xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision d2522c19e8fa9bca20aaca277941d9a63e71db6a)
1 
2 /*
3   Defines matrix-matrix product routines for pairs of MPIAIJ matrices
4           C = A * B
5 */
6 #include <../src/mat/impls/aij/seq/aij.h> /*I "petscmat.h" I*/
7 #include <../src/mat/utils/freespace.h>
8 #include <../src/mat/impls/aij/mpi/mpiaij.h>
9 #include <petscbt.h>
10 #include <../src/mat/impls/dense/mpi/mpidense.h>
11 #include <petsc/private/vecimpl.h>
12 #include <petsc/private/sfimpl.h>
13 
14 #if defined(PETSC_HAVE_HYPRE)
15 PETSC_INTERN PetscErrorCode MatMatMultSymbolic_AIJ_AIJ_wHYPRE(Mat, Mat, PetscReal, Mat);
16 #endif
17 
18 PETSC_INTERN PetscErrorCode MatProductSymbolic_ABt_MPIAIJ_MPIAIJ(Mat C) {
19   Mat_Product *product = C->product;
20   Mat          B       = product->B;
21 
22   PetscFunctionBegin;
23   PetscCall(MatTranspose(B, MAT_INITIAL_MATRIX, &product->B));
24   PetscCall(MatDestroy(&B));
25   PetscCall(MatProductSymbolic_AB_MPIAIJ_MPIAIJ(C));
26   PetscFunctionReturn(0);
27 }
28 
29 PETSC_INTERN PetscErrorCode MatProductSymbolic_AB_MPIAIJ_MPIAIJ(Mat C) {
30   Mat_Product        *product = C->product;
31   Mat                 A = product->A, B = product->B;
32   MatProductAlgorithm alg  = product->alg;
33   PetscReal           fill = product->fill;
34   PetscBool           flg;
35 
36   PetscFunctionBegin;
37   /* scalable */
38   PetscCall(PetscStrcmp(alg, "scalable", &flg));
39   if (flg) {
40     PetscCall(MatMatMultSymbolic_MPIAIJ_MPIAIJ(A, B, fill, C));
41     PetscFunctionReturn(0);
42   }
43 
44   /* nonscalable */
45   PetscCall(PetscStrcmp(alg, "nonscalable", &flg));
46   if (flg) {
47     PetscCall(MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(A, B, fill, C));
48     PetscFunctionReturn(0);
49   }
50 
51   /* seqmpi */
52   PetscCall(PetscStrcmp(alg, "seqmpi", &flg));
53   if (flg) {
54     PetscCall(MatMatMultSymbolic_MPIAIJ_MPIAIJ_seqMPI(A, B, fill, C));
55     PetscFunctionReturn(0);
56   }
57 
58   /* backend general code */
59   PetscCall(PetscStrcmp(alg, "backend", &flg));
60   if (flg) {
61     PetscCall(MatProductSymbolic_MPIAIJBACKEND(C));
62     PetscFunctionReturn(0);
63   }
64 
65 #if defined(PETSC_HAVE_HYPRE)
66   PetscCall(PetscStrcmp(alg, "hypre", &flg));
67   if (flg) {
68     PetscCall(MatMatMultSymbolic_AIJ_AIJ_wHYPRE(A, B, fill, C));
69     PetscFunctionReturn(0);
70   }
71 #endif
72   SETERRQ(PetscObjectComm((PetscObject)C), PETSC_ERR_SUP, "Mat Product Algorithm is not supported");
73 }
74 
75 PetscErrorCode MatDestroy_MPIAIJ_MatMatMult(void *data) {
76   Mat_APMPI *ptap = (Mat_APMPI *)data;
77 
78   PetscFunctionBegin;
79   PetscCall(PetscFree2(ptap->startsj_s, ptap->startsj_r));
80   PetscCall(PetscFree(ptap->bufa));
81   PetscCall(MatDestroy(&ptap->P_loc));
82   PetscCall(MatDestroy(&ptap->P_oth));
83   PetscCall(MatDestroy(&ptap->Pt));
84   PetscCall(PetscFree(ptap->api));
85   PetscCall(PetscFree(ptap->apj));
86   PetscCall(PetscFree(ptap->apa));
87   PetscCall(PetscFree(ptap));
88   PetscFunctionReturn(0);
89 }
90 
91 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat A, Mat P, Mat C) {
92   Mat_MPIAIJ        *a = (Mat_MPIAIJ *)A->data, *c = (Mat_MPIAIJ *)C->data;
93   Mat_SeqAIJ        *ad = (Mat_SeqAIJ *)(a->A)->data, *ao = (Mat_SeqAIJ *)(a->B)->data;
94   Mat_SeqAIJ        *cd = (Mat_SeqAIJ *)(c->A)->data, *co = (Mat_SeqAIJ *)(c->B)->data;
95   PetscScalar       *cda = cd->a, *coa = co->a;
96   Mat_SeqAIJ        *p_loc, *p_oth;
97   PetscScalar       *apa, *ca;
98   PetscInt           cm = C->rmap->n;
99   Mat_APMPI         *ptap;
100   PetscInt          *api, *apj, *apJ, i, k;
101   PetscInt           cstart = C->cmap->rstart;
102   PetscInt           cdnz, conz, k0, k1;
103   const PetscScalar *dummy;
104   MPI_Comm           comm;
105   PetscMPIInt        size;
106 
107   PetscFunctionBegin;
108   MatCheckProduct(C, 3);
109   ptap = (Mat_APMPI *)C->product->data;
110   PetscCheck(ptap, PetscObjectComm((PetscObject)C), PETSC_ERR_ARG_WRONGSTATE, "PtAP cannot be computed. Missing data");
111   PetscCall(PetscObjectGetComm((PetscObject)A, &comm));
112   PetscCallMPI(MPI_Comm_size(comm, &size));
113   PetscCheck(ptap->P_oth || size <= 1, PetscObjectComm((PetscObject)C), PETSC_ERR_ARG_WRONGSTATE, "AP cannot be reused. Do not call MatProductClear()");
114 
115   /* flag CPU mask for C */
116 #if defined(PETSC_HAVE_DEVICE)
117   if (C->offloadmask != PETSC_OFFLOAD_UNALLOCATED) C->offloadmask = PETSC_OFFLOAD_CPU;
118   if (c->A->offloadmask != PETSC_OFFLOAD_UNALLOCATED) c->A->offloadmask = PETSC_OFFLOAD_CPU;
119   if (c->B->offloadmask != PETSC_OFFLOAD_UNALLOCATED) c->B->offloadmask = PETSC_OFFLOAD_CPU;
120 #endif
121 
122   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
123   /*-----------------------------------------------------*/
124   /* update numerical values of P_oth and P_loc */
125   PetscCall(MatGetBrowsOfAoCols_MPIAIJ(A, P, MAT_REUSE_MATRIX, &ptap->startsj_s, &ptap->startsj_r, &ptap->bufa, &ptap->P_oth));
126   PetscCall(MatMPIAIJGetLocalMat(P, MAT_REUSE_MATRIX, &ptap->P_loc));
127 
128   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
129   /*----------------------------------------------------------*/
130   /* get data from symbolic products */
131   p_loc = (Mat_SeqAIJ *)(ptap->P_loc)->data;
132   p_oth = NULL;
133   if (size > 1) { p_oth = (Mat_SeqAIJ *)(ptap->P_oth)->data; }
134 
135   /* get apa for storing dense row A[i,:]*P */
136   apa = ptap->apa;
137 
138   api = ptap->api;
139   apj = ptap->apj;
140   /* trigger copy to CPU */
141   PetscCall(MatSeqAIJGetArrayRead(a->A, &dummy));
142   PetscCall(MatSeqAIJRestoreArrayRead(a->A, &dummy));
143   PetscCall(MatSeqAIJGetArrayRead(a->B, &dummy));
144   PetscCall(MatSeqAIJRestoreArrayRead(a->B, &dummy));
145   for (i = 0; i < cm; i++) {
146     /* compute apa = A[i,:]*P */
147     AProw_nonscalable(i, ad, ao, p_loc, p_oth, apa);
148 
149     /* set values in C */
150     apJ  = apj + api[i];
151     cdnz = cd->i[i + 1] - cd->i[i];
152     conz = co->i[i + 1] - co->i[i];
153 
154     /* 1st off-diagonal part of C */
155     ca = coa + co->i[i];
156     k  = 0;
157     for (k0 = 0; k0 < conz; k0++) {
158       if (apJ[k] >= cstart) break;
159       ca[k0]        = apa[apJ[k]];
160       apa[apJ[k++]] = 0.0;
161     }
162 
163     /* diagonal part of C */
164     ca = cda + cd->i[i];
165     for (k1 = 0; k1 < cdnz; k1++) {
166       ca[k1]        = apa[apJ[k]];
167       apa[apJ[k++]] = 0.0;
168     }
169 
170     /* 2nd off-diagonal part of C */
171     ca = coa + co->i[i];
172     for (; k0 < conz; k0++) {
173       ca[k0]        = apa[apJ[k]];
174       apa[apJ[k++]] = 0.0;
175     }
176   }
177   PetscCall(MatAssemblyBegin(C, MAT_FINAL_ASSEMBLY));
178   PetscCall(MatAssemblyEnd(C, MAT_FINAL_ASSEMBLY));
179   PetscFunctionReturn(0);
180 }
181 
182 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat A, Mat P, PetscReal fill, Mat C) {
183   MPI_Comm           comm;
184   PetscMPIInt        size;
185   Mat_APMPI         *ptap;
186   PetscFreeSpaceList free_space = NULL, current_space = NULL;
187   Mat_MPIAIJ        *a  = (Mat_MPIAIJ *)A->data;
188   Mat_SeqAIJ        *ad = (Mat_SeqAIJ *)(a->A)->data, *ao = (Mat_SeqAIJ *)(a->B)->data, *p_loc, *p_oth;
189   PetscInt          *pi_loc, *pj_loc, *pi_oth, *pj_oth, *dnz, *onz;
190   PetscInt          *adi = ad->i, *adj = ad->j, *aoi = ao->i, *aoj = ao->j, rstart = A->rmap->rstart;
191   PetscInt          *lnk, i, pnz, row, *api, *apj, *Jptr, apnz, nspacedouble = 0, j, nzi;
192   PetscInt           am = A->rmap->n, pN = P->cmap->N, pn = P->cmap->n, pm = P->rmap->n;
193   PetscBT            lnkbt;
194   PetscReal          afill;
195   MatType            mtype;
196 
197   PetscFunctionBegin;
198   MatCheckProduct(C, 4);
199   PetscCheck(!C->product->data, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Extra product struct not empty");
200   PetscCall(PetscObjectGetComm((PetscObject)A, &comm));
201   PetscCallMPI(MPI_Comm_size(comm, &size));
202 
203   /* create struct Mat_APMPI and attached it to C later */
204   PetscCall(PetscNew(&ptap));
205 
206   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
207   PetscCall(MatGetBrowsOfAoCols_MPIAIJ(A, P, MAT_INITIAL_MATRIX, &ptap->startsj_s, &ptap->startsj_r, &ptap->bufa, &ptap->P_oth));
208 
209   /* get P_loc by taking all local rows of P */
210   PetscCall(MatMPIAIJGetLocalMat(P, MAT_INITIAL_MATRIX, &ptap->P_loc));
211 
212   p_loc  = (Mat_SeqAIJ *)(ptap->P_loc)->data;
213   pi_loc = p_loc->i;
214   pj_loc = p_loc->j;
215   if (size > 1) {
216     p_oth  = (Mat_SeqAIJ *)(ptap->P_oth)->data;
217     pi_oth = p_oth->i;
218     pj_oth = p_oth->j;
219   } else {
220     p_oth  = NULL;
221     pi_oth = NULL;
222     pj_oth = NULL;
223   }
224 
225   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
226   /*-------------------------------------------------------------------*/
227   PetscCall(PetscMalloc1(am + 2, &api));
228   ptap->api = api;
229   api[0]    = 0;
230 
231   /* create and initialize a linked list */
232   PetscCall(PetscLLCondensedCreate(pN, pN, &lnk, &lnkbt));
233 
234   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
235   PetscCall(PetscFreeSpaceGet(PetscRealIntMultTruncate(fill, PetscIntSumTruncate(adi[am], PetscIntSumTruncate(aoi[am], pi_loc[pm]))), &free_space));
236   current_space = free_space;
237 
238   MatPreallocateBegin(comm, am, pn, dnz, onz);
239   for (i = 0; i < am; i++) {
240     /* diagonal portion of A */
241     nzi = adi[i + 1] - adi[i];
242     for (j = 0; j < nzi; j++) {
243       row  = *adj++;
244       pnz  = pi_loc[row + 1] - pi_loc[row];
245       Jptr = pj_loc + pi_loc[row];
246       /* add non-zero cols of P into the sorted linked list lnk */
247       PetscCall(PetscLLCondensedAddSorted(pnz, Jptr, lnk, lnkbt));
248     }
249     /* off-diagonal portion of A */
250     nzi = aoi[i + 1] - aoi[i];
251     for (j = 0; j < nzi; j++) {
252       row  = *aoj++;
253       pnz  = pi_oth[row + 1] - pi_oth[row];
254       Jptr = pj_oth + pi_oth[row];
255       PetscCall(PetscLLCondensedAddSorted(pnz, Jptr, lnk, lnkbt));
256     }
257     /* add possible missing diagonal entry */
258     if (C->force_diagonals) {
259       j = i + rstart; /* column index */
260       PetscCall(PetscLLCondensedAddSorted(1, &j, lnk, lnkbt));
261     }
262 
263     apnz       = lnk[0];
264     api[i + 1] = api[i] + apnz;
265 
266     /* if free space is not available, double the total space in the list */
267     if (current_space->local_remaining < apnz) {
268       PetscCall(PetscFreeSpaceGet(PetscIntSumTruncate(apnz, current_space->total_array_size), &current_space));
269       nspacedouble++;
270     }
271 
272     /* Copy data into free space, then initialize lnk */
273     PetscCall(PetscLLCondensedClean(pN, apnz, current_space->array, lnk, lnkbt));
274     PetscCall(MatPreallocateSet(i + rstart, apnz, current_space->array, dnz, onz));
275 
276     current_space->array += apnz;
277     current_space->local_used += apnz;
278     current_space->local_remaining -= apnz;
279   }
280 
281   /* Allocate space for apj, initialize apj, and */
282   /* destroy list of free space and other temporary array(s) */
283   PetscCall(PetscMalloc1(api[am] + 1, &ptap->apj));
284   apj = ptap->apj;
285   PetscCall(PetscFreeSpaceContiguous(&free_space, ptap->apj));
286   PetscCall(PetscLLDestroy(lnk, lnkbt));
287 
288   /* malloc apa to store dense row A[i,:]*P */
289   PetscCall(PetscCalloc1(pN, &ptap->apa));
290 
291   /* set and assemble symbolic parallel matrix C */
292   /*---------------------------------------------*/
293   PetscCall(MatSetSizes(C, am, pn, PETSC_DETERMINE, PETSC_DETERMINE));
294   PetscCall(MatSetBlockSizesFromMats(C, A, P));
295 
296   PetscCall(MatGetType(A, &mtype));
297   PetscCall(MatSetType(C, mtype));
298   PetscCall(MatMPIAIJSetPreallocation(C, 0, dnz, 0, onz));
299   MatPreallocateEnd(dnz, onz);
300 
301   PetscCall(MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(C, apj, api));
302   PetscCall(MatAssemblyBegin(C, MAT_FINAL_ASSEMBLY));
303   PetscCall(MatAssemblyEnd(C, MAT_FINAL_ASSEMBLY));
304   PetscCall(MatSetOption(C, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
305 
306   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
307   C->ops->productnumeric = MatProductNumeric_AB;
308 
309   /* attach the supporting struct to C for reuse */
310   C->product->data    = ptap;
311   C->product->destroy = MatDestroy_MPIAIJ_MatMatMult;
312 
313   /* set MatInfo */
314   afill = (PetscReal)api[am] / (adi[am] + aoi[am] + pi_loc[pm] + 1) + 1.e-5;
315   if (afill < 1.0) afill = 1.0;
316   C->info.mallocs           = nspacedouble;
317   C->info.fill_ratio_given  = fill;
318   C->info.fill_ratio_needed = afill;
319 
320 #if defined(PETSC_USE_INFO)
321   if (api[am]) {
322     PetscCall(PetscInfo(C, "Reallocs %" PetscInt_FMT "; Fill ratio: given %g needed %g.\n", nspacedouble, (double)fill, (double)afill));
323     PetscCall(PetscInfo(C, "Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n", (double)afill));
324   } else {
325     PetscCall(PetscInfo(C, "Empty matrix product\n"));
326   }
327 #endif
328   PetscFunctionReturn(0);
329 }
330 
331 /* ------------------------------------------------------- */
332 static PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat, Mat, PetscReal, Mat);
333 static PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat, Mat, Mat);
334 
335 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_MPIDense_AB(Mat C) {
336   Mat_Product *product = C->product;
337   Mat          A = product->A, B = product->B;
338 
339   PetscFunctionBegin;
340   if (A->cmap->rstart != B->rmap->rstart || A->cmap->rend != B->rmap->rend)
341     SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Matrix local dimensions are incompatible, (%" PetscInt_FMT ", %" PetscInt_FMT ") != (%" PetscInt_FMT ",%" PetscInt_FMT ")", A->cmap->rstart, A->cmap->rend, B->rmap->rstart, B->rmap->rend);
342 
343   C->ops->matmultsymbolic = MatMatMultSymbolic_MPIAIJ_MPIDense;
344   C->ops->productsymbolic = MatProductSymbolic_AB;
345   PetscFunctionReturn(0);
346 }
347 /* -------------------------------------------------------------------- */
348 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_MPIDense_AtB(Mat C) {
349   Mat_Product *product = C->product;
350   Mat          A = product->A, B = product->B;
351 
352   PetscFunctionBegin;
353   if (A->rmap->rstart != B->rmap->rstart || A->rmap->rend != B->rmap->rend)
354     SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Matrix local dimensions are incompatible, (%" PetscInt_FMT ", %" PetscInt_FMT ") != (%" PetscInt_FMT ",%" PetscInt_FMT ")", A->rmap->rstart, A->rmap->rend, B->rmap->rstart, B->rmap->rend);
355 
356   C->ops->transposematmultsymbolic = MatTransposeMatMultSymbolic_MPIAIJ_MPIDense;
357   C->ops->productsymbolic          = MatProductSymbolic_AtB;
358   PetscFunctionReturn(0);
359 }
360 
361 /* --------------------------------------------------------------------- */
362 PETSC_INTERN PetscErrorCode MatProductSetFromOptions_MPIAIJ_MPIDense(Mat C) {
363   Mat_Product *product = C->product;
364 
365   PetscFunctionBegin;
366   switch (product->type) {
367   case MATPRODUCT_AB: PetscCall(MatProductSetFromOptions_MPIAIJ_MPIDense_AB(C)); break;
368   case MATPRODUCT_AtB: PetscCall(MatProductSetFromOptions_MPIAIJ_MPIDense_AtB(C)); break;
369   default: break;
370   }
371   PetscFunctionReturn(0);
372 }
373 /* ------------------------------------------------------- */
374 
375 typedef struct {
376   Mat           workB, workB1;
377   MPI_Request  *rwaits, *swaits;
378   PetscInt      nsends, nrecvs;
379   MPI_Datatype *stype, *rtype;
380   PetscInt      blda;
381 } MPIAIJ_MPIDense;
382 
383 PetscErrorCode MatMPIAIJ_MPIDenseDestroy(void *ctx) {
384   MPIAIJ_MPIDense *contents = (MPIAIJ_MPIDense *)ctx;
385   PetscInt         i;
386 
387   PetscFunctionBegin;
388   PetscCall(MatDestroy(&contents->workB));
389   PetscCall(MatDestroy(&contents->workB1));
390   for (i = 0; i < contents->nsends; i++) { PetscCallMPI(MPI_Type_free(&contents->stype[i])); }
391   for (i = 0; i < contents->nrecvs; i++) { PetscCallMPI(MPI_Type_free(&contents->rtype[i])); }
392   PetscCall(PetscFree4(contents->stype, contents->rtype, contents->rwaits, contents->swaits));
393   PetscCall(PetscFree(contents));
394   PetscFunctionReturn(0);
395 }
396 
397 static PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat A, Mat B, PetscReal fill, Mat C) {
398   Mat_MPIAIJ      *aij = (Mat_MPIAIJ *)A->data;
399   PetscInt         nz  = aij->B->cmap->n, nsends, nrecvs, i, nrows_to, j, blda, m, M, n, N;
400   MPIAIJ_MPIDense *contents;
401   VecScatter       ctx = aij->Mvctx;
402   PetscInt         Am = A->rmap->n, Bm = B->rmap->n, BN = B->cmap->N, Bbn, Bbn1, bs, nrows_from, numBb;
403   MPI_Comm         comm;
404   MPI_Datatype     type1, *stype, *rtype;
405   const PetscInt  *sindices, *sstarts, *rstarts;
406   PetscMPIInt     *disp;
407   PetscBool        cisdense;
408 
409   PetscFunctionBegin;
410   MatCheckProduct(C, 4);
411   PetscCheck(!C->product->data, PetscObjectComm((PetscObject)C), PETSC_ERR_PLIB, "Product data not empty");
412   PetscCall(PetscObjectGetComm((PetscObject)A, &comm));
413   PetscCall(PetscObjectBaseTypeCompare((PetscObject)C, MATMPIDENSE, &cisdense));
414   if (!cisdense) { PetscCall(MatSetType(C, ((PetscObject)B)->type_name)); }
415   PetscCall(MatGetLocalSize(C, &m, &n));
416   PetscCall(MatGetSize(C, &M, &N));
417   if (m == PETSC_DECIDE || n == PETSC_DECIDE || M == PETSC_DECIDE || N == PETSC_DECIDE) { PetscCall(MatSetSizes(C, Am, B->cmap->n, A->rmap->N, BN)); }
418   PetscCall(MatSetBlockSizesFromMats(C, A, B));
419   PetscCall(MatSetUp(C));
420   PetscCall(MatDenseGetLDA(B, &blda));
421   PetscCall(PetscNew(&contents));
422 
423   PetscCall(VecScatterGetRemote_Private(ctx, PETSC_TRUE /*send*/, &nsends, &sstarts, &sindices, NULL, NULL));
424   PetscCall(VecScatterGetRemoteOrdered_Private(ctx, PETSC_FALSE /*recv*/, &nrecvs, &rstarts, NULL, NULL, NULL));
425 
426   /* Create column block of B and C for memory scalability when BN is too large */
427   /* Estimate Bbn, column size of Bb */
428   if (nz) {
429     Bbn1 = 2 * Am * BN / nz;
430     if (!Bbn1) Bbn1 = 1;
431   } else Bbn1 = BN;
432 
433   bs   = PetscAbs(B->cmap->bs);
434   Bbn1 = Bbn1 / bs * bs; /* Bbn1 is a multiple of bs */
435   if (Bbn1 > BN) Bbn1 = BN;
436   PetscCallMPI(MPI_Allreduce(&Bbn1, &Bbn, 1, MPIU_INT, MPI_MAX, comm));
437 
438   /* Enable runtime option for Bbn */
439   PetscOptionsBegin(comm, ((PetscObject)C)->prefix, "MatMatMult", "Mat");
440   PetscCall(PetscOptionsInt("-matmatmult_Bbn", "Number of columns in Bb", "MatMatMult", Bbn, &Bbn, NULL));
441   PetscOptionsEnd();
442   Bbn = PetscMin(Bbn, BN);
443 
444   if (Bbn > 0 && Bbn < BN) {
445     numBb = BN / Bbn;
446     Bbn1  = BN - numBb * Bbn;
447   } else numBb = 0;
448 
449   if (numBb) {
450     PetscCall(PetscInfo(C, "use Bb, BN=%" PetscInt_FMT ", Bbn=%" PetscInt_FMT "; numBb=%" PetscInt_FMT "\n", BN, Bbn, numBb));
451     if (Bbn1) { /* Create workB1 for the remaining columns */
452       PetscCall(PetscInfo(C, "use Bb1, BN=%" PetscInt_FMT ", Bbn1=%" PetscInt_FMT "\n", BN, Bbn1));
453       /* Create work matrix used to store off processor rows of B needed for local product */
454       PetscCall(MatCreateSeqDense(PETSC_COMM_SELF, nz, Bbn1, NULL, &contents->workB1));
455     } else contents->workB1 = NULL;
456   }
457 
458   /* Create work matrix used to store off processor rows of B needed for local product */
459   PetscCall(MatCreateSeqDense(PETSC_COMM_SELF, nz, Bbn, NULL, &contents->workB));
460 
461   /* Use MPI derived data type to reduce memory required by the send/recv buffers */
462   PetscCall(PetscMalloc4(nsends, &stype, nrecvs, &rtype, nrecvs, &contents->rwaits, nsends, &contents->swaits));
463   contents->stype  = stype;
464   contents->nsends = nsends;
465 
466   contents->rtype  = rtype;
467   contents->nrecvs = nrecvs;
468   contents->blda   = blda;
469 
470   PetscCall(PetscMalloc1(Bm + 1, &disp));
471   for (i = 0; i < nsends; i++) {
472     nrows_to = sstarts[i + 1] - sstarts[i];
473     for (j = 0; j < nrows_to; j++) disp[j] = sindices[sstarts[i] + j]; /* rowB to be sent */
474     PetscCallMPI(MPI_Type_create_indexed_block(nrows_to, 1, disp, MPIU_SCALAR, &type1));
475     PetscCallMPI(MPI_Type_create_resized(type1, 0, blda * sizeof(PetscScalar), &stype[i]));
476     PetscCallMPI(MPI_Type_commit(&stype[i]));
477     PetscCallMPI(MPI_Type_free(&type1));
478   }
479 
480   for (i = 0; i < nrecvs; i++) {
481     /* received values from a process form a (nrows_from x Bbn) row block in workB (column-wise) */
482     nrows_from = rstarts[i + 1] - rstarts[i];
483     disp[0]    = 0;
484     PetscCallMPI(MPI_Type_create_indexed_block(1, nrows_from, disp, MPIU_SCALAR, &type1));
485     PetscCallMPI(MPI_Type_create_resized(type1, 0, nz * sizeof(PetscScalar), &rtype[i]));
486     PetscCallMPI(MPI_Type_commit(&rtype[i]));
487     PetscCallMPI(MPI_Type_free(&type1));
488   }
489 
490   PetscCall(PetscFree(disp));
491   PetscCall(VecScatterRestoreRemote_Private(ctx, PETSC_TRUE /*send*/, &nsends, &sstarts, &sindices, NULL, NULL));
492   PetscCall(VecScatterRestoreRemoteOrdered_Private(ctx, PETSC_FALSE /*recv*/, &nrecvs, &rstarts, NULL, NULL, NULL));
493   PetscCall(MatSetOption(C, MAT_NO_OFF_PROC_ENTRIES, PETSC_TRUE));
494   PetscCall(MatAssemblyBegin(C, MAT_FINAL_ASSEMBLY));
495   PetscCall(MatAssemblyEnd(C, MAT_FINAL_ASSEMBLY));
496   PetscCall(MatSetOption(C, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
497 
498   C->product->data       = contents;
499   C->product->destroy    = MatMPIAIJ_MPIDenseDestroy;
500   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
501   PetscFunctionReturn(0);
502 }
503 
504 PETSC_INTERN PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat, Mat, Mat, const PetscBool);
505 /*
506     Performs an efficient scatter on the rows of B needed by this process; this is
507     a modification of the VecScatterBegin_() routines.
508 
509     Input: Bbidx = 0: B = Bb
510                  = 1: B = Bb1, see MatMatMultSymbolic_MPIAIJ_MPIDense()
511 */
512 PetscErrorCode              MatMPIDenseScatter(Mat A, Mat B, PetscInt Bbidx, Mat C, Mat *outworkB) {
513                Mat_MPIAIJ        *aij = (Mat_MPIAIJ *)A->data;
514                const PetscScalar *b;
515                PetscScalar       *rvalues;
516                VecScatter         ctx = aij->Mvctx;
517                const PetscInt    *sindices, *sstarts, *rstarts;
518                const PetscMPIInt *sprocs, *rprocs;
519                PetscInt           i, nsends, nrecvs;
520                MPI_Request       *swaits, *rwaits;
521                MPI_Comm           comm;
522                PetscMPIInt        tag = ((PetscObject)ctx)->tag, ncols = B->cmap->N, nrows = aij->B->cmap->n, nsends_mpi, nrecvs_mpi;
523                MPIAIJ_MPIDense   *contents;
524                Mat                workB;
525                MPI_Datatype      *stype, *rtype;
526                PetscInt           blda;
527 
528                PetscFunctionBegin;
529                MatCheckProduct(C, 4);
530                PetscCheck(C->product->data, PetscObjectComm((PetscObject)C), PETSC_ERR_PLIB, "Product data empty");
531                contents = (MPIAIJ_MPIDense *)C->product->data;
532                PetscCall(VecScatterGetRemote_Private(ctx, PETSC_TRUE /*send*/, &nsends, &sstarts, &sindices, &sprocs, NULL /*bs*/));
533                PetscCall(VecScatterGetRemoteOrdered_Private(ctx, PETSC_FALSE /*recv*/, &nrecvs, &rstarts, NULL, &rprocs, NULL /*bs*/));
534                PetscCall(PetscMPIIntCast(nsends, &nsends_mpi));
535                PetscCall(PetscMPIIntCast(nrecvs, &nrecvs_mpi));
536                if (Bbidx == 0) workB = *outworkB = contents->workB;
537   else workB = *outworkB = contents->workB1;
538   PetscCheck(nrows == workB->rmap->n, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Number of rows of workB %" PetscInt_FMT " not equal to columns of aij->B %d", workB->cmap->n, nrows);
539                swaits = contents->swaits;
540                rwaits = contents->rwaits;
541 
542                PetscCall(MatDenseGetArrayRead(B, &b));
543                PetscCall(MatDenseGetLDA(B, &blda));
544                PetscCheck(blda == contents->blda, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Cannot reuse an input matrix with lda %" PetscInt_FMT " != %" PetscInt_FMT, blda, contents->blda);
545                PetscCall(MatDenseGetArray(workB, &rvalues));
546 
547                /* Post recv, use MPI derived data type to save memory */
548                PetscCall(PetscObjectGetComm((PetscObject)C, &comm));
549                rtype = contents->rtype;
550                for (i = 0; i < nrecvs; i++) { PetscCallMPI(MPI_Irecv(rvalues + (rstarts[i] - rstarts[0]), ncols, rtype[i], rprocs[i], tag, comm, rwaits + i)); }
551 
552                stype = contents->stype;
553                for (i = 0; i < nsends; i++) { PetscCallMPI(MPI_Isend(b, ncols, stype[i], sprocs[i], tag, comm, swaits + i)); }
554 
555                if (nrecvs) PetscCallMPI(MPI_Waitall(nrecvs_mpi, rwaits, MPI_STATUSES_IGNORE));
556   if (nsends) PetscCallMPI(MPI_Waitall(nsends_mpi, swaits, MPI_STATUSES_IGNORE));
557 
558   PetscCall(VecScatterRestoreRemote_Private(ctx, PETSC_TRUE /*send*/, &nsends, &sstarts, &sindices, &sprocs, NULL));
559                PetscCall(VecScatterRestoreRemoteOrdered_Private(ctx, PETSC_FALSE /*recv*/, &nrecvs, &rstarts, NULL, &rprocs, NULL));
560                PetscCall(MatDenseRestoreArrayRead(B, &b));
561                PetscCall(MatDenseRestoreArray(workB, &rvalues));
562                PetscFunctionReturn(0);
563 }
564 
565 static PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat A, Mat B, Mat C) {
566   Mat_MPIAIJ      *aij    = (Mat_MPIAIJ *)A->data;
567   Mat_MPIDense    *bdense = (Mat_MPIDense *)B->data;
568   Mat_MPIDense    *cdense = (Mat_MPIDense *)C->data;
569   Mat              workB;
570   MPIAIJ_MPIDense *contents;
571 
572   PetscFunctionBegin;
573   MatCheckProduct(C, 3);
574   PetscCheck(C->product->data, PetscObjectComm((PetscObject)C), PETSC_ERR_PLIB, "Product data empty");
575   contents = (MPIAIJ_MPIDense *)C->product->data;
576   /* diagonal block of A times all local rows of B */
577   /* TODO: this calls a symbolic multiplication every time, which could be avoided */
578   PetscCall(MatMatMult(aij->A, bdense->A, MAT_REUSE_MATRIX, PETSC_DEFAULT, &cdense->A));
579   if (contents->workB->cmap->n == B->cmap->N) {
580     /* get off processor parts of B needed to complete C=A*B */
581     PetscCall(MatMPIDenseScatter(A, B, 0, C, &workB));
582 
583     /* off-diagonal block of A times nonlocal rows of B */
584     PetscCall(MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B, workB, cdense->A, PETSC_TRUE));
585   } else {
586     Mat       Bb, Cb;
587     PetscInt  BN = B->cmap->N, n = contents->workB->cmap->n, i;
588     PetscBool ccpu;
589 
590     PetscCheck(n > 0, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Column block size %" PetscInt_FMT " must be positive", n);
591     /* Prevent from unneeded copies back and forth from the GPU
592        when getting and restoring the submatrix
593        We need a proper GPU code for AIJ * dense in parallel */
594     PetscCall(MatBoundToCPU(C, &ccpu));
595     PetscCall(MatBindToCPU(C, PETSC_TRUE));
596     for (i = 0; i < BN; i += n) {
597       PetscCall(MatDenseGetSubMatrix(B, PETSC_DECIDE, PETSC_DECIDE, i, PetscMin(i + n, BN), &Bb));
598       PetscCall(MatDenseGetSubMatrix(C, PETSC_DECIDE, PETSC_DECIDE, i, PetscMin(i + n, BN), &Cb));
599 
600       /* get off processor parts of B needed to complete C=A*B */
601       PetscCall(MatMPIDenseScatter(A, Bb, (i + n) > BN, C, &workB));
602 
603       /* off-diagonal block of A times nonlocal rows of B */
604       cdense = (Mat_MPIDense *)Cb->data;
605       PetscCall(MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B, workB, cdense->A, PETSC_TRUE));
606       PetscCall(MatDenseRestoreSubMatrix(B, &Bb));
607       PetscCall(MatDenseRestoreSubMatrix(C, &Cb));
608     }
609     PetscCall(MatBindToCPU(C, ccpu));
610   }
611   PetscFunctionReturn(0);
612 }
613 
614 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ(Mat A, Mat P, Mat C) {
615   Mat_MPIAIJ        *a = (Mat_MPIAIJ *)A->data, *c = (Mat_MPIAIJ *)C->data;
616   Mat_SeqAIJ        *ad = (Mat_SeqAIJ *)(a->A)->data, *ao = (Mat_SeqAIJ *)(a->B)->data;
617   Mat_SeqAIJ        *cd = (Mat_SeqAIJ *)(c->A)->data, *co = (Mat_SeqAIJ *)(c->B)->data;
618   PetscInt          *adi = ad->i, *adj, *aoi = ao->i, *aoj;
619   PetscScalar       *ada, *aoa, *cda = cd->a, *coa = co->a;
620   Mat_SeqAIJ        *p_loc, *p_oth;
621   PetscInt          *pi_loc, *pj_loc, *pi_oth, *pj_oth, *pj;
622   PetscScalar       *pa_loc, *pa_oth, *pa, valtmp, *ca;
623   PetscInt           cm = C->rmap->n, anz, pnz;
624   Mat_APMPI         *ptap;
625   PetscScalar       *apa_sparse;
626   const PetscScalar *dummy;
627   PetscInt          *api, *apj, *apJ, i, j, k, row;
628   PetscInt           cstart = C->cmap->rstart;
629   PetscInt           cdnz, conz, k0, k1, nextp;
630   MPI_Comm           comm;
631   PetscMPIInt        size;
632 
633   PetscFunctionBegin;
634   MatCheckProduct(C, 3);
635   ptap = (Mat_APMPI *)C->product->data;
636   PetscCheck(ptap, PetscObjectComm((PetscObject)C), PETSC_ERR_ARG_WRONGSTATE, "PtAP cannot be computed. Missing data");
637   PetscCall(PetscObjectGetComm((PetscObject)C, &comm));
638   PetscCallMPI(MPI_Comm_size(comm, &size));
639   PetscCheck(ptap->P_oth || size <= 1, PetscObjectComm((PetscObject)C), PETSC_ERR_ARG_WRONGSTATE, "AP cannot be reused. Do not call MatProductClear()");
640 
641   /* flag CPU mask for C */
642 #if defined(PETSC_HAVE_DEVICE)
643   if (C->offloadmask != PETSC_OFFLOAD_UNALLOCATED) C->offloadmask = PETSC_OFFLOAD_CPU;
644   if (c->A->offloadmask != PETSC_OFFLOAD_UNALLOCATED) c->A->offloadmask = PETSC_OFFLOAD_CPU;
645   if (c->B->offloadmask != PETSC_OFFLOAD_UNALLOCATED) c->B->offloadmask = PETSC_OFFLOAD_CPU;
646 #endif
647   apa_sparse = ptap->apa;
648 
649   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
650   /*-----------------------------------------------------*/
651   /* update numerical values of P_oth and P_loc */
652   PetscCall(MatGetBrowsOfAoCols_MPIAIJ(A, P, MAT_REUSE_MATRIX, &ptap->startsj_s, &ptap->startsj_r, &ptap->bufa, &ptap->P_oth));
653   PetscCall(MatMPIAIJGetLocalMat(P, MAT_REUSE_MATRIX, &ptap->P_loc));
654 
655   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
656   /*----------------------------------------------------------*/
657   /* get data from symbolic products */
658   p_loc  = (Mat_SeqAIJ *)(ptap->P_loc)->data;
659   pi_loc = p_loc->i;
660   pj_loc = p_loc->j;
661   pa_loc = p_loc->a;
662   if (size > 1) {
663     p_oth  = (Mat_SeqAIJ *)(ptap->P_oth)->data;
664     pi_oth = p_oth->i;
665     pj_oth = p_oth->j;
666     pa_oth = p_oth->a;
667   } else {
668     p_oth  = NULL;
669     pi_oth = NULL;
670     pj_oth = NULL;
671     pa_oth = NULL;
672   }
673 
674   /* trigger copy to CPU */
675   PetscCall(MatSeqAIJGetArrayRead(a->A, &dummy));
676   PetscCall(MatSeqAIJRestoreArrayRead(a->A, &dummy));
677   PetscCall(MatSeqAIJGetArrayRead(a->B, &dummy));
678   PetscCall(MatSeqAIJRestoreArrayRead(a->B, &dummy));
679   api = ptap->api;
680   apj = ptap->apj;
681   for (i = 0; i < cm; i++) {
682     apJ = apj + api[i];
683 
684     /* diagonal portion of A */
685     anz = adi[i + 1] - adi[i];
686     adj = ad->j + adi[i];
687     ada = ad->a + adi[i];
688     for (j = 0; j < anz; j++) {
689       row    = adj[j];
690       pnz    = pi_loc[row + 1] - pi_loc[row];
691       pj     = pj_loc + pi_loc[row];
692       pa     = pa_loc + pi_loc[row];
693       /* perform sparse axpy */
694       valtmp = ada[j];
695       nextp  = 0;
696       for (k = 0; nextp < pnz; k++) {
697         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
698           apa_sparse[k] += valtmp * pa[nextp++];
699         }
700       }
701       PetscCall(PetscLogFlops(2.0 * pnz));
702     }
703 
704     /* off-diagonal portion of A */
705     anz = aoi[i + 1] - aoi[i];
706     aoj = ao->j + aoi[i];
707     aoa = ao->a + aoi[i];
708     for (j = 0; j < anz; j++) {
709       row    = aoj[j];
710       pnz    = pi_oth[row + 1] - pi_oth[row];
711       pj     = pj_oth + pi_oth[row];
712       pa     = pa_oth + pi_oth[row];
713       /* perform sparse axpy */
714       valtmp = aoa[j];
715       nextp  = 0;
716       for (k = 0; nextp < pnz; k++) {
717         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
718           apa_sparse[k] += valtmp * pa[nextp++];
719         }
720       }
721       PetscCall(PetscLogFlops(2.0 * pnz));
722     }
723 
724     /* set values in C */
725     cdnz = cd->i[i + 1] - cd->i[i];
726     conz = co->i[i + 1] - co->i[i];
727 
728     /* 1st off-diagonal part of C */
729     ca = coa + co->i[i];
730     k  = 0;
731     for (k0 = 0; k0 < conz; k0++) {
732       if (apJ[k] >= cstart) break;
733       ca[k0]        = apa_sparse[k];
734       apa_sparse[k] = 0.0;
735       k++;
736     }
737 
738     /* diagonal part of C */
739     ca = cda + cd->i[i];
740     for (k1 = 0; k1 < cdnz; k1++) {
741       ca[k1]        = apa_sparse[k];
742       apa_sparse[k] = 0.0;
743       k++;
744     }
745 
746     /* 2nd off-diagonal part of C */
747     ca = coa + co->i[i];
748     for (; k0 < conz; k0++) {
749       ca[k0]        = apa_sparse[k];
750       apa_sparse[k] = 0.0;
751       k++;
752     }
753   }
754   PetscCall(MatAssemblyBegin(C, MAT_FINAL_ASSEMBLY));
755   PetscCall(MatAssemblyEnd(C, MAT_FINAL_ASSEMBLY));
756   PetscFunctionReturn(0);
757 }
758 
759 /* same as MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(), except using LLCondensed to avoid O(BN) memory requirement */
760 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ(Mat A, Mat P, PetscReal fill, Mat C) {
761   MPI_Comm           comm;
762   PetscMPIInt        size;
763   Mat_APMPI         *ptap;
764   PetscFreeSpaceList free_space = NULL, current_space = NULL;
765   Mat_MPIAIJ        *a  = (Mat_MPIAIJ *)A->data;
766   Mat_SeqAIJ        *ad = (Mat_SeqAIJ *)(a->A)->data, *ao = (Mat_SeqAIJ *)(a->B)->data, *p_loc, *p_oth;
767   PetscInt          *pi_loc, *pj_loc, *pi_oth, *pj_oth, *dnz, *onz;
768   PetscInt          *adi = ad->i, *adj = ad->j, *aoi = ao->i, *aoj = ao->j, rstart = A->rmap->rstart;
769   PetscInt           i, pnz, row, *api, *apj, *Jptr, apnz, nspacedouble = 0, j, nzi, *lnk, apnz_max = 1;
770   PetscInt           am = A->rmap->n, pn = P->cmap->n, pm = P->rmap->n, lsize = pn + 20;
771   PetscReal          afill;
772   MatType            mtype;
773 
774   PetscFunctionBegin;
775   MatCheckProduct(C, 4);
776   PetscCheck(!C->product->data, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Extra product struct not empty");
777   PetscCall(PetscObjectGetComm((PetscObject)A, &comm));
778   PetscCallMPI(MPI_Comm_size(comm, &size));
779 
780   /* create struct Mat_APMPI and attached it to C later */
781   PetscCall(PetscNew(&ptap));
782 
783   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
784   PetscCall(MatGetBrowsOfAoCols_MPIAIJ(A, P, MAT_INITIAL_MATRIX, &ptap->startsj_s, &ptap->startsj_r, &ptap->bufa, &ptap->P_oth));
785 
786   /* get P_loc by taking all local rows of P */
787   PetscCall(MatMPIAIJGetLocalMat(P, MAT_INITIAL_MATRIX, &ptap->P_loc));
788 
789   p_loc  = (Mat_SeqAIJ *)(ptap->P_loc)->data;
790   pi_loc = p_loc->i;
791   pj_loc = p_loc->j;
792   if (size > 1) {
793     p_oth  = (Mat_SeqAIJ *)(ptap->P_oth)->data;
794     pi_oth = p_oth->i;
795     pj_oth = p_oth->j;
796   } else {
797     p_oth  = NULL;
798     pi_oth = NULL;
799     pj_oth = NULL;
800   }
801 
802   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
803   /*-------------------------------------------------------------------*/
804   PetscCall(PetscMalloc1(am + 2, &api));
805   ptap->api = api;
806   api[0]    = 0;
807 
808   PetscCall(PetscLLCondensedCreate_Scalable(lsize, &lnk));
809 
810   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
811   PetscCall(PetscFreeSpaceGet(PetscRealIntMultTruncate(fill, PetscIntSumTruncate(adi[am], PetscIntSumTruncate(aoi[am], pi_loc[pm]))), &free_space));
812   current_space = free_space;
813   MatPreallocateBegin(comm, am, pn, dnz, onz);
814   for (i = 0; i < am; i++) {
815     /* diagonal portion of A */
816     nzi = adi[i + 1] - adi[i];
817     for (j = 0; j < nzi; j++) {
818       row  = *adj++;
819       pnz  = pi_loc[row + 1] - pi_loc[row];
820       Jptr = pj_loc + pi_loc[row];
821       /* Expand list if it is not long enough */
822       if (pnz + apnz_max > lsize) {
823         lsize = pnz + apnz_max;
824         PetscCall(PetscLLCondensedExpand_Scalable(lsize, &lnk));
825       }
826       /* add non-zero cols of P into the sorted linked list lnk */
827       PetscCall(PetscLLCondensedAddSorted_Scalable(pnz, Jptr, lnk));
828       apnz       = *lnk; /* The first element in the list is the number of items in the list */
829       api[i + 1] = api[i] + apnz;
830       if (apnz > apnz_max) apnz_max = apnz + 1; /* '1' for diagonal entry */
831     }
832     /* off-diagonal portion of A */
833     nzi = aoi[i + 1] - aoi[i];
834     for (j = 0; j < nzi; j++) {
835       row  = *aoj++;
836       pnz  = pi_oth[row + 1] - pi_oth[row];
837       Jptr = pj_oth + pi_oth[row];
838       /* Expand list if it is not long enough */
839       if (pnz + apnz_max > lsize) {
840         lsize = pnz + apnz_max;
841         PetscCall(PetscLLCondensedExpand_Scalable(lsize, &lnk));
842       }
843       /* add non-zero cols of P into the sorted linked list lnk */
844       PetscCall(PetscLLCondensedAddSorted_Scalable(pnz, Jptr, lnk));
845       apnz       = *lnk; /* The first element in the list is the number of items in the list */
846       api[i + 1] = api[i] + apnz;
847       if (apnz > apnz_max) apnz_max = apnz + 1; /* '1' for diagonal entry */
848     }
849 
850     /* add missing diagonal entry */
851     if (C->force_diagonals) {
852       j = i + rstart; /* column index */
853       PetscCall(PetscLLCondensedAddSorted_Scalable(1, &j, lnk));
854     }
855 
856     apnz       = *lnk;
857     api[i + 1] = api[i] + apnz;
858     if (apnz > apnz_max) apnz_max = apnz;
859 
860     /* if free space is not available, double the total space in the list */
861     if (current_space->local_remaining < apnz) {
862       PetscCall(PetscFreeSpaceGet(PetscIntSumTruncate(apnz, current_space->total_array_size), &current_space));
863       nspacedouble++;
864     }
865 
866     /* Copy data into free space, then initialize lnk */
867     PetscCall(PetscLLCondensedClean_Scalable(apnz, current_space->array, lnk));
868     PetscCall(MatPreallocateSet(i + rstart, apnz, current_space->array, dnz, onz));
869 
870     current_space->array += apnz;
871     current_space->local_used += apnz;
872     current_space->local_remaining -= apnz;
873   }
874 
875   /* Allocate space for apj, initialize apj, and */
876   /* destroy list of free space and other temporary array(s) */
877   PetscCall(PetscMalloc1(api[am] + 1, &ptap->apj));
878   apj = ptap->apj;
879   PetscCall(PetscFreeSpaceContiguous(&free_space, ptap->apj));
880   PetscCall(PetscLLCondensedDestroy_Scalable(lnk));
881 
882   /* create and assemble symbolic parallel matrix C */
883   /*----------------------------------------------------*/
884   PetscCall(MatSetSizes(C, am, pn, PETSC_DETERMINE, PETSC_DETERMINE));
885   PetscCall(MatSetBlockSizesFromMats(C, A, P));
886   PetscCall(MatGetType(A, &mtype));
887   PetscCall(MatSetType(C, mtype));
888   PetscCall(MatMPIAIJSetPreallocation(C, 0, dnz, 0, onz));
889   MatPreallocateEnd(dnz, onz);
890 
891   /* malloc apa for assembly C */
892   PetscCall(PetscCalloc1(apnz_max, &ptap->apa));
893 
894   PetscCall(MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(C, apj, api));
895   PetscCall(MatAssemblyBegin(C, MAT_FINAL_ASSEMBLY));
896   PetscCall(MatAssemblyEnd(C, MAT_FINAL_ASSEMBLY));
897   PetscCall(MatSetOption(C, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
898 
899   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ;
900   C->ops->productnumeric = MatProductNumeric_AB;
901 
902   /* attach the supporting struct to C for reuse */
903   C->product->data    = ptap;
904   C->product->destroy = MatDestroy_MPIAIJ_MatMatMult;
905 
906   /* set MatInfo */
907   afill = (PetscReal)api[am] / (adi[am] + aoi[am] + pi_loc[pm] + 1) + 1.e-5;
908   if (afill < 1.0) afill = 1.0;
909   C->info.mallocs           = nspacedouble;
910   C->info.fill_ratio_given  = fill;
911   C->info.fill_ratio_needed = afill;
912 
913 #if defined(PETSC_USE_INFO)
914   if (api[am]) {
915     PetscCall(PetscInfo(C, "Reallocs %" PetscInt_FMT "; Fill ratio: given %g needed %g.\n", nspacedouble, (double)fill, (double)afill));
916     PetscCall(PetscInfo(C, "Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n", (double)afill));
917   } else {
918     PetscCall(PetscInfo(C, "Empty matrix product\n"));
919   }
920 #endif
921   PetscFunctionReturn(0);
922 }
923 
924 /* This function is needed for the seqMPI matrix-matrix multiplication.  */
925 /* Three input arrays are merged to one output array. The size of the    */
926 /* output array is also output. Duplicate entries only show up once.     */
927 static void Merge3SortedArrays(PetscInt size1, PetscInt *in1, PetscInt size2, PetscInt *in2, PetscInt size3, PetscInt *in3, PetscInt *size4, PetscInt *out) {
928   int i = 0, j = 0, k = 0, l = 0;
929 
930   /* Traverse all three arrays */
931   while (i < size1 && j < size2 && k < size3) {
932     if (in1[i] < in2[j] && in1[i] < in3[k]) {
933       out[l++] = in1[i++];
934     } else if (in2[j] < in1[i] && in2[j] < in3[k]) {
935       out[l++] = in2[j++];
936     } else if (in3[k] < in1[i] && in3[k] < in2[j]) {
937       out[l++] = in3[k++];
938     } else if (in1[i] == in2[j] && in1[i] < in3[k]) {
939       out[l++] = in1[i];
940       i++, j++;
941     } else if (in1[i] == in3[k] && in1[i] < in2[j]) {
942       out[l++] = in1[i];
943       i++, k++;
944     } else if (in3[k] == in2[j] && in2[j] < in1[i]) {
945       out[l++] = in2[j];
946       k++, j++;
947     } else if (in1[i] == in2[j] && in1[i] == in3[k]) {
948       out[l++] = in1[i];
949       i++, j++, k++;
950     }
951   }
952 
953   /* Traverse two remaining arrays */
954   while (i < size1 && j < size2) {
955     if (in1[i] < in2[j]) {
956       out[l++] = in1[i++];
957     } else if (in1[i] > in2[j]) {
958       out[l++] = in2[j++];
959     } else {
960       out[l++] = in1[i];
961       i++, j++;
962     }
963   }
964 
965   while (i < size1 && k < size3) {
966     if (in1[i] < in3[k]) {
967       out[l++] = in1[i++];
968     } else if (in1[i] > in3[k]) {
969       out[l++] = in3[k++];
970     } else {
971       out[l++] = in1[i];
972       i++, k++;
973     }
974   }
975 
976   while (k < size3 && j < size2) {
977     if (in3[k] < in2[j]) {
978       out[l++] = in3[k++];
979     } else if (in3[k] > in2[j]) {
980       out[l++] = in2[j++];
981     } else {
982       out[l++] = in3[k];
983       k++, j++;
984     }
985   }
986 
987   /* Traverse one remaining array */
988   while (i < size1) out[l++] = in1[i++];
989   while (j < size2) out[l++] = in2[j++];
990   while (k < size3) out[l++] = in3[k++];
991 
992   *size4 = l;
993 }
994 
995 /* This matrix-matrix multiplication algorithm divides the multiplication into three multiplications and  */
996 /* adds up the products. Two of these three multiplications are performed with existing (sequential)      */
997 /* matrix-matrix multiplications.  */
998 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_seqMPI(Mat A, Mat P, PetscReal fill, Mat C) {
999   MPI_Comm           comm;
1000   PetscMPIInt        size;
1001   Mat_APMPI         *ptap;
1002   PetscFreeSpaceList free_space_diag = NULL, current_space = NULL;
1003   Mat_MPIAIJ        *a  = (Mat_MPIAIJ *)A->data;
1004   Mat_SeqAIJ        *ad = (Mat_SeqAIJ *)(a->A)->data, *ao = (Mat_SeqAIJ *)(a->B)->data, *p_loc;
1005   Mat_MPIAIJ        *p = (Mat_MPIAIJ *)P->data;
1006   Mat_SeqAIJ        *adpd_seq, *p_off, *aopoth_seq;
1007   PetscInt           adponz, adpdnz;
1008   PetscInt          *pi_loc, *dnz, *onz;
1009   PetscInt          *adi = ad->i, *adj = ad->j, *aoi = ao->i, rstart = A->rmap->rstart;
1010   PetscInt          *lnk, i, i1 = 0, pnz, row, *adpoi, *adpoj, *api, *adpoJ, *aopJ, *apJ, *Jptr, aopnz, nspacedouble = 0, j, nzi, *apj, apnz, *adpdi, *adpdj, *adpdJ, *poff_i, *poff_j, *j_temp, *aopothi, *aopothj;
1011   PetscInt           am = A->rmap->n, pN = P->cmap->N, pn = P->cmap->n, pm = P->rmap->n, p_colstart, p_colend;
1012   PetscBT            lnkbt;
1013   PetscReal          afill;
1014   PetscMPIInt        rank;
1015   Mat                adpd, aopoth;
1016   MatType            mtype;
1017   const char        *prefix;
1018 
1019   PetscFunctionBegin;
1020   MatCheckProduct(C, 4);
1021   PetscCheck(!C->product->data, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Extra product struct not empty");
1022   PetscCall(PetscObjectGetComm((PetscObject)A, &comm));
1023   PetscCallMPI(MPI_Comm_size(comm, &size));
1024   PetscCallMPI(MPI_Comm_rank(comm, &rank));
1025   PetscCall(MatGetOwnershipRangeColumn(P, &p_colstart, &p_colend));
1026 
1027   /* create struct Mat_APMPI and attached it to C later */
1028   PetscCall(PetscNew(&ptap));
1029 
1030   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
1031   PetscCall(MatGetBrowsOfAoCols_MPIAIJ(A, P, MAT_INITIAL_MATRIX, &ptap->startsj_s, &ptap->startsj_r, &ptap->bufa, &ptap->P_oth));
1032 
1033   /* get P_loc by taking all local rows of P */
1034   PetscCall(MatMPIAIJGetLocalMat(P, MAT_INITIAL_MATRIX, &ptap->P_loc));
1035 
1036   p_loc  = (Mat_SeqAIJ *)(ptap->P_loc)->data;
1037   pi_loc = p_loc->i;
1038 
1039   /* Allocate memory for the i arrays of the matrices A*P, A_diag*P_off and A_offd * P */
1040   PetscCall(PetscMalloc1(am + 2, &api));
1041   PetscCall(PetscMalloc1(am + 2, &adpoi));
1042 
1043   adpoi[0]  = 0;
1044   ptap->api = api;
1045   api[0]    = 0;
1046 
1047   /* create and initialize a linked list, will be used for both A_diag * P_loc_off and A_offd * P_oth */
1048   PetscCall(PetscLLCondensedCreate(pN, pN, &lnk, &lnkbt));
1049   MatPreallocateBegin(comm, am, pn, dnz, onz);
1050 
1051   /* Symbolic calc of A_loc_diag * P_loc_diag */
1052   PetscCall(MatGetOptionsPrefix(A, &prefix));
1053   PetscCall(MatProductCreate(a->A, p->A, NULL, &adpd));
1054   PetscCall(MatGetOptionsPrefix(A, &prefix));
1055   PetscCall(MatSetOptionsPrefix(adpd, prefix));
1056   PetscCall(MatAppendOptionsPrefix(adpd, "inner_diag_"));
1057 
1058   PetscCall(MatProductSetType(adpd, MATPRODUCT_AB));
1059   PetscCall(MatProductSetAlgorithm(adpd, "sorted"));
1060   PetscCall(MatProductSetFill(adpd, fill));
1061   PetscCall(MatProductSetFromOptions(adpd));
1062 
1063   adpd->force_diagonals = C->force_diagonals;
1064   PetscCall(MatProductSymbolic(adpd));
1065 
1066   adpd_seq = (Mat_SeqAIJ *)((adpd)->data);
1067   adpdi    = adpd_seq->i;
1068   adpdj    = adpd_seq->j;
1069   p_off    = (Mat_SeqAIJ *)((p->B)->data);
1070   poff_i   = p_off->i;
1071   poff_j   = p_off->j;
1072 
1073   /* j_temp stores indices of a result row before they are added to the linked list */
1074   PetscCall(PetscMalloc1(pN + 2, &j_temp));
1075 
1076   /* Symbolic calc of the A_diag * p_loc_off */
1077   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
1078   PetscCall(PetscFreeSpaceGet(PetscRealIntMultTruncate(fill, PetscIntSumTruncate(adi[am], PetscIntSumTruncate(aoi[am], pi_loc[pm]))), &free_space_diag));
1079   current_space = free_space_diag;
1080 
1081   for (i = 0; i < am; i++) {
1082     /* A_diag * P_loc_off */
1083     nzi = adi[i + 1] - adi[i];
1084     for (j = 0; j < nzi; j++) {
1085       row  = *adj++;
1086       pnz  = poff_i[row + 1] - poff_i[row];
1087       Jptr = poff_j + poff_i[row];
1088       for (i1 = 0; i1 < pnz; i1++) { j_temp[i1] = p->garray[Jptr[i1]]; }
1089       /* add non-zero cols of P into the sorted linked list lnk */
1090       PetscCall(PetscLLCondensedAddSorted(pnz, j_temp, lnk, lnkbt));
1091     }
1092 
1093     adponz       = lnk[0];
1094     adpoi[i + 1] = adpoi[i] + adponz;
1095 
1096     /* if free space is not available, double the total space in the list */
1097     if (current_space->local_remaining < adponz) {
1098       PetscCall(PetscFreeSpaceGet(PetscIntSumTruncate(adponz, current_space->total_array_size), &current_space));
1099       nspacedouble++;
1100     }
1101 
1102     /* Copy data into free space, then initialize lnk */
1103     PetscCall(PetscLLCondensedClean(pN, adponz, current_space->array, lnk, lnkbt));
1104 
1105     current_space->array += adponz;
1106     current_space->local_used += adponz;
1107     current_space->local_remaining -= adponz;
1108   }
1109 
1110   /* Symbolic calc of A_off * P_oth */
1111   PetscCall(MatSetOptionsPrefix(a->B, prefix));
1112   PetscCall(MatAppendOptionsPrefix(a->B, "inner_offdiag_"));
1113   PetscCall(MatCreate(PETSC_COMM_SELF, &aopoth));
1114   PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ(a->B, ptap->P_oth, fill, aopoth));
1115   aopoth_seq = (Mat_SeqAIJ *)((aopoth)->data);
1116   aopothi    = aopoth_seq->i;
1117   aopothj    = aopoth_seq->j;
1118 
1119   /* Allocate space for apj, adpj, aopj, ... */
1120   /* destroy lists of free space and other temporary array(s) */
1121 
1122   PetscCall(PetscMalloc1(aopothi[am] + adpoi[am] + adpdi[am] + 2, &ptap->apj));
1123   PetscCall(PetscMalloc1(adpoi[am] + 2, &adpoj));
1124 
1125   /* Copy from linked list to j-array */
1126   PetscCall(PetscFreeSpaceContiguous(&free_space_diag, adpoj));
1127   PetscCall(PetscLLDestroy(lnk, lnkbt));
1128 
1129   adpoJ = adpoj;
1130   adpdJ = adpdj;
1131   aopJ  = aopothj;
1132   apj   = ptap->apj;
1133   apJ   = apj; /* still empty */
1134 
1135   /* Merge j-arrays of A_off * P, A_diag * P_loc_off, and */
1136   /* A_diag * P_loc_diag to get A*P */
1137   for (i = 0; i < am; i++) {
1138     aopnz  = aopothi[i + 1] - aopothi[i];
1139     adponz = adpoi[i + 1] - adpoi[i];
1140     adpdnz = adpdi[i + 1] - adpdi[i];
1141 
1142     /* Correct indices from A_diag*P_diag */
1143     for (i1 = 0; i1 < adpdnz; i1++) { adpdJ[i1] += p_colstart; }
1144     /* Merge j-arrays of A_diag * P_loc_off and A_diag * P_loc_diag and A_off * P_oth */
1145     Merge3SortedArrays(adponz, adpoJ, adpdnz, adpdJ, aopnz, aopJ, &apnz, apJ);
1146     PetscCall(MatPreallocateSet(i + rstart, apnz, apJ, dnz, onz));
1147 
1148     aopJ += aopnz;
1149     adpoJ += adponz;
1150     adpdJ += adpdnz;
1151     apJ += apnz;
1152     api[i + 1] = api[i] + apnz;
1153   }
1154 
1155   /* malloc apa to store dense row A[i,:]*P */
1156   PetscCall(PetscCalloc1(pN + 2, &ptap->apa));
1157 
1158   /* create and assemble symbolic parallel matrix C */
1159   PetscCall(MatSetSizes(C, am, pn, PETSC_DETERMINE, PETSC_DETERMINE));
1160   PetscCall(MatSetBlockSizesFromMats(C, A, P));
1161   PetscCall(MatGetType(A, &mtype));
1162   PetscCall(MatSetType(C, mtype));
1163   PetscCall(MatMPIAIJSetPreallocation(C, 0, dnz, 0, onz));
1164   MatPreallocateEnd(dnz, onz);
1165 
1166   PetscCall(MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(C, apj, api));
1167   PetscCall(MatAssemblyBegin(C, MAT_FINAL_ASSEMBLY));
1168   PetscCall(MatAssemblyEnd(C, MAT_FINAL_ASSEMBLY));
1169   PetscCall(MatSetOption(C, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
1170 
1171   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1172   C->ops->productnumeric = MatProductNumeric_AB;
1173 
1174   /* attach the supporting struct to C for reuse */
1175   C->product->data    = ptap;
1176   C->product->destroy = MatDestroy_MPIAIJ_MatMatMult;
1177 
1178   /* set MatInfo */
1179   afill = (PetscReal)api[am] / (adi[am] + aoi[am] + pi_loc[pm] + 1) + 1.e-5;
1180   if (afill < 1.0) afill = 1.0;
1181   C->info.mallocs           = nspacedouble;
1182   C->info.fill_ratio_given  = fill;
1183   C->info.fill_ratio_needed = afill;
1184 
1185 #if defined(PETSC_USE_INFO)
1186   if (api[am]) {
1187     PetscCall(PetscInfo(C, "Reallocs %" PetscInt_FMT "; Fill ratio: given %g needed %g.\n", nspacedouble, (double)fill, (double)afill));
1188     PetscCall(PetscInfo(C, "Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n", (double)afill));
1189   } else {
1190     PetscCall(PetscInfo(C, "Empty matrix product\n"));
1191   }
1192 #endif
1193 
1194   PetscCall(MatDestroy(&aopoth));
1195   PetscCall(MatDestroy(&adpd));
1196   PetscCall(PetscFree(j_temp));
1197   PetscCall(PetscFree(adpoj));
1198   PetscCall(PetscFree(adpoi));
1199   PetscFunctionReturn(0);
1200 }
1201 
1202 /*-------------------------------------------------------------------------*/
1203 /* This routine only works when scall=MAT_REUSE_MATRIX! */
1204 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult(Mat P, Mat A, Mat C) {
1205   Mat_APMPI *ptap;
1206   Mat        Pt;
1207 
1208   PetscFunctionBegin;
1209   MatCheckProduct(C, 3);
1210   ptap = (Mat_APMPI *)C->product->data;
1211   PetscCheck(ptap, PetscObjectComm((PetscObject)C), PETSC_ERR_ARG_WRONGSTATE, "PtAP cannot be computed. Missing data");
1212   PetscCheck(ptap->Pt, PetscObjectComm((PetscObject)C), PETSC_ERR_ARG_WRONGSTATE, "PtA cannot be reused. Do not call MatProductClear()");
1213 
1214   Pt = ptap->Pt;
1215   PetscCall(MatTransposeSetPrecursor(P, Pt));
1216   PetscCall(MatTranspose(P, MAT_REUSE_MATRIX, &Pt));
1217   PetscCall(MatMatMultNumeric_MPIAIJ_MPIAIJ(Pt, A, C));
1218   PetscFunctionReturn(0);
1219 }
1220 
1221 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1222 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat P, Mat A, PetscReal fill, Mat C) {
1223   Mat_APMPI               *ptap;
1224   Mat_MPIAIJ              *p = (Mat_MPIAIJ *)P->data;
1225   MPI_Comm                 comm;
1226   PetscMPIInt              size, rank;
1227   PetscFreeSpaceList       free_space = NULL, current_space = NULL;
1228   PetscInt                 pn = P->cmap->n, aN = A->cmap->N, an = A->cmap->n;
1229   PetscInt                *lnk, i, k, nsend, rstart;
1230   PetscBT                  lnkbt;
1231   PetscMPIInt              tagi, tagj, *len_si, *len_s, *len_ri, nrecv;
1232   PETSC_UNUSED PetscMPIInt icompleted = 0;
1233   PetscInt               **buf_rj, **buf_ri, **buf_ri_k, row, ncols, *cols;
1234   PetscInt                 len, proc, *dnz, *onz, *owners, nzi;
1235   PetscInt                 nrows, *buf_s, *buf_si, *buf_si_i, **nextrow, **nextci;
1236   MPI_Request             *swaits, *rwaits;
1237   MPI_Status              *sstatus, rstatus;
1238   PetscLayout              rowmap;
1239   PetscInt                *owners_co, *coi, *coj; /* i and j array of (p->B)^T*A*P - used in the communication */
1240   PetscMPIInt             *len_r, *id_r;          /* array of length of comm->size, store send/recv matrix values */
1241   PetscInt                *Jptr, *prmap = p->garray, con, j, Crmax;
1242   Mat_SeqAIJ              *a_loc, *c_loc, *c_oth;
1243   PetscTable               ta;
1244   MatType                  mtype;
1245   const char              *prefix;
1246 
1247   PetscFunctionBegin;
1248   PetscCall(PetscObjectGetComm((PetscObject)A, &comm));
1249   PetscCallMPI(MPI_Comm_size(comm, &size));
1250   PetscCallMPI(MPI_Comm_rank(comm, &rank));
1251 
1252   /* create symbolic parallel matrix C */
1253   PetscCall(MatGetType(A, &mtype));
1254   PetscCall(MatSetType(C, mtype));
1255 
1256   C->ops->transposematmultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1257 
1258   /* create struct Mat_APMPI and attached it to C later */
1259   PetscCall(PetscNew(&ptap));
1260 
1261   /* (0) compute Rd = Pd^T, Ro = Po^T  */
1262   /* --------------------------------- */
1263   PetscCall(MatTranspose(p->A, MAT_INITIAL_MATRIX, &ptap->Rd));
1264   PetscCall(MatTranspose(p->B, MAT_INITIAL_MATRIX, &ptap->Ro));
1265 
1266   /* (1) compute symbolic A_loc */
1267   /* ---------------------------*/
1268   PetscCall(MatMPIAIJGetLocalMat(A, MAT_INITIAL_MATRIX, &ptap->A_loc));
1269 
1270   /* (2-1) compute symbolic C_oth = Ro*A_loc  */
1271   /* ------------------------------------ */
1272   PetscCall(MatGetOptionsPrefix(A, &prefix));
1273   PetscCall(MatSetOptionsPrefix(ptap->Ro, prefix));
1274   PetscCall(MatAppendOptionsPrefix(ptap->Ro, "inner_offdiag_"));
1275   PetscCall(MatCreate(PETSC_COMM_SELF, &ptap->C_oth));
1276   PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Ro, ptap->A_loc, fill, ptap->C_oth));
1277 
1278   /* (3) send coj of C_oth to other processors  */
1279   /* ------------------------------------------ */
1280   /* determine row ownership */
1281   PetscCall(PetscLayoutCreate(comm, &rowmap));
1282   rowmap->n  = pn;
1283   rowmap->bs = 1;
1284   PetscCall(PetscLayoutSetUp(rowmap));
1285   owners = rowmap->range;
1286 
1287   /* determine the number of messages to send, their lengths */
1288   PetscCall(PetscMalloc4(size, &len_s, size, &len_si, size, &sstatus, size + 2, &owners_co));
1289   PetscCall(PetscArrayzero(len_s, size));
1290   PetscCall(PetscArrayzero(len_si, size));
1291 
1292   c_oth = (Mat_SeqAIJ *)ptap->C_oth->data;
1293   coi   = c_oth->i;
1294   coj   = c_oth->j;
1295   con   = ptap->C_oth->rmap->n;
1296   proc  = 0;
1297   for (i = 0; i < con; i++) {
1298     while (prmap[i] >= owners[proc + 1]) proc++;
1299     len_si[proc]++;                     /* num of rows in Co(=Pt*A) to be sent to [proc] */
1300     len_s[proc] += coi[i + 1] - coi[i]; /* num of nonzeros in Co to be sent to [proc] */
1301   }
1302 
1303   len          = 0; /* max length of buf_si[], see (4) */
1304   owners_co[0] = 0;
1305   nsend        = 0;
1306   for (proc = 0; proc < size; proc++) {
1307     owners_co[proc + 1] = owners_co[proc] + len_si[proc];
1308     if (len_s[proc]) {
1309       nsend++;
1310       len_si[proc] = 2 * (len_si[proc] + 1); /* length of buf_si to be sent to [proc] */
1311       len += len_si[proc];
1312     }
1313   }
1314 
1315   /* determine the number and length of messages to receive for coi and coj  */
1316   PetscCall(PetscGatherNumberOfMessages(comm, NULL, len_s, &nrecv));
1317   PetscCall(PetscGatherMessageLengths2(comm, nsend, nrecv, len_s, len_si, &id_r, &len_r, &len_ri));
1318 
1319   /* post the Irecv and Isend of coj */
1320   PetscCall(PetscCommGetNewTag(comm, &tagj));
1321   PetscCall(PetscPostIrecvInt(comm, tagj, nrecv, id_r, len_r, &buf_rj, &rwaits));
1322   PetscCall(PetscMalloc1(nsend + 1, &swaits));
1323   for (proc = 0, k = 0; proc < size; proc++) {
1324     if (!len_s[proc]) continue;
1325     i = owners_co[proc];
1326     PetscCallMPI(MPI_Isend(coj + coi[i], len_s[proc], MPIU_INT, proc, tagj, comm, swaits + k));
1327     k++;
1328   }
1329 
1330   /* (2-2) compute symbolic C_loc = Rd*A_loc */
1331   /* ---------------------------------------- */
1332   PetscCall(MatSetOptionsPrefix(ptap->Rd, prefix));
1333   PetscCall(MatAppendOptionsPrefix(ptap->Rd, "inner_diag_"));
1334   PetscCall(MatCreate(PETSC_COMM_SELF, &ptap->C_loc));
1335   PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Rd, ptap->A_loc, fill, ptap->C_loc));
1336   c_loc = (Mat_SeqAIJ *)ptap->C_loc->data;
1337 
1338   /* receives coj are complete */
1339   for (i = 0; i < nrecv; i++) { PetscCallMPI(MPI_Waitany(nrecv, rwaits, &icompleted, &rstatus)); }
1340   PetscCall(PetscFree(rwaits));
1341   if (nsend) PetscCallMPI(MPI_Waitall(nsend, swaits, sstatus));
1342 
1343   /* add received column indices into ta to update Crmax */
1344   a_loc = (Mat_SeqAIJ *)(ptap->A_loc)->data;
1345 
1346   /* create and initialize a linked list */
1347   PetscCall(PetscTableCreate(an, aN, &ta)); /* for compute Crmax */
1348   MatRowMergeMax_SeqAIJ(a_loc, ptap->A_loc->rmap->N, ta);
1349 
1350   for (k = 0; k < nrecv; k++) { /* k-th received message */
1351     Jptr = buf_rj[k];
1352     for (j = 0; j < len_r[k]; j++) { PetscCall(PetscTableAdd(ta, *(Jptr + j) + 1, 1, INSERT_VALUES)); }
1353   }
1354   PetscCall(PetscTableGetCount(ta, &Crmax));
1355   PetscCall(PetscTableDestroy(&ta));
1356 
1357   /* (4) send and recv coi */
1358   /*-----------------------*/
1359   PetscCall(PetscCommGetNewTag(comm, &tagi));
1360   PetscCall(PetscPostIrecvInt(comm, tagi, nrecv, id_r, len_ri, &buf_ri, &rwaits));
1361   PetscCall(PetscMalloc1(len + 1, &buf_s));
1362   buf_si = buf_s; /* points to the beginning of k-th msg to be sent */
1363   for (proc = 0, k = 0; proc < size; proc++) {
1364     if (!len_s[proc]) continue;
1365     /* form outgoing message for i-structure:
1366          buf_si[0]:                 nrows to be sent
1367                [1:nrows]:           row index (global)
1368                [nrows+1:2*nrows+1]: i-structure index
1369     */
1370     /*-------------------------------------------*/
1371     nrows       = len_si[proc] / 2 - 1; /* num of rows in Co to be sent to [proc] */
1372     buf_si_i    = buf_si + nrows + 1;
1373     buf_si[0]   = nrows;
1374     buf_si_i[0] = 0;
1375     nrows       = 0;
1376     for (i = owners_co[proc]; i < owners_co[proc + 1]; i++) {
1377       nzi                 = coi[i + 1] - coi[i];
1378       buf_si_i[nrows + 1] = buf_si_i[nrows] + nzi;   /* i-structure */
1379       buf_si[nrows + 1]   = prmap[i] - owners[proc]; /* local row index */
1380       nrows++;
1381     }
1382     PetscCallMPI(MPI_Isend(buf_si, len_si[proc], MPIU_INT, proc, tagi, comm, swaits + k));
1383     k++;
1384     buf_si += len_si[proc];
1385   }
1386   for (i = 0; i < nrecv; i++) { PetscCallMPI(MPI_Waitany(nrecv, rwaits, &icompleted, &rstatus)); }
1387   PetscCall(PetscFree(rwaits));
1388   if (nsend) PetscCallMPI(MPI_Waitall(nsend, swaits, sstatus));
1389 
1390   PetscCall(PetscFree4(len_s, len_si, sstatus, owners_co));
1391   PetscCall(PetscFree(len_ri));
1392   PetscCall(PetscFree(swaits));
1393   PetscCall(PetscFree(buf_s));
1394 
1395   /* (5) compute the local portion of C      */
1396   /* ------------------------------------------ */
1397   /* set initial free space to be Crmax, sufficient for holding nozeros in each row of C */
1398   PetscCall(PetscFreeSpaceGet(Crmax, &free_space));
1399   current_space = free_space;
1400 
1401   PetscCall(PetscMalloc3(nrecv, &buf_ri_k, nrecv, &nextrow, nrecv, &nextci));
1402   for (k = 0; k < nrecv; k++) {
1403     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1404     nrows       = *buf_ri_k[k];
1405     nextrow[k]  = buf_ri_k[k] + 1;           /* next row number of k-th recved i-structure */
1406     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th recved i-structure  */
1407   }
1408 
1409   MatPreallocateBegin(comm, pn, an, dnz, onz);
1410   PetscCall(PetscLLCondensedCreate(Crmax, aN, &lnk, &lnkbt));
1411   for (i = 0; i < pn; i++) { /* for each local row of C */
1412     /* add C_loc into C */
1413     nzi  = c_loc->i[i + 1] - c_loc->i[i];
1414     Jptr = c_loc->j + c_loc->i[i];
1415     PetscCall(PetscLLCondensedAddSorted(nzi, Jptr, lnk, lnkbt));
1416 
1417     /* add received col data into lnk */
1418     for (k = 0; k < nrecv; k++) { /* k-th received message */
1419       if (i == *nextrow[k]) {     /* i-th row */
1420         nzi  = *(nextci[k] + 1) - *nextci[k];
1421         Jptr = buf_rj[k] + *nextci[k];
1422         PetscCall(PetscLLCondensedAddSorted(nzi, Jptr, lnk, lnkbt));
1423         nextrow[k]++;
1424         nextci[k]++;
1425       }
1426     }
1427 
1428     /* add missing diagonal entry */
1429     if (C->force_diagonals) {
1430       k = i + owners[rank]; /* column index */
1431       PetscCall(PetscLLCondensedAddSorted(1, &k, lnk, lnkbt));
1432     }
1433 
1434     nzi = lnk[0];
1435 
1436     /* copy data into free space, then initialize lnk */
1437     PetscCall(PetscLLCondensedClean(aN, nzi, current_space->array, lnk, lnkbt));
1438     PetscCall(MatPreallocateSet(i + owners[rank], nzi, current_space->array, dnz, onz));
1439   }
1440   PetscCall(PetscFree3(buf_ri_k, nextrow, nextci));
1441   PetscCall(PetscLLDestroy(lnk, lnkbt));
1442   PetscCall(PetscFreeSpaceDestroy(free_space));
1443 
1444   /* local sizes and preallocation */
1445   PetscCall(MatSetSizes(C, pn, an, PETSC_DETERMINE, PETSC_DETERMINE));
1446   if (P->cmap->bs > 0) PetscCall(PetscLayoutSetBlockSize(C->rmap, P->cmap->bs));
1447   if (A->cmap->bs > 0) PetscCall(PetscLayoutSetBlockSize(C->cmap, A->cmap->bs));
1448   PetscCall(MatMPIAIJSetPreallocation(C, 0, dnz, 0, onz));
1449   MatPreallocateEnd(dnz, onz);
1450 
1451   /* add C_loc and C_oth to C */
1452   PetscCall(MatGetOwnershipRange(C, &rstart, NULL));
1453   for (i = 0; i < pn; i++) {
1454     ncols = c_loc->i[i + 1] - c_loc->i[i];
1455     cols  = c_loc->j + c_loc->i[i];
1456     row   = rstart + i;
1457     PetscCall(MatSetValues(C, 1, (const PetscInt *)&row, ncols, (const PetscInt *)cols, NULL, INSERT_VALUES));
1458 
1459     if (C->force_diagonals) { PetscCall(MatSetValues(C, 1, (const PetscInt *)&row, 1, (const PetscInt *)&row, NULL, INSERT_VALUES)); }
1460   }
1461   for (i = 0; i < con; i++) {
1462     ncols = c_oth->i[i + 1] - c_oth->i[i];
1463     cols  = c_oth->j + c_oth->i[i];
1464     row   = prmap[i];
1465     PetscCall(MatSetValues(C, 1, (const PetscInt *)&row, ncols, (const PetscInt *)cols, NULL, INSERT_VALUES));
1466   }
1467   PetscCall(MatAssemblyBegin(C, MAT_FINAL_ASSEMBLY));
1468   PetscCall(MatAssemblyEnd(C, MAT_FINAL_ASSEMBLY));
1469   PetscCall(MatSetOption(C, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
1470 
1471   /* members in merge */
1472   PetscCall(PetscFree(id_r));
1473   PetscCall(PetscFree(len_r));
1474   PetscCall(PetscFree(buf_ri[0]));
1475   PetscCall(PetscFree(buf_ri));
1476   PetscCall(PetscFree(buf_rj[0]));
1477   PetscCall(PetscFree(buf_rj));
1478   PetscCall(PetscLayoutDestroy(&rowmap));
1479 
1480   /* attach the supporting struct to C for reuse */
1481   C->product->data    = ptap;
1482   C->product->destroy = MatDestroy_MPIAIJ_PtAP;
1483   PetscFunctionReturn(0);
1484 }
1485 
1486 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat P, Mat A, Mat C) {
1487   Mat_MPIAIJ        *p = (Mat_MPIAIJ *)P->data;
1488   Mat_SeqAIJ        *c_seq;
1489   Mat_APMPI         *ptap;
1490   Mat                A_loc, C_loc, C_oth;
1491   PetscInt           i, rstart, rend, cm, ncols, row;
1492   const PetscInt    *cols;
1493   const PetscScalar *vals;
1494 
1495   PetscFunctionBegin;
1496   MatCheckProduct(C, 3);
1497   ptap = (Mat_APMPI *)C->product->data;
1498   PetscCheck(ptap, PetscObjectComm((PetscObject)C), PETSC_ERR_ARG_WRONGSTATE, "PtAP cannot be computed. Missing data");
1499   PetscCheck(ptap->A_loc, PetscObjectComm((PetscObject)C), PETSC_ERR_ARG_WRONGSTATE, "PtA cannot be reused. Do not call MatProductClear()");
1500   PetscCall(MatZeroEntries(C));
1501 
1502   /* These matrices are obtained in MatTransposeMatMultSymbolic() */
1503   /* 1) get R = Pd^T, Ro = Po^T */
1504   /*----------------------------*/
1505   PetscCall(MatTransposeSetPrecursor(p->A, ptap->Rd));
1506   PetscCall(MatTranspose(p->A, MAT_REUSE_MATRIX, &ptap->Rd));
1507   PetscCall(MatTransposeSetPrecursor(p->B, ptap->Ro));
1508   PetscCall(MatTranspose(p->B, MAT_REUSE_MATRIX, &ptap->Ro));
1509 
1510   /* 2) compute numeric A_loc */
1511   /*--------------------------*/
1512   PetscCall(MatMPIAIJGetLocalMat(A, MAT_REUSE_MATRIX, &ptap->A_loc));
1513 
1514   /* 3) C_loc = Rd*A_loc, C_oth = Ro*A_loc */
1515   A_loc = ptap->A_loc;
1516   PetscCall(((ptap->C_loc)->ops->matmultnumeric)(ptap->Rd, A_loc, ptap->C_loc));
1517   PetscCall(((ptap->C_oth)->ops->matmultnumeric)(ptap->Ro, A_loc, ptap->C_oth));
1518   C_loc = ptap->C_loc;
1519   C_oth = ptap->C_oth;
1520 
1521   /* add C_loc and C_oth to C */
1522   PetscCall(MatGetOwnershipRange(C, &rstart, &rend));
1523 
1524   /* C_loc -> C */
1525   cm    = C_loc->rmap->N;
1526   c_seq = (Mat_SeqAIJ *)C_loc->data;
1527   cols  = c_seq->j;
1528   vals  = c_seq->a;
1529   for (i = 0; i < cm; i++) {
1530     ncols = c_seq->i[i + 1] - c_seq->i[i];
1531     row   = rstart + i;
1532     PetscCall(MatSetValues(C, 1, &row, ncols, cols, vals, ADD_VALUES));
1533     cols += ncols;
1534     vals += ncols;
1535   }
1536 
1537   /* Co -> C, off-processor part */
1538   cm    = C_oth->rmap->N;
1539   c_seq = (Mat_SeqAIJ *)C_oth->data;
1540   cols  = c_seq->j;
1541   vals  = c_seq->a;
1542   for (i = 0; i < cm; i++) {
1543     ncols = c_seq->i[i + 1] - c_seq->i[i];
1544     row   = p->garray[i];
1545     PetscCall(MatSetValues(C, 1, &row, ncols, cols, vals, ADD_VALUES));
1546     cols += ncols;
1547     vals += ncols;
1548   }
1549   PetscCall(MatAssemblyBegin(C, MAT_FINAL_ASSEMBLY));
1550   PetscCall(MatAssemblyEnd(C, MAT_FINAL_ASSEMBLY));
1551   PetscCall(MatSetOption(C, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
1552   PetscFunctionReturn(0);
1553 }
1554 
1555 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ(Mat P, Mat A, Mat C) {
1556   Mat_Merge_SeqsToMPI *merge;
1557   Mat_MPIAIJ          *p  = (Mat_MPIAIJ *)P->data;
1558   Mat_SeqAIJ          *pd = (Mat_SeqAIJ *)(p->A)->data, *po = (Mat_SeqAIJ *)(p->B)->data;
1559   Mat_APMPI           *ap;
1560   PetscInt            *adj;
1561   PetscInt             i, j, k, anz, pnz, row, *cj, nexta;
1562   MatScalar           *ada, *ca, valtmp;
1563   PetscInt             am = A->rmap->n, cm = C->rmap->n, pon = (p->B)->cmap->n;
1564   MPI_Comm             comm;
1565   PetscMPIInt          size, rank, taga, *len_s;
1566   PetscInt            *owners, proc, nrows, **buf_ri_k, **nextrow, **nextci;
1567   PetscInt           **buf_ri, **buf_rj;
1568   PetscInt             cnz = 0, *bj_i, *bi, *bj, bnz, nextcj; /* bi,bj,ba: local array of C(mpi mat) */
1569   MPI_Request         *s_waits, *r_waits;
1570   MPI_Status          *status;
1571   MatScalar          **abuf_r, *ba_i, *pA, *coa, *ba;
1572   const PetscScalar   *dummy;
1573   PetscInt            *ai, *aj, *coi, *coj, *poJ, *pdJ;
1574   Mat                  A_loc;
1575   Mat_SeqAIJ          *a_loc;
1576 
1577   PetscFunctionBegin;
1578   MatCheckProduct(C, 3);
1579   ap = (Mat_APMPI *)C->product->data;
1580   PetscCheck(ap, PetscObjectComm((PetscObject)C), PETSC_ERR_ARG_WRONGSTATE, "PtA cannot be computed. Missing data");
1581   PetscCheck(ap->A_loc, PetscObjectComm((PetscObject)C), PETSC_ERR_ARG_WRONGSTATE, "PtA cannot be reused. Do not call MatProductClear()");
1582   PetscCall(PetscObjectGetComm((PetscObject)C, &comm));
1583   PetscCallMPI(MPI_Comm_size(comm, &size));
1584   PetscCallMPI(MPI_Comm_rank(comm, &rank));
1585 
1586   merge = ap->merge;
1587 
1588   /* 2) compute numeric C_seq = P_loc^T*A_loc */
1589   /*------------------------------------------*/
1590   /* get data from symbolic products */
1591   coi = merge->coi;
1592   coj = merge->coj;
1593   PetscCall(PetscCalloc1(coi[pon] + 1, &coa));
1594   bi     = merge->bi;
1595   bj     = merge->bj;
1596   owners = merge->rowmap->range;
1597   PetscCall(PetscCalloc1(bi[cm] + 1, &ba));
1598 
1599   /* get A_loc by taking all local rows of A */
1600   A_loc = ap->A_loc;
1601   PetscCall(MatMPIAIJGetLocalMat(A, MAT_REUSE_MATRIX, &A_loc));
1602   a_loc = (Mat_SeqAIJ *)(A_loc)->data;
1603   ai    = a_loc->i;
1604   aj    = a_loc->j;
1605 
1606   /* trigger copy to CPU */
1607   PetscCall(MatSeqAIJGetArrayRead(p->A, &dummy));
1608   PetscCall(MatSeqAIJRestoreArrayRead(p->A, &dummy));
1609   PetscCall(MatSeqAIJGetArrayRead(p->B, &dummy));
1610   PetscCall(MatSeqAIJRestoreArrayRead(p->B, &dummy));
1611   for (i = 0; i < am; i++) {
1612     anz = ai[i + 1] - ai[i];
1613     adj = aj + ai[i];
1614     ada = a_loc->a + ai[i];
1615 
1616     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1617     /*-------------------------------------------------------------*/
1618     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1619     pnz = po->i[i + 1] - po->i[i];
1620     poJ = po->j + po->i[i];
1621     pA  = po->a + po->i[i];
1622     for (j = 0; j < pnz; j++) {
1623       row    = poJ[j];
1624       cj     = coj + coi[row];
1625       ca     = coa + coi[row];
1626       /* perform sparse axpy */
1627       nexta  = 0;
1628       valtmp = pA[j];
1629       for (k = 0; nexta < anz; k++) {
1630         if (cj[k] == adj[nexta]) {
1631           ca[k] += valtmp * ada[nexta];
1632           nexta++;
1633         }
1634       }
1635       PetscCall(PetscLogFlops(2.0 * anz));
1636     }
1637 
1638     /* put the value into Cd (diagonal part) */
1639     pnz = pd->i[i + 1] - pd->i[i];
1640     pdJ = pd->j + pd->i[i];
1641     pA  = pd->a + pd->i[i];
1642     for (j = 0; j < pnz; j++) {
1643       row    = pdJ[j];
1644       cj     = bj + bi[row];
1645       ca     = ba + bi[row];
1646       /* perform sparse axpy */
1647       nexta  = 0;
1648       valtmp = pA[j];
1649       for (k = 0; nexta < anz; k++) {
1650         if (cj[k] == adj[nexta]) {
1651           ca[k] += valtmp * ada[nexta];
1652           nexta++;
1653         }
1654       }
1655       PetscCall(PetscLogFlops(2.0 * anz));
1656     }
1657   }
1658 
1659   /* 3) send and recv matrix values coa */
1660   /*------------------------------------*/
1661   buf_ri = merge->buf_ri;
1662   buf_rj = merge->buf_rj;
1663   len_s  = merge->len_s;
1664   PetscCall(PetscCommGetNewTag(comm, &taga));
1665   PetscCall(PetscPostIrecvScalar(comm, taga, merge->nrecv, merge->id_r, merge->len_r, &abuf_r, &r_waits));
1666 
1667   PetscCall(PetscMalloc2(merge->nsend + 1, &s_waits, size, &status));
1668   for (proc = 0, k = 0; proc < size; proc++) {
1669     if (!len_s[proc]) continue;
1670     i = merge->owners_co[proc];
1671     PetscCallMPI(MPI_Isend(coa + coi[i], len_s[proc], MPIU_MATSCALAR, proc, taga, comm, s_waits + k));
1672     k++;
1673   }
1674   if (merge->nrecv) PetscCallMPI(MPI_Waitall(merge->nrecv, r_waits, status));
1675   if (merge->nsend) PetscCallMPI(MPI_Waitall(merge->nsend, s_waits, status));
1676 
1677   PetscCall(PetscFree2(s_waits, status));
1678   PetscCall(PetscFree(r_waits));
1679   PetscCall(PetscFree(coa));
1680 
1681   /* 4) insert local Cseq and received values into Cmpi */
1682   /*----------------------------------------------------*/
1683   PetscCall(PetscMalloc3(merge->nrecv, &buf_ri_k, merge->nrecv, &nextrow, merge->nrecv, &nextci));
1684   for (k = 0; k < merge->nrecv; k++) {
1685     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1686     nrows       = *(buf_ri_k[k]);
1687     nextrow[k]  = buf_ri_k[k] + 1;           /* next row number of k-th recved i-structure */
1688     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th recved i-structure  */
1689   }
1690 
1691   for (i = 0; i < cm; i++) {
1692     row  = owners[rank] + i; /* global row index of C_seq */
1693     bj_i = bj + bi[i];       /* col indices of the i-th row of C */
1694     ba_i = ba + bi[i];
1695     bnz  = bi[i + 1] - bi[i];
1696     /* add received vals into ba */
1697     for (k = 0; k < merge->nrecv; k++) { /* k-th received message */
1698       /* i-th row */
1699       if (i == *nextrow[k]) {
1700         cnz    = *(nextci[k] + 1) - *nextci[k];
1701         cj     = buf_rj[k] + *(nextci[k]);
1702         ca     = abuf_r[k] + *(nextci[k]);
1703         nextcj = 0;
1704         for (j = 0; nextcj < cnz; j++) {
1705           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1706             ba_i[j] += ca[nextcj++];
1707           }
1708         }
1709         nextrow[k]++;
1710         nextci[k]++;
1711         PetscCall(PetscLogFlops(2.0 * cnz));
1712       }
1713     }
1714     PetscCall(MatSetValues(C, 1, &row, bnz, bj_i, ba_i, INSERT_VALUES));
1715   }
1716   PetscCall(MatAssemblyBegin(C, MAT_FINAL_ASSEMBLY));
1717   PetscCall(MatAssemblyEnd(C, MAT_FINAL_ASSEMBLY));
1718 
1719   PetscCall(PetscFree(ba));
1720   PetscCall(PetscFree(abuf_r[0]));
1721   PetscCall(PetscFree(abuf_r));
1722   PetscCall(PetscFree3(buf_ri_k, nextrow, nextci));
1723   PetscFunctionReturn(0);
1724 }
1725 
1726 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P, Mat A, PetscReal fill, Mat C) {
1727   Mat                  A_loc;
1728   Mat_APMPI           *ap;
1729   PetscFreeSpaceList   free_space = NULL, current_space = NULL;
1730   Mat_MPIAIJ          *p = (Mat_MPIAIJ *)P->data, *a = (Mat_MPIAIJ *)A->data;
1731   PetscInt            *pdti, *pdtj, *poti, *potj, *ptJ;
1732   PetscInt             nnz;
1733   PetscInt            *lnk, *owners_co, *coi, *coj, i, k, pnz, row;
1734   PetscInt             am = A->rmap->n, pn = P->cmap->n;
1735   MPI_Comm             comm;
1736   PetscMPIInt          size, rank, tagi, tagj, *len_si, *len_s, *len_ri;
1737   PetscInt           **buf_rj, **buf_ri, **buf_ri_k;
1738   PetscInt             len, proc, *dnz, *onz, *owners;
1739   PetscInt             nzi, *bi, *bj;
1740   PetscInt             nrows, *buf_s, *buf_si, *buf_si_i, **nextrow, **nextci;
1741   MPI_Request         *swaits, *rwaits;
1742   MPI_Status          *sstatus, rstatus;
1743   Mat_Merge_SeqsToMPI *merge;
1744   PetscInt            *ai, *aj, *Jptr, anz, *prmap = p->garray, pon, nspacedouble = 0, j;
1745   PetscReal            afill  = 1.0, afill_tmp;
1746   PetscInt             rstart = P->cmap->rstart, rmax, aN = A->cmap->N, Armax;
1747   Mat_SeqAIJ          *a_loc;
1748   PetscTable           ta;
1749   MatType              mtype;
1750 
1751   PetscFunctionBegin;
1752   PetscCall(PetscObjectGetComm((PetscObject)A, &comm));
1753   /* check if matrix local sizes are compatible */
1754   PetscCheck(A->rmap->rstart == P->rmap->rstart && A->rmap->rend == P->rmap->rend, comm, PETSC_ERR_ARG_SIZ, "Matrix local dimensions are incompatible, A (%" PetscInt_FMT ", %" PetscInt_FMT ") != P (%" PetscInt_FMT ",%" PetscInt_FMT ")", A->rmap->rstart,
1755              A->rmap->rend, P->rmap->rstart, P->rmap->rend);
1756 
1757   PetscCallMPI(MPI_Comm_size(comm, &size));
1758   PetscCallMPI(MPI_Comm_rank(comm, &rank));
1759 
1760   /* create struct Mat_APMPI and attached it to C later */
1761   PetscCall(PetscNew(&ap));
1762 
1763   /* get A_loc by taking all local rows of A */
1764   PetscCall(MatMPIAIJGetLocalMat(A, MAT_INITIAL_MATRIX, &A_loc));
1765 
1766   ap->A_loc = A_loc;
1767   a_loc     = (Mat_SeqAIJ *)(A_loc)->data;
1768   ai        = a_loc->i;
1769   aj        = a_loc->j;
1770 
1771   /* determine symbolic Co=(p->B)^T*A - send to others */
1772   /*----------------------------------------------------*/
1773   PetscCall(MatGetSymbolicTranspose_SeqAIJ(p->A, &pdti, &pdtj));
1774   PetscCall(MatGetSymbolicTranspose_SeqAIJ(p->B, &poti, &potj));
1775   pon = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1776                          >= (num of nonzero rows of C_seq) - pn */
1777   PetscCall(PetscMalloc1(pon + 1, &coi));
1778   coi[0] = 0;
1779 
1780   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1781   nnz = PetscRealIntMultTruncate(fill, PetscIntSumTruncate(poti[pon], ai[am]));
1782   PetscCall(PetscFreeSpaceGet(nnz, &free_space));
1783   current_space = free_space;
1784 
1785   /* create and initialize a linked list */
1786   PetscCall(PetscTableCreate(A->cmap->n + a->B->cmap->N, aN, &ta));
1787   MatRowMergeMax_SeqAIJ(a_loc, am, ta);
1788   PetscCall(PetscTableGetCount(ta, &Armax));
1789 
1790   PetscCall(PetscLLCondensedCreate_Scalable(Armax, &lnk));
1791 
1792   for (i = 0; i < pon; i++) {
1793     pnz = poti[i + 1] - poti[i];
1794     ptJ = potj + poti[i];
1795     for (j = 0; j < pnz; j++) {
1796       row  = ptJ[j]; /* row of A_loc == col of Pot */
1797       anz  = ai[row + 1] - ai[row];
1798       Jptr = aj + ai[row];
1799       /* add non-zero cols of AP into the sorted linked list lnk */
1800       PetscCall(PetscLLCondensedAddSorted_Scalable(anz, Jptr, lnk));
1801     }
1802     nnz = lnk[0];
1803 
1804     /* If free space is not available, double the total space in the list */
1805     if (current_space->local_remaining < nnz) {
1806       PetscCall(PetscFreeSpaceGet(PetscIntSumTruncate(nnz, current_space->total_array_size), &current_space));
1807       nspacedouble++;
1808     }
1809 
1810     /* Copy data into free space, and zero out denserows */
1811     PetscCall(PetscLLCondensedClean_Scalable(nnz, current_space->array, lnk));
1812 
1813     current_space->array += nnz;
1814     current_space->local_used += nnz;
1815     current_space->local_remaining -= nnz;
1816 
1817     coi[i + 1] = coi[i] + nnz;
1818   }
1819 
1820   PetscCall(PetscMalloc1(coi[pon] + 1, &coj));
1821   PetscCall(PetscFreeSpaceContiguous(&free_space, coj));
1822   PetscCall(PetscLLCondensedDestroy_Scalable(lnk)); /* must destroy to get a new one for C */
1823 
1824   afill_tmp = (PetscReal)coi[pon] / (poti[pon] + ai[am] + 1);
1825   if (afill_tmp > afill) afill = afill_tmp;
1826 
1827   /* send j-array (coj) of Co to other processors */
1828   /*----------------------------------------------*/
1829   /* determine row ownership */
1830   PetscCall(PetscNew(&merge));
1831   PetscCall(PetscLayoutCreate(comm, &merge->rowmap));
1832 
1833   merge->rowmap->n  = pn;
1834   merge->rowmap->bs = 1;
1835 
1836   PetscCall(PetscLayoutSetUp(merge->rowmap));
1837   owners = merge->rowmap->range;
1838 
1839   /* determine the number of messages to send, their lengths */
1840   PetscCall(PetscCalloc1(size, &len_si));
1841   PetscCall(PetscCalloc1(size, &merge->len_s));
1842 
1843   len_s        = merge->len_s;
1844   merge->nsend = 0;
1845 
1846   PetscCall(PetscMalloc1(size + 2, &owners_co));
1847 
1848   proc = 0;
1849   for (i = 0; i < pon; i++) {
1850     while (prmap[i] >= owners[proc + 1]) proc++;
1851     len_si[proc]++; /* num of rows in Co to be sent to [proc] */
1852     len_s[proc] += coi[i + 1] - coi[i];
1853   }
1854 
1855   len          = 0; /* max length of buf_si[] */
1856   owners_co[0] = 0;
1857   for (proc = 0; proc < size; proc++) {
1858     owners_co[proc + 1] = owners_co[proc] + len_si[proc];
1859     if (len_si[proc]) {
1860       merge->nsend++;
1861       len_si[proc] = 2 * (len_si[proc] + 1);
1862       len += len_si[proc];
1863     }
1864   }
1865 
1866   /* determine the number and length of messages to receive for coi and coj  */
1867   PetscCall(PetscGatherNumberOfMessages(comm, NULL, len_s, &merge->nrecv));
1868   PetscCall(PetscGatherMessageLengths2(comm, merge->nsend, merge->nrecv, len_s, len_si, &merge->id_r, &merge->len_r, &len_ri));
1869 
1870   /* post the Irecv and Isend of coj */
1871   PetscCall(PetscCommGetNewTag(comm, &tagj));
1872   PetscCall(PetscPostIrecvInt(comm, tagj, merge->nrecv, merge->id_r, merge->len_r, &buf_rj, &rwaits));
1873   PetscCall(PetscMalloc1(merge->nsend + 1, &swaits));
1874   for (proc = 0, k = 0; proc < size; proc++) {
1875     if (!len_s[proc]) continue;
1876     i = owners_co[proc];
1877     PetscCallMPI(MPI_Isend(coj + coi[i], len_s[proc], MPIU_INT, proc, tagj, comm, swaits + k));
1878     k++;
1879   }
1880 
1881   /* receives and sends of coj are complete */
1882   PetscCall(PetscMalloc1(size, &sstatus));
1883   for (i = 0; i < merge->nrecv; i++) {
1884     PETSC_UNUSED PetscMPIInt icompleted;
1885     PetscCallMPI(MPI_Waitany(merge->nrecv, rwaits, &icompleted, &rstatus));
1886   }
1887   PetscCall(PetscFree(rwaits));
1888   if (merge->nsend) PetscCallMPI(MPI_Waitall(merge->nsend, swaits, sstatus));
1889 
1890   /* add received column indices into table to update Armax */
1891   /* Armax can be as large as aN if a P[row,:] is dense, see src/ksp/ksp/tutorials/ex56.c! */
1892   for (k = 0; k < merge->nrecv; k++) { /* k-th received message */
1893     Jptr = buf_rj[k];
1894     for (j = 0; j < merge->len_r[k]; j++) { PetscCall(PetscTableAdd(ta, *(Jptr + j) + 1, 1, INSERT_VALUES)); }
1895   }
1896   PetscCall(PetscTableGetCount(ta, &Armax));
1897 
1898   /* send and recv coi */
1899   /*-------------------*/
1900   PetscCall(PetscCommGetNewTag(comm, &tagi));
1901   PetscCall(PetscPostIrecvInt(comm, tagi, merge->nrecv, merge->id_r, len_ri, &buf_ri, &rwaits));
1902   PetscCall(PetscMalloc1(len + 1, &buf_s));
1903   buf_si = buf_s; /* points to the beginning of k-th msg to be sent */
1904   for (proc = 0, k = 0; proc < size; proc++) {
1905     if (!len_s[proc]) continue;
1906     /* form outgoing message for i-structure:
1907          buf_si[0]:                 nrows to be sent
1908                [1:nrows]:           row index (global)
1909                [nrows+1:2*nrows+1]: i-structure index
1910     */
1911     /*-------------------------------------------*/
1912     nrows       = len_si[proc] / 2 - 1;
1913     buf_si_i    = buf_si + nrows + 1;
1914     buf_si[0]   = nrows;
1915     buf_si_i[0] = 0;
1916     nrows       = 0;
1917     for (i = owners_co[proc]; i < owners_co[proc + 1]; i++) {
1918       nzi                 = coi[i + 1] - coi[i];
1919       buf_si_i[nrows + 1] = buf_si_i[nrows] + nzi;   /* i-structure */
1920       buf_si[nrows + 1]   = prmap[i] - owners[proc]; /* local row index */
1921       nrows++;
1922     }
1923     PetscCallMPI(MPI_Isend(buf_si, len_si[proc], MPIU_INT, proc, tagi, comm, swaits + k));
1924     k++;
1925     buf_si += len_si[proc];
1926   }
1927   i = merge->nrecv;
1928   while (i--) {
1929     PETSC_UNUSED PetscMPIInt icompleted;
1930     PetscCallMPI(MPI_Waitany(merge->nrecv, rwaits, &icompleted, &rstatus));
1931   }
1932   PetscCall(PetscFree(rwaits));
1933   if (merge->nsend) PetscCallMPI(MPI_Waitall(merge->nsend, swaits, sstatus));
1934   PetscCall(PetscFree(len_si));
1935   PetscCall(PetscFree(len_ri));
1936   PetscCall(PetscFree(swaits));
1937   PetscCall(PetscFree(sstatus));
1938   PetscCall(PetscFree(buf_s));
1939 
1940   /* compute the local portion of C (mpi mat) */
1941   /*------------------------------------------*/
1942   /* allocate bi array and free space for accumulating nonzero column info */
1943   PetscCall(PetscMalloc1(pn + 1, &bi));
1944   bi[0] = 0;
1945 
1946   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
1947   nnz = PetscRealIntMultTruncate(fill, PetscIntSumTruncate(pdti[pn], PetscIntSumTruncate(poti[pon], ai[am])));
1948   PetscCall(PetscFreeSpaceGet(nnz, &free_space));
1949   current_space = free_space;
1950 
1951   PetscCall(PetscMalloc3(merge->nrecv, &buf_ri_k, merge->nrecv, &nextrow, merge->nrecv, &nextci));
1952   for (k = 0; k < merge->nrecv; k++) {
1953     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1954     nrows       = *buf_ri_k[k];
1955     nextrow[k]  = buf_ri_k[k] + 1;           /* next row number of k-th recved i-structure */
1956     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th received i-structure  */
1957   }
1958 
1959   PetscCall(PetscLLCondensedCreate_Scalable(Armax, &lnk));
1960   MatPreallocateBegin(comm, pn, A->cmap->n, dnz, onz);
1961   rmax = 0;
1962   for (i = 0; i < pn; i++) {
1963     /* add pdt[i,:]*AP into lnk */
1964     pnz = pdti[i + 1] - pdti[i];
1965     ptJ = pdtj + pdti[i];
1966     for (j = 0; j < pnz; j++) {
1967       row  = ptJ[j]; /* row of AP == col of Pt */
1968       anz  = ai[row + 1] - ai[row];
1969       Jptr = aj + ai[row];
1970       /* add non-zero cols of AP into the sorted linked list lnk */
1971       PetscCall(PetscLLCondensedAddSorted_Scalable(anz, Jptr, lnk));
1972     }
1973 
1974     /* add received col data into lnk */
1975     for (k = 0; k < merge->nrecv; k++) { /* k-th received message */
1976       if (i == *nextrow[k]) {            /* i-th row */
1977         nzi  = *(nextci[k] + 1) - *nextci[k];
1978         Jptr = buf_rj[k] + *nextci[k];
1979         PetscCall(PetscLLCondensedAddSorted_Scalable(nzi, Jptr, lnk));
1980         nextrow[k]++;
1981         nextci[k]++;
1982       }
1983     }
1984 
1985     /* add missing diagonal entry */
1986     if (C->force_diagonals) {
1987       k = i + owners[rank]; /* column index */
1988       PetscCall(PetscLLCondensedAddSorted_Scalable(1, &k, lnk));
1989     }
1990 
1991     nnz = lnk[0];
1992 
1993     /* if free space is not available, make more free space */
1994     if (current_space->local_remaining < nnz) {
1995       PetscCall(PetscFreeSpaceGet(PetscIntSumTruncate(nnz, current_space->total_array_size), &current_space));
1996       nspacedouble++;
1997     }
1998     /* copy data into free space, then initialize lnk */
1999     PetscCall(PetscLLCondensedClean_Scalable(nnz, current_space->array, lnk));
2000     PetscCall(MatPreallocateSet(i + owners[rank], nnz, current_space->array, dnz, onz));
2001 
2002     current_space->array += nnz;
2003     current_space->local_used += nnz;
2004     current_space->local_remaining -= nnz;
2005 
2006     bi[i + 1] = bi[i] + nnz;
2007     if (nnz > rmax) rmax = nnz;
2008   }
2009   PetscCall(PetscFree3(buf_ri_k, nextrow, nextci));
2010 
2011   PetscCall(PetscMalloc1(bi[pn] + 1, &bj));
2012   PetscCall(PetscFreeSpaceContiguous(&free_space, bj));
2013   afill_tmp = (PetscReal)bi[pn] / (pdti[pn] + poti[pon] + ai[am] + 1);
2014   if (afill_tmp > afill) afill = afill_tmp;
2015   PetscCall(PetscLLCondensedDestroy_Scalable(lnk));
2016   PetscCall(PetscTableDestroy(&ta));
2017   PetscCall(MatRestoreSymbolicTranspose_SeqAIJ(p->A, &pdti, &pdtj));
2018   PetscCall(MatRestoreSymbolicTranspose_SeqAIJ(p->B, &poti, &potj));
2019 
2020   /* create symbolic parallel matrix C - why cannot be assembled in Numeric part   */
2021   /*-------------------------------------------------------------------------------*/
2022   PetscCall(MatSetSizes(C, pn, A->cmap->n, PETSC_DETERMINE, PETSC_DETERMINE));
2023   PetscCall(MatSetBlockSizes(C, PetscAbs(P->cmap->bs), PetscAbs(A->cmap->bs)));
2024   PetscCall(MatGetType(A, &mtype));
2025   PetscCall(MatSetType(C, mtype));
2026   PetscCall(MatMPIAIJSetPreallocation(C, 0, dnz, 0, onz));
2027   MatPreallocateEnd(dnz, onz);
2028   PetscCall(MatSetBlockSize(C, 1));
2029   PetscCall(MatSetOption(C, MAT_NO_OFF_PROC_ENTRIES, PETSC_TRUE));
2030   for (i = 0; i < pn; i++) {
2031     row  = i + rstart;
2032     nnz  = bi[i + 1] - bi[i];
2033     Jptr = bj + bi[i];
2034     PetscCall(MatSetValues(C, 1, &row, nnz, Jptr, NULL, INSERT_VALUES));
2035   }
2036   PetscCall(MatAssemblyBegin(C, MAT_FINAL_ASSEMBLY));
2037   PetscCall(MatAssemblyEnd(C, MAT_FINAL_ASSEMBLY));
2038   PetscCall(MatSetOption(C, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
2039   merge->bi        = bi;
2040   merge->bj        = bj;
2041   merge->coi       = coi;
2042   merge->coj       = coj;
2043   merge->buf_ri    = buf_ri;
2044   merge->buf_rj    = buf_rj;
2045   merge->owners_co = owners_co;
2046 
2047   /* attach the supporting struct to C for reuse */
2048   C->product->data    = ap;
2049   C->product->destroy = MatDestroy_MPIAIJ_PtAP;
2050   ap->merge           = merge;
2051 
2052   C->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ;
2053 
2054 #if defined(PETSC_USE_INFO)
2055   if (bi[pn] != 0) {
2056     PetscCall(PetscInfo(C, "Reallocs %" PetscInt_FMT "; Fill ratio: given %g needed %g.\n", nspacedouble, (double)fill, (double)afill));
2057     PetscCall(PetscInfo(C, "Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n", (double)afill));
2058   } else {
2059     PetscCall(PetscInfo(C, "Empty matrix product\n"));
2060   }
2061 #endif
2062   PetscFunctionReturn(0);
2063 }
2064 
2065 /* ---------------------------------------------------------------- */
2066 static PetscErrorCode MatProductSymbolic_AtB_MPIAIJ_MPIAIJ(Mat C) {
2067   Mat_Product *product = C->product;
2068   Mat          A = product->A, B = product->B;
2069   PetscReal    fill = product->fill;
2070   PetscBool    flg;
2071 
2072   PetscFunctionBegin;
2073   /* scalable */
2074   PetscCall(PetscStrcmp(product->alg, "scalable", &flg));
2075   if (flg) {
2076     PetscCall(MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(A, B, fill, C));
2077     goto next;
2078   }
2079 
2080   /* nonscalable */
2081   PetscCall(PetscStrcmp(product->alg, "nonscalable", &flg));
2082   if (flg) {
2083     PetscCall(MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(A, B, fill, C));
2084     goto next;
2085   }
2086 
2087   /* matmatmult */
2088   PetscCall(PetscStrcmp(product->alg, "at*b", &flg));
2089   if (flg) {
2090     Mat        At;
2091     Mat_APMPI *ptap;
2092 
2093     PetscCall(MatTranspose(A, MAT_INITIAL_MATRIX, &At));
2094     PetscCall(MatMatMultSymbolic_MPIAIJ_MPIAIJ(At, B, fill, C));
2095     ptap = (Mat_APMPI *)C->product->data;
2096     if (ptap) {
2097       ptap->Pt            = At;
2098       C->product->destroy = MatDestroy_MPIAIJ_PtAP;
2099     }
2100     C->ops->transposematmultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult;
2101     goto next;
2102   }
2103 
2104   /* backend general code */
2105   PetscCall(PetscStrcmp(product->alg, "backend", &flg));
2106   if (flg) {
2107     PetscCall(MatProductSymbolic_MPIAIJBACKEND(C));
2108     PetscFunctionReturn(0);
2109   }
2110 
2111   SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "MatProduct type is not supported");
2112 
2113 next:
2114   C->ops->productnumeric = MatProductNumeric_AtB;
2115   PetscFunctionReturn(0);
2116 }
2117 
2118 /* ---------------------------------------------------------------- */
2119 /* Set options for MatMatMultxxx_MPIAIJ_MPIAIJ */
2120 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_AB(Mat C) {
2121   Mat_Product *product = C->product;
2122   Mat          A = product->A, B = product->B;
2123 #if defined(PETSC_HAVE_HYPRE)
2124   const char *algTypes[5] = {"scalable", "nonscalable", "seqmpi", "backend", "hypre"};
2125   PetscInt    nalg        = 5;
2126 #else
2127   const char *algTypes[4] = {
2128     "scalable",
2129     "nonscalable",
2130     "seqmpi",
2131     "backend",
2132   };
2133   PetscInt    nalg        = 4;
2134 #endif
2135   PetscInt  alg = 1; /* set nonscalable algorithm as default */
2136   PetscBool flg;
2137   MPI_Comm  comm;
2138 
2139   PetscFunctionBegin;
2140   PetscCall(PetscObjectGetComm((PetscObject)C, &comm));
2141 
2142   /* Set "nonscalable" as default algorithm */
2143   PetscCall(PetscStrcmp(C->product->alg, "default", &flg));
2144   if (flg) {
2145     PetscCall(MatProductSetAlgorithm(C, (MatProductAlgorithm)algTypes[alg]));
2146 
2147     /* Set "scalable" as default if BN and local nonzeros of A and B are large */
2148     if (B->cmap->N > 100000) { /* may switch to scalable algorithm as default */
2149       MatInfo   Ainfo, Binfo;
2150       PetscInt  nz_local;
2151       PetscBool alg_scalable_loc = PETSC_FALSE, alg_scalable;
2152 
2153       PetscCall(MatGetInfo(A, MAT_LOCAL, &Ainfo));
2154       PetscCall(MatGetInfo(B, MAT_LOCAL, &Binfo));
2155       nz_local = (PetscInt)(Ainfo.nz_allocated + Binfo.nz_allocated);
2156 
2157       if (B->cmap->N > product->fill * nz_local) alg_scalable_loc = PETSC_TRUE;
2158       PetscCall(MPIU_Allreduce(&alg_scalable_loc, &alg_scalable, 1, MPIU_BOOL, MPI_LOR, comm));
2159 
2160       if (alg_scalable) {
2161         alg = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
2162         PetscCall(MatProductSetAlgorithm(C, (MatProductAlgorithm)algTypes[alg]));
2163         PetscCall(PetscInfo(B, "Use scalable algorithm, BN %" PetscInt_FMT ", fill*nz_allocated %g\n", B->cmap->N, (double)(product->fill * nz_local)));
2164       }
2165     }
2166   }
2167 
2168   /* Get runtime option */
2169   if (product->api_user) {
2170     PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatMatMult", "Mat");
2171     PetscCall(PetscOptionsEList("-matmatmult_via", "Algorithmic approach", "MatMatMult", algTypes, nalg, algTypes[alg], &alg, &flg));
2172     PetscOptionsEnd();
2173   } else {
2174     PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatProduct_AB", "Mat");
2175     PetscCall(PetscOptionsEList("-mat_product_algorithm", "Algorithmic approach", "MatMatMult", algTypes, nalg, algTypes[alg], &alg, &flg));
2176     PetscOptionsEnd();
2177   }
2178   if (flg) { PetscCall(MatProductSetAlgorithm(C, (MatProductAlgorithm)algTypes[alg])); }
2179 
2180   C->ops->productsymbolic = MatProductSymbolic_AB_MPIAIJ_MPIAIJ;
2181   PetscFunctionReturn(0);
2182 }
2183 
2184 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_ABt(Mat C) {
2185   PetscFunctionBegin;
2186   PetscCall(MatProductSetFromOptions_MPIAIJ_AB(C));
2187   C->ops->productsymbolic = MatProductSymbolic_ABt_MPIAIJ_MPIAIJ;
2188   PetscFunctionReturn(0);
2189 }
2190 
2191 /* Set options for MatTransposeMatMultXXX_MPIAIJ_MPIAIJ */
2192 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_AtB(Mat C) {
2193   Mat_Product *product = C->product;
2194   Mat          A = product->A, B = product->B;
2195   const char  *algTypes[4] = {"scalable", "nonscalable", "at*b", "backend"};
2196   PetscInt     nalg        = 4;
2197   PetscInt     alg         = 1; /* set default algorithm  */
2198   PetscBool    flg;
2199   MPI_Comm     comm;
2200 
2201   PetscFunctionBegin;
2202   /* Check matrix local sizes */
2203   PetscCall(PetscObjectGetComm((PetscObject)C, &comm));
2204   PetscCheck(A->rmap->rstart == B->rmap->rstart && A->rmap->rend == B->rmap->rend, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Matrix local dimensions are incompatible, A (%" PetscInt_FMT ", %" PetscInt_FMT ") != B (%" PetscInt_FMT ",%" PetscInt_FMT ")",
2205              A->rmap->rstart, A->rmap->rend, B->rmap->rstart, B->rmap->rend);
2206 
2207   /* Set default algorithm */
2208   PetscCall(PetscStrcmp(C->product->alg, "default", &flg));
2209   if (flg) { PetscCall(MatProductSetAlgorithm(C, (MatProductAlgorithm)algTypes[alg])); }
2210 
2211   /* Set "scalable" as default if BN and local nonzeros of A and B are large */
2212   if (alg && B->cmap->N > 100000) { /* may switch to scalable algorithm as default */
2213     MatInfo   Ainfo, Binfo;
2214     PetscInt  nz_local;
2215     PetscBool alg_scalable_loc = PETSC_FALSE, alg_scalable;
2216 
2217     PetscCall(MatGetInfo(A, MAT_LOCAL, &Ainfo));
2218     PetscCall(MatGetInfo(B, MAT_LOCAL, &Binfo));
2219     nz_local = (PetscInt)(Ainfo.nz_allocated + Binfo.nz_allocated);
2220 
2221     if (B->cmap->N > product->fill * nz_local) alg_scalable_loc = PETSC_TRUE;
2222     PetscCall(MPIU_Allreduce(&alg_scalable_loc, &alg_scalable, 1, MPIU_BOOL, MPI_LOR, comm));
2223 
2224     if (alg_scalable) {
2225       alg = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
2226       PetscCall(MatProductSetAlgorithm(C, (MatProductAlgorithm)algTypes[alg]));
2227       PetscCall(PetscInfo(B, "Use scalable algorithm, BN %" PetscInt_FMT ", fill*nz_allocated %g\n", B->cmap->N, (double)(product->fill * nz_local)));
2228     }
2229   }
2230 
2231   /* Get runtime option */
2232   if (product->api_user) {
2233     PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatTransposeMatMult", "Mat");
2234     PetscCall(PetscOptionsEList("-mattransposematmult_via", "Algorithmic approach", "MatTransposeMatMult", algTypes, nalg, algTypes[alg], &alg, &flg));
2235     PetscOptionsEnd();
2236   } else {
2237     PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatProduct_AtB", "Mat");
2238     PetscCall(PetscOptionsEList("-mat_product_algorithm", "Algorithmic approach", "MatTransposeMatMult", algTypes, nalg, algTypes[alg], &alg, &flg));
2239     PetscOptionsEnd();
2240   }
2241   if (flg) { PetscCall(MatProductSetAlgorithm(C, (MatProductAlgorithm)algTypes[alg])); }
2242 
2243   C->ops->productsymbolic = MatProductSymbolic_AtB_MPIAIJ_MPIAIJ;
2244   PetscFunctionReturn(0);
2245 }
2246 
2247 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_PtAP(Mat C) {
2248   Mat_Product *product = C->product;
2249   Mat          A = product->A, P = product->B;
2250   MPI_Comm     comm;
2251   PetscBool    flg;
2252   PetscInt     alg = 1; /* set default algorithm */
2253 #if !defined(PETSC_HAVE_HYPRE)
2254   const char *algTypes[5] = {"scalable", "nonscalable", "allatonce", "allatonce_merged", "backend"};
2255   PetscInt    nalg        = 5;
2256 #else
2257   const char *algTypes[6] = {"scalable", "nonscalable", "allatonce", "allatonce_merged", "backend", "hypre"};
2258   PetscInt    nalg        = 6;
2259 #endif
2260   PetscInt pN = P->cmap->N;
2261 
2262   PetscFunctionBegin;
2263   /* Check matrix local sizes */
2264   PetscCall(PetscObjectGetComm((PetscObject)C, &comm));
2265   PetscCheck(A->rmap->rstart == P->rmap->rstart && A->rmap->rend == P->rmap->rend, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Matrix local dimensions are incompatible, Arow (%" PetscInt_FMT ", %" PetscInt_FMT ") != Prow (%" PetscInt_FMT ",%" PetscInt_FMT ")",
2266              A->rmap->rstart, A->rmap->rend, P->rmap->rstart, P->rmap->rend);
2267   PetscCheck(A->cmap->rstart == P->rmap->rstart && A->cmap->rend == P->rmap->rend, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Matrix local dimensions are incompatible, Acol (%" PetscInt_FMT ", %" PetscInt_FMT ") != Prow (%" PetscInt_FMT ",%" PetscInt_FMT ")",
2268              A->cmap->rstart, A->cmap->rend, P->rmap->rstart, P->rmap->rend);
2269 
2270   /* Set "nonscalable" as default algorithm */
2271   PetscCall(PetscStrcmp(C->product->alg, "default", &flg));
2272   if (flg) {
2273     PetscCall(MatProductSetAlgorithm(C, (MatProductAlgorithm)algTypes[alg]));
2274 
2275     /* Set "scalable" as default if BN and local nonzeros of A and B are large */
2276     if (pN > 100000) {
2277       MatInfo   Ainfo, Pinfo;
2278       PetscInt  nz_local;
2279       PetscBool alg_scalable_loc = PETSC_FALSE, alg_scalable;
2280 
2281       PetscCall(MatGetInfo(A, MAT_LOCAL, &Ainfo));
2282       PetscCall(MatGetInfo(P, MAT_LOCAL, &Pinfo));
2283       nz_local = (PetscInt)(Ainfo.nz_allocated + Pinfo.nz_allocated);
2284 
2285       if (pN > product->fill * nz_local) alg_scalable_loc = PETSC_TRUE;
2286       PetscCall(MPIU_Allreduce(&alg_scalable_loc, &alg_scalable, 1, MPIU_BOOL, MPI_LOR, comm));
2287 
2288       if (alg_scalable) {
2289         alg = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
2290         PetscCall(MatProductSetAlgorithm(C, (MatProductAlgorithm)algTypes[alg]));
2291       }
2292     }
2293   }
2294 
2295   /* Get runtime option */
2296   if (product->api_user) {
2297     PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatPtAP", "Mat");
2298     PetscCall(PetscOptionsEList("-matptap_via", "Algorithmic approach", "MatPtAP", algTypes, nalg, algTypes[alg], &alg, &flg));
2299     PetscOptionsEnd();
2300   } else {
2301     PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatProduct_PtAP", "Mat");
2302     PetscCall(PetscOptionsEList("-mat_product_algorithm", "Algorithmic approach", "MatPtAP", algTypes, nalg, algTypes[alg], &alg, &flg));
2303     PetscOptionsEnd();
2304   }
2305   if (flg) { PetscCall(MatProductSetAlgorithm(C, (MatProductAlgorithm)algTypes[alg])); }
2306 
2307   C->ops->productsymbolic = MatProductSymbolic_PtAP_MPIAIJ_MPIAIJ;
2308   PetscFunctionReturn(0);
2309 }
2310 
2311 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_RARt(Mat C) {
2312   Mat_Product *product = C->product;
2313   Mat          A = product->A, R = product->B;
2314 
2315   PetscFunctionBegin;
2316   /* Check matrix local sizes */
2317   PetscCheck(A->cmap->n == R->cmap->n && A->rmap->n == R->cmap->n, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Matrix local dimensions are incompatible, A local (%" PetscInt_FMT ", %" PetscInt_FMT "), R local (%" PetscInt_FMT ",%" PetscInt_FMT ")", A->rmap->n,
2318              A->rmap->n, R->rmap->n, R->cmap->n);
2319 
2320   C->ops->productsymbolic = MatProductSymbolic_RARt_MPIAIJ_MPIAIJ;
2321   PetscFunctionReturn(0);
2322 }
2323 
2324 /*
2325  Set options for ABC = A*B*C = A*(B*C); ABC's algorithm must be chosen from AB's algorithm
2326 */
2327 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_ABC(Mat C) {
2328   Mat_Product *product     = C->product;
2329   PetscBool    flg         = PETSC_FALSE;
2330   PetscInt     alg         = 1; /* default algorithm */
2331   const char  *algTypes[3] = {"scalable", "nonscalable", "seqmpi"};
2332   PetscInt     nalg        = 3;
2333 
2334   PetscFunctionBegin;
2335   /* Set default algorithm */
2336   PetscCall(PetscStrcmp(C->product->alg, "default", &flg));
2337   if (flg) { PetscCall(MatProductSetAlgorithm(C, (MatProductAlgorithm)algTypes[alg])); }
2338 
2339   /* Get runtime option */
2340   if (product->api_user) {
2341     PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatMatMatMult", "Mat");
2342     PetscCall(PetscOptionsEList("-matmatmatmult_via", "Algorithmic approach", "MatMatMatMult", algTypes, nalg, algTypes[alg], &alg, &flg));
2343     PetscOptionsEnd();
2344   } else {
2345     PetscOptionsBegin(PetscObjectComm((PetscObject)C), ((PetscObject)C)->prefix, "MatProduct_ABC", "Mat");
2346     PetscCall(PetscOptionsEList("-mat_product_algorithm", "Algorithmic approach", "MatProduct_ABC", algTypes, nalg, algTypes[alg], &alg, &flg));
2347     PetscOptionsEnd();
2348   }
2349   if (flg) { PetscCall(MatProductSetAlgorithm(C, (MatProductAlgorithm)algTypes[alg])); }
2350 
2351   C->ops->matmatmultsymbolic = MatMatMatMultSymbolic_MPIAIJ_MPIAIJ_MPIAIJ;
2352   C->ops->productsymbolic    = MatProductSymbolic_ABC;
2353   PetscFunctionReturn(0);
2354 }
2355 
2356 PETSC_INTERN PetscErrorCode MatProductSetFromOptions_MPIAIJ(Mat C) {
2357   Mat_Product *product = C->product;
2358 
2359   PetscFunctionBegin;
2360   switch (product->type) {
2361   case MATPRODUCT_AB: PetscCall(MatProductSetFromOptions_MPIAIJ_AB(C)); break;
2362   case MATPRODUCT_ABt: PetscCall(MatProductSetFromOptions_MPIAIJ_ABt(C)); break;
2363   case MATPRODUCT_AtB: PetscCall(MatProductSetFromOptions_MPIAIJ_AtB(C)); break;
2364   case MATPRODUCT_PtAP: PetscCall(MatProductSetFromOptions_MPIAIJ_PtAP(C)); break;
2365   case MATPRODUCT_RARt: PetscCall(MatProductSetFromOptions_MPIAIJ_RARt(C)); break;
2366   case MATPRODUCT_ABC: PetscCall(MatProductSetFromOptions_MPIAIJ_ABC(C)); break;
2367   default: break;
2368   }
2369   PetscFunctionReturn(0);
2370 }
2371