xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision efa12513287cff49a2b9648ae83199dcbfaad71a)
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   PetscErrorCode      ierr;
21   Mat_Product         *product = C->product;
22   Mat                 A=product->A,B=product->B;
23   MatProductAlgorithm alg=product->alg;
24   PetscReal           fill=product->fill;
25   PetscBool           flg;
26 
27   PetscFunctionBegin;
28   /* scalable */
29   ierr = PetscStrcmp(alg,"scalable",&flg);CHKERRQ(ierr);
30   if (flg) {
31     ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ(A,B,fill,C);CHKERRQ(ierr);
32     PetscFunctionReturn(0);
33   }
34 
35   /* nonscalable */
36   ierr = PetscStrcmp(alg,"nonscalable",&flg);CHKERRQ(ierr);
37   if (flg) {
38     ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(A,B,fill,C);CHKERRQ(ierr);
39     PetscFunctionReturn(0);
40   }
41 
42   /* seqmpi */
43   ierr = PetscStrcmp(alg,"seqmpi",&flg);CHKERRQ(ierr);
44   if (flg) {
45     ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ_seqMPI(A,B,fill,C);CHKERRQ(ierr);
46     PetscFunctionReturn(0);
47   }
48 
49   /* backend general code */
50   ierr = PetscStrcmp(alg,"backend",&flg);CHKERRQ(ierr);
51   if (flg) {
52     ierr = MatProductSymbolic_MPIAIJBACKEND(C);CHKERRQ(ierr);
53     PetscFunctionReturn(0);
54   }
55 
56 #if defined(PETSC_HAVE_HYPRE)
57   ierr = PetscStrcmp(alg,"hypre",&flg);CHKERRQ(ierr);
58   if (flg) {
59     ierr = MatMatMultSymbolic_AIJ_AIJ_wHYPRE(A,B,fill,C);CHKERRQ(ierr);
60     PetscFunctionReturn(0);
61   }
62 #endif
63   SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_SUP,"Mat Product Algorithm is not supported");
64 }
65 
66 PetscErrorCode MatDestroy_MPIAIJ_MatMatMult(void *data)
67 {
68   PetscErrorCode ierr;
69   Mat_APMPI      *ptap = (Mat_APMPI*)data;
70 
71   PetscFunctionBegin;
72   ierr = PetscFree2(ptap->startsj_s,ptap->startsj_r);CHKERRQ(ierr);
73   ierr = PetscFree(ptap->bufa);CHKERRQ(ierr);
74   ierr = MatDestroy(&ptap->P_loc);CHKERRQ(ierr);
75   ierr = MatDestroy(&ptap->P_oth);CHKERRQ(ierr);
76   ierr = MatDestroy(&ptap->Pt);CHKERRQ(ierr);
77   ierr = PetscFree(ptap->api);CHKERRQ(ierr);
78   ierr = PetscFree(ptap->apj);CHKERRQ(ierr);
79   ierr = PetscFree(ptap->apa);CHKERRQ(ierr);
80   ierr = PetscFree(ptap);CHKERRQ(ierr);
81   PetscFunctionReturn(0);
82 }
83 
84 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,Mat C)
85 {
86   PetscErrorCode    ierr;
87   Mat_MPIAIJ        *a  =(Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
88   Mat_SeqAIJ        *ad =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
89   Mat_SeqAIJ        *cd =(Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
90   PetscScalar       *cda=cd->a,*coa=co->a;
91   Mat_SeqAIJ        *p_loc,*p_oth;
92   PetscScalar       *apa,*ca;
93   PetscInt          cm =C->rmap->n;
94   Mat_APMPI         *ptap;
95   PetscInt          *api,*apj,*apJ,i,k;
96   PetscInt          cstart=C->cmap->rstart;
97   PetscInt          cdnz,conz,k0,k1;
98   const PetscScalar *dummy;
99   MPI_Comm          comm;
100   PetscMPIInt       size;
101 
102   PetscFunctionBegin;
103   MatCheckProduct(C,3);
104   ptap = (Mat_APMPI*)C->product->data;
105   if (!ptap) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be computed. Missing data");
106   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
107   ierr = MPI_Comm_size(comm,&size);CHKERRMPI(ierr);
108 
109   if (!ptap->P_oth && size>1) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"AP cannot be reused. Do not call MatProductClear()");
110 
111   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
112   /*-----------------------------------------------------*/
113   /* update numerical values of P_oth and P_loc */
114   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
115   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
116 
117   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
118   /*----------------------------------------------------------*/
119   /* get data from symbolic products */
120   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
121   p_oth = NULL;
122   if (size >1) {
123     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
124   }
125 
126   /* get apa for storing dense row A[i,:]*P */
127   apa = ptap->apa;
128 
129   api = ptap->api;
130   apj = ptap->apj;
131   /* trigger copy to CPU */
132   ierr = MatSeqAIJGetArrayRead(a->A,&dummy);CHKERRQ(ierr);
133   ierr = MatSeqAIJRestoreArrayRead(a->A,&dummy);CHKERRQ(ierr);
134   ierr = MatSeqAIJGetArrayRead(a->B,&dummy);CHKERRQ(ierr);
135   ierr = MatSeqAIJRestoreArrayRead(a->B,&dummy);CHKERRQ(ierr);
136   for (i=0; i<cm; i++) {
137     /* compute apa = A[i,:]*P */
138     AProw_nonscalable(i,ad,ao,p_loc,p_oth,apa);
139 
140     /* set values in C */
141     apJ  = apj + api[i];
142     cdnz = cd->i[i+1] - cd->i[i];
143     conz = co->i[i+1] - co->i[i];
144 
145     /* 1st off-diagonal part of C */
146     ca = coa + co->i[i];
147     k  = 0;
148     for (k0=0; k0<conz; k0++) {
149       if (apJ[k] >= cstart) break;
150       ca[k0]      = apa[apJ[k]];
151       apa[apJ[k++]] = 0.0;
152     }
153 
154     /* diagonal part of C */
155     ca = cda + cd->i[i];
156     for (k1=0; k1<cdnz; k1++) {
157       ca[k1]      = apa[apJ[k]];
158       apa[apJ[k++]] = 0.0;
159     }
160 
161     /* 2nd off-diagonal part of C */
162     ca = coa + co->i[i];
163     for (; k0<conz; k0++) {
164       ca[k0]      = apa[apJ[k]];
165       apa[apJ[k++]] = 0.0;
166     }
167   }
168   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
169   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
170   PetscFunctionReturn(0);
171 }
172 
173 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,PetscReal fill,Mat C)
174 {
175   PetscErrorCode     ierr;
176   MPI_Comm           comm;
177   PetscMPIInt        size;
178   Mat_APMPI          *ptap;
179   PetscFreeSpaceList free_space=NULL,current_space=NULL;
180   Mat_MPIAIJ         *a=(Mat_MPIAIJ*)A->data;
181   Mat_SeqAIJ         *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
182   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
183   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
184   PetscInt           *lnk,i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi;
185   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
186   PetscBT            lnkbt;
187   PetscReal          afill;
188   MatType            mtype;
189 
190   PetscFunctionBegin;
191   MatCheckProduct(C,4);
192   if (C->product->data) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Extra product struct not empty");
193   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
194   ierr = MPI_Comm_size(comm,&size);CHKERRMPI(ierr);
195 
196   /* create struct Mat_APMPI and attached it to C later */
197   ierr = PetscNew(&ptap);CHKERRQ(ierr);
198 
199   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
200   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
201 
202   /* get P_loc by taking all local rows of P */
203   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
204 
205   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
206   pi_loc = p_loc->i; pj_loc = p_loc->j;
207   if (size > 1) {
208     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
209     pi_oth = p_oth->i; pj_oth = p_oth->j;
210   } else {
211     p_oth = NULL;
212     pi_oth = NULL; pj_oth = NULL;
213   }
214 
215   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
216   /*-------------------------------------------------------------------*/
217   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
218   ptap->api = api;
219   api[0]    = 0;
220 
221   /* create and initialize a linked list */
222   ierr = PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt);CHKERRQ(ierr);
223 
224   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
225   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space);CHKERRQ(ierr);
226   current_space = free_space;
227 
228   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
229   for (i=0; i<am; i++) {
230     /* diagonal portion of A */
231     nzi = adi[i+1] - adi[i];
232     for (j=0; j<nzi; j++) {
233       row  = *adj++;
234       pnz  = pi_loc[row+1] - pi_loc[row];
235       Jptr = pj_loc + pi_loc[row];
236       /* add non-zero cols of P into the sorted linked list lnk */
237       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
238     }
239     /* off-diagonal portion of A */
240     nzi = aoi[i+1] - aoi[i];
241     for (j=0; j<nzi; j++) {
242       row  = *aoj++;
243       pnz  = pi_oth[row+1] - pi_oth[row];
244       Jptr = pj_oth + pi_oth[row];
245       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
246     }
247     /* add possible missing diagonal entry */
248     if (C->force_diagonals) {
249       j = i + rstart; /* column index */
250       ierr = PetscLLCondensedAddSorted(1,&j,lnk,lnkbt);CHKERRQ(ierr);
251     }
252 
253     apnz     = lnk[0];
254     api[i+1] = api[i] + apnz;
255 
256     /* if free space is not available, double the total space in the list */
257     if (current_space->local_remaining<apnz) {
258       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
259       nspacedouble++;
260     }
261 
262     /* Copy data into free space, then initialize lnk */
263     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
264     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
265 
266     current_space->array           += apnz;
267     current_space->local_used      += apnz;
268     current_space->local_remaining -= apnz;
269   }
270 
271   /* Allocate space for apj, initialize apj, and */
272   /* destroy list of free space and other temporary array(s) */
273   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
274   apj  = ptap->apj;
275   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
276   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
277 
278   /* malloc apa to store dense row A[i,:]*P */
279   ierr = PetscCalloc1(pN,&ptap->apa);CHKERRQ(ierr);
280 
281   /* set and assemble symbolic parallel matrix C */
282   /*---------------------------------------------*/
283   ierr = MatSetSizes(C,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
284   ierr = MatSetBlockSizesFromMats(C,A,P);CHKERRQ(ierr);
285 
286   ierr = MatGetType(A,&mtype);CHKERRQ(ierr);
287   ierr = MatSetType(C,mtype);CHKERRQ(ierr);
288   ierr = MatMPIAIJSetPreallocation(C,0,dnz,0,onz);CHKERRQ(ierr);
289   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
290 
291   ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(C, apj, api);CHKERRQ(ierr);
292   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
293   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
294   ierr = MatSetOption(C,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);CHKERRQ(ierr);
295 
296   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
297   C->ops->productnumeric = MatProductNumeric_AB;
298 
299   /* attach the supporting struct to C for reuse */
300   C->product->data    = ptap;
301   C->product->destroy = MatDestroy_MPIAIJ_MatMatMult;
302 
303   /* set MatInfo */
304   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
305   if (afill < 1.0) afill = 1.0;
306   C->info.mallocs           = nspacedouble;
307   C->info.fill_ratio_given  = fill;
308   C->info.fill_ratio_needed = afill;
309 
310 #if defined(PETSC_USE_INFO)
311   if (api[am]) {
312     ierr = PetscInfo3(C,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
313     ierr = PetscInfo1(C,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
314   } else {
315     ierr = PetscInfo(C,"Empty matrix product\n");CHKERRQ(ierr);
316   }
317 #endif
318   PetscFunctionReturn(0);
319 }
320 
321 /* ------------------------------------------------------- */
322 static PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat,Mat,PetscReal,Mat);
323 static PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat,Mat,Mat);
324 
325 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_MPIDense_AB(Mat C)
326 {
327   Mat_Product *product = C->product;
328   Mat         A = product->A,B=product->B;
329 
330   PetscFunctionBegin;
331   if (A->cmap->rstart != B->rmap->rstart || A->cmap->rend != B->rmap->rend)
332     SETERRQ4(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, (%D, %D) != (%D,%D)",A->cmap->rstart,A->cmap->rend,B->rmap->rstart,B->rmap->rend);
333 
334   C->ops->matmultsymbolic = MatMatMultSymbolic_MPIAIJ_MPIDense;
335   C->ops->productsymbolic = MatProductSymbolic_AB;
336   PetscFunctionReturn(0);
337 }
338 /* -------------------------------------------------------------------- */
339 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_MPIDense_AtB(Mat C)
340 {
341   Mat_Product *product = C->product;
342   Mat         A = product->A,B=product->B;
343 
344   PetscFunctionBegin;
345   if (A->rmap->rstart != B->rmap->rstart || A->rmap->rend != B->rmap->rend)
346     SETERRQ4(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, (%D, %D) != (%D,%D)",A->rmap->rstart,A->rmap->rend,B->rmap->rstart,B->rmap->rend);
347 
348   C->ops->transposematmultsymbolic = MatTransposeMatMultSymbolic_MPIAIJ_MPIDense;
349   C->ops->productsymbolic          = MatProductSymbolic_AtB;
350   PetscFunctionReturn(0);
351 }
352 
353 /* --------------------------------------------------------------------- */
354 PETSC_INTERN PetscErrorCode MatProductSetFromOptions_MPIAIJ_MPIDense(Mat C)
355 {
356   PetscErrorCode ierr;
357   Mat_Product    *product = C->product;
358 
359   PetscFunctionBegin;
360   switch (product->type) {
361   case MATPRODUCT_AB:
362     ierr = MatProductSetFromOptions_MPIAIJ_MPIDense_AB(C);CHKERRQ(ierr);
363     break;
364   case MATPRODUCT_AtB:
365     ierr = MatProductSetFromOptions_MPIAIJ_MPIDense_AtB(C);CHKERRQ(ierr);
366     break;
367   default:
368     break;
369   }
370   PetscFunctionReturn(0);
371 }
372 /* ------------------------------------------------------- */
373 
374 typedef struct {
375   Mat          workB,workB1;
376   MPI_Request  *rwaits,*swaits;
377   PetscInt     nsends,nrecvs;
378   MPI_Datatype *stype,*rtype;
379   PetscInt     blda;
380 } MPIAIJ_MPIDense;
381 
382 PetscErrorCode MatMPIAIJ_MPIDenseDestroy(void *ctx)
383 {
384   MPIAIJ_MPIDense *contents = (MPIAIJ_MPIDense*)ctx;
385   PetscErrorCode  ierr;
386   PetscInt        i;
387 
388   PetscFunctionBegin;
389   ierr = MatDestroy(&contents->workB);CHKERRQ(ierr);
390   ierr = MatDestroy(&contents->workB1);CHKERRQ(ierr);
391   for (i=0; i<contents->nsends; i++) {
392     ierr = MPI_Type_free(&contents->stype[i]);CHKERRMPI(ierr);
393   }
394   for (i=0; i<contents->nrecvs; i++) {
395     ierr = MPI_Type_free(&contents->rtype[i]);CHKERRMPI(ierr);
396   }
397   ierr = PetscFree4(contents->stype,contents->rtype,contents->rwaits,contents->swaits);CHKERRQ(ierr);
398   ierr = PetscFree(contents);CHKERRQ(ierr);
399   PetscFunctionReturn(0);
400 }
401 
402 static PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat A,Mat B,PetscReal fill,Mat C)
403 {
404   PetscErrorCode  ierr;
405   Mat_MPIAIJ      *aij=(Mat_MPIAIJ*)A->data;
406   PetscInt        nz=aij->B->cmap->n,nsends,nrecvs,i,nrows_to,j,blda,clda;
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   if (C->product->data) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_PLIB,"Product data not empty");
419   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
420   ierr = PetscObjectBaseTypeCompare((PetscObject)C,MATMPIDENSE,&cisdense);CHKERRQ(ierr);
421   if (!cisdense) {
422     ierr = MatSetType(C,((PetscObject)B)->type_name);CHKERRQ(ierr);
423   }
424   ierr = MatSetSizes(C,Am,B->cmap->n,A->rmap->N,BN);CHKERRQ(ierr);
425   ierr = MatSetBlockSizesFromMats(C,A,B);CHKERRQ(ierr);
426   ierr = MatSetUp(C);CHKERRQ(ierr);
427   ierr = MatDenseGetLDA(B,&blda);CHKERRQ(ierr);
428   ierr = MatDenseGetLDA(C,&clda);CHKERRQ(ierr);
429   ierr = PetscNew(&contents);CHKERRQ(ierr);
430 
431   ierr = VecScatterGetRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,NULL,NULL);CHKERRQ(ierr);
432   ierr = VecScatterGetRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,NULL,NULL);CHKERRQ(ierr);
433 
434   /* Create column block of B and C for memory scalability when BN is too large */
435   /* Estimate Bbn, column size of Bb */
436   if (nz) {
437     Bbn1 = 2*Am*BN/nz;
438   } else Bbn1 = BN;
439 
440   bs = PetscAbs(B->cmap->bs);
441   Bbn1 = Bbn1/bs *bs; /* Bbn1 is a multiple of bs */
442   if (Bbn1 > BN) Bbn1 = BN;
443   ierr = MPI_Allreduce(&Bbn1,&Bbn,1,MPIU_INT,MPI_MAX,comm);CHKERRMPI(ierr);
444 
445   /* Enable runtime option for Bbn */
446   ierr = PetscOptionsBegin(comm,((PetscObject)C)->prefix,"MatMatMult","Mat");CHKERRQ(ierr);
447   ierr = PetscOptionsInt("-matmatmult_Bbn","Number of columns in Bb","MatMatMult",Bbn,&Bbn,NULL);CHKERRQ(ierr);
448   ierr = PetscOptionsEnd();CHKERRQ(ierr);
449   Bbn  = PetscMin(Bbn,BN);
450 
451   if (Bbn > 0 && Bbn < BN) {
452     numBb = BN/Bbn;
453     Bbn1 = BN - numBb*Bbn;
454   } else numBb = 0;
455 
456   if (numBb) {
457     ierr = PetscInfo3(C,"use Bb, BN=%D, Bbn=%D; numBb=%D\n",BN,Bbn,numBb);CHKERRQ(ierr);
458     if (Bbn1) { /* Create workB1 for the remaining columns */
459       ierr = PetscInfo2(C,"use Bb1, BN=%D, Bbn1=%D\n",BN,Bbn1);CHKERRQ(ierr);
460       /* Create work matrix used to store off processor rows of B needed for local product */
461       ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,Bbn1,NULL,&contents->workB1);CHKERRQ(ierr);
462     } else contents->workB1 = NULL;
463   }
464 
465   /* Create work matrix used to store off processor rows of B needed for local product */
466   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,Bbn,NULL,&contents->workB);CHKERRQ(ierr);
467 
468   /* Use MPI derived data type to reduce memory required by the send/recv buffers */
469   ierr = PetscMalloc4(nsends,&stype,nrecvs,&rtype,nrecvs,&contents->rwaits,nsends,&contents->swaits);CHKERRQ(ierr);
470   contents->stype  = stype;
471   contents->nsends = nsends;
472 
473   contents->rtype  = rtype;
474   contents->nrecvs = nrecvs;
475   contents->blda   = blda;
476 
477   ierr = PetscMalloc1(Bm+1,&disp);CHKERRQ(ierr);
478   for (i=0; i<nsends; i++) {
479     nrows_to = sstarts[i+1]-sstarts[i];
480     for (j=0; j<nrows_to; j++){
481       disp[j] = sindices[sstarts[i]+j]; /* rowB to be sent */
482     }
483     ierr = MPI_Type_create_indexed_block(nrows_to,1,(const PetscMPIInt *)disp,MPIU_SCALAR,&type1);CHKERRMPI(ierr);
484 
485     ierr = MPI_Type_create_resized(type1,0,blda*sizeof(PetscScalar),&stype[i]);CHKERRMPI(ierr);
486     ierr = MPI_Type_commit(&stype[i]);CHKERRMPI(ierr);
487     ierr = MPI_Type_free(&type1);CHKERRMPI(ierr);
488   }
489 
490   for (i=0; i<nrecvs; i++) {
491     /* received values from a process form a (nrows_from x Bbn) row block in workB (column-wise) */
492     nrows_from = rstarts[i+1]-rstarts[i];
493     disp[0] = 0;
494     ierr = MPI_Type_create_indexed_block(1, nrows_from, (const PetscMPIInt *)disp, MPIU_SCALAR, &type1);CHKERRMPI(ierr);
495     ierr = MPI_Type_create_resized(type1, 0, nz*sizeof(PetscScalar), &rtype[i]);CHKERRMPI(ierr);
496     ierr = MPI_Type_commit(&rtype[i]);CHKERRMPI(ierr);
497     ierr = MPI_Type_free(&type1);CHKERRMPI(ierr);
498   }
499 
500   ierr = PetscFree(disp);CHKERRQ(ierr);
501   ierr = VecScatterRestoreRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,NULL,NULL);CHKERRQ(ierr);
502   ierr = VecScatterRestoreRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,NULL,NULL);CHKERRQ(ierr);
503   ierr = MatSetOption(C,MAT_NO_OFF_PROC_ENTRIES,PETSC_TRUE);CHKERRQ(ierr);
504   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
505   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
506   ierr = MatSetOption(C,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);CHKERRQ(ierr);
507 
508   C->product->data = contents;
509   C->product->destroy = MatMPIAIJ_MPIDenseDestroy;
510   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
511   PetscFunctionReturn(0);
512 }
513 
514 PETSC_INTERN PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat,Mat,Mat,const PetscBool);
515 /*
516     Performs an efficient scatter on the rows of B needed by this process; this is
517     a modification of the VecScatterBegin_() routines.
518 
519     Input: Bbidx = 0: B = Bb
520                  = 1: B = Bb1, see MatMatMultSymbolic_MPIAIJ_MPIDense()
521 */
522 PetscErrorCode MatMPIDenseScatter(Mat A,Mat B,PetscInt Bbidx,Mat C,Mat *outworkB)
523 {
524   Mat_MPIAIJ        *aij = (Mat_MPIAIJ*)A->data;
525   PetscErrorCode    ierr;
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   if (!C->product->data) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_PLIB,"Product data empty");
543   contents = (MPIAIJ_MPIDense*)C->product->data;
544   ierr = VecScatterGetRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,&sprocs,NULL/*bs*/);CHKERRQ(ierr);
545   ierr = VecScatterGetRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,&rprocs,NULL/*bs*/);CHKERRQ(ierr);
546   ierr = PetscMPIIntCast(nsends,&nsends_mpi);CHKERRQ(ierr);
547   ierr = PetscMPIIntCast(nrecvs,&nrecvs_mpi);CHKERRQ(ierr);
548   if (Bbidx == 0) {
549     workB = *outworkB = contents->workB;
550   } else {
551     workB = *outworkB = contents->workB1;
552   }
553   if (nrows != workB->rmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Number of rows of workB %D not equal to columns of aij->B %D",workB->cmap->n,nrows);
554   swaits = contents->swaits;
555   rwaits = contents->rwaits;
556 
557   ierr = MatDenseGetArrayRead(B,&b);CHKERRQ(ierr);
558   ierr = MatDenseGetLDA(B,&blda);CHKERRQ(ierr);
559   if (blda != contents->blda) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Cannot reuse an input matrix with lda %D != %D",blda,contents->blda);
560   ierr = MatDenseGetArray(workB,&rvalues);CHKERRQ(ierr);
561 
562   /* Post recv, use MPI derived data type to save memory */
563   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
564   rtype = contents->rtype;
565   for (i=0; i<nrecvs; i++) {
566     ierr = MPI_Irecv(rvalues+(rstarts[i]-rstarts[0]),ncols,rtype[i],rprocs[i],tag,comm,rwaits+i);CHKERRMPI(ierr);
567   }
568 
569   stype = contents->stype;
570   for (i=0; i<nsends; i++) {
571     ierr = MPI_Isend(b,ncols,stype[i],sprocs[i],tag,comm,swaits+i);CHKERRMPI(ierr);
572   }
573 
574   if (nrecvs) {ierr = MPI_Waitall(nrecvs_mpi,rwaits,MPI_STATUSES_IGNORE);CHKERRMPI(ierr);}
575   if (nsends) {ierr = MPI_Waitall(nsends_mpi,swaits,MPI_STATUSES_IGNORE);CHKERRMPI(ierr);}
576 
577   ierr = VecScatterRestoreRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,&sprocs,NULL);CHKERRQ(ierr);
578   ierr = VecScatterRestoreRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,&rprocs,NULL);CHKERRQ(ierr);
579   ierr = MatDenseRestoreArrayRead(B,&b);CHKERRQ(ierr);
580   ierr = MatDenseRestoreArray(workB,&rvalues);CHKERRQ(ierr);
581   PetscFunctionReturn(0);
582 }
583 
584 static PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat A,Mat B,Mat C)
585 {
586   PetscErrorCode  ierr;
587   Mat_MPIAIJ      *aij    = (Mat_MPIAIJ*)A->data;
588   Mat_MPIDense    *bdense = (Mat_MPIDense*)B->data;
589   Mat_MPIDense    *cdense = (Mat_MPIDense*)C->data;
590   Mat             workB;
591   MPIAIJ_MPIDense *contents;
592 
593   PetscFunctionBegin;
594   MatCheckProduct(C,3);
595   if (!C->product->data) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_PLIB,"Product data empty");
596   contents = (MPIAIJ_MPIDense*)C->product->data;
597   /* diagonal block of A times all local rows of B */
598   /* TODO: this calls a symbolic multiplication every time, which could be avoided */
599   ierr = MatMatMult(aij->A,bdense->A,MAT_REUSE_MATRIX,PETSC_DEFAULT,&cdense->A);CHKERRQ(ierr);
600   if (contents->workB->cmap->n == B->cmap->N) {
601     /* get off processor parts of B needed to complete C=A*B */
602     ierr = MatMPIDenseScatter(A,B,0,C,&workB);CHKERRQ(ierr);
603 
604     /* off-diagonal block of A times nonlocal rows of B */
605     ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A,PETSC_TRUE);CHKERRQ(ierr);
606   } else {
607     Mat      Bb,Cb;
608     PetscInt BN=B->cmap->N,n=contents->workB->cmap->n,i;
609 
610     for (i=0; i<BN; i+=n) {
611       ierr = MatDenseGetSubMatrix(B,i,PetscMin(i+n,BN),&Bb);CHKERRQ(ierr);
612       ierr = MatDenseGetSubMatrix(C,i,PetscMin(i+n,BN),&Cb);CHKERRQ(ierr);
613 
614       /* get off processor parts of B needed to complete C=A*B */
615       ierr = MatMPIDenseScatter(A,Bb,i+n>BN,C,&workB);CHKERRQ(ierr);
616 
617       /* off-diagonal block of A times nonlocal rows of B */
618       cdense = (Mat_MPIDense*)Cb->data;
619       ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A,PETSC_TRUE);CHKERRQ(ierr);
620 
621       ierr = MatDenseRestoreSubMatrix(B,&Bb);CHKERRQ(ierr);
622       ierr = MatDenseRestoreSubMatrix(C,&Cb);CHKERRQ(ierr);
623     }
624   }
625   PetscFunctionReturn(0);
626 }
627 
628 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
629 {
630   PetscErrorCode    ierr;
631   Mat_MPIAIJ        *a   = (Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
632   Mat_SeqAIJ        *ad  = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
633   Mat_SeqAIJ        *cd  = (Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
634   PetscInt          *adi = ad->i,*adj,*aoi=ao->i,*aoj;
635   PetscScalar       *ada,*aoa,*cda=cd->a,*coa=co->a;
636   Mat_SeqAIJ        *p_loc,*p_oth;
637   PetscInt          *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
638   PetscScalar       *pa_loc,*pa_oth,*pa,valtmp,*ca;
639   PetscInt          cm    = C->rmap->n,anz,pnz;
640   Mat_APMPI         *ptap;
641   PetscScalar       *apa_sparse;
642   const PetscScalar *dummy;
643   PetscInt          *api,*apj,*apJ,i,j,k,row;
644   PetscInt          cstart = C->cmap->rstart;
645   PetscInt          cdnz,conz,k0,k1,nextp;
646   MPI_Comm          comm;
647   PetscMPIInt       size;
648 
649   PetscFunctionBegin;
650   MatCheckProduct(C,3);
651   ptap = (Mat_APMPI*)C->product->data;
652   if (!ptap) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be computed. Missing data");
653   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
654   ierr = MPI_Comm_size(comm,&size);CHKERRMPI(ierr);
655   if (!ptap->P_oth && size>1) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"AP cannot be reused. Do not call MatProductClear()");
656 
657   apa_sparse = ptap->apa;
658 
659   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
660   /*-----------------------------------------------------*/
661   /* update numerical values of P_oth and P_loc */
662   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
663   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
664 
665   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
666   /*----------------------------------------------------------*/
667   /* get data from symbolic products */
668   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
669   pi_loc = p_loc->i; pj_loc = p_loc->j; pa_loc = p_loc->a;
670   if (size >1) {
671     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
672     pi_oth = p_oth->i; pj_oth = p_oth->j; pa_oth = p_oth->a;
673   } else {
674     p_oth = NULL; pi_oth = NULL; pj_oth = NULL; pa_oth = NULL;
675   }
676 
677   /* trigger copy to CPU */
678   ierr = MatSeqAIJGetArrayRead(a->A,&dummy);CHKERRQ(ierr);
679   ierr = MatSeqAIJRestoreArrayRead(a->A,&dummy);CHKERRQ(ierr);
680   ierr = MatSeqAIJGetArrayRead(a->B,&dummy);CHKERRQ(ierr);
681   ierr = MatSeqAIJRestoreArrayRead(a->B,&dummy);CHKERRQ(ierr);
682   api = ptap->api;
683   apj = ptap->apj;
684   for (i=0; i<cm; i++) {
685     apJ = apj + api[i];
686 
687     /* diagonal portion of A */
688     anz = adi[i+1] - adi[i];
689     adj = ad->j + adi[i];
690     ada = ad->a + adi[i];
691     for (j=0; j<anz; j++) {
692       row = adj[j];
693       pnz = pi_loc[row+1] - pi_loc[row];
694       pj  = pj_loc + pi_loc[row];
695       pa  = pa_loc + pi_loc[row];
696       /* perform sparse axpy */
697       valtmp = ada[j];
698       nextp  = 0;
699       for (k=0; nextp<pnz; k++) {
700         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
701           apa_sparse[k] += valtmp*pa[nextp++];
702         }
703       }
704       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
705     }
706 
707     /* off-diagonal portion of A */
708     anz = aoi[i+1] - aoi[i];
709     aoj = ao->j + aoi[i];
710     aoa = ao->a + aoi[i];
711     for (j=0; j<anz; j++) {
712       row = aoj[j];
713       pnz = pi_oth[row+1] - pi_oth[row];
714       pj  = pj_oth + pi_oth[row];
715       pa  = pa_oth + pi_oth[row];
716       /* perform sparse axpy */
717       valtmp = aoa[j];
718       nextp  = 0;
719       for (k=0; nextp<pnz; k++) {
720         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
721           apa_sparse[k] += valtmp*pa[nextp++];
722         }
723       }
724       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
725     }
726 
727     /* set values in C */
728     cdnz = cd->i[i+1] - cd->i[i];
729     conz = co->i[i+1] - co->i[i];
730 
731     /* 1st off-diagonal part of C */
732     ca = coa + co->i[i];
733     k  = 0;
734     for (k0=0; k0<conz; k0++) {
735       if (apJ[k] >= cstart) break;
736       ca[k0]        = apa_sparse[k];
737       apa_sparse[k] = 0.0;
738       k++;
739     }
740 
741     /* diagonal part of C */
742     ca = cda + cd->i[i];
743     for (k1=0; k1<cdnz; k1++) {
744       ca[k1]        = apa_sparse[k];
745       apa_sparse[k] = 0.0;
746       k++;
747     }
748 
749     /* 2nd off-diagonal part of C */
750     ca = coa + co->i[i];
751     for (; k0<conz; k0++) {
752       ca[k0]        = apa_sparse[k];
753       apa_sparse[k] = 0.0;
754       k++;
755     }
756   }
757   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
758   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
759   PetscFunctionReturn(0);
760 }
761 
762 /* same as MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(), except using LLCondensed to avoid O(BN) memory requirement */
763 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat C)
764 {
765   PetscErrorCode     ierr;
766   MPI_Comm           comm;
767   PetscMPIInt        size;
768   Mat_APMPI          *ptap;
769   PetscFreeSpaceList free_space = NULL,current_space=NULL;
770   Mat_MPIAIJ         *a  = (Mat_MPIAIJ*)A->data;
771   Mat_SeqAIJ         *ad = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
772   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
773   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
774   PetscInt           i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi,*lnk,apnz_max=1;
775   PetscInt           am=A->rmap->n,pn=P->cmap->n,pm=P->rmap->n,lsize=pn+20;
776   PetscReal          afill;
777   MatType            mtype;
778 
779   PetscFunctionBegin;
780   MatCheckProduct(C,4);
781   if (C->product->data) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Extra product struct not empty");
782   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
783   ierr = MPI_Comm_size(comm,&size);CHKERRMPI(ierr);
784 
785   /* create struct Mat_APMPI and attached it to C later */
786   ierr = PetscNew(&ptap);CHKERRQ(ierr);
787 
788   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
789   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
790 
791   /* get P_loc by taking all local rows of P */
792   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
793 
794   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
795   pi_loc = p_loc->i; pj_loc = p_loc->j;
796   if (size > 1) {
797     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
798     pi_oth = p_oth->i; pj_oth = p_oth->j;
799   } else {
800     p_oth  = NULL;
801     pi_oth = NULL; pj_oth = NULL;
802   }
803 
804   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
805   /*-------------------------------------------------------------------*/
806   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
807   ptap->api = api;
808   api[0]    = 0;
809 
810   ierr = PetscLLCondensedCreate_Scalable(lsize,&lnk);CHKERRQ(ierr);
811 
812   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
813   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space);CHKERRQ(ierr);
814   current_space = free_space;
815   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
816   for (i=0; i<am; i++) {
817     /* diagonal portion of A */
818     nzi = adi[i+1] - adi[i];
819     for (j=0; j<nzi; j++) {
820       row  = *adj++;
821       pnz  = pi_loc[row+1] - pi_loc[row];
822       Jptr = pj_loc + pi_loc[row];
823       /* Expand list if it is not long enough */
824       if (pnz+apnz_max > lsize) {
825         lsize = pnz+apnz_max;
826         ierr = PetscLLCondensedExpand_Scalable(lsize, &lnk);CHKERRQ(ierr);
827       }
828       /* add non-zero cols of P into the sorted linked list lnk */
829       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
830       apnz     = *lnk; /* The first element in the list is the number of items in the list */
831       api[i+1] = api[i] + apnz;
832       if (apnz > apnz_max) apnz_max = apnz + 1; /* '1' for diagonal entry */
833     }
834     /* off-diagonal portion of A */
835     nzi = aoi[i+1] - aoi[i];
836     for (j=0; j<nzi; j++) {
837       row  = *aoj++;
838       pnz  = pi_oth[row+1] - pi_oth[row];
839       Jptr = pj_oth + pi_oth[row];
840       /* Expand list if it is not long enough */
841       if (pnz+apnz_max > lsize) {
842         lsize = pnz + apnz_max;
843         ierr = PetscLLCondensedExpand_Scalable(lsize, &lnk);CHKERRQ(ierr);
844       }
845       /* add non-zero cols of P into the sorted linked list lnk */
846       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
847       apnz     = *lnk;  /* The first element in the list is the number of items in the list */
848       api[i+1] = api[i] + apnz;
849       if (apnz > apnz_max) apnz_max = apnz + 1; /* '1' for diagonal entry */
850     }
851 
852     /* add missing diagonal entry */
853     if (C->force_diagonals) {
854       j = i + rstart; /* column index */
855       ierr = PetscLLCondensedAddSorted_Scalable(1,&j,lnk);CHKERRQ(ierr);
856     }
857 
858     apnz     = *lnk;
859     api[i+1] = api[i] + apnz;
860     if (apnz > apnz_max) apnz_max = apnz;
861 
862     /* if free space is not available, double the total space in the list */
863     if (current_space->local_remaining<apnz) {
864       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
865       nspacedouble++;
866     }
867 
868     /* Copy data into free space, then initialize lnk */
869     ierr = PetscLLCondensedClean_Scalable(apnz,current_space->array,lnk);CHKERRQ(ierr);
870     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
871 
872     current_space->array           += apnz;
873     current_space->local_used      += apnz;
874     current_space->local_remaining -= apnz;
875   }
876 
877   /* Allocate space for apj, initialize apj, and */
878   /* destroy list of free space and other temporary array(s) */
879   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
880   apj  = ptap->apj;
881   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
882   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
883 
884   /* create and assemble symbolic parallel matrix C */
885   /*----------------------------------------------------*/
886   ierr = MatSetSizes(C,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
887   ierr = MatSetBlockSizesFromMats(C,A,P);CHKERRQ(ierr);
888   ierr = MatGetType(A,&mtype);CHKERRQ(ierr);
889   ierr = MatSetType(C,mtype);CHKERRQ(ierr);
890   ierr = MatMPIAIJSetPreallocation(C,0,dnz,0,onz);CHKERRQ(ierr);
891   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
892 
893   /* malloc apa for assembly C */
894   ierr = PetscCalloc1(apnz_max,&ptap->apa);CHKERRQ(ierr);
895 
896   ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(C, apj, api);CHKERRQ(ierr);
897   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
898   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
899   ierr = MatSetOption(C,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);CHKERRQ(ierr);
900 
901   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ;
902   C->ops->productnumeric = MatProductNumeric_AB;
903 
904   /* attach the supporting struct to C for reuse */
905   C->product->data    = ptap;
906   C->product->destroy = MatDestroy_MPIAIJ_MatMatMult;
907 
908   /* set MatInfo */
909   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
910   if (afill < 1.0) afill = 1.0;
911   C->info.mallocs           = nspacedouble;
912   C->info.fill_ratio_given  = fill;
913   C->info.fill_ratio_needed = afill;
914 
915 #if defined(PETSC_USE_INFO)
916   if (api[am]) {
917     ierr = PetscInfo3(C,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
918     ierr = PetscInfo1(C,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
919   } else {
920     ierr = PetscInfo(C,"Empty matrix product\n");CHKERRQ(ierr);
921   }
922 #endif
923   PetscFunctionReturn(0);
924 }
925 
926 /* This function is needed for the seqMPI matrix-matrix multiplication.  */
927 /* Three input arrays are merged to one output array. The size of the    */
928 /* output array is also output. Duplicate entries only show up once.     */
929 static void Merge3SortedArrays(PetscInt  size1, PetscInt *in1,
930                                PetscInt  size2, PetscInt *in2,
931                                PetscInt  size3, PetscInt *in3,
932                                PetscInt *size4, PetscInt *out)
933 {
934   int i = 0, j = 0, k = 0, l = 0;
935 
936   /* Traverse all three arrays */
937   while (i<size1 && j<size2 && k<size3) {
938     if (in1[i] < in2[j] && in1[i] < in3[k]) {
939       out[l++] = in1[i++];
940     }
941     else if (in2[j] < in1[i] && in2[j] < in3[k]) {
942       out[l++] = in2[j++];
943     }
944     else if (in3[k] < in1[i] && in3[k] < in2[j]) {
945       out[l++] = in3[k++];
946     }
947     else if (in1[i] == in2[j] && in1[i] < in3[k]) {
948       out[l++] = in1[i];
949       i++, j++;
950     }
951     else if (in1[i] == in3[k] && in1[i] < in2[j]) {
952       out[l++] = in1[i];
953       i++, k++;
954     }
955     else if (in3[k] == in2[j] && in2[j] < in1[i])  {
956       out[l++] = in2[j];
957       k++, j++;
958     }
959     else if (in1[i] == in2[j] && in1[i] == in3[k]) {
960       out[l++] = in1[i];
961       i++, j++, k++;
962     }
963   }
964 
965   /* Traverse two remaining arrays */
966   while (i<size1 && j<size2) {
967     if (in1[i] < in2[j]) {
968       out[l++] = in1[i++];
969     }
970     else if (in1[i] > in2[j]) {
971       out[l++] = in2[j++];
972     }
973     else {
974       out[l++] = in1[i];
975       i++, j++;
976     }
977   }
978 
979   while (i<size1 && k<size3) {
980     if (in1[i] < in3[k]) {
981       out[l++] = in1[i++];
982     }
983     else if (in1[i] > in3[k]) {
984       out[l++] = in3[k++];
985     }
986     else {
987       out[l++] = in1[i];
988       i++, k++;
989     }
990   }
991 
992   while (k<size3 && j<size2)  {
993     if (in3[k] < in2[j]) {
994       out[l++] = in3[k++];
995     }
996     else if (in3[k] > in2[j]) {
997       out[l++] = in2[j++];
998     }
999     else {
1000       out[l++] = in3[k];
1001       k++, j++;
1002     }
1003   }
1004 
1005   /* Traverse one remaining array */
1006   while (i<size1) out[l++] = in1[i++];
1007   while (j<size2) out[l++] = in2[j++];
1008   while (k<size3) out[l++] = in3[k++];
1009 
1010   *size4 = l;
1011 }
1012 
1013 /* This matrix-matrix multiplication algorithm divides the multiplication into three multiplications and  */
1014 /* adds up the products. Two of these three multiplications are performed with existing (sequential)      */
1015 /* matrix-matrix multiplications.  */
1016 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_seqMPI(Mat A, Mat P, PetscReal fill, Mat C)
1017 {
1018   PetscErrorCode     ierr;
1019   MPI_Comm           comm;
1020   PetscMPIInt        size;
1021   Mat_APMPI          *ptap;
1022   PetscFreeSpaceList free_space_diag=NULL, current_space=NULL;
1023   Mat_MPIAIJ         *a  =(Mat_MPIAIJ*)A->data;
1024   Mat_SeqAIJ         *ad =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc;
1025   Mat_MPIAIJ         *p  =(Mat_MPIAIJ*)P->data;
1026   Mat_SeqAIJ         *adpd_seq, *p_off, *aopoth_seq;
1027   PetscInt           adponz, adpdnz;
1028   PetscInt           *pi_loc,*dnz,*onz;
1029   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,rstart=A->rmap->rstart;
1030   PetscInt           *lnk,i, i1=0,pnz,row,*adpoi,*adpoj, *api, *adpoJ, *aopJ, *apJ,*Jptr, aopnz, nspacedouble=0,j,nzi,
1031                      *apj,apnz, *adpdi, *adpdj, *adpdJ, *poff_i, *poff_j, *j_temp, *aopothi, *aopothj;
1032   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n, p_colstart, p_colend;
1033   PetscBT            lnkbt;
1034   PetscReal          afill;
1035   PetscMPIInt        rank;
1036   Mat                adpd, aopoth;
1037   MatType            mtype;
1038   const char         *prefix;
1039 
1040   PetscFunctionBegin;
1041   MatCheckProduct(C,4);
1042   if (C->product->data) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Extra product struct not empty");
1043   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1044   ierr = MPI_Comm_size(comm,&size);CHKERRMPI(ierr);
1045   ierr = MPI_Comm_rank(comm, &rank);CHKERRMPI(ierr);
1046   ierr = MatGetOwnershipRangeColumn(P, &p_colstart, &p_colend);CHKERRQ(ierr);
1047 
1048   /* create struct Mat_APMPI and attached it to C later */
1049   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1050 
1051   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
1052   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
1053 
1054   /* get P_loc by taking all local rows of P */
1055   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
1056 
1057 
1058   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
1059   pi_loc = p_loc->i;
1060 
1061   /* Allocate memory for the i arrays of the matrices A*P, A_diag*P_off and A_offd * P */
1062   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
1063   ierr      = PetscMalloc1(am+2,&adpoi);CHKERRQ(ierr);
1064 
1065   adpoi[0]    = 0;
1066   ptap->api = api;
1067   api[0] = 0;
1068 
1069   /* create and initialize a linked list, will be used for both A_diag * P_loc_off and A_offd * P_oth */
1070   ierr = PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt);CHKERRQ(ierr);
1071   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
1072 
1073   /* Symbolic calc of A_loc_diag * P_loc_diag */
1074   ierr = MatGetOptionsPrefix(A,&prefix);CHKERRQ(ierr);
1075   ierr = MatProductCreate(a->A,p->A,NULL,&adpd);CHKERRQ(ierr);
1076   ierr = MatGetOptionsPrefix(A,&prefix);CHKERRQ(ierr);
1077   ierr = MatSetOptionsPrefix(adpd,prefix);CHKERRQ(ierr);
1078   ierr = MatAppendOptionsPrefix(adpd,"inner_diag_");CHKERRQ(ierr);
1079 
1080   ierr = MatProductSetType(adpd,MATPRODUCT_AB);CHKERRQ(ierr);
1081   ierr = MatProductSetAlgorithm(adpd,"sorted");CHKERRQ(ierr);
1082   ierr = MatProductSetFill(adpd,fill);CHKERRQ(ierr);
1083   ierr = MatProductSetFromOptions(adpd);CHKERRQ(ierr);
1084 
1085   adpd->force_diagonals = C->force_diagonals;
1086   ierr = MatProductSymbolic(adpd);CHKERRQ(ierr);
1087 
1088   adpd_seq = (Mat_SeqAIJ*)((adpd)->data);
1089   adpdi = adpd_seq->i; adpdj = adpd_seq->j;
1090   p_off = (Mat_SeqAIJ*)((p->B)->data);
1091   poff_i = p_off->i; poff_j = p_off->j;
1092 
1093   /* j_temp stores indices of a result row before they are added to the linked list */
1094   ierr = PetscMalloc1(pN+2,&j_temp);CHKERRQ(ierr);
1095 
1096 
1097   /* Symbolic calc of the A_diag * p_loc_off */
1098   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
1099   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space_diag);CHKERRQ(ierr);
1100   current_space = free_space_diag;
1101 
1102   for (i=0; i<am; i++) {
1103     /* A_diag * P_loc_off */
1104     nzi = adi[i+1] - adi[i];
1105     for (j=0; j<nzi; j++) {
1106       row  = *adj++;
1107       pnz  = poff_i[row+1] - poff_i[row];
1108       Jptr = poff_j + poff_i[row];
1109       for (i1 = 0; i1 < pnz; i1++) {
1110         j_temp[i1] = p->garray[Jptr[i1]];
1111       }
1112       /* add non-zero cols of P into the sorted linked list lnk */
1113       ierr = PetscLLCondensedAddSorted(pnz,j_temp,lnk,lnkbt);CHKERRQ(ierr);
1114     }
1115 
1116     adponz     = lnk[0];
1117     adpoi[i+1] = adpoi[i] + adponz;
1118 
1119     /* if free space is not available, double the total space in the list */
1120     if (current_space->local_remaining<adponz) {
1121       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(adponz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1122       nspacedouble++;
1123     }
1124 
1125     /* Copy data into free space, then initialize lnk */
1126     ierr = PetscLLCondensedClean(pN,adponz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1127 
1128     current_space->array           += adponz;
1129     current_space->local_used      += adponz;
1130     current_space->local_remaining -= adponz;
1131   }
1132 
1133   /* Symbolic calc of A_off * P_oth */
1134   ierr = MatSetOptionsPrefix(a->B,prefix);CHKERRQ(ierr);
1135   ierr = MatAppendOptionsPrefix(a->B,"inner_offdiag_");CHKERRQ(ierr);
1136   ierr = MatCreate(PETSC_COMM_SELF,&aopoth);CHKERRQ(ierr);
1137   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(a->B, ptap->P_oth, fill, aopoth);CHKERRQ(ierr);
1138   aopoth_seq = (Mat_SeqAIJ*)((aopoth)->data);
1139   aopothi = aopoth_seq->i; aopothj = aopoth_seq->j;
1140 
1141   /* Allocate space for apj, adpj, aopj, ... */
1142   /* destroy lists of free space and other temporary array(s) */
1143 
1144   ierr = PetscMalloc1(aopothi[am] + adpoi[am] + adpdi[am]+2, &ptap->apj);CHKERRQ(ierr);
1145   ierr = PetscMalloc1(adpoi[am]+2, &adpoj);CHKERRQ(ierr);
1146 
1147   /* Copy from linked list to j-array */
1148   ierr = PetscFreeSpaceContiguous(&free_space_diag,adpoj);CHKERRQ(ierr);
1149   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1150 
1151   adpoJ = adpoj;
1152   adpdJ = adpdj;
1153   aopJ = aopothj;
1154   apj  = ptap->apj;
1155   apJ = apj; /* still empty */
1156 
1157   /* Merge j-arrays of A_off * P, A_diag * P_loc_off, and */
1158   /* A_diag * P_loc_diag to get A*P */
1159   for (i = 0; i < am; i++) {
1160     aopnz  =  aopothi[i+1] -  aopothi[i];
1161     adponz = adpoi[i+1] - adpoi[i];
1162     adpdnz = adpdi[i+1] - adpdi[i];
1163 
1164     /* Correct indices from A_diag*P_diag */
1165     for (i1 = 0; i1 < adpdnz; i1++) {
1166       adpdJ[i1] += p_colstart;
1167     }
1168     /* Merge j-arrays of A_diag * P_loc_off and A_diag * P_loc_diag and A_off * P_oth */
1169     Merge3SortedArrays(adponz, adpoJ, adpdnz, adpdJ, aopnz, aopJ, &apnz, apJ);
1170     ierr = MatPreallocateSet(i+rstart, apnz, apJ, dnz, onz);CHKERRQ(ierr);
1171 
1172     aopJ += aopnz;
1173     adpoJ += adponz;
1174     adpdJ += adpdnz;
1175     apJ += apnz;
1176     api[i+1] = api[i] + apnz;
1177   }
1178 
1179   /* malloc apa to store dense row A[i,:]*P */
1180   ierr = PetscCalloc1(pN+2,&ptap->apa);CHKERRQ(ierr);
1181 
1182   /* create and assemble symbolic parallel matrix C */
1183   ierr = MatSetSizes(C,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1184   ierr = MatSetBlockSizesFromMats(C,A,P);CHKERRQ(ierr);
1185   ierr = MatGetType(A,&mtype);CHKERRQ(ierr);
1186   ierr = MatSetType(C,mtype);CHKERRQ(ierr);
1187   ierr = MatMPIAIJSetPreallocation(C,0,dnz,0,onz);CHKERRQ(ierr);
1188   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1189 
1190   ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(C, apj, api);CHKERRQ(ierr);
1191   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1192   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1193   ierr = MatSetOption(C,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);CHKERRQ(ierr);
1194 
1195   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1196   C->ops->productnumeric = MatProductNumeric_AB;
1197 
1198   /* attach the supporting struct to C for reuse */
1199   C->product->data    = ptap;
1200   C->product->destroy = MatDestroy_MPIAIJ_MatMatMult;
1201 
1202   /* set MatInfo */
1203   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
1204   if (afill < 1.0) afill = 1.0;
1205   C->info.mallocs           = nspacedouble;
1206   C->info.fill_ratio_given  = fill;
1207   C->info.fill_ratio_needed = afill;
1208 
1209 #if defined(PETSC_USE_INFO)
1210   if (api[am]) {
1211     ierr = PetscInfo3(C,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1212     ierr = PetscInfo1(C,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
1213   } else {
1214     ierr = PetscInfo(C,"Empty matrix product\n");CHKERRQ(ierr);
1215   }
1216 #endif
1217 
1218   ierr = MatDestroy(&aopoth);CHKERRQ(ierr);
1219   ierr = MatDestroy(&adpd);CHKERRQ(ierr);
1220   ierr = PetscFree(j_temp);CHKERRQ(ierr);
1221   ierr = PetscFree(adpoj);CHKERRQ(ierr);
1222   ierr = PetscFree(adpoi);CHKERRQ(ierr);
1223   PetscFunctionReturn(0);
1224 }
1225 
1226 /*-------------------------------------------------------------------------*/
1227 /* This routine only works when scall=MAT_REUSE_MATRIX! */
1228 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult(Mat P,Mat A,Mat C)
1229 {
1230   PetscErrorCode ierr;
1231   Mat_APMPI      *ptap;
1232   Mat            Pt;
1233 
1234   PetscFunctionBegin;
1235   MatCheckProduct(C,3);
1236   ptap = (Mat_APMPI*)C->product->data;
1237   if (!ptap) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be computed. Missing data");
1238   if (!ptap->Pt) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtA cannot be reused. Do not call MatProductClear()");
1239 
1240   Pt   = ptap->Pt;
1241   ierr = MatTranspose(P,MAT_REUSE_MATRIX,&Pt);CHKERRQ(ierr);
1242   ierr = MatMatMultNumeric_MPIAIJ_MPIAIJ(Pt,A,C);CHKERRQ(ierr);
1243   PetscFunctionReturn(0);
1244 }
1245 
1246 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1247 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,PetscReal fill,Mat C)
1248 {
1249   PetscErrorCode      ierr;
1250   Mat_APMPI           *ptap;
1251   Mat_MPIAIJ          *p=(Mat_MPIAIJ*)P->data;
1252   MPI_Comm            comm;
1253   PetscMPIInt         size,rank;
1254   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1255   PetscInt            pn=P->cmap->n,aN=A->cmap->N,an=A->cmap->n;
1256   PetscInt            *lnk,i,k,nsend,rstart;
1257   PetscBT             lnkbt;
1258   PetscMPIInt         tagi,tagj,*len_si,*len_s,*len_ri,nrecv;
1259   PETSC_UNUSED PetscMPIInt icompleted=0;
1260   PetscInt            **buf_rj,**buf_ri,**buf_ri_k,row,ncols,*cols;
1261   PetscInt            len,proc,*dnz,*onz,*owners,nzi;
1262   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1263   MPI_Request         *swaits,*rwaits;
1264   MPI_Status          *sstatus,rstatus;
1265   PetscLayout         rowmap;
1266   PetscInt            *owners_co,*coi,*coj;    /* i and j array of (p->B)^T*A*P - used in the communication */
1267   PetscMPIInt         *len_r,*id_r;    /* array of length of comm->size, store send/recv matrix values */
1268   PetscInt            *Jptr,*prmap=p->garray,con,j,Crmax;
1269   Mat_SeqAIJ          *a_loc,*c_loc,*c_oth;
1270   PetscTable          ta;
1271   MatType             mtype;
1272   const char          *prefix;
1273 
1274   PetscFunctionBegin;
1275   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1276   ierr = MPI_Comm_size(comm,&size);CHKERRMPI(ierr);
1277   ierr = MPI_Comm_rank(comm,&rank);CHKERRMPI(ierr);
1278 
1279   /* create symbolic parallel matrix C */
1280   ierr = MatGetType(A,&mtype);CHKERRQ(ierr);
1281   ierr = MatSetType(C,mtype);CHKERRQ(ierr);
1282 
1283   C->ops->transposematmultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1284 
1285   /* create struct Mat_APMPI and attached it to C later */
1286   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1287   ptap->reuse = MAT_INITIAL_MATRIX;
1288 
1289   /* (0) compute Rd = Pd^T, Ro = Po^T  */
1290   /* --------------------------------- */
1291   ierr = MatTranspose_SeqAIJ(p->A,MAT_INITIAL_MATRIX,&ptap->Rd);CHKERRQ(ierr);
1292   ierr = MatTranspose_SeqAIJ(p->B,MAT_INITIAL_MATRIX,&ptap->Ro);CHKERRQ(ierr);
1293 
1294   /* (1) compute symbolic A_loc */
1295   /* ---------------------------*/
1296   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&ptap->A_loc);CHKERRQ(ierr);
1297 
1298   /* (2-1) compute symbolic C_oth = Ro*A_loc  */
1299   /* ------------------------------------ */
1300   ierr = MatGetOptionsPrefix(A,&prefix);CHKERRQ(ierr);
1301   ierr = MatSetOptionsPrefix(ptap->Ro,prefix);CHKERRQ(ierr);
1302   ierr = MatAppendOptionsPrefix(ptap->Ro,"inner_offdiag_");CHKERRQ(ierr);
1303   ierr = MatCreate(PETSC_COMM_SELF,&ptap->C_oth);CHKERRQ(ierr);
1304   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Ro,ptap->A_loc,fill,ptap->C_oth);CHKERRQ(ierr);
1305 
1306   /* (3) send coj of C_oth to other processors  */
1307   /* ------------------------------------------ */
1308   /* determine row ownership */
1309   ierr = PetscLayoutCreate(comm,&rowmap);CHKERRQ(ierr);
1310   rowmap->n  = pn;
1311   rowmap->bs = 1;
1312   ierr   = PetscLayoutSetUp(rowmap);CHKERRQ(ierr);
1313   owners = rowmap->range;
1314 
1315   /* determine the number of messages to send, their lengths */
1316   ierr = PetscMalloc4(size,&len_s,size,&len_si,size,&sstatus,size+2,&owners_co);CHKERRQ(ierr);
1317   ierr = PetscArrayzero(len_s,size);CHKERRQ(ierr);
1318   ierr = PetscArrayzero(len_si,size);CHKERRQ(ierr);
1319 
1320   c_oth = (Mat_SeqAIJ*)ptap->C_oth->data;
1321   coi   = c_oth->i; coj = c_oth->j;
1322   con   = ptap->C_oth->rmap->n;
1323   proc  = 0;
1324   for (i=0; i<con; i++) {
1325     while (prmap[i] >= owners[proc+1]) proc++;
1326     len_si[proc]++;               /* num of rows in Co(=Pt*A) to be sent to [proc] */
1327     len_s[proc] += coi[i+1] - coi[i]; /* num of nonzeros in Co to be sent to [proc] */
1328   }
1329 
1330   len          = 0; /* max length of buf_si[], see (4) */
1331   owners_co[0] = 0;
1332   nsend        = 0;
1333   for (proc=0; proc<size; proc++) {
1334     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1335     if (len_s[proc]) {
1336       nsend++;
1337       len_si[proc] = 2*(len_si[proc] + 1); /* length of buf_si to be sent to [proc] */
1338       len         += len_si[proc];
1339     }
1340   }
1341 
1342   /* determine the number and length of messages to receive for coi and coj  */
1343   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&nrecv);CHKERRQ(ierr);
1344   ierr = PetscGatherMessageLengths2(comm,nsend,nrecv,len_s,len_si,&id_r,&len_r,&len_ri);CHKERRQ(ierr);
1345 
1346   /* post the Irecv and Isend of coj */
1347   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1348   ierr = PetscPostIrecvInt(comm,tagj,nrecv,id_r,len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1349   ierr = PetscMalloc1(nsend+1,&swaits);CHKERRQ(ierr);
1350   for (proc=0, k=0; proc<size; proc++) {
1351     if (!len_s[proc]) continue;
1352     i    = owners_co[proc];
1353     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRMPI(ierr);
1354     k++;
1355   }
1356 
1357   /* (2-2) compute symbolic C_loc = Rd*A_loc */
1358   /* ---------------------------------------- */
1359   ierr = MatSetOptionsPrefix(ptap->Rd,prefix);CHKERRQ(ierr);
1360   ierr = MatAppendOptionsPrefix(ptap->Rd,"inner_diag_");CHKERRQ(ierr);
1361   ierr = MatCreate(PETSC_COMM_SELF,&ptap->C_loc);CHKERRQ(ierr);
1362   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Rd,ptap->A_loc,fill,ptap->C_loc);CHKERRQ(ierr);
1363   c_loc = (Mat_SeqAIJ*)ptap->C_loc->data;
1364 
1365   /* receives coj are complete */
1366   for (i=0; i<nrecv; i++) {
1367     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRMPI(ierr);
1368   }
1369   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1370   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRMPI(ierr);}
1371 
1372   /* add received column indices into ta to update Crmax */
1373   a_loc = (Mat_SeqAIJ*)(ptap->A_loc)->data;
1374 
1375   /* create and initialize a linked list */
1376   ierr = PetscTableCreate(an,aN,&ta);CHKERRQ(ierr); /* for compute Crmax */
1377   MatRowMergeMax_SeqAIJ(a_loc,ptap->A_loc->rmap->N,ta);
1378 
1379   for (k=0; k<nrecv; k++) {/* k-th received message */
1380     Jptr = buf_rj[k];
1381     for (j=0; j<len_r[k]; j++) {
1382       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
1383     }
1384   }
1385   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
1386   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
1387 
1388   /* (4) send and recv coi */
1389   /*-----------------------*/
1390   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1391   ierr   = PetscPostIrecvInt(comm,tagi,nrecv,id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1392   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1393   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1394   for (proc=0,k=0; proc<size; proc++) {
1395     if (!len_s[proc]) continue;
1396     /* form outgoing message for i-structure:
1397          buf_si[0]:                 nrows to be sent
1398                [1:nrows]:           row index (global)
1399                [nrows+1:2*nrows+1]: i-structure index
1400     */
1401     /*-------------------------------------------*/
1402     nrows       = len_si[proc]/2 - 1; /* num of rows in Co to be sent to [proc] */
1403     buf_si_i    = buf_si + nrows+1;
1404     buf_si[0]   = nrows;
1405     buf_si_i[0] = 0;
1406     nrows       = 0;
1407     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1408       nzi = coi[i+1] - coi[i];
1409       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1410       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1411       nrows++;
1412     }
1413     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRMPI(ierr);
1414     k++;
1415     buf_si += len_si[proc];
1416   }
1417   for (i=0; i<nrecv; i++) {
1418     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRMPI(ierr);
1419   }
1420   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1421   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRMPI(ierr);}
1422 
1423   ierr = PetscFree4(len_s,len_si,sstatus,owners_co);CHKERRQ(ierr);
1424   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1425   ierr = PetscFree(swaits);CHKERRQ(ierr);
1426   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1427 
1428   /* (5) compute the local portion of C      */
1429   /* ------------------------------------------ */
1430   /* set initial free space to be Crmax, sufficient for holding nozeros in each row of C */
1431   ierr          = PetscFreeSpaceGet(Crmax,&free_space);CHKERRQ(ierr);
1432   current_space = free_space;
1433 
1434   ierr = PetscMalloc3(nrecv,&buf_ri_k,nrecv,&nextrow,nrecv,&nextci);CHKERRQ(ierr);
1435   for (k=0; k<nrecv; k++) {
1436     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1437     nrows       = *buf_ri_k[k];
1438     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1439     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1440   }
1441 
1442   ierr = MatPreallocateInitialize(comm,pn,an,dnz,onz);CHKERRQ(ierr);
1443   ierr = PetscLLCondensedCreate(Crmax,aN,&lnk,&lnkbt);CHKERRQ(ierr);
1444   for (i=0; i<pn; i++) { /* for each local row of C */
1445     /* add C_loc into C */
1446     nzi  = c_loc->i[i+1] - c_loc->i[i];
1447     Jptr = c_loc->j + c_loc->i[i];
1448     ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1449 
1450     /* add received col data into lnk */
1451     for (k=0; k<nrecv; k++) { /* k-th received message */
1452       if (i == *nextrow[k]) { /* i-th row */
1453         nzi  = *(nextci[k]+1) - *nextci[k];
1454         Jptr = buf_rj[k] + *nextci[k];
1455         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1456         nextrow[k]++; nextci[k]++;
1457       }
1458     }
1459 
1460     /* add missing diagonal entry */
1461     if (C->force_diagonals) {
1462       k = i + owners[rank]; /* column index */
1463       ierr = PetscLLCondensedAddSorted(1,&k,lnk,lnkbt);CHKERRQ(ierr);
1464     }
1465 
1466     nzi = lnk[0];
1467 
1468     /* copy data into free space, then initialize lnk */
1469     ierr = PetscLLCondensedClean(aN,nzi,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1470     ierr = MatPreallocateSet(i+owners[rank],nzi,current_space->array,dnz,onz);CHKERRQ(ierr);
1471   }
1472   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1473   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1474   ierr = PetscFreeSpaceDestroy(free_space);CHKERRQ(ierr);
1475 
1476   /* local sizes and preallocation */
1477   ierr = MatSetSizes(C,pn,an,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1478   if (P->cmap->bs > 0) {ierr = PetscLayoutSetBlockSize(C->rmap,P->cmap->bs);CHKERRQ(ierr);}
1479   if (A->cmap->bs > 0) {ierr = PetscLayoutSetBlockSize(C->cmap,A->cmap->bs);CHKERRQ(ierr);}
1480   ierr = MatMPIAIJSetPreallocation(C,0,dnz,0,onz);CHKERRQ(ierr);
1481   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1482 
1483   /* add C_loc and C_oth to C */
1484   ierr = MatGetOwnershipRange(C,&rstart,NULL);CHKERRQ(ierr);
1485   for (i=0; i<pn; i++) {
1486     ncols = c_loc->i[i+1] - c_loc->i[i];
1487     cols  = c_loc->j + c_loc->i[i];
1488     row   = rstart + i;
1489     ierr = MatSetValues(C,1,(const PetscInt*)&row,ncols,(const PetscInt*)cols,NULL,INSERT_VALUES);CHKERRQ(ierr);
1490 
1491     if (C->force_diagonals) {
1492       ierr = MatSetValues(C,1,(const PetscInt*)&row,1,(const PetscInt*)&row,NULL,INSERT_VALUES);CHKERRQ(ierr);
1493     }
1494   }
1495   for (i=0; i<con; i++) {
1496     ncols = c_oth->i[i+1] - c_oth->i[i];
1497     cols  = c_oth->j + c_oth->i[i];
1498     row   = prmap[i];
1499     ierr = MatSetValues(C,1,(const PetscInt*)&row,ncols,(const PetscInt*)cols,NULL,INSERT_VALUES);CHKERRQ(ierr);
1500   }
1501   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1502   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1503   ierr = MatSetOption(C,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);CHKERRQ(ierr);
1504 
1505   /* members in merge */
1506   ierr = PetscFree(id_r);CHKERRQ(ierr);
1507   ierr = PetscFree(len_r);CHKERRQ(ierr);
1508   ierr = PetscFree(buf_ri[0]);CHKERRQ(ierr);
1509   ierr = PetscFree(buf_ri);CHKERRQ(ierr);
1510   ierr = PetscFree(buf_rj[0]);CHKERRQ(ierr);
1511   ierr = PetscFree(buf_rj);CHKERRQ(ierr);
1512   ierr = PetscLayoutDestroy(&rowmap);CHKERRQ(ierr);
1513 
1514   /* attach the supporting struct to C for reuse */
1515   C->product->data    = ptap;
1516   C->product->destroy = MatDestroy_MPIAIJ_PtAP;
1517   PetscFunctionReturn(0);
1518 }
1519 
1520 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,Mat C)
1521 {
1522   PetscErrorCode    ierr;
1523   Mat_MPIAIJ        *p=(Mat_MPIAIJ*)P->data;
1524   Mat_SeqAIJ        *c_seq;
1525   Mat_APMPI         *ptap;
1526   Mat               A_loc,C_loc,C_oth;
1527   PetscInt          i,rstart,rend,cm,ncols,row;
1528   const PetscInt    *cols;
1529   const PetscScalar *vals;
1530 
1531   PetscFunctionBegin;
1532   MatCheckProduct(C,3);
1533   ptap = (Mat_APMPI*)C->product->data;
1534   if (!ptap) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be computed. Missing data");
1535   if (!ptap->A_loc) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtA cannot be reused. Do not call MatProductClear()");
1536   ierr = MatZeroEntries(C);CHKERRQ(ierr);
1537 
1538   if (ptap->reuse == MAT_REUSE_MATRIX) {
1539     /* These matrices are obtained in MatTransposeMatMultSymbolic() */
1540     /* 1) get R = Pd^T, Ro = Po^T */
1541     /*----------------------------*/
1542     ierr = MatTranspose_SeqAIJ(p->A,MAT_REUSE_MATRIX,&ptap->Rd);CHKERRQ(ierr);
1543     ierr = MatTranspose_SeqAIJ(p->B,MAT_REUSE_MATRIX,&ptap->Ro);CHKERRQ(ierr);
1544 
1545     /* 2) compute numeric A_loc */
1546     /*--------------------------*/
1547     ierr = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&ptap->A_loc);CHKERRQ(ierr);
1548   }
1549 
1550   /* 3) C_loc = Rd*A_loc, C_oth = Ro*A_loc */
1551   A_loc = ptap->A_loc;
1552   ierr = ((ptap->C_loc)->ops->matmultnumeric)(ptap->Rd,A_loc,ptap->C_loc);CHKERRQ(ierr);
1553   ierr = ((ptap->C_oth)->ops->matmultnumeric)(ptap->Ro,A_loc,ptap->C_oth);CHKERRQ(ierr);
1554   C_loc = ptap->C_loc;
1555   C_oth = ptap->C_oth;
1556 
1557   /* add C_loc and C_oth to C */
1558   ierr = MatGetOwnershipRange(C,&rstart,&rend);CHKERRQ(ierr);
1559 
1560   /* C_loc -> C */
1561   cm    = C_loc->rmap->N;
1562   c_seq = (Mat_SeqAIJ*)C_loc->data;
1563   cols = c_seq->j;
1564   vals = c_seq->a;
1565   for (i=0; i<cm; i++) {
1566     ncols = c_seq->i[i+1] - c_seq->i[i];
1567     row = rstart + i;
1568     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1569     cols += ncols; vals += ncols;
1570   }
1571 
1572   /* Co -> C, off-processor part */
1573   cm    = C_oth->rmap->N;
1574   c_seq = (Mat_SeqAIJ*)C_oth->data;
1575   cols  = c_seq->j;
1576   vals  = c_seq->a;
1577   for (i=0; i<cm; i++) {
1578     ncols = c_seq->i[i+1] - c_seq->i[i];
1579     row = p->garray[i];
1580     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1581     cols += ncols; vals += ncols;
1582   }
1583   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1584   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1585   ierr = MatSetOption(C,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);CHKERRQ(ierr);
1586 
1587   ptap->reuse = MAT_REUSE_MATRIX;
1588   PetscFunctionReturn(0);
1589 }
1590 
1591 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ(Mat P,Mat A,Mat C)
1592 {
1593   PetscErrorCode      ierr;
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   if (!ptap) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be computed. Missing data");
1619   if (!ptap->A_loc) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtA cannot be reused. Do not call MatProductClear()");
1620   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1621   ierr = MPI_Comm_size(comm,&size);CHKERRMPI(ierr);
1622   ierr = MPI_Comm_rank(comm,&rank);CHKERRMPI(ierr);
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   ierr   = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
1631   bi     = merge->bi; bj = merge->bj;
1632   owners = merge->rowmap->range;
1633   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
1634 
1635   /* get A_loc by taking all local rows of A */
1636   A_loc = ptap->A_loc;
1637   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
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   ierr = MatSeqAIJGetArrayRead(p->A,&dummy);CHKERRQ(ierr);
1644   ierr = MatSeqAIJRestoreArrayRead(p->A,&dummy);CHKERRQ(ierr);
1645   ierr = MatSeqAIJGetArrayRead(p->B,&dummy);CHKERRQ(ierr);
1646   ierr = MatSeqAIJRestoreArrayRead(p->B,&dummy);CHKERRQ(ierr);
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       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
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       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
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   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1701   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1702 
1703   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1704   for (proc=0,k=0; proc<size; proc++) {
1705     if (!len_s[proc]) continue;
1706     i    = merge->owners_co[proc];
1707     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRMPI(ierr);
1708     k++;
1709   }
1710   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRMPI(ierr);}
1711   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRMPI(ierr);}
1712 
1713   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1714   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1715   ierr = PetscFree(coa);CHKERRQ(ierr);
1716 
1717   /* 4) insert local Cseq and received values into Cmpi */
1718   /*----------------------------------------------------*/
1719   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
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); /* poins 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         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1747       }
1748     }
1749     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1750   }
1751   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1752   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1753 
1754   ierr = PetscFree(ba);CHKERRQ(ierr);
1755   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1756   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1757   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1758   PetscFunctionReturn(0);
1759 }
1760 
1761 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P,Mat A,PetscReal fill,Mat C)
1762 {
1763   PetscErrorCode      ierr;
1764   Mat                 A_loc;
1765   Mat_APMPI           *ptap;
1766   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1767   Mat_MPIAIJ          *p=(Mat_MPIAIJ*)P->data,*a=(Mat_MPIAIJ*)A->data;
1768   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1769   PetscInt            nnz;
1770   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1771   PetscInt            am  =A->rmap->n,pn=P->cmap->n;
1772   MPI_Comm            comm;
1773   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1774   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1775   PetscInt            len,proc,*dnz,*onz,*owners;
1776   PetscInt            nzi,*bi,*bj;
1777   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1778   MPI_Request         *swaits,*rwaits;
1779   MPI_Status          *sstatus,rstatus;
1780   Mat_Merge_SeqsToMPI *merge;
1781   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1782   PetscReal           afill  =1.0,afill_tmp;
1783   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Armax;
1784   Mat_SeqAIJ          *a_loc;
1785   PetscTable          ta;
1786   MatType             mtype;
1787 
1788   PetscFunctionBegin;
1789   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1790   /* check if matrix local sizes are compatible */
1791   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend) SETERRQ4(comm,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, A (%D, %D) != P (%D,%D)",A->rmap->rstart,A->rmap->rend,P->rmap->rstart,P->rmap->rend);
1792 
1793   ierr = MPI_Comm_size(comm,&size);CHKERRMPI(ierr);
1794   ierr = MPI_Comm_rank(comm,&rank);CHKERRMPI(ierr);
1795 
1796   /* create struct Mat_APMPI and attached it to C later */
1797   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1798 
1799   /* get A_loc by taking all local rows of A */
1800   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1801 
1802   ptap->A_loc = A_loc;
1803   a_loc       = (Mat_SeqAIJ*)(A_loc)->data;
1804   ai          = a_loc->i;
1805   aj          = a_loc->j;
1806 
1807   /* determine symbolic Co=(p->B)^T*A - send to others */
1808   /*----------------------------------------------------*/
1809   ierr = MatGetSymbolicTranspose_SeqAIJ(p->A,&pdti,&pdtj);CHKERRQ(ierr);
1810   ierr = MatGetSymbolicTranspose_SeqAIJ(p->B,&poti,&potj);CHKERRQ(ierr);
1811   pon = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1812                          >= (num of nonzero rows of C_seq) - pn */
1813   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
1814   coi[0] = 0;
1815 
1816   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1817   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(poti[pon],ai[am]));
1818   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1819   current_space = free_space;
1820 
1821   /* create and initialize a linked list */
1822   ierr = PetscTableCreate(A->cmap->n + a->B->cmap->N,aN,&ta);CHKERRQ(ierr);
1823   MatRowMergeMax_SeqAIJ(a_loc,am,ta);
1824   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
1825 
1826   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
1827 
1828   for (i=0; i<pon; i++) {
1829     pnz = poti[i+1] - poti[i];
1830     ptJ = potj + poti[i];
1831     for (j=0; j<pnz; j++) {
1832       row  = ptJ[j]; /* row of A_loc == col of Pot */
1833       anz  = ai[row+1] - ai[row];
1834       Jptr = aj + ai[row];
1835       /* add non-zero cols of AP into the sorted linked list lnk */
1836       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1837     }
1838     nnz = lnk[0];
1839 
1840     /* If free space is not available, double the total space in the list */
1841     if (current_space->local_remaining<nnz) {
1842       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1843       nspacedouble++;
1844     }
1845 
1846     /* Copy data into free space, and zero out denserows */
1847     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1848 
1849     current_space->array           += nnz;
1850     current_space->local_used      += nnz;
1851     current_space->local_remaining -= nnz;
1852 
1853     coi[i+1] = coi[i] + nnz;
1854   }
1855 
1856   ierr = PetscMalloc1(coi[pon]+1,&coj);CHKERRQ(ierr);
1857   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1858   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr); /* must destroy to get a new one for C */
1859 
1860   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1861   if (afill_tmp > afill) afill = afill_tmp;
1862 
1863   /* send j-array (coj) of Co to other processors */
1864   /*----------------------------------------------*/
1865   /* determine row ownership */
1866   ierr = PetscNew(&merge);CHKERRQ(ierr);
1867   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1868 
1869   merge->rowmap->n  = pn;
1870   merge->rowmap->bs = 1;
1871 
1872   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1873   owners = merge->rowmap->range;
1874 
1875   /* determine the number of messages to send, their lengths */
1876   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
1877   ierr = PetscCalloc1(size,&merge->len_s);CHKERRQ(ierr);
1878 
1879   len_s        = merge->len_s;
1880   merge->nsend = 0;
1881 
1882   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
1883 
1884   proc = 0;
1885   for (i=0; i<pon; i++) {
1886     while (prmap[i] >= owners[proc+1]) proc++;
1887     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1888     len_s[proc] += coi[i+1] - coi[i];
1889   }
1890 
1891   len          = 0; /* max length of buf_si[] */
1892   owners_co[0] = 0;
1893   for (proc=0; proc<size; proc++) {
1894     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1895     if (len_si[proc]) {
1896       merge->nsend++;
1897       len_si[proc] = 2*(len_si[proc] + 1);
1898       len         += len_si[proc];
1899     }
1900   }
1901 
1902   /* determine the number and length of messages to receive for coi and coj  */
1903   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1904   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1905 
1906   /* post the Irecv and Isend of coj */
1907   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1908   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1909   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
1910   for (proc=0, k=0; proc<size; proc++) {
1911     if (!len_s[proc]) continue;
1912     i    = owners_co[proc];
1913     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRMPI(ierr);
1914     k++;
1915   }
1916 
1917   /* receives and sends of coj are complete */
1918   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
1919   for (i=0; i<merge->nrecv; i++) {
1920     PETSC_UNUSED PetscMPIInt icompleted;
1921     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRMPI(ierr);
1922   }
1923   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1924   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRMPI(ierr);}
1925 
1926   /* add received column indices into table to update Armax */
1927   /* Armax can be as large as aN if a P[row,:] is dense, see src/ksp/ksp/tutorials/ex56.c! */
1928   for (k=0; k<merge->nrecv; k++) {/* k-th received message */
1929     Jptr = buf_rj[k];
1930     for (j=0; j<merge->len_r[k]; j++) {
1931       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
1932     }
1933   }
1934   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
1935   /* printf("Armax %d, an %d + Bn %d = %d, aN %d\n",Armax,A->cmap->n,a->B->cmap->N,A->cmap->n+a->B->cmap->N,aN); */
1936 
1937   /* send and recv coi */
1938   /*-------------------*/
1939   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1940   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1941   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1942   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1943   for (proc=0,k=0; proc<size; proc++) {
1944     if (!len_s[proc]) continue;
1945     /* form outgoing message for i-structure:
1946          buf_si[0]:                 nrows to be sent
1947                [1:nrows]:           row index (global)
1948                [nrows+1:2*nrows+1]: i-structure index
1949     */
1950     /*-------------------------------------------*/
1951     nrows       = len_si[proc]/2 - 1;
1952     buf_si_i    = buf_si + nrows+1;
1953     buf_si[0]   = nrows;
1954     buf_si_i[0] = 0;
1955     nrows       = 0;
1956     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1957       nzi               = coi[i+1] - coi[i];
1958       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1959       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1960       nrows++;
1961     }
1962     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRMPI(ierr);
1963     k++;
1964     buf_si += len_si[proc];
1965   }
1966   i = merge->nrecv;
1967   while (i--) {
1968     PETSC_UNUSED PetscMPIInt icompleted;
1969     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRMPI(ierr);
1970   }
1971   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1972   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRMPI(ierr);}
1973   ierr = PetscFree(len_si);CHKERRQ(ierr);
1974   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1975   ierr = PetscFree(swaits);CHKERRQ(ierr);
1976   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1977   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1978 
1979   /* compute the local portion of C (mpi mat) */
1980   /*------------------------------------------*/
1981   /* allocate bi array and free space for accumulating nonzero column info */
1982   ierr  = PetscMalloc1(pn+1,&bi);CHKERRQ(ierr);
1983   bi[0] = 0;
1984 
1985   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
1986   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(pdti[pn],PetscIntSumTruncate(poti[pon],ai[am])));
1987   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1988   current_space = free_space;
1989 
1990   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1991   for (k=0; k<merge->nrecv; k++) {
1992     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1993     nrows       = *buf_ri_k[k];
1994     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1995     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th received i-structure  */
1996   }
1997 
1998   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
1999   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
2000   rmax = 0;
2001   for (i=0; i<pn; i++) {
2002     /* add pdt[i,:]*AP into lnk */
2003     pnz = pdti[i+1] - pdti[i];
2004     ptJ = pdtj + pdti[i];
2005     for (j=0; j<pnz; j++) {
2006       row  = ptJ[j];  /* row of AP == col of Pt */
2007       anz  = ai[row+1] - ai[row];
2008       Jptr = aj + ai[row];
2009       /* add non-zero cols of AP into the sorted linked list lnk */
2010       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
2011     }
2012 
2013     /* add received col data into lnk */
2014     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
2015       if (i == *nextrow[k]) { /* i-th row */
2016         nzi  = *(nextci[k]+1) - *nextci[k];
2017         Jptr = buf_rj[k] + *nextci[k];
2018         ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
2019         nextrow[k]++; nextci[k]++;
2020       }
2021     }
2022 
2023     /* add missing diagonal entry */
2024     if (C->force_diagonals) {
2025       k = i + owners[rank]; /* column index */
2026       ierr = PetscLLCondensedAddSorted_Scalable(1,&k,lnk);CHKERRQ(ierr);
2027     }
2028 
2029     nnz = lnk[0];
2030 
2031     /* if free space is not available, make more free space */
2032     if (current_space->local_remaining<nnz) {
2033       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
2034       nspacedouble++;
2035     }
2036     /* copy data into free space, then initialize lnk */
2037     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
2038     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
2039 
2040     current_space->array           += nnz;
2041     current_space->local_used      += nnz;
2042     current_space->local_remaining -= nnz;
2043 
2044     bi[i+1] = bi[i] + nnz;
2045     if (nnz > rmax) rmax = nnz;
2046   }
2047   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
2048 
2049   ierr      = PetscMalloc1(bi[pn]+1,&bj);CHKERRQ(ierr);
2050   ierr      = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
2051   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
2052   if (afill_tmp > afill) afill = afill_tmp;
2053   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
2054   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
2055   ierr = MatRestoreSymbolicTranspose_SeqAIJ(p->A,&pdti,&pdtj);CHKERRQ(ierr);
2056   ierr = MatRestoreSymbolicTranspose_SeqAIJ(p->B,&poti,&potj);CHKERRQ(ierr);
2057 
2058   /* create symbolic parallel matrix C - why cannot be assembled in Numeric part   */
2059   /*-------------------------------------------------------------------------------*/
2060   ierr = MatSetSizes(C,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
2061   ierr = MatSetBlockSizes(C,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
2062   ierr = MatGetType(A,&mtype);CHKERRQ(ierr);
2063   ierr = MatSetType(C,mtype);CHKERRQ(ierr);
2064   ierr = MatMPIAIJSetPreallocation(C,0,dnz,0,onz);CHKERRQ(ierr);
2065   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
2066   ierr = MatSetBlockSize(C,1);CHKERRQ(ierr);
2067   ierr = MatSetOption(C,MAT_NO_OFF_PROC_ENTRIES,PETSC_TRUE);CHKERRQ(ierr);
2068   for (i=0; i<pn; i++) {
2069     row  = i + rstart;
2070     nnz  = bi[i+1] - bi[i];
2071     Jptr = bj + bi[i];
2072     ierr = MatSetValues(C,1,&row,nnz,Jptr,NULL,INSERT_VALUES);CHKERRQ(ierr);
2073   }
2074   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2075   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2076   ierr = MatSetOption(C,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);CHKERRQ(ierr);
2077   merge->bi        = bi;
2078   merge->bj        = bj;
2079   merge->coi       = coi;
2080   merge->coj       = coj;
2081   merge->buf_ri    = buf_ri;
2082   merge->buf_rj    = buf_rj;
2083   merge->owners_co = owners_co;
2084 
2085   /* attach the supporting struct to C for reuse */
2086   C->product->data    = ptap;
2087   C->product->destroy = MatDestroy_MPIAIJ_PtAP;
2088   ptap->merge         = merge;
2089 
2090   C->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ;
2091 
2092 #if defined(PETSC_USE_INFO)
2093   if (bi[pn] != 0) {
2094     ierr = PetscInfo3(C,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
2095     ierr = PetscInfo1(C,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
2096   } else {
2097     ierr = PetscInfo(C,"Empty matrix product\n");CHKERRQ(ierr);
2098   }
2099 #endif
2100   PetscFunctionReturn(0);
2101 }
2102 
2103 /* ---------------------------------------------------------------- */
2104 static PetscErrorCode MatProductSymbolic_AtB_MPIAIJ_MPIAIJ(Mat C)
2105 {
2106   PetscErrorCode ierr;
2107   Mat_Product    *product = C->product;
2108   Mat            A=product->A,B=product->B;
2109   PetscReal      fill=product->fill;
2110   PetscBool      flg;
2111 
2112   PetscFunctionBegin;
2113   /* scalable */
2114   ierr = PetscStrcmp(product->alg,"scalable",&flg);CHKERRQ(ierr);
2115   if (flg) {
2116     ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(A,B,fill,C);CHKERRQ(ierr);
2117     goto next;
2118   }
2119 
2120   /* nonscalable */
2121   ierr = PetscStrcmp(product->alg,"nonscalable",&flg);CHKERRQ(ierr);
2122   if (flg) {
2123     ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(A,B,fill,C);CHKERRQ(ierr);
2124     goto next;
2125   }
2126 
2127   /* matmatmult */
2128   ierr = PetscStrcmp(product->alg,"at*b",&flg);CHKERRQ(ierr);
2129   if (flg) {
2130     Mat       At;
2131     Mat_APMPI *ptap;
2132 
2133     ierr = MatTranspose(A,MAT_INITIAL_MATRIX,&At);CHKERRQ(ierr);
2134     ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ(At,B,fill,C);CHKERRQ(ierr);
2135     ptap = (Mat_APMPI*)C->product->data;
2136     if (ptap) {
2137       ptap->Pt = At;
2138       C->product->destroy = MatDestroy_MPIAIJ_PtAP;
2139     }
2140     C->ops->transposematmultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult;
2141     goto next;
2142   }
2143 
2144   /* backend general code */
2145   ierr = PetscStrcmp(product->alg,"backend",&flg);CHKERRQ(ierr);
2146   if (flg) {
2147     ierr = MatProductSymbolic_MPIAIJBACKEND(C);CHKERRQ(ierr);
2148     PetscFunctionReturn(0);
2149   }
2150 
2151   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"MatProduct type is not supported");
2152 
2153 next:
2154   C->ops->productnumeric = MatProductNumeric_AtB;
2155   PetscFunctionReturn(0);
2156 }
2157 
2158 /* ---------------------------------------------------------------- */
2159 /* Set options for MatMatMultxxx_MPIAIJ_MPIAIJ */
2160 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_AB(Mat C)
2161 {
2162   PetscErrorCode ierr;
2163   Mat_Product    *product = C->product;
2164   Mat            A=product->A,B=product->B;
2165 #if defined(PETSC_HAVE_HYPRE)
2166   const char     *algTypes[5] = {"scalable","nonscalable","seqmpi","backend","hypre"};
2167   PetscInt       nalg = 5;
2168 #else
2169   const char     *algTypes[4] = {"scalable","nonscalable","seqmpi","backend",};
2170   PetscInt       nalg = 4;
2171 #endif
2172   PetscInt       alg = 1; /* set nonscalable algorithm as default */
2173   PetscBool      flg;
2174   MPI_Comm       comm;
2175 
2176   PetscFunctionBegin;
2177   /* Check matrix local sizes */
2178   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
2179   if (A->cmap->rstart != B->rmap->rstart || A->cmap->rend != B->rmap->rend) SETERRQ4(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, (%D, %D) != (%D,%D)",A->cmap->rstart,A->cmap->rend,B->rmap->rstart,B->rmap->rend);
2180 
2181   /* Set "nonscalable" as default algorithm */
2182   ierr = PetscStrcmp(C->product->alg,"default",&flg);CHKERRQ(ierr);
2183   if (flg) {
2184     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2185 
2186     /* Set "scalable" as default if BN and local nonzeros of A and B are large */
2187     if (B->cmap->N > 100000) { /* may switch to scalable algorithm as default */
2188       MatInfo     Ainfo,Binfo;
2189       PetscInt    nz_local;
2190       PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
2191 
2192       ierr = MatGetInfo(A,MAT_LOCAL,&Ainfo);CHKERRQ(ierr);
2193       ierr = MatGetInfo(B,MAT_LOCAL,&Binfo);CHKERRQ(ierr);
2194       nz_local = (PetscInt)(Ainfo.nz_allocated + Binfo.nz_allocated);
2195 
2196       if (B->cmap->N > product->fill*nz_local) alg_scalable_loc = PETSC_TRUE;
2197       ierr = MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm);CHKERRQ(ierr);
2198 
2199       if (alg_scalable) {
2200         alg  = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
2201         ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2202         ierr = PetscInfo2(B,"Use scalable algorithm, BN %D, fill*nz_allocated %g\n",B->cmap->N,product->fill*nz_local);CHKERRQ(ierr);
2203       }
2204     }
2205   }
2206 
2207   /* Get runtime option */
2208   if (product->api_user) {
2209     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatMatMult","Mat");CHKERRQ(ierr);
2210     ierr = PetscOptionsEList("-matmatmult_via","Algorithmic approach","MatMatMult",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2211     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2212   } else {
2213     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_AB","Mat");CHKERRQ(ierr);
2214     ierr = PetscOptionsEList("-matproduct_ab_via","Algorithmic approach","MatMatMult",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2215     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2216   }
2217   if (flg) {
2218     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2219   }
2220 
2221   C->ops->productsymbolic = MatProductSymbolic_AB_MPIAIJ_MPIAIJ;
2222   PetscFunctionReturn(0);
2223 }
2224 
2225 /* Set options for MatTransposeMatMultXXX_MPIAIJ_MPIAIJ */
2226 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_AtB(Mat C)
2227 {
2228   PetscErrorCode ierr;
2229   Mat_Product    *product = C->product;
2230   Mat            A=product->A,B=product->B;
2231   const char     *algTypes[4] = {"scalable","nonscalable","at*b","backend"};
2232   PetscInt       nalg = 4;
2233   PetscInt       alg = 1; /* set default algorithm  */
2234   PetscBool      flg;
2235   MPI_Comm       comm;
2236 
2237   PetscFunctionBegin;
2238   /* Check matrix local sizes */
2239   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
2240   if (A->rmap->rstart != B->rmap->rstart || A->rmap->rend != B->rmap->rend) SETERRQ4(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, A (%D, %D) != B (%D,%D)",A->rmap->rstart,A->rmap->rend,B->rmap->rstart,B->rmap->rend);
2241 
2242   /* Set default algorithm */
2243   ierr = PetscStrcmp(C->product->alg,"default",&flg);CHKERRQ(ierr);
2244   if (flg) {
2245     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2246   }
2247 
2248   /* Set "scalable" as default if BN and local nonzeros of A and B are large */
2249   if (alg && B->cmap->N > 100000) { /* may switch to scalable algorithm as default */
2250     MatInfo     Ainfo,Binfo;
2251     PetscInt    nz_local;
2252     PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
2253 
2254     ierr = MatGetInfo(A,MAT_LOCAL,&Ainfo);CHKERRQ(ierr);
2255     ierr = MatGetInfo(B,MAT_LOCAL,&Binfo);CHKERRQ(ierr);
2256     nz_local = (PetscInt)(Ainfo.nz_allocated + Binfo.nz_allocated);
2257 
2258     if (B->cmap->N > product->fill*nz_local) alg_scalable_loc = PETSC_TRUE;
2259     ierr = MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm);CHKERRQ(ierr);
2260 
2261     if (alg_scalable) {
2262       alg  = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
2263       ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2264       ierr = PetscInfo2(B,"Use scalable algorithm, BN %D, fill*nz_allocated %g\n",B->cmap->N,product->fill*nz_local);CHKERRQ(ierr);
2265     }
2266   }
2267 
2268   /* Get runtime option */
2269   if (product->api_user) {
2270     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatTransposeMatMult","Mat");CHKERRQ(ierr);
2271     ierr = PetscOptionsEList("-mattransposematmult_via","Algorithmic approach","MatTransposeMatMult",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2272     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2273   } else {
2274     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_AtB","Mat");CHKERRQ(ierr);
2275     ierr = PetscOptionsEList("-matproduct_atb_via","Algorithmic approach","MatTransposeMatMult",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2276     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2277   }
2278   if (flg) {
2279     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2280   }
2281 
2282   C->ops->productsymbolic = MatProductSymbolic_AtB_MPIAIJ_MPIAIJ;
2283   PetscFunctionReturn(0);
2284 }
2285 
2286 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_PtAP(Mat C)
2287 {
2288   PetscErrorCode ierr;
2289   Mat_Product    *product = C->product;
2290   Mat            A=product->A,P=product->B;
2291   MPI_Comm       comm;
2292   PetscBool      flg;
2293   PetscInt       alg=1; /* set default algorithm */
2294 #if !defined(PETSC_HAVE_HYPRE)
2295   const char     *algTypes[5] = {"scalable","nonscalable","allatonce","allatonce_merged","backend"};
2296   PetscInt       nalg=5;
2297 #else
2298   const char     *algTypes[6] = {"scalable","nonscalable","allatonce","allatonce_merged","backend","hypre"};
2299   PetscInt       nalg=6;
2300 #endif
2301   PetscInt       pN=P->cmap->N;
2302 
2303   PetscFunctionBegin;
2304   /* Check matrix local sizes */
2305   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
2306   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend) SETERRQ4(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, Arow (%D, %D) != Prow (%D,%D)",A->rmap->rstart,A->rmap->rend,P->rmap->rstart,P->rmap->rend);
2307   if (A->cmap->rstart != P->rmap->rstart || A->cmap->rend != P->rmap->rend) SETERRQ4(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, Acol (%D, %D) != Prow (%D,%D)",A->cmap->rstart,A->cmap->rend,P->rmap->rstart,P->rmap->rend);
2308 
2309   /* Set "nonscalable" as default algorithm */
2310   ierr = PetscStrcmp(C->product->alg,"default",&flg);CHKERRQ(ierr);
2311   if (flg) {
2312     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2313 
2314     /* Set "scalable" as default if BN and local nonzeros of A and B are large */
2315     if (pN > 100000) {
2316       MatInfo     Ainfo,Pinfo;
2317       PetscInt    nz_local;
2318       PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
2319 
2320       ierr = MatGetInfo(A,MAT_LOCAL,&Ainfo);CHKERRQ(ierr);
2321       ierr = MatGetInfo(P,MAT_LOCAL,&Pinfo);CHKERRQ(ierr);
2322       nz_local = (PetscInt)(Ainfo.nz_allocated + Pinfo.nz_allocated);
2323 
2324       if (pN > product->fill*nz_local) alg_scalable_loc = PETSC_TRUE;
2325       ierr = MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm);CHKERRQ(ierr);
2326 
2327       if (alg_scalable) {
2328         alg = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
2329         ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2330       }
2331     }
2332   }
2333 
2334   /* Get runtime option */
2335   if (product->api_user) {
2336     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatPtAP","Mat");CHKERRQ(ierr);
2337     ierr = PetscOptionsEList("-matptap_via","Algorithmic approach","MatPtAP",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2338     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2339   } else {
2340     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_PtAP","Mat");CHKERRQ(ierr);
2341     ierr = PetscOptionsEList("-matproduct_ptap_via","Algorithmic approach","MatPtAP",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2342     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2343   }
2344   if (flg) {
2345     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2346   }
2347 
2348   C->ops->productsymbolic = MatProductSymbolic_PtAP_MPIAIJ_MPIAIJ;
2349   PetscFunctionReturn(0);
2350 }
2351 
2352 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_RARt(Mat C)
2353 {
2354   Mat_Product *product = C->product;
2355   Mat         A = product->A,R=product->B;
2356 
2357   PetscFunctionBegin;
2358   /* Check matrix local sizes */
2359   if (A->cmap->n != R->cmap->n || A->rmap->n != R->cmap->n) SETERRQ4(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, A local (%D, %D), R local (%D,%D)",A->rmap->n,A->rmap->n,R->rmap->n,R->cmap->n);
2360 
2361   C->ops->productsymbolic = MatProductSymbolic_RARt_MPIAIJ_MPIAIJ;
2362   PetscFunctionReturn(0);
2363 }
2364 
2365 /*
2366  Set options for ABC = A*B*C = A*(B*C); ABC's algorithm must be chosen from AB's algorithm
2367 */
2368 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_ABC(Mat C)
2369 {
2370   PetscErrorCode ierr;
2371   Mat_Product    *product = C->product;
2372   PetscBool      flg = PETSC_FALSE;
2373   PetscInt       alg = 1; /* default algorithm */
2374   const char     *algTypes[3] = {"scalable","nonscalable","seqmpi"};
2375   PetscInt       nalg = 3;
2376 
2377   PetscFunctionBegin;
2378   /* Set default algorithm */
2379   ierr = PetscStrcmp(C->product->alg,"default",&flg);CHKERRQ(ierr);
2380   if (flg) {
2381     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2382   }
2383 
2384   /* Get runtime option */
2385   if (product->api_user) {
2386     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatMatMatMult","Mat");CHKERRQ(ierr);
2387     ierr = PetscOptionsEList("-matmatmatmult_via","Algorithmic approach","MatMatMatMult",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2388     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2389   } else {
2390     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_ABC","Mat");CHKERRQ(ierr);
2391     ierr = PetscOptionsEList("-matproduct_abc_via","Algorithmic approach","MatProduct_ABC",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2392     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2393   }
2394   if (flg) {
2395     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2396   }
2397 
2398   C->ops->matmatmultsymbolic = MatMatMatMultSymbolic_MPIAIJ_MPIAIJ_MPIAIJ;
2399   C->ops->productsymbolic    = MatProductSymbolic_ABC;
2400   PetscFunctionReturn(0);
2401 }
2402 
2403 PETSC_INTERN PetscErrorCode MatProductSetFromOptions_MPIAIJ(Mat C)
2404 {
2405   PetscErrorCode ierr;
2406   Mat_Product    *product = C->product;
2407 
2408   PetscFunctionBegin;
2409   switch (product->type) {
2410   case MATPRODUCT_AB:
2411     ierr = MatProductSetFromOptions_MPIAIJ_AB(C);CHKERRQ(ierr);
2412     break;
2413   case MATPRODUCT_AtB:
2414     ierr = MatProductSetFromOptions_MPIAIJ_AtB(C);CHKERRQ(ierr);
2415     break;
2416   case MATPRODUCT_PtAP:
2417     ierr = MatProductSetFromOptions_MPIAIJ_PtAP(C);CHKERRQ(ierr);
2418     break;
2419   case MATPRODUCT_RARt:
2420     ierr = MatProductSetFromOptions_MPIAIJ_RARt(C);CHKERRQ(ierr);
2421     break;
2422   case MATPRODUCT_ABC:
2423     ierr = MatProductSetFromOptions_MPIAIJ_ABC(C);CHKERRQ(ierr);
2424     break;
2425   default:
2426     break;
2427   }
2428   PetscFunctionReturn(0);
2429 }
2430