xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision ef46b1a67e276116c83b5d4ce8efc2932ea4fc0a) !
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_AB_MPIAIJ_MPIAIJ(Mat C)
19 {
20   Mat_Product         *product = C->product;
21   Mat                 A=product->A,B=product->B;
22   MatProductAlgorithm alg=product->alg;
23   PetscReal           fill=product->fill;
24   PetscBool           flg;
25 
26   PetscFunctionBegin;
27   /* scalable */
28   PetscCall(PetscStrcmp(alg,"scalable",&flg));
29   if (flg) {
30     PetscCall(MatMatMultSymbolic_MPIAIJ_MPIAIJ(A,B,fill,C));
31     PetscFunctionReturn(0);
32   }
33 
34   /* nonscalable */
35   PetscCall(PetscStrcmp(alg,"nonscalable",&flg));
36   if (flg) {
37     PetscCall(MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(A,B,fill,C));
38     PetscFunctionReturn(0);
39   }
40 
41   /* seqmpi */
42   PetscCall(PetscStrcmp(alg,"seqmpi",&flg));
43   if (flg) {
44     PetscCall(MatMatMultSymbolic_MPIAIJ_MPIAIJ_seqMPI(A,B,fill,C));
45     PetscFunctionReturn(0);
46   }
47 
48   /* backend general code */
49   PetscCall(PetscStrcmp(alg,"backend",&flg));
50   if (flg) {
51     PetscCall(MatProductSymbolic_MPIAIJBACKEND(C));
52     PetscFunctionReturn(0);
53   }
54 
55 #if defined(PETSC_HAVE_HYPRE)
56   PetscCall(PetscStrcmp(alg,"hypre",&flg));
57   if (flg) {
58     PetscCall(MatMatMultSymbolic_AIJ_AIJ_wHYPRE(A,B,fill,C));
59     PetscFunctionReturn(0);
60   }
61 #endif
62   SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_SUP,"Mat Product Algorithm is not supported");
63 }
64 
65 PetscErrorCode MatDestroy_MPIAIJ_MatMatMult(void *data)
66 {
67   Mat_APMPI      *ptap = (Mat_APMPI*)data;
68 
69   PetscFunctionBegin;
70   PetscCall(PetscFree2(ptap->startsj_s,ptap->startsj_r));
71   PetscCall(PetscFree(ptap->bufa));
72   PetscCall(MatDestroy(&ptap->P_loc));
73   PetscCall(MatDestroy(&ptap->P_oth));
74   PetscCall(MatDestroy(&ptap->Pt));
75   PetscCall(PetscFree(ptap->api));
76   PetscCall(PetscFree(ptap->apj));
77   PetscCall(PetscFree(ptap->apa));
78   PetscCall(PetscFree(ptap));
79   PetscFunctionReturn(0);
80 }
81 
82 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,Mat C)
83 {
84   Mat_MPIAIJ        *a  =(Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
85   Mat_SeqAIJ        *ad =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
86   Mat_SeqAIJ        *cd =(Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
87   PetscScalar       *cda=cd->a,*coa=co->a;
88   Mat_SeqAIJ        *p_loc,*p_oth;
89   PetscScalar       *apa,*ca;
90   PetscInt          cm =C->rmap->n;
91   Mat_APMPI         *ptap;
92   PetscInt          *api,*apj,*apJ,i,k;
93   PetscInt          cstart=C->cmap->rstart;
94   PetscInt          cdnz,conz,k0,k1;
95   const PetscScalar *dummy;
96   MPI_Comm          comm;
97   PetscMPIInt       size;
98 
99   PetscFunctionBegin;
100   MatCheckProduct(C,3);
101   ptap = (Mat_APMPI*)C->product->data;
102   PetscCheck(ptap,PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be computed. Missing data");
103   PetscCall(PetscObjectGetComm((PetscObject)A,&comm));
104   PetscCallMPI(MPI_Comm_size(comm,&size));
105 
106   PetscCheckFalse(!ptap->P_oth && size>1,PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"AP cannot be reused. Do not call MatProductClear()");
107 
108   /* flag CPU mask for C */
109 #if defined(PETSC_HAVE_DEVICE)
110   if (C->offloadmask != PETSC_OFFLOAD_UNALLOCATED) C->offloadmask = PETSC_OFFLOAD_CPU;
111   if (c->A->offloadmask != PETSC_OFFLOAD_UNALLOCATED) c->A->offloadmask = PETSC_OFFLOAD_CPU;
112   if (c->B->offloadmask != PETSC_OFFLOAD_UNALLOCATED) c->B->offloadmask = PETSC_OFFLOAD_CPU;
113 #endif
114 
115   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
116   /*-----------------------------------------------------*/
117   /* update numerical values of P_oth and P_loc */
118   PetscCall(MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth));
119   PetscCall(MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc));
120 
121   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
122   /*----------------------------------------------------------*/
123   /* get data from symbolic products */
124   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
125   p_oth = NULL;
126   if (size >1) {
127     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
128   }
129 
130   /* get apa for storing dense row A[i,:]*P */
131   apa = ptap->apa;
132 
133   api = ptap->api;
134   apj = ptap->apj;
135   /* trigger copy to CPU */
136   PetscCall(MatSeqAIJGetArrayRead(a->A,&dummy));
137   PetscCall(MatSeqAIJRestoreArrayRead(a->A,&dummy));
138   PetscCall(MatSeqAIJGetArrayRead(a->B,&dummy));
139   PetscCall(MatSeqAIJRestoreArrayRead(a->B,&dummy));
140   for (i=0; i<cm; i++) {
141     /* compute apa = A[i,:]*P */
142     AProw_nonscalable(i,ad,ao,p_loc,p_oth,apa);
143 
144     /* set values in C */
145     apJ  = apj + api[i];
146     cdnz = cd->i[i+1] - cd->i[i];
147     conz = co->i[i+1] - co->i[i];
148 
149     /* 1st off-diagonal part of C */
150     ca = coa + co->i[i];
151     k  = 0;
152     for (k0=0; k0<conz; k0++) {
153       if (apJ[k] >= cstart) break;
154       ca[k0]      = apa[apJ[k]];
155       apa[apJ[k++]] = 0.0;
156     }
157 
158     /* diagonal part of C */
159     ca = cda + cd->i[i];
160     for (k1=0; k1<cdnz; k1++) {
161       ca[k1]      = apa[apJ[k]];
162       apa[apJ[k++]] = 0.0;
163     }
164 
165     /* 2nd off-diagonal part of C */
166     ca = coa + co->i[i];
167     for (; k0<conz; k0++) {
168       ca[k0]      = apa[apJ[k]];
169       apa[apJ[k++]] = 0.0;
170     }
171   }
172   PetscCall(MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY));
173   PetscCall(MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY));
174   PetscFunctionReturn(0);
175 }
176 
177 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,PetscReal fill,Mat C)
178 {
179   MPI_Comm           comm;
180   PetscMPIInt        size;
181   Mat_APMPI          *ptap;
182   PetscFreeSpaceList free_space=NULL,current_space=NULL;
183   Mat_MPIAIJ         *a=(Mat_MPIAIJ*)A->data;
184   Mat_SeqAIJ         *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
185   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
186   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
187   PetscInt           *lnk,i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi;
188   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
189   PetscBT            lnkbt;
190   PetscReal          afill;
191   MatType            mtype;
192 
193   PetscFunctionBegin;
194   MatCheckProduct(C,4);
195   PetscCheck(!C->product->data,PETSC_COMM_SELF,PETSC_ERR_PLIB,"Extra product struct not empty");
196   PetscCall(PetscObjectGetComm((PetscObject)A,&comm));
197   PetscCallMPI(MPI_Comm_size(comm,&size));
198 
199   /* create struct Mat_APMPI and attached it to C later */
200   PetscCall(PetscNew(&ptap));
201 
202   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
203   PetscCall(MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth));
204 
205   /* get P_loc by taking all local rows of P */
206   PetscCall(MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc));
207 
208   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
209   pi_loc = p_loc->i; pj_loc = p_loc->j;
210   if (size > 1) {
211     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
212     pi_oth = p_oth->i; pj_oth = p_oth->j;
213   } else {
214     p_oth = NULL;
215     pi_oth = NULL; pj_oth = NULL;
216   }
217 
218   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
219   /*-------------------------------------------------------------------*/
220   PetscCall(PetscMalloc1(am+2,&api));
221   ptap->api = api;
222   api[0]    = 0;
223 
224   /* create and initialize a linked list */
225   PetscCall(PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt));
226 
227   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
228   PetscCall(PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space));
229   current_space = free_space;
230 
231   MatPreallocateBegin(comm,am,pn,dnz,onz);
232   for (i=0; i<am; i++) {
233     /* diagonal portion of A */
234     nzi = adi[i+1] - adi[i];
235     for (j=0; j<nzi; j++) {
236       row  = *adj++;
237       pnz  = pi_loc[row+1] - pi_loc[row];
238       Jptr = pj_loc + pi_loc[row];
239       /* add non-zero cols of P into the sorted linked list lnk */
240       PetscCall(PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt));
241     }
242     /* off-diagonal portion of A */
243     nzi = aoi[i+1] - aoi[i];
244     for (j=0; j<nzi; j++) {
245       row  = *aoj++;
246       pnz  = pi_oth[row+1] - pi_oth[row];
247       Jptr = pj_oth + pi_oth[row];
248       PetscCall(PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt));
249     }
250     /* add possible missing diagonal entry */
251     if (C->force_diagonals) {
252       j = i + rstart; /* column index */
253       PetscCall(PetscLLCondensedAddSorted(1,&j,lnk,lnkbt));
254     }
255 
256     apnz     = lnk[0];
257     api[i+1] = api[i] + apnz;
258 
259     /* if free space is not available, double the total space in the list */
260     if (current_space->local_remaining<apnz) {
261       PetscCall(PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space));
262       nspacedouble++;
263     }
264 
265     /* Copy data into free space, then initialize lnk */
266     PetscCall(PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt));
267     PetscCall(MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz));
268 
269     current_space->array           += apnz;
270     current_space->local_used      += apnz;
271     current_space->local_remaining -= apnz;
272   }
273 
274   /* Allocate space for apj, initialize apj, and */
275   /* destroy list of free space and other temporary array(s) */
276   PetscCall(PetscMalloc1(api[am]+1,&ptap->apj));
277   apj  = ptap->apj;
278   PetscCall(PetscFreeSpaceContiguous(&free_space,ptap->apj));
279   PetscCall(PetscLLDestroy(lnk,lnkbt));
280 
281   /* malloc apa to store dense row A[i,:]*P */
282   PetscCall(PetscCalloc1(pN,&ptap->apa));
283 
284   /* set and assemble symbolic parallel matrix C */
285   /*---------------------------------------------*/
286   PetscCall(MatSetSizes(C,am,pn,PETSC_DETERMINE,PETSC_DETERMINE));
287   PetscCall(MatSetBlockSizesFromMats(C,A,P));
288 
289   PetscCall(MatGetType(A,&mtype));
290   PetscCall(MatSetType(C,mtype));
291   PetscCall(MatMPIAIJSetPreallocation(C,0,dnz,0,onz));
292   MatPreallocateEnd(dnz,onz);
293 
294   PetscCall(MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(C, apj, api));
295   PetscCall(MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY));
296   PetscCall(MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY));
297   PetscCall(MatSetOption(C,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE));
298 
299   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
300   C->ops->productnumeric = MatProductNumeric_AB;
301 
302   /* attach the supporting struct to C for reuse */
303   C->product->data    = ptap;
304   C->product->destroy = MatDestroy_MPIAIJ_MatMatMult;
305 
306   /* set MatInfo */
307   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
308   if (afill < 1.0) afill = 1.0;
309   C->info.mallocs           = nspacedouble;
310   C->info.fill_ratio_given  = fill;
311   C->info.fill_ratio_needed = afill;
312 
313 #if defined(PETSC_USE_INFO)
314   if (api[am]) {
315     PetscCall(PetscInfo(C,"Reallocs %" PetscInt_FMT "; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill));
316     PetscCall(PetscInfo(C,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill));
317   } else {
318     PetscCall(PetscInfo(C,"Empty matrix product\n"));
319   }
320 #endif
321   PetscFunctionReturn(0);
322 }
323 
324 /* ------------------------------------------------------- */
325 static PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat,Mat,PetscReal,Mat);
326 static PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat,Mat,Mat);
327 
328 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_MPIDense_AB(Mat C)
329 {
330   Mat_Product *product = C->product;
331   Mat         A = product->A,B=product->B;
332 
333   PetscFunctionBegin;
334   if (A->cmap->rstart != B->rmap->rstart || A->cmap->rend != B->rmap->rend)
335     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);
336 
337   C->ops->matmultsymbolic = MatMatMultSymbolic_MPIAIJ_MPIDense;
338   C->ops->productsymbolic = MatProductSymbolic_AB;
339   PetscFunctionReturn(0);
340 }
341 /* -------------------------------------------------------------------- */
342 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_MPIDense_AtB(Mat C)
343 {
344   Mat_Product *product = C->product;
345   Mat         A = product->A,B=product->B;
346 
347   PetscFunctionBegin;
348   if (A->rmap->rstart != B->rmap->rstart || A->rmap->rend != B->rmap->rend)
349     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);
350 
351   C->ops->transposematmultsymbolic = MatTransposeMatMultSymbolic_MPIAIJ_MPIDense;
352   C->ops->productsymbolic          = MatProductSymbolic_AtB;
353   PetscFunctionReturn(0);
354 }
355 
356 /* --------------------------------------------------------------------- */
357 PETSC_INTERN PetscErrorCode MatProductSetFromOptions_MPIAIJ_MPIDense(Mat C)
358 {
359   Mat_Product    *product = C->product;
360 
361   PetscFunctionBegin;
362   switch (product->type) {
363   case MATPRODUCT_AB:
364     PetscCall(MatProductSetFromOptions_MPIAIJ_MPIDense_AB(C));
365     break;
366   case MATPRODUCT_AtB:
367     PetscCall(MatProductSetFromOptions_MPIAIJ_MPIDense_AtB(C));
368     break;
369   default:
370     break;
371   }
372   PetscFunctionReturn(0);
373 }
374 /* ------------------------------------------------------- */
375 
376 typedef struct {
377   Mat          workB,workB1;
378   MPI_Request  *rwaits,*swaits;
379   PetscInt     nsends,nrecvs;
380   MPI_Datatype *stype,*rtype;
381   PetscInt     blda;
382 } MPIAIJ_MPIDense;
383 
384 PetscErrorCode MatMPIAIJ_MPIDenseDestroy(void *ctx)
385 {
386   MPIAIJ_MPIDense *contents = (MPIAIJ_MPIDense*)ctx;
387   PetscInt        i;
388 
389   PetscFunctionBegin;
390   PetscCall(MatDestroy(&contents->workB));
391   PetscCall(MatDestroy(&contents->workB1));
392   for (i=0; i<contents->nsends; i++) {
393     PetscCallMPI(MPI_Type_free(&contents->stype[i]));
394   }
395   for (i=0; i<contents->nrecvs; i++) {
396     PetscCallMPI(MPI_Type_free(&contents->rtype[i]));
397   }
398   PetscCall(PetscFree4(contents->stype,contents->rtype,contents->rwaits,contents->swaits));
399   PetscCall(PetscFree(contents));
400   PetscFunctionReturn(0);
401 }
402 
403 static PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat A,Mat B,PetscReal fill,Mat C)
404 {
405   Mat_MPIAIJ      *aij=(Mat_MPIAIJ*)A->data;
406   PetscInt        nz=aij->B->cmap->n,nsends,nrecvs,i,nrows_to,j,blda,m,M,n,N;
407   MPIAIJ_MPIDense *contents;
408   VecScatter      ctx=aij->Mvctx;
409   PetscInt        Am=A->rmap->n,Bm=B->rmap->n,BN=B->cmap->N,Bbn,Bbn1,bs,nrows_from,numBb;
410   MPI_Comm        comm;
411   MPI_Datatype    type1,*stype,*rtype;
412   const PetscInt  *sindices,*sstarts,*rstarts;
413   PetscMPIInt     *disp;
414   PetscBool       cisdense;
415 
416   PetscFunctionBegin;
417   MatCheckProduct(C,4);
418   PetscCheck(!C->product->data,PetscObjectComm((PetscObject)C),PETSC_ERR_PLIB,"Product data not empty");
419   PetscCall(PetscObjectGetComm((PetscObject)A,&comm));
420   PetscCall(PetscObjectBaseTypeCompare((PetscObject)C,MATMPIDENSE,&cisdense));
421   if (!cisdense) {
422     PetscCall(MatSetType(C,((PetscObject)B)->type_name));
423   }
424   PetscCall(MatGetLocalSize(C,&m,&n));
425   PetscCall(MatGetSize(C,&M,&N));
426   if (m == PETSC_DECIDE || n == PETSC_DECIDE || M == PETSC_DECIDE || N == PETSC_DECIDE) {
427     PetscCall(MatSetSizes(C,Am,B->cmap->n,A->rmap->N,BN));
428   }
429   PetscCall(MatSetBlockSizesFromMats(C,A,B));
430   PetscCall(MatSetUp(C));
431   PetscCall(MatDenseGetLDA(B,&blda));
432   PetscCall(PetscNew(&contents));
433 
434   PetscCall(VecScatterGetRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,NULL,NULL));
435   PetscCall(VecScatterGetRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,NULL,NULL));
436 
437   /* Create column block of B and C for memory scalability when BN is too large */
438   /* Estimate Bbn, column size of Bb */
439   if (nz) {
440     Bbn1 = 2*Am*BN/nz;
441     if (!Bbn1) Bbn1 = 1;
442   } else Bbn1 = BN;
443 
444   bs = PetscAbs(B->cmap->bs);
445   Bbn1 = Bbn1/bs *bs; /* Bbn1 is a multiple of bs */
446   if (Bbn1 > BN) Bbn1 = BN;
447   PetscCallMPI(MPI_Allreduce(&Bbn1,&Bbn,1,MPIU_INT,MPI_MAX,comm));
448 
449   /* Enable runtime option for Bbn */
450   PetscOptionsBegin(comm,((PetscObject)C)->prefix,"MatMatMult","Mat");
451   PetscCall(PetscOptionsInt("-matmatmult_Bbn","Number of columns in Bb","MatMatMult",Bbn,&Bbn,NULL));
452   PetscOptionsEnd();
453   Bbn  = PetscMin(Bbn,BN);
454 
455   if (Bbn > 0 && Bbn < BN) {
456     numBb = BN/Bbn;
457     Bbn1 = BN - numBb*Bbn;
458   } else numBb = 0;
459 
460   if (numBb) {
461     PetscCall(PetscInfo(C,"use Bb, BN=%" PetscInt_FMT ", Bbn=%" PetscInt_FMT "; numBb=%" PetscInt_FMT "\n",BN,Bbn,numBb));
462     if (Bbn1) { /* Create workB1 for the remaining columns */
463       PetscCall(PetscInfo(C,"use Bb1, BN=%" PetscInt_FMT ", Bbn1=%" PetscInt_FMT "\n",BN,Bbn1));
464       /* Create work matrix used to store off processor rows of B needed for local product */
465       PetscCall(MatCreateSeqDense(PETSC_COMM_SELF,nz,Bbn1,NULL,&contents->workB1));
466     } else contents->workB1 = NULL;
467   }
468 
469   /* Create work matrix used to store off processor rows of B needed for local product */
470   PetscCall(MatCreateSeqDense(PETSC_COMM_SELF,nz,Bbn,NULL,&contents->workB));
471 
472   /* Use MPI derived data type to reduce memory required by the send/recv buffers */
473   PetscCall(PetscMalloc4(nsends,&stype,nrecvs,&rtype,nrecvs,&contents->rwaits,nsends,&contents->swaits));
474   contents->stype  = stype;
475   contents->nsends = nsends;
476 
477   contents->rtype  = rtype;
478   contents->nrecvs = nrecvs;
479   contents->blda   = blda;
480 
481   PetscCall(PetscMalloc1(Bm+1,&disp));
482   for (i=0; i<nsends; i++) {
483     nrows_to = sstarts[i+1]-sstarts[i];
484     for (j=0; j<nrows_to; j++) disp[j] = sindices[sstarts[i]+j]; /* rowB to be sent */
485     PetscCallMPI(MPI_Type_create_indexed_block(nrows_to,1,disp,MPIU_SCALAR,&type1));
486     PetscCallMPI(MPI_Type_create_resized(type1,0,blda*sizeof(PetscScalar),&stype[i]));
487     PetscCallMPI(MPI_Type_commit(&stype[i]));
488     PetscCallMPI(MPI_Type_free(&type1));
489   }
490 
491   for (i=0; i<nrecvs; i++) {
492     /* received values from a process form a (nrows_from x Bbn) row block in workB (column-wise) */
493     nrows_from = rstarts[i+1]-rstarts[i];
494     disp[0] = 0;
495     PetscCallMPI(MPI_Type_create_indexed_block(1,nrows_from,disp,MPIU_SCALAR,&type1));
496     PetscCallMPI(MPI_Type_create_resized(type1,0,nz*sizeof(PetscScalar),&rtype[i]));
497     PetscCallMPI(MPI_Type_commit(&rtype[i]));
498     PetscCallMPI(MPI_Type_free(&type1));
499   }
500 
501   PetscCall(PetscFree(disp));
502   PetscCall(VecScatterRestoreRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,NULL,NULL));
503   PetscCall(VecScatterRestoreRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,NULL,NULL));
504   PetscCall(MatSetOption(C,MAT_NO_OFF_PROC_ENTRIES,PETSC_TRUE));
505   PetscCall(MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY));
506   PetscCall(MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY));
507   PetscCall(MatSetOption(C,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE));
508 
509   C->product->data = contents;
510   C->product->destroy = MatMPIAIJ_MPIDenseDestroy;
511   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
512   PetscFunctionReturn(0);
513 }
514 
515 PETSC_INTERN PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat,Mat,Mat,const PetscBool);
516 /*
517     Performs an efficient scatter on the rows of B needed by this process; this is
518     a modification of the VecScatterBegin_() routines.
519 
520     Input: Bbidx = 0: B = Bb
521                  = 1: B = Bb1, see MatMatMultSymbolic_MPIAIJ_MPIDense()
522 */
523 PetscErrorCode MatMPIDenseScatter(Mat A,Mat B,PetscInt Bbidx,Mat C,Mat *outworkB)
524 {
525   Mat_MPIAIJ        *aij = (Mat_MPIAIJ*)A->data;
526   const PetscScalar *b;
527   PetscScalar       *rvalues;
528   VecScatter        ctx = aij->Mvctx;
529   const PetscInt    *sindices,*sstarts,*rstarts;
530   const PetscMPIInt *sprocs,*rprocs;
531   PetscInt          i,nsends,nrecvs;
532   MPI_Request       *swaits,*rwaits;
533   MPI_Comm          comm;
534   PetscMPIInt       tag=((PetscObject)ctx)->tag,ncols=B->cmap->N,nrows=aij->B->cmap->n,nsends_mpi,nrecvs_mpi;
535   MPIAIJ_MPIDense   *contents;
536   Mat               workB;
537   MPI_Datatype      *stype,*rtype;
538   PetscInt          blda;
539 
540   PetscFunctionBegin;
541   MatCheckProduct(C,4);
542   PetscCheck(C->product->data,PetscObjectComm((PetscObject)C),PETSC_ERR_PLIB,"Product data empty");
543   contents = (MPIAIJ_MPIDense*)C->product->data;
544   PetscCall(VecScatterGetRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,&sprocs,NULL/*bs*/));
545   PetscCall(VecScatterGetRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,&rprocs,NULL/*bs*/));
546   PetscCall(PetscMPIIntCast(nsends,&nsends_mpi));
547   PetscCall(PetscMPIIntCast(nrecvs,&nrecvs_mpi));
548   if (Bbidx == 0) workB = *outworkB = contents->workB;
549   else workB = *outworkB = contents->workB1;
550   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);
551   swaits = contents->swaits;
552   rwaits = contents->rwaits;
553 
554   PetscCall(MatDenseGetArrayRead(B,&b));
555   PetscCall(MatDenseGetLDA(B,&blda));
556   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);
557   PetscCall(MatDenseGetArray(workB,&rvalues));
558 
559   /* Post recv, use MPI derived data type to save memory */
560   PetscCall(PetscObjectGetComm((PetscObject)C,&comm));
561   rtype = contents->rtype;
562   for (i=0; i<nrecvs; i++) {
563     PetscCallMPI(MPI_Irecv(rvalues+(rstarts[i]-rstarts[0]),ncols,rtype[i],rprocs[i],tag,comm,rwaits+i));
564   }
565 
566   stype = contents->stype;
567   for (i=0; i<nsends; i++) {
568     PetscCallMPI(MPI_Isend(b,ncols,stype[i],sprocs[i],tag,comm,swaits+i));
569   }
570 
571   if (nrecvs) PetscCallMPI(MPI_Waitall(nrecvs_mpi,rwaits,MPI_STATUSES_IGNORE));
572   if (nsends) PetscCallMPI(MPI_Waitall(nsends_mpi,swaits,MPI_STATUSES_IGNORE));
573 
574   PetscCall(VecScatterRestoreRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,&sprocs,NULL));
575   PetscCall(VecScatterRestoreRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,&rprocs,NULL));
576   PetscCall(MatDenseRestoreArrayRead(B,&b));
577   PetscCall(MatDenseRestoreArray(workB,&rvalues));
578   PetscFunctionReturn(0);
579 }
580 
581 static PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat A,Mat B,Mat C)
582 {
583   Mat_MPIAIJ      *aij    = (Mat_MPIAIJ*)A->data;
584   Mat_MPIDense    *bdense = (Mat_MPIDense*)B->data;
585   Mat_MPIDense    *cdense = (Mat_MPIDense*)C->data;
586   Mat             workB;
587   MPIAIJ_MPIDense *contents;
588 
589   PetscFunctionBegin;
590   MatCheckProduct(C,3);
591   PetscCheck(C->product->data,PetscObjectComm((PetscObject)C),PETSC_ERR_PLIB,"Product data empty");
592   contents = (MPIAIJ_MPIDense*)C->product->data;
593   /* diagonal block of A times all local rows of B */
594   /* TODO: this calls a symbolic multiplication every time, which could be avoided */
595   PetscCall(MatMatMult(aij->A,bdense->A,MAT_REUSE_MATRIX,PETSC_DEFAULT,&cdense->A));
596   if (contents->workB->cmap->n == B->cmap->N) {
597     /* get off processor parts of B needed to complete C=A*B */
598     PetscCall(MatMPIDenseScatter(A,B,0,C,&workB));
599 
600     /* off-diagonal block of A times nonlocal rows of B */
601     PetscCall(MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A,PETSC_TRUE));
602   } else {
603     Mat       Bb,Cb;
604     PetscInt  BN=B->cmap->N,n=contents->workB->cmap->n,i;
605     PetscBool ccpu;
606 
607     PetscCheck(n > 0,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Column block size %" PetscInt_FMT " must be positive",n);
608     /* Prevent from unneeded copies back and forth from the GPU
609        when getting and restoring the submatrix
610        We need a proper GPU code for AIJ * dense in parallel */
611     PetscCall(MatBoundToCPU(C,&ccpu));
612     PetscCall(MatBindToCPU(C,PETSC_TRUE));
613     for (i=0; i<BN; i+=n) {
614       PetscCall(MatDenseGetSubMatrix(B,i,PetscMin(i+n,BN),&Bb));
615       PetscCall(MatDenseGetSubMatrix(C,i,PetscMin(i+n,BN),&Cb));
616 
617       /* get off processor parts of B needed to complete C=A*B */
618       PetscCall(MatMPIDenseScatter(A,Bb,(i+n)>BN,C,&workB));
619 
620       /* off-diagonal block of A times nonlocal rows of B */
621       cdense = (Mat_MPIDense*)Cb->data;
622       PetscCall(MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A,PETSC_TRUE));
623       PetscCall(MatDenseRestoreSubMatrix(B,&Bb));
624       PetscCall(MatDenseRestoreSubMatrix(C,&Cb));
625     }
626     PetscCall(MatBindToCPU(C,ccpu));
627   }
628   PetscFunctionReturn(0);
629 }
630 
631 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
632 {
633   Mat_MPIAIJ        *a   = (Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
634   Mat_SeqAIJ        *ad  = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
635   Mat_SeqAIJ        *cd  = (Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
636   PetscInt          *adi = ad->i,*adj,*aoi=ao->i,*aoj;
637   PetscScalar       *ada,*aoa,*cda=cd->a,*coa=co->a;
638   Mat_SeqAIJ        *p_loc,*p_oth;
639   PetscInt          *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
640   PetscScalar       *pa_loc,*pa_oth,*pa,valtmp,*ca;
641   PetscInt          cm    = C->rmap->n,anz,pnz;
642   Mat_APMPI         *ptap;
643   PetscScalar       *apa_sparse;
644   const PetscScalar *dummy;
645   PetscInt          *api,*apj,*apJ,i,j,k,row;
646   PetscInt          cstart = C->cmap->rstart;
647   PetscInt          cdnz,conz,k0,k1,nextp;
648   MPI_Comm          comm;
649   PetscMPIInt       size;
650 
651   PetscFunctionBegin;
652   MatCheckProduct(C,3);
653   ptap = (Mat_APMPI*)C->product->data;
654   PetscCheck(ptap,PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be computed. Missing data");
655   PetscCall(PetscObjectGetComm((PetscObject)C,&comm));
656   PetscCallMPI(MPI_Comm_size(comm,&size));
657   PetscCheckFalse(!ptap->P_oth && size>1,PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"AP cannot be reused. Do not call MatProductClear()");
658 
659   /* flag CPU mask for C */
660 #if defined(PETSC_HAVE_DEVICE)
661   if (C->offloadmask != PETSC_OFFLOAD_UNALLOCATED) C->offloadmask = PETSC_OFFLOAD_CPU;
662   if (c->A->offloadmask != PETSC_OFFLOAD_UNALLOCATED) c->A->offloadmask = PETSC_OFFLOAD_CPU;
663   if (c->B->offloadmask != PETSC_OFFLOAD_UNALLOCATED) c->B->offloadmask = PETSC_OFFLOAD_CPU;
664 #endif
665   apa_sparse = ptap->apa;
666 
667   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
668   /*-----------------------------------------------------*/
669   /* update numerical values of P_oth and P_loc */
670   PetscCall(MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth));
671   PetscCall(MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc));
672 
673   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
674   /*----------------------------------------------------------*/
675   /* get data from symbolic products */
676   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
677   pi_loc = p_loc->i; pj_loc = p_loc->j; pa_loc = p_loc->a;
678   if (size >1) {
679     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
680     pi_oth = p_oth->i; pj_oth = p_oth->j; pa_oth = p_oth->a;
681   } else {
682     p_oth = NULL; pi_oth = NULL; pj_oth = NULL; pa_oth = NULL;
683   }
684 
685   /* trigger copy to CPU */
686   PetscCall(MatSeqAIJGetArrayRead(a->A,&dummy));
687   PetscCall(MatSeqAIJRestoreArrayRead(a->A,&dummy));
688   PetscCall(MatSeqAIJGetArrayRead(a->B,&dummy));
689   PetscCall(MatSeqAIJRestoreArrayRead(a->B,&dummy));
690   api = ptap->api;
691   apj = ptap->apj;
692   for (i=0; i<cm; i++) {
693     apJ = apj + api[i];
694 
695     /* diagonal portion of A */
696     anz = adi[i+1] - adi[i];
697     adj = ad->j + adi[i];
698     ada = ad->a + adi[i];
699     for (j=0; j<anz; j++) {
700       row = adj[j];
701       pnz = pi_loc[row+1] - pi_loc[row];
702       pj  = pj_loc + pi_loc[row];
703       pa  = pa_loc + pi_loc[row];
704       /* perform sparse axpy */
705       valtmp = ada[j];
706       nextp  = 0;
707       for (k=0; nextp<pnz; k++) {
708         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
709           apa_sparse[k] += valtmp*pa[nextp++];
710         }
711       }
712       PetscCall(PetscLogFlops(2.0*pnz));
713     }
714 
715     /* off-diagonal portion of A */
716     anz = aoi[i+1] - aoi[i];
717     aoj = ao->j + aoi[i];
718     aoa = ao->a + aoi[i];
719     for (j=0; j<anz; j++) {
720       row = aoj[j];
721       pnz = pi_oth[row+1] - pi_oth[row];
722       pj  = pj_oth + pi_oth[row];
723       pa  = pa_oth + pi_oth[row];
724       /* perform sparse axpy */
725       valtmp = aoa[j];
726       nextp  = 0;
727       for (k=0; nextp<pnz; k++) {
728         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
729           apa_sparse[k] += valtmp*pa[nextp++];
730         }
731       }
732       PetscCall(PetscLogFlops(2.0*pnz));
733     }
734 
735     /* set values in C */
736     cdnz = cd->i[i+1] - cd->i[i];
737     conz = co->i[i+1] - co->i[i];
738 
739     /* 1st off-diagonal part of C */
740     ca = coa + co->i[i];
741     k  = 0;
742     for (k0=0; k0<conz; k0++) {
743       if (apJ[k] >= cstart) break;
744       ca[k0]        = apa_sparse[k];
745       apa_sparse[k] = 0.0;
746       k++;
747     }
748 
749     /* diagonal part of C */
750     ca = cda + cd->i[i];
751     for (k1=0; k1<cdnz; k1++) {
752       ca[k1]        = apa_sparse[k];
753       apa_sparse[k] = 0.0;
754       k++;
755     }
756 
757     /* 2nd off-diagonal part of C */
758     ca = coa + co->i[i];
759     for (; k0<conz; k0++) {
760       ca[k0]        = apa_sparse[k];
761       apa_sparse[k] = 0.0;
762       k++;
763     }
764   }
765   PetscCall(MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY));
766   PetscCall(MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY));
767   PetscFunctionReturn(0);
768 }
769 
770 /* same as MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(), except using LLCondensed to avoid O(BN) memory requirement */
771 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat C)
772 {
773   MPI_Comm           comm;
774   PetscMPIInt        size;
775   Mat_APMPI          *ptap;
776   PetscFreeSpaceList free_space = NULL,current_space=NULL;
777   Mat_MPIAIJ         *a  = (Mat_MPIAIJ*)A->data;
778   Mat_SeqAIJ         *ad = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
779   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
780   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
781   PetscInt           i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi,*lnk,apnz_max=1;
782   PetscInt           am=A->rmap->n,pn=P->cmap->n,pm=P->rmap->n,lsize=pn+20;
783   PetscReal          afill;
784   MatType            mtype;
785 
786   PetscFunctionBegin;
787   MatCheckProduct(C,4);
788   PetscCheck(!C->product->data,PETSC_COMM_SELF,PETSC_ERR_PLIB,"Extra product struct not empty");
789   PetscCall(PetscObjectGetComm((PetscObject)A,&comm));
790   PetscCallMPI(MPI_Comm_size(comm,&size));
791 
792   /* create struct Mat_APMPI and attached it to C later */
793   PetscCall(PetscNew(&ptap));
794 
795   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
796   PetscCall(MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth));
797 
798   /* get P_loc by taking all local rows of P */
799   PetscCall(MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc));
800 
801   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
802   pi_loc = p_loc->i; pj_loc = p_loc->j;
803   if (size > 1) {
804     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
805     pi_oth = p_oth->i; pj_oth = p_oth->j;
806   } else {
807     p_oth  = NULL;
808     pi_oth = NULL; pj_oth = NULL;
809   }
810 
811   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
812   /*-------------------------------------------------------------------*/
813   PetscCall(PetscMalloc1(am+2,&api));
814   ptap->api = api;
815   api[0]    = 0;
816 
817   PetscCall(PetscLLCondensedCreate_Scalable(lsize,&lnk));
818 
819   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
820   PetscCall(PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space));
821   current_space = free_space;
822   MatPreallocateBegin(comm,am,pn,dnz,onz);
823   for (i=0; i<am; i++) {
824     /* diagonal portion of A */
825     nzi = adi[i+1] - adi[i];
826     for (j=0; j<nzi; j++) {
827       row  = *adj++;
828       pnz  = pi_loc[row+1] - pi_loc[row];
829       Jptr = pj_loc + pi_loc[row];
830       /* Expand list if it is not long enough */
831       if (pnz+apnz_max > lsize) {
832         lsize = pnz+apnz_max;
833         PetscCall(PetscLLCondensedExpand_Scalable(lsize, &lnk));
834       }
835       /* add non-zero cols of P into the sorted linked list lnk */
836       PetscCall(PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk));
837       apnz     = *lnk; /* The first element in the list is the number of items in the list */
838       api[i+1] = api[i] + apnz;
839       if (apnz > apnz_max) apnz_max = apnz + 1; /* '1' for diagonal entry */
840     }
841     /* off-diagonal portion of A */
842     nzi = aoi[i+1] - aoi[i];
843     for (j=0; j<nzi; j++) {
844       row  = *aoj++;
845       pnz  = pi_oth[row+1] - pi_oth[row];
846       Jptr = pj_oth + pi_oth[row];
847       /* Expand list if it is not long enough */
848       if (pnz+apnz_max > lsize) {
849         lsize = pnz + apnz_max;
850         PetscCall(PetscLLCondensedExpand_Scalable(lsize, &lnk));
851       }
852       /* add non-zero cols of P into the sorted linked list lnk */
853       PetscCall(PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk));
854       apnz     = *lnk;  /* The first element in the list is the number of items in the list */
855       api[i+1] = api[i] + apnz;
856       if (apnz > apnz_max) apnz_max = apnz + 1; /* '1' for diagonal entry */
857     }
858 
859     /* add missing diagonal entry */
860     if (C->force_diagonals) {
861       j = i + rstart; /* column index */
862       PetscCall(PetscLLCondensedAddSorted_Scalable(1,&j,lnk));
863     }
864 
865     apnz     = *lnk;
866     api[i+1] = api[i] + apnz;
867     if (apnz > apnz_max) apnz_max = apnz;
868 
869     /* if free space is not available, double the total space in the list */
870     if (current_space->local_remaining<apnz) {
871       PetscCall(PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space));
872       nspacedouble++;
873     }
874 
875     /* Copy data into free space, then initialize lnk */
876     PetscCall(PetscLLCondensedClean_Scalable(apnz,current_space->array,lnk));
877     PetscCall(MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz));
878 
879     current_space->array           += apnz;
880     current_space->local_used      += apnz;
881     current_space->local_remaining -= apnz;
882   }
883 
884   /* Allocate space for apj, initialize apj, and */
885   /* destroy list of free space and other temporary array(s) */
886   PetscCall(PetscMalloc1(api[am]+1,&ptap->apj));
887   apj  = ptap->apj;
888   PetscCall(PetscFreeSpaceContiguous(&free_space,ptap->apj));
889   PetscCall(PetscLLCondensedDestroy_Scalable(lnk));
890 
891   /* create and assemble symbolic parallel matrix C */
892   /*----------------------------------------------------*/
893   PetscCall(MatSetSizes(C,am,pn,PETSC_DETERMINE,PETSC_DETERMINE));
894   PetscCall(MatSetBlockSizesFromMats(C,A,P));
895   PetscCall(MatGetType(A,&mtype));
896   PetscCall(MatSetType(C,mtype));
897   PetscCall(MatMPIAIJSetPreallocation(C,0,dnz,0,onz));
898   MatPreallocateEnd(dnz,onz);
899 
900   /* malloc apa for assembly C */
901   PetscCall(PetscCalloc1(apnz_max,&ptap->apa));
902 
903   PetscCall(MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(C, apj, api));
904   PetscCall(MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY));
905   PetscCall(MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY));
906   PetscCall(MatSetOption(C,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE));
907 
908   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ;
909   C->ops->productnumeric = MatProductNumeric_AB;
910 
911   /* attach the supporting struct to C for reuse */
912   C->product->data    = ptap;
913   C->product->destroy = MatDestroy_MPIAIJ_MatMatMult;
914 
915   /* set MatInfo */
916   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
917   if (afill < 1.0) afill = 1.0;
918   C->info.mallocs           = nspacedouble;
919   C->info.fill_ratio_given  = fill;
920   C->info.fill_ratio_needed = afill;
921 
922 #if defined(PETSC_USE_INFO)
923   if (api[am]) {
924     PetscCall(PetscInfo(C,"Reallocs %" PetscInt_FMT "; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill));
925     PetscCall(PetscInfo(C,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill));
926   } else {
927     PetscCall(PetscInfo(C,"Empty matrix product\n"));
928   }
929 #endif
930   PetscFunctionReturn(0);
931 }
932 
933 /* This function is needed for the seqMPI matrix-matrix multiplication.  */
934 /* Three input arrays are merged to one output array. The size of the    */
935 /* output array is also output. Duplicate entries only show up once.     */
936 static void Merge3SortedArrays(PetscInt  size1, PetscInt *in1,
937                                PetscInt  size2, PetscInt *in2,
938                                PetscInt  size3, PetscInt *in3,
939                                PetscInt *size4, PetscInt *out)
940 {
941   int i = 0, j = 0, k = 0, l = 0;
942 
943   /* Traverse all three arrays */
944   while (i<size1 && j<size2 && k<size3) {
945     if (in1[i] < in2[j] && in1[i] < in3[k]) {
946       out[l++] = in1[i++];
947     }
948     else if (in2[j] < in1[i] && in2[j] < in3[k]) {
949       out[l++] = in2[j++];
950     }
951     else if (in3[k] < in1[i] && in3[k] < in2[j]) {
952       out[l++] = in3[k++];
953     }
954     else if (in1[i] == in2[j] && in1[i] < in3[k]) {
955       out[l++] = in1[i];
956       i++, j++;
957     }
958     else if (in1[i] == in3[k] && in1[i] < in2[j]) {
959       out[l++] = in1[i];
960       i++, k++;
961     }
962     else if (in3[k] == in2[j] && in2[j] < in1[i])  {
963       out[l++] = in2[j];
964       k++, j++;
965     }
966     else if (in1[i] == in2[j] && in1[i] == in3[k]) {
967       out[l++] = in1[i];
968       i++, j++, k++;
969     }
970   }
971 
972   /* Traverse two remaining arrays */
973   while (i<size1 && j<size2) {
974     if (in1[i] < in2[j]) {
975       out[l++] = in1[i++];
976     }
977     else if (in1[i] > in2[j]) {
978       out[l++] = in2[j++];
979     }
980     else {
981       out[l++] = in1[i];
982       i++, j++;
983     }
984   }
985 
986   while (i<size1 && k<size3) {
987     if (in1[i] < in3[k]) {
988       out[l++] = in1[i++];
989     }
990     else if (in1[i] > in3[k]) {
991       out[l++] = in3[k++];
992     }
993     else {
994       out[l++] = in1[i];
995       i++, k++;
996     }
997   }
998 
999   while (k<size3 && j<size2)  {
1000     if (in3[k] < in2[j]) {
1001       out[l++] = in3[k++];
1002     }
1003     else if (in3[k] > in2[j]) {
1004       out[l++] = in2[j++];
1005     }
1006     else {
1007       out[l++] = in3[k];
1008       k++, j++;
1009     }
1010   }
1011 
1012   /* Traverse one remaining array */
1013   while (i<size1) out[l++] = in1[i++];
1014   while (j<size2) out[l++] = in2[j++];
1015   while (k<size3) out[l++] = in3[k++];
1016 
1017   *size4 = l;
1018 }
1019 
1020 /* This matrix-matrix multiplication algorithm divides the multiplication into three multiplications and  */
1021 /* adds up the products. Two of these three multiplications are performed with existing (sequential)      */
1022 /* matrix-matrix multiplications.  */
1023 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_seqMPI(Mat A, Mat P, PetscReal fill, Mat C)
1024 {
1025   MPI_Comm           comm;
1026   PetscMPIInt        size;
1027   Mat_APMPI          *ptap;
1028   PetscFreeSpaceList free_space_diag=NULL, current_space=NULL;
1029   Mat_MPIAIJ         *a  =(Mat_MPIAIJ*)A->data;
1030   Mat_SeqAIJ         *ad =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc;
1031   Mat_MPIAIJ         *p  =(Mat_MPIAIJ*)P->data;
1032   Mat_SeqAIJ         *adpd_seq, *p_off, *aopoth_seq;
1033   PetscInt           adponz, adpdnz;
1034   PetscInt           *pi_loc,*dnz,*onz;
1035   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,rstart=A->rmap->rstart;
1036   PetscInt           *lnk,i, i1=0,pnz,row,*adpoi,*adpoj, *api, *adpoJ, *aopJ, *apJ,*Jptr, aopnz, nspacedouble=0,j,nzi,
1037                      *apj,apnz, *adpdi, *adpdj, *adpdJ, *poff_i, *poff_j, *j_temp, *aopothi, *aopothj;
1038   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n, p_colstart, p_colend;
1039   PetscBT            lnkbt;
1040   PetscReal          afill;
1041   PetscMPIInt        rank;
1042   Mat                adpd, aopoth;
1043   MatType            mtype;
1044   const char         *prefix;
1045 
1046   PetscFunctionBegin;
1047   MatCheckProduct(C,4);
1048   PetscCheck(!C->product->data,PETSC_COMM_SELF,PETSC_ERR_PLIB,"Extra product struct not empty");
1049   PetscCall(PetscObjectGetComm((PetscObject)A,&comm));
1050   PetscCallMPI(MPI_Comm_size(comm,&size));
1051   PetscCallMPI(MPI_Comm_rank(comm, &rank));
1052   PetscCall(MatGetOwnershipRangeColumn(P, &p_colstart, &p_colend));
1053 
1054   /* create struct Mat_APMPI and attached it to C later */
1055   PetscCall(PetscNew(&ptap));
1056 
1057   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
1058   PetscCall(MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth));
1059 
1060   /* get P_loc by taking all local rows of P */
1061   PetscCall(MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc));
1062 
1063   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
1064   pi_loc = p_loc->i;
1065 
1066   /* Allocate memory for the i arrays of the matrices A*P, A_diag*P_off and A_offd * P */
1067   PetscCall(PetscMalloc1(am+2,&api));
1068   PetscCall(PetscMalloc1(am+2,&adpoi));
1069 
1070   adpoi[0]    = 0;
1071   ptap->api = api;
1072   api[0] = 0;
1073 
1074   /* create and initialize a linked list, will be used for both A_diag * P_loc_off and A_offd * P_oth */
1075   PetscCall(PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt));
1076   MatPreallocateBegin(comm,am,pn,dnz,onz);
1077 
1078   /* Symbolic calc of A_loc_diag * P_loc_diag */
1079   PetscCall(MatGetOptionsPrefix(A,&prefix));
1080   PetscCall(MatProductCreate(a->A,p->A,NULL,&adpd));
1081   PetscCall(MatGetOptionsPrefix(A,&prefix));
1082   PetscCall(MatSetOptionsPrefix(adpd,prefix));
1083   PetscCall(MatAppendOptionsPrefix(adpd,"inner_diag_"));
1084 
1085   PetscCall(MatProductSetType(adpd,MATPRODUCT_AB));
1086   PetscCall(MatProductSetAlgorithm(adpd,"sorted"));
1087   PetscCall(MatProductSetFill(adpd,fill));
1088   PetscCall(MatProductSetFromOptions(adpd));
1089 
1090   adpd->force_diagonals = C->force_diagonals;
1091   PetscCall(MatProductSymbolic(adpd));
1092 
1093   adpd_seq = (Mat_SeqAIJ*)((adpd)->data);
1094   adpdi = adpd_seq->i; adpdj = adpd_seq->j;
1095   p_off = (Mat_SeqAIJ*)((p->B)->data);
1096   poff_i = p_off->i; poff_j = p_off->j;
1097 
1098   /* j_temp stores indices of a result row before they are added to the linked list */
1099   PetscCall(PetscMalloc1(pN+2,&j_temp));
1100 
1101   /* Symbolic calc of the A_diag * p_loc_off */
1102   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
1103   PetscCall(PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space_diag));
1104   current_space = free_space_diag;
1105 
1106   for (i=0; i<am; i++) {
1107     /* A_diag * P_loc_off */
1108     nzi = adi[i+1] - adi[i];
1109     for (j=0; j<nzi; j++) {
1110       row  = *adj++;
1111       pnz  = poff_i[row+1] - poff_i[row];
1112       Jptr = poff_j + poff_i[row];
1113       for (i1 = 0; i1 < pnz; i1++) {
1114         j_temp[i1] = p->garray[Jptr[i1]];
1115       }
1116       /* add non-zero cols of P into the sorted linked list lnk */
1117       PetscCall(PetscLLCondensedAddSorted(pnz,j_temp,lnk,lnkbt));
1118     }
1119 
1120     adponz     = lnk[0];
1121     adpoi[i+1] = adpoi[i] + adponz;
1122 
1123     /* if free space is not available, double the total space in the list */
1124     if (current_space->local_remaining<adponz) {
1125       PetscCall(PetscFreeSpaceGet(PetscIntSumTruncate(adponz,current_space->total_array_size),&current_space));
1126       nspacedouble++;
1127     }
1128 
1129     /* Copy data into free space, then initialize lnk */
1130     PetscCall(PetscLLCondensedClean(pN,adponz,current_space->array,lnk,lnkbt));
1131 
1132     current_space->array           += adponz;
1133     current_space->local_used      += adponz;
1134     current_space->local_remaining -= adponz;
1135   }
1136 
1137   /* Symbolic calc of A_off * P_oth */
1138   PetscCall(MatSetOptionsPrefix(a->B,prefix));
1139   PetscCall(MatAppendOptionsPrefix(a->B,"inner_offdiag_"));
1140   PetscCall(MatCreate(PETSC_COMM_SELF,&aopoth));
1141   PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ(a->B, ptap->P_oth, fill, aopoth));
1142   aopoth_seq = (Mat_SeqAIJ*)((aopoth)->data);
1143   aopothi = aopoth_seq->i; aopothj = aopoth_seq->j;
1144 
1145   /* Allocate space for apj, adpj, aopj, ... */
1146   /* destroy lists of free space and other temporary array(s) */
1147 
1148   PetscCall(PetscMalloc1(aopothi[am] + adpoi[am] + adpdi[am]+2, &ptap->apj));
1149   PetscCall(PetscMalloc1(adpoi[am]+2, &adpoj));
1150 
1151   /* Copy from linked list to j-array */
1152   PetscCall(PetscFreeSpaceContiguous(&free_space_diag,adpoj));
1153   PetscCall(PetscLLDestroy(lnk,lnkbt));
1154 
1155   adpoJ = adpoj;
1156   adpdJ = adpdj;
1157   aopJ = aopothj;
1158   apj  = ptap->apj;
1159   apJ = apj; /* still empty */
1160 
1161   /* Merge j-arrays of A_off * P, A_diag * P_loc_off, and */
1162   /* A_diag * P_loc_diag to get A*P */
1163   for (i = 0; i < am; i++) {
1164     aopnz  =  aopothi[i+1] -  aopothi[i];
1165     adponz = adpoi[i+1] - adpoi[i];
1166     adpdnz = adpdi[i+1] - adpdi[i];
1167 
1168     /* Correct indices from A_diag*P_diag */
1169     for (i1 = 0; i1 < adpdnz; i1++) {
1170       adpdJ[i1] += p_colstart;
1171     }
1172     /* Merge j-arrays of A_diag * P_loc_off and A_diag * P_loc_diag and A_off * P_oth */
1173     Merge3SortedArrays(adponz, adpoJ, adpdnz, adpdJ, aopnz, aopJ, &apnz, apJ);
1174     PetscCall(MatPreallocateSet(i+rstart, apnz, apJ, dnz, onz));
1175 
1176     aopJ += aopnz;
1177     adpoJ += adponz;
1178     adpdJ += adpdnz;
1179     apJ += apnz;
1180     api[i+1] = api[i] + apnz;
1181   }
1182 
1183   /* malloc apa to store dense row A[i,:]*P */
1184   PetscCall(PetscCalloc1(pN+2,&ptap->apa));
1185 
1186   /* create and assemble symbolic parallel matrix C */
1187   PetscCall(MatSetSizes(C,am,pn,PETSC_DETERMINE,PETSC_DETERMINE));
1188   PetscCall(MatSetBlockSizesFromMats(C,A,P));
1189   PetscCall(MatGetType(A,&mtype));
1190   PetscCall(MatSetType(C,mtype));
1191   PetscCall(MatMPIAIJSetPreallocation(C,0,dnz,0,onz));
1192   MatPreallocateEnd(dnz,onz);
1193 
1194   PetscCall(MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(C, apj, api));
1195   PetscCall(MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY));
1196   PetscCall(MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY));
1197   PetscCall(MatSetOption(C,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE));
1198 
1199   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1200   C->ops->productnumeric = MatProductNumeric_AB;
1201 
1202   /* attach the supporting struct to C for reuse */
1203   C->product->data    = ptap;
1204   C->product->destroy = MatDestroy_MPIAIJ_MatMatMult;
1205 
1206   /* set MatInfo */
1207   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
1208   if (afill < 1.0) afill = 1.0;
1209   C->info.mallocs           = nspacedouble;
1210   C->info.fill_ratio_given  = fill;
1211   C->info.fill_ratio_needed = afill;
1212 
1213 #if defined(PETSC_USE_INFO)
1214   if (api[am]) {
1215     PetscCall(PetscInfo(C,"Reallocs %" PetscInt_FMT "; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill));
1216     PetscCall(PetscInfo(C,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill));
1217   } else {
1218     PetscCall(PetscInfo(C,"Empty matrix product\n"));
1219   }
1220 #endif
1221 
1222   PetscCall(MatDestroy(&aopoth));
1223   PetscCall(MatDestroy(&adpd));
1224   PetscCall(PetscFree(j_temp));
1225   PetscCall(PetscFree(adpoj));
1226   PetscCall(PetscFree(adpoi));
1227   PetscFunctionReturn(0);
1228 }
1229 
1230 /*-------------------------------------------------------------------------*/
1231 /* This routine only works when scall=MAT_REUSE_MATRIX! */
1232 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult(Mat P,Mat A,Mat C)
1233 {
1234   Mat_APMPI      *ptap;
1235   Mat            Pt;
1236 
1237   PetscFunctionBegin;
1238   MatCheckProduct(C,3);
1239   ptap = (Mat_APMPI*)C->product->data;
1240   PetscCheck(ptap,PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be computed. Missing data");
1241   PetscCheck(ptap->Pt,PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtA cannot be reused. Do not call MatProductClear()");
1242 
1243   Pt   = ptap->Pt;
1244   PetscCall(MatTranspose(P,MAT_REUSE_MATRIX,&Pt));
1245   PetscCall(MatMatMultNumeric_MPIAIJ_MPIAIJ(Pt,A,C));
1246   PetscFunctionReturn(0);
1247 }
1248 
1249 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1250 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,PetscReal fill,Mat C)
1251 {
1252   Mat_APMPI           *ptap;
1253   Mat_MPIAIJ          *p=(Mat_MPIAIJ*)P->data;
1254   MPI_Comm            comm;
1255   PetscMPIInt         size,rank;
1256   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1257   PetscInt            pn=P->cmap->n,aN=A->cmap->N,an=A->cmap->n;
1258   PetscInt            *lnk,i,k,nsend,rstart;
1259   PetscBT             lnkbt;
1260   PetscMPIInt         tagi,tagj,*len_si,*len_s,*len_ri,nrecv;
1261   PETSC_UNUSED PetscMPIInt icompleted=0;
1262   PetscInt            **buf_rj,**buf_ri,**buf_ri_k,row,ncols,*cols;
1263   PetscInt            len,proc,*dnz,*onz,*owners,nzi;
1264   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1265   MPI_Request         *swaits,*rwaits;
1266   MPI_Status          *sstatus,rstatus;
1267   PetscLayout         rowmap;
1268   PetscInt            *owners_co,*coi,*coj;    /* i and j array of (p->B)^T*A*P - used in the communication */
1269   PetscMPIInt         *len_r,*id_r;    /* array of length of comm->size, store send/recv matrix values */
1270   PetscInt            *Jptr,*prmap=p->garray,con,j,Crmax;
1271   Mat_SeqAIJ          *a_loc,*c_loc,*c_oth;
1272   PetscTable          ta;
1273   MatType             mtype;
1274   const char          *prefix;
1275 
1276   PetscFunctionBegin;
1277   PetscCall(PetscObjectGetComm((PetscObject)A,&comm));
1278   PetscCallMPI(MPI_Comm_size(comm,&size));
1279   PetscCallMPI(MPI_Comm_rank(comm,&rank));
1280 
1281   /* create symbolic parallel matrix C */
1282   PetscCall(MatGetType(A,&mtype));
1283   PetscCall(MatSetType(C,mtype));
1284 
1285   C->ops->transposematmultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1286 
1287   /* create struct Mat_APMPI and attached it to C later */
1288   PetscCall(PetscNew(&ptap));
1289   ptap->reuse = MAT_INITIAL_MATRIX;
1290 
1291   /* (0) compute Rd = Pd^T, Ro = Po^T  */
1292   /* --------------------------------- */
1293   PetscCall(MatTranspose_SeqAIJ(p->A,MAT_INITIAL_MATRIX,&ptap->Rd));
1294   PetscCall(MatTranspose_SeqAIJ(p->B,MAT_INITIAL_MATRIX,&ptap->Ro));
1295 
1296   /* (1) compute symbolic A_loc */
1297   /* ---------------------------*/
1298   PetscCall(MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&ptap->A_loc));
1299 
1300   /* (2-1) compute symbolic C_oth = Ro*A_loc  */
1301   /* ------------------------------------ */
1302   PetscCall(MatGetOptionsPrefix(A,&prefix));
1303   PetscCall(MatSetOptionsPrefix(ptap->Ro,prefix));
1304   PetscCall(MatAppendOptionsPrefix(ptap->Ro,"inner_offdiag_"));
1305   PetscCall(MatCreate(PETSC_COMM_SELF,&ptap->C_oth));
1306   PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Ro,ptap->A_loc,fill,ptap->C_oth));
1307 
1308   /* (3) send coj of C_oth to other processors  */
1309   /* ------------------------------------------ */
1310   /* determine row ownership */
1311   PetscCall(PetscLayoutCreate(comm,&rowmap));
1312   rowmap->n  = pn;
1313   rowmap->bs = 1;
1314   PetscCall(PetscLayoutSetUp(rowmap));
1315   owners = rowmap->range;
1316 
1317   /* determine the number of messages to send, their lengths */
1318   PetscCall(PetscMalloc4(size,&len_s,size,&len_si,size,&sstatus,size+2,&owners_co));
1319   PetscCall(PetscArrayzero(len_s,size));
1320   PetscCall(PetscArrayzero(len_si,size));
1321 
1322   c_oth = (Mat_SeqAIJ*)ptap->C_oth->data;
1323   coi   = c_oth->i; coj = c_oth->j;
1324   con   = ptap->C_oth->rmap->n;
1325   proc  = 0;
1326   for (i=0; i<con; i++) {
1327     while (prmap[i] >= owners[proc+1]) proc++;
1328     len_si[proc]++;               /* num of rows in Co(=Pt*A) to be sent to [proc] */
1329     len_s[proc] += coi[i+1] - coi[i]; /* num of nonzeros in Co to be sent to [proc] */
1330   }
1331 
1332   len          = 0; /* max length of buf_si[], see (4) */
1333   owners_co[0] = 0;
1334   nsend        = 0;
1335   for (proc=0; proc<size; proc++) {
1336     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1337     if (len_s[proc]) {
1338       nsend++;
1339       len_si[proc] = 2*(len_si[proc] + 1); /* length of buf_si to be sent to [proc] */
1340       len         += len_si[proc];
1341     }
1342   }
1343 
1344   /* determine the number and length of messages to receive for coi and coj  */
1345   PetscCall(PetscGatherNumberOfMessages(comm,NULL,len_s,&nrecv));
1346   PetscCall(PetscGatherMessageLengths2(comm,nsend,nrecv,len_s,len_si,&id_r,&len_r,&len_ri));
1347 
1348   /* post the Irecv and Isend of coj */
1349   PetscCall(PetscCommGetNewTag(comm,&tagj));
1350   PetscCall(PetscPostIrecvInt(comm,tagj,nrecv,id_r,len_r,&buf_rj,&rwaits));
1351   PetscCall(PetscMalloc1(nsend+1,&swaits));
1352   for (proc=0, k=0; proc<size; proc++) {
1353     if (!len_s[proc]) continue;
1354     i    = owners_co[proc];
1355     PetscCallMPI(MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k));
1356     k++;
1357   }
1358 
1359   /* (2-2) compute symbolic C_loc = Rd*A_loc */
1360   /* ---------------------------------------- */
1361   PetscCall(MatSetOptionsPrefix(ptap->Rd,prefix));
1362   PetscCall(MatAppendOptionsPrefix(ptap->Rd,"inner_diag_"));
1363   PetscCall(MatCreate(PETSC_COMM_SELF,&ptap->C_loc));
1364   PetscCall(MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Rd,ptap->A_loc,fill,ptap->C_loc));
1365   c_loc = (Mat_SeqAIJ*)ptap->C_loc->data;
1366 
1367   /* receives coj are complete */
1368   for (i=0; i<nrecv; i++) {
1369     PetscCallMPI(MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus));
1370   }
1371   PetscCall(PetscFree(rwaits));
1372   if (nsend) PetscCallMPI(MPI_Waitall(nsend,swaits,sstatus));
1373 
1374   /* add received column indices into ta to update Crmax */
1375   a_loc = (Mat_SeqAIJ*)(ptap->A_loc)->data;
1376 
1377   /* create and initialize a linked list */
1378   PetscCall(PetscTableCreate(an,aN,&ta)); /* for compute Crmax */
1379   MatRowMergeMax_SeqAIJ(a_loc,ptap->A_loc->rmap->N,ta);
1380 
1381   for (k=0; k<nrecv; k++) {/* k-th received message */
1382     Jptr = buf_rj[k];
1383     for (j=0; j<len_r[k]; j++) {
1384       PetscCall(PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES));
1385     }
1386   }
1387   PetscCall(PetscTableGetCount(ta,&Crmax));
1388   PetscCall(PetscTableDestroy(&ta));
1389 
1390   /* (4) send and recv coi */
1391   /*-----------------------*/
1392   PetscCall(PetscCommGetNewTag(comm,&tagi));
1393   PetscCall(PetscPostIrecvInt(comm,tagi,nrecv,id_r,len_ri,&buf_ri,&rwaits));
1394   PetscCall(PetscMalloc1(len+1,&buf_s));
1395   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1396   for (proc=0,k=0; proc<size; proc++) {
1397     if (!len_s[proc]) continue;
1398     /* form outgoing message for i-structure:
1399          buf_si[0]:                 nrows to be sent
1400                [1:nrows]:           row index (global)
1401                [nrows+1:2*nrows+1]: i-structure index
1402     */
1403     /*-------------------------------------------*/
1404     nrows       = len_si[proc]/2 - 1; /* num of rows in Co to be sent to [proc] */
1405     buf_si_i    = buf_si + nrows+1;
1406     buf_si[0]   = nrows;
1407     buf_si_i[0] = 0;
1408     nrows       = 0;
1409     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1410       nzi = coi[i+1] - coi[i];
1411       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1412       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1413       nrows++;
1414     }
1415     PetscCallMPI(MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k));
1416     k++;
1417     buf_si += len_si[proc];
1418   }
1419   for (i=0; i<nrecv; i++) {
1420     PetscCallMPI(MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus));
1421   }
1422   PetscCall(PetscFree(rwaits));
1423   if (nsend) PetscCallMPI(MPI_Waitall(nsend,swaits,sstatus));
1424 
1425   PetscCall(PetscFree4(len_s,len_si,sstatus,owners_co));
1426   PetscCall(PetscFree(len_ri));
1427   PetscCall(PetscFree(swaits));
1428   PetscCall(PetscFree(buf_s));
1429 
1430   /* (5) compute the local portion of C      */
1431   /* ------------------------------------------ */
1432   /* set initial free space to be Crmax, sufficient for holding nozeros in each row of C */
1433   PetscCall(PetscFreeSpaceGet(Crmax,&free_space));
1434   current_space = free_space;
1435 
1436   PetscCall(PetscMalloc3(nrecv,&buf_ri_k,nrecv,&nextrow,nrecv,&nextci));
1437   for (k=0; k<nrecv; k++) {
1438     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1439     nrows       = *buf_ri_k[k];
1440     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1441     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th recved i-structure  */
1442   }
1443 
1444   MatPreallocateBegin(comm,pn,an,dnz,onz);
1445   PetscCall(PetscLLCondensedCreate(Crmax,aN,&lnk,&lnkbt));
1446   for (i=0; i<pn; i++) { /* for each local row of C */
1447     /* add C_loc into C */
1448     nzi  = c_loc->i[i+1] - c_loc->i[i];
1449     Jptr = c_loc->j + c_loc->i[i];
1450     PetscCall(PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt));
1451 
1452     /* add received col data into lnk */
1453     for (k=0; k<nrecv; k++) { /* k-th received message */
1454       if (i == *nextrow[k]) { /* i-th row */
1455         nzi  = *(nextci[k]+1) - *nextci[k];
1456         Jptr = buf_rj[k] + *nextci[k];
1457         PetscCall(PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt));
1458         nextrow[k]++; nextci[k]++;
1459       }
1460     }
1461 
1462     /* add missing diagonal entry */
1463     if (C->force_diagonals) {
1464       k = i + owners[rank]; /* column index */
1465       PetscCall(PetscLLCondensedAddSorted(1,&k,lnk,lnkbt));
1466     }
1467 
1468     nzi = lnk[0];
1469 
1470     /* copy data into free space, then initialize lnk */
1471     PetscCall(PetscLLCondensedClean(aN,nzi,current_space->array,lnk,lnkbt));
1472     PetscCall(MatPreallocateSet(i+owners[rank],nzi,current_space->array,dnz,onz));
1473   }
1474   PetscCall(PetscFree3(buf_ri_k,nextrow,nextci));
1475   PetscCall(PetscLLDestroy(lnk,lnkbt));
1476   PetscCall(PetscFreeSpaceDestroy(free_space));
1477 
1478   /* local sizes and preallocation */
1479   PetscCall(MatSetSizes(C,pn,an,PETSC_DETERMINE,PETSC_DETERMINE));
1480   if (P->cmap->bs > 0) PetscCall(PetscLayoutSetBlockSize(C->rmap,P->cmap->bs));
1481   if (A->cmap->bs > 0) PetscCall(PetscLayoutSetBlockSize(C->cmap,A->cmap->bs));
1482   PetscCall(MatMPIAIJSetPreallocation(C,0,dnz,0,onz));
1483   MatPreallocateEnd(dnz,onz);
1484 
1485   /* add C_loc and C_oth to C */
1486   PetscCall(MatGetOwnershipRange(C,&rstart,NULL));
1487   for (i=0; i<pn; i++) {
1488     ncols = c_loc->i[i+1] - c_loc->i[i];
1489     cols  = c_loc->j + c_loc->i[i];
1490     row   = rstart + i;
1491     PetscCall(MatSetValues(C,1,(const PetscInt*)&row,ncols,(const PetscInt*)cols,NULL,INSERT_VALUES));
1492 
1493     if (C->force_diagonals) {
1494       PetscCall(MatSetValues(C,1,(const PetscInt*)&row,1,(const PetscInt*)&row,NULL,INSERT_VALUES));
1495     }
1496   }
1497   for (i=0; i<con; i++) {
1498     ncols = c_oth->i[i+1] - c_oth->i[i];
1499     cols  = c_oth->j + c_oth->i[i];
1500     row   = prmap[i];
1501     PetscCall(MatSetValues(C,1,(const PetscInt*)&row,ncols,(const PetscInt*)cols,NULL,INSERT_VALUES));
1502   }
1503   PetscCall(MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY));
1504   PetscCall(MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY));
1505   PetscCall(MatSetOption(C,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE));
1506 
1507   /* members in merge */
1508   PetscCall(PetscFree(id_r));
1509   PetscCall(PetscFree(len_r));
1510   PetscCall(PetscFree(buf_ri[0]));
1511   PetscCall(PetscFree(buf_ri));
1512   PetscCall(PetscFree(buf_rj[0]));
1513   PetscCall(PetscFree(buf_rj));
1514   PetscCall(PetscLayoutDestroy(&rowmap));
1515 
1516   /* attach the supporting struct to C for reuse */
1517   C->product->data    = ptap;
1518   C->product->destroy = MatDestroy_MPIAIJ_PtAP;
1519   PetscFunctionReturn(0);
1520 }
1521 
1522 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,Mat C)
1523 {
1524   Mat_MPIAIJ        *p=(Mat_MPIAIJ*)P->data;
1525   Mat_SeqAIJ        *c_seq;
1526   Mat_APMPI         *ptap;
1527   Mat               A_loc,C_loc,C_oth;
1528   PetscInt          i,rstart,rend,cm,ncols,row;
1529   const PetscInt    *cols;
1530   const PetscScalar *vals;
1531 
1532   PetscFunctionBegin;
1533   MatCheckProduct(C,3);
1534   ptap = (Mat_APMPI*)C->product->data;
1535   PetscCheck(ptap,PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be computed. Missing data");
1536   PetscCheck(ptap->A_loc,PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtA cannot be reused. Do not call MatProductClear()");
1537   PetscCall(MatZeroEntries(C));
1538 
1539   if (ptap->reuse == MAT_REUSE_MATRIX) {
1540     /* These matrices are obtained in MatTransposeMatMultSymbolic() */
1541     /* 1) get R = Pd^T, Ro = Po^T */
1542     /*----------------------------*/
1543     PetscCall(MatTranspose_SeqAIJ(p->A,MAT_REUSE_MATRIX,&ptap->Rd));
1544     PetscCall(MatTranspose_SeqAIJ(p->B,MAT_REUSE_MATRIX,&ptap->Ro));
1545 
1546     /* 2) compute numeric A_loc */
1547     /*--------------------------*/
1548     PetscCall(MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&ptap->A_loc));
1549   }
1550 
1551   /* 3) C_loc = Rd*A_loc, C_oth = Ro*A_loc */
1552   A_loc = ptap->A_loc;
1553   PetscCall(((ptap->C_loc)->ops->matmultnumeric)(ptap->Rd,A_loc,ptap->C_loc));
1554   PetscCall(((ptap->C_oth)->ops->matmultnumeric)(ptap->Ro,A_loc,ptap->C_oth));
1555   C_loc = ptap->C_loc;
1556   C_oth = ptap->C_oth;
1557 
1558   /* add C_loc and C_oth to C */
1559   PetscCall(MatGetOwnershipRange(C,&rstart,&rend));
1560 
1561   /* C_loc -> C */
1562   cm    = C_loc->rmap->N;
1563   c_seq = (Mat_SeqAIJ*)C_loc->data;
1564   cols = c_seq->j;
1565   vals = c_seq->a;
1566   for (i=0; i<cm; i++) {
1567     ncols = c_seq->i[i+1] - c_seq->i[i];
1568     row = rstart + i;
1569     PetscCall(MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES));
1570     cols += ncols; vals += ncols;
1571   }
1572 
1573   /* Co -> C, off-processor part */
1574   cm    = C_oth->rmap->N;
1575   c_seq = (Mat_SeqAIJ*)C_oth->data;
1576   cols  = c_seq->j;
1577   vals  = c_seq->a;
1578   for (i=0; i<cm; i++) {
1579     ncols = c_seq->i[i+1] - c_seq->i[i];
1580     row = p->garray[i];
1581     PetscCall(MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES));
1582     cols += ncols; vals += ncols;
1583   }
1584   PetscCall(MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY));
1585   PetscCall(MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY));
1586   PetscCall(MatSetOption(C,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE));
1587 
1588   ptap->reuse = MAT_REUSE_MATRIX;
1589   PetscFunctionReturn(0);
1590 }
1591 
1592 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ(Mat P,Mat A,Mat C)
1593 {
1594   Mat_Merge_SeqsToMPI *merge;
1595   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data;
1596   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
1597   Mat_APMPI           *ptap;
1598   PetscInt            *adj;
1599   PetscInt            i,j,k,anz,pnz,row,*cj,nexta;
1600   MatScalar           *ada,*ca,valtmp;
1601   PetscInt            am=A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
1602   MPI_Comm            comm;
1603   PetscMPIInt         size,rank,taga,*len_s;
1604   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
1605   PetscInt            **buf_ri,**buf_rj;
1606   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
1607   MPI_Request         *s_waits,*r_waits;
1608   MPI_Status          *status;
1609   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
1610   const PetscScalar   *dummy;
1611   PetscInt            *ai,*aj,*coi,*coj,*poJ,*pdJ;
1612   Mat                 A_loc;
1613   Mat_SeqAIJ          *a_loc;
1614 
1615   PetscFunctionBegin;
1616   MatCheckProduct(C,3);
1617   ptap = (Mat_APMPI*)C->product->data;
1618   PetscCheck(ptap,PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be computed. Missing data");
1619   PetscCheck(ptap->A_loc,PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtA cannot be reused. Do not call MatProductClear()");
1620   PetscCall(PetscObjectGetComm((PetscObject)C,&comm));
1621   PetscCallMPI(MPI_Comm_size(comm,&size));
1622   PetscCallMPI(MPI_Comm_rank(comm,&rank));
1623 
1624   merge = ptap->merge;
1625 
1626   /* 2) compute numeric C_seq = P_loc^T*A_loc */
1627   /*------------------------------------------*/
1628   /* get data from symbolic products */
1629   coi    = merge->coi; coj = merge->coj;
1630   PetscCall(PetscCalloc1(coi[pon]+1,&coa));
1631   bi     = merge->bi; bj = merge->bj;
1632   owners = merge->rowmap->range;
1633   PetscCall(PetscCalloc1(bi[cm]+1,&ba));
1634 
1635   /* get A_loc by taking all local rows of A */
1636   A_loc = ptap->A_loc;
1637   PetscCall(MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc));
1638   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1639   ai    = a_loc->i;
1640   aj    = a_loc->j;
1641 
1642   /* trigger copy to CPU */
1643   PetscCall(MatSeqAIJGetArrayRead(p->A,&dummy));
1644   PetscCall(MatSeqAIJRestoreArrayRead(p->A,&dummy));
1645   PetscCall(MatSeqAIJGetArrayRead(p->B,&dummy));
1646   PetscCall(MatSeqAIJRestoreArrayRead(p->B,&dummy));
1647   for (i=0; i<am; i++) {
1648     anz = ai[i+1] - ai[i];
1649     adj = aj + ai[i];
1650     ada = a_loc->a + ai[i];
1651 
1652     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1653     /*-------------------------------------------------------------*/
1654     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1655     pnz = po->i[i+1] - po->i[i];
1656     poJ = po->j + po->i[i];
1657     pA  = po->a + po->i[i];
1658     for (j=0; j<pnz; j++) {
1659       row = poJ[j];
1660       cj  = coj + coi[row];
1661       ca  = coa + coi[row];
1662       /* perform sparse axpy */
1663       nexta  = 0;
1664       valtmp = pA[j];
1665       for (k=0; nexta<anz; k++) {
1666         if (cj[k] == adj[nexta]) {
1667           ca[k] += valtmp*ada[nexta];
1668           nexta++;
1669         }
1670       }
1671       PetscCall(PetscLogFlops(2.0*anz));
1672     }
1673 
1674     /* put the value into Cd (diagonal part) */
1675     pnz = pd->i[i+1] - pd->i[i];
1676     pdJ = pd->j + pd->i[i];
1677     pA  = pd->a + pd->i[i];
1678     for (j=0; j<pnz; j++) {
1679       row = pdJ[j];
1680       cj  = bj + bi[row];
1681       ca  = ba + bi[row];
1682       /* perform sparse axpy */
1683       nexta  = 0;
1684       valtmp = pA[j];
1685       for (k=0; nexta<anz; k++) {
1686         if (cj[k] == adj[nexta]) {
1687           ca[k] += valtmp*ada[nexta];
1688           nexta++;
1689         }
1690       }
1691       PetscCall(PetscLogFlops(2.0*anz));
1692     }
1693   }
1694 
1695   /* 3) send and recv matrix values coa */
1696   /*------------------------------------*/
1697   buf_ri = merge->buf_ri;
1698   buf_rj = merge->buf_rj;
1699   len_s  = merge->len_s;
1700   PetscCall(PetscCommGetNewTag(comm,&taga));
1701   PetscCall(PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits));
1702 
1703   PetscCall(PetscMalloc2(merge->nsend+1,&s_waits,size,&status));
1704   for (proc=0,k=0; proc<size; proc++) {
1705     if (!len_s[proc]) continue;
1706     i    = merge->owners_co[proc];
1707     PetscCallMPI(MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k));
1708     k++;
1709   }
1710   if (merge->nrecv) PetscCallMPI(MPI_Waitall(merge->nrecv,r_waits,status));
1711   if (merge->nsend) PetscCallMPI(MPI_Waitall(merge->nsend,s_waits,status));
1712 
1713   PetscCall(PetscFree2(s_waits,status));
1714   PetscCall(PetscFree(r_waits));
1715   PetscCall(PetscFree(coa));
1716 
1717   /* 4) insert local Cseq and received values into Cmpi */
1718   /*----------------------------------------------------*/
1719   PetscCall(PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci));
1720   for (k=0; k<merge->nrecv; k++) {
1721     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1722     nrows       = *(buf_ri_k[k]);
1723     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1724     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th recved i-structure  */
1725   }
1726 
1727   for (i=0; i<cm; i++) {
1728     row  = owners[rank] + i; /* global row index of C_seq */
1729     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1730     ba_i = ba + bi[i];
1731     bnz  = bi[i+1] - bi[i];
1732     /* add received vals into ba */
1733     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1734       /* i-th row */
1735       if (i == *nextrow[k]) {
1736         cnz    = *(nextci[k]+1) - *nextci[k];
1737         cj     = buf_rj[k] + *(nextci[k]);
1738         ca     = abuf_r[k] + *(nextci[k]);
1739         nextcj = 0;
1740         for (j=0; nextcj<cnz; j++) {
1741           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1742             ba_i[j] += ca[nextcj++];
1743           }
1744         }
1745         nextrow[k]++; nextci[k]++;
1746         PetscCall(PetscLogFlops(2.0*cnz));
1747       }
1748     }
1749     PetscCall(MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES));
1750   }
1751   PetscCall(MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY));
1752   PetscCall(MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY));
1753 
1754   PetscCall(PetscFree(ba));
1755   PetscCall(PetscFree(abuf_r[0]));
1756   PetscCall(PetscFree(abuf_r));
1757   PetscCall(PetscFree3(buf_ri_k,nextrow,nextci));
1758   PetscFunctionReturn(0);
1759 }
1760 
1761 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P,Mat A,PetscReal fill,Mat C)
1762 {
1763   Mat                 A_loc;
1764   Mat_APMPI           *ptap;
1765   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1766   Mat_MPIAIJ          *p=(Mat_MPIAIJ*)P->data,*a=(Mat_MPIAIJ*)A->data;
1767   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1768   PetscInt            nnz;
1769   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1770   PetscInt            am  =A->rmap->n,pn=P->cmap->n;
1771   MPI_Comm            comm;
1772   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1773   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1774   PetscInt            len,proc,*dnz,*onz,*owners;
1775   PetscInt            nzi,*bi,*bj;
1776   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1777   MPI_Request         *swaits,*rwaits;
1778   MPI_Status          *sstatus,rstatus;
1779   Mat_Merge_SeqsToMPI *merge;
1780   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1781   PetscReal           afill  =1.0,afill_tmp;
1782   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Armax;
1783   Mat_SeqAIJ          *a_loc;
1784   PetscTable          ta;
1785   MatType             mtype;
1786 
1787   PetscFunctionBegin;
1788   PetscCall(PetscObjectGetComm((PetscObject)A,&comm));
1789   /* check if matrix local sizes are compatible */
1790   PetscCheckFalse(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,A->rmap->rend,P->rmap->rstart,P->rmap->rend);
1791 
1792   PetscCallMPI(MPI_Comm_size(comm,&size));
1793   PetscCallMPI(MPI_Comm_rank(comm,&rank));
1794 
1795   /* create struct Mat_APMPI and attached it to C later */
1796   PetscCall(PetscNew(&ptap));
1797 
1798   /* get A_loc by taking all local rows of A */
1799   PetscCall(MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc));
1800 
1801   ptap->A_loc = A_loc;
1802   a_loc       = (Mat_SeqAIJ*)(A_loc)->data;
1803   ai          = a_loc->i;
1804   aj          = a_loc->j;
1805 
1806   /* determine symbolic Co=(p->B)^T*A - send to others */
1807   /*----------------------------------------------------*/
1808   PetscCall(MatGetSymbolicTranspose_SeqAIJ(p->A,&pdti,&pdtj));
1809   PetscCall(MatGetSymbolicTranspose_SeqAIJ(p->B,&poti,&potj));
1810   pon = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1811                          >= (num of nonzero rows of C_seq) - pn */
1812   PetscCall(PetscMalloc1(pon+1,&coi));
1813   coi[0] = 0;
1814 
1815   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1816   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(poti[pon],ai[am]));
1817   PetscCall(PetscFreeSpaceGet(nnz,&free_space));
1818   current_space = free_space;
1819 
1820   /* create and initialize a linked list */
1821   PetscCall(PetscTableCreate(A->cmap->n + a->B->cmap->N,aN,&ta));
1822   MatRowMergeMax_SeqAIJ(a_loc,am,ta);
1823   PetscCall(PetscTableGetCount(ta,&Armax));
1824 
1825   PetscCall(PetscLLCondensedCreate_Scalable(Armax,&lnk));
1826 
1827   for (i=0; i<pon; i++) {
1828     pnz = poti[i+1] - poti[i];
1829     ptJ = potj + poti[i];
1830     for (j=0; j<pnz; j++) {
1831       row  = ptJ[j]; /* row of A_loc == col of Pot */
1832       anz  = ai[row+1] - ai[row];
1833       Jptr = aj + ai[row];
1834       /* add non-zero cols of AP into the sorted linked list lnk */
1835       PetscCall(PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk));
1836     }
1837     nnz = lnk[0];
1838 
1839     /* If free space is not available, double the total space in the list */
1840     if (current_space->local_remaining<nnz) {
1841       PetscCall(PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space));
1842       nspacedouble++;
1843     }
1844 
1845     /* Copy data into free space, and zero out denserows */
1846     PetscCall(PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk));
1847 
1848     current_space->array           += nnz;
1849     current_space->local_used      += nnz;
1850     current_space->local_remaining -= nnz;
1851 
1852     coi[i+1] = coi[i] + nnz;
1853   }
1854 
1855   PetscCall(PetscMalloc1(coi[pon]+1,&coj));
1856   PetscCall(PetscFreeSpaceContiguous(&free_space,coj));
1857   PetscCall(PetscLLCondensedDestroy_Scalable(lnk)); /* must destroy to get a new one for C */
1858 
1859   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1860   if (afill_tmp > afill) afill = afill_tmp;
1861 
1862   /* send j-array (coj) of Co to other processors */
1863   /*----------------------------------------------*/
1864   /* determine row ownership */
1865   PetscCall(PetscNew(&merge));
1866   PetscCall(PetscLayoutCreate(comm,&merge->rowmap));
1867 
1868   merge->rowmap->n  = pn;
1869   merge->rowmap->bs = 1;
1870 
1871   PetscCall(PetscLayoutSetUp(merge->rowmap));
1872   owners = merge->rowmap->range;
1873 
1874   /* determine the number of messages to send, their lengths */
1875   PetscCall(PetscCalloc1(size,&len_si));
1876   PetscCall(PetscCalloc1(size,&merge->len_s));
1877 
1878   len_s        = merge->len_s;
1879   merge->nsend = 0;
1880 
1881   PetscCall(PetscMalloc1(size+2,&owners_co));
1882 
1883   proc = 0;
1884   for (i=0; i<pon; i++) {
1885     while (prmap[i] >= owners[proc+1]) proc++;
1886     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1887     len_s[proc] += coi[i+1] - coi[i];
1888   }
1889 
1890   len          = 0; /* max length of buf_si[] */
1891   owners_co[0] = 0;
1892   for (proc=0; proc<size; proc++) {
1893     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1894     if (len_si[proc]) {
1895       merge->nsend++;
1896       len_si[proc] = 2*(len_si[proc] + 1);
1897       len         += len_si[proc];
1898     }
1899   }
1900 
1901   /* determine the number and length of messages to receive for coi and coj  */
1902   PetscCall(PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv));
1903   PetscCall(PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri));
1904 
1905   /* post the Irecv and Isend of coj */
1906   PetscCall(PetscCommGetNewTag(comm,&tagj));
1907   PetscCall(PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits));
1908   PetscCall(PetscMalloc1(merge->nsend+1,&swaits));
1909   for (proc=0, k=0; proc<size; proc++) {
1910     if (!len_s[proc]) continue;
1911     i    = owners_co[proc];
1912     PetscCallMPI(MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k));
1913     k++;
1914   }
1915 
1916   /* receives and sends of coj are complete */
1917   PetscCall(PetscMalloc1(size,&sstatus));
1918   for (i=0; i<merge->nrecv; i++) {
1919     PETSC_UNUSED PetscMPIInt icompleted;
1920     PetscCallMPI(MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus));
1921   }
1922   PetscCall(PetscFree(rwaits));
1923   if (merge->nsend) PetscCallMPI(MPI_Waitall(merge->nsend,swaits,sstatus));
1924 
1925   /* add received column indices into table to update Armax */
1926   /* Armax can be as large as aN if a P[row,:] is dense, see src/ksp/ksp/tutorials/ex56.c! */
1927   for (k=0; k<merge->nrecv; k++) {/* k-th received message */
1928     Jptr = buf_rj[k];
1929     for (j=0; j<merge->len_r[k]; j++) {
1930       PetscCall(PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES));
1931     }
1932   }
1933   PetscCall(PetscTableGetCount(ta,&Armax));
1934 
1935   /* send and recv coi */
1936   /*-------------------*/
1937   PetscCall(PetscCommGetNewTag(comm,&tagi));
1938   PetscCall(PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits));
1939   PetscCall(PetscMalloc1(len+1,&buf_s));
1940   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1941   for (proc=0,k=0; proc<size; proc++) {
1942     if (!len_s[proc]) continue;
1943     /* form outgoing message for i-structure:
1944          buf_si[0]:                 nrows to be sent
1945                [1:nrows]:           row index (global)
1946                [nrows+1:2*nrows+1]: i-structure index
1947     */
1948     /*-------------------------------------------*/
1949     nrows       = len_si[proc]/2 - 1;
1950     buf_si_i    = buf_si + nrows+1;
1951     buf_si[0]   = nrows;
1952     buf_si_i[0] = 0;
1953     nrows       = 0;
1954     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1955       nzi               = coi[i+1] - coi[i];
1956       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1957       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1958       nrows++;
1959     }
1960     PetscCallMPI(MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k));
1961     k++;
1962     buf_si += len_si[proc];
1963   }
1964   i = merge->nrecv;
1965   while (i--) {
1966     PETSC_UNUSED PetscMPIInt icompleted;
1967     PetscCallMPI(MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus));
1968   }
1969   PetscCall(PetscFree(rwaits));
1970   if (merge->nsend) PetscCallMPI(MPI_Waitall(merge->nsend,swaits,sstatus));
1971   PetscCall(PetscFree(len_si));
1972   PetscCall(PetscFree(len_ri));
1973   PetscCall(PetscFree(swaits));
1974   PetscCall(PetscFree(sstatus));
1975   PetscCall(PetscFree(buf_s));
1976 
1977   /* compute the local portion of C (mpi mat) */
1978   /*------------------------------------------*/
1979   /* allocate bi array and free space for accumulating nonzero column info */
1980   PetscCall(PetscMalloc1(pn+1,&bi));
1981   bi[0] = 0;
1982 
1983   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
1984   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(pdti[pn],PetscIntSumTruncate(poti[pon],ai[am])));
1985   PetscCall(PetscFreeSpaceGet(nnz,&free_space));
1986   current_space = free_space;
1987 
1988   PetscCall(PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci));
1989   for (k=0; k<merge->nrecv; k++) {
1990     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1991     nrows       = *buf_ri_k[k];
1992     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1993     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th received i-structure  */
1994   }
1995 
1996   PetscCall(PetscLLCondensedCreate_Scalable(Armax,&lnk));
1997   MatPreallocateBegin(comm,pn,A->cmap->n,dnz,onz);
1998   rmax = 0;
1999   for (i=0; i<pn; i++) {
2000     /* add pdt[i,:]*AP into lnk */
2001     pnz = pdti[i+1] - pdti[i];
2002     ptJ = pdtj + pdti[i];
2003     for (j=0; j<pnz; j++) {
2004       row  = ptJ[j];  /* row of AP == col of Pt */
2005       anz  = ai[row+1] - ai[row];
2006       Jptr = aj + ai[row];
2007       /* add non-zero cols of AP into the sorted linked list lnk */
2008       PetscCall(PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk));
2009     }
2010 
2011     /* add received col data into lnk */
2012     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
2013       if (i == *nextrow[k]) { /* i-th row */
2014         nzi  = *(nextci[k]+1) - *nextci[k];
2015         Jptr = buf_rj[k] + *nextci[k];
2016         PetscCall(PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk));
2017         nextrow[k]++; nextci[k]++;
2018       }
2019     }
2020 
2021     /* add missing diagonal entry */
2022     if (C->force_diagonals) {
2023       k = i + owners[rank]; /* column index */
2024       PetscCall(PetscLLCondensedAddSorted_Scalable(1,&k,lnk));
2025     }
2026 
2027     nnz = lnk[0];
2028 
2029     /* if free space is not available, make more free space */
2030     if (current_space->local_remaining<nnz) {
2031       PetscCall(PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space));
2032       nspacedouble++;
2033     }
2034     /* copy data into free space, then initialize lnk */
2035     PetscCall(PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk));
2036     PetscCall(MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz));
2037 
2038     current_space->array           += nnz;
2039     current_space->local_used      += nnz;
2040     current_space->local_remaining -= nnz;
2041 
2042     bi[i+1] = bi[i] + nnz;
2043     if (nnz > rmax) rmax = nnz;
2044   }
2045   PetscCall(PetscFree3(buf_ri_k,nextrow,nextci));
2046 
2047   PetscCall(PetscMalloc1(bi[pn]+1,&bj));
2048   PetscCall(PetscFreeSpaceContiguous(&free_space,bj));
2049   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
2050   if (afill_tmp > afill) afill = afill_tmp;
2051   PetscCall(PetscLLCondensedDestroy_Scalable(lnk));
2052   PetscCall(PetscTableDestroy(&ta));
2053   PetscCall(MatRestoreSymbolicTranspose_SeqAIJ(p->A,&pdti,&pdtj));
2054   PetscCall(MatRestoreSymbolicTranspose_SeqAIJ(p->B,&poti,&potj));
2055 
2056   /* create symbolic parallel matrix C - why cannot be assembled in Numeric part   */
2057   /*-------------------------------------------------------------------------------*/
2058   PetscCall(MatSetSizes(C,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE));
2059   PetscCall(MatSetBlockSizes(C,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs)));
2060   PetscCall(MatGetType(A,&mtype));
2061   PetscCall(MatSetType(C,mtype));
2062   PetscCall(MatMPIAIJSetPreallocation(C,0,dnz,0,onz));
2063   MatPreallocateEnd(dnz,onz);
2064   PetscCall(MatSetBlockSize(C,1));
2065   PetscCall(MatSetOption(C,MAT_NO_OFF_PROC_ENTRIES,PETSC_TRUE));
2066   for (i=0; i<pn; i++) {
2067     row  = i + rstart;
2068     nnz  = bi[i+1] - bi[i];
2069     Jptr = bj + bi[i];
2070     PetscCall(MatSetValues(C,1,&row,nnz,Jptr,NULL,INSERT_VALUES));
2071   }
2072   PetscCall(MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY));
2073   PetscCall(MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY));
2074   PetscCall(MatSetOption(C,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE));
2075   merge->bi        = bi;
2076   merge->bj        = bj;
2077   merge->coi       = coi;
2078   merge->coj       = coj;
2079   merge->buf_ri    = buf_ri;
2080   merge->buf_rj    = buf_rj;
2081   merge->owners_co = owners_co;
2082 
2083   /* attach the supporting struct to C for reuse */
2084   C->product->data    = ptap;
2085   C->product->destroy = MatDestroy_MPIAIJ_PtAP;
2086   ptap->merge         = merge;
2087 
2088   C->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ;
2089 
2090 #if defined(PETSC_USE_INFO)
2091   if (bi[pn] != 0) {
2092     PetscCall(PetscInfo(C,"Reallocs %" PetscInt_FMT "; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill));
2093     PetscCall(PetscInfo(C,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill));
2094   } else {
2095     PetscCall(PetscInfo(C,"Empty matrix product\n"));
2096   }
2097 #endif
2098   PetscFunctionReturn(0);
2099 }
2100 
2101 /* ---------------------------------------------------------------- */
2102 static PetscErrorCode MatProductSymbolic_AtB_MPIAIJ_MPIAIJ(Mat C)
2103 {
2104   Mat_Product    *product = C->product;
2105   Mat            A=product->A,B=product->B;
2106   PetscReal      fill=product->fill;
2107   PetscBool      flg;
2108 
2109   PetscFunctionBegin;
2110   /* scalable */
2111   PetscCall(PetscStrcmp(product->alg,"scalable",&flg));
2112   if (flg) {
2113     PetscCall(MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(A,B,fill,C));
2114     goto next;
2115   }
2116 
2117   /* nonscalable */
2118   PetscCall(PetscStrcmp(product->alg,"nonscalable",&flg));
2119   if (flg) {
2120     PetscCall(MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(A,B,fill,C));
2121     goto next;
2122   }
2123 
2124   /* matmatmult */
2125   PetscCall(PetscStrcmp(product->alg,"at*b",&flg));
2126   if (flg) {
2127     Mat       At;
2128     Mat_APMPI *ptap;
2129 
2130     PetscCall(MatTranspose(A,MAT_INITIAL_MATRIX,&At));
2131     PetscCall(MatMatMultSymbolic_MPIAIJ_MPIAIJ(At,B,fill,C));
2132     ptap = (Mat_APMPI*)C->product->data;
2133     if (ptap) {
2134       ptap->Pt = At;
2135       C->product->destroy = MatDestroy_MPIAIJ_PtAP;
2136     }
2137     C->ops->transposematmultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult;
2138     goto next;
2139   }
2140 
2141   /* backend general code */
2142   PetscCall(PetscStrcmp(product->alg,"backend",&flg));
2143   if (flg) {
2144     PetscCall(MatProductSymbolic_MPIAIJBACKEND(C));
2145     PetscFunctionReturn(0);
2146   }
2147 
2148   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"MatProduct type is not supported");
2149 
2150 next:
2151   C->ops->productnumeric = MatProductNumeric_AtB;
2152   PetscFunctionReturn(0);
2153 }
2154 
2155 /* ---------------------------------------------------------------- */
2156 /* Set options for MatMatMultxxx_MPIAIJ_MPIAIJ */
2157 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_AB(Mat C)
2158 {
2159   Mat_Product    *product = C->product;
2160   Mat            A=product->A,B=product->B;
2161 #if defined(PETSC_HAVE_HYPRE)
2162   const char     *algTypes[5] = {"scalable","nonscalable","seqmpi","backend","hypre"};
2163   PetscInt       nalg = 5;
2164 #else
2165   const char     *algTypes[4] = {"scalable","nonscalable","seqmpi","backend",};
2166   PetscInt       nalg = 4;
2167 #endif
2168   PetscInt       alg = 1; /* set nonscalable algorithm as default */
2169   PetscBool      flg;
2170   MPI_Comm       comm;
2171 
2172   PetscFunctionBegin;
2173   /* Check matrix local sizes */
2174   PetscCall(PetscObjectGetComm((PetscObject)C,&comm));
2175   PetscCheckFalse(A->cmap->rstart != B->rmap->rstart || A->cmap->rend != B->rmap->rend,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);
2176 
2177   /* Set "nonscalable" as default algorithm */
2178   PetscCall(PetscStrcmp(C->product->alg,"default",&flg));
2179   if (flg) {
2180     PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));
2181 
2182     /* Set "scalable" as default if BN and local nonzeros of A and B are large */
2183     if (B->cmap->N > 100000) { /* may switch to scalable algorithm as default */
2184       MatInfo     Ainfo,Binfo;
2185       PetscInt    nz_local;
2186       PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
2187 
2188       PetscCall(MatGetInfo(A,MAT_LOCAL,&Ainfo));
2189       PetscCall(MatGetInfo(B,MAT_LOCAL,&Binfo));
2190       nz_local = (PetscInt)(Ainfo.nz_allocated + Binfo.nz_allocated);
2191 
2192       if (B->cmap->N > product->fill*nz_local) alg_scalable_loc = PETSC_TRUE;
2193       PetscCall(MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm));
2194 
2195       if (alg_scalable) {
2196         alg  = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
2197         PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));
2198         PetscCall(PetscInfo(B,"Use scalable algorithm, BN %" PetscInt_FMT ", fill*nz_allocated %g\n",B->cmap->N,(double)(product->fill*nz_local)));
2199       }
2200     }
2201   }
2202 
2203   /* Get runtime option */
2204   if (product->api_user) {
2205     PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatMatMult","Mat");
2206     PetscCall(PetscOptionsEList("-matmatmult_via","Algorithmic approach","MatMatMult",algTypes,nalg,algTypes[alg],&alg,&flg));
2207     PetscOptionsEnd();
2208   } else {
2209     PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_AB","Mat");
2210     PetscCall(PetscOptionsEList("-mat_product_algorithm","Algorithmic approach","MatMatMult",algTypes,nalg,algTypes[alg],&alg,&flg));
2211     PetscOptionsEnd();
2212   }
2213   if (flg) {
2214     PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));
2215   }
2216 
2217   C->ops->productsymbolic = MatProductSymbolic_AB_MPIAIJ_MPIAIJ;
2218   PetscFunctionReturn(0);
2219 }
2220 
2221 /* Set options for MatTransposeMatMultXXX_MPIAIJ_MPIAIJ */
2222 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_AtB(Mat C)
2223 {
2224   Mat_Product    *product = C->product;
2225   Mat            A=product->A,B=product->B;
2226   const char     *algTypes[4] = {"scalable","nonscalable","at*b","backend"};
2227   PetscInt       nalg = 4;
2228   PetscInt       alg = 1; /* set default algorithm  */
2229   PetscBool      flg;
2230   MPI_Comm       comm;
2231 
2232   PetscFunctionBegin;
2233   /* Check matrix local sizes */
2234   PetscCall(PetscObjectGetComm((PetscObject)C,&comm));
2235   PetscCheckFalse(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 ")",A->rmap->rstart,A->rmap->rend,B->rmap->rstart,B->rmap->rend);
2236 
2237   /* Set default algorithm */
2238   PetscCall(PetscStrcmp(C->product->alg,"default",&flg));
2239   if (flg) {
2240     PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));
2241   }
2242 
2243   /* Set "scalable" as default if BN and local nonzeros of A and B are large */
2244   if (alg && B->cmap->N > 100000) { /* may switch to scalable algorithm as default */
2245     MatInfo     Ainfo,Binfo;
2246     PetscInt    nz_local;
2247     PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
2248 
2249     PetscCall(MatGetInfo(A,MAT_LOCAL,&Ainfo));
2250     PetscCall(MatGetInfo(B,MAT_LOCAL,&Binfo));
2251     nz_local = (PetscInt)(Ainfo.nz_allocated + Binfo.nz_allocated);
2252 
2253     if (B->cmap->N > product->fill*nz_local) alg_scalable_loc = PETSC_TRUE;
2254     PetscCall(MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm));
2255 
2256     if (alg_scalable) {
2257       alg  = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
2258       PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));
2259       PetscCall(PetscInfo(B,"Use scalable algorithm, BN %" PetscInt_FMT ", fill*nz_allocated %g\n",B->cmap->N,(double)(product->fill*nz_local)));
2260     }
2261   }
2262 
2263   /* Get runtime option */
2264   if (product->api_user) {
2265     PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatTransposeMatMult","Mat");
2266     PetscCall(PetscOptionsEList("-mattransposematmult_via","Algorithmic approach","MatTransposeMatMult",algTypes,nalg,algTypes[alg],&alg,&flg));
2267     PetscOptionsEnd();
2268   } else {
2269     PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_AtB","Mat");
2270     PetscCall(PetscOptionsEList("-mat_product_algorithm","Algorithmic approach","MatTransposeMatMult",algTypes,nalg,algTypes[alg],&alg,&flg));
2271     PetscOptionsEnd();
2272   }
2273   if (flg) {
2274     PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));
2275   }
2276 
2277   C->ops->productsymbolic = MatProductSymbolic_AtB_MPIAIJ_MPIAIJ;
2278   PetscFunctionReturn(0);
2279 }
2280 
2281 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_PtAP(Mat C)
2282 {
2283   Mat_Product    *product = C->product;
2284   Mat            A=product->A,P=product->B;
2285   MPI_Comm       comm;
2286   PetscBool      flg;
2287   PetscInt       alg=1; /* set default algorithm */
2288 #if !defined(PETSC_HAVE_HYPRE)
2289   const char     *algTypes[5] = {"scalable","nonscalable","allatonce","allatonce_merged","backend"};
2290   PetscInt       nalg=5;
2291 #else
2292   const char     *algTypes[6] = {"scalable","nonscalable","allatonce","allatonce_merged","backend","hypre"};
2293   PetscInt       nalg=6;
2294 #endif
2295   PetscInt       pN=P->cmap->N;
2296 
2297   PetscFunctionBegin;
2298   /* Check matrix local sizes */
2299   PetscCall(PetscObjectGetComm((PetscObject)C,&comm));
2300   PetscCheckFalse(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 ")",A->rmap->rstart,A->rmap->rend,P->rmap->rstart,P->rmap->rend);
2301   PetscCheckFalse(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 ")",A->cmap->rstart,A->cmap->rend,P->rmap->rstart,P->rmap->rend);
2302 
2303   /* Set "nonscalable" as default algorithm */
2304   PetscCall(PetscStrcmp(C->product->alg,"default",&flg));
2305   if (flg) {
2306     PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));
2307 
2308     /* Set "scalable" as default if BN and local nonzeros of A and B are large */
2309     if (pN > 100000) {
2310       MatInfo     Ainfo,Pinfo;
2311       PetscInt    nz_local;
2312       PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
2313 
2314       PetscCall(MatGetInfo(A,MAT_LOCAL,&Ainfo));
2315       PetscCall(MatGetInfo(P,MAT_LOCAL,&Pinfo));
2316       nz_local = (PetscInt)(Ainfo.nz_allocated + Pinfo.nz_allocated);
2317 
2318       if (pN > product->fill*nz_local) alg_scalable_loc = PETSC_TRUE;
2319       PetscCall(MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm));
2320 
2321       if (alg_scalable) {
2322         alg = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
2323         PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));
2324       }
2325     }
2326   }
2327 
2328   /* Get runtime option */
2329   if (product->api_user) {
2330     PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatPtAP","Mat");
2331     PetscCall(PetscOptionsEList("-matptap_via","Algorithmic approach","MatPtAP",algTypes,nalg,algTypes[alg],&alg,&flg));
2332     PetscOptionsEnd();
2333   } else {
2334     PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_PtAP","Mat");
2335     PetscCall(PetscOptionsEList("-mat_product_algorithm","Algorithmic approach","MatPtAP",algTypes,nalg,algTypes[alg],&alg,&flg));
2336     PetscOptionsEnd();
2337   }
2338   if (flg) {
2339     PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));
2340   }
2341 
2342   C->ops->productsymbolic = MatProductSymbolic_PtAP_MPIAIJ_MPIAIJ;
2343   PetscFunctionReturn(0);
2344 }
2345 
2346 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_RARt(Mat C)
2347 {
2348   Mat_Product *product = C->product;
2349   Mat         A = product->A,R=product->B;
2350 
2351   PetscFunctionBegin;
2352   /* Check matrix local sizes */
2353   PetscCheckFalse(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,A->rmap->n,R->rmap->n,R->cmap->n);
2354 
2355   C->ops->productsymbolic = MatProductSymbolic_RARt_MPIAIJ_MPIAIJ;
2356   PetscFunctionReturn(0);
2357 }
2358 
2359 /*
2360  Set options for ABC = A*B*C = A*(B*C); ABC's algorithm must be chosen from AB's algorithm
2361 */
2362 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_ABC(Mat C)
2363 {
2364   Mat_Product    *product = C->product;
2365   PetscBool      flg = PETSC_FALSE;
2366   PetscInt       alg = 1; /* default algorithm */
2367   const char     *algTypes[3] = {"scalable","nonscalable","seqmpi"};
2368   PetscInt       nalg = 3;
2369 
2370   PetscFunctionBegin;
2371   /* Set default algorithm */
2372   PetscCall(PetscStrcmp(C->product->alg,"default",&flg));
2373   if (flg) {
2374     PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));
2375   }
2376 
2377   /* Get runtime option */
2378   if (product->api_user) {
2379     PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatMatMatMult","Mat");
2380     PetscCall(PetscOptionsEList("-matmatmatmult_via","Algorithmic approach","MatMatMatMult",algTypes,nalg,algTypes[alg],&alg,&flg));
2381     PetscOptionsEnd();
2382   } else {
2383     PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_ABC","Mat");
2384     PetscCall(PetscOptionsEList("-mat_product_algorithm","Algorithmic approach","MatProduct_ABC",algTypes,nalg,algTypes[alg],&alg,&flg));
2385     PetscOptionsEnd();
2386   }
2387   if (flg) {
2388     PetscCall(MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]));
2389   }
2390 
2391   C->ops->matmatmultsymbolic = MatMatMatMultSymbolic_MPIAIJ_MPIAIJ_MPIAIJ;
2392   C->ops->productsymbolic    = MatProductSymbolic_ABC;
2393   PetscFunctionReturn(0);
2394 }
2395 
2396 PETSC_INTERN PetscErrorCode MatProductSetFromOptions_MPIAIJ(Mat C)
2397 {
2398   Mat_Product    *product = C->product;
2399 
2400   PetscFunctionBegin;
2401   switch (product->type) {
2402   case MATPRODUCT_AB:
2403     PetscCall(MatProductSetFromOptions_MPIAIJ_AB(C));
2404     break;
2405   case MATPRODUCT_AtB:
2406     PetscCall(MatProductSetFromOptions_MPIAIJ_AtB(C));
2407     break;
2408   case MATPRODUCT_PtAP:
2409     PetscCall(MatProductSetFromOptions_MPIAIJ_PtAP(C));
2410     break;
2411   case MATPRODUCT_RARt:
2412     PetscCall(MatProductSetFromOptions_MPIAIJ_RARt(C));
2413     break;
2414   case MATPRODUCT_ABC:
2415     PetscCall(MatProductSetFromOptions_MPIAIJ_ABC(C));
2416     break;
2417   default:
2418     break;
2419   }
2420   PetscFunctionReturn(0);
2421 }
2422