xref: /petsc/src/mat/impls/aij/seq/matmatmult.c (revision d52a580b706c59ca78066c1e38754e45b6b56e2b)
1 /*
2   Defines matrix-matrix product routines for pairs of SeqAIJ matrices
3           C = A * B
4 */
5 
6 #include <../src/mat/impls/aij/seq/aij.h> /*I "petscmat.h" I*/
7 #include <../src/mat/utils/freespace.h>
8 #include <petscbt.h>
9 #include <petsc/private/isimpl.h>
10 #include <../src/mat/impls/dense/seq/dense.h>
11 
12 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ(Mat A, Mat B, Mat C)
13 {
14   PetscFunctionBegin;
15   if (C->ops->matmultnumeric) PetscCall((*C->ops->matmultnumeric)(A, B, C));
16   else PetscCall(MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted(A, B, C));
17   PetscFunctionReturn(PETSC_SUCCESS);
18 }
19 
20 /* Modified from MatCreateSeqAIJWithArrays() */
21 PETSC_INTERN PetscErrorCode MatSetSeqAIJWithArrays_private(MPI_Comm comm, PetscInt m, PetscInt n, PetscInt i[], PetscInt j[], PetscScalar a[], MatType mtype, Mat mat)
22 {
23   PetscInt    ii;
24   Mat_SeqAIJ *aij;
25   PetscBool   isseqaij, ofree_a, ofree_ij;
26 
27   PetscFunctionBegin;
28   PetscCheck(m <= 0 || !i[0], PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "i (row indices) must start with 0");
29   PetscCall(MatSetSizes(mat, m, n, m, n));
30 
31   if (!mtype) {
32     PetscCall(PetscObjectBaseTypeCompare((PetscObject)mat, MATSEQAIJ, &isseqaij));
33     if (!isseqaij) PetscCall(MatSetType(mat, MATSEQAIJ));
34   } else {
35     PetscCall(MatSetType(mat, mtype));
36   }
37 
38   aij      = (Mat_SeqAIJ *)mat->data;
39   ofree_a  = aij->free_a;
40   ofree_ij = aij->free_ij;
41   /* changes the free flags */
42   PetscCall(MatSeqAIJSetPreallocation_SeqAIJ(mat, MAT_SKIP_ALLOCATION, NULL));
43 
44   PetscCall(PetscFree(aij->ilen));
45   PetscCall(PetscFree(aij->imax));
46   PetscCall(PetscMalloc1(m, &aij->imax));
47   PetscCall(PetscMalloc1(m, &aij->ilen));
48   for (ii = 0, aij->nonzerorowcnt = 0, aij->rmax = 0; ii < m; ii++) {
49     const PetscInt rnz = i[ii + 1] - i[ii];
50     aij->nonzerorowcnt += !!rnz;
51     aij->rmax     = PetscMax(aij->rmax, rnz);
52     aij->ilen[ii] = aij->imax[ii] = i[ii + 1] - i[ii];
53   }
54   aij->maxnz = i[m];
55   aij->nz    = i[m];
56 
57   if (ofree_a) PetscCall(PetscShmgetDeallocateArray((void **)&aij->a));
58   if (ofree_ij) PetscCall(PetscShmgetDeallocateArray((void **)&aij->j));
59   if (ofree_ij) PetscCall(PetscShmgetDeallocateArray((void **)&aij->i));
60 
61   aij->i       = i;
62   aij->j       = j;
63   aij->a       = a;
64   aij->nonew   = -1; /* this indicates that inserting a new value in the matrix that generates a new nonzero is an error */
65   aij->free_a  = PETSC_FALSE;
66   aij->free_ij = PETSC_FALSE;
67   PetscCall(MatCheckCompressedRow(mat, aij->nonzerorowcnt, &aij->compressedrow, aij->i, m, 0.6));
68   // Always build the diag info when i, j are set
69   PetscFunctionReturn(PETSC_SUCCESS);
70 }
71 
72 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ(Mat A, Mat B, PetscReal fill, Mat C)
73 {
74   Mat_Product        *product = C->product;
75   MatProductAlgorithm alg;
76   PetscBool           flg;
77 
78   PetscFunctionBegin;
79   if (product) {
80     alg = product->alg;
81   } else {
82     alg = "sorted";
83   }
84   /* sorted */
85   PetscCall(PetscStrcmp(alg, "sorted", &flg));
86   if (flg) {
87     PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ_Sorted(A, B, fill, C));
88     PetscFunctionReturn(PETSC_SUCCESS);
89   }
90 
91   /* scalable */
92   PetscCall(PetscStrcmp(alg, "scalable", &flg));
93   if (flg) {
94     PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable(A, B, fill, C));
95     PetscFunctionReturn(PETSC_SUCCESS);
96   }
97 
98   /* scalable_fast */
99   PetscCall(PetscStrcmp(alg, "scalable_fast", &flg));
100   if (flg) {
101     PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable_fast(A, B, fill, C));
102     PetscFunctionReturn(PETSC_SUCCESS);
103   }
104 
105   /* heap */
106   PetscCall(PetscStrcmp(alg, "heap", &flg));
107   if (flg) {
108     PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ_Heap(A, B, fill, C));
109     PetscFunctionReturn(PETSC_SUCCESS);
110   }
111 
112   /* btheap */
113   PetscCall(PetscStrcmp(alg, "btheap", &flg));
114   if (flg) {
115     PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ_BTHeap(A, B, fill, C));
116     PetscFunctionReturn(PETSC_SUCCESS);
117   }
118 
119   /* llcondensed */
120   PetscCall(PetscStrcmp(alg, "llcondensed", &flg));
121   if (flg) {
122     PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ_LLCondensed(A, B, fill, C));
123     PetscFunctionReturn(PETSC_SUCCESS);
124   }
125 
126   /* rowmerge */
127   PetscCall(PetscStrcmp(alg, "rowmerge", &flg));
128   if (flg) {
129     PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ_RowMerge(A, B, fill, C));
130     PetscFunctionReturn(PETSC_SUCCESS);
131   }
132 
133 #if defined(PETSC_HAVE_HYPRE)
134   PetscCall(PetscStrcmp(alg, "hypre", &flg));
135   if (flg) {
136     PetscCall(MatMatMultSymbolic_AIJ_AIJ_wHYPRE(A, B, fill, C));
137     PetscFunctionReturn(PETSC_SUCCESS);
138   }
139 #endif
140 
141   SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Mat Product Algorithm is not supported");
142 }
143 
144 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_LLCondensed(Mat A, Mat B, PetscReal fill, Mat C)
145 {
146   Mat_SeqAIJ        *a = (Mat_SeqAIJ *)A->data, *b = (Mat_SeqAIJ *)B->data, *c;
147   PetscInt          *ai = a->i, *bi = b->i, *ci, *cj;
148   PetscInt           am = A->rmap->N, bn = B->cmap->N, bm = B->rmap->N;
149   PetscReal          afill;
150   PetscInt           i, j, anzi, brow, bnzj, cnzi, *bj, *aj, *lnk, ndouble = 0, Crmax;
151   PetscHMapI         ta;
152   PetscBT            lnkbt;
153   PetscFreeSpaceList free_space = NULL, current_space = NULL;
154 
155   PetscFunctionBegin;
156   /* Get ci and cj */
157   /* Allocate ci array, arrays for fill computation and */
158   /* free space for accumulating nonzero column info */
159   PetscCall(PetscMalloc1(am + 2, &ci));
160   ci[0] = 0;
161 
162   /* create and initialize a linked list */
163   PetscCall(PetscHMapICreateWithSize(bn, &ta));
164   MatRowMergeMax_SeqAIJ(b, bm, ta);
165   PetscCall(PetscHMapIGetSize(ta, &Crmax));
166   PetscCall(PetscHMapIDestroy(&ta));
167 
168   PetscCall(PetscLLCondensedCreate(Crmax, bn, &lnk, &lnkbt));
169 
170   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
171   PetscCall(PetscFreeSpaceGet(PetscRealIntMultTruncate(fill, PetscIntSumTruncate(ai[am], bi[bm])), &free_space));
172 
173   current_space = free_space;
174 
175   /* Determine ci and cj */
176   for (i = 0; i < am; i++) {
177     anzi = ai[i + 1] - ai[i];
178     aj   = a->j + ai[i];
179     for (j = 0; j < anzi; j++) {
180       brow = aj[j];
181       bnzj = bi[brow + 1] - bi[brow];
182       bj   = b->j + bi[brow];
183       /* add non-zero cols of B into the sorted linked list lnk */
184       PetscCall(PetscLLCondensedAddSorted(bnzj, bj, lnk, lnkbt));
185     }
186     /* add possible missing diagonal entry */
187     if (C->force_diagonals) PetscCall(PetscLLCondensedAddSorted(1, &i, lnk, lnkbt));
188     cnzi = lnk[0];
189 
190     /* If free space is not available, make more free space */
191     /* Double the amount of total space in the list */
192     if (current_space->local_remaining < cnzi) {
193       PetscCall(PetscFreeSpaceGet(PetscIntSumTruncate(cnzi, current_space->total_array_size), &current_space));
194       ndouble++;
195     }
196 
197     /* Copy data into free space, then initialize lnk */
198     PetscCall(PetscLLCondensedClean(bn, cnzi, current_space->array, lnk, lnkbt));
199 
200     current_space->array += cnzi;
201     current_space->local_used += cnzi;
202     current_space->local_remaining -= cnzi;
203 
204     ci[i + 1] = ci[i] + cnzi;
205   }
206 
207   /* Column indices are in the list of free space */
208   /* Allocate space for cj, initialize cj, and */
209   /* destroy list of free space and other temporary array(s) */
210   PetscCall(PetscMalloc1(ci[am] + 1, &cj));
211   PetscCall(PetscFreeSpaceContiguous(&free_space, cj));
212   PetscCall(PetscLLCondensedDestroy(lnk, lnkbt));
213 
214   /* put together the new symbolic matrix */
215   PetscCall(MatSetSeqAIJWithArrays_private(PetscObjectComm((PetscObject)A), am, bn, ci, cj, NULL, ((PetscObject)A)->type_name, C));
216   PetscCall(MatSetBlockSizesFromMats(C, A, B));
217 
218   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
219   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
220   c          = (Mat_SeqAIJ *)C->data;
221   c->free_a  = PETSC_FALSE;
222   c->free_ij = PETSC_TRUE;
223   c->nonew   = 0;
224 
225   /* fast, needs non-scalable O(bn) array 'abdense' */
226   C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;
227 
228   /* set MatInfo */
229   afill = (PetscReal)ci[am] / (ai[am] + bi[bm]) + 1.e-5;
230   if (afill < 1.0) afill = 1.0;
231   C->info.mallocs           = ndouble;
232   C->info.fill_ratio_given  = fill;
233   C->info.fill_ratio_needed = afill;
234 
235 #if defined(PETSC_USE_INFO)
236   if (ci[am]) {
237     PetscCall(PetscInfo(C, "Reallocs %" PetscInt_FMT "; Fill ratio: given %g needed %g.\n", ndouble, (double)fill, (double)afill));
238     PetscCall(PetscInfo(C, "Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n", (double)afill));
239   } else {
240     PetscCall(PetscInfo(C, "Empty matrix product\n"));
241   }
242 #endif
243   PetscFunctionReturn(PETSC_SUCCESS);
244 }
245 
246 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted(Mat A, Mat B, Mat C)
247 {
248   PetscLogDouble     flops = 0.0;
249   Mat_SeqAIJ        *a     = (Mat_SeqAIJ *)A->data;
250   Mat_SeqAIJ        *b     = (Mat_SeqAIJ *)B->data;
251   Mat_SeqAIJ        *c     = (Mat_SeqAIJ *)C->data;
252   PetscInt          *ai = a->i, *aj = a->j, *bi = b->i, *bj = b->j, *bjj, *ci = c->i, *cj = c->j;
253   PetscInt           am = A->rmap->n, cm = C->rmap->n;
254   PetscInt           i, j, k, anzi, bnzi, cnzi, brow;
255   PetscScalar       *ca, valtmp;
256   PetscScalar       *ab_dense;
257   PetscContainer     cab_dense;
258   const PetscScalar *aa, *ba, *baj;
259 
260   PetscFunctionBegin;
261   PetscCall(MatSeqAIJGetArrayRead(A, &aa));
262   PetscCall(MatSeqAIJGetArrayRead(B, &ba));
263   if (!c->a) { /* first call of MatMatMultNumeric_SeqAIJ_SeqAIJ, allocate ca and matmult_abdense */
264     PetscCall(PetscMalloc1(ci[cm] + 1, &ca));
265     c->a      = ca;
266     c->free_a = PETSC_TRUE;
267   } else ca = c->a;
268 
269   /* TODO this should be done in the symbolic phase */
270   /* However, this function is so heavily used (sometimes in an hidden way through multnumeric function pointers
271      that is hard to eradicate) */
272   PetscCall(PetscObjectQuery((PetscObject)C, "__PETSc__ab_dense", (PetscObject *)&cab_dense));
273   if (!cab_dense) {
274     PetscCall(PetscMalloc1(B->cmap->N, &ab_dense));
275     PetscCall(PetscObjectContainerCompose((PetscObject)C, "__PETSc__ab_dense", ab_dense, PetscCtxDestroyDefault));
276   } else PetscCall(PetscContainerGetPointer(cab_dense, &ab_dense));
277   PetscCall(PetscArrayzero(ab_dense, B->cmap->N));
278 
279   /* clean old values in C */
280   PetscCall(PetscArrayzero(ca, ci[cm]));
281   /* Traverse A row-wise. */
282   /* Build the ith row in C by summing over nonzero columns in A, */
283   /* the rows of B corresponding to nonzeros of A. */
284   for (i = 0; i < am; i++) {
285     anzi = ai[i + 1] - ai[i];
286     for (j = 0; j < anzi; j++) {
287       brow = aj[j];
288       bnzi = bi[brow + 1] - bi[brow];
289       bjj  = PetscSafePointerPlusOffset(bj, bi[brow]);
290       baj  = PetscSafePointerPlusOffset(ba, bi[brow]);
291       /* perform dense axpy */
292       valtmp = aa[j];
293       for (k = 0; k < bnzi; k++) ab_dense[bjj[k]] += valtmp * baj[k];
294       flops += 2 * bnzi;
295     }
296     aj = PetscSafePointerPlusOffset(aj, anzi);
297     aa = PetscSafePointerPlusOffset(aa, anzi);
298 
299     cnzi = ci[i + 1] - ci[i];
300     for (k = 0; k < cnzi; k++) {
301       ca[k] += ab_dense[cj[k]];
302       ab_dense[cj[k]] = 0.0; /* zero ab_dense */
303     }
304     flops += cnzi;
305     cj = PetscSafePointerPlusOffset(cj, cnzi);
306     ca += cnzi;
307   }
308 #if defined(PETSC_HAVE_DEVICE)
309   if (C->offloadmask != PETSC_OFFLOAD_UNALLOCATED) C->offloadmask = PETSC_OFFLOAD_CPU;
310 #endif
311   PetscCall(MatAssemblyBegin(C, MAT_FINAL_ASSEMBLY));
312   PetscCall(MatAssemblyEnd(C, MAT_FINAL_ASSEMBLY));
313   PetscCall(PetscLogFlops(flops));
314   PetscCall(MatSeqAIJRestoreArrayRead(A, &aa));
315   PetscCall(MatSeqAIJRestoreArrayRead(B, &ba));
316   PetscFunctionReturn(PETSC_SUCCESS);
317 }
318 
319 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable(Mat A, Mat B, Mat C)
320 {
321   PetscLogDouble     flops = 0.0;
322   Mat_SeqAIJ        *a     = (Mat_SeqAIJ *)A->data;
323   Mat_SeqAIJ        *b     = (Mat_SeqAIJ *)B->data;
324   Mat_SeqAIJ        *c     = (Mat_SeqAIJ *)C->data;
325   PetscInt          *ai = a->i, *aj = a->j, *bi = b->i, *bj = b->j, *bjj, *ci = c->i, *cj = c->j;
326   PetscInt           am = A->rmap->N, cm = C->rmap->N;
327   PetscInt           i, j, k, anzi, bnzi, cnzi, brow;
328   PetscScalar       *ca = c->a, valtmp;
329   const PetscScalar *aa, *ba, *baj;
330   PetscInt           nextb;
331 
332   PetscFunctionBegin;
333   PetscCall(MatSeqAIJGetArrayRead(A, &aa));
334   PetscCall(MatSeqAIJGetArrayRead(B, &ba));
335   if (!ca) { /* first call of MatMatMultNumeric_SeqAIJ_SeqAIJ, allocate ca and matmult_abdense */
336     PetscCall(PetscMalloc1(ci[cm] + 1, &ca));
337     c->a      = ca;
338     c->free_a = PETSC_TRUE;
339   }
340 
341   /* clean old values in C */
342   PetscCall(PetscArrayzero(ca, ci[cm]));
343   /* Traverse A row-wise. */
344   /* Build the ith row in C by summing over nonzero columns in A, */
345   /* the rows of B corresponding to nonzeros of A. */
346   for (i = 0; i < am; i++) {
347     anzi = ai[i + 1] - ai[i];
348     cnzi = ci[i + 1] - ci[i];
349     for (j = 0; j < anzi; j++) {
350       brow = aj[j];
351       bnzi = bi[brow + 1] - bi[brow];
352       bjj  = bj + bi[brow];
353       baj  = ba + bi[brow];
354       /* perform sparse axpy */
355       valtmp = aa[j];
356       nextb  = 0;
357       for (k = 0; nextb < bnzi; k++) {
358         if (cj[k] == bjj[nextb]) { /* ccol == bcol */
359           ca[k] += valtmp * baj[nextb++];
360         }
361       }
362       flops += 2 * bnzi;
363     }
364     aj += anzi;
365     aa += anzi;
366     cj += cnzi;
367     ca += cnzi;
368   }
369 #if defined(PETSC_HAVE_DEVICE)
370   if (C->offloadmask != PETSC_OFFLOAD_UNALLOCATED) C->offloadmask = PETSC_OFFLOAD_CPU;
371 #endif
372   PetscCall(MatAssemblyBegin(C, MAT_FINAL_ASSEMBLY));
373   PetscCall(MatAssemblyEnd(C, MAT_FINAL_ASSEMBLY));
374   PetscCall(PetscLogFlops(flops));
375   PetscCall(MatSeqAIJRestoreArrayRead(A, &aa));
376   PetscCall(MatSeqAIJRestoreArrayRead(B, &ba));
377   PetscFunctionReturn(PETSC_SUCCESS);
378 }
379 
380 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable_fast(Mat A, Mat B, PetscReal fill, Mat C)
381 {
382   Mat_SeqAIJ        *a = (Mat_SeqAIJ *)A->data, *b = (Mat_SeqAIJ *)B->data, *c;
383   PetscInt          *ai = a->i, *bi = b->i, *ci, *cj;
384   PetscInt           am = A->rmap->N, bn = B->cmap->N, bm = B->rmap->N;
385   MatScalar         *ca;
386   PetscReal          afill;
387   PetscInt           i, j, anzi, brow, bnzj, cnzi, *bj, *aj, *lnk, ndouble = 0, Crmax;
388   PetscHMapI         ta;
389   PetscFreeSpaceList free_space = NULL, current_space = NULL;
390 
391   PetscFunctionBegin;
392   /* Get ci and cj - same as MatMatMultSymbolic_SeqAIJ_SeqAIJ except using PetscLLxxx_fast() */
393   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
394   PetscCall(PetscMalloc1(am + 2, &ci));
395   ci[0] = 0;
396 
397   /* create and initialize a linked list */
398   PetscCall(PetscHMapICreateWithSize(bn, &ta));
399   MatRowMergeMax_SeqAIJ(b, bm, ta);
400   PetscCall(PetscHMapIGetSize(ta, &Crmax));
401   PetscCall(PetscHMapIDestroy(&ta));
402 
403   PetscCall(PetscLLCondensedCreate_fast(Crmax, &lnk));
404 
405   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
406   PetscCall(PetscFreeSpaceGet(PetscRealIntMultTruncate(fill, PetscIntSumTruncate(ai[am], bi[bm])), &free_space));
407   current_space = free_space;
408 
409   /* Determine ci and cj */
410   for (i = 0; i < am; i++) {
411     anzi = ai[i + 1] - ai[i];
412     aj   = a->j + ai[i];
413     for (j = 0; j < anzi; j++) {
414       brow = aj[j];
415       bnzj = bi[brow + 1] - bi[brow];
416       bj   = b->j + bi[brow];
417       /* add non-zero cols of B into the sorted linked list lnk */
418       PetscCall(PetscLLCondensedAddSorted_fast(bnzj, bj, lnk));
419     }
420     /* add possible missing diagonal entry */
421     if (C->force_diagonals) PetscCall(PetscLLCondensedAddSorted_fast(1, &i, lnk));
422     cnzi = lnk[1];
423 
424     /* If free space is not available, make more free space */
425     /* Double the amount of total space in the list */
426     if (current_space->local_remaining < cnzi) {
427       PetscCall(PetscFreeSpaceGet(PetscIntSumTruncate(cnzi, current_space->total_array_size), &current_space));
428       ndouble++;
429     }
430 
431     /* Copy data into free space, then initialize lnk */
432     PetscCall(PetscLLCondensedClean_fast(cnzi, current_space->array, lnk));
433 
434     current_space->array += cnzi;
435     current_space->local_used += cnzi;
436     current_space->local_remaining -= cnzi;
437 
438     ci[i + 1] = ci[i] + cnzi;
439   }
440 
441   /* Column indices are in the list of free space */
442   /* Allocate space for cj, initialize cj, and */
443   /* destroy list of free space and other temporary array(s) */
444   PetscCall(PetscMalloc1(ci[am] + 1, &cj));
445   PetscCall(PetscFreeSpaceContiguous(&free_space, cj));
446   PetscCall(PetscLLCondensedDestroy_fast(lnk));
447 
448   /* Allocate space for ca */
449   PetscCall(PetscCalloc1(ci[am] + 1, &ca));
450 
451   /* put together the new symbolic matrix */
452   PetscCall(MatSetSeqAIJWithArrays_private(PetscObjectComm((PetscObject)A), am, bn, ci, cj, ca, ((PetscObject)A)->type_name, C));
453   PetscCall(MatSetBlockSizesFromMats(C, A, B));
454 
455   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
456   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
457   c          = (Mat_SeqAIJ *)C->data;
458   c->free_a  = PETSC_TRUE;
459   c->free_ij = PETSC_TRUE;
460   c->nonew   = 0;
461 
462   /* slower, less memory */
463   C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable;
464 
465   /* set MatInfo */
466   afill = (PetscReal)ci[am] / (ai[am] + bi[bm]) + 1.e-5;
467   if (afill < 1.0) afill = 1.0;
468   C->info.mallocs           = ndouble;
469   C->info.fill_ratio_given  = fill;
470   C->info.fill_ratio_needed = afill;
471 
472 #if defined(PETSC_USE_INFO)
473   if (ci[am]) {
474     PetscCall(PetscInfo(C, "Reallocs %" PetscInt_FMT "; Fill ratio: given %g needed %g.\n", ndouble, (double)fill, (double)afill));
475     PetscCall(PetscInfo(C, "Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n", (double)afill));
476   } else {
477     PetscCall(PetscInfo(C, "Empty matrix product\n"));
478   }
479 #endif
480   PetscFunctionReturn(PETSC_SUCCESS);
481 }
482 
483 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable(Mat A, Mat B, PetscReal fill, Mat C)
484 {
485   Mat_SeqAIJ        *a = (Mat_SeqAIJ *)A->data, *b = (Mat_SeqAIJ *)B->data, *c;
486   PetscInt          *ai = a->i, *bi = b->i, *ci, *cj;
487   PetscInt           am = A->rmap->N, bn = B->cmap->N, bm = B->rmap->N;
488   MatScalar         *ca;
489   PetscReal          afill;
490   PetscInt           i, j, anzi, brow, bnzj, cnzi, *bj, *aj, *lnk, ndouble = 0, Crmax;
491   PetscHMapI         ta;
492   PetscFreeSpaceList free_space = NULL, current_space = NULL;
493 
494   PetscFunctionBegin;
495   /* Get ci and cj - same as MatMatMultSymbolic_SeqAIJ_SeqAIJ except using PetscLLxxx_Scalalbe() */
496   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
497   PetscCall(PetscMalloc1(am + 2, &ci));
498   ci[0] = 0;
499 
500   /* create and initialize a linked list */
501   PetscCall(PetscHMapICreateWithSize(bn, &ta));
502   MatRowMergeMax_SeqAIJ(b, bm, ta);
503   PetscCall(PetscHMapIGetSize(ta, &Crmax));
504   PetscCall(PetscHMapIDestroy(&ta));
505   PetscCall(PetscLLCondensedCreate_Scalable(Crmax, &lnk));
506 
507   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
508   PetscCall(PetscFreeSpaceGet(PetscRealIntMultTruncate(fill, PetscIntSumTruncate(ai[am], bi[bm])), &free_space));
509   current_space = free_space;
510 
511   /* Determine ci and cj */
512   for (i = 0; i < am; i++) {
513     anzi = ai[i + 1] - ai[i];
514     aj   = a->j + ai[i];
515     for (j = 0; j < anzi; j++) {
516       brow = aj[j];
517       bnzj = bi[brow + 1] - bi[brow];
518       bj   = b->j + bi[brow];
519       /* add non-zero cols of B into the sorted linked list lnk */
520       PetscCall(PetscLLCondensedAddSorted_Scalable(bnzj, bj, lnk));
521     }
522     /* add possible missing diagonal entry */
523     if (C->force_diagonals) PetscCall(PetscLLCondensedAddSorted_Scalable(1, &i, lnk));
524 
525     cnzi = lnk[0];
526 
527     /* If free space is not available, make more free space */
528     /* Double the amount of total space in the list */
529     if (current_space->local_remaining < cnzi) {
530       PetscCall(PetscFreeSpaceGet(PetscIntSumTruncate(cnzi, current_space->total_array_size), &current_space));
531       ndouble++;
532     }
533 
534     /* Copy data into free space, then initialize lnk */
535     PetscCall(PetscLLCondensedClean_Scalable(cnzi, current_space->array, lnk));
536 
537     current_space->array += cnzi;
538     current_space->local_used += cnzi;
539     current_space->local_remaining -= cnzi;
540 
541     ci[i + 1] = ci[i] + cnzi;
542   }
543 
544   /* Column indices are in the list of free space */
545   /* Allocate space for cj, initialize cj, and */
546   /* destroy list of free space and other temporary array(s) */
547   PetscCall(PetscMalloc1(ci[am] + 1, &cj));
548   PetscCall(PetscFreeSpaceContiguous(&free_space, cj));
549   PetscCall(PetscLLCondensedDestroy_Scalable(lnk));
550 
551   /* Allocate space for ca */
552   PetscCall(PetscCalloc1(ci[am] + 1, &ca));
553 
554   /* put together the new symbolic matrix */
555   PetscCall(MatSetSeqAIJWithArrays_private(PetscObjectComm((PetscObject)A), am, bn, ci, cj, ca, ((PetscObject)A)->type_name, C));
556   PetscCall(MatSetBlockSizesFromMats(C, A, B));
557 
558   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
559   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
560   c          = (Mat_SeqAIJ *)C->data;
561   c->free_a  = PETSC_TRUE;
562   c->free_ij = PETSC_TRUE;
563   c->nonew   = 0;
564 
565   /* slower, less memory */
566   C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable;
567 
568   /* set MatInfo */
569   afill = (PetscReal)ci[am] / (ai[am] + bi[bm]) + 1.e-5;
570   if (afill < 1.0) afill = 1.0;
571   C->info.mallocs           = ndouble;
572   C->info.fill_ratio_given  = fill;
573   C->info.fill_ratio_needed = afill;
574 
575 #if defined(PETSC_USE_INFO)
576   if (ci[am]) {
577     PetscCall(PetscInfo(C, "Reallocs %" PetscInt_FMT "; Fill ratio: given %g needed %g.\n", ndouble, (double)fill, (double)afill));
578     PetscCall(PetscInfo(C, "Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n", (double)afill));
579   } else {
580     PetscCall(PetscInfo(C, "Empty matrix product\n"));
581   }
582 #endif
583   PetscFunctionReturn(PETSC_SUCCESS);
584 }
585 
586 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Heap(Mat A, Mat B, PetscReal fill, Mat C)
587 {
588   Mat_SeqAIJ        *a = (Mat_SeqAIJ *)A->data, *b = (Mat_SeqAIJ *)B->data, *c;
589   const PetscInt    *ai = a->i, *bi = b->i, *aj = a->j, *bj = b->j;
590   PetscInt          *ci, *cj, *bb;
591   PetscInt           am = A->rmap->N, bn = B->cmap->N, bm = B->rmap->N;
592   PetscReal          afill;
593   PetscInt           i, j, col, ndouble = 0;
594   PetscFreeSpaceList free_space = NULL, current_space = NULL;
595   PetscHeap          h;
596 
597   PetscFunctionBegin;
598   /* Get ci and cj - by merging sorted rows using a heap */
599   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
600   PetscCall(PetscMalloc1(am + 2, &ci));
601   ci[0] = 0;
602 
603   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
604   PetscCall(PetscFreeSpaceGet(PetscRealIntMultTruncate(fill, PetscIntSumTruncate(ai[am], bi[bm])), &free_space));
605   current_space = free_space;
606 
607   PetscCall(PetscHeapCreate(a->rmax, &h));
608   PetscCall(PetscMalloc1(a->rmax, &bb));
609 
610   /* Determine ci and cj */
611   for (i = 0; i < am; i++) {
612     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 */
613     const PetscInt *acol = aj + ai[i];        /* column indices of nonzero entries in this row */
614     ci[i + 1]            = ci[i];
615     /* Populate the min heap */
616     for (j = 0; j < anzi; j++) {
617       bb[j] = bi[acol[j]];           /* bb points at the start of the row */
618       if (bb[j] < bi[acol[j] + 1]) { /* Add if row is nonempty */
619         PetscCall(PetscHeapAdd(h, j, bj[bb[j]++]));
620       }
621     }
622     /* Pick off the min element, adding it to free space */
623     PetscCall(PetscHeapPop(h, &j, &col));
624     while (j >= 0) {
625       if (current_space->local_remaining < 1) { /* double the size, but don't exceed 16 MiB */
626         PetscCall(PetscFreeSpaceGet(PetscMin(PetscIntMultTruncate(2, current_space->total_array_size), 16 << 20), &current_space));
627         ndouble++;
628       }
629       *(current_space->array++) = col;
630       current_space->local_used++;
631       current_space->local_remaining--;
632       ci[i + 1]++;
633 
634       /* stash if anything else remains in this row of B */
635       if (bb[j] < bi[acol[j] + 1]) PetscCall(PetscHeapStash(h, j, bj[bb[j]++]));
636       while (1) { /* pop and stash any other rows of B that also had an entry in this column */
637         PetscInt j2, col2;
638         PetscCall(PetscHeapPeek(h, &j2, &col2));
639         if (col2 != col) break;
640         PetscCall(PetscHeapPop(h, &j2, &col2));
641         if (bb[j2] < bi[acol[j2] + 1]) PetscCall(PetscHeapStash(h, j2, bj[bb[j2]++]));
642       }
643       /* Put any stashed elements back into the min heap */
644       PetscCall(PetscHeapUnstash(h));
645       PetscCall(PetscHeapPop(h, &j, &col));
646     }
647   }
648   PetscCall(PetscFree(bb));
649   PetscCall(PetscHeapDestroy(&h));
650 
651   /* Column indices are in the list of free space */
652   /* Allocate space for cj, initialize cj, and */
653   /* destroy list of free space and other temporary array(s) */
654   PetscCall(PetscMalloc1(ci[am], &cj));
655   PetscCall(PetscFreeSpaceContiguous(&free_space, cj));
656 
657   /* put together the new symbolic matrix */
658   PetscCall(MatSetSeqAIJWithArrays_private(PetscObjectComm((PetscObject)A), am, bn, ci, cj, NULL, ((PetscObject)A)->type_name, C));
659   PetscCall(MatSetBlockSizesFromMats(C, A, B));
660 
661   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
662   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
663   c          = (Mat_SeqAIJ *)C->data;
664   c->free_a  = PETSC_TRUE;
665   c->free_ij = PETSC_TRUE;
666   c->nonew   = 0;
667 
668   C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;
669 
670   /* set MatInfo */
671   afill = (PetscReal)ci[am] / (ai[am] + bi[bm]) + 1.e-5;
672   if (afill < 1.0) afill = 1.0;
673   C->info.mallocs           = ndouble;
674   C->info.fill_ratio_given  = fill;
675   C->info.fill_ratio_needed = afill;
676 
677 #if defined(PETSC_USE_INFO)
678   if (ci[am]) {
679     PetscCall(PetscInfo(C, "Reallocs %" PetscInt_FMT "; Fill ratio: given %g needed %g.\n", ndouble, (double)fill, (double)afill));
680     PetscCall(PetscInfo(C, "Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n", (double)afill));
681   } else {
682     PetscCall(PetscInfo(C, "Empty matrix product\n"));
683   }
684 #endif
685   PetscFunctionReturn(PETSC_SUCCESS);
686 }
687 
688 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_BTHeap(Mat A, Mat B, PetscReal fill, Mat C)
689 {
690   Mat_SeqAIJ        *a = (Mat_SeqAIJ *)A->data, *b = (Mat_SeqAIJ *)B->data, *c;
691   const PetscInt    *ai = a->i, *bi = b->i, *aj = a->j, *bj = b->j;
692   PetscInt          *ci, *cj, *bb;
693   PetscInt           am = A->rmap->N, bn = B->cmap->N, bm = B->rmap->N;
694   PetscReal          afill;
695   PetscInt           i, j, col, ndouble = 0;
696   PetscFreeSpaceList free_space = NULL, current_space = NULL;
697   PetscHeap          h;
698   PetscBT            bt;
699 
700   PetscFunctionBegin;
701   /* Get ci and cj - using a heap for the sorted rows, but use BT so that each index is only added once */
702   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
703   PetscCall(PetscMalloc1(am + 2, &ci));
704   ci[0] = 0;
705 
706   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
707   PetscCall(PetscFreeSpaceGet(PetscRealIntMultTruncate(fill, PetscIntSumTruncate(ai[am], bi[bm])), &free_space));
708 
709   current_space = free_space;
710 
711   PetscCall(PetscHeapCreate(a->rmax, &h));
712   PetscCall(PetscMalloc1(a->rmax, &bb));
713   PetscCall(PetscBTCreate(bn, &bt));
714 
715   /* Determine ci and cj */
716   for (i = 0; i < am; i++) {
717     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 */
718     const PetscInt *acol = aj + ai[i];           /* column indices of nonzero entries in this row */
719     const PetscInt *fptr = current_space->array; /* Save beginning of the row so we can clear the BT later */
720     ci[i + 1]            = ci[i];
721     /* Populate the min heap */
722     for (j = 0; j < anzi; j++) {
723       PetscInt brow = acol[j];
724       for (bb[j] = bi[brow]; bb[j] < bi[brow + 1]; bb[j]++) {
725         PetscInt bcol = bj[bb[j]];
726         if (!PetscBTLookupSet(bt, bcol)) { /* new entry */
727           PetscCall(PetscHeapAdd(h, j, bcol));
728           bb[j]++;
729           break;
730         }
731       }
732     }
733     /* Pick off the min element, adding it to free space */
734     PetscCall(PetscHeapPop(h, &j, &col));
735     while (j >= 0) {
736       if (current_space->local_remaining < 1) { /* double the size, but don't exceed 16 MiB */
737         fptr = NULL;                            /* need PetscBTMemzero */
738         PetscCall(PetscFreeSpaceGet(PetscMin(PetscIntMultTruncate(2, current_space->total_array_size), 16 << 20), &current_space));
739         ndouble++;
740       }
741       *(current_space->array++) = col;
742       current_space->local_used++;
743       current_space->local_remaining--;
744       ci[i + 1]++;
745 
746       /* stash if anything else remains in this row of B */
747       for (; bb[j] < bi[acol[j] + 1]; bb[j]++) {
748         PetscInt bcol = bj[bb[j]];
749         if (!PetscBTLookupSet(bt, bcol)) { /* new entry */
750           PetscCall(PetscHeapAdd(h, j, bcol));
751           bb[j]++;
752           break;
753         }
754       }
755       PetscCall(PetscHeapPop(h, &j, &col));
756     }
757     if (fptr) { /* Clear the bits for this row */
758       for (; fptr < current_space->array; fptr++) PetscCall(PetscBTClear(bt, *fptr));
759     } else { /* We reallocated so we don't remember (easily) how to clear only the bits we changed */
760       PetscCall(PetscBTMemzero(bn, bt));
761     }
762   }
763   PetscCall(PetscFree(bb));
764   PetscCall(PetscHeapDestroy(&h));
765   PetscCall(PetscBTDestroy(&bt));
766 
767   /* Column indices are in the list of free space */
768   /* Allocate space for cj, initialize cj, and */
769   /* destroy list of free space and other temporary array(s) */
770   PetscCall(PetscMalloc1(ci[am], &cj));
771   PetscCall(PetscFreeSpaceContiguous(&free_space, cj));
772 
773   /* put together the new symbolic matrix */
774   PetscCall(MatSetSeqAIJWithArrays_private(PetscObjectComm((PetscObject)A), am, bn, ci, cj, NULL, ((PetscObject)A)->type_name, C));
775   PetscCall(MatSetBlockSizesFromMats(C, A, B));
776 
777   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
778   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
779   c          = (Mat_SeqAIJ *)C->data;
780   c->free_a  = PETSC_TRUE;
781   c->free_ij = PETSC_TRUE;
782   c->nonew   = 0;
783 
784   C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;
785 
786   /* set MatInfo */
787   afill = (PetscReal)ci[am] / (ai[am] + bi[bm]) + 1.e-5;
788   if (afill < 1.0) afill = 1.0;
789   C->info.mallocs           = ndouble;
790   C->info.fill_ratio_given  = fill;
791   C->info.fill_ratio_needed = afill;
792 
793 #if defined(PETSC_USE_INFO)
794   if (ci[am]) {
795     PetscCall(PetscInfo(C, "Reallocs %" PetscInt_FMT "; Fill ratio: given %g needed %g.\n", ndouble, (double)fill, (double)afill));
796     PetscCall(PetscInfo(C, "Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n", (double)afill));
797   } else {
798     PetscCall(PetscInfo(C, "Empty matrix product\n"));
799   }
800 #endif
801   PetscFunctionReturn(PETSC_SUCCESS);
802 }
803 
804 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_RowMerge(Mat A, Mat B, PetscReal fill, Mat C)
805 {
806   Mat_SeqAIJ     *a = (Mat_SeqAIJ *)A->data, *b = (Mat_SeqAIJ *)B->data, *c;
807   const PetscInt *ai = a->i, *bi = b->i, *aj = a->j, *bj = b->j, *inputi, *inputj, *inputcol, *inputcol_L1;
808   PetscInt       *ci, *cj, *outputj, worki_L1[9], worki_L2[9];
809   PetscInt        c_maxmem, a_maxrownnz = 0, a_rownnz;
810   const PetscInt  workcol[8] = {0, 1, 2, 3, 4, 5, 6, 7};
811   const PetscInt  am = A->rmap->N, bn = B->cmap->N, bm = B->rmap->N;
812   const PetscInt *brow_ptr[8], *brow_end[8];
813   PetscInt        window[8];
814   PetscInt        window_min, old_window_min, ci_nnz, outputi_nnz = 0, L1_nrows, L2_nrows;
815   PetscInt        i, k, ndouble = 0, L1_rowsleft, rowsleft;
816   PetscReal       afill;
817   PetscInt       *workj_L1, *workj_L2, *workj_L3;
818   PetscInt        L1_nnz, L2_nnz;
819 
820   /* Step 1: Get upper bound on memory required for allocation.
821              Because of the way virtual memory works,
822              only the memory pages that are actually needed will be physically allocated. */
823   PetscFunctionBegin;
824   PetscCall(PetscMalloc1(am + 1, &ci));
825   for (i = 0; i < am; i++) {
826     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 */
827     const PetscInt *acol = aj + ai[i];        /* column indices of nonzero entries in this row */
828     a_rownnz             = 0;
829     for (k = 0; k < anzi; ++k) {
830       a_rownnz += bi[acol[k] + 1] - bi[acol[k]];
831       if (a_rownnz > bn) {
832         a_rownnz = bn;
833         break;
834       }
835     }
836     a_maxrownnz = PetscMax(a_maxrownnz, a_rownnz);
837   }
838   /* temporary work areas for merging rows */
839   PetscCall(PetscMalloc1(a_maxrownnz * 8, &workj_L1));
840   PetscCall(PetscMalloc1(a_maxrownnz * 8, &workj_L2));
841   PetscCall(PetscMalloc1(a_maxrownnz, &workj_L3));
842 
843   /* This should be enough for almost all matrices. If not, memory is reallocated later. */
844   c_maxmem = 8 * (ai[am] + bi[bm]);
845   /* Step 2: Populate pattern for C */
846   PetscCall(PetscMalloc1(c_maxmem, &cj));
847 
848   ci_nnz      = 0;
849   ci[0]       = 0;
850   worki_L1[0] = 0;
851   worki_L2[0] = 0;
852   for (i = 0; i < am; i++) {
853     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 */
854     const PetscInt *acol = aj + ai[i];        /* column indices of nonzero entries in this row */
855     rowsleft             = anzi;
856     inputcol_L1          = acol;
857     L2_nnz               = 0;
858     L2_nrows             = 1; /* Number of rows to be merged on Level 3. output of L3 already exists -> initial value 1   */
859     worki_L2[1]          = 0;
860     outputi_nnz          = 0;
861 
862     /* If the number of indices in C so far + the max number of columns in the next row > c_maxmem  -> allocate more memory */
863     while (ci_nnz + a_maxrownnz > c_maxmem) {
864       c_maxmem *= 2;
865       ndouble++;
866       PetscCall(PetscRealloc(sizeof(PetscInt) * c_maxmem, &cj));
867     }
868 
869     while (rowsleft) {
870       L1_rowsleft = PetscMin(64, rowsleft); /* In the inner loop max 64 rows of B can be merged */
871       L1_nrows    = 0;
872       L1_nnz      = 0;
873       inputcol    = inputcol_L1;
874       inputi      = bi;
875       inputj      = bj;
876 
877       /* The following macro is used to specialize for small rows in A.
878          This helps with compiler unrolling, improving performance substantially.
879           Input:  inputj   inputi  inputcol  bn
880           Output: outputj  outputi_nnz                       */
881 #define MatMatMultSymbolic_RowMergeMacro(ANNZ) \
882   do { \
883     window_min  = bn; \
884     outputi_nnz = 0; \
885     for (k = 0; k < ANNZ; ++k) { \
886       brow_ptr[k] = inputj + inputi[inputcol[k]]; \
887       brow_end[k] = inputj + inputi[inputcol[k] + 1]; \
888       window[k]   = (brow_ptr[k] != brow_end[k]) ? *brow_ptr[k] : bn; \
889       window_min  = PetscMin(window[k], window_min); \
890     } \
891     while (window_min < bn) { \
892       outputj[outputi_nnz++] = window_min; \
893       /* advance front and compute new minimum */ \
894       old_window_min = window_min; \
895       window_min     = bn; \
896       for (k = 0; k < ANNZ; ++k) { \
897         if (window[k] == old_window_min) { \
898           brow_ptr[k]++; \
899           window[k] = (brow_ptr[k] != brow_end[k]) ? *brow_ptr[k] : bn; \
900         } \
901         window_min = PetscMin(window[k], window_min); \
902       } \
903     } \
904   } while (0)
905 
906       /************** L E V E L  1 ***************/
907       /* Merge up to 8 rows of B to L1 work array*/
908       while (L1_rowsleft) {
909         outputi_nnz = 0;
910         if (anzi > 8) outputj = workj_L1 + L1_nnz; /* Level 1 rowmerge*/
911         else outputj = cj + ci_nnz;                /* Merge directly to C */
912 
913         switch (L1_rowsleft) {
914         case 1:
915           brow_ptr[0] = inputj + inputi[inputcol[0]];
916           brow_end[0] = inputj + inputi[inputcol[0] + 1];
917           for (; brow_ptr[0] != brow_end[0]; ++brow_ptr[0]) outputj[outputi_nnz++] = *brow_ptr[0]; /* copy row in b over */
918           inputcol += L1_rowsleft;
919           rowsleft -= L1_rowsleft;
920           L1_rowsleft = 0;
921           break;
922         case 2:
923           MatMatMultSymbolic_RowMergeMacro(2);
924           inputcol += L1_rowsleft;
925           rowsleft -= L1_rowsleft;
926           L1_rowsleft = 0;
927           break;
928         case 3:
929           MatMatMultSymbolic_RowMergeMacro(3);
930           inputcol += L1_rowsleft;
931           rowsleft -= L1_rowsleft;
932           L1_rowsleft = 0;
933           break;
934         case 4:
935           MatMatMultSymbolic_RowMergeMacro(4);
936           inputcol += L1_rowsleft;
937           rowsleft -= L1_rowsleft;
938           L1_rowsleft = 0;
939           break;
940         case 5:
941           MatMatMultSymbolic_RowMergeMacro(5);
942           inputcol += L1_rowsleft;
943           rowsleft -= L1_rowsleft;
944           L1_rowsleft = 0;
945           break;
946         case 6:
947           MatMatMultSymbolic_RowMergeMacro(6);
948           inputcol += L1_rowsleft;
949           rowsleft -= L1_rowsleft;
950           L1_rowsleft = 0;
951           break;
952         case 7:
953           MatMatMultSymbolic_RowMergeMacro(7);
954           inputcol += L1_rowsleft;
955           rowsleft -= L1_rowsleft;
956           L1_rowsleft = 0;
957           break;
958         default:
959           MatMatMultSymbolic_RowMergeMacro(8);
960           inputcol += 8;
961           rowsleft -= 8;
962           L1_rowsleft -= 8;
963           break;
964         }
965         inputcol_L1 = inputcol;
966         L1_nnz += outputi_nnz;
967         worki_L1[++L1_nrows] = L1_nnz;
968       }
969 
970       /********************** L E V E L  2 ************************/
971       /* Merge from L1 work array to either C or to L2 work array */
972       if (anzi > 8) {
973         inputi      = worki_L1;
974         inputj      = workj_L1;
975         inputcol    = workcol;
976         outputi_nnz = 0;
977 
978         if (anzi <= 64) outputj = cj + ci_nnz; /* Merge from L1 work array to C */
979         else outputj = workj_L2 + L2_nnz;      /* Merge from L1 work array to L2 work array */
980 
981         switch (L1_nrows) {
982         case 1:
983           brow_ptr[0] = inputj + inputi[inputcol[0]];
984           brow_end[0] = inputj + inputi[inputcol[0] + 1];
985           for (; brow_ptr[0] != brow_end[0]; ++brow_ptr[0]) outputj[outputi_nnz++] = *brow_ptr[0]; /* copy row in b over */
986           break;
987         case 2:
988           MatMatMultSymbolic_RowMergeMacro(2);
989           break;
990         case 3:
991           MatMatMultSymbolic_RowMergeMacro(3);
992           break;
993         case 4:
994           MatMatMultSymbolic_RowMergeMacro(4);
995           break;
996         case 5:
997           MatMatMultSymbolic_RowMergeMacro(5);
998           break;
999         case 6:
1000           MatMatMultSymbolic_RowMergeMacro(6);
1001           break;
1002         case 7:
1003           MatMatMultSymbolic_RowMergeMacro(7);
1004           break;
1005         case 8:
1006           MatMatMultSymbolic_RowMergeMacro(8);
1007           break;
1008         default:
1009           SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "MatMatMult logic error: Not merging 1-8 rows from L1 work array!");
1010         }
1011         L2_nnz += outputi_nnz;
1012         worki_L2[++L2_nrows] = L2_nnz;
1013 
1014         /************************ L E V E L  3 **********************/
1015         /* Merge from L2 work array to either C or to L2 work array */
1016         if (anzi > 64 && (L2_nrows == 8 || rowsleft == 0)) {
1017           inputi      = worki_L2;
1018           inputj      = workj_L2;
1019           inputcol    = workcol;
1020           outputi_nnz = 0;
1021           if (rowsleft) outputj = workj_L3;
1022           else outputj = cj + ci_nnz;
1023           switch (L2_nrows) {
1024           case 1:
1025             brow_ptr[0] = inputj + inputi[inputcol[0]];
1026             brow_end[0] = inputj + inputi[inputcol[0] + 1];
1027             for (; brow_ptr[0] != brow_end[0]; ++brow_ptr[0]) outputj[outputi_nnz++] = *brow_ptr[0]; /* copy row in b over */
1028             break;
1029           case 2:
1030             MatMatMultSymbolic_RowMergeMacro(2);
1031             break;
1032           case 3:
1033             MatMatMultSymbolic_RowMergeMacro(3);
1034             break;
1035           case 4:
1036             MatMatMultSymbolic_RowMergeMacro(4);
1037             break;
1038           case 5:
1039             MatMatMultSymbolic_RowMergeMacro(5);
1040             break;
1041           case 6:
1042             MatMatMultSymbolic_RowMergeMacro(6);
1043             break;
1044           case 7:
1045             MatMatMultSymbolic_RowMergeMacro(7);
1046             break;
1047           case 8:
1048             MatMatMultSymbolic_RowMergeMacro(8);
1049             break;
1050           default:
1051             SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "MatMatMult logic error: Not merging 1-8 rows from L2 work array!");
1052           }
1053           L2_nrows    = 1;
1054           L2_nnz      = outputi_nnz;
1055           worki_L2[1] = outputi_nnz;
1056           /* Copy to workj_L2 */
1057           if (rowsleft) {
1058             for (k = 0; k < outputi_nnz; ++k) workj_L2[k] = outputj[k];
1059           }
1060         }
1061       }
1062     } /* while (rowsleft) */
1063 #undef MatMatMultSymbolic_RowMergeMacro
1064 
1065     /* terminate current row */
1066     ci_nnz += outputi_nnz;
1067     ci[i + 1] = ci_nnz;
1068   }
1069 
1070   /* Step 3: Create the new symbolic matrix */
1071   PetscCall(MatSetSeqAIJWithArrays_private(PetscObjectComm((PetscObject)A), am, bn, ci, cj, NULL, ((PetscObject)A)->type_name, C));
1072   PetscCall(MatSetBlockSizesFromMats(C, A, B));
1073 
1074   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
1075   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
1076   c          = (Mat_SeqAIJ *)C->data;
1077   c->free_a  = PETSC_TRUE;
1078   c->free_ij = PETSC_TRUE;
1079   c->nonew   = 0;
1080 
1081   C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;
1082 
1083   /* set MatInfo */
1084   afill = (PetscReal)ci[am] / (ai[am] + bi[bm]) + 1.e-5;
1085   if (afill < 1.0) afill = 1.0;
1086   C->info.mallocs           = ndouble;
1087   C->info.fill_ratio_given  = fill;
1088   C->info.fill_ratio_needed = afill;
1089 
1090 #if defined(PETSC_USE_INFO)
1091   if (ci[am]) {
1092     PetscCall(PetscInfo(C, "Reallocs %" PetscInt_FMT "; Fill ratio: given %g needed %g.\n", ndouble, (double)fill, (double)afill));
1093     PetscCall(PetscInfo(C, "Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n", (double)afill));
1094   } else {
1095     PetscCall(PetscInfo(C, "Empty matrix product\n"));
1096   }
1097 #endif
1098 
1099   /* Step 4: Free temporary work areas */
1100   PetscCall(PetscFree(workj_L1));
1101   PetscCall(PetscFree(workj_L2));
1102   PetscCall(PetscFree(workj_L3));
1103   PetscFunctionReturn(PETSC_SUCCESS);
1104 }
1105 
1106 /* concatenate unique entries and then sort */
1107 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Sorted(Mat A, Mat B, PetscReal fill, Mat C)
1108 {
1109   Mat_SeqAIJ     *a = (Mat_SeqAIJ *)A->data, *b = (Mat_SeqAIJ *)B->data, *c;
1110   const PetscInt *ai = a->i, *bi = b->i, *aj = a->j, *bj = b->j;
1111   PetscInt       *ci, *cj, bcol;
1112   PetscInt        am = A->rmap->N, bn = B->cmap->N, bm = B->rmap->N;
1113   PetscReal       afill;
1114   PetscInt        i, j, ndouble = 0;
1115   PetscSegBuffer  seg, segrow;
1116   char           *seen;
1117 
1118   PetscFunctionBegin;
1119   PetscCall(PetscMalloc1(am + 1, &ci));
1120   ci[0] = 0;
1121 
1122   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
1123   PetscCall(PetscSegBufferCreate(sizeof(PetscInt), (PetscInt)(fill * (ai[am] + bi[bm])), &seg));
1124   PetscCall(PetscSegBufferCreate(sizeof(PetscInt), 100, &segrow));
1125   PetscCall(PetscCalloc1(bn, &seen));
1126 
1127   /* Determine ci and cj */
1128   for (i = 0; i < am; i++) {
1129     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 */
1130     const PetscInt *acol = PetscSafePointerPlusOffset(aj, ai[i]); /* column indices of nonzero entries in this row */
1131     PetscInt packlen     = 0, *PETSC_RESTRICT crow;
1132 
1133     /* Pack segrow */
1134     for (j = 0; j < anzi; j++) {
1135       PetscInt brow = acol[j], bjstart = bi[brow], bjend = bi[brow + 1], k;
1136       for (k = bjstart; k < bjend; k++) {
1137         bcol = bj[k];
1138         if (!seen[bcol]) { /* new entry */
1139           PetscInt *PETSC_RESTRICT slot;
1140           PetscCall(PetscSegBufferGetInts(segrow, 1, &slot));
1141           *slot      = bcol;
1142           seen[bcol] = 1;
1143           packlen++;
1144         }
1145       }
1146     }
1147 
1148     /* Check i-th diagonal entry */
1149     if (C->force_diagonals && !seen[i]) {
1150       PetscInt *PETSC_RESTRICT slot;
1151       PetscCall(PetscSegBufferGetInts(segrow, 1, &slot));
1152       *slot   = i;
1153       seen[i] = 1;
1154       packlen++;
1155     }
1156 
1157     PetscCall(PetscSegBufferGetInts(seg, packlen, &crow));
1158     PetscCall(PetscSegBufferExtractTo(segrow, crow));
1159     PetscCall(PetscSortInt(packlen, crow));
1160     ci[i + 1] = ci[i] + packlen;
1161     for (j = 0; j < packlen; j++) seen[crow[j]] = 0;
1162   }
1163   PetscCall(PetscSegBufferDestroy(&segrow));
1164   PetscCall(PetscFree(seen));
1165 
1166   /* Column indices are in the segmented buffer */
1167   PetscCall(PetscSegBufferExtractAlloc(seg, &cj));
1168   PetscCall(PetscSegBufferDestroy(&seg));
1169 
1170   /* put together the new symbolic matrix */
1171   PetscCall(MatSetSeqAIJWithArrays_private(PetscObjectComm((PetscObject)A), am, bn, ci, cj, NULL, ((PetscObject)A)->type_name, C));
1172   PetscCall(MatSetBlockSizesFromMats(C, A, B));
1173 
1174   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
1175   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
1176   c          = (Mat_SeqAIJ *)C->data;
1177   c->free_a  = PETSC_TRUE;
1178   c->free_ij = PETSC_TRUE;
1179   c->nonew   = 0;
1180 
1181   C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;
1182 
1183   /* set MatInfo */
1184   afill = (PetscReal)ci[am] / PetscMax(ai[am] + bi[bm], 1) + 1.e-5;
1185   if (afill < 1.0) afill = 1.0;
1186   C->info.mallocs           = ndouble;
1187   C->info.fill_ratio_given  = fill;
1188   C->info.fill_ratio_needed = afill;
1189 
1190 #if defined(PETSC_USE_INFO)
1191   if (ci[am]) {
1192     PetscCall(PetscInfo(C, "Reallocs %" PetscInt_FMT "; Fill ratio: given %g needed %g.\n", ndouble, (double)fill, (double)afill));
1193     PetscCall(PetscInfo(C, "Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n", (double)afill));
1194   } else {
1195     PetscCall(PetscInfo(C, "Empty matrix product\n"));
1196   }
1197 #endif
1198   PetscFunctionReturn(PETSC_SUCCESS);
1199 }
1200 
1201 static PetscErrorCode MatProductCtxDestroy_SeqAIJ_MatMatMultTrans(PetscCtxRt data)
1202 {
1203   MatProductCtx_MatMatTransMult *abt = *(MatProductCtx_MatMatTransMult **)data;
1204 
1205   PetscFunctionBegin;
1206   PetscCall(MatTransposeColoringDestroy(&abt->matcoloring));
1207   PetscCall(MatDestroy(&abt->Bt_den));
1208   PetscCall(MatDestroy(&abt->ABt_den));
1209   PetscCall(PetscFree(abt));
1210   PetscFunctionReturn(PETSC_SUCCESS);
1211 }
1212 
1213 PetscErrorCode MatMatTransposeMultSymbolic_SeqAIJ_SeqAIJ(Mat A, Mat B, PetscReal fill, Mat C)
1214 {
1215   Mat                            Bt;
1216   MatProductCtx_MatMatTransMult *abt;
1217   Mat_Product                   *product = C->product;
1218   char                          *alg;
1219 
1220   PetscFunctionBegin;
1221   PetscCheck(product, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Missing product struct");
1222   PetscCheck(!product->data, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Extra product struct not empty");
1223 
1224   /* create symbolic Bt */
1225   PetscCall(MatTransposeSymbolic(B, &Bt));
1226   PetscCall(MatSetBlockSizes(Bt, A->cmap->bs, B->cmap->bs));
1227   PetscCall(MatSetType(Bt, ((PetscObject)A)->type_name));
1228 
1229   /* get symbolic C=A*Bt */
1230   PetscCall(PetscStrallocpy(product->alg, &alg));
1231   PetscCall(MatProductSetAlgorithm(C, "sorted")); /* set algorithm for C = A*Bt */
1232   PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ(A, Bt, fill, C));
1233   PetscCall(MatProductSetAlgorithm(C, alg)); /* resume original algorithm for ABt product */
1234   PetscCall(PetscFree(alg));
1235 
1236   /* create a supporting struct for reuse intermediate dense matrices with matcoloring */
1237   PetscCall(PetscNew(&abt));
1238 
1239   product->data    = abt;
1240   product->destroy = MatProductCtxDestroy_SeqAIJ_MatMatMultTrans;
1241 
1242   C->ops->mattransposemultnumeric = MatMatTransposeMultNumeric_SeqAIJ_SeqAIJ;
1243 
1244   abt->usecoloring = PETSC_FALSE;
1245   PetscCall(PetscStrcmp(product->alg, "color", &abt->usecoloring));
1246   if (abt->usecoloring) {
1247     /* Create MatTransposeColoring from symbolic C=A*B^T */
1248     MatTransposeColoring matcoloring;
1249     MatColoring          coloring;
1250     ISColoring           iscoloring;
1251     Mat                  Bt_dense, C_dense;
1252 
1253     /* inode causes memory problem */
1254     PetscCall(MatSetOption(C, MAT_USE_INODES, PETSC_FALSE));
1255 
1256     PetscCall(MatColoringCreate(C, &coloring));
1257     PetscCall(MatColoringSetDistance(coloring, 2));
1258     PetscCall(MatColoringSetType(coloring, MATCOLORINGSL));
1259     PetscCall(MatColoringSetFromOptions(coloring));
1260     PetscCall(MatColoringApply(coloring, &iscoloring));
1261     PetscCall(MatColoringDestroy(&coloring));
1262     PetscCall(MatTransposeColoringCreate(C, iscoloring, &matcoloring));
1263 
1264     abt->matcoloring = matcoloring;
1265 
1266     PetscCall(ISColoringDestroy(&iscoloring));
1267 
1268     /* Create Bt_dense and C_dense = A*Bt_dense */
1269     PetscCall(MatCreate(PETSC_COMM_SELF, &Bt_dense));
1270     PetscCall(MatSetSizes(Bt_dense, A->cmap->n, matcoloring->ncolors, A->cmap->n, matcoloring->ncolors));
1271     PetscCall(MatSetType(Bt_dense, MATSEQDENSE));
1272     PetscCall(MatSeqDenseSetPreallocation(Bt_dense, NULL));
1273 
1274     Bt_dense->assembled = PETSC_TRUE;
1275     abt->Bt_den         = Bt_dense;
1276 
1277     PetscCall(MatCreate(PETSC_COMM_SELF, &C_dense));
1278     PetscCall(MatSetSizes(C_dense, A->rmap->n, matcoloring->ncolors, A->rmap->n, matcoloring->ncolors));
1279     PetscCall(MatSetType(C_dense, MATSEQDENSE));
1280     PetscCall(MatSeqDenseSetPreallocation(C_dense, NULL));
1281 
1282     Bt_dense->assembled = PETSC_TRUE;
1283     abt->ABt_den        = C_dense;
1284 
1285 #if defined(PETSC_USE_INFO)
1286     {
1287       Mat_SeqAIJ *c = (Mat_SeqAIJ *)C->data;
1288       PetscCall(PetscInfo(C, "Use coloring of C=A*B^T; B^T: %" PetscInt_FMT " %" PetscInt_FMT ", Bt_dense: %" PetscInt_FMT ",%" PetscInt_FMT "; Cnz %" PetscInt_FMT " / (cm*ncolors %" PetscInt_FMT ") = %g\n", B->cmap->n, B->rmap->n, Bt_dense->rmap->n,
1289                           Bt_dense->cmap->n, c->nz, A->rmap->n * matcoloring->ncolors, (double)(((PetscReal)c->nz) / ((PetscReal)(A->rmap->n * matcoloring->ncolors)))));
1290     }
1291 #endif
1292   }
1293   /* clean up */
1294   PetscCall(MatDestroy(&Bt));
1295   PetscFunctionReturn(PETSC_SUCCESS);
1296 }
1297 
1298 PetscErrorCode MatMatTransposeMultNumeric_SeqAIJ_SeqAIJ(Mat A, Mat B, Mat C)
1299 {
1300   Mat_SeqAIJ                    *a = (Mat_SeqAIJ *)A->data, *b = (Mat_SeqAIJ *)B->data, *c = (Mat_SeqAIJ *)C->data;
1301   PetscInt                      *ai = a->i, *aj = a->j, *bi = b->i, *bj = b->j, anzi, bnzj, nexta, nextb, *acol, *bcol, brow;
1302   PetscInt                       cm = C->rmap->n, *ci = c->i, *cj = c->j, i, j, cnzi, *ccol;
1303   PetscLogDouble                 flops = 0.0;
1304   MatScalar                     *aa = a->a, *aval, *ba = b->a, *bval, *ca, *cval;
1305   MatProductCtx_MatMatTransMult *abt;
1306   Mat_Product                   *product = C->product;
1307 
1308   PetscFunctionBegin;
1309   PetscCheck(product, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Missing product struct");
1310   abt = (MatProductCtx_MatMatTransMult *)product->data;
1311   PetscCheck(abt, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Missing product struct");
1312   /* clear old values in C */
1313   if (!c->a) {
1314     PetscCall(PetscCalloc1(ci[cm] + 1, &ca));
1315     c->a      = ca;
1316     c->free_a = PETSC_TRUE;
1317   } else {
1318     ca = c->a;
1319     PetscCall(PetscArrayzero(ca, ci[cm] + 1));
1320   }
1321 
1322   if (abt->usecoloring) {
1323     MatTransposeColoring matcoloring = abt->matcoloring;
1324     Mat                  Bt_dense, C_dense = abt->ABt_den;
1325 
1326     /* Get Bt_dense by Apply MatTransposeColoring to B */
1327     Bt_dense = abt->Bt_den;
1328     PetscCall(MatTransColoringApplySpToDen(matcoloring, B, Bt_dense));
1329 
1330     /* C_dense = A*Bt_dense */
1331     PetscCall(MatMatMultNumeric_SeqAIJ_SeqDense(A, Bt_dense, C_dense));
1332 
1333     /* Recover C from C_dense */
1334     PetscCall(MatTransColoringApplyDenToSp(matcoloring, C_dense, C));
1335     PetscFunctionReturn(PETSC_SUCCESS);
1336   }
1337 
1338   for (i = 0; i < cm; i++) {
1339     anzi = ai[i + 1] - ai[i];
1340     acol = PetscSafePointerPlusOffset(aj, ai[i]);
1341     aval = PetscSafePointerPlusOffset(aa, ai[i]);
1342     cnzi = ci[i + 1] - ci[i];
1343     ccol = PetscSafePointerPlusOffset(cj, ci[i]);
1344     cval = ca + ci[i];
1345     for (j = 0; j < cnzi; j++) {
1346       brow = ccol[j];
1347       bnzj = bi[brow + 1] - bi[brow];
1348       bcol = bj + bi[brow];
1349       bval = ba + bi[brow];
1350 
1351       /* perform sparse inner-product c(i,j)=A[i,:]*B[j,:]^T */
1352       nexta = 0;
1353       nextb = 0;
1354       while (nexta < anzi && nextb < bnzj) {
1355         while (nexta < anzi && acol[nexta] < bcol[nextb]) nexta++;
1356         if (nexta == anzi) break;
1357         while (nextb < bnzj && acol[nexta] > bcol[nextb]) nextb++;
1358         if (nextb == bnzj) break;
1359         if (acol[nexta] == bcol[nextb]) {
1360           cval[j] += aval[nexta] * bval[nextb];
1361           nexta++;
1362           nextb++;
1363           flops += 2;
1364         }
1365       }
1366     }
1367   }
1368   PetscCall(MatAssemblyBegin(C, MAT_FINAL_ASSEMBLY));
1369   PetscCall(MatAssemblyEnd(C, MAT_FINAL_ASSEMBLY));
1370   PetscCall(PetscLogFlops(flops));
1371   PetscFunctionReturn(PETSC_SUCCESS);
1372 }
1373 
1374 PetscErrorCode MatProductCtxDestroy_SeqAIJ_MatTransMatMult(PetscCtxRt data)
1375 {
1376   MatProductCtx_MatTransMatMult *atb = *(MatProductCtx_MatTransMatMult **)data;
1377 
1378   PetscFunctionBegin;
1379   PetscCall(MatDestroy(&atb->At));
1380   if (atb->destroy) PetscCall((*atb->destroy)(&atb->data));
1381   PetscCall(PetscFree(atb));
1382   PetscFunctionReturn(PETSC_SUCCESS);
1383 }
1384 
1385 PetscErrorCode MatTransposeMatMultSymbolic_SeqAIJ_SeqAIJ(Mat A, Mat B, PetscReal fill, Mat C)
1386 {
1387   Mat          At      = NULL;
1388   Mat_Product *product = C->product;
1389   PetscBool    flg, def, square;
1390 
1391   PetscFunctionBegin;
1392   MatCheckProduct(C, 4);
1393   square = (PetscBool)(A == B && A->symmetric == PETSC_BOOL3_TRUE);
1394   /* outerproduct */
1395   PetscCall(PetscStrcmp(product->alg, "outerproduct", &flg));
1396   if (flg) {
1397     /* create symbolic At */
1398     if (!square) {
1399       PetscCall(MatTransposeSymbolic(A, &At));
1400       PetscCall(MatSetBlockSizes(At, A->cmap->bs, B->cmap->bs));
1401       PetscCall(MatSetType(At, ((PetscObject)A)->type_name));
1402     }
1403     /* get symbolic C=At*B */
1404     PetscCall(MatProductSetAlgorithm(C, "sorted"));
1405     PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ(square ? A : At, B, fill, C));
1406 
1407     /* clean up */
1408     if (!square) PetscCall(MatDestroy(&At));
1409 
1410     C->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_SeqAIJ_SeqAIJ; /* outerproduct */
1411     PetscCall(MatProductSetAlgorithm(C, "outerproduct"));
1412     PetscFunctionReturn(PETSC_SUCCESS);
1413   }
1414 
1415   /* matmatmult */
1416   PetscCall(PetscStrcmp(product->alg, "default", &def));
1417   PetscCall(PetscStrcmp(product->alg, "at*b", &flg));
1418   if (flg || def) {
1419     MatProductCtx_MatTransMatMult *atb;
1420 
1421     PetscCheck(!product->data, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Extra product struct not empty");
1422     PetscCall(PetscNew(&atb));
1423     if (!square) PetscCall(MatTranspose(A, MAT_INITIAL_MATRIX, &At));
1424     PetscCall(MatProductSetAlgorithm(C, "sorted"));
1425     PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ(square ? A : At, B, fill, C));
1426     PetscCall(MatProductSetAlgorithm(C, "at*b"));
1427     product->data    = atb;
1428     product->destroy = MatProductCtxDestroy_SeqAIJ_MatTransMatMult;
1429     atb->At          = At;
1430 
1431     C->ops->mattransposemultnumeric = NULL; /* see MatProductNumeric_AtB_SeqAIJ_SeqAIJ */
1432     PetscFunctionReturn(PETSC_SUCCESS);
1433   }
1434 
1435   SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Mat Product Algorithm is not supported");
1436 }
1437 
1438 PetscErrorCode MatTransposeMatMultNumeric_SeqAIJ_SeqAIJ(Mat A, Mat B, Mat C)
1439 {
1440   Mat_SeqAIJ    *a = (Mat_SeqAIJ *)A->data, *b = (Mat_SeqAIJ *)B->data, *c = (Mat_SeqAIJ *)C->data;
1441   PetscInt       am = A->rmap->n, anzi, *ai = a->i, *aj = a->j, *bi = b->i, *bj, bnzi, nextb;
1442   PetscInt       cm = C->rmap->n, *ci = c->i, *cj = c->j, crow, *cjj, i, j, k;
1443   PetscLogDouble flops = 0.0;
1444   MatScalar     *aa    = a->a, *ba, *ca, *caj;
1445 
1446   PetscFunctionBegin;
1447   if (!c->a) {
1448     PetscCall(PetscCalloc1(ci[cm] + 1, &ca));
1449 
1450     c->a      = ca;
1451     c->free_a = PETSC_TRUE;
1452   } else {
1453     ca = c->a;
1454     PetscCall(PetscArrayzero(ca, ci[cm]));
1455   }
1456 
1457   /* compute A^T*B using outer product (A^T)[:,i]*B[i,:] */
1458   for (i = 0; i < am; i++) {
1459     bj   = b->j + bi[i];
1460     ba   = b->a + bi[i];
1461     bnzi = bi[i + 1] - bi[i];
1462     anzi = ai[i + 1] - ai[i];
1463     for (j = 0; j < anzi; j++) {
1464       nextb = 0;
1465       crow  = *aj++;
1466       cjj   = cj + ci[crow];
1467       caj   = ca + ci[crow];
1468       /* perform sparse axpy operation.  Note cjj includes bj. */
1469       for (k = 0; nextb < bnzi; k++) {
1470         if (cjj[k] == *(bj + nextb)) { /* ccol == bcol */
1471           caj[k] += (*aa) * (*(ba + nextb));
1472           nextb++;
1473         }
1474       }
1475       flops += 2 * bnzi;
1476       aa++;
1477     }
1478   }
1479 
1480   /* Assemble the final matrix and clean up */
1481   PetscCall(MatAssemblyBegin(C, MAT_FINAL_ASSEMBLY));
1482   PetscCall(MatAssemblyEnd(C, MAT_FINAL_ASSEMBLY));
1483   PetscCall(PetscLogFlops(flops));
1484   PetscFunctionReturn(PETSC_SUCCESS);
1485 }
1486 
1487 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqDense(Mat A, Mat B, PetscReal fill, Mat C)
1488 {
1489   PetscFunctionBegin;
1490   PetscCall(MatMatMultSymbolic_SeqDense_SeqDense(A, B, 0.0, C));
1491   C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqDense;
1492   PetscFunctionReturn(PETSC_SUCCESS);
1493 }
1494 
1495 PETSC_INTERN PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat A, Mat B, Mat C, const PetscBool add)
1496 {
1497   Mat_SeqAIJ        *a = (Mat_SeqAIJ *)A->data;
1498   PetscScalar       *c, r1, r2, r3, r4, *c1, *c2, *c3, *c4;
1499   const PetscScalar *aa, *b, *b1, *b2, *b3, *b4, *av;
1500   const PetscInt    *aj;
1501   PetscInt           cm = C->rmap->n, cn = B->cmap->n, bm, am = A->rmap->n;
1502   PetscInt           clda;
1503   PetscInt           am4, bm4, col, i, j, n;
1504 
1505   PetscFunctionBegin;
1506   if (!cm || !cn) PetscFunctionReturn(PETSC_SUCCESS);
1507   PetscCall(MatSeqAIJGetArrayRead(A, &av));
1508   if (add) {
1509     PetscCall(MatDenseGetArray(C, &c));
1510   } else {
1511     PetscCall(MatDenseGetArrayWrite(C, &c));
1512   }
1513   PetscCall(MatDenseGetArrayRead(B, &b));
1514   PetscCall(MatDenseGetLDA(B, &bm));
1515   PetscCall(MatDenseGetLDA(C, &clda));
1516   am4 = 4 * clda;
1517   bm4 = 4 * bm;
1518   if (b) {
1519     b1 = b;
1520     b2 = b1 + bm;
1521     b3 = b2 + bm;
1522     b4 = b3 + bm;
1523   } else b1 = b2 = b3 = b4 = NULL;
1524   c1 = c;
1525   c2 = c1 + clda;
1526   c3 = c2 + clda;
1527   c4 = c3 + clda;
1528   for (col = 0; col < (cn / 4) * 4; col += 4) { /* over columns of C */
1529     for (i = 0; i < am; i++) {                  /* over rows of A in those columns */
1530       r1 = r2 = r3 = r4 = 0.0;
1531       n                 = a->i[i + 1] - a->i[i];
1532       aj                = PetscSafePointerPlusOffset(a->j, a->i[i]);
1533       aa                = PetscSafePointerPlusOffset(av, a->i[i]);
1534       for (j = 0; j < n; j++) {
1535         const PetscScalar aatmp = aa[j];
1536         const PetscInt    ajtmp = aj[j];
1537         r1 += aatmp * b1[ajtmp];
1538         r2 += aatmp * b2[ajtmp];
1539         r3 += aatmp * b3[ajtmp];
1540         r4 += aatmp * b4[ajtmp];
1541       }
1542       if (add) {
1543         c1[i] += r1;
1544         c2[i] += r2;
1545         c3[i] += r3;
1546         c4[i] += r4;
1547       } else {
1548         c1[i] = r1;
1549         c2[i] = r2;
1550         c3[i] = r3;
1551         c4[i] = r4;
1552       }
1553     }
1554     if (b) {
1555       b1 += bm4;
1556       b2 += bm4;
1557       b3 += bm4;
1558       b4 += bm4;
1559     }
1560     c1 += am4;
1561     c2 += am4;
1562     c3 += am4;
1563     c4 += am4;
1564   }
1565   /* process remaining columns */
1566   if (col != cn) {
1567     PetscInt rc = cn - col;
1568 
1569     if (rc == 1) {
1570       for (i = 0; i < am; i++) {
1571         r1 = 0.0;
1572         n  = a->i[i + 1] - a->i[i];
1573         aj = PetscSafePointerPlusOffset(a->j, a->i[i]);
1574         aa = PetscSafePointerPlusOffset(av, a->i[i]);
1575         for (j = 0; j < n; j++) r1 += aa[j] * b1[aj[j]];
1576         if (add) c1[i] += r1;
1577         else c1[i] = r1;
1578       }
1579     } else if (rc == 2) {
1580       for (i = 0; i < am; i++) {
1581         r1 = r2 = 0.0;
1582         n       = a->i[i + 1] - a->i[i];
1583         aj      = PetscSafePointerPlusOffset(a->j, a->i[i]);
1584         aa      = PetscSafePointerPlusOffset(av, a->i[i]);
1585         for (j = 0; j < n; j++) {
1586           const PetscScalar aatmp = aa[j];
1587           const PetscInt    ajtmp = aj[j];
1588           r1 += aatmp * b1[ajtmp];
1589           r2 += aatmp * b2[ajtmp];
1590         }
1591         if (add) {
1592           c1[i] += r1;
1593           c2[i] += r2;
1594         } else {
1595           c1[i] = r1;
1596           c2[i] = r2;
1597         }
1598       }
1599     } else {
1600       for (i = 0; i < am; i++) {
1601         r1 = r2 = r3 = 0.0;
1602         n            = a->i[i + 1] - a->i[i];
1603         aj           = PetscSafePointerPlusOffset(a->j, a->i[i]);
1604         aa           = PetscSafePointerPlusOffset(av, a->i[i]);
1605         for (j = 0; j < n; j++) {
1606           const PetscScalar aatmp = aa[j];
1607           const PetscInt    ajtmp = aj[j];
1608           r1 += aatmp * b1[ajtmp];
1609           r2 += aatmp * b2[ajtmp];
1610           r3 += aatmp * b3[ajtmp];
1611         }
1612         if (add) {
1613           c1[i] += r1;
1614           c2[i] += r2;
1615           c3[i] += r3;
1616         } else {
1617           c1[i] = r1;
1618           c2[i] = r2;
1619           c3[i] = r3;
1620         }
1621       }
1622     }
1623   }
1624   PetscCall(PetscLogFlops(cn * (2.0 * a->nz)));
1625   if (add) {
1626     PetscCall(MatDenseRestoreArray(C, &c));
1627   } else {
1628     PetscCall(MatDenseRestoreArrayWrite(C, &c));
1629   }
1630   PetscCall(MatDenseRestoreArrayRead(B, &b));
1631   PetscCall(MatSeqAIJRestoreArrayRead(A, &av));
1632   PetscFunctionReturn(PETSC_SUCCESS);
1633 }
1634 
1635 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqDense(Mat A, Mat B, Mat C)
1636 {
1637   PetscFunctionBegin;
1638   PetscCheck(B->rmap->n == A->cmap->n, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Number columns in A %" PetscInt_FMT " not equal rows in B %" PetscInt_FMT, A->cmap->n, B->rmap->n);
1639   PetscCheck(A->rmap->n == C->rmap->n, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Number rows in C %" PetscInt_FMT " not equal rows in A %" PetscInt_FMT, C->rmap->n, A->rmap->n);
1640   PetscCheck(B->cmap->n == C->cmap->n, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Number columns in B %" PetscInt_FMT " not equal columns in C %" PetscInt_FMT, B->cmap->n, C->cmap->n);
1641 
1642   PetscCall(MatMatMultNumericAdd_SeqAIJ_SeqDense(A, B, C, PETSC_FALSE));
1643   PetscFunctionReturn(PETSC_SUCCESS);
1644 }
1645 
1646 static PetscErrorCode MatProductSetFromOptions_SeqAIJ_SeqDense_AB(Mat C)
1647 {
1648   PetscFunctionBegin;
1649   C->ops->matmultsymbolic = MatMatMultSymbolic_SeqAIJ_SeqDense;
1650   C->ops->productsymbolic = MatProductSymbolic_AB;
1651   PetscFunctionReturn(PETSC_SUCCESS);
1652 }
1653 
1654 PETSC_INTERN PetscErrorCode MatTMatTMultSymbolic_SeqAIJ_SeqDense(Mat, Mat, PetscReal, Mat);
1655 
1656 static PetscErrorCode MatProductSetFromOptions_SeqAIJ_SeqDense_AtB(Mat C)
1657 {
1658   PetscFunctionBegin;
1659   C->ops->transposematmultsymbolic = MatTMatTMultSymbolic_SeqAIJ_SeqDense;
1660   C->ops->productsymbolic          = MatProductSymbolic_AtB;
1661   PetscFunctionReturn(PETSC_SUCCESS);
1662 }
1663 
1664 static PetscErrorCode MatProductSetFromOptions_SeqAIJ_SeqDense_ABt(Mat C)
1665 {
1666   PetscFunctionBegin;
1667   C->ops->mattransposemultsymbolic = MatTMatTMultSymbolic_SeqAIJ_SeqDense;
1668   C->ops->productsymbolic          = MatProductSymbolic_ABt;
1669   PetscFunctionReturn(PETSC_SUCCESS);
1670 }
1671 
1672 PETSC_INTERN PetscErrorCode MatProductSetFromOptions_SeqAIJ_SeqDense(Mat C)
1673 {
1674   Mat_Product *product = C->product;
1675 
1676   PetscFunctionBegin;
1677   switch (product->type) {
1678   case MATPRODUCT_AB:
1679     PetscCall(MatProductSetFromOptions_SeqAIJ_SeqDense_AB(C));
1680     break;
1681   case MATPRODUCT_AtB:
1682     PetscCall(MatProductSetFromOptions_SeqAIJ_SeqDense_AtB(C));
1683     break;
1684   case MATPRODUCT_ABt:
1685     PetscCall(MatProductSetFromOptions_SeqAIJ_SeqDense_ABt(C));
1686     break;
1687   default:
1688     break;
1689   }
1690   PetscFunctionReturn(PETSC_SUCCESS);
1691 }
1692 
1693 static PetscErrorCode MatProductSetFromOptions_SeqXBAIJ_SeqDense_AB(Mat C)
1694 {
1695   Mat_Product *product = C->product;
1696   Mat          A       = product->A;
1697   PetscBool    baij;
1698 
1699   PetscFunctionBegin;
1700   PetscCall(PetscObjectTypeCompare((PetscObject)A, MATSEQBAIJ, &baij));
1701   if (!baij) { /* A is seqsbaij */
1702     PetscBool sbaij;
1703     PetscCall(PetscObjectTypeCompare((PetscObject)A, MATSEQSBAIJ, &sbaij));
1704     PetscCheck(sbaij, PetscObjectComm((PetscObject)C), PETSC_ERR_ARG_WRONGSTATE, "Mat must be either seqbaij or seqsbaij format");
1705 
1706     C->ops->matmultsymbolic = MatMatMultSymbolic_SeqSBAIJ_SeqDense;
1707   } else { /* A is seqbaij */
1708     C->ops->matmultsymbolic = MatMatMultSymbolic_SeqBAIJ_SeqDense;
1709   }
1710 
1711   C->ops->productsymbolic = MatProductSymbolic_AB;
1712   PetscFunctionReturn(PETSC_SUCCESS);
1713 }
1714 
1715 PETSC_INTERN PetscErrorCode MatProductSetFromOptions_SeqXBAIJ_SeqDense(Mat C)
1716 {
1717   Mat_Product *product = C->product;
1718 
1719   PetscFunctionBegin;
1720   MatCheckProduct(C, 1);
1721   PetscCheck(product->A, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Missing A");
1722   if (product->type == MATPRODUCT_AB || (product->type == MATPRODUCT_AtB && product->A->symmetric == PETSC_BOOL3_TRUE)) PetscCall(MatProductSetFromOptions_SeqXBAIJ_SeqDense_AB(C));
1723   else if (product->type == MATPRODUCT_AtB) {
1724     PetscBool flg;
1725 
1726     PetscCall(PetscObjectTypeCompare((PetscObject)product->A, MATSEQBAIJ, &flg));
1727     if (flg) {
1728       C->ops->transposematmultsymbolic = MatTransposeMatMultSymbolic_SeqBAIJ_SeqDense;
1729       C->ops->productsymbolic          = MatProductSymbolic_AtB;
1730     }
1731   }
1732   PetscFunctionReturn(PETSC_SUCCESS);
1733 }
1734 
1735 static PetscErrorCode MatProductSetFromOptions_SeqDense_SeqAIJ_AB(Mat C)
1736 {
1737   PetscFunctionBegin;
1738   C->ops->matmultsymbolic = MatMatMultSymbolic_SeqDense_SeqAIJ;
1739   C->ops->productsymbolic = MatProductSymbolic_AB;
1740   PetscFunctionReturn(PETSC_SUCCESS);
1741 }
1742 
1743 PETSC_INTERN PetscErrorCode MatProductSetFromOptions_SeqDense_SeqAIJ(Mat C)
1744 {
1745   Mat_Product *product = C->product;
1746 
1747   PetscFunctionBegin;
1748   if (product->type == MATPRODUCT_AB) PetscCall(MatProductSetFromOptions_SeqDense_SeqAIJ_AB(C));
1749   PetscFunctionReturn(PETSC_SUCCESS);
1750 }
1751 
1752 PetscErrorCode MatTransColoringApplySpToDen_SeqAIJ(MatTransposeColoring coloring, Mat B, Mat Btdense)
1753 {
1754   Mat_SeqAIJ   *b       = (Mat_SeqAIJ *)B->data;
1755   Mat_SeqDense *btdense = (Mat_SeqDense *)Btdense->data;
1756   PetscInt     *bi = b->i, *bj = b->j;
1757   PetscInt      m = Btdense->rmap->n, n = Btdense->cmap->n, j, k, l, col, anz, *btcol, brow, ncolumns;
1758   MatScalar    *btval, *btval_den, *ba = b->a;
1759   PetscInt     *columns = coloring->columns, *colorforcol = coloring->colorforcol, ncolors = coloring->ncolors;
1760 
1761   PetscFunctionBegin;
1762   btval_den = btdense->v;
1763   PetscCall(PetscArrayzero(btval_den, m * n));
1764   for (k = 0; k < ncolors; k++) {
1765     ncolumns = coloring->ncolumns[k];
1766     for (l = 0; l < ncolumns; l++) { /* insert a row of B to a column of Btdense */
1767       col   = *(columns + colorforcol[k] + l);
1768       btcol = bj + bi[col];
1769       btval = ba + bi[col];
1770       anz   = bi[col + 1] - bi[col];
1771       for (j = 0; j < anz; j++) {
1772         brow            = btcol[j];
1773         btval_den[brow] = btval[j];
1774       }
1775     }
1776     btval_den += m;
1777   }
1778   PetscFunctionReturn(PETSC_SUCCESS);
1779 }
1780 
1781 PetscErrorCode MatTransColoringApplyDenToSp_SeqAIJ(MatTransposeColoring matcoloring, Mat Cden, Mat Csp)
1782 {
1783   Mat_SeqAIJ        *csp = (Mat_SeqAIJ *)Csp->data;
1784   const PetscScalar *ca_den, *ca_den_ptr;
1785   PetscScalar       *ca = csp->a;
1786   PetscInt           k, l, m = Cden->rmap->n, ncolors = matcoloring->ncolors;
1787   PetscInt           brows = matcoloring->brows, *den2sp = matcoloring->den2sp;
1788   PetscInt           nrows, *row, *idx;
1789   PetscInt          *rows = matcoloring->rows, *colorforrow = matcoloring->colorforrow;
1790 
1791   PetscFunctionBegin;
1792   PetscCall(MatDenseGetArrayRead(Cden, &ca_den));
1793 
1794   if (brows > 0) {
1795     PetscInt *lstart, row_end, row_start;
1796     lstart = matcoloring->lstart;
1797     PetscCall(PetscArrayzero(lstart, ncolors));
1798 
1799     row_end = brows;
1800     if (row_end > m) row_end = m;
1801     for (row_start = 0; row_start < m; row_start += brows) { /* loop over row blocks of Csp */
1802       ca_den_ptr = ca_den;
1803       for (k = 0; k < ncolors; k++) { /* loop over colors (columns of Cden) */
1804         nrows = matcoloring->nrows[k];
1805         row   = rows + colorforrow[k];
1806         idx   = den2sp + colorforrow[k];
1807         for (l = lstart[k]; l < nrows; l++) {
1808           if (row[l] >= row_end) {
1809             lstart[k] = l;
1810             break;
1811           } else {
1812             ca[idx[l]] = ca_den_ptr[row[l]];
1813           }
1814         }
1815         ca_den_ptr += m;
1816       }
1817       row_end += brows;
1818       if (row_end > m) row_end = m;
1819     }
1820   } else { /* non-blocked impl: loop over columns of Csp - slow if Csp is large */
1821     ca_den_ptr = ca_den;
1822     for (k = 0; k < ncolors; k++) {
1823       nrows = matcoloring->nrows[k];
1824       row   = rows + colorforrow[k];
1825       idx   = den2sp + colorforrow[k];
1826       for (l = 0; l < nrows; l++) ca[idx[l]] = ca_den_ptr[row[l]];
1827       ca_den_ptr += m;
1828     }
1829   }
1830 
1831   PetscCall(MatDenseRestoreArrayRead(Cden, &ca_den));
1832 #if defined(PETSC_USE_INFO)
1833   if (matcoloring->brows > 0) {
1834     PetscCall(PetscInfo(Csp, "Loop over %" PetscInt_FMT " row blocks for den2sp\n", brows));
1835   } else {
1836     PetscCall(PetscInfo(Csp, "Loop over colors/columns of Cden, inefficient for large sparse matrix product \n"));
1837   }
1838 #endif
1839   PetscFunctionReturn(PETSC_SUCCESS);
1840 }
1841 
1842 PetscErrorCode MatTransposeColoringCreate_SeqAIJ(Mat mat, ISColoring iscoloring, MatTransposeColoring c)
1843 {
1844   PetscInt        i, n, nrows, Nbs, j, k, m, ncols, col, cm;
1845   const PetscInt *is, *ci, *cj, *row_idx;
1846   PetscInt        nis = iscoloring->n, *rowhit, bs = 1;
1847   IS             *isa;
1848   Mat_SeqAIJ     *csp = (Mat_SeqAIJ *)mat->data;
1849   PetscInt       *colorforrow, *rows, *rows_i, *idxhit, *spidx, *den2sp, *den2sp_i;
1850   PetscInt       *colorforcol, *columns, *columns_i, brows;
1851   PetscBool       flg;
1852 
1853   PetscFunctionBegin;
1854   PetscCall(ISColoringGetIS(iscoloring, PETSC_USE_POINTER, PETSC_IGNORE, &isa));
1855 
1856   /* bs > 1 is not being tested yet! */
1857   Nbs       = mat->cmap->N / bs;
1858   c->M      = mat->rmap->N / bs; /* set total rows, columns and local rows */
1859   c->N      = Nbs;
1860   c->m      = c->M;
1861   c->rstart = 0;
1862   c->brows  = 100;
1863 
1864   c->ncolors = nis;
1865   PetscCall(PetscMalloc3(nis, &c->ncolumns, nis, &c->nrows, nis + 1, &colorforrow));
1866   PetscCall(PetscMalloc1(csp->nz + 1, &rows));
1867   PetscCall(PetscMalloc1(csp->nz + 1, &den2sp));
1868 
1869   brows = c->brows;
1870   PetscCall(PetscOptionsGetInt(NULL, NULL, "-matden2sp_brows", &brows, &flg));
1871   if (flg) c->brows = brows;
1872   if (brows > 0) PetscCall(PetscMalloc1(nis + 1, &c->lstart));
1873 
1874   colorforrow[0] = 0;
1875   rows_i         = rows;
1876   den2sp_i       = den2sp;
1877 
1878   PetscCall(PetscMalloc1(nis + 1, &colorforcol));
1879   PetscCall(PetscMalloc1(Nbs + 1, &columns));
1880 
1881   colorforcol[0] = 0;
1882   columns_i      = columns;
1883 
1884   /* get column-wise storage of mat */
1885   PetscCall(MatGetColumnIJ_SeqAIJ_Color(mat, 0, PETSC_FALSE, PETSC_FALSE, &ncols, &ci, &cj, &spidx, NULL));
1886 
1887   cm = c->m;
1888   PetscCall(PetscMalloc1(cm + 1, &rowhit));
1889   PetscCall(PetscMalloc1(cm + 1, &idxhit));
1890   for (i = 0; i < nis; i++) { /* loop over color */
1891     PetscCall(ISGetLocalSize(isa[i], &n));
1892     PetscCall(ISGetIndices(isa[i], &is));
1893 
1894     c->ncolumns[i] = n;
1895     if (n) PetscCall(PetscArraycpy(columns_i, is, n));
1896     colorforcol[i + 1] = colorforcol[i] + n;
1897     columns_i += n;
1898 
1899     /* fast, crude version requires O(N*N) work */
1900     PetscCall(PetscArrayzero(rowhit, cm));
1901 
1902     for (j = 0; j < n; j++) { /* loop over columns*/
1903       col     = is[j];
1904       row_idx = cj + ci[col];
1905       m       = ci[col + 1] - ci[col];
1906       for (k = 0; k < m; k++) { /* loop over columns marking them in rowhit */
1907         idxhit[*row_idx]   = spidx[ci[col] + k];
1908         rowhit[*row_idx++] = col + 1;
1909       }
1910     }
1911     /* count the number of hits */
1912     nrows = 0;
1913     for (j = 0; j < cm; j++) {
1914       if (rowhit[j]) nrows++;
1915     }
1916     c->nrows[i]        = nrows;
1917     colorforrow[i + 1] = colorforrow[i] + nrows;
1918 
1919     nrows = 0;
1920     for (j = 0; j < cm; j++) { /* loop over rows */
1921       if (rowhit[j]) {
1922         rows_i[nrows]   = j;
1923         den2sp_i[nrows] = idxhit[j];
1924         nrows++;
1925       }
1926     }
1927     den2sp_i += nrows;
1928 
1929     PetscCall(ISRestoreIndices(isa[i], &is));
1930     rows_i += nrows;
1931   }
1932   PetscCall(MatRestoreColumnIJ_SeqAIJ_Color(mat, 0, PETSC_FALSE, PETSC_FALSE, &ncols, &ci, &cj, &spidx, NULL));
1933   PetscCall(PetscFree(rowhit));
1934   PetscCall(ISColoringRestoreIS(iscoloring, PETSC_USE_POINTER, &isa));
1935   PetscCheck(csp->nz == colorforrow[nis], PETSC_COMM_SELF, PETSC_ERR_PLIB, "csp->nz %" PetscInt_FMT " != colorforrow[nis] %" PetscInt_FMT, csp->nz, colorforrow[nis]);
1936 
1937   c->colorforrow = colorforrow;
1938   c->rows        = rows;
1939   c->den2sp      = den2sp;
1940   c->colorforcol = colorforcol;
1941   c->columns     = columns;
1942 
1943   PetscCall(PetscFree(idxhit));
1944   PetscFunctionReturn(PETSC_SUCCESS);
1945 }
1946 
1947 static PetscErrorCode MatProductNumeric_AtB_SeqAIJ_SeqAIJ(Mat C)
1948 {
1949   Mat_Product *product = C->product;
1950   Mat          A = product->A, B = product->B;
1951 
1952   PetscFunctionBegin;
1953   if (C->ops->mattransposemultnumeric) {
1954     /* Alg: "outerproduct" */
1955     PetscCall((*C->ops->mattransposemultnumeric)(A, B, C));
1956   } else {
1957     /* Alg: "matmatmult" -- C = At*B */
1958     MatProductCtx_MatTransMatMult *atb = (MatProductCtx_MatTransMatMult *)product->data;
1959 
1960     PetscCheck(atb, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Missing product struct");
1961     if (atb->At) {
1962       /* At is computed in MatTransposeMatMultSymbolic_SeqAIJ_SeqAIJ();
1963          user may have called MatProductReplaceMats() to get this A=product->A */
1964       PetscCall(MatTransposeSetPrecursor(A, atb->At));
1965       PetscCall(MatTranspose(A, MAT_REUSE_MATRIX, &atb->At));
1966     }
1967     PetscCall(MatMatMultNumeric_SeqAIJ_SeqAIJ(atb->At ? atb->At : A, B, C));
1968   }
1969   PetscFunctionReturn(PETSC_SUCCESS);
1970 }
1971 
1972 static PetscErrorCode MatProductSymbolic_AtB_SeqAIJ_SeqAIJ(Mat C)
1973 {
1974   Mat_Product *product = C->product;
1975   Mat          A = product->A, B = product->B;
1976   PetscReal    fill = product->fill;
1977 
1978   PetscFunctionBegin;
1979   PetscCall(MatTransposeMatMultSymbolic_SeqAIJ_SeqAIJ(A, B, fill, C));
1980 
1981   C->ops->productnumeric = MatProductNumeric_AtB_SeqAIJ_SeqAIJ;
1982   PetscFunctionReturn(PETSC_SUCCESS);
1983 }
1984 
1985 static PetscErrorCode MatProductSetFromOptions_SeqAIJ_AB(Mat C)
1986 {
1987   Mat_Product *product = C->product;
1988   PetscInt     alg     = 0; /* default algorithm */
1989   PetscBool    flg     = PETSC_FALSE;
1990 #if !defined(PETSC_HAVE_HYPRE)
1991   const char *algTypes[7] = {"sorted", "scalable", "scalable_fast", "heap", "btheap", "llcondensed", "rowmerge"};
1992   PetscInt    nalg        = 7;
1993 #else
1994   const char *algTypes[8] = {"sorted", "scalable", "scalable_fast", "heap", "btheap", "llcondensed", "rowmerge", "hypre"};
1995   PetscInt    nalg        = 8;
1996 #endif
1997 
1998   PetscFunctionBegin;
1999   /* Set default algorithm */
2000   PetscCall(PetscStrcmp(C->product->alg, "default", &flg));
2001   if (flg) PetscCall(MatProductSetAlgorithm(C, algTypes[alg]));
2002 
2003   /* Get runtime option */
2004   if (product->api_user) {
2005     PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatMatMult", "Mat");
2006     PetscCall(PetscOptionsEList("-matmatmult_via", "Algorithmic approach", "MatMatMult", algTypes, nalg, algTypes[0], &alg, &flg));
2007     PetscOptionsEnd();
2008   } else {
2009     PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatProduct_AB", "Mat");
2010     PetscCall(PetscOptionsEList("-mat_product_algorithm", "Algorithmic approach", "MatProduct_AB", algTypes, nalg, algTypes[0], &alg, &flg));
2011     PetscOptionsEnd();
2012   }
2013   if (flg) PetscCall(MatProductSetAlgorithm(C, algTypes[alg]));
2014 
2015   C->ops->productsymbolic = MatProductSymbolic_AB;
2016   C->ops->matmultsymbolic = MatMatMultSymbolic_SeqAIJ_SeqAIJ;
2017   PetscFunctionReturn(PETSC_SUCCESS);
2018 }
2019 
2020 static PetscErrorCode MatProductSetFromOptions_SeqAIJ_AtB(Mat C)
2021 {
2022   Mat_Product *product     = C->product;
2023   PetscInt     alg         = 0; /* default algorithm */
2024   PetscBool    flg         = PETSC_FALSE;
2025   const char  *algTypes[3] = {"default", "at*b", "outerproduct"};
2026   PetscInt     nalg        = 3;
2027 
2028   PetscFunctionBegin;
2029   /* Get runtime option */
2030   if (product->api_user) {
2031     PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatTransposeMatMult", "Mat");
2032     PetscCall(PetscOptionsEList("-mattransposematmult_via", "Algorithmic approach", "MatTransposeMatMult", algTypes, nalg, algTypes[alg], &alg, &flg));
2033     PetscOptionsEnd();
2034   } else {
2035     PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatProduct_AtB", "Mat");
2036     PetscCall(PetscOptionsEList("-mat_product_algorithm", "Algorithmic approach", "MatProduct_AtB", algTypes, nalg, algTypes[alg], &alg, &flg));
2037     PetscOptionsEnd();
2038   }
2039   if (flg) PetscCall(MatProductSetAlgorithm(C, algTypes[alg]));
2040 
2041   C->ops->productsymbolic = MatProductSymbolic_AtB_SeqAIJ_SeqAIJ;
2042   PetscFunctionReturn(PETSC_SUCCESS);
2043 }
2044 
2045 static PetscErrorCode MatProductSetFromOptions_SeqAIJ_ABt(Mat C)
2046 {
2047   Mat_Product *product     = C->product;
2048   PetscInt     alg         = 0; /* default algorithm */
2049   PetscBool    flg         = PETSC_FALSE;
2050   const char  *algTypes[2] = {"default", "color"};
2051   PetscInt     nalg        = 2;
2052 
2053   PetscFunctionBegin;
2054   /* Set default algorithm */
2055   PetscCall(PetscStrcmp(C->product->alg, "default", &flg));
2056   if (!flg) {
2057     alg = 1;
2058     PetscCall(MatProductSetAlgorithm(C, algTypes[alg]));
2059   }
2060 
2061   /* Get runtime option */
2062   if (product->api_user) {
2063     PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatMatTransposeMult", "Mat");
2064     PetscCall(PetscOptionsEList("-matmattransmult_via", "Algorithmic approach", "MatMatTransposeMult", algTypes, nalg, algTypes[alg], &alg, &flg));
2065     PetscOptionsEnd();
2066   } else {
2067     PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatProduct_ABt", "Mat");
2068     PetscCall(PetscOptionsEList("-mat_product_algorithm", "Algorithmic approach", "MatProduct_ABt", algTypes, nalg, algTypes[alg], &alg, &flg));
2069     PetscOptionsEnd();
2070   }
2071   if (flg) PetscCall(MatProductSetAlgorithm(C, algTypes[alg]));
2072 
2073   C->ops->mattransposemultsymbolic = MatMatTransposeMultSymbolic_SeqAIJ_SeqAIJ;
2074   C->ops->productsymbolic          = MatProductSymbolic_ABt;
2075   PetscFunctionReturn(PETSC_SUCCESS);
2076 }
2077 
2078 static PetscErrorCode MatProductSetFromOptions_SeqAIJ_PtAP(Mat C)
2079 {
2080   Mat_Product *product = C->product;
2081   PetscBool    flg     = PETSC_FALSE;
2082   PetscInt     alg     = 0; /* default algorithm -- alg=1 should be default!!! */
2083 #if !defined(PETSC_HAVE_HYPRE)
2084   const char *algTypes[2] = {"scalable", "rap"};
2085   PetscInt    nalg        = 2;
2086 #else
2087   const char *algTypes[3] = {"scalable", "rap", "hypre"};
2088   PetscInt    nalg        = 3;
2089 #endif
2090 
2091   PetscFunctionBegin;
2092   /* Set default algorithm */
2093   PetscCall(PetscStrcmp(product->alg, "default", &flg));
2094   if (flg) PetscCall(MatProductSetAlgorithm(C, algTypes[alg]));
2095 
2096   /* Get runtime option */
2097   if (product->api_user) {
2098     PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatPtAP", "Mat");
2099     PetscCall(PetscOptionsEList("-matptap_via", "Algorithmic approach", "MatPtAP", algTypes, nalg, algTypes[0], &alg, &flg));
2100     PetscOptionsEnd();
2101   } else {
2102     PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatProduct_PtAP", "Mat");
2103     PetscCall(PetscOptionsEList("-mat_product_algorithm", "Algorithmic approach", "MatProduct_PtAP", algTypes, nalg, algTypes[0], &alg, &flg));
2104     PetscOptionsEnd();
2105   }
2106   if (flg) PetscCall(MatProductSetAlgorithm(C, algTypes[alg]));
2107 
2108   C->ops->productsymbolic = MatProductSymbolic_PtAP_SeqAIJ_SeqAIJ;
2109   PetscFunctionReturn(PETSC_SUCCESS);
2110 }
2111 
2112 static PetscErrorCode MatProductSetFromOptions_SeqAIJ_RARt(Mat C)
2113 {
2114   Mat_Product *product     = C->product;
2115   PetscBool    flg         = PETSC_FALSE;
2116   PetscInt     alg         = 0; /* default algorithm */
2117   const char  *algTypes[3] = {"r*a*rt", "r*art", "coloring_rart"};
2118   PetscInt     nalg        = 3;
2119 
2120   PetscFunctionBegin;
2121   /* Set default algorithm */
2122   PetscCall(PetscStrcmp(product->alg, "default", &flg));
2123   if (flg) PetscCall(MatProductSetAlgorithm(C, algTypes[alg]));
2124 
2125   /* Get runtime option */
2126   if (product->api_user) {
2127     PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatRARt", "Mat");
2128     PetscCall(PetscOptionsEList("-matrart_via", "Algorithmic approach", "MatRARt", algTypes, nalg, algTypes[0], &alg, &flg));
2129     PetscOptionsEnd();
2130   } else {
2131     PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatProduct_RARt", "Mat");
2132     PetscCall(PetscOptionsEList("-mat_product_algorithm", "Algorithmic approach", "MatProduct_RARt", algTypes, nalg, algTypes[0], &alg, &flg));
2133     PetscOptionsEnd();
2134   }
2135   if (flg) PetscCall(MatProductSetAlgorithm(C, algTypes[alg]));
2136 
2137   C->ops->productsymbolic = MatProductSymbolic_RARt_SeqAIJ_SeqAIJ;
2138   PetscFunctionReturn(PETSC_SUCCESS);
2139 }
2140 
2141 /* ABC = A*B*C = A*(B*C); ABC's algorithm must be chosen from AB's algorithm */
2142 static PetscErrorCode MatProductSetFromOptions_SeqAIJ_ABC(Mat C)
2143 {
2144   Mat_Product *product     = C->product;
2145   PetscInt     alg         = 0; /* default algorithm */
2146   PetscBool    flg         = PETSC_FALSE;
2147   const char  *algTypes[7] = {"sorted", "scalable", "scalable_fast", "heap", "btheap", "llcondensed", "rowmerge"};
2148   PetscInt     nalg        = 7;
2149 
2150   PetscFunctionBegin;
2151   /* Set default algorithm */
2152   PetscCall(PetscStrcmp(product->alg, "default", &flg));
2153   if (flg) PetscCall(MatProductSetAlgorithm(C, algTypes[alg]));
2154 
2155   /* Get runtime option */
2156   if (product->api_user) {
2157     PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatMatMatMult", "Mat");
2158     PetscCall(PetscOptionsEList("-matmatmatmult_via", "Algorithmic approach", "MatMatMatMult", algTypes, nalg, algTypes[alg], &alg, &flg));
2159     PetscOptionsEnd();
2160   } else {
2161     PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatProduct_ABC", "Mat");
2162     PetscCall(PetscOptionsEList("-mat_product_algorithm", "Algorithmic approach", "MatProduct_ABC", algTypes, nalg, algTypes[alg], &alg, &flg));
2163     PetscOptionsEnd();
2164   }
2165   if (flg) PetscCall(MatProductSetAlgorithm(C, algTypes[alg]));
2166 
2167   C->ops->matmatmultsymbolic = MatMatMatMultSymbolic_SeqAIJ_SeqAIJ_SeqAIJ;
2168   C->ops->productsymbolic    = MatProductSymbolic_ABC;
2169   PetscFunctionReturn(PETSC_SUCCESS);
2170 }
2171 
2172 PetscErrorCode MatProductSetFromOptions_SeqAIJ(Mat C)
2173 {
2174   Mat_Product *product = C->product;
2175 
2176   PetscFunctionBegin;
2177   switch (product->type) {
2178   case MATPRODUCT_AB:
2179     PetscCall(MatProductSetFromOptions_SeqAIJ_AB(C));
2180     break;
2181   case MATPRODUCT_AtB:
2182     PetscCall(MatProductSetFromOptions_SeqAIJ_AtB(C));
2183     break;
2184   case MATPRODUCT_ABt:
2185     PetscCall(MatProductSetFromOptions_SeqAIJ_ABt(C));
2186     break;
2187   case MATPRODUCT_PtAP:
2188     PetscCall(MatProductSetFromOptions_SeqAIJ_PtAP(C));
2189     break;
2190   case MATPRODUCT_RARt:
2191     PetscCall(MatProductSetFromOptions_SeqAIJ_RARt(C));
2192     break;
2193   case MATPRODUCT_ABC:
2194     PetscCall(MatProductSetFromOptions_SeqAIJ_ABC(C));
2195     break;
2196   default:
2197     break;
2198   }
2199   PetscFunctionReturn(PETSC_SUCCESS);
2200 }
2201