(bb);CHKERRQ(ierr);
698   ierr = PetscHeapDestroy(&h);CHKERRQ(ierr);
699   ierr = PetscBTDestroy(&bt);CHKERRQ(ierr);
700 
701   /* Column indices are in the list of free space */
702   /* Allocate space for cj, initialize cj, and */
703   /* destroy list of free space and other temporary array(s) */
704   ierr = PetscMalloc1(ci[am],&cj);CHKERRQ(ierr);
705   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
706 
707   /* put together the new symbolic matrix */
708   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
709   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
710   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
711 
712   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
713   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
714   c          = (Mat_SeqAIJ*)((*C)->data);
715   c->free_a  = PETSC_TRUE;
716   c->free_ij = PETSC_TRUE;
717   c->nonew   = 0;
718 
719   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ;
720 
721   /* set MatInfo */
722   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
723   if (afill < 1.0) afill = 1.0;
724   c->maxnz                     = ci[am];
725   c->nz                        = ci[am];
726   (*C)->info.mallocs           = ndouble;
727   (*C)->info.fill_ratio_given  = fill;
728   (*C)->info.fill_ratio_needed = afill;
729 
730 #if defined(PETSC_USE_INFO)
731   if (ci[am]) {
732     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
733     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
734   } else {
735     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
736   }
737 #endif
738   PetscFunctionReturn(0);
739 }
740 
741 
742 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_RowMerge(Mat A,Mat B,PetscReal fill,Mat *C)
743 {
744   PetscErrorCode     ierr;
745   Mat_SeqAIJ         *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
746   const PetscInt     *ai=a->i,*bi=b->i,*aj=a->j,*bj=b->j,*inputi,*inputj,*inputcol,*inputcol_L1;
747   PetscInt           *ci,*cj,*outputj,worki_L1[9],worki_L2[9];
748   PetscInt           c_maxmem,a_maxrownnz=0,a_rownnz;
749   const PetscInt     workcol[8]={0,1,2,3,4,5,6,7};
750   const PetscInt     am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
751   const PetscInt     *brow_ptr[8],*brow_end[8];
752   PetscInt           window[8];
753   PetscInt           window_min,old_window_min,ci_nnz,outputi_nnz=0,L1_nrows,L2_nrows;
754   PetscInt           i,k,ndouble=0,L1_rowsleft,rowsleft;
755   PetscReal          afill;
756   PetscInt           *workj_L1,*workj_L2,*workj_L3;
757   PetscInt           L1_nnz,L2_nnz;
758 
759   /* Step 1: Get upper bound on memory required for allocation.
760              Because of the way virtual memory works,
761              only the memory pages that are actually needed will be physically allocated. */
762   PetscFunctionBegin;
763   ierr = PetscMalloc1(am+1,&ci);CHKERRQ(ierr);
764   for (i=0; i<am; i++) {
765     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 */
766     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
767     a_rownnz = 0;
768     for (k=0; k<anzi; ++k) {
769       a_rownnz += bi[acol[k]+1] - bi[acol[k]];
770       if (a_rownnz > bn) {
771         a_rownnz = bn;
772         break;
773       }
774     }
775     a_maxrownnz = PetscMax(a_maxrownnz, a_rownnz);
776   }
777   /* temporary work areas for merging rows */
778   ierr = PetscMalloc1(a_maxrownnz*8,&workj_L1);CHKERRQ(ierr);
779   ierr = PetscMalloc1(a_maxrownnz*8,&workj_L2);CHKERRQ(ierr);
780   ierr = PetscMalloc1(a_maxrownnz,&workj_L3);CHKERRQ(ierr);
781 
782   /* This should be enough for almost all matrices. If not, memory is reallocated later. */
783   c_maxmem = 8*(ai[am]+bi[bm]);
784   /* Step 2: Populate pattern for C */
785   ierr  = PetscMalloc1(c_maxmem,&cj);CHKERRQ(ierr);
786 
787   ci_nnz       = 0;
788   ci[0]        = 0;
789   worki_L1[0]  = 0;
790   worki_L2[0]  = 0;
791   for (i=0; i<am; i++) {
792     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 */
793     const PetscInt *acol = aj + ai[i];      /* column indices of nonzero entries in this row */
794     rowsleft             = anzi;
795     inputcol_L1          = acol;
796     L2_nnz               = 0;
797     L2_nrows             = 1;  /* Number of rows to be merged on Level 3. output of L3 already exists -> initial value 1   */
798     worki_L2[1]          = 0;
799     outputi_nnz          = 0;
800 
801     /* If the number of indices in C so far + the max number of columns in the next row > c_maxmem  -> allocate more memory */
802     while (ci_nnz+a_maxrownnz > c_maxmem) {
803       c_maxmem *= 2;
804       ndouble++;
805       ierr = PetscRealloc(sizeof(PetscInt)*c_maxmem,&cj);CHKERRQ(ierr);
806     }
807 
808     while (rowsleft) {
809       L1_rowsleft = PetscMin(64, rowsleft); /* In the inner loop max 64 rows of B can be merged */
810       L1_nrows    = 0;
811       L1_nnz      = 0;
812       inputcol    = inputcol_L1;
813       inputi      = bi;
814       inputj      = bj;
815 
816       /* The following macro is used to specialize for small rows in A.
817          This helps with compiler unrolling, improving performance substantially.
818           Input:  inputj   inputi  inputcol  bn
819           Output: outputj  outputi_nnz                       */
820        #define MatMatMultSymbolic_RowMergeMacro(ANNZ)                        \
821          window_min  = bn;                                                   \
822          outputi_nnz = 0;                                                    \
823          for (k=0; k<ANNZ; ++k) {                                            \
824            brow_ptr[k] = inputj + inputi[inputcol[k]];                       \
825            brow_end[k] = inputj + inputi[inputcol[k]+1];                     \
826            window[k]   = (brow_ptr[k] != brow_end[k]) ? *brow_ptr[k] : bn;   \
827            window_min  = PetscMin(window[k], window_min);                    \
828          }                                                                   \
829          while (window_min < bn) {                                           \
830            outputj[outputi_nnz++] = window_min;                              \
831            /* advance front and compute new minimum */                       \
832            old_window_min = window_min;                                      \
833            window_min = bn;                                                  \
834            for (k=0; k<ANNZ; ++k) {                                          \
835              if (window[k] == old_window_min) {                              \
836                brow_ptr[k]++;                                                \
837                window[k] = (brow_ptr[k] != brow_end[k]) ? *brow_ptr[k] : bn; \
838              }                                                               \
839              window_min = PetscMin(window[k], window_min);                   \
840            }                                                                 \
841          }
842 
843       /************** L E V E L  1 ***************/
844       /* Merge up to 8 rows of B to L1 work array*/
845       while (L1_rowsleft) {
846         outputi_nnz = 0;
847         if (anzi > 8)  outputj = workj_L1 + L1_nnz;     /* Level 1 rowmerge*/
848         else           outputj = cj + ci_nnz;           /* Merge directly to C */
849 
850         switch (L1_rowsleft) {
851         case 1:  brow_ptr[0] = inputj + inputi[inputcol[0]];
852                  brow_end[0] = inputj + inputi[inputcol[0]+1];
853                  for (; brow_ptr[0] != brow_end[0]; ++brow_ptr[0]) outputj[outputi_nnz++] = *brow_ptr[0]; /* copy row in b over */
854                  inputcol    += L1_rowsleft;
855                  rowsleft    -= L1_rowsleft;
856                  L1_rowsleft  = 0;
857                  break;
858         case 2:  MatMatMultSymbolic_RowMergeMacro(2);
859                  inputcol    += L1_rowsleft;
860                  rowsleft    -= L1_rowsleft;
861                  L1_rowsleft  = 0;
862                  break;
863         case 3: MatMatMultSymbolic_RowMergeMacro(3);
864                  inputcol    += L1_rowsleft;
865                  rowsleft    -= L1_rowsleft;
866                  L1_rowsleft  = 0;
867                  break;
868         case 4:  MatMatMultSymbolic_RowMergeMacro(4);
869                  inputcol    += L1_rowsleft;
870                  rowsleft    -= L1_rowsleft;
871                  L1_rowsleft  = 0;
872                  break;
873         case 5:  MatMatMultSymbolic_RowMergeMacro(5);
874                  inputcol    += L1_rowsleft;
875                  rowsleft    -= L1_rowsleft;
876                  L1_rowsleft  = 0;
877                  break;
878         case 6:  MatMatMultSymbolic_RowMergeMacro(6);
879                  inputcol    += L1_rowsleft;
880                  rowsleft    -= L1_rowsleft;
881                  L1_rowsleft  = 0;
882                  break;
883         case 7:  MatMatMultSymbolic_RowMergeMacro(7);
884                  inputcol    += L1_rowsleft;
885                  rowsleft    -= L1_rowsleft;
886                  L1_rowsleft  = 0;
887                  break;
888         default: MatMatMultSymbolic_RowMergeMacro(8);
889                  inputcol    += 8;
890                  rowsleft    -= 8;
891                  L1_rowsleft -= 8;
892                  break;
893         }
894         inputcol_L1           = inputcol;
895         L1_nnz               += outputi_nnz;
896         worki_L1[++L1_nrows]  = L1_nnz;
897       }
898 
899       /********************** L E V E L  2 ************************/
900       /* Merge from L1 work array to either C or to L2 work array */
901       if (anzi > 8) {
902         inputi      = worki_L1;
903         inputj      = workj_L1;
904         inputcol    = workcol;
905         outputi_nnz = 0;
906 
907         if (anzi <= 64) outputj = cj + ci_nnz;        /* Merge from L1 work array to C */
908         else            outputj = workj_L2 + L2_nnz;  /* Merge from L1 work array to L2 work array */
909 
910         switch (L1_nrows) {
911         case 1:  brow_ptr[0] = inputj + inputi[inputcol[0]];
912                  brow_end[0] = inputj + inputi[inputcol[0]+1];
913                  for (; brow_ptr[0] != brow_end[0]; ++brow_ptr[0]) outputj[outputi_nnz++] = *brow_ptr[0]; /* copy row in b over */
914                  break;
915         case 2:  MatMatMultSymbolic_RowMergeMacro(2); break;
916         case 3:  MatMatMultSymbolic_RowMergeMacro(3); break;
917         case 4:  MatMatMultSymbolic_RowMergeMacro(4); break;
918         case 5:  MatMatMultSymbolic_RowMergeMacro(5); break;
919         case 6:  MatMatMultSymbolic_RowMergeMacro(6); break;
920         case 7:  MatMatMultSymbolic_RowMergeMacro(7); break;
921         case 8:  MatMatMultSymbolic_RowMergeMacro(8); break;
922         default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"MatMatMult logic error: Not merging 1-8 rows from L1 work array!");
923         }
924         L2_nnz               += outputi_nnz;
925         worki_L2[++L2_nrows]  = L2_nnz;
926 
927         /************************ L E V E L  3 **********************/
928         /* Merge from L2 work array to either C or to L2 work array */
929         if (anzi > 64 && (L2_nrows == 8 || rowsleft == 0)) {
930           inputi      = worki_L2;
931           inputj      = workj_L2;
932           inputcol    = workcol;
933           outputi_nnz = 0;
934           if (rowsleft) outputj = workj_L3;
935           else          outputj = cj + ci_nnz;
936           switch (L2_nrows) {
937           case 1:  brow_ptr[0] = inputj + inputi[inputcol[0]];
938                    brow_end[0] = inputj + inputi[inputcol[0]+1];
939                    for (; brow_ptr[0] != brow_end[0]; ++brow_ptr[0]) outputj[outputi_nnz++] = *brow_ptr[0]; /* copy row in b over */
940                    break;
941           case 2:  MatMatMultSymbolic_RowMergeMacro(2); break;
942           case 3:  MatMatMultSymbolic_RowMergeMacro(3); break;
943           case 4:  MatMatMultSymbolic_RowMergeMacro(4); break;
944           case 5:  MatMatMultSymbolic_RowMergeMacro(5); break;
945           case 6:  MatMatMultSymbolic_RowMergeMacro(6); break;
946           case 7:  MatMatMultSymbolic_RowMergeMacro(7); break;
947           case 8:  MatMatMultSymbolic_RowMergeMacro(8); break;
948           default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"MatMatMult logic error: Not merging 1-8 rows from L2 work array!");
949           }
950           L2_nrows    = 1;
951           L2_nnz      = outputi_nnz;
952           worki_L2[1] = outputi_nnz;
953           /* Copy to workj_L2 */
954           if (rowsleft) {
955             for (k=0; k<outputi_nnz; ++k)  workj_L2[k] = outputj[k];
956           }
957         }
958       }
959     }  /* while (rowsleft) */
960 #undef MatMatMultSymbolic_RowMergeMacro
961 
962     /* terminate current row */
963     ci_nnz += outputi_nnz;
964     ci[i+1] = ci_nnz;
965   }
966 
967   /* Step 3: Create the new symbolic matrix */
968   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
969   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
970   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
971 
972   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
973   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
974   c          = (Mat_SeqAIJ*)((*C)->data);
975   c->free_a  = PETSC_TRUE;
976   c->free_ij = PETSC_TRUE;
977   c->nonew   = 0;
978 
979   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ;
980 
981   /* set MatInfo */
982   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
983   if (afill < 1.0) afill = 1.0;
984   c->maxnz                     = ci[am];
985   c->nz                        = ci[am];
986   (*C)->info.mallocs           = ndouble;
987   (*C)->info.fill_ratio_given  = fill;
988   (*C)->info.fill_ratio_needed = afill;
989 
990 #if defined(PETSC_USE_INFO)
991   if (ci[am]) {
992     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
993     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
994   } else {
995     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
996   }
997 #endif
998 
999   /* Step 4: Free temporary work areas */
1000   ierr = PetscFree(workj_L1);CHKERRQ(ierr);
1001   ierr = PetscFree(workj_L2);CHKERRQ(ierr);
1002   ierr = PetscFree(workj_L3);CHKERRQ(ierr);
1003   PetscFunctionReturn(0);
1004 }
1005 
1006 /* concatenate unique entries and then sort */
1007 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ(Mat A,Mat B,PetscReal fill,Mat *C)
1008 {
1009   PetscErrorCode     ierr;
1010   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
1011   const PetscInt     *ai = a->i,*bi=b->i,*aj=a->j,*bj=b->j;
1012   PetscInt           *ci,*cj;
1013   PetscInt           am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
1014   PetscReal          afill;
1015   PetscInt           i,j,ndouble = 0;
1016   PetscSegBuffer     seg,segrow;
1017   char               *seen;
1018 
1019   PetscFunctionBegin;
1020   ierr  = PetscMalloc1(am+1,&ci);CHKERRQ(ierr);
1021   ci[0] = 0;
1022 
1023   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
1024   ierr = PetscSegBufferCreate(sizeof(PetscInt),(PetscInt)(fill*(ai[am]+bi[bm])),&seg);CHKERRQ(ierr);
1025   ierr = PetscSegBufferCreate(sizeof(PetscInt),100,&segrow);CHKERRQ(ierr);
1026   ierr = PetscMalloc1(bn,&seen);CHKERRQ(ierr);
1027   ierr = PetscMemzero(seen,bn*sizeof(char));CHKERRQ(ierr);
1028 
1029   /* Determine ci and cj */
1030   for (i=0; i<am; i++) {
1031     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 */
1032     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
1033     PetscInt packlen = 0,*PETSC_RESTRICT crow;
1034     /* Pack segrow */
1035     for (j=0; j<anzi; j++) {
1036       PetscInt brow = acol[j],bjstart = bi[brow],bjend = bi[brow+1],k;
1037       for (k=bjstart; k<bjend; k++) {
1038         PetscInt bcol = bj[k];
1039         if (!seen[bcol]) { /* new entry */
1040           PetscInt *PETSC_RESTRICT slot;
1041           ierr = PetscSegBufferGetInts(segrow,1,&slot);CHKERRQ(ierr);
1042           *slot = bcol;
1043           seen[bcol] = 1;
1044           packlen++;
1045         }
1046       }
1047     }
1048     ierr = PetscSegBufferGetInts(seg,packlen,&crow);CHKERRQ(ierr);
1049     ierr = PetscSegBufferExtractTo(segrow,crow);CHKERRQ(ierr);
1050     ierr = PetscSortInt(packlen,crow);CHKERRQ(ierr);
1051     ci[i+1] = ci[i] + packlen;
1052     for (j=0; j<packlen; j++) seen[crow[j]] = 0;
1053   }
1054   ierr = PetscSegBufferDestroy(&segrow);CHKERRQ(ierr);
1055   ierr = PetscFree(seen);CHKERRQ(ierr);
1056 
1057   /* Column indices are in the segmented buffer */
1058   ierr = PetscSegBufferExtractAlloc(seg,&cj);CHKERRQ(ierr);
1059   ierr = PetscSegBufferDestroy(&seg);CHKERRQ(ierr);
1060 
1061   /* put together the new symbolic matrix */
1062   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
1063   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
1064   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
1065 
1066   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
1067   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
1068   c          = (Mat_SeqAIJ*)((*C)->data);
1069   c->free_a  = PETSC_TRUE;
1070   c->free_ij = PETSC_TRUE;
1071   c->nonew   = 0;
1072 
1073   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ;
1074 
1075   /* set MatInfo */
1076   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
1077   if (afill < 1.0) afill = 1.0;
1078   c->maxnz                     = ci[am];
1079   c->nz                        = ci[am];
1080   (*C)->info.mallocs           = ndouble;
1081   (*C)->info.fill_ratio_given  = fill;
1082   (*C)->info.fill_ratio_needed = afill;
1083 
1084 #if defined(PETSC_USE_INFO)
1085   if (ci[am]) {
1086     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
1087     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
1088   } else {
1089     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
1090   }
1091 #endif
1092   PetscFunctionReturn(0);
1093 }
1094 
1095 /* This routine is not used. Should be removed! */
1096 PetscErrorCode MatMatTransposeMult_SeqAIJ_SeqAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
1097 {
1098   PetscErrorCode ierr;
1099 
1100   PetscFunctionBegin;
1101   if (scall == MAT_INITIAL_MATRIX) {
1102     ierr = PetscLogEventBegin(MAT_MatTransposeMultSymbolic,A,B,0,0);CHKERRQ(ierr);
1103     ierr = MatMatTransposeMultSymbolic_SeqAIJ_SeqAIJ(A,B,fill,C);CHKERRQ(ierr);
1104     ierr = PetscLogEventEnd(MAT_MatTransposeMultSymbolic,A,B,0,0);CHKERRQ(ierr);
1105   }
1106   ierr = PetscLogEventBegin(MAT_MatTransposeMultNumeric,A,B,0,0);CHKERRQ(ierr);
1107   ierr = MatMatTransposeMultNumeric_SeqAIJ_SeqAIJ(A,B,*C);CHKERRQ(ierr);
1108   ierr = PetscLogEventEnd(MAT_MatTransposeMultNumeric,A,B,0,0);CHKERRQ(ierr);
1109   PetscFunctionReturn(0);
1110 }
1111 
1112 PetscErrorCode MatDestroy_SeqAIJ_MatMatMultTrans(Mat A)
1113 {
1114   PetscErrorCode      ierr;
1115   Mat_SeqAIJ          *a=(Mat_SeqAIJ*)A->data;
1116   Mat_MatMatTransMult *abt=a->abt;
1117 
1118   PetscFunctionBegin;
1119   ierr = (abt->destroy)(A);CHKERRQ(ierr);
1120   ierr = MatTransposeColoringDestroy(&abt->matcoloring);CHKERRQ(ierr);
1121   ierr = MatDestroy(&abt->Bt_den);CHKERRQ(ierr);
1122   ierr = MatDestroy(&abt->ABt_den);CHKERRQ(ierr);
1123   ierr = PetscFree(abt);CHKERRQ(ierr);
1124   PetscFunctionReturn(0);
1125 }
1126 
1127 PetscErrorCode MatMatTransposeMultSymbolic_SeqAIJ_SeqAIJ(Mat A,Mat B,PetscReal fill,Mat *C)
1128 {
1129   PetscErrorCode      ierr;
1130   Mat                 Bt;
1131   PetscInt            *bti,*btj;
1132   Mat_MatMatTransMult *abt;
1133   Mat_SeqAIJ          *c;
1134 
1135   PetscFunctionBegin;
1136   /* create symbolic Bt */
1137   ierr = MatGetSymbolicTranspose_SeqAIJ(B,&bti,&btj);CHKERRQ(ierr);
1138   ierr = MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,B->cmap->n,B->rmap->n,bti,btj,NULL,&Bt);CHKERRQ(ierr);
1139   ierr = MatSetBlockSizes(Bt,PetscAbs(A->cmap->bs),PetscAbs(B->cmap->bs));CHKERRQ(ierr);
1140   ierr = MatSetType(Bt,((PetscObject)A)->type_name);CHKERRQ(ierr);
1141 
1142   /* get symbolic C=A*Bt */
1143   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(A,Bt,fill,C);CHKERRQ(ierr);
1144 
1145   /* create a supporting struct for reuse intermidiate dense matrices with matcoloring */
1146   ierr   = PetscNew(&abt);CHKERRQ(ierr);
1147   c      = (Mat_SeqAIJ*)(*C)->data;
1148   c->abt = abt;
1149 
1150   abt->usecoloring = PETSC_FALSE;
1151   abt->destroy     = (*C)->ops->destroy;
1152   (*C)->ops->destroy     = MatDestroy_SeqAIJ_MatMatMultTrans;
1153 
1154   ierr = PetscOptionsGetBool(((PetscObject)A)->options,NULL,"-matmattransmult_color",&abt->usecoloring,NULL);CHKERRQ(ierr);
1155   if (abt->usecoloring) {
1156     /* Create MatTransposeColoring from symbolic C=A*B^T */
1157     MatTransposeColoring matcoloring;
1158     MatColoring          coloring;
1159     ISColoring           iscoloring;
1160     Mat                  Bt_dense,C_dense;
1161     Mat_SeqAIJ           *c=(Mat_SeqAIJ*)(*C)->data;
1162     /* inode causes memory problem, don't know why */
1163     if (c->inode.use) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"MAT_USE_INODES is not supported. Use '-mat_no_inode'");
1164 
1165     ierr = MatColoringCreate(*C,&coloring);CHKERRQ(ierr);
1166     ierr = MatColoringSetDistance(coloring,2);CHKERRQ(ierr);
1167     ierr = MatColoringSetType(coloring,MATCOLORINGSL);CHKERRQ(ierr);
1168     ierr = MatColoringSetFromOptions(coloring);CHKERRQ(ierr);
1169     ierr = MatColoringApply(coloring,&iscoloring);CHKERRQ(ierr);
1170     ierr = MatColoringDestroy(&coloring);CHKERRQ(ierr);
1171     ierr = MatTransposeColoringCreate(*C,iscoloring,&matcoloring);CHKERRQ(ierr);
1172 
1173     abt->matcoloring = matcoloring;
1174 
1175     ierr = ISColoringDestroy(&iscoloring);CHKERRQ(ierr);
1176 
1177     /* Create Bt_dense and C_dense = A*Bt_dense */
1178     ierr = MatCreate(PETSC_COMM_SELF,&Bt_dense);CHKERRQ(ierr);
1179     ierr = MatSetSizes(Bt_dense,A->cmap->n,matcoloring->ncolors,A->cmap->n,matcoloring->ncolors);CHKERRQ(ierr);
1180     ierr = MatSetType(Bt_dense,MATSEQDENSE);CHKERRQ(ierr);
1181     ierr = MatSeqDenseSetPreallocation(Bt_dense,NULL);CHKERRQ(ierr);
1182 
1183     Bt_dense->assembled = PETSC_TRUE;
1184     abt->Bt_den   = Bt_dense;
1185 
1186     ierr = MatCreate(PETSC_COMM_SELF,&C_dense);CHKERRQ(ierr);
1187     ierr = MatSetSizes(C_dense,A->rmap->n,matcoloring->ncolors,A->rmap->n,matcoloring->ncolors);CHKERRQ(ierr);
1188     ierr = MatSetType(C_dense,MATSEQDENSE);CHKERRQ(ierr);
1189     ierr = MatSeqDenseSetPreallocation(C_dense,NULL);CHKERRQ(ierr);
1190 
1191     Bt_dense->assembled = PETSC_TRUE;
1192     abt->ABt_den  = C_dense;
1193 
1194 #if defined(PETSC_USE_INFO)
1195     {
1196       Mat_SeqAIJ *c = (Mat_SeqAIJ*)(*C)->data;
1197       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);
1198     }
1199 #endif
1200   }
1201   /* clean up */
1202   ierr = MatDestroy(&Bt);CHKERRQ(ierr);
1203   ierr = MatRestoreSymbolicTranspose_SeqAIJ(B,&bti,&btj);CHKERRQ(ierr);
1204   PetscFunctionReturn(0);
1205 }
1206 
1207 PetscErrorCode MatMatTransposeMultNumeric_SeqAIJ_SeqAIJ(Mat A,Mat B,Mat C)
1208 {
1209   PetscErrorCode      ierr;
1210   Mat_SeqAIJ          *a   =(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c=(Mat_SeqAIJ*)C->data;
1211   PetscInt            *ai  =a->i,*aj=a->j,*bi=b->i,*bj=b->j,anzi,bnzj,nexta,nextb,*acol,*bcol,brow;
1212   PetscInt            cm   =C->rmap->n,*ci=c->i,*cj=c->j,i,j,cnzi,*ccol;
1213   PetscLogDouble      flops=0.0;
1214   MatScalar           *aa  =a->a,*aval,*ba=b->a,*bval,*ca,*cval;
1215   Mat_MatMatTransMult *abt = c->abt;
1216 
1217   PetscFunctionBegin;
1218   /* clear old values in C */
1219   if (!c->a) {
1220     ierr      = PetscMalloc1(ci[cm]+1,&ca);CHKERRQ(ierr);
1221     c->a      = ca;
1222     c->free_a = PETSC_TRUE;
1223   } else {
1224     ca =  c->a;
1225   }
1226   ierr = PetscMemzero(ca,ci[cm]*sizeof(MatScalar));CHKERRQ(ierr);
1227 
1228   if (abt->usecoloring) {
1229     MatTransposeColoring matcoloring = abt->matcoloring;
1230     Mat                  Bt_dense,C_dense = abt->ABt_den;
1231 
1232     /* Get Bt_dense by Apply MatTransposeColoring to B */
1233     Bt_dense = abt->Bt_den;
1234     ierr = MatTransColoringApplySpToDen(matcoloring,B,Bt_dense);CHKERRQ(ierr);
1235 
1236     /* C_dense = A*Bt_dense */
1237     ierr = MatMatMultNumeric_SeqAIJ_SeqDense(A,Bt_dense,C_dense);CHKERRQ(ierr);
1238 
1239     /* Recover C from C_dense */
1240     ierr = MatTransColoringApplyDenToSp(matcoloring,C_dense,C);CHKERRQ(ierr);
1241     PetscFunctionReturn(0);
1242   }
1243 
1244   for (i=0; i<cm; i++) {
1245     anzi = ai[i+1] - ai[i];
1246     acol = aj + ai[i];
1247     aval = aa + ai[i];
1248     cnzi = ci[i+1] - ci[i];
1249     ccol = cj + ci[i];
1250     cval = ca + ci[i];
1251     for (j=0; j<cnzi; j++) {
1252       brow = ccol[j];
1253       bnzj = bi[brow+1] - bi[brow];
1254       bcol = bj + bi[brow];
1255       bval = ba + bi[brow];
1256 
1257       /* perform sparse inner-product c(i,j)=A[i,:]*B[j,:]^T */
1258       nexta = 0; nextb = 0;
1259       while (nexta<anzi && nextb<bnzj) {
1260         while (nexta < anzi && acol[nexta] < bcol[nextb]) nexta++;
1261         if (nexta == anzi) break;
1262         while (nextb < bnzj && acol[nexta] > bcol[nextb]) nextb++;
1263         if (nextb == bnzj) break;
1264         if (acol[nexta] == bcol[nextb]) {
1265           cval[j] += aval[nexta]*bval[nextb];
1266           nexta++; nextb++;
1267           flops += 2;
1268         }
1269       }
1270     }
1271   }
1272   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1273   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1274   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
1275   PetscFunctionReturn(0);
1276 }
1277 
1278 PetscErrorCode MatDestroy_SeqAIJ_MatTransMatMult(Mat A)
1279 {
1280   PetscErrorCode      ierr;
1281   Mat_SeqAIJ          *a = (Mat_SeqAIJ*)A->data;
1282   Mat_MatTransMatMult *atb = a->atb;
1283 
1284   PetscFunctionBegin;
1285   if (atb) {
1286     ierr = MatDestroy(&atb->At);CHKERRQ(ierr);
1287     ierr = (*atb->destroy)(A);CHKERRQ(ierr);
1288   }
1289   ierr = PetscFree(atb);CHKERRQ(ierr);
1290   PetscFunctionReturn(0);
1291 }
1292 
1293 PetscErrorCode MatTransposeMatMult_SeqAIJ_SeqAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
1294 {
1295   PetscErrorCode      ierr;
1296   const char          *algTypes[2] = {"matmatmult","outerproduct"};
1297   PetscInt            alg=0; /* set default algorithm */
1298   Mat                 At;
1299   Mat_MatTransMatMult *atb;
1300   Mat_SeqAIJ          *c;
1301 
1302   PetscFunctionBegin;
1303   if (scall == MAT_INITIAL_MATRIX) {
1304     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)A),((PetscObject)A)->prefix,"MatTransposeMatMult","Mat");CHKERRQ(ierr);
1305     ierr = PetscOptionsEList("-mattransposematmult_via","Algorithmic approach","MatTransposeMatMult",algTypes,2,algTypes[0],&alg,NULL);CHKERRQ(ierr);
1306     ierr = PetscOptionsEnd();CHKERRQ(ierr);
1307 
1308     switch (alg) {
1309     case 1:
1310       ierr = MatTransposeMatMultSymbolic_SeqAIJ_SeqAIJ(A,B,fill,C);CHKERRQ(ierr);
1311       break;
1312     default:
1313       ierr = PetscNew(&atb);CHKERRQ(ierr);
1314       ierr = MatTranspose_SeqAIJ(A,MAT_INITIAL_MATRIX,&At);CHKERRQ(ierr);
1315       ierr = MatMatMult_SeqAIJ_SeqAIJ(At,B,MAT_INITIAL_MATRIX,fill,C);CHKERRQ(ierr);
1316 
1317       c                  = (Mat_SeqAIJ*)(*C)->data;
1318       c->atb             = atb;
1319       atb->At            = At;
1320       atb->destroy       = (*C)->ops->destroy;
1321       (*C)->ops->destroy = MatDestroy_SeqAIJ_MatTransMatMult;
1322 
1323       break;
1324     }
1325   }
1326   if (alg) {
1327     ierr = (*(*C)->ops->mattransposemultnumeric)(A,B,*C);CHKERRQ(ierr);
1328   } else if (!alg && scall == MAT_REUSE_MATRIX) {
1329     c   = (Mat_SeqAIJ*)(*C)->data;
1330     atb = c->atb;
1331     At  = atb->At;
1332     ierr = MatTranspose_SeqAIJ(A,MAT_REUSE_MATRIX,&At);CHKERRQ(ierr);
1333     ierr = MatMatMult_SeqAIJ_SeqAIJ(At,B,MAT_REUSE_MATRIX,fill,C);CHKERRQ(ierr);
1334   }
1335   PetscFunctionReturn(0);
1336 }
1337 
1338 PetscErrorCode MatTransposeMatMultSymbolic_SeqAIJ_SeqAIJ(Mat A,Mat B,PetscReal fill,Mat *C)
1339 {
1340   PetscErrorCode ierr;
1341   Mat            At;
1342   PetscInt       *ati,*atj;
1343 
1344   PetscFunctionBegin;
1345   /* create symbolic At */
1346   ierr = MatGetSymbolicTranspose_SeqAIJ(A,&ati,&atj);CHKERRQ(ierr);
1347   ierr = MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,A->cmap->n,A->rmap->n,ati,atj,NULL,&At);CHKERRQ(ierr);
1348   ierr = MatSetBlockSizes(At,PetscAbs(A->cmap->bs),PetscAbs(B->cmap->bs));CHKERRQ(ierr);
1349   ierr = MatSetType(At,((PetscObject)A)->type_name);CHKERRQ(ierr);
1350 
1351   /* get symbolic C=At*B */
1352   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(At,B,fill,C);CHKERRQ(ierr);
1353 
1354   /* clean up */
1355   ierr = MatDestroy(&At);CHKERRQ(ierr);
1356   ierr = MatRestoreSymbolicTranspose_SeqAIJ(A,&ati,&atj);CHKERRQ(ierr);
1357 
1358   (*C)->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_SeqAIJ_SeqAIJ;
1359   PetscFunctionReturn(0);
1360 }
1361 
1362 PetscErrorCode MatTransposeMatMultNumeric_SeqAIJ_SeqAIJ(Mat A,Mat B,Mat C)
1363 {
1364   PetscErrorCode ierr;
1365   Mat_SeqAIJ     *a   =(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c=(Mat_SeqAIJ*)C->data;
1366   PetscInt       am   =A->rmap->n,anzi,*ai=a->i,*aj=a->j,*bi=b->i,*bj,bnzi,nextb;
1367   PetscInt       cm   =C->rmap->n,*ci=c->i,*cj=c->j,crow,*cjj,i,j,k;
1368   PetscLogDouble flops=0.0;
1369   MatScalar      *aa  =a->a,*ba,*ca,*caj;
1370 
1371   PetscFunctionBegin;
1372   if (!c->a) {
1373     ierr = PetscMalloc1(ci[cm]+1,&ca);CHKERRQ(ierr);
1374 
1375     c->a      = ca;
1376     c->free_a = PETSC_TRUE;
1377   } else {
1378     ca = c->a;
1379   }
1380   /* clear old values in C */
1381   ierr = PetscMemzero(ca,ci[cm]*sizeof(MatScalar));CHKERRQ(ierr);
1382 
1383   /* compute A^T*B using outer product (A^T)[:,i]*B[i,:] */
1384   for (i=0; i<am; i++) {
1385     bj   = b->j + bi[i];
1386     ba   = b->a + bi[i];
1387     bnzi = bi[i+1] - bi[i];
1388     anzi = ai[i+1] - ai[i];
1389     for (j=0; j<anzi; j++) {
1390       nextb = 0;
1391       crow  = *aj++;
1392       cjj   = cj + ci[crow];
1393       caj   = ca + ci[crow];
1394       /* perform sparse axpy operation.  Note cjj includes bj. */
1395       for (k=0; nextb<bnzi; k++) {
1396         if (cjj[k] == *(bj+nextb)) { /* ccol == bcol */
1397           caj[k] += (*aa)*(*(ba+nextb));
1398           nextb++;
1399         }
1400       }
1401       flops += 2*bnzi;
1402       aa++;
1403     }
1404   }
1405 
1406   /* Assemble the final matrix and clean up */
1407   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1408   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1409   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
1410   PetscFunctionReturn(0);
1411 }
1412 
1413 PETSC_INTERN PetscErrorCode MatMatMult_SeqAIJ_SeqDense(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
1414 {
1415   PetscErrorCode ierr;
1416 
1417   PetscFunctionBegin;
1418   if (scall == MAT_INITIAL_MATRIX) {
1419     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
1420     ierr = MatMatMultSymbolic_SeqAIJ_SeqDense(A,B,fill,C);CHKERRQ(ierr);
1421     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
1422   }
1423   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
1424   ierr = MatMatMultNumeric_SeqAIJ_SeqDense(A,B,*C);CHKERRQ(ierr);
1425   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
1426   PetscFunctionReturn(0);
1427 }
1428 
1429 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqDense(Mat A,Mat B,PetscReal fill,Mat *C)
1430 {
1431   PetscErrorCode ierr;
1432 
1433   PetscFunctionBegin;
1434   ierr = MatMatMultSymbolic_SeqDense_SeqDense(A,B,0.0,C);CHKERRQ(ierr);
1435 
1436   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqDense;
1437   PetscFunctionReturn(0);
1438 }
1439 
1440 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqDense(Mat A,Mat B,Mat C)
1441 {
1442   Mat_SeqAIJ        *a=(Mat_SeqAIJ*)A->data;
1443   Mat_SeqDense      *bd = (Mat_SeqDense*)B->data;
1444   PetscErrorCode    ierr;
1445   PetscScalar       *c,*b,r1,r2,r3,r4,*c1,*c2,*c3,*c4,aatmp;
1446   const PetscScalar *aa,*b1,*b2,*b3,*b4;
1447   const PetscInt    *aj;
1448   PetscInt          cm=C->rmap->n,cn=B->cmap->n,bm=bd->lda,am=A->rmap->n;
1449   PetscInt          am4=4*am,bm4=4*bm,col,i,j,n,ajtmp;
1450 
1451   PetscFunctionBegin;
1452   if (!cm || !cn) PetscFunctionReturn(0);
1453   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);
1454   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);
1455   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);
1456   b = bd->v;
1457   ierr = MatDenseGetArray(C,&c);CHKERRQ(ierr);
1458   b1 = b; b2 = b1 + bm; b3 = b2 + bm; b4 = b3 + bm;
1459   c1 = c; c2 = c1 + am; c3 = c2 + am; c4 = c3 + am;
1460   for (col=0; col<cn-4; col += 4) {  /* over columns of C */
1461     for (i=0; i<am; i++) {        /* over rows of C in those columns */
1462       r1 = r2 = r3 = r4 = 0.0;
1463       n  = a->i[i+1] - a->i[i];
1464       aj = a->j + a->i[i];
1465       aa = a->a + a->i[i];
1466       for (j=0; j<n; j++) {
1467         aatmp = aa[j]; ajtmp = aj[j];
1468         r1 += aatmp*b1[ajtmp];
1469         r2 += aatmp*b2[ajtmp];
1470         r3 += aatmp*b3[ajtmp];
1471         r4 += aatmp*b4[ajtmp];
1472       }
1473       c1[i] = r1;
1474       c2[i] = r2;
1475       c3[i] = r3;
1476       c4[i] = r4;
1477     }
1478     b1 += bm4; b2 += bm4; b3 += bm4; b4 += bm4;
1479     c1 += am4; c2 += am4; c3 += am4; c4 += am4;
1480   }
1481   for (; col<cn; col++) {   /* over extra columns of C */
1482     for (i=0; i<am; i++) {  /* over rows of C in those columns */
1483       r1 = 0.0;
1484       n  = a->i[i+1] - a->i[i];
1485       aj = a->j + a->i[i];
1486       aa = a->a + a->i[i];
1487       for (j=0; j<n; j++) {
1488         r1 += aa[j]*b1[aj[j]];
1489       }
1490       c1[i] = r1;
1491     }
1492     b1 += bm;
1493     c1 += am;
1494   }
1495   ierr = PetscLogFlops(cn*(2.0*a->nz));CHKERRQ(ierr);
1496   ierr = MatDenseRestoreArray(C,&c);CHKERRQ(ierr);
1497   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1498   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1499   PetscFunctionReturn(0);
1500 }
1501 
1502 /*
1503    Note very similar to MatMult_SeqAIJ(), should generate both codes from same base
1504 */
1505 PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat A,Mat B,Mat C)
1506 {
1507   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data;
1508   Mat_SeqDense   *bd = (Mat_SeqDense*)B->data;
1509   PetscErrorCode ierr;
1510   PetscScalar    *b,*c,r1,r2,r3,r4,*b1,*b2,*b3,*b4;
1511   MatScalar      *aa;
1512   PetscInt       cm  = C->rmap->n, cn=B->cmap->n, bm=bd->lda, col, i,j,n,*aj, am = A->rmap->n,*ii,arm;
1513   PetscInt       am2 = 2*am, am3 = 3*am,  bm4 = 4*bm,colam,*ridx;
1514 
1515   PetscFunctionBegin;
1516   if (!cm || !cn) PetscFunctionReturn(0);
1517   b = bd->v;
1518   ierr = MatDenseGetArray(C,&c);CHKERRQ(ierr);
1519   b1   = b; b2 = b1 + bm; b3 = b2 + bm; b4 = b3 + bm;
1520 
1521   if (a->compressedrow.use) { /* use compressed row format */
1522     for (col=0; col<cn-4; col += 4) {  /* over columns of C */
1523       colam = col*am;
1524       arm   = a->compressedrow.nrows;
1525       ii    = a->compressedrow.i;
1526       ridx  = a->compressedrow.rindex;
1527       for (i=0; i<arm; i++) {        /* over rows of C in those columns */
1528         r1 = r2 = r3 = r4 = 0.0;
1529         n  = ii[i+1] - ii[i];
1530         aj = a->j + ii[i];
1531         aa = a->a + ii[i];
1532         for (j=0; j<n; j++) {
1533           r1 += (*aa)*b1[*aj];
1534           r2 += (*aa)*b2[*aj];
1535           r3 += (*aa)*b3[*aj];
1536           r4 += (*aa++)*b4[*aj++];
1537         }
1538         c[colam       + ridx[i]] += r1;
1539         c[colam + am  + ridx[i]] += r2;
1540         c[colam + am2 + ridx[i]] += r3;
1541         c[colam + am3 + ridx[i]] += r4;
1542       }
1543       b1 += bm4;
1544       b2 += bm4;
1545       b3 += bm4;
1546       b4 += bm4;
1547     }
1548     for (; col<cn; col++) {     /* over extra columns of C */
1549       colam = col*am;
1550       arm   = a->compressedrow.nrows;
1551       ii    = a->compressedrow.i;
1552       ridx  = a->compressedrow.rindex;
1553       for (i=0; i<arm; i++) {  /* over rows of C in those columns */
1554         r1 = 0.0;
1555         n  = ii[i+1] - ii[i];
1556         aj = a->j + ii[i];
1557         aa = a->a + ii[i];
1558 
1559         for (j=0; j<n; j++) {
1560           r1 += (*aa++)*b1[*aj++];
1561         }
1562         c[colam + ridx[i]] += r1;
1563       }
1564       b1 += bm;
1565     }
1566   } else {
1567     for (col=0; col<cn-4; col += 4) {  /* over columns of C */
1568       colam = col*am;
1569       for (i=0; i<am; i++) {        /* over rows of C in those columns */
1570         r1 = r2 = r3 = r4 = 0.0;
1571         n  = a->i[i+1] - a->i[i];
1572         aj = a->j + a->i[i];
1573         aa = a->a + a->i[i];
1574         for (j=0; j<n; j++) {
1575           r1 += (*aa)*b1[*aj];
1576           r2 += (*aa)*b2[*aj];
1577           r3 += (*aa)*b3[*aj];
1578           r4 += (*aa++)*b4[*aj++];
1579         }
1580         c[colam + i]       += r1;
1581         c[colam + am + i]  += r2;
1582         c[colam + am2 + i] += r3;
1583         c[colam + am3 + i] += r4;
1584       }
1585       b1 += bm4;
1586       b2 += bm4;
1587       b3 += bm4;
1588       b4 += bm4;
1589     }
1590     for (; col<cn; col++) {     /* over extra columns of C */
1591       colam = col*am;
1592       for (i=0; i<am; i++) {  /* over rows of C in those columns */
1593         r1 = 0.0;
1594         n  = a->i[i+1] - a->i[i];
1595         aj = a->j + a->i[i];
1596         aa = a->a + a->i[i];
1597 
1598         for (j=0; j<n; j++) {
1599           r1 += (*aa++)*b1[*aj++];
1600         }
1601         c[colam + i] += r1;
1602       }
1603       b1 += bm;
1604     }
1605   }
1606   ierr = PetscLogFlops(cn*2.0*a->nz);CHKERRQ(ierr);
1607   ierr = MatDenseRestoreArray(C,&c);CHKERRQ(ierr);
1608   PetscFunctionReturn(0);
1609 }
1610 
1611 PetscErrorCode  MatTransColoringApplySpToDen_SeqAIJ(MatTransposeColoring coloring,Mat B,Mat Btdense)
1612 {
1613   PetscErrorCode ierr;
1614   Mat_SeqAIJ     *b       = (Mat_SeqAIJ*)B->data;
1615   Mat_SeqDense   *btdense = (Mat_SeqDense*)Btdense->data;
1616   PetscInt       *bi      = b->i,*bj=b->j;
1617   PetscInt       m        = Btdense->rmap->n,n=Btdense->cmap->n,j,k,l,col,anz,*btcol,brow,ncolumns;
1618   MatScalar      *btval,*btval_den,*ba=b->a;
1619   PetscInt       *columns=coloring->columns,*colorforcol=coloring->colorforcol,ncolors=coloring->ncolors;
1620 
1621   PetscFunctionBegin;
1622   btval_den=btdense->v;
1623   ierr     = PetscMemzero(btval_den,(m*n)*sizeof(MatScalar));CHKERRQ(ierr);
1624   for (k=0; k<ncolors; k++) {
1625     ncolumns = coloring->ncolumns[k];
1626     for (l=0; l<ncolumns; l++) { /* insert a row of B to a column of Btdense */
1627       col   = *(columns + colorforcol[k] + l);
1628       btcol = bj + bi[col];
1629       btval = ba + bi[col];
1630       anz   = bi[col+1] - bi[col];
1631       for (j=0; j<anz; j++) {
1632         brow            = btcol[j];
1633         btval_den[brow] = btval[j];
1634       }
1635     }
1636     btval_den += m;
1637   }
1638   PetscFunctionReturn(0);
1639 }
1640 
1641 PetscErrorCode MatTransColoringApplyDenToSp_SeqAIJ(MatTransposeColoring matcoloring,Mat Cden,Mat Csp)
1642 {
1643   PetscErrorCode ierr;
1644   Mat_SeqAIJ     *csp = (Mat_SeqAIJ*)Csp->data;
1645   PetscScalar    *ca_den,*ca_den_ptr,*ca=csp->a;
1646   PetscInt       k,l,m=Cden->rmap->n,ncolors=matcoloring->ncolors;
1647   PetscInt       brows=matcoloring->brows,*den2sp=matcoloring->den2sp;
1648   PetscInt       nrows,*row,*idx;
1649   PetscInt       *rows=matcoloring->rows,*colorforrow=matcoloring->colorforrow;
1650 
1651   PetscFunctionBegin;
1652   ierr   = MatDenseGetArray(Cden,&ca_den);CHKERRQ(ierr);
1653 
1654   if (brows > 0) {
1655     PetscInt *lstart,row_end,row_start;
1656     lstart = matcoloring->lstart;
1657     ierr = PetscMemzero(lstart,ncolors*sizeof(PetscInt));CHKERRQ(ierr);
1658 
1659     row_end = brows;
1660     if (row_end > m) row_end = m;
1661     for (row_start=0; row_start<m; row_start+=brows) { /* loop over row blocks of Csp */
1662       ca_den_ptr = ca_den;
1663       for (k=0; k<ncolors; k++) { /* loop over colors (columns of Cden) */
1664         nrows = matcoloring->nrows[k];
1665         row   = rows  + colorforrow[k];
1666         idx   = den2sp + colorforrow[k];
1667         for (l=lstart[k]; l<nrows; l++) {
1668           if (row[l] >= row_end) {
1669             lstart[k] = l;
1670             break;
1671           } else {
1672             ca[idx[l]] = ca_den_ptr[row[l]];
1673           }
1674         }
1675         ca_den_ptr += m;
1676       }
1677       row_end += brows;
1678       if (row_end > m) row_end = m;
1679     }
1680   } else { /* non-blocked impl: loop over columns of Csp - slow if Csp is large */
1681     ca_den_ptr = ca_den;
1682     for (k=0; k<ncolors; k++) {
1683       nrows = matcoloring->nrows[k];
1684       row   = rows  + colorforrow[k];
1685       idx   = den2sp + colorforrow[k];
1686       for (l=0; l<nrows; l++) {
1687         ca[idx[l]] = ca_den_ptr[row[l]];
1688       }
1689       ca_den_ptr += m;
1690     }
1691   }
1692 
1693   ierr = MatDenseRestoreArray(Cden,&ca_den);CHKERRQ(ierr);
1694 #if defined(PETSC_USE_INFO)
1695   if (matcoloring->brows > 0) {
1696     ierr = PetscInfo1(Csp,"Loop over %D row blocks for den2sp\n",brows);CHKERRQ(ierr);
1697   } else {
1698     ierr = PetscInfo(Csp,"Loop over colors/columns of Cden, inefficient for large sparse matrix product \n");CHKERRQ(ierr);
1699   }
1700 #endif
1701   PetscFunctionReturn(0);
1702 }
1703 
1704 PetscErrorCode MatTransposeColoringCreate_SeqAIJ(Mat mat,ISColoring iscoloring,MatTransposeColoring c)
1705 {
1706   PetscErrorCode ierr;
1707   PetscInt       i,n,nrows,Nbs,j,k,m,ncols,col,cm;
1708   const PetscInt *is,*ci,*cj,*row_idx;
1709   PetscInt       nis = iscoloring->n,*rowhit,bs = 1;
1710   IS             *isa;
1711   Mat_SeqAIJ     *csp = (Mat_SeqAIJ*)mat->data;
1712   PetscInt       *colorforrow,*rows,*rows_i,*idxhit,*spidx,*den2sp,*den2sp_i;
1713   PetscInt       *colorforcol,*columns,*columns_i,brows;
1714   PetscBool      flg;
1715 
1716   PetscFunctionBegin;
1717   ierr = ISColoringGetIS(iscoloring,PETSC_IGNORE,&isa);CHKERRQ(ierr);
1718 
1719   /* bs >1 is not being tested yet! */
1720   Nbs       = mat->cmap->N/bs;
1721   c->M      = mat->rmap->N/bs;  /* set total rows, columns and local rows */
1722   c->N      = Nbs;
1723   c->m      = c->M;
1724   c->rstart = 0;
1725   c->brows  = 100;
1726 
1727   c->ncolors = nis;
1728   ierr = PetscMalloc3(nis,&c->ncolumns,nis,&c->nrows,nis+1,&colorforrow);CHKERRQ(ierr);
1729   ierr = PetscMalloc1(csp->nz+1,&rows);CHKERRQ(ierr);
1730   ierr = PetscMalloc1(csp->nz+1,&den2sp);CHKERRQ(ierr);
1731 
1732   brows = c->brows;
1733   ierr = PetscOptionsGetInt(NULL,NULL,"-matden2sp_brows",&brows,&flg);CHKERRQ(ierr);
1734   if (flg) c->brows = brows;
1735   if (brows > 0) {
1736     ierr = PetscMalloc1(nis+1,&c->lstart);CHKERRQ(ierr);
1737   }
1738 
1739   colorforrow[0] = 0;
1740   rows_i         = rows;
1741   den2sp_i       = den2sp;
1742 
1743   ierr = PetscMalloc1(nis+1,&colorforcol);CHKERRQ(ierr);
1744   ierr = PetscMalloc1(Nbs+1,&columns);CHKERRQ(ierr);
1745 
1746   colorforcol[0] = 0;
1747   columns_i      = columns;
1748 
1749   /* get column-wise storage of mat */
1750   ierr = MatGetColumnIJ_SeqAIJ_Color(mat,0,PETSC_FALSE,PETSC_FALSE,&ncols,&ci,&cj,&spidx,NULL);CHKERRQ(ierr);
1751 
1752   cm   = c->m;
1753   ierr = PetscMalloc1(cm+1,&rowhit);CHKERRQ(ierr);
1754   ierr = PetscMalloc1(cm+1,&idxhit);CHKERRQ(ierr);
1755   for (i=0; i<nis; i++) { /* loop over color */
1756     ierr = ISGetLocalSize(isa[i],&n);CHKERRQ(ierr);
1757     ierr = ISGetIndices(isa[i],&is);CHKERRQ(ierr);
1758 
1759     c->ncolumns[i] = n;
1760     if (n) {
1761       ierr = PetscMemcpy(columns_i,is,n*sizeof(PetscInt));CHKERRQ(ierr);
1762     }
1763     colorforcol[i+1] = colorforcol[i] + n;
1764     columns_i       += n;
1765 
1766     /* fast, crude version requires O(N*N) work */
1767     ierr = PetscMemzero(rowhit,cm*sizeof(PetscInt));CHKERRQ(ierr);
1768 
1769     for (j=0; j<n; j++) { /* loop over columns*/
1770       col     = is[j];
1771       row_idx = cj + ci[col];
1772       m       = ci[col+1] - ci[col];
1773       for (k=0; k<m; k++) { /* loop over columns marking them in rowhit */
1774         idxhit[*row_idx]   = spidx[ci[col] + k];
1775         rowhit[*row_idx++] = col + 1;
1776       }
1777     }
1778     /* count the number of hits */
1779     nrows = 0;
1780     for (j=0; j<cm; j++) {
1781       if (rowhit[j]) nrows++;
1782     }
1783     c->nrows[i]      = nrows;
1784     colorforrow[i+1] = colorforrow[i] + nrows;
1785 
1786     nrows = 0;
1787     for (j=0; j<cm; j++) { /* loop over rows */
1788       if (rowhit[j]) {
1789         rows_i[nrows]   = j;
1790         den2sp_i[nrows] = idxhit[j];
1791         nrows++;
1792       }
1793     }
1794     den2sp_i += nrows;
1795 
1796     ierr    = ISRestoreIndices(isa[i],&is);CHKERRQ(ierr);
1797     rows_i += nrows;
1798   }
1799   ierr = MatRestoreColumnIJ_SeqAIJ_Color(mat,0,PETSC_FALSE,PETSC_FALSE,&ncols,&ci,&cj,&spidx,NULL);CHKERRQ(ierr);
1800   ierr = PetscFree(rowhit);CHKERRQ(ierr);
1801   ierr = ISColoringRestoreIS(iscoloring,&isa);CHKERRQ(ierr);
1802   if (csp->nz != colorforrow[nis]) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_PLIB,"csp->nz %d != colorforrow[nis] %d",csp->nz,colorforrow[nis]);
1803 
1804   c->colorforrow = colorforrow;
1805   c->rows        = rows;
1806   c->den2sp      = den2sp;
1807   c->colorforcol = colorforcol;
1808   c->columns     = columns;
1809 
1810   ierr = PetscFree(idxhit);CHKERRQ(ierr);
1811   PetscFunctionReturn(0);
1812 }
1813 
1814 /* This algorithm combines the symbolic and numeric phase of matrix-matrix multiplication. */
1815 PetscErrorCode MatMatMult_SeqAIJ_SeqAIJ_Combined(Mat A,Mat B,PetscReal fill,Mat *C)
1816 {
1817   PetscErrorCode     ierr;
1818   PetscLogDouble     flops=0.0;
1819   Mat_SeqAIJ         *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
1820   const PetscInt     *ai=a->i,*bi=b->i;
1821   PetscInt           *ci,*cj,*cj_i;
1822   PetscScalar        *ca,*ca_i;
1823   PetscInt           b_maxmemrow,c_maxmem,a_col;
1824   PetscInt           am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
1825   PetscInt           i,k,ndouble=0;
1826   PetscReal          afill;
1827   PetscScalar        *c_row_val_dense;
1828   PetscBool          *c_row_idx_flags;
1829   PetscInt           *aj_i=a->j;
1830   PetscScalar        *aa_i=a->a;
1831 
1832   PetscFunctionBegin;
1833 
1834   /* Step 1: Determine upper bounds on memory for C and allocate memory */
1835   /* This should be enough for almost all matrices. If still more memory is needed, it is reallocated later. */
1836   c_maxmem    = 8*(ai[am]+bi[bm]);
1837   b_maxmemrow = PetscMin(bi[bm],bn);
1838   ierr  = PetscMalloc1(am+1,&ci);CHKERRQ(ierr);
1839   ierr  = PetscMalloc1(bn,&c_row_val_dense);CHKERRQ(ierr);
1840   ierr  = PetscMalloc1(bn,&c_row_idx_flags);CHKERRQ(ierr);
1841   ierr  = PetscMalloc1(c_maxmem,&cj);CHKERRQ(ierr);
1842   ierr  = PetscMalloc1(c_maxmem,&ca);CHKERRQ(ierr);
1843   ca_i  = ca;
1844   cj_i  = cj;
1845   ci[0] = 0;
1846   ierr  = PetscMemzero(c_row_val_dense,bn*sizeof(PetscScalar));CHKERRQ(ierr);
1847   ierr  = PetscMemzero(c_row_idx_flags,bn*sizeof(PetscBool));CHKERRQ(ierr);
1848   for (i=0; i<am; i++) {
1849     /* Step 2: Initialize the dense row vector for C  */
1850     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 */
1851     PetscInt       cnzi = 0;
1852     PetscInt       *bj_i;
1853     PetscScalar    *ba_i;
1854     /* If the number of indices in C so far + the max number of columns in the next row > c_maxmem  -> allocate more memory
1855        Usually, there is enough memory in the first place, so this is not executed. */
1856     while (ci[i] + b_maxmemrow > c_maxmem) {
1857       c_maxmem *= 2;
1858       ndouble++;
1859       ierr = PetscRealloc(sizeof(PetscInt)*c_maxmem,&cj);CHKERRQ(ierr);
1860       ierr = PetscRealloc(sizeof(PetscScalar)*c_maxmem,&ca);CHKERRQ(ierr);
1861     }
1862 
1863     /* Step 3: Do the numerical calculations */
1864     for (a_col=0; a_col<anzi; a_col++) {          /* iterate over all non zero values in a row of A */
1865       PetscInt       a_col_index = aj_i[a_col];
1866       const PetscInt bnzi        = bi[a_col_index+1] - bi[a_col_index];
1867       flops += 2*bnzi;
1868       bj_i   = b->j + bi[a_col_index];   /* points to the current row in bj */
1869       ba_i   = b->a + bi[a_col_index];   /* points to the current row in ba */
1870       for (k=0; k<bnzi; ++k) { /* iterate over all non zeros of this row in B */
1871         if (c_row_idx_flags[bj_i[k]] == PETSC_FALSE) {
1872           cj_i[cnzi++]             = bj_i[k];
1873           c_row_idx_flags[bj_i[k]] = PETSC_TRUE;
1874         }
1875         c_row_val_dense[bj_i[k]] += aa_i[a_col] * ba_i[k];
1876       }
1877     }
1878 
1879     /* Sort array */
1880     ierr = PetscSortInt(cnzi,cj_i);CHKERRQ(ierr);
1881     /* Step 4 */
1882     for (k=0; k<cnzi; k++) {
1883       ca_i[k]                  = c_row_val_dense[cj_i[k]];
1884       c_row_val_dense[cj_i[k]] = 0.;
1885       c_row_idx_flags[cj_i[k]] = PETSC_FALSE;
1886     }
1887     /* terminate current row */
1888     aa_i   += anzi;
1889     aj_i   += anzi;
1890     ca_i   += cnzi;
1891     cj_i   += cnzi;
1892     ci[i+1] = ci[i] + cnzi;
1893     flops  += cnzi;
1894   }
1895 
1896   /* Step 5 */
1897   /* Create the new matrix */
1898   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
1899   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
1900   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
1901 
1902   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
1903   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
1904   c          = (Mat_SeqAIJ*)((*C)->data);
1905   c->a       = ca;
1906   c->free_a  = PETSC_TRUE;
1907   c->free_ij = PETSC_TRUE;
1908   c->nonew   = 0;
1909 
1910   /* set MatInfo */
1911   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
1912   if (afill < 1.0) afill = 1.0;
1913   c->maxnz                     = ci[am];
1914   c->nz                        = ci[am];
1915   (*C)->info.mallocs           = ndouble;
1916   (*C)->info.fill_ratio_given  = fill;
1917   (*C)->info.fill_ratio_needed = afill;
1918 
1919   ierr = PetscFree(c_row_val_dense);CHKERRQ(ierr);
1920   ierr = PetscFree(c_row_idx_flags);CHKERRQ(ierr);
1921 
1922   ierr = MatAssemblyBegin(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1923   ierr = MatAssemblyEnd(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1924   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
1925   PetscFunctionReturn(0);
1926 }
1927