xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision 03e6d32953ff6d123393c3df7e15df579e2665b4)
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 
13 #if defined(PETSC_HAVE_HYPRE)
14 PETSC_INTERN PetscErrorCode MatMatMultSymbolic_AIJ_AIJ_wHYPRE(Mat,Mat,PetscReal,Mat*);
15 #endif
16 
17 #undef __FUNCT__
18 #define __FUNCT__ "MatMatMult_MPIAIJ_MPIAIJ"
19 PETSC_INTERN PetscErrorCode MatMatMult_MPIAIJ_MPIAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill, Mat *C)
20 {
21   PetscErrorCode ierr;
22 #if defined(PETSC_HAVE_HYPRE)
23   const char     *algTypes[3] = {"scalable","nonscalable","hypre"};
24   PetscInt       nalg = 3;
25 #else
26   const char     *algTypes[2] = {"scalable","nonscalable"};
27   PetscInt       nalg = 2;
28 #endif
29   PetscInt       alg = 1; /* set default algorithm */
30   MPI_Comm       comm;
31 
32   PetscFunctionBegin;
33   if (scall == MAT_INITIAL_MATRIX) {
34     ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
35     if (A->cmap->rstart != B->rmap->rstart || A->cmap->rend != B->rmap->rend) SETERRQ4(comm,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);
36 
37     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
38     ierr = PetscOptionsEList("-matmatmult_via","Algorithmic approach","MatMatMult",algTypes,nalg,algTypes[1],&alg,NULL);CHKERRQ(ierr);
39     ierr = PetscOptionsEnd();CHKERRQ(ierr);
40 
41     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
42     switch (alg) {
43     case 1:
44       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(A,B,fill,C);CHKERRQ(ierr);
45       break;
46 #if defined(PETSC_HAVE_HYPRE)
47     case 2:
48       ierr = MatMatMultSymbolic_AIJ_AIJ_wHYPRE(A,B,fill,C);CHKERRQ(ierr);
49       break;
50 #endif
51     default:
52       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ(A,B,fill,C);CHKERRQ(ierr);
53       break;
54     }
55     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
56   }
57   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
58   ierr = (*(*C)->ops->matmultnumeric)(A,B,*C);CHKERRQ(ierr);
59   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
60   PetscFunctionReturn(0);
61 }
62 
63 PetscErrorCode MatDestroy_MPIAIJ_MatMatMult(Mat A)
64 {
65   PetscErrorCode ierr;
66   Mat_MPIAIJ     *a    = (Mat_MPIAIJ*)A->data;
67   Mat_PtAPMPI    *ptap = a->ptap;
68 
69   PetscFunctionBegin;
70   ierr = PetscFree2(ptap->startsj_s,ptap->startsj_r);CHKERRQ(ierr);
71   ierr = PetscFree(ptap->bufa);CHKERRQ(ierr);
72   ierr = MatDestroy(&ptap->P_loc);CHKERRQ(ierr);
73   ierr = MatDestroy(&ptap->P_oth);CHKERRQ(ierr);
74   ierr = MatDestroy(&ptap->Pt);CHKERRQ(ierr);
75   ierr = PetscFree(ptap->api);CHKERRQ(ierr);
76   ierr = PetscFree(ptap->apj);CHKERRQ(ierr);
77   ierr = PetscFree(ptap->apa);CHKERRQ(ierr);
78   ierr = ptap->destroy(A);CHKERRQ(ierr);
79   ierr = PetscFree(ptap);CHKERRQ(ierr);
80   PetscFunctionReturn(0);
81 }
82 
83 PetscErrorCode MatDuplicate_MPIAIJ_MatMatMult(Mat A, MatDuplicateOption op, Mat *M)
84 {
85   PetscErrorCode ierr;
86   Mat_MPIAIJ     *a    = (Mat_MPIAIJ*)A->data;
87   Mat_PtAPMPI    *ptap = a->ptap;
88 
89   PetscFunctionBegin;
90   ierr = (*ptap->duplicate)(A,op,M);CHKERRQ(ierr);
91 
92   (*M)->ops->destroy   = ptap->destroy;   /* = MatDestroy_MPIAIJ, *M doesn't duplicate A's special structure! */
93   (*M)->ops->duplicate = ptap->duplicate; /* = MatDuplicate_MPIAIJ */
94   PetscFunctionReturn(0);
95 }
96 
97 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,Mat C)
98 {
99   PetscErrorCode ierr;
100   Mat_MPIAIJ     *a  =(Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
101   Mat_SeqAIJ     *ad =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
102   Mat_SeqAIJ     *cd =(Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
103   PetscScalar    *cda=cd->a,*coa=co->a;
104   Mat_SeqAIJ     *p_loc,*p_oth;
105   PetscScalar    *apa,*ca;
106   PetscInt       cm   =C->rmap->n;
107   Mat_PtAPMPI    *ptap=c->ptap;
108   PetscInt       *api,*apj,*apJ,i,k;
109   PetscInt       cstart=C->cmap->rstart;
110   PetscInt       cdnz,conz,k0,k1;
111   MPI_Comm       comm;
112   PetscMPIInt    size;
113 
114   PetscFunctionBegin;
115   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
116   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
117 
118   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
119   /*-----------------------------------------------------*/
120   /* update numerical values of P_oth and P_loc */
121   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
122   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
123 
124   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
125   /*----------------------------------------------------------*/
126   /* get data from symbolic products */
127   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
128   p_oth = NULL;
129   if (size >1) {
130     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
131   }
132 
133   /* get apa for storing dense row A[i,:]*P */
134   apa = ptap->apa;
135 
136   api = ptap->api;
137   apj = ptap->apj;
138   for (i=0; i<cm; i++) {
139     /* compute apa = A[i,:]*P */
140     AProw_nonscalable(i,ad,ao,p_loc,p_oth,apa);
141 
142     /* set values in C */
143     apJ  = apj + api[i];
144     cdnz = cd->i[i+1] - cd->i[i];
145     conz = co->i[i+1] - co->i[i];
146 
147     /* 1st off-diagoanl part of C */
148     ca = coa + co->i[i];
149     k  = 0;
150     for (k0=0; k0<conz; k0++) {
151       if (apJ[k] >= cstart) break;
152       ca[k0]      = apa[apJ[k]];
153       apa[apJ[k]] = 0.0;
154       k++;
155     }
156 
157     /* diagonal part of C */
158     ca = cda + cd->i[i];
159     for (k1=0; k1<cdnz; k1++) {
160       ca[k1]      = apa[apJ[k]];
161       apa[apJ[k]] = 0.0;
162       k++;
163     }
164 
165     /* 2nd off-diagoanl part of C */
166     ca = coa + co->i[i];
167     for (; k0<conz; k0++) {
168       ca[k0]      = apa[apJ[k]];
169       apa[apJ[k]] = 0.0;
170       k++;
171     }
172   }
173   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
174   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
175   PetscFunctionReturn(0);
176 }
177 
178 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,PetscReal fill,Mat *C)
179 {
180   PetscErrorCode     ierr;
181   MPI_Comm           comm;
182   PetscMPIInt        size;
183   Mat                Cmpi;
184   Mat_PtAPMPI        *ptap;
185   PetscFreeSpaceList free_space=NULL,current_space=NULL;
186   Mat_MPIAIJ         *a        =(Mat_MPIAIJ*)A->data,*c;
187   Mat_SeqAIJ         *ad       =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
188   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
189   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
190   PetscInt           *lnk,i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi;
191   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n,Crmax;
192   PetscBT            lnkbt;
193   PetscScalar        *apa;
194   PetscReal          afill;
195   PetscTable         ta;
196 
197   PetscFunctionBegin;
198   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
199   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
200 
201   /* create struct Mat_PtAPMPI and attached it to C later */
202   ierr = PetscNew(&ptap);CHKERRQ(ierr);
203 
204   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
205   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
206 
207   /* get P_loc by taking all local rows of P */
208   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
209 
210   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
211   pi_loc = p_loc->i; pj_loc = p_loc->j;
212   if (size > 1) {
213     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
214     pi_oth = p_oth->i; pj_oth = p_oth->j;
215   } else {
216     p_oth = NULL;
217     pi_oth = NULL; pj_oth = NULL;
218   }
219 
220   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
221   /*-------------------------------------------------------------------*/
222   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
223   ptap->api = api;
224   api[0]    = 0;
225 
226   /* create and initialize a linked list */
227   ierr = PetscTableCreate(pN,pN,&ta);CHKERRQ(ierr);
228   MatRowMergeMax_SeqAIJ(p_loc,ptap->P_loc->rmap->N,ta);
229   MatRowMergeMax_SeqAIJ(p_oth,ptap->P_oth->rmap->N,ta);
230   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
231   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
232 
233   ierr = PetscLLCondensedCreate(Crmax,pN,&lnk,&lnkbt);CHKERRQ(ierr);
234 
235   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
236   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space);CHKERRQ(ierr);
237   current_space = free_space;
238 
239   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
240   for (i=0; i<am; i++) {
241     /* diagonal portion of A */
242     nzi = adi[i+1] - adi[i];
243     for (j=0; j<nzi; j++) {
244       row  = *adj++;
245       pnz  = pi_loc[row+1] - pi_loc[row];
246       Jptr = pj_loc + pi_loc[row];
247       /* add non-zero cols of P into the sorted linked list lnk */
248       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
249     }
250     /* off-diagonal portion of A */
251     nzi = aoi[i+1] - aoi[i];
252     for (j=0; j<nzi; j++) {
253       row  = *aoj++;
254       pnz  = pi_oth[row+1] - pi_oth[row];
255       Jptr = pj_oth + pi_oth[row];
256       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
257     }
258 
259     apnz     = lnk[0];
260     api[i+1] = api[i] + apnz;
261 
262     /* if free space is not available, double the total space in the list */
263     if (current_space->local_remaining<apnz) {
264       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
265       nspacedouble++;
266     }
267 
268     /* Copy data into free space, then initialize lnk */
269     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
270     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
271 
272     current_space->array           += apnz;
273     current_space->local_used      += apnz;
274     current_space->local_remaining -= apnz;
275   }
276 
277   /* Allocate space for apj, initialize apj, and */
278   /* destroy list of free space and other temporary array(s) */
279   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
280   apj  = ptap->apj;
281   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
282   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
283 
284   /* malloc apa to store dense row A[i,:]*P */
285   ierr = PetscCalloc1(pN,&apa);CHKERRQ(ierr);
286 
287   ptap->apa = apa;
288 
289   /* create and assemble symbolic parallel matrix Cmpi */
290   /*----------------------------------------------------*/
291   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
292   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
293   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
294 
295   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
296   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
297   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
298   for (i=0; i<am; i++) {
299     row  = i + rstart;
300     apnz = api[i+1] - api[i];
301     ierr = MatSetValues(Cmpi,1,&row,apnz,apj,apa,INSERT_VALUES);CHKERRQ(ierr);
302     apj += apnz;
303   }
304   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
305   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
306 
307   ptap->destroy        = Cmpi->ops->destroy;
308   ptap->duplicate      = Cmpi->ops->duplicate;
309   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
310   Cmpi->ops->destroy   = MatDestroy_MPIAIJ_MatMatMult;
311   Cmpi->ops->duplicate = MatDuplicate_MPIAIJ_MatMatMult;
312 
313   /* attach the supporting struct to Cmpi for reuse */
314   c       = (Mat_MPIAIJ*)Cmpi->data;
315   c->ptap = ptap;
316 
317   *C = Cmpi;
318 
319   /* set MatInfo */
320   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
321   if (afill < 1.0) afill = 1.0;
322   Cmpi->info.mallocs           = nspacedouble;
323   Cmpi->info.fill_ratio_given  = fill;
324   Cmpi->info.fill_ratio_needed = afill;
325 
326 #if defined(PETSC_USE_INFO)
327   if (api[am]) {
328     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
329     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
330   } else {
331     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
332   }
333 #endif
334   PetscFunctionReturn(0);
335 }
336 
337 PETSC_INTERN PetscErrorCode MatMatMult_MPIAIJ_MPIDense(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
338 {
339   PetscErrorCode ierr;
340 
341   PetscFunctionBegin;
342   if (scall == MAT_INITIAL_MATRIX) {
343     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
344     ierr = MatMatMultSymbolic_MPIAIJ_MPIDense(A,B,fill,C);CHKERRQ(ierr);
345     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
346   }
347   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
348   ierr = MatMatMultNumeric_MPIAIJ_MPIDense(A,B,*C);CHKERRQ(ierr);
349   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
350   PetscFunctionReturn(0);
351 }
352 
353 typedef struct {
354   Mat         workB;
355   PetscScalar *rvalues,*svalues;
356   MPI_Request *rwaits,*swaits;
357 } MPIAIJ_MPIDense;
358 
359 PetscErrorCode MatMPIAIJ_MPIDenseDestroy(void *ctx)
360 {
361   MPIAIJ_MPIDense *contents = (MPIAIJ_MPIDense*) ctx;
362   PetscErrorCode  ierr;
363 
364   PetscFunctionBegin;
365   ierr = MatDestroy(&contents->workB);CHKERRQ(ierr);
366   ierr = PetscFree4(contents->rvalues,contents->svalues,contents->rwaits,contents->swaits);CHKERRQ(ierr);
367   ierr = PetscFree(contents);CHKERRQ(ierr);
368   PetscFunctionReturn(0);
369 }
370 
371 /*
372     This is a "dummy function" that handles the case where matrix C was created as a dense matrix
373   directly by the user and passed to MatMatMult() with the MAT_REUSE_MATRIX option
374 
375   It is the same as MatMatMultSymbolic_MPIAIJ_MPIDense() except does not create C
376 */
377 PetscErrorCode MatMatMultNumeric_MPIDense(Mat A,Mat B,Mat C)
378 {
379   PetscErrorCode         ierr;
380   PetscBool              flg;
381   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*) A->data;
382   PetscInt               nz   = aij->B->cmap->n;
383   PetscContainer         container;
384   MPIAIJ_MPIDense        *contents;
385   VecScatter             ctx   = aij->Mvctx;
386   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
387   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
388 
389   PetscFunctionBegin;
390   ierr = PetscObjectTypeCompare((PetscObject)B,MATMPIDENSE,&flg);CHKERRQ(ierr);
391   if (!flg) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Second matrix must be mpidense");
392 
393   /* Handle case where where user provided the final C matrix rather than calling MatMatMult() with MAT_INITIAL_MATRIX*/
394   ierr = PetscObjectTypeCompare((PetscObject)A,MATMPIAIJ,&flg);CHKERRQ(ierr);
395   if (!flg) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"First matrix must be MPIAIJ");
396 
397   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
398 
399   ierr = PetscNew(&contents);CHKERRQ(ierr);
400   /* Create work matrix used to store off processor rows of B needed for local product */
401   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,B->cmap->N,NULL,&contents->workB);CHKERRQ(ierr);
402   /* Create work arrays needed */
403   ierr = PetscMalloc4(B->cmap->N*from->starts[from->n],&contents->rvalues,
404                       B->cmap->N*to->starts[to->n],&contents->svalues,
405                       from->n,&contents->rwaits,
406                       to->n,&contents->swaits);CHKERRQ(ierr);
407 
408   ierr = PetscContainerCreate(PetscObjectComm((PetscObject)A),&container);CHKERRQ(ierr);
409   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
410   ierr = PetscContainerSetUserDestroy(container,MatMPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
411   ierr = PetscObjectCompose((PetscObject)C,"workB",(PetscObject)container);CHKERRQ(ierr);
412   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
413 
414   ierr = (*C->ops->matmultnumeric)(A,B,C);CHKERRQ(ierr);
415   PetscFunctionReturn(0);
416 }
417 
418 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat A,Mat B,PetscReal fill,Mat *C)
419 {
420   PetscErrorCode         ierr;
421   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*) A->data;
422   PetscInt               nz   = aij->B->cmap->n;
423   PetscContainer         container;
424   MPIAIJ_MPIDense        *contents;
425   VecScatter             ctx   = aij->Mvctx;
426   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
427   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
428   PetscInt               m     = A->rmap->n,n=B->cmap->n;
429 
430   PetscFunctionBegin;
431   ierr = MatCreate(PetscObjectComm((PetscObject)B),C);CHKERRQ(ierr);
432   ierr = MatSetSizes(*C,m,n,A->rmap->N,B->cmap->N);CHKERRQ(ierr);
433   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
434   ierr = MatSetType(*C,MATMPIDENSE);CHKERRQ(ierr);
435   ierr = MatMPIDenseSetPreallocation(*C,NULL);CHKERRQ(ierr);
436   ierr = MatAssemblyBegin(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
437   ierr = MatAssemblyEnd(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
438 
439   (*C)->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
440 
441   ierr = PetscNew(&contents);CHKERRQ(ierr);
442   /* Create work matrix used to store off processor rows of B needed for local product */
443   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,B->cmap->N,NULL,&contents->workB);CHKERRQ(ierr);
444   /* Create work arrays needed */
445   ierr = PetscMalloc4(B->cmap->N*from->starts[from->n],&contents->rvalues,
446                       B->cmap->N*to->starts[to->n],&contents->svalues,
447                       from->n,&contents->rwaits,
448                       to->n,&contents->swaits);CHKERRQ(ierr);
449 
450   ierr = PetscContainerCreate(PetscObjectComm((PetscObject)A),&container);CHKERRQ(ierr);
451   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
452   ierr = PetscContainerSetUserDestroy(container,MatMPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
453   ierr = PetscObjectCompose((PetscObject)(*C),"workB",(PetscObject)container);CHKERRQ(ierr);
454   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
455   PetscFunctionReturn(0);
456 }
457 
458 /*
459     Performs an efficient scatter on the rows of B needed by this process; this is
460     a modification of the VecScatterBegin_() routines.
461 */
462 PetscErrorCode MatMPIDenseScatter(Mat A,Mat B,Mat C,Mat *outworkB)
463 {
464   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*)A->data;
465   PetscErrorCode         ierr;
466   PetscScalar            *b,*w,*svalues,*rvalues;
467   VecScatter             ctx   = aij->Mvctx;
468   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
469   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
470   PetscInt               i,j,k;
471   PetscInt               *sindices,*sstarts,*rindices,*rstarts;
472   PetscMPIInt            *sprocs,*rprocs,nrecvs;
473   MPI_Request            *swaits,*rwaits;
474   MPI_Comm               comm;
475   PetscMPIInt            tag  = ((PetscObject)ctx)->tag,ncols = B->cmap->N, nrows = aij->B->cmap->n,imdex,nrowsB = B->rmap->n;
476   MPI_Status             status;
477   MPIAIJ_MPIDense        *contents;
478   PetscContainer         container;
479   Mat                    workB;
480 
481   PetscFunctionBegin;
482   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
483   ierr = PetscObjectQuery((PetscObject)C,"workB",(PetscObject*)&container);CHKERRQ(ierr);
484   if (!container) SETERRQ(comm,PETSC_ERR_PLIB,"Container does not exist");
485   ierr = PetscContainerGetPointer(container,(void**)&contents);CHKERRQ(ierr);
486 
487   workB = *outworkB = contents->workB;
488   if (nrows != workB->rmap->n) SETERRQ2(comm,PETSC_ERR_PLIB,"Number of rows of workB %D not equal to columns of aij->B %D",nrows,workB->cmap->n);
489   sindices = to->indices;
490   sstarts  = to->starts;
491   sprocs   = to->procs;
492   swaits   = contents->swaits;
493   svalues  = contents->svalues;
494 
495   rindices = from->indices;
496   rstarts  = from->starts;
497   rprocs   = from->procs;
498   rwaits   = contents->rwaits;
499   rvalues  = contents->rvalues;
500 
501   ierr = MatDenseGetArray(B,&b);CHKERRQ(ierr);
502   ierr = MatDenseGetArray(workB,&w);CHKERRQ(ierr);
503 
504   for (i=0; i<from->n; i++) {
505     ierr = MPI_Irecv(rvalues+ncols*rstarts[i],ncols*(rstarts[i+1]-rstarts[i]),MPIU_SCALAR,rprocs[i],tag,comm,rwaits+i);CHKERRQ(ierr);
506   }
507 
508   for (i=0; i<to->n; i++) {
509     /* pack a message at a time */
510     for (j=0; j<sstarts[i+1]-sstarts[i]; j++) {
511       for (k=0; k<ncols; k++) {
512         svalues[ncols*(sstarts[i] + j) + k] = b[sindices[sstarts[i]+j] + nrowsB*k];
513       }
514     }
515     ierr = MPI_Isend(svalues+ncols*sstarts[i],ncols*(sstarts[i+1]-sstarts[i]),MPIU_SCALAR,sprocs[i],tag,comm,swaits+i);CHKERRQ(ierr);
516   }
517 
518   nrecvs = from->n;
519   while (nrecvs) {
520     ierr = MPI_Waitany(from->n,rwaits,&imdex,&status);CHKERRQ(ierr);
521     nrecvs--;
522     /* unpack a message at a time */
523     for (j=0; j<rstarts[imdex+1]-rstarts[imdex]; j++) {
524       for (k=0; k<ncols; k++) {
525         w[rindices[rstarts[imdex]+j] + nrows*k] = rvalues[ncols*(rstarts[imdex] + j) + k];
526       }
527     }
528   }
529   if (to->n) {ierr = MPI_Waitall(to->n,swaits,to->sstatus);CHKERRQ(ierr);}
530 
531   ierr = MatDenseRestoreArray(B,&b);CHKERRQ(ierr);
532   ierr = MatDenseRestoreArray(workB,&w);CHKERRQ(ierr);
533   ierr = MatAssemblyBegin(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
534   ierr = MatAssemblyEnd(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
535   PetscFunctionReturn(0);
536 }
537 extern PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat,Mat,Mat);
538 
539 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat A,Mat B,Mat C)
540 {
541   PetscErrorCode ierr;
542   Mat_MPIAIJ     *aij    = (Mat_MPIAIJ*)A->data;
543   Mat_MPIDense   *bdense = (Mat_MPIDense*)B->data;
544   Mat_MPIDense   *cdense = (Mat_MPIDense*)C->data;
545   Mat            workB;
546 
547   PetscFunctionBegin;
548   /* diagonal block of A times all local rows of B*/
549   ierr = MatMatMultNumeric_SeqAIJ_SeqDense(aij->A,bdense->A,cdense->A);CHKERRQ(ierr);
550 
551   /* get off processor parts of B needed to complete the product */
552   ierr = MatMPIDenseScatter(A,B,C,&workB);CHKERRQ(ierr);
553 
554   /* off-diagonal block of A times nonlocal rows of B */
555   ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A);CHKERRQ(ierr);
556   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
557   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
558   PetscFunctionReturn(0);
559 }
560 
561 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
562 {
563   PetscErrorCode ierr;
564   Mat_MPIAIJ     *a   = (Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
565   Mat_SeqAIJ     *ad  = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
566   Mat_SeqAIJ     *cd  = (Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
567   PetscInt       *adi = ad->i,*adj,*aoi=ao->i,*aoj;
568   PetscScalar    *ada,*aoa,*cda=cd->a,*coa=co->a;
569   Mat_SeqAIJ     *p_loc,*p_oth;
570   PetscInt       *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
571   PetscScalar    *pa_loc,*pa_oth,*pa,valtmp,*ca;
572   PetscInt       cm          = C->rmap->n,anz,pnz;
573   Mat_PtAPMPI    *ptap       = c->ptap;
574   PetscScalar    *apa_sparse = ptap->apa;
575   PetscInt       *api,*apj,*apJ,i,j,k,row;
576   PetscInt       cstart = C->cmap->rstart;
577   PetscInt       cdnz,conz,k0,k1,nextp;
578   MPI_Comm       comm;
579   PetscMPIInt    size;
580 
581   PetscFunctionBegin;
582   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
583   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
584 
585   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
586   /*-----------------------------------------------------*/
587   /* update numerical values of P_oth and P_loc */
588   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
589   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
590 
591   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
592   /*----------------------------------------------------------*/
593   /* get data from symbolic products */
594   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
595   pi_loc = p_loc->i; pj_loc = p_loc->j; pa_loc = p_loc->a;
596   if (size >1) {
597     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
598     pi_oth = p_oth->i; pj_oth = p_oth->j; pa_oth = p_oth->a;
599   } else {
600     p_oth = NULL; pi_oth = NULL; pj_oth = NULL; pa_oth = NULL;
601   }
602 
603   api = ptap->api;
604   apj = ptap->apj;
605   for (i=0; i<cm; i++) {
606     apJ = apj + api[i];
607 
608     /* diagonal portion of A */
609     anz = adi[i+1] - adi[i];
610     adj = ad->j + adi[i];
611     ada = ad->a + adi[i];
612     for (j=0; j<anz; j++) {
613       row = adj[j];
614       pnz = pi_loc[row+1] - pi_loc[row];
615       pj  = pj_loc + pi_loc[row];
616       pa  = pa_loc + pi_loc[row];
617       /* perform sparse axpy */
618       valtmp = ada[j];
619       nextp  = 0;
620       for (k=0; nextp<pnz; k++) {
621         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
622           apa_sparse[k] += valtmp*pa[nextp++];
623         }
624       }
625       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
626     }
627 
628     /* off-diagonal portion of A */
629     anz = aoi[i+1] - aoi[i];
630     aoj = ao->j + aoi[i];
631     aoa = ao->a + aoi[i];
632     for (j=0; j<anz; j++) {
633       row = aoj[j];
634       pnz = pi_oth[row+1] - pi_oth[row];
635       pj  = pj_oth + pi_oth[row];
636       pa  = pa_oth + pi_oth[row];
637       /* perform sparse axpy */
638       valtmp = aoa[j];
639       nextp  = 0;
640       for (k=0; nextp<pnz; k++) {
641         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
642           apa_sparse[k] += valtmp*pa[nextp++];
643         }
644       }
645       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
646     }
647 
648     /* set values in C */
649     cdnz = cd->i[i+1] - cd->i[i];
650     conz = co->i[i+1] - co->i[i];
651 
652     /* 1st off-diagoanl part of C */
653     ca = coa + co->i[i];
654     k  = 0;
655     for (k0=0; k0<conz; k0++) {
656       if (apJ[k] >= cstart) break;
657       ca[k0]        = apa_sparse[k];
658       apa_sparse[k] = 0.0;
659       k++;
660     }
661 
662     /* diagonal part of C */
663     ca = cda + cd->i[i];
664     for (k1=0; k1<cdnz; k1++) {
665       ca[k1]        = apa_sparse[k];
666       apa_sparse[k] = 0.0;
667       k++;
668     }
669 
670     /* 2nd off-diagoanl part of C */
671     ca = coa + co->i[i];
672     for (; k0<conz; k0++) {
673       ca[k0]        = apa_sparse[k];
674       apa_sparse[k] = 0.0;
675       k++;
676     }
677   }
678   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
679   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
680   PetscFunctionReturn(0);
681 }
682 
683 /* same as MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(), except using LLCondensed to avoid O(BN) memory requirement */
684 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat *C)
685 {
686   PetscErrorCode     ierr;
687   MPI_Comm           comm;
688   PetscMPIInt        size;
689   Mat                Cmpi;
690   Mat_PtAPMPI        *ptap;
691   PetscFreeSpaceList free_space = NULL,current_space=NULL;
692   Mat_MPIAIJ         *a         = (Mat_MPIAIJ*)A->data,*c;
693   Mat_SeqAIJ         *ad        = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
694   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
695   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
696   PetscInt           i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi,*lnk,apnz_max;
697   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
698   PetscReal          afill;
699   PetscScalar        *apa;
700   PetscTable         ta;
701 
702   PetscFunctionBegin;
703   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
704   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
705 
706   /* create struct Mat_PtAPMPI and attached it to C later */
707   ierr = PetscNew(&ptap);CHKERRQ(ierr);
708 
709   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
710   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
711 
712   /* get P_loc by taking all local rows of P */
713   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
714 
715   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
716   pi_loc = p_loc->i; pj_loc = p_loc->j;
717   if (size > 1) {
718     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
719     pi_oth = p_oth->i; pj_oth = p_oth->j;
720   } else {
721     p_oth  = NULL;
722     pi_oth = NULL; pj_oth = NULL;
723   }
724 
725   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
726   /*-------------------------------------------------------------------*/
727   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
728   ptap->api = api;
729   api[0]    = 0;
730 
731   /* create and initialize a linked list */
732   apnz_max = 6*(p_loc->rmax + (PetscInt)(1.e-2*pN)); /* expected apnz_max */
733   if (apnz_max > pN) apnz_max = pN;
734   ierr = PetscTableCreate(apnz_max,pN,&ta);CHKERRQ(ierr);
735 
736   /* Calculate apnz_max */
737   apnz_max = 0;
738   for (i=0; i<am; i++) {
739     ierr = PetscTableRemoveAll(ta);CHKERRQ(ierr);
740     /* diagonal portion of A */
741     nzi  = adi[i+1] - adi[i];
742     Jptr = adj+adi[i];  /* cols of A_diag */
743     MatMergeRows_SeqAIJ(p_loc,nzi,Jptr,ta);
744     ierr = PetscTableGetCount(ta,&apnz);CHKERRQ(ierr);
745     if (apnz_max < apnz) apnz_max = apnz;
746 
747     /*  off-diagonal portion of A */
748     nzi = aoi[i+1] - aoi[i];
749     Jptr = aoj+aoi[i];  /* cols of A_off */
750     MatMergeRows_SeqAIJ(p_oth,nzi,Jptr,ta);
751     ierr = PetscTableGetCount(ta,&apnz);CHKERRQ(ierr);
752     if (apnz_max < apnz) apnz_max = apnz;
753   }
754   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
755 
756   ierr = PetscLLCondensedCreate_Scalable(apnz_max,&lnk);CHKERRQ(ierr);
757 
758   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
759   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space);CHKERRQ(ierr);
760   current_space = free_space;
761   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
762   for (i=0; i<am; i++) {
763     /* diagonal portion of A */
764     nzi = adi[i+1] - adi[i];
765     for (j=0; j<nzi; j++) {
766       row  = *adj++;
767       pnz  = pi_loc[row+1] - pi_loc[row];
768       Jptr = pj_loc + pi_loc[row];
769       /* add non-zero cols of P into the sorted linked list lnk */
770       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
771     }
772     /* off-diagonal portion of A */
773     nzi = aoi[i+1] - aoi[i];
774     for (j=0; j<nzi; j++) {
775       row  = *aoj++;
776       pnz  = pi_oth[row+1] - pi_oth[row];
777       Jptr = pj_oth + pi_oth[row];
778       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
779     }
780 
781     apnz     = *lnk;
782     api[i+1] = api[i] + apnz;
783 
784     /* if free space is not available, double the total space in the list */
785     if (current_space->local_remaining<apnz) {
786       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
787       nspacedouble++;
788     }
789 
790     /* Copy data into free space, then initialize lnk */
791     ierr = PetscLLCondensedClean_Scalable(apnz,current_space->array,lnk);CHKERRQ(ierr);
792     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
793 
794     current_space->array           += apnz;
795     current_space->local_used      += apnz;
796     current_space->local_remaining -= apnz;
797   }
798 
799   /* Allocate space for apj, initialize apj, and */
800   /* destroy list of free space and other temporary array(s) */
801   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
802   apj  = ptap->apj;
803   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
804   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
805 
806   /* create and assemble symbolic parallel matrix Cmpi */
807   /*----------------------------------------------------*/
808   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
809   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
810   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
811   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
812   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
813   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
814 
815   /* malloc apa for assembly Cmpi */
816   ierr = PetscCalloc1(apnz_max,&apa);CHKERRQ(ierr);
817 
818   ptap->apa = apa;
819   for (i=0; i<am; i++) {
820     row  = i + rstart;
821     apnz = api[i+1] - api[i];
822     ierr = MatSetValues(Cmpi,1,&row,apnz,apj,apa,INSERT_VALUES);CHKERRQ(ierr);
823     apj += apnz;
824   }
825   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
826   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
827 
828   ptap->destroy             = Cmpi->ops->destroy;
829   ptap->duplicate           = Cmpi->ops->duplicate;
830   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ;
831   Cmpi->ops->destroy        = MatDestroy_MPIAIJ_MatMatMult;
832   Cmpi->ops->duplicate      = MatDuplicate_MPIAIJ_MatMatMult;
833 
834   /* attach the supporting struct to Cmpi for reuse */
835   c       = (Mat_MPIAIJ*)Cmpi->data;
836   c->ptap = ptap;
837 
838   *C = Cmpi;
839 
840   /* set MatInfo */
841   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
842   if (afill < 1.0) afill = 1.0;
843   Cmpi->info.mallocs           = nspacedouble;
844   Cmpi->info.fill_ratio_given  = fill;
845   Cmpi->info.fill_ratio_needed = afill;
846 
847 #if defined(PETSC_USE_INFO)
848   if (api[am]) {
849     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
850     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
851   } else {
852     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
853   }
854 #endif
855   PetscFunctionReturn(0);
856 }
857 
858 /*-------------------------------------------------------------------------*/
859 PetscErrorCode MatTransposeMatMult_MPIAIJ_MPIAIJ(Mat P,Mat A,MatReuse scall,PetscReal fill,Mat *C)
860 {
861   PetscErrorCode ierr;
862   const char     *algTypes[3] = {"scalable","nonscalable","matmatmult"};
863   PetscInt       alg=0; /* set default algorithm */
864 
865   PetscFunctionBegin;
866   if (scall == MAT_INITIAL_MATRIX) {
867     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
868     ierr = PetscOptionsEList("-mattransposematmult_via","Algorithmic approach","MatTransposeMatMult",algTypes,3,algTypes[0],&alg,NULL);CHKERRQ(ierr);
869     ierr = PetscOptionsEnd();CHKERRQ(ierr);
870 
871     ierr = PetscLogEventBegin(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
872     switch (alg) {
873     case 1:
874       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(P,A,fill,C);CHKERRQ(ierr);
875       break;
876     case 2:
877     {
878       Mat         Pt;
879       Mat_PtAPMPI *ptap;
880       Mat_MPIAIJ  *c;
881       ierr = MatTranspose(P,MAT_INITIAL_MATRIX,&Pt);CHKERRQ(ierr);
882       ierr = MatMatMult(Pt,A,MAT_INITIAL_MATRIX,fill,C);CHKERRQ(ierr);
883       c        = (Mat_MPIAIJ*)(*C)->data;
884       ptap     = c->ptap;
885       ptap->Pt = Pt;
886       (*C)->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult;
887       PetscFunctionReturn(0);
888     }
889       break;
890     default:
891       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(P,A,fill,C);CHKERRQ(ierr);
892       break;
893     }
894     ierr = PetscLogEventEnd(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
895   }
896   ierr = PetscLogEventBegin(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
897   ierr = (*(*C)->ops->mattransposemultnumeric)(P,A,*C);CHKERRQ(ierr);
898   ierr = PetscLogEventEnd(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
899   PetscFunctionReturn(0);
900 }
901 
902 /* This routine only works when scall=MAT_REUSE_MATRIX! */
903 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult(Mat P,Mat A,Mat C)
904 {
905   PetscErrorCode ierr;
906   Mat_MPIAIJ     *c=(Mat_MPIAIJ*)C->data;
907   Mat_PtAPMPI    *ptap= c->ptap;
908   Mat            Pt=ptap->Pt;
909 
910   PetscFunctionBegin;
911   ierr = MatTranspose(P,MAT_REUSE_MATRIX,&Pt);CHKERRQ(ierr);
912   ierr = MatMatMultNumeric(Pt,A,C);CHKERRQ(ierr);
913   PetscFunctionReturn(0);
914 }
915 
916 /* Non-scalable version, use dense axpy */
917 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,Mat C)
918 {
919   PetscErrorCode      ierr;
920   Mat_Merge_SeqsToMPI *merge;
921   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
922   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
923   Mat_PtAPMPI         *ptap;
924   PetscInt            *adj,*aJ;
925   PetscInt            i,j,k,anz,pnz,row,*cj;
926   MatScalar           *ada,*aval,*ca,valtmp;
927   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
928   MPI_Comm            comm;
929   PetscMPIInt         size,rank,taga,*len_s;
930   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
931   PetscInt            **buf_ri,**buf_rj;
932   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
933   MPI_Request         *s_waits,*r_waits;
934   MPI_Status          *status;
935   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
936   PetscInt            *ai,*aj,*coi,*coj;
937   PetscInt            *poJ,*pdJ;
938   Mat                 A_loc;
939   Mat_SeqAIJ          *a_loc;
940 
941   PetscFunctionBegin;
942   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
943   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
944   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
945 
946   ptap  = c->ptap;
947   merge = ptap->merge;
948 
949   /* 2) compute numeric C_seq = P_loc^T*A_loc*P - dominating part */
950   /*--------------------------------------------------------------*/
951   /* get data from symbolic products */
952   coi  = merge->coi; coj = merge->coj;
953   ierr = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
954 
955   bi     = merge->bi; bj = merge->bj;
956   owners = merge->rowmap->range;
957   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
958 
959   /* get A_loc by taking all local rows of A */
960   A_loc = ptap->A_loc;
961   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
962   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
963   ai    = a_loc->i;
964   aj    = a_loc->j;
965 
966   ierr = PetscCalloc1(A->cmap->N,&aval);CHKERRQ(ierr); /* non-scalable!!! */
967 
968   for (i=0; i<am; i++) {
969     /* 2-a) put A[i,:] to dense array aval */
970     anz = ai[i+1] - ai[i];
971     adj = aj + ai[i];
972     ada = a_loc->a + ai[i];
973     for (j=0; j<anz; j++) {
974       aval[adj[j]] = ada[j];
975     }
976 
977     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
978     /*--------------------------------------------------------------*/
979     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
980     pnz = po->i[i+1] - po->i[i];
981     poJ = po->j + po->i[i];
982     pA  = po->a + po->i[i];
983     for (j=0; j<pnz; j++) {
984       row = poJ[j];
985       cnz = coi[row+1] - coi[row];
986       cj  = coj + coi[row];
987       ca  = coa + coi[row];
988       /* perform dense axpy */
989       valtmp = pA[j];
990       for (k=0; k<cnz; k++) {
991         ca[k] += valtmp*aval[cj[k]];
992       }
993       ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
994     }
995 
996     /* put the value into Cd (diagonal part) */
997     pnz = pd->i[i+1] - pd->i[i];
998     pdJ = pd->j + pd->i[i];
999     pA  = pd->a + pd->i[i];
1000     for (j=0; j<pnz; j++) {
1001       row = pdJ[j];
1002       cnz = bi[row+1] - bi[row];
1003       cj  = bj + bi[row];
1004       ca  = ba + bi[row];
1005       /* perform dense axpy */
1006       valtmp = pA[j];
1007       for (k=0; k<cnz; k++) {
1008         ca[k] += valtmp*aval[cj[k]];
1009       }
1010       ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1011     }
1012 
1013     /* zero the current row of Pt*A */
1014     aJ = aj + ai[i];
1015     for (k=0; k<anz; k++) aval[aJ[k]] = 0.0;
1016   }
1017 
1018   /* 3) send and recv matrix values coa */
1019   /*------------------------------------*/
1020   buf_ri = merge->buf_ri;
1021   buf_rj = merge->buf_rj;
1022   len_s  = merge->len_s;
1023   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1024   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1025 
1026   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1027   for (proc=0,k=0; proc<size; proc++) {
1028     if (!len_s[proc]) continue;
1029     i    = merge->owners_co[proc];
1030     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1031     k++;
1032   }
1033   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1034   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1035 
1036   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1037   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1038   ierr = PetscFree(coa);CHKERRQ(ierr);
1039 
1040   /* 4) insert local Cseq and received values into Cmpi */
1041   /*----------------------------------------------------*/
1042   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1043   for (k=0; k<merge->nrecv; k++) {
1044     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1045     nrows       = *(buf_ri_k[k]);
1046     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1047     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1048   }
1049 
1050   for (i=0; i<cm; i++) {
1051     row  = owners[rank] + i; /* global row index of C_seq */
1052     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1053     ba_i = ba + bi[i];
1054     bnz  = bi[i+1] - bi[i];
1055     /* add received vals into ba */
1056     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1057       /* i-th row */
1058       if (i == *nextrow[k]) {
1059         cnz    = *(nextci[k]+1) - *nextci[k];
1060         cj     = buf_rj[k] + *(nextci[k]);
1061         ca     = abuf_r[k] + *(nextci[k]);
1062         nextcj = 0;
1063         for (j=0; nextcj<cnz; j++) {
1064           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1065             ba_i[j] += ca[nextcj++];
1066           }
1067         }
1068         nextrow[k]++; nextci[k]++;
1069         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1070       }
1071     }
1072     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1073   }
1074   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1075   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1076 
1077   ierr = PetscFree(ba);CHKERRQ(ierr);
1078   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1079   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1080   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1081   ierr = PetscFree(aval);CHKERRQ(ierr);
1082   PetscFunctionReturn(0);
1083 }
1084 
1085 PetscErrorCode MatDuplicate_MPIAIJ_MatPtAP(Mat, MatDuplicateOption,Mat*);
1086 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1087 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,PetscReal fill,Mat *C)
1088 {
1089   PetscErrorCode      ierr;
1090   Mat                 Cmpi,A_loc,POt,PDt;
1091   Mat_PtAPMPI         *ptap;
1092   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1093   Mat_MPIAIJ          *p        =(Mat_MPIAIJ*)P->data,*c;
1094   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1095   PetscInt            nnz;
1096   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1097   PetscInt            am=A->rmap->n,pn=P->cmap->n;
1098   PetscBT             lnkbt;
1099   MPI_Comm            comm;
1100   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1101   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1102   PetscInt            len,proc,*dnz,*onz,*owners;
1103   PetscInt            nzi,*bi,*bj;
1104   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1105   MPI_Request         *swaits,*rwaits;
1106   MPI_Status          *sstatus,rstatus;
1107   Mat_Merge_SeqsToMPI *merge;
1108   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1109   PetscReal           afill  =1.0,afill_tmp;
1110   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N;
1111   PetscScalar         *vals;
1112   Mat_SeqAIJ          *a_loc, *pdt,*pot;
1113 
1114   PetscFunctionBegin;
1115   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1116   /* check if matrix local sizes are compatible */
1117   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);
1118 
1119   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1120   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1121 
1122   /* create struct Mat_PtAPMPI and attached it to C later */
1123   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1124 
1125   /* get A_loc by taking all local rows of A */
1126   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1127 
1128   ptap->A_loc = A_loc;
1129 
1130   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1131   ai    = a_loc->i;
1132   aj    = a_loc->j;
1133 
1134   /* determine symbolic Co=(p->B)^T*A - send to others */
1135   /*----------------------------------------------------*/
1136   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1137   pdt  = (Mat_SeqAIJ*)PDt->data;
1138   pdti = pdt->i; pdtj = pdt->j;
1139 
1140   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1141   pot  = (Mat_SeqAIJ*)POt->data;
1142   poti = pot->i; potj = pot->j;
1143 
1144   /* then, compute symbolic Co = (p->B)^T*A */
1145   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors >= (num of nonzero rows of C_seq) - pn */
1146   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
1147   coi[0] = 0;
1148 
1149   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1150   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(poti[pon],ai[am]));
1151   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1152   current_space = free_space;
1153 
1154   /* create and initialize a linked list */
1155   ierr = PetscLLCondensedCreate(aN,aN,&lnk,&lnkbt);CHKERRQ(ierr);
1156 
1157   for (i=0; i<pon; i++) {
1158     pnz = poti[i+1] - poti[i];
1159     ptJ = potj + poti[i];
1160     for (j=0; j<pnz; j++) {
1161       row  = ptJ[j]; /* row of A_loc == col of Pot */
1162       anz  = ai[row+1] - ai[row];
1163       Jptr = aj + ai[row];
1164       /* add non-zero cols of AP into the sorted linked list lnk */
1165       ierr = PetscLLCondensedAddSorted(anz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1166     }
1167     nnz = lnk[0];
1168 
1169     /* If free space is not available, double the total space in the list */
1170     if (current_space->local_remaining<nnz) {
1171       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1172       nspacedouble++;
1173     }
1174 
1175     /* Copy data into free space, and zero out denserows */
1176     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1177 
1178     current_space->array           += nnz;
1179     current_space->local_used      += nnz;
1180     current_space->local_remaining -= nnz;
1181 
1182     coi[i+1] = coi[i] + nnz;
1183   }
1184 
1185   ierr = PetscMalloc1(coi[pon]+1,&coj);CHKERRQ(ierr);
1186   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1187 
1188   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1189   if (afill_tmp > afill) afill = afill_tmp;
1190 
1191   /* send j-array (coj) of Co to other processors */
1192   /*----------------------------------------------*/
1193   /* determine row ownership */
1194   ierr = PetscNew(&merge);CHKERRQ(ierr);
1195   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1196 
1197   merge->rowmap->n  = pn;
1198   merge->rowmap->bs = 1;
1199 
1200   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1201   owners = merge->rowmap->range;
1202 
1203   /* determine the number of messages to send, their lengths */
1204   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
1205   ierr = PetscMalloc1(size,&merge->len_s);CHKERRQ(ierr);
1206 
1207   len_s        = merge->len_s;
1208   merge->nsend = 0;
1209 
1210   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
1211   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1212 
1213   proc = 0;
1214   for (i=0; i<pon; i++) {
1215     while (prmap[i] >= owners[proc+1]) proc++;
1216     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1217     len_s[proc] += coi[i+1] - coi[i];
1218   }
1219 
1220   len          = 0; /* max length of buf_si[] */
1221   owners_co[0] = 0;
1222   for (proc=0; proc<size; proc++) {
1223     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1224     if (len_si[proc]) {
1225       merge->nsend++;
1226       len_si[proc] = 2*(len_si[proc] + 1);
1227       len         += len_si[proc];
1228     }
1229   }
1230 
1231   /* determine the number and length of messages to receive for coi and coj  */
1232   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1233   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1234 
1235   /* post the Irecv and Isend of coj */
1236   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1237   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1238   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
1239   for (proc=0, k=0; proc<size; proc++) {
1240     if (!len_s[proc]) continue;
1241     i    = owners_co[proc];
1242     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1243     k++;
1244   }
1245 
1246   /* receives and sends of coj are complete */
1247   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
1248   for (i=0; i<merge->nrecv; i++) {
1249     PetscMPIInt icompleted;
1250     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1251   }
1252   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1253   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1254 
1255   /* send and recv coi */
1256   /*-------------------*/
1257   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1258   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1259   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1260   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1261   for (proc=0,k=0; proc<size; proc++) {
1262     if (!len_s[proc]) continue;
1263     /* form outgoing message for i-structure:
1264          buf_si[0]:                 nrows to be sent
1265                [1:nrows]:           row index (global)
1266                [nrows+1:2*nrows+1]: i-structure index
1267     */
1268     /*-------------------------------------------*/
1269     nrows       = len_si[proc]/2 - 1;
1270     buf_si_i    = buf_si + nrows+1;
1271     buf_si[0]   = nrows;
1272     buf_si_i[0] = 0;
1273     nrows       = 0;
1274     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1275       nzi               = coi[i+1] - coi[i];
1276       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi; /* i-structure */
1277       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1278       nrows++;
1279     }
1280     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1281     k++;
1282     buf_si += len_si[proc];
1283   }
1284   i = merge->nrecv;
1285   while (i--) {
1286     PetscMPIInt icompleted;
1287     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1288   }
1289   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1290   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1291   ierr = PetscFree(len_si);CHKERRQ(ierr);
1292   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1293   ierr = PetscFree(swaits);CHKERRQ(ierr);
1294   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1295   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1296 
1297   /* compute the local portion of C (mpi mat) */
1298   /*------------------------------------------*/
1299   /* allocate bi array and free space for accumulating nonzero column info */
1300   ierr  = PetscMalloc1(pn+1,&bi);CHKERRQ(ierr);
1301   bi[0] = 0;
1302 
1303   /* set initial free space to be fill*(nnz(P) + nnz(A)) */
1304   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(pdti[pn],PetscIntSumTruncate(poti[pon],ai[am])));
1305   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1306   current_space = free_space;
1307 
1308   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1309   for (k=0; k<merge->nrecv; k++) {
1310     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1311     nrows       = *buf_ri_k[k];
1312     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1313     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1314   }
1315 
1316   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1317   rmax = 0;
1318   for (i=0; i<pn; i++) {
1319     /* add pdt[i,:]*AP into lnk */
1320     pnz = pdti[i+1] - pdti[i];
1321     ptJ = pdtj + pdti[i];
1322     for (j=0; j<pnz; j++) {
1323       row  = ptJ[j];  /* row of AP == col of Pt */
1324       anz  = ai[row+1] - ai[row];
1325       Jptr = aj + ai[row];
1326       /* add non-zero cols of AP into the sorted linked list lnk */
1327       ierr = PetscLLCondensedAddSorted(anz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1328     }
1329 
1330     /* add received col data into lnk */
1331     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1332       if (i == *nextrow[k]) { /* i-th row */
1333         nzi  = *(nextci[k]+1) - *nextci[k];
1334         Jptr = buf_rj[k] + *nextci[k];
1335         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1336         nextrow[k]++; nextci[k]++;
1337       }
1338     }
1339     nnz = lnk[0];
1340 
1341     /* if free space is not available, make more free space */
1342     if (current_space->local_remaining<nnz) {
1343       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1344       nspacedouble++;
1345     }
1346     /* copy data into free space, then initialize lnk */
1347     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1348     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1349 
1350     current_space->array           += nnz;
1351     current_space->local_used      += nnz;
1352     current_space->local_remaining -= nnz;
1353 
1354     bi[i+1] = bi[i] + nnz;
1355     if (nnz > rmax) rmax = nnz;
1356   }
1357   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1358 
1359   ierr = PetscMalloc1(bi[pn]+1,&bj);CHKERRQ(ierr);
1360   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1361 
1362   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
1363   if (afill_tmp > afill) afill = afill_tmp;
1364   ierr = PetscLLCondensedDestroy(lnk,lnkbt);CHKERRQ(ierr);
1365   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1366   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1367 
1368   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
1369   /*----------------------------------------------------------------------------------*/
1370   ierr = PetscCalloc1(rmax+1,&vals);CHKERRQ(ierr);
1371 
1372   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1373   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1374   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
1375   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1376   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1377   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1378   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
1379   for (i=0; i<pn; i++) {
1380     row  = i + rstart;
1381     nnz  = bi[i+1] - bi[i];
1382     Jptr = bj + bi[i];
1383     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
1384   }
1385   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1386   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1387   ierr = PetscFree(vals);CHKERRQ(ierr);
1388 
1389   merge->bi        = bi;
1390   merge->bj        = bj;
1391   merge->coi       = coi;
1392   merge->coj       = coj;
1393   merge->buf_ri    = buf_ri;
1394   merge->buf_rj    = buf_rj;
1395   merge->owners_co = owners_co;
1396 
1397   /* attach the supporting struct to Cmpi for reuse */
1398   c           = (Mat_MPIAIJ*)Cmpi->data;
1399   c->ptap     = ptap;
1400   ptap->api   = NULL;
1401   ptap->apj   = NULL;
1402   ptap->merge = merge;
1403   ptap->destroy   = Cmpi->ops->destroy;
1404   ptap->duplicate = Cmpi->ops->duplicate;
1405 
1406   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1407   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
1408   Cmpi->ops->duplicate               = MatDuplicate_MPIAIJ_MatPtAP;
1409 
1410   *C = Cmpi;
1411 #if defined(PETSC_USE_INFO)
1412   if (bi[pn] != 0) {
1413     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1414     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
1415   } else {
1416     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1417   }
1418 #endif
1419   PetscFunctionReturn(0);
1420 }
1421 
1422 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ(Mat P,Mat A,Mat C)
1423 {
1424   PetscErrorCode      ierr;
1425   Mat_Merge_SeqsToMPI *merge;
1426   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1427   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
1428   Mat_PtAPMPI         *ptap;
1429   PetscInt            *adj;
1430   PetscInt            i,j,k,anz,pnz,row,*cj,nexta;
1431   MatScalar           *ada,*ca,valtmp;
1432   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
1433   MPI_Comm            comm;
1434   PetscMPIInt         size,rank,taga,*len_s;
1435   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
1436   PetscInt            **buf_ri,**buf_rj;
1437   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
1438   MPI_Request         *s_waits,*r_waits;
1439   MPI_Status          *status;
1440   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
1441   PetscInt            *ai,*aj,*coi,*coj;
1442   PetscInt            *poJ,*pdJ;
1443   Mat                 A_loc;
1444   Mat_SeqAIJ          *a_loc;
1445 
1446   PetscFunctionBegin;
1447   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1448   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1449   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1450 
1451   ptap  = c->ptap;
1452   merge = ptap->merge;
1453 
1454   /* 2) compute numeric C_seq = P_loc^T*A_loc */
1455   /*------------------------------------------*/
1456   /* get data from symbolic products */
1457   coi    = merge->coi; coj = merge->coj;
1458   ierr   = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
1459   bi     = merge->bi; bj = merge->bj;
1460   owners = merge->rowmap->range;
1461   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
1462 
1463   /* get A_loc by taking all local rows of A */
1464   A_loc = ptap->A_loc;
1465   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
1466   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1467   ai    = a_loc->i;
1468   aj    = a_loc->j;
1469 
1470   for (i=0; i<am; i++) {
1471     anz = ai[i+1] - ai[i];
1472     adj = aj + ai[i];
1473     ada = a_loc->a + ai[i];
1474 
1475     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1476     /*-------------------------------------------------------------*/
1477     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1478     pnz = po->i[i+1] - po->i[i];
1479     poJ = po->j + po->i[i];
1480     pA  = po->a + po->i[i];
1481     for (j=0; j<pnz; j++) {
1482       row = poJ[j];
1483       cj  = coj + coi[row];
1484       ca  = coa + coi[row];
1485       /* perform sparse axpy */
1486       nexta  = 0;
1487       valtmp = pA[j];
1488       for (k=0; nexta<anz; k++) {
1489         if (cj[k] == adj[nexta]) {
1490           ca[k] += valtmp*ada[nexta];
1491           nexta++;
1492         }
1493       }
1494       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1495     }
1496 
1497     /* put the value into Cd (diagonal part) */
1498     pnz = pd->i[i+1] - pd->i[i];
1499     pdJ = pd->j + pd->i[i];
1500     pA  = pd->a + pd->i[i];
1501     for (j=0; j<pnz; j++) {
1502       row = pdJ[j];
1503       cj  = bj + bi[row];
1504       ca  = ba + bi[row];
1505       /* perform sparse axpy */
1506       nexta  = 0;
1507       valtmp = pA[j];
1508       for (k=0; nexta<anz; k++) {
1509         if (cj[k] == adj[nexta]) {
1510           ca[k] += valtmp*ada[nexta];
1511           nexta++;
1512         }
1513       }
1514       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1515     }
1516   }
1517 
1518   /* 3) send and recv matrix values coa */
1519   /*------------------------------------*/
1520   buf_ri = merge->buf_ri;
1521   buf_rj = merge->buf_rj;
1522   len_s  = merge->len_s;
1523   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1524   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1525 
1526   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1527   for (proc=0,k=0; proc<size; proc++) {
1528     if (!len_s[proc]) continue;
1529     i    = merge->owners_co[proc];
1530     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1531     k++;
1532   }
1533   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1534   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1535 
1536   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1537   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1538   ierr = PetscFree(coa);CHKERRQ(ierr);
1539 
1540   /* 4) insert local Cseq and received values into Cmpi */
1541   /*----------------------------------------------------*/
1542   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1543   for (k=0; k<merge->nrecv; k++) {
1544     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1545     nrows       = *(buf_ri_k[k]);
1546     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1547     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1548   }
1549 
1550   for (i=0; i<cm; i++) {
1551     row  = owners[rank] + i; /* global row index of C_seq */
1552     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1553     ba_i = ba + bi[i];
1554     bnz  = bi[i+1] - bi[i];
1555     /* add received vals into ba */
1556     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1557       /* i-th row */
1558       if (i == *nextrow[k]) {
1559         cnz    = *(nextci[k]+1) - *nextci[k];
1560         cj     = buf_rj[k] + *(nextci[k]);
1561         ca     = abuf_r[k] + *(nextci[k]);
1562         nextcj = 0;
1563         for (j=0; nextcj<cnz; j++) {
1564           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1565             ba_i[j] += ca[nextcj++];
1566           }
1567         }
1568         nextrow[k]++; nextci[k]++;
1569         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1570       }
1571     }
1572     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1573   }
1574   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1575   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1576 
1577   ierr = PetscFree(ba);CHKERRQ(ierr);
1578   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1579   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1580   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1581   PetscFunctionReturn(0);
1582 }
1583 
1584 PetscErrorCode MatDuplicate_MPIAIJ_MatPtAP(Mat, MatDuplicateOption,Mat*);
1585 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ();
1586    differ from MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable in using LLCondensedCreate_Scalable() */
1587 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P,Mat A,PetscReal fill,Mat *C)
1588 {
1589   PetscErrorCode      ierr;
1590   Mat                 Cmpi,A_loc,POt,PDt;
1591   Mat_PtAPMPI         *ptap;
1592   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1593   Mat_MPIAIJ          *p        =(Mat_MPIAIJ*)P->data,*c;
1594   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1595   PetscInt            nnz;
1596   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1597   PetscInt            am  =A->rmap->n,pn=P->cmap->n;
1598   MPI_Comm            comm;
1599   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1600   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1601   PetscInt            len,proc,*dnz,*onz,*owners;
1602   PetscInt            nzi,*bi,*bj;
1603   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1604   MPI_Request         *swaits,*rwaits;
1605   MPI_Status          *sstatus,rstatus;
1606   Mat_Merge_SeqsToMPI *merge;
1607   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1608   PetscReal           afill  =1.0,afill_tmp;
1609   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Armax;
1610   PetscScalar         *vals;
1611   Mat_SeqAIJ          *a_loc,*pdt,*pot;
1612   PetscTable          ta;
1613 
1614   PetscFunctionBegin;
1615   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1616   /* check if matrix local sizes are compatible */
1617   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);
1618 
1619   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1620   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1621 
1622   /* create struct Mat_PtAPMPI and attached it to C later */
1623   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1624 
1625   /* get A_loc by taking all local rows of A */
1626   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1627 
1628   ptap->A_loc = A_loc;
1629   a_loc       = (Mat_SeqAIJ*)(A_loc)->data;
1630   ai          = a_loc->i;
1631   aj          = a_loc->j;
1632 
1633   /* determine symbolic Co=(p->B)^T*A - send to others */
1634   /*----------------------------------------------------*/
1635   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1636   pdt  = (Mat_SeqAIJ*)PDt->data;
1637   pdti = pdt->i; pdtj = pdt->j;
1638 
1639   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1640   pot  = (Mat_SeqAIJ*)POt->data;
1641   poti = pot->i; potj = pot->j;
1642 
1643   /* then, compute symbolic Co = (p->B)^T*A */
1644   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1645                          >= (num of nonzero rows of C_seq) - pn */
1646   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
1647   coi[0] = 0;
1648 
1649   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1650   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(poti[pon],ai[am]));
1651   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1652   current_space = free_space;
1653 
1654   /* create and initialize a linked list */
1655   ierr = PetscTableCreate(aN,aN,&ta);CHKERRQ(ierr);
1656   MatRowMergeMax_SeqAIJ(a_loc,am,ta);
1657   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
1658   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
1659 
1660   for (i=0; i<pon; i++) {
1661     pnz = poti[i+1] - poti[i];
1662     ptJ = potj + poti[i];
1663     for (j=0; j<pnz; j++) {
1664       row  = ptJ[j]; /* row of A_loc == col of Pot */
1665       anz  = ai[row+1] - ai[row];
1666       Jptr = aj + ai[row];
1667       /* add non-zero cols of AP into the sorted linked list lnk */
1668       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1669     }
1670     nnz = lnk[0];
1671 
1672     /* If free space is not available, double the total space in the list */
1673     if (current_space->local_remaining<nnz) {
1674       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1675       nspacedouble++;
1676     }
1677 
1678     /* Copy data into free space, and zero out denserows */
1679     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1680 
1681     current_space->array           += nnz;
1682     current_space->local_used      += nnz;
1683     current_space->local_remaining -= nnz;
1684 
1685     coi[i+1] = coi[i] + nnz;
1686   }
1687 
1688   ierr = PetscMalloc1(coi[pon]+1,&coj);CHKERRQ(ierr);
1689   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1690   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr); /* must destroy to get a new one for C */
1691 
1692   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1693   if (afill_tmp > afill) afill = afill_tmp;
1694 
1695   /* send j-array (coj) of Co to other processors */
1696   /*----------------------------------------------*/
1697   /* determine row ownership */
1698   ierr = PetscNew(&merge);CHKERRQ(ierr);
1699   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1700 
1701   merge->rowmap->n  = pn;
1702   merge->rowmap->bs = 1;
1703 
1704   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1705   owners = merge->rowmap->range;
1706 
1707   /* determine the number of messages to send, their lengths */
1708   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
1709   ierr = PetscMalloc1(size,&merge->len_s);CHKERRQ(ierr);
1710 
1711   len_s        = merge->len_s;
1712   merge->nsend = 0;
1713 
1714   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
1715   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1716 
1717   proc = 0;
1718   for (i=0; i<pon; i++) {
1719     while (prmap[i] >= owners[proc+1]) proc++;
1720     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1721     len_s[proc] += coi[i+1] - coi[i];
1722   }
1723 
1724   len          = 0; /* max length of buf_si[] */
1725   owners_co[0] = 0;
1726   for (proc=0; proc<size; proc++) {
1727     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1728     if (len_si[proc]) {
1729       merge->nsend++;
1730       len_si[proc] = 2*(len_si[proc] + 1);
1731       len         += len_si[proc];
1732     }
1733   }
1734 
1735   /* determine the number and length of messages to receive for coi and coj  */
1736   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1737   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1738 
1739   /* post the Irecv and Isend of coj */
1740   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1741   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1742   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
1743   for (proc=0, k=0; proc<size; proc++) {
1744     if (!len_s[proc]) continue;
1745     i    = owners_co[proc];
1746     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1747     k++;
1748   }
1749 
1750   /* receives and sends of coj are complete */
1751   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
1752   for (i=0; i<merge->nrecv; i++) {
1753     PetscMPIInt icompleted;
1754     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1755   }
1756   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1757   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1758 
1759   /* add received column indices into table to update Armax */
1760   for (k=0; k<merge->nrecv; k++) {/* k-th received message */
1761     Jptr = buf_rj[k];
1762     for (j=0; j<merge->len_r[k]; j++) {
1763       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
1764     }
1765   }
1766   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
1767 
1768   /* send and recv coi */
1769   /*-------------------*/
1770   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1771   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1772   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1773   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1774   for (proc=0,k=0; proc<size; proc++) {
1775     if (!len_s[proc]) continue;
1776     /* form outgoing message for i-structure:
1777          buf_si[0]:                 nrows to be sent
1778                [1:nrows]:           row index (global)
1779                [nrows+1:2*nrows+1]: i-structure index
1780     */
1781     /*-------------------------------------------*/
1782     nrows       = len_si[proc]/2 - 1;
1783     buf_si_i    = buf_si + nrows+1;
1784     buf_si[0]   = nrows;
1785     buf_si_i[0] = 0;
1786     nrows       = 0;
1787     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1788       nzi               = coi[i+1] - coi[i];
1789       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1790       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1791       nrows++;
1792     }
1793     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1794     k++;
1795     buf_si += len_si[proc];
1796   }
1797   i = merge->nrecv;
1798   while (i--) {
1799     PetscMPIInt icompleted;
1800     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1801   }
1802   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1803   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1804   ierr = PetscFree(len_si);CHKERRQ(ierr);
1805   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1806   ierr = PetscFree(swaits);CHKERRQ(ierr);
1807   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1808   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1809 
1810   /* compute the local portion of C (mpi mat) */
1811   /*------------------------------------------*/
1812   /* allocate bi array and free space for accumulating nonzero column info */
1813   ierr  = PetscMalloc1(pn+1,&bi);CHKERRQ(ierr);
1814   bi[0] = 0;
1815 
1816   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
1817   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(pdti[pn],PetscIntSumTruncate(poti[pon],ai[am])));
1818   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1819   current_space = free_space;
1820 
1821   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1822   for (k=0; k<merge->nrecv; k++) {
1823     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1824     nrows       = *buf_ri_k[k];
1825     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1826     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th recieved i-structure  */
1827   }
1828 
1829   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
1830   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1831   rmax = 0;
1832   for (i=0; i<pn; i++) {
1833     /* add pdt[i,:]*AP into lnk */
1834     pnz = pdti[i+1] - pdti[i];
1835     ptJ = pdtj + pdti[i];
1836     for (j=0; j<pnz; j++) {
1837       row  = ptJ[j];  /* row of AP == col of Pt */
1838       anz  = ai[row+1] - ai[row];
1839       Jptr = aj + ai[row];
1840       /* add non-zero cols of AP into the sorted linked list lnk */
1841       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1842     }
1843 
1844     /* add received col data into lnk */
1845     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1846       if (i == *nextrow[k]) { /* i-th row */
1847         nzi  = *(nextci[k]+1) - *nextci[k];
1848         Jptr = buf_rj[k] + *nextci[k];
1849         ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
1850         nextrow[k]++; nextci[k]++;
1851       }
1852     }
1853     nnz = lnk[0];
1854 
1855     /* if free space is not available, make more free space */
1856     if (current_space->local_remaining<nnz) {
1857       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1858       nspacedouble++;
1859     }
1860     /* copy data into free space, then initialize lnk */
1861     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1862     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1863 
1864     current_space->array           += nnz;
1865     current_space->local_used      += nnz;
1866     current_space->local_remaining -= nnz;
1867 
1868     bi[i+1] = bi[i] + nnz;
1869     if (nnz > rmax) rmax = nnz;
1870   }
1871   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1872 
1873   ierr      = PetscMalloc1(bi[pn]+1,&bj);CHKERRQ(ierr);
1874   ierr      = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1875   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
1876   if (afill_tmp > afill) afill = afill_tmp;
1877   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
1878   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
1879 
1880   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1881   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1882 
1883   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
1884   /*----------------------------------------------------------------------------------*/
1885   ierr = PetscCalloc1(rmax+1,&vals);CHKERRQ(ierr);
1886 
1887   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1888   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1889   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
1890   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1891   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1892   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1893   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
1894   for (i=0; i<pn; i++) {
1895     row  = i + rstart;
1896     nnz  = bi[i+1] - bi[i];
1897     Jptr = bj + bi[i];
1898     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
1899   }
1900   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1901   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1902   ierr = PetscFree(vals);CHKERRQ(ierr);
1903 
1904   merge->bi        = bi;
1905   merge->bj        = bj;
1906   merge->coi       = coi;
1907   merge->coj       = coj;
1908   merge->buf_ri    = buf_ri;
1909   merge->buf_rj    = buf_rj;
1910   merge->owners_co = owners_co;
1911 
1912   /* attach the supporting struct to Cmpi for reuse */
1913   c = (Mat_MPIAIJ*)Cmpi->data;
1914 
1915   c->ptap     = ptap;
1916   ptap->api   = NULL;
1917   ptap->apj   = NULL;
1918   ptap->merge = merge;
1919   ptap->apa   = NULL;
1920   ptap->destroy   = Cmpi->ops->destroy;
1921   ptap->duplicate = Cmpi->ops->duplicate;
1922 
1923   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ;
1924   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
1925   Cmpi->ops->duplicate               = MatDuplicate_MPIAIJ_MatPtAP;
1926 
1927   *C = Cmpi;
1928 #if defined(PETSC_USE_INFO)
1929   if (bi[pn] != 0) {
1930     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1931     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
1932   } else {
1933     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1934   }
1935 #endif
1936   PetscFunctionReturn(0);
1937 }
1938