xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision eca6b86a16371f6fe3b252d05990b562dc47a2ae)
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 #undef __FUNCT__
14 #define __FUNCT__ "MatMatMult_MPIAIJ_MPIAIJ"
15 PetscErrorCode MatMatMult_MPIAIJ_MPIAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill, Mat *C)
16 {
17   PetscErrorCode ierr;
18   const char     *algTypes[2] = {"scalable","nonscalable"};
19   PetscInt       alg=1; /* set default algorithm */
20   MPI_Comm       comm;
21 
22   PetscFunctionBegin;
23   if (scall == MAT_INITIAL_MATRIX) {
24     ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
25     if (A->cmap->rstart != B->rmap->rstart || A->cmap->rend != B->rmap->rend) {
26       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);
27     }
28 
29     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
30     ierr = PetscOptionsEList("-matmatmult_via","Algorithmic approach","MatMatMult",algTypes,2,algTypes[1],&alg,NULL);CHKERRQ(ierr);
31     ierr = PetscOptionsEnd();CHKERRQ(ierr);
32 
33     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
34     switch (alg) {
35     case 1:
36       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(A,B,fill,C);CHKERRQ(ierr);
37       break;
38     default:
39       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ(A,B,fill,C);CHKERRQ(ierr);
40       break;
41     }
42     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
43   }
44   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
45   ierr = (*(*C)->ops->matmultnumeric)(A,B,*C);CHKERRQ(ierr);
46   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
47   PetscFunctionReturn(0);
48 }
49 
50 #undef __FUNCT__
51 #define __FUNCT__ "MatDestroy_MPIAIJ_MatMatMult"
52 PetscErrorCode MatDestroy_MPIAIJ_MatMatMult(Mat A)
53 {
54   PetscErrorCode ierr;
55   Mat_MPIAIJ     *a    = (Mat_MPIAIJ*)A->data;
56   Mat_PtAPMPI    *ptap = a->ptap;
57 
58   PetscFunctionBegin;
59   ierr = PetscFree2(ptap->startsj_s,ptap->startsj_r);CHKERRQ(ierr);
60   ierr = PetscFree(ptap->bufa);CHKERRQ(ierr);
61   ierr = MatDestroy(&ptap->P_loc);CHKERRQ(ierr);
62   ierr = MatDestroy(&ptap->P_oth);CHKERRQ(ierr);
63   ierr = MatDestroy(&ptap->Pt);CHKERRQ(ierr);
64   ierr = PetscFree(ptap->api);CHKERRQ(ierr);
65   ierr = PetscFree(ptap->apj);CHKERRQ(ierr);
66   ierr = PetscFree(ptap->apa);CHKERRQ(ierr);
67   ierr = ptap->destroy(A);CHKERRQ(ierr);
68   ierr = PetscFree(ptap);CHKERRQ(ierr);
69   PetscFunctionReturn(0);
70 }
71 
72 #undef __FUNCT__
73 #define __FUNCT__ "MatDuplicate_MPIAIJ_MatMatMult"
74 PetscErrorCode MatDuplicate_MPIAIJ_MatMatMult(Mat A, MatDuplicateOption op, Mat *M)
75 {
76   PetscErrorCode ierr;
77   Mat_MPIAIJ     *a    = (Mat_MPIAIJ*)A->data;
78   Mat_PtAPMPI    *ptap = a->ptap;
79 
80   PetscFunctionBegin;
81   ierr = (*ptap->duplicate)(A,op,M);CHKERRQ(ierr);
82 
83   (*M)->ops->destroy   = ptap->destroy;   /* = MatDestroy_MPIAIJ, *M doesn't duplicate A's special structure! */
84   (*M)->ops->duplicate = ptap->duplicate; /* = MatDuplicate_MPIAIJ */
85   PetscFunctionReturn(0);
86 }
87 
88 #undef __FUNCT__
89 #define __FUNCT__ "MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable"
90 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,Mat C)
91 {
92   PetscErrorCode ierr;
93   Mat_MPIAIJ     *a  =(Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
94   Mat_SeqAIJ     *ad =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
95   Mat_SeqAIJ     *cd =(Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
96   PetscScalar    *cda=cd->a,*coa=co->a;
97   Mat_SeqAIJ     *p_loc,*p_oth;
98   PetscScalar    *apa,*ca;
99   PetscInt       cm   =C->rmap->n;
100   Mat_PtAPMPI    *ptap=c->ptap;
101   PetscInt       *api,*apj,*apJ,i,k;
102   PetscInt       cstart=C->cmap->rstart;
103   PetscInt       cdnz,conz,k0,k1;
104   MPI_Comm       comm;
105   PetscMPIInt    size;
106 
107   PetscFunctionBegin;
108   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
109   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
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   for (i=0; i<cm; i++) {
132     /* compute apa = A[i,:]*P */
133     AProw_nonscalable(i,ad,ao,p_loc,p_oth,apa);
134 
135     /* set values in C */
136     apJ  = apj + api[i];
137     cdnz = cd->i[i+1] - cd->i[i];
138     conz = co->i[i+1] - co->i[i];
139 
140     /* 1st off-diagoanl part of C */
141     ca = coa + co->i[i];
142     k  = 0;
143     for (k0=0; k0<conz; k0++) {
144       if (apJ[k] >= cstart) break;
145       ca[k0]      = apa[apJ[k]];
146       apa[apJ[k]] = 0.0;
147       k++;
148     }
149 
150     /* diagonal part of C */
151     ca = cda + cd->i[i];
152     for (k1=0; k1<cdnz; k1++) {
153       ca[k1]      = apa[apJ[k]];
154       apa[apJ[k]] = 0.0;
155       k++;
156     }
157 
158     /* 2nd off-diagoanl part of C */
159     ca = coa + co->i[i];
160     for (; k0<conz; k0++) {
161       ca[k0]      = apa[apJ[k]];
162       apa[apJ[k]] = 0.0;
163       k++;
164     }
165   }
166   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
167   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
168   PetscFunctionReturn(0);
169 }
170 
171 #undef __FUNCT__
172 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable"
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                Cmpi;
179   Mat_PtAPMPI        *ptap;
180   PetscFreeSpaceList free_space=NULL,current_space=NULL;
181   Mat_MPIAIJ         *a        =(Mat_MPIAIJ*)A->data,*c;
182   Mat_SeqAIJ         *ad       =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
183   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
184   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
185   PetscInt           *lnk,i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi;
186   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n,Crmax;
187   PetscBT            lnkbt;
188   PetscScalar        *apa;
189   PetscReal          afill;
190   PetscTable         ta;
191 
192   PetscFunctionBegin;
193   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
194   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
195 
196   /* create struct Mat_PtAPMPI 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   Crmax = p_loc->rmax+p_oth->rmax;
223   if (Crmax > pN) Crmax = pN;
224   ierr = PetscTableCreate(Crmax,pN,&ta);CHKERRQ(ierr);
225   MatRowMergeMax_SeqAIJ(p_loc,ptap->P_loc->rmap->N,ta);
226   MatRowMergeMax_SeqAIJ(p_oth,ptap->P_oth->rmap->N,ta);
227   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
228   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
229 
230   ierr = PetscLLCondensedCreate(Crmax,pN,&lnk,&lnkbt);CHKERRQ(ierr);
231 
232   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
233   ierr = PetscFreeSpaceGet((PetscInt)(fill*(adi[am]+aoi[am]+pi_loc[pm])),&free_space);CHKERRQ(ierr);
234   current_space = free_space;
235 
236   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
237   for (i=0; i<am; i++) {
238     /* diagonal portion of A */
239     nzi = adi[i+1] - adi[i];
240     for (j=0; j<nzi; j++) {
241       row  = *adj++;
242       pnz  = pi_loc[row+1] - pi_loc[row];
243       Jptr = pj_loc + pi_loc[row];
244       /* add non-zero cols of P into the sorted linked list lnk */
245       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
246     }
247     /* off-diagonal portion of A */
248     nzi = aoi[i+1] - aoi[i];
249     for (j=0; j<nzi; j++) {
250       row  = *aoj++;
251       pnz  = pi_oth[row+1] - pi_oth[row];
252       Jptr = pj_oth + pi_oth[row];
253       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
254     }
255 
256     apnz     = lnk[0];
257     api[i+1] = api[i] + apnz;
258 
259     /* if free space is not available, double the total space in the list */
260     if (current_space->local_remaining<apnz) {
261       ierr = PetscFreeSpaceGet(apnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
262       nspacedouble++;
263     }
264 
265     /* Copy data into free space, then initialize lnk */
266     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
267     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
268 
269     current_space->array           += apnz;
270     current_space->local_used      += apnz;
271     current_space->local_remaining -= apnz;
272   }
273 
274   /* Allocate space for apj, initialize apj, and */
275   /* destroy list of free space and other temporary array(s) */
276   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
277   apj  = ptap->apj;
278   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
279   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
280 
281   /* malloc apa to store dense row A[i,:]*P */
282   ierr = PetscCalloc1(pN,&apa);CHKERRQ(ierr);
283 
284   ptap->apa = apa;
285 
286   /* create and assemble symbolic parallel matrix Cmpi */
287   /*----------------------------------------------------*/
288   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
289   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
290   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
291 
292   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
293   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
294   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
295   for (i=0; i<am; i++) {
296     row  = i + rstart;
297     apnz = api[i+1] - api[i];
298     ierr = MatSetValues(Cmpi,1,&row,apnz,apj,apa,INSERT_VALUES);CHKERRQ(ierr);
299     apj += apnz;
300   }
301   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
302   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
303 
304   ptap->destroy        = Cmpi->ops->destroy;
305   ptap->duplicate      = Cmpi->ops->duplicate;
306   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
307   Cmpi->ops->destroy   = MatDestroy_MPIAIJ_MatMatMult;
308   Cmpi->ops->duplicate = MatDuplicate_MPIAIJ_MatMatMult;
309 
310   /* attach the supporting struct to Cmpi for reuse */
311   c       = (Mat_MPIAIJ*)Cmpi->data;
312   c->ptap = ptap;
313 
314   *C = Cmpi;
315 
316   /* set MatInfo */
317   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
318   if (afill < 1.0) afill = 1.0;
319   Cmpi->info.mallocs           = nspacedouble;
320   Cmpi->info.fill_ratio_given  = fill;
321   Cmpi->info.fill_ratio_needed = afill;
322 
323 #if defined(PETSC_USE_INFO)
324   if (api[am]) {
325     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
326     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
327   } else {
328     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
329   }
330 #endif
331   PetscFunctionReturn(0);
332 }
333 
334 #undef __FUNCT__
335 #define __FUNCT__ "MatMatMult_MPIAIJ_MPIDense"
336 PetscErrorCode MatMatMult_MPIAIJ_MPIDense(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
337 {
338   PetscErrorCode ierr;
339 
340   PetscFunctionBegin;
341   if (scall == MAT_INITIAL_MATRIX) {
342     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
343     ierr = MatMatMultSymbolic_MPIAIJ_MPIDense(A,B,fill,C);CHKERRQ(ierr);
344     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
345   }
346   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
347   ierr = MatMatMultNumeric_MPIAIJ_MPIDense(A,B,*C);CHKERRQ(ierr);
348   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
349   PetscFunctionReturn(0);
350 }
351 
352 typedef struct {
353   Mat         workB;
354   PetscScalar *rvalues,*svalues;
355   MPI_Request *rwaits,*swaits;
356 } MPIAIJ_MPIDense;
357 
358 #undef __FUNCT__
359 #define __FUNCT__ "MatMPIAIJ_MPIDenseDestroy"
360 PetscErrorCode MatMPIAIJ_MPIDenseDestroy(void *ctx)
361 {
362   MPIAIJ_MPIDense *contents = (MPIAIJ_MPIDense*) ctx;
363   PetscErrorCode  ierr;
364 
365   PetscFunctionBegin;
366   ierr = MatDestroy(&contents->workB);CHKERRQ(ierr);
367   ierr = PetscFree4(contents->rvalues,contents->svalues,contents->rwaits,contents->swaits);CHKERRQ(ierr);
368   ierr = PetscFree(contents);CHKERRQ(ierr);
369   PetscFunctionReturn(0);
370 }
371 
372 #undef __FUNCT__
373 #define __FUNCT__ "MatMatMultNumeric_MPIDense"
374 /*
375     This is a "dummy function" that handles the case where matrix C was created as a dense matrix
376   directly by the user and passed to MatMatMult() with the MAT_REUSE_MATRIX option
377 
378   It is the same as MatMatMultSymbolic_MPIAIJ_MPIDense() except does not create C
379 */
380 PetscErrorCode MatMatMultNumeric_MPIDense(Mat A,Mat B,Mat C)
381 {
382   PetscErrorCode         ierr;
383   PetscBool              flg;
384   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*) A->data;
385   PetscInt               nz   = aij->B->cmap->n;
386   PetscContainer         container;
387   MPIAIJ_MPIDense        *contents;
388   VecScatter             ctx   = aij->Mvctx;
389   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
390   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
391 
392   PetscFunctionBegin;
393   ierr = PetscObjectTypeCompare((PetscObject)B,MATMPIDENSE,&flg);CHKERRQ(ierr);
394   if (!flg) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Second matrix must be mpidense");
395 
396   /* Handle case where where user provided the final C matrix rather than calling MatMatMult() with MAT_INITIAL_MATRIX*/
397   ierr = PetscObjectTypeCompare((PetscObject)A,MATMPIAIJ,&flg);CHKERRQ(ierr);
398   if (!flg) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"First matrix must be MPIAIJ");
399 
400   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
401 
402   ierr = PetscNew(&contents);CHKERRQ(ierr);
403   /* Create work matrix used to store off processor rows of B needed for local product */
404   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,B->cmap->N,NULL,&contents->workB);CHKERRQ(ierr);
405   /* Create work arrays needed */
406   ierr = PetscMalloc4(B->cmap->N*from->starts[from->n],&contents->rvalues,
407                       B->cmap->N*to->starts[to->n],&contents->svalues,
408                       from->n,&contents->rwaits,
409                       to->n,&contents->swaits);CHKERRQ(ierr);
410 
411   ierr = PetscContainerCreate(PetscObjectComm((PetscObject)A),&container);CHKERRQ(ierr);
412   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
413   ierr = PetscContainerSetUserDestroy(container,MatMPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
414   ierr = PetscObjectCompose((PetscObject)C,"workB",(PetscObject)container);CHKERRQ(ierr);
415   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
416 
417   ierr = (*C->ops->matmultnumeric)(A,B,C);CHKERRQ(ierr);
418   PetscFunctionReturn(0);
419 }
420 
421 #undef __FUNCT__
422 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIDense"
423 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat A,Mat B,PetscReal fill,Mat *C)
424 {
425   PetscErrorCode         ierr;
426   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*) A->data;
427   PetscInt               nz   = aij->B->cmap->n;
428   PetscContainer         container;
429   MPIAIJ_MPIDense        *contents;
430   VecScatter             ctx   = aij->Mvctx;
431   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
432   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
433   PetscInt               m     = A->rmap->n,n=B->cmap->n;
434 
435   PetscFunctionBegin;
436   ierr = MatCreate(PetscObjectComm((PetscObject)B),C);CHKERRQ(ierr);
437   ierr = MatSetSizes(*C,m,n,A->rmap->N,B->cmap->N);CHKERRQ(ierr);
438   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
439   ierr = MatSetType(*C,MATMPIDENSE);CHKERRQ(ierr);
440   ierr = MatMPIDenseSetPreallocation(*C,NULL);CHKERRQ(ierr);
441   ierr = MatAssemblyBegin(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
442   ierr = MatAssemblyEnd(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
443 
444   (*C)->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
445 
446   ierr = PetscNew(&contents);CHKERRQ(ierr);
447   /* Create work matrix used to store off processor rows of B needed for local product */
448   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,B->cmap->N,NULL,&contents->workB);CHKERRQ(ierr);
449   /* Create work arrays needed */
450   ierr = PetscMalloc4(B->cmap->N*from->starts[from->n],&contents->rvalues,
451                       B->cmap->N*to->starts[to->n],&contents->svalues,
452                       from->n,&contents->rwaits,
453                       to->n,&contents->swaits);CHKERRQ(ierr);
454 
455   ierr = PetscContainerCreate(PetscObjectComm((PetscObject)A),&container);CHKERRQ(ierr);
456   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
457   ierr = PetscContainerSetUserDestroy(container,MatMPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
458   ierr = PetscObjectCompose((PetscObject)(*C),"workB",(PetscObject)container);CHKERRQ(ierr);
459   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
460   PetscFunctionReturn(0);
461 }
462 
463 #undef __FUNCT__
464 #define __FUNCT__ "MatMPIDenseScatter"
465 /*
466     Performs an efficient scatter on the rows of B needed by this process; this is
467     a modification of the VecScatterBegin_() routines.
468 */
469 PetscErrorCode MatMPIDenseScatter(Mat A,Mat B,Mat C,Mat *outworkB)
470 {
471   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*)A->data;
472   PetscErrorCode         ierr;
473   PetscScalar            *b,*w,*svalues,*rvalues;
474   VecScatter             ctx   = aij->Mvctx;
475   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
476   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
477   PetscInt               i,j,k;
478   PetscInt               *sindices,*sstarts,*rindices,*rstarts;
479   PetscMPIInt            *sprocs,*rprocs,nrecvs;
480   MPI_Request            *swaits,*rwaits;
481   MPI_Comm               comm;
482   PetscMPIInt            tag  = ((PetscObject)ctx)->tag,ncols = B->cmap->N, nrows = aij->B->cmap->n,imdex,nrowsB = B->rmap->n;
483   MPI_Status             status;
484   MPIAIJ_MPIDense        *contents;
485   PetscContainer         container;
486   Mat                    workB;
487 
488   PetscFunctionBegin;
489   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
490   ierr = PetscObjectQuery((PetscObject)C,"workB",(PetscObject*)&container);CHKERRQ(ierr);
491   if (!container) SETERRQ(comm,PETSC_ERR_PLIB,"Container does not exist");
492   ierr = PetscContainerGetPointer(container,(void**)&contents);CHKERRQ(ierr);
493 
494   workB = *outworkB = contents->workB;
495   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);
496   sindices = to->indices;
497   sstarts  = to->starts;
498   sprocs   = to->procs;
499   swaits   = contents->swaits;
500   svalues  = contents->svalues;
501 
502   rindices = from->indices;
503   rstarts  = from->starts;
504   rprocs   = from->procs;
505   rwaits   = contents->rwaits;
506   rvalues  = contents->rvalues;
507 
508   ierr = MatDenseGetArray(B,&b);CHKERRQ(ierr);
509   ierr = MatDenseGetArray(workB,&w);CHKERRQ(ierr);
510 
511   for (i=0; i<from->n; i++) {
512     ierr = MPI_Irecv(rvalues+ncols*rstarts[i],ncols*(rstarts[i+1]-rstarts[i]),MPIU_SCALAR,rprocs[i],tag,comm,rwaits+i);CHKERRQ(ierr);
513   }
514 
515   for (i=0; i<to->n; i++) {
516     /* pack a message at a time */
517     for (j=0; j<sstarts[i+1]-sstarts[i]; j++) {
518       for (k=0; k<ncols; k++) {
519         svalues[ncols*(sstarts[i] + j) + k] = b[sindices[sstarts[i]+j] + nrowsB*k];
520       }
521     }
522     ierr = MPI_Isend(svalues+ncols*sstarts[i],ncols*(sstarts[i+1]-sstarts[i]),MPIU_SCALAR,sprocs[i],tag,comm,swaits+i);CHKERRQ(ierr);
523   }
524 
525   nrecvs = from->n;
526   while (nrecvs) {
527     ierr = MPI_Waitany(from->n,rwaits,&imdex,&status);CHKERRQ(ierr);
528     nrecvs--;
529     /* unpack a message at a time */
530     for (j=0; j<rstarts[imdex+1]-rstarts[imdex]; j++) {
531       for (k=0; k<ncols; k++) {
532         w[rindices[rstarts[imdex]+j] + nrows*k] = rvalues[ncols*(rstarts[imdex] + j) + k];
533       }
534     }
535   }
536   if (to->n) {ierr = MPI_Waitall(to->n,swaits,to->sstatus);CHKERRQ(ierr);}
537 
538   ierr = MatDenseRestoreArray(B,&b);CHKERRQ(ierr);
539   ierr = MatDenseRestoreArray(workB,&w);CHKERRQ(ierr);
540   ierr = MatAssemblyBegin(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
541   ierr = MatAssemblyEnd(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
542   PetscFunctionReturn(0);
543 }
544 extern PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat,Mat,Mat);
545 
546 #undef __FUNCT__
547 #define __FUNCT__ "MatMatMultNumeric_MPIAIJ_MPIDense"
548 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat A,Mat B,Mat C)
549 {
550   PetscErrorCode ierr;
551   Mat_MPIAIJ     *aij    = (Mat_MPIAIJ*)A->data;
552   Mat_MPIDense   *bdense = (Mat_MPIDense*)B->data;
553   Mat_MPIDense   *cdense = (Mat_MPIDense*)C->data;
554   Mat            workB;
555 
556   PetscFunctionBegin;
557   /* diagonal block of A times all local rows of B*/
558   ierr = MatMatMultNumeric_SeqAIJ_SeqDense(aij->A,bdense->A,cdense->A);CHKERRQ(ierr);
559 
560   /* get off processor parts of B needed to complete the product */
561   ierr = MatMPIDenseScatter(A,B,C,&workB);CHKERRQ(ierr);
562 
563   /* off-diagonal block of A times nonlocal rows of B */
564   ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A);CHKERRQ(ierr);
565   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
566   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
567   PetscFunctionReturn(0);
568 }
569 
570 #undef __FUNCT__
571 #define __FUNCT__ "MatMatMultNumeric_MPIAIJ_MPIAIJ"
572 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
573 {
574   PetscErrorCode ierr;
575   Mat_MPIAIJ     *a   = (Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
576   Mat_SeqAIJ     *ad  = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
577   Mat_SeqAIJ     *cd  = (Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
578   PetscInt       *adi = ad->i,*adj,*aoi=ao->i,*aoj;
579   PetscScalar    *ada,*aoa,*cda=cd->a,*coa=co->a;
580   Mat_SeqAIJ     *p_loc,*p_oth;
581   PetscInt       *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
582   PetscScalar    *pa_loc,*pa_oth,*pa,valtmp,*ca;
583   PetscInt       cm          = C->rmap->n,anz,pnz;
584   Mat_PtAPMPI    *ptap       = c->ptap;
585   PetscScalar    *apa_sparse = ptap->apa;
586   PetscInt       *api,*apj,*apJ,i,j,k,row;
587   PetscInt       cstart = C->cmap->rstart;
588   PetscInt       cdnz,conz,k0,k1,nextp;
589   MPI_Comm       comm;
590   PetscMPIInt    size;
591 
592   PetscFunctionBegin;
593   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
594   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
595 
596   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
597   /*-----------------------------------------------------*/
598   /* update numerical values of P_oth and P_loc */
599   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
600   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
601 
602   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
603   /*----------------------------------------------------------*/
604   /* get data from symbolic products */
605   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
606   pi_loc = p_loc->i; pj_loc = p_loc->j; pa_loc = p_loc->a;
607   if (size >1) {
608     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
609     pi_oth = p_oth->i; pj_oth = p_oth->j; pa_oth = p_oth->a;
610   } else {
611     p_oth = NULL; pi_oth = NULL; pj_oth = NULL; pa_oth = NULL;
612   }
613 
614   api = ptap->api;
615   apj = ptap->apj;
616   for (i=0; i<cm; i++) {
617     apJ = apj + api[i];
618 
619     /* diagonal portion of A */
620     anz = adi[i+1] - adi[i];
621     adj = ad->j + adi[i];
622     ada = ad->a + adi[i];
623     for (j=0; j<anz; j++) {
624       row = adj[j];
625       pnz = pi_loc[row+1] - pi_loc[row];
626       pj  = pj_loc + pi_loc[row];
627       pa  = pa_loc + pi_loc[row];
628       /* perform sparse axpy */
629       valtmp = ada[j];
630       nextp  = 0;
631       for (k=0; nextp<pnz; k++) {
632         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
633           apa_sparse[k] += valtmp*pa[nextp++];
634         }
635       }
636       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
637     }
638 
639     /* off-diagonal portion of A */
640     anz = aoi[i+1] - aoi[i];
641     aoj = ao->j + aoi[i];
642     aoa = ao->a + aoi[i];
643     for (j=0; j<anz; j++) {
644       row = aoj[j];
645       pnz = pi_oth[row+1] - pi_oth[row];
646       pj  = pj_oth + pi_oth[row];
647       pa  = pa_oth + pi_oth[row];
648       /* perform sparse axpy */
649       valtmp = aoa[j];
650       nextp  = 0;
651       for (k=0; nextp<pnz; k++) {
652         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
653           apa_sparse[k] += valtmp*pa[nextp++];
654         }
655       }
656       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
657     }
658 
659     /* set values in C */
660     cdnz = cd->i[i+1] - cd->i[i];
661     conz = co->i[i+1] - co->i[i];
662 
663     /* 1st off-diagoanl part of C */
664     ca = coa + co->i[i];
665     k  = 0;
666     for (k0=0; k0<conz; k0++) {
667       if (apJ[k] >= cstart) break;
668       ca[k0]        = apa_sparse[k];
669       apa_sparse[k] = 0.0;
670       k++;
671     }
672 
673     /* diagonal part of C */
674     ca = cda + cd->i[i];
675     for (k1=0; k1<cdnz; k1++) {
676       ca[k1]        = apa_sparse[k];
677       apa_sparse[k] = 0.0;
678       k++;
679     }
680 
681     /* 2nd off-diagoanl part of C */
682     ca = coa + co->i[i];
683     for (; k0<conz; k0++) {
684       ca[k0]        = apa_sparse[k];
685       apa_sparse[k] = 0.0;
686       k++;
687     }
688   }
689   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
690   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
691   PetscFunctionReturn(0);
692 }
693 
694 /* same as MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(), except using LLCondensed to avoid O(BN) memory requirement */
695 #undef __FUNCT__
696 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIAIJ"
697 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat *C)
698 {
699   PetscErrorCode     ierr;
700   MPI_Comm           comm;
701   PetscMPIInt        size;
702   Mat                Cmpi;
703   Mat_PtAPMPI        *ptap;
704   PetscFreeSpaceList free_space = NULL,current_space=NULL;
705   Mat_MPIAIJ         *a         = (Mat_MPIAIJ*)A->data,*c;
706   Mat_SeqAIJ         *ad        = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
707   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
708   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
709   PetscInt           i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi,*lnk,apnz_max=0;
710   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n,Crmax;
711   PetscInt           nlnk_max,armax,prmax;
712   PetscReal          afill;
713   PetscScalar        *apa;
714   PetscTable         ta;
715 
716   PetscFunctionBegin;
717   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
718   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
719 
720   /* create struct Mat_PtAPMPI and attached it to C later */
721   ierr = PetscNew(&ptap);CHKERRQ(ierr);
722 
723   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
724   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
725 
726   /* get P_loc by taking all local rows of P */
727   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
728 
729   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
730   pi_loc = p_loc->i; pj_loc = p_loc->j;
731   if (size > 1) {
732     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
733     pi_oth = p_oth->i; pj_oth = p_oth->j;
734   } else {
735     p_oth  = NULL;
736     pi_oth = NULL; pj_oth = NULL;
737   }
738 
739   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
740   /*-------------------------------------------------------------------*/
741   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
742   ptap->api = api;
743   api[0]    = 0;
744 
745   /* create and initialize a linked list */
746   Crmax = p_loc->rmax+p_oth->rmax;
747   if (Crmax > pN) Crmax = pN;
748   ierr = PetscTableCreate(Crmax,pN,&ta);CHKERRQ(ierr);
749   MatRowMergeMax_SeqAIJ(p_loc,ptap->P_loc->rmap->N,ta);
750   MatRowMergeMax_SeqAIJ(p_oth,ptap->P_oth->rmap->N,ta);
751   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
752   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
753 
754   ierr = PetscLLCondensedCreate_Scalable(Crmax,&lnk);CHKERRQ(ierr);
755 
756   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
757   ierr = PetscFreeSpaceGet((PetscInt)(fill*(adi[am]+aoi[am]+pi_loc[pm])),&free_space);CHKERRQ(ierr);
758 
759   current_space = free_space;
760 
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     if (apnz > apnz_max) apnz_max = apnz;
784 
785     /* if free space is not available, double the total space in the list */
786     if (current_space->local_remaining<apnz) {
787       ierr = PetscFreeSpaceGet(apnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
788       nspacedouble++;
789     }
790 
791     /* Copy data into free space, then initialize lnk */
792     ierr = PetscLLCondensedClean_Scalable(apnz,current_space->array,lnk);CHKERRQ(ierr);
793     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
794 
795     current_space->array           += apnz;
796     current_space->local_used      += apnz;
797     current_space->local_remaining -= apnz;
798   }
799 
800   /* Allocate space for apj, initialize apj, and */
801   /* destroy list of free space and other temporary array(s) */
802   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
803   apj  = ptap->apj;
804   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
805   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
806 
807   /* create and assemble symbolic parallel matrix Cmpi */
808   /*----------------------------------------------------*/
809   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
810   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
811   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
812   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
813   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
814   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
815 
816   /* malloc apa for assembly Cmpi */
817   ierr = PetscCalloc1(apnz_max,&apa);CHKERRQ(ierr);
818 
819   ptap->apa = apa;
820   for (i=0; i<am; i++) {
821     row  = i + rstart;
822     apnz = api[i+1] - api[i];
823     ierr = MatSetValues(Cmpi,1,&row,apnz,apj,apa,INSERT_VALUES);CHKERRQ(ierr);
824     apj += apnz;
825   }
826   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
827   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
828 
829   ptap->destroy             = Cmpi->ops->destroy;
830   ptap->duplicate           = Cmpi->ops->duplicate;
831   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ;
832   Cmpi->ops->destroy        = MatDestroy_MPIAIJ_MatMatMult;
833   Cmpi->ops->duplicate      = MatDuplicate_MPIAIJ_MatMatMult;
834 
835   /* attach the supporting struct to Cmpi for reuse */
836   c       = (Mat_MPIAIJ*)Cmpi->data;
837   c->ptap = ptap;
838 
839   *C = Cmpi;
840 
841   /* set MatInfo */
842   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
843   if (afill < 1.0) afill = 1.0;
844   Cmpi->info.mallocs           = nspacedouble;
845   Cmpi->info.fill_ratio_given  = fill;
846   Cmpi->info.fill_ratio_needed = afill;
847 
848 #if defined(PETSC_USE_INFO)
849   if (api[am]) {
850     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
851     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
852   } else {
853     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
854   }
855 #endif
856   PetscFunctionReturn(0);
857 }
858 
859 /*-------------------------------------------------------------------------*/
860 #undef __FUNCT__
861 #define __FUNCT__ "MatTransposeMatMult_MPIAIJ_MPIAIJ"
862 PetscErrorCode MatTransposeMatMult_MPIAIJ_MPIAIJ(Mat P,Mat A,MatReuse scall,PetscReal fill,Mat *C)
863 {
864   PetscErrorCode ierr;
865   const char     *algTypes[3] = {"scalable","nonscalable","matmatmult"};
866   PetscInt       alg=0; /* set default algorithm */
867 
868   PetscFunctionBegin;
869   if (scall == MAT_INITIAL_MATRIX) {
870     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
871     ierr = PetscOptionsEList("-mattransposematmult_via","Algorithmic approach","MatTransposeMatMult",algTypes,3,algTypes[0],&alg,NULL);CHKERRQ(ierr);
872     ierr = PetscOptionsEnd();CHKERRQ(ierr);
873 
874     ierr = PetscLogEventBegin(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
875     switch (alg) {
876     case 1:
877       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(P,A,fill,C);CHKERRQ(ierr);
878       break;
879     case 2:
880     {
881       Mat         Pt;
882       Mat_PtAPMPI *ptap;
883       Mat_MPIAIJ  *c;
884       ierr = MatTranspose(P,MAT_INITIAL_MATRIX,&Pt);CHKERRQ(ierr);
885       ierr = MatMatMult(Pt,A,MAT_INITIAL_MATRIX,fill,C);CHKERRQ(ierr);
886       c        = (Mat_MPIAIJ*)(*C)->data;
887       ptap     = c->ptap;
888       ptap->Pt = Pt;
889       (*C)->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult;
890       PetscFunctionReturn(0);
891     }
892       break;
893     default:
894       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(P,A,fill,C);CHKERRQ(ierr);
895       break;
896     }
897     ierr = PetscLogEventEnd(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
898   }
899   ierr = PetscLogEventBegin(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
900   ierr = (*(*C)->ops->mattransposemultnumeric)(P,A,*C);CHKERRQ(ierr);
901   ierr = PetscLogEventEnd(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
902   PetscFunctionReturn(0);
903 }
904 
905 /* This routine only works when scall=MAT_REUSE_MATRIX! */
906 #undef __FUNCT__
907 #define __FUNCT__ "MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult"
908 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult(Mat P,Mat A,Mat C)
909 {
910   PetscErrorCode ierr;
911   Mat_MPIAIJ     *c=(Mat_MPIAIJ*)C->data;
912   Mat_PtAPMPI    *ptap= c->ptap;
913   Mat            Pt=ptap->Pt;
914 
915   PetscFunctionBegin;
916   ierr = MatTranspose(P,MAT_REUSE_MATRIX,&Pt);CHKERRQ(ierr);
917   ierr = MatMatMultNumeric(Pt,A,C);CHKERRQ(ierr);
918   PetscFunctionReturn(0);
919 }
920 
921 /* Non-scalable version, use dense axpy */
922 #undef __FUNCT__
923 #define __FUNCT__ "MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable"
924 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,Mat C)
925 {
926   PetscErrorCode      ierr;
927   Mat_Merge_SeqsToMPI *merge;
928   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
929   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
930   Mat_PtAPMPI         *ptap;
931   PetscInt            *adj,*aJ;
932   PetscInt            i,j,k,anz,pnz,row,*cj;
933   MatScalar           *ada,*aval,*ca,valtmp;
934   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
935   MPI_Comm            comm;
936   PetscMPIInt         size,rank,taga,*len_s;
937   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
938   PetscInt            **buf_ri,**buf_rj;
939   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
940   MPI_Request         *s_waits,*r_waits;
941   MPI_Status          *status;
942   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
943   PetscInt            *ai,*aj,*coi,*coj;
944   PetscInt            *poJ,*pdJ;
945   Mat                 A_loc;
946   Mat_SeqAIJ          *a_loc;
947 
948   PetscFunctionBegin;
949   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
950   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
951   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
952 
953   ptap  = c->ptap;
954   merge = ptap->merge;
955 
956   /* 2) compute numeric C_seq = P_loc^T*A_loc*P - dominating part */
957   /*--------------------------------------------------------------*/
958   /* get data from symbolic products */
959   coi  = merge->coi; coj = merge->coj;
960   ierr = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
961 
962   bi     = merge->bi; bj = merge->bj;
963   owners = merge->rowmap->range;
964   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
965 
966   /* get A_loc by taking all local rows of A */
967   A_loc = ptap->A_loc;
968   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
969   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
970   ai    = a_loc->i;
971   aj    = a_loc->j;
972 
973   ierr = PetscCalloc1(A->cmap->N,&aval);CHKERRQ(ierr); /* non-scalable!!! */
974 
975   for (i=0; i<am; i++) {
976     /* 2-a) put A[i,:] to dense array aval */
977     anz = ai[i+1] - ai[i];
978     adj = aj + ai[i];
979     ada = a_loc->a + ai[i];
980     for (j=0; j<anz; j++) {
981       aval[adj[j]] = ada[j];
982     }
983 
984     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
985     /*--------------------------------------------------------------*/
986     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
987     pnz = po->i[i+1] - po->i[i];
988     poJ = po->j + po->i[i];
989     pA  = po->a + po->i[i];
990     for (j=0; j<pnz; j++) {
991       row = poJ[j];
992       cnz = coi[row+1] - coi[row];
993       cj  = coj + coi[row];
994       ca  = coa + coi[row];
995       /* perform dense axpy */
996       valtmp = pA[j];
997       for (k=0; k<cnz; k++) {
998         ca[k] += valtmp*aval[cj[k]];
999       }
1000       ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1001     }
1002 
1003     /* put the value into Cd (diagonal part) */
1004     pnz = pd->i[i+1] - pd->i[i];
1005     pdJ = pd->j + pd->i[i];
1006     pA  = pd->a + pd->i[i];
1007     for (j=0; j<pnz; j++) {
1008       row = pdJ[j];
1009       cnz = bi[row+1] - bi[row];
1010       cj  = bj + bi[row];
1011       ca  = ba + bi[row];
1012       /* perform dense axpy */
1013       valtmp = pA[j];
1014       for (k=0; k<cnz; k++) {
1015         ca[k] += valtmp*aval[cj[k]];
1016       }
1017       ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1018     }
1019 
1020     /* zero the current row of Pt*A */
1021     aJ = aj + ai[i];
1022     for (k=0; k<anz; k++) aval[aJ[k]] = 0.0;
1023   }
1024 
1025   /* 3) send and recv matrix values coa */
1026   /*------------------------------------*/
1027   buf_ri = merge->buf_ri;
1028   buf_rj = merge->buf_rj;
1029   len_s  = merge->len_s;
1030   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1031   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1032 
1033   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1034   for (proc=0,k=0; proc<size; proc++) {
1035     if (!len_s[proc]) continue;
1036     i    = merge->owners_co[proc];
1037     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1038     k++;
1039   }
1040   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1041   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1042 
1043   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1044   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1045   ierr = PetscFree(coa);CHKERRQ(ierr);
1046 
1047   /* 4) insert local Cseq and received values into Cmpi */
1048   /*----------------------------------------------------*/
1049   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1050   for (k=0; k<merge->nrecv; k++) {
1051     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1052     nrows       = *(buf_ri_k[k]);
1053     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1054     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1055   }
1056 
1057   for (i=0; i<cm; i++) {
1058     row  = owners[rank] + i; /* global row index of C_seq */
1059     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1060     ba_i = ba + bi[i];
1061     bnz  = bi[i+1] - bi[i];
1062     /* add received vals into ba */
1063     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1064       /* i-th row */
1065       if (i == *nextrow[k]) {
1066         cnz    = *(nextci[k]+1) - *nextci[k];
1067         cj     = buf_rj[k] + *(nextci[k]);
1068         ca     = abuf_r[k] + *(nextci[k]);
1069         nextcj = 0;
1070         for (j=0; nextcj<cnz; j++) {
1071           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1072             ba_i[j] += ca[nextcj++];
1073           }
1074         }
1075         nextrow[k]++; nextci[k]++;
1076         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1077       }
1078     }
1079     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1080   }
1081   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1082   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1083 
1084   ierr = PetscFree(ba);CHKERRQ(ierr);
1085   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1086   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1087   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1088   ierr = PetscFree(aval);CHKERRQ(ierr);
1089   PetscFunctionReturn(0);
1090 }
1091 
1092 PetscErrorCode MatDuplicate_MPIAIJ_MatPtAP(Mat, MatDuplicateOption,Mat*);
1093 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1094 #undef __FUNCT__
1095 #define __FUNCT__ "MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable"
1096 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,PetscReal fill,Mat *C)
1097 {
1098   PetscErrorCode      ierr;
1099   Mat                 Cmpi,A_loc,POt,PDt;
1100   Mat_PtAPMPI         *ptap;
1101   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1102   Mat_MPIAIJ          *p        =(Mat_MPIAIJ*)P->data,*c;
1103   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1104   PetscInt            nnz;
1105   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1106   PetscInt            am=A->rmap->n,pn=P->cmap->n;
1107   PetscBT             lnkbt;
1108   MPI_Comm            comm;
1109   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1110   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1111   PetscInt            len,proc,*dnz,*onz,*owners;
1112   PetscInt            nzi,*bi,*bj;
1113   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1114   MPI_Request         *swaits,*rwaits;
1115   MPI_Status          *sstatus,rstatus;
1116   Mat_Merge_SeqsToMPI *merge;
1117   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1118   PetscReal           afill  =1.0,afill_tmp;
1119   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N;
1120   PetscScalar         *vals;
1121   Mat_SeqAIJ          *a_loc, *pdt,*pot;
1122 
1123   PetscFunctionBegin;
1124   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1125   /* check if matrix local sizes are compatible */
1126   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend) {
1127     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);
1128   }
1129 
1130   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1131   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1132 
1133   /* create struct Mat_PtAPMPI and attached it to C later */
1134   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1135 
1136   /* get A_loc by taking all local rows of A */
1137   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1138 
1139   ptap->A_loc = A_loc;
1140 
1141   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1142   ai    = a_loc->i;
1143   aj    = a_loc->j;
1144 
1145   /* determine symbolic Co=(p->B)^T*A - send to others */
1146   /*----------------------------------------------------*/
1147   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1148   pdt  = (Mat_SeqAIJ*)PDt->data;
1149   pdti = pdt->i; pdtj = pdt->j;
1150 
1151   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1152   pot  = (Mat_SeqAIJ*)POt->data;
1153   poti = pot->i; potj = pot->j;
1154 
1155   /* then, compute symbolic Co = (p->B)^T*A */
1156   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors >= (num of nonzero rows of C_seq) - pn */
1157   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
1158   coi[0] = 0;
1159 
1160   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1161   nnz           = fill*(poti[pon] + ai[am]);
1162   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1163   current_space = free_space;
1164 
1165   /* create and initialize a linked list */
1166   ierr = PetscLLCondensedCreate(aN,aN,&lnk,&lnkbt);CHKERRQ(ierr);
1167 
1168   for (i=0; i<pon; i++) {
1169     pnz = poti[i+1] - poti[i];
1170     ptJ = potj + poti[i];
1171     for (j=0; j<pnz; j++) {
1172       row  = ptJ[j]; /* row of A_loc == col of Pot */
1173       anz  = ai[row+1] - ai[row];
1174       Jptr = aj + ai[row];
1175       /* add non-zero cols of AP into the sorted linked list lnk */
1176       ierr = PetscLLCondensedAddSorted(anz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1177     }
1178     nnz = lnk[0];
1179 
1180     /* If free space is not available, double the total space in the list */
1181     if (current_space->local_remaining<nnz) {
1182       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1183       nspacedouble++;
1184     }
1185 
1186     /* Copy data into free space, and zero out denserows */
1187     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1188 
1189     current_space->array           += nnz;
1190     current_space->local_used      += nnz;
1191     current_space->local_remaining -= nnz;
1192 
1193     coi[i+1] = coi[i] + nnz;
1194   }
1195 
1196   ierr = PetscMalloc1(coi[pon]+1,&coj);CHKERRQ(ierr);
1197   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1198 
1199   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1200   if (afill_tmp > afill) afill = afill_tmp;
1201 
1202   /* send j-array (coj) of Co to other processors */
1203   /*----------------------------------------------*/
1204   /* determine row ownership */
1205   ierr = PetscNew(&merge);CHKERRQ(ierr);
1206   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1207 
1208   merge->rowmap->n  = pn;
1209   merge->rowmap->bs = 1;
1210 
1211   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1212   owners = merge->rowmap->range;
1213 
1214   /* determine the number of messages to send, their lengths */
1215   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
1216   ierr = PetscMalloc1(size,&merge->len_s);CHKERRQ(ierr);
1217 
1218   len_s        = merge->len_s;
1219   merge->nsend = 0;
1220 
1221   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
1222   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1223 
1224   proc = 0;
1225   for (i=0; i<pon; i++) {
1226     while (prmap[i] >= owners[proc+1]) proc++;
1227     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1228     len_s[proc] += coi[i+1] - coi[i];
1229   }
1230 
1231   len          = 0; /* max length of buf_si[] */
1232   owners_co[0] = 0;
1233   for (proc=0; proc<size; proc++) {
1234     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1235     if (len_si[proc]) {
1236       merge->nsend++;
1237       len_si[proc] = 2*(len_si[proc] + 1);
1238       len         += len_si[proc];
1239     }
1240   }
1241 
1242   /* determine the number and length of messages to receive for coi and coj  */
1243   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1244   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1245 
1246   /* post the Irecv and Isend of coj */
1247   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1248   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1249   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
1250   for (proc=0, k=0; proc<size; proc++) {
1251     if (!len_s[proc]) continue;
1252     i    = owners_co[proc];
1253     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1254     k++;
1255   }
1256 
1257   /* receives and sends of coj are complete */
1258   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
1259   for (i=0; i<merge->nrecv; i++) {
1260     PetscMPIInt icompleted;
1261     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1262   }
1263   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1264   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1265 
1266   /* send and recv coi */
1267   /*-------------------*/
1268   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1269   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1270   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1271   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1272   for (proc=0,k=0; proc<size; proc++) {
1273     if (!len_s[proc]) continue;
1274     /* form outgoing message for i-structure:
1275          buf_si[0]:                 nrows to be sent
1276                [1:nrows]:           row index (global)
1277                [nrows+1:2*nrows+1]: i-structure index
1278     */
1279     /*-------------------------------------------*/
1280     nrows       = len_si[proc]/2 - 1;
1281     buf_si_i    = buf_si + nrows+1;
1282     buf_si[0]   = nrows;
1283     buf_si_i[0] = 0;
1284     nrows       = 0;
1285     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1286       nzi               = coi[i+1] - coi[i];
1287       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi; /* i-structure */
1288       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1289       nrows++;
1290     }
1291     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1292     k++;
1293     buf_si += len_si[proc];
1294   }
1295   i = merge->nrecv;
1296   while (i--) {
1297     PetscMPIInt icompleted;
1298     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1299   }
1300   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1301   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1302   ierr = PetscFree(len_si);CHKERRQ(ierr);
1303   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1304   ierr = PetscFree(swaits);CHKERRQ(ierr);
1305   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1306   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1307 
1308   /* compute the local portion of C (mpi mat) */
1309   /*------------------------------------------*/
1310   /* allocate bi array and free space for accumulating nonzero column info */
1311   ierr  = PetscMalloc1(pn+1,&bi);CHKERRQ(ierr);
1312   bi[0] = 0;
1313 
1314   /* set initial free space to be fill*(nnz(P) + nnz(A)) */
1315   nnz           = fill*(pdti[pn] + poti[pon] + ai[am]);
1316   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1317   current_space = free_space;
1318 
1319   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1320   for (k=0; k<merge->nrecv; k++) {
1321     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1322     nrows       = *buf_ri_k[k];
1323     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1324     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1325   }
1326 
1327   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1328   rmax = 0;
1329   for (i=0; i<pn; i++) {
1330     /* add pdt[i,:]*AP into lnk */
1331     pnz = pdti[i+1] - pdti[i];
1332     ptJ = pdtj + pdti[i];
1333     for (j=0; j<pnz; j++) {
1334       row  = ptJ[j];  /* row of AP == col of Pt */
1335       anz  = ai[row+1] - ai[row];
1336       Jptr = aj + ai[row];
1337       /* add non-zero cols of AP into the sorted linked list lnk */
1338       ierr = PetscLLCondensedAddSorted(anz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1339     }
1340 
1341     /* add received col data into lnk */
1342     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1343       if (i == *nextrow[k]) { /* i-th row */
1344         nzi  = *(nextci[k]+1) - *nextci[k];
1345         Jptr = buf_rj[k] + *nextci[k];
1346         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1347         nextrow[k]++; nextci[k]++;
1348       }
1349     }
1350     nnz = lnk[0];
1351 
1352     /* if free space is not available, make more free space */
1353     if (current_space->local_remaining<nnz) {
1354       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1355       nspacedouble++;
1356     }
1357     /* copy data into free space, then initialize lnk */
1358     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1359     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1360 
1361     current_space->array           += nnz;
1362     current_space->local_used      += nnz;
1363     current_space->local_remaining -= nnz;
1364 
1365     bi[i+1] = bi[i] + nnz;
1366     if (nnz > rmax) rmax = nnz;
1367   }
1368   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1369 
1370   ierr = PetscMalloc1(bi[pn]+1,&bj);CHKERRQ(ierr);
1371   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1372 
1373   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
1374   if (afill_tmp > afill) afill = afill_tmp;
1375   ierr = PetscLLCondensedDestroy(lnk,lnkbt);CHKERRQ(ierr);
1376   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1377   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1378 
1379   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
1380   /*----------------------------------------------------------------------------------*/
1381   ierr = PetscCalloc1(rmax+1,&vals);CHKERRQ(ierr);
1382 
1383   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1384   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1385   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
1386   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1387   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1388   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1389   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
1390   for (i=0; i<pn; i++) {
1391     row  = i + rstart;
1392     nnz  = bi[i+1] - bi[i];
1393     Jptr = bj + bi[i];
1394     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
1395   }
1396   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1397   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1398   ierr = PetscFree(vals);CHKERRQ(ierr);
1399 
1400   merge->bi        = bi;
1401   merge->bj        = bj;
1402   merge->coi       = coi;
1403   merge->coj       = coj;
1404   merge->buf_ri    = buf_ri;
1405   merge->buf_rj    = buf_rj;
1406   merge->owners_co = owners_co;
1407   merge->destroy   = Cmpi->ops->destroy;
1408   merge->duplicate = Cmpi->ops->duplicate;
1409 
1410   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1411   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
1412   Cmpi->ops->duplicate               = MatDuplicate_MPIAIJ_MatPtAP;
1413 
1414   /* attach the supporting struct to Cmpi for reuse */
1415   c           = (Mat_MPIAIJ*)Cmpi->data;
1416   c->ptap     = ptap;
1417   ptap->api   = NULL;
1418   ptap->apj   = NULL;
1419   ptap->merge = merge;
1420   ptap->rmax  = rmax;
1421 
1422   *C = Cmpi;
1423 #if defined(PETSC_USE_INFO)
1424   if (bi[pn] != 0) {
1425     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1426     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
1427   } else {
1428     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1429   }
1430 #endif
1431   PetscFunctionReturn(0);
1432 }
1433 
1434 #undef __FUNCT__
1435 #define __FUNCT__ "MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ"
1436 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ(Mat P,Mat A,Mat C)
1437 {
1438   PetscErrorCode      ierr;
1439   Mat_Merge_SeqsToMPI *merge;
1440   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1441   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
1442   Mat_PtAPMPI         *ptap;
1443   PetscInt            *adj;
1444   PetscInt            i,j,k,anz,pnz,row,*cj,nexta;
1445   MatScalar           *ada,*ca,valtmp;
1446   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
1447   MPI_Comm            comm;
1448   PetscMPIInt         size,rank,taga,*len_s;
1449   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
1450   PetscInt            **buf_ri,**buf_rj;
1451   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
1452   MPI_Request         *s_waits,*r_waits;
1453   MPI_Status          *status;
1454   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
1455   PetscInt            *ai,*aj,*coi,*coj;
1456   PetscInt            *poJ,*pdJ;
1457   Mat                 A_loc;
1458   Mat_SeqAIJ          *a_loc;
1459 
1460   PetscFunctionBegin;
1461   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1462   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1463   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1464 
1465   ptap  = c->ptap;
1466   merge = ptap->merge;
1467 
1468   /* 2) compute numeric C_seq = P_loc^T*A_loc */
1469   /*------------------------------------------*/
1470   /* get data from symbolic products */
1471   coi    = merge->coi; coj = merge->coj;
1472   ierr   = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
1473   bi     = merge->bi; bj = merge->bj;
1474   owners = merge->rowmap->range;
1475   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
1476 
1477   /* get A_loc by taking all local rows of A */
1478   A_loc = ptap->A_loc;
1479   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
1480   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1481   ai    = a_loc->i;
1482   aj    = a_loc->j;
1483 
1484   for (i=0; i<am; i++) {
1485     anz = ai[i+1] - ai[i];
1486     adj = aj + ai[i];
1487     ada = a_loc->a + ai[i];
1488 
1489     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1490     /*-------------------------------------------------------------*/
1491     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1492     pnz = po->i[i+1] - po->i[i];
1493     poJ = po->j + po->i[i];
1494     pA  = po->a + po->i[i];
1495     for (j=0; j<pnz; j++) {
1496       row = poJ[j];
1497       cj  = coj + coi[row];
1498       ca  = coa + coi[row];
1499       /* perform sparse axpy */
1500       nexta  = 0;
1501       valtmp = pA[j];
1502       for (k=0; nexta<anz; k++) {
1503         if (cj[k] == adj[nexta]) {
1504           ca[k] += valtmp*ada[nexta];
1505           nexta++;
1506         }
1507       }
1508       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1509     }
1510 
1511     /* put the value into Cd (diagonal part) */
1512     pnz = pd->i[i+1] - pd->i[i];
1513     pdJ = pd->j + pd->i[i];
1514     pA  = pd->a + pd->i[i];
1515     for (j=0; j<pnz; j++) {
1516       row = pdJ[j];
1517       cj  = bj + bi[row];
1518       ca  = ba + bi[row];
1519       /* perform sparse axpy */
1520       nexta  = 0;
1521       valtmp = pA[j];
1522       for (k=0; nexta<anz; k++) {
1523         if (cj[k] == adj[nexta]) {
1524           ca[k] += valtmp*ada[nexta];
1525           nexta++;
1526         }
1527       }
1528       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1529     }
1530   }
1531 
1532   /* 3) send and recv matrix values coa */
1533   /*------------------------------------*/
1534   buf_ri = merge->buf_ri;
1535   buf_rj = merge->buf_rj;
1536   len_s  = merge->len_s;
1537   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1538   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1539 
1540   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1541   for (proc=0,k=0; proc<size; proc++) {
1542     if (!len_s[proc]) continue;
1543     i    = merge->owners_co[proc];
1544     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1545     k++;
1546   }
1547   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1548   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1549 
1550   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1551   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1552   ierr = PetscFree(coa);CHKERRQ(ierr);
1553 
1554   /* 4) insert local Cseq and received values into Cmpi */
1555   /*----------------------------------------------------*/
1556   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1557   for (k=0; k<merge->nrecv; k++) {
1558     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1559     nrows       = *(buf_ri_k[k]);
1560     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1561     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1562   }
1563 
1564   for (i=0; i<cm; i++) {
1565     row  = owners[rank] + i; /* global row index of C_seq */
1566     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1567     ba_i = ba + bi[i];
1568     bnz  = bi[i+1] - bi[i];
1569     /* add received vals into ba */
1570     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1571       /* i-th row */
1572       if (i == *nextrow[k]) {
1573         cnz    = *(nextci[k]+1) - *nextci[k];
1574         cj     = buf_rj[k] + *(nextci[k]);
1575         ca     = abuf_r[k] + *(nextci[k]);
1576         nextcj = 0;
1577         for (j=0; nextcj<cnz; j++) {
1578           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1579             ba_i[j] += ca[nextcj++];
1580           }
1581         }
1582         nextrow[k]++; nextci[k]++;
1583         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1584       }
1585     }
1586     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1587   }
1588   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1589   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1590 
1591   ierr = PetscFree(ba);CHKERRQ(ierr);
1592   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1593   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1594   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1595   PetscFunctionReturn(0);
1596 }
1597 
1598 PetscErrorCode MatDuplicate_MPIAIJ_MatPtAP(Mat, MatDuplicateOption,Mat*);
1599 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ();
1600    differ from MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable in using LLCondensedCreate_Scalable() */
1601 #undef __FUNCT__
1602 #define __FUNCT__ "MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ"
1603 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P,Mat A,PetscReal fill,Mat *C)
1604 {
1605   PetscErrorCode      ierr;
1606   Mat                 Cmpi,A_loc,POt,PDt;
1607   Mat_PtAPMPI         *ptap;
1608   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1609   Mat_MPIAIJ          *p        =(Mat_MPIAIJ*)P->data,*c;
1610   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1611   PetscInt            nnz;
1612   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1613   PetscInt            am  =A->rmap->n,pn=P->cmap->n;
1614   MPI_Comm            comm;
1615   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1616   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1617   PetscInt            len,proc,*dnz,*onz,*owners;
1618   PetscInt            nzi,*bi,*bj;
1619   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1620   MPI_Request         *swaits,*rwaits;
1621   MPI_Status          *sstatus,rstatus;
1622   Mat_Merge_SeqsToMPI *merge;
1623   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1624   PetscReal           afill  =1.0,afill_tmp;
1625   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Armax;
1626   PetscScalar         *vals;
1627   Mat_SeqAIJ          *a_loc,*pdt,*pot;
1628   PetscTable          ta;
1629 
1630   PetscFunctionBegin;
1631   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1632   /* check if matrix local sizes are compatible */
1633   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend) {
1634     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);
1635   }
1636 
1637   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1638   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1639 
1640   /* create struct Mat_PtAPMPI and attached it to C later */
1641   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1642 
1643   /* get A_loc by taking all local rows of A */
1644   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1645 
1646   ptap->A_loc = A_loc;
1647   a_loc       = (Mat_SeqAIJ*)(A_loc)->data;
1648   ai          = a_loc->i;
1649   aj          = a_loc->j;
1650 
1651   /* determine symbolic Co=(p->B)^T*A - send to others */
1652   /*----------------------------------------------------*/
1653   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1654   pdt  = (Mat_SeqAIJ*)PDt->data;
1655   pdti = pdt->i; pdtj = pdt->j;
1656 
1657   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1658   pot  = (Mat_SeqAIJ*)POt->data;
1659   poti = pot->i; potj = pot->j;
1660 
1661   /* then, compute symbolic Co = (p->B)^T*A */
1662   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1663                          >= (num of nonzero rows of C_seq) - pn */
1664   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
1665   coi[0] = 0;
1666 
1667   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1668   nnz           = fill*(poti[pon] + ai[am]);
1669   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1670   current_space = free_space;
1671 
1672   /* create and initialize a linked list */
1673   ierr = PetscTableCreate(2*a_loc->rmax,aN,&ta);CHKERRQ(ierr);
1674   MatRowMergeMax_SeqAIJ(a_loc,am,ta);
1675   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
1676   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
1677 
1678   for (i=0; i<pon; i++) {
1679     pnz = poti[i+1] - poti[i];
1680     ptJ = potj + poti[i];
1681     for (j=0; j<pnz; j++) {
1682       row  = ptJ[j]; /* row of A_loc == col of Pot */
1683       anz  = ai[row+1] - ai[row];
1684       Jptr = aj + ai[row];
1685       /* add non-zero cols of AP into the sorted linked list lnk */
1686       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1687     }
1688     nnz = lnk[0];
1689 
1690     /* If free space is not available, double the total space in the list */
1691     if (current_space->local_remaining<nnz) {
1692       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1693       nspacedouble++;
1694     }
1695 
1696     /* Copy data into free space, and zero out denserows */
1697     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1698 
1699     current_space->array           += nnz;
1700     current_space->local_used      += nnz;
1701     current_space->local_remaining -= nnz;
1702 
1703     coi[i+1] = coi[i] + nnz;
1704   }
1705 
1706   ierr = PetscMalloc1(coi[pon]+1,&coj);CHKERRQ(ierr);
1707   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1708   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr); /* must destroy to get a new one for C */
1709 
1710   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1711   if (afill_tmp > afill) afill = afill_tmp;
1712 
1713   /* send j-array (coj) of Co to other processors */
1714   /*----------------------------------------------*/
1715   /* determine row ownership */
1716   ierr = PetscNew(&merge);CHKERRQ(ierr);
1717   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1718 
1719   merge->rowmap->n  = pn;
1720   merge->rowmap->bs = 1;
1721 
1722   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1723   owners = merge->rowmap->range;
1724 
1725   /* determine the number of messages to send, their lengths */
1726   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
1727   ierr = PetscMalloc1(size,&merge->len_s);CHKERRQ(ierr);
1728 
1729   len_s        = merge->len_s;
1730   merge->nsend = 0;
1731 
1732   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
1733   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1734 
1735   proc = 0;
1736   for (i=0; i<pon; i++) {
1737     while (prmap[i] >= owners[proc+1]) proc++;
1738     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1739     len_s[proc] += coi[i+1] - coi[i];
1740   }
1741 
1742   len          = 0; /* max length of buf_si[] */
1743   owners_co[0] = 0;
1744   for (proc=0; proc<size; proc++) {
1745     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1746     if (len_si[proc]) {
1747       merge->nsend++;
1748       len_si[proc] = 2*(len_si[proc] + 1);
1749       len         += len_si[proc];
1750     }
1751   }
1752 
1753   /* determine the number and length of messages to receive for coi and coj  */
1754   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1755   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1756 
1757   /* post the Irecv and Isend of coj */
1758   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1759   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1760   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
1761   for (proc=0, k=0; proc<size; proc++) {
1762     if (!len_s[proc]) continue;
1763     i    = owners_co[proc];
1764     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1765     k++;
1766   }
1767 
1768   /* receives and sends of coj are complete */
1769   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
1770   for (i=0; i<merge->nrecv; i++) {
1771     PetscMPIInt icompleted;
1772     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1773   }
1774   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1775   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1776 
1777   /* add received column indices into table to update Armax */
1778   for (k=0; k<merge->nrecv; k++) {/* k-th received message */
1779     Jptr = buf_rj[k];
1780     for (j=0; j<merge->len_r[k]; j++) {
1781       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
1782     }
1783   }
1784   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
1785 
1786   /* send and recv coi */
1787   /*-------------------*/
1788   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1789   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1790   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1791   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1792   for (proc=0,k=0; proc<size; proc++) {
1793     if (!len_s[proc]) continue;
1794     /* form outgoing message for i-structure:
1795          buf_si[0]:                 nrows to be sent
1796                [1:nrows]:           row index (global)
1797                [nrows+1:2*nrows+1]: i-structure index
1798     */
1799     /*-------------------------------------------*/
1800     nrows       = len_si[proc]/2 - 1;
1801     buf_si_i    = buf_si + nrows+1;
1802     buf_si[0]   = nrows;
1803     buf_si_i[0] = 0;
1804     nrows       = 0;
1805     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1806       nzi               = coi[i+1] - coi[i];
1807       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1808       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1809       nrows++;
1810     }
1811     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1812     k++;
1813     buf_si += len_si[proc];
1814   }
1815   i = merge->nrecv;
1816   while (i--) {
1817     PetscMPIInt icompleted;
1818     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1819   }
1820   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1821   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1822   ierr = PetscFree(len_si);CHKERRQ(ierr);
1823   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1824   ierr = PetscFree(swaits);CHKERRQ(ierr);
1825   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1826   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1827 
1828   /* compute the local portion of C (mpi mat) */
1829   /*------------------------------------------*/
1830   /* allocate bi array and free space for accumulating nonzero column info */
1831   ierr  = PetscMalloc1(pn+1,&bi);CHKERRQ(ierr);
1832   bi[0] = 0;
1833 
1834   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
1835   nnz           = fill*(pdti[pn] + poti[pon] + ai[am]);
1836   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1837   current_space = free_space;
1838 
1839   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1840   for (k=0; k<merge->nrecv; k++) {
1841     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1842     nrows       = *buf_ri_k[k];
1843     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1844     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th recieved i-structure  */
1845   }
1846 
1847   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
1848   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1849   rmax = 0;
1850   for (i=0; i<pn; i++) {
1851     /* add pdt[i,:]*AP into lnk */
1852     pnz = pdti[i+1] - pdti[i];
1853     ptJ = pdtj + pdti[i];
1854     for (j=0; j<pnz; j++) {
1855       row  = ptJ[j];  /* row of AP == col of Pt */
1856       anz  = ai[row+1] - ai[row];
1857       Jptr = aj + ai[row];
1858       /* add non-zero cols of AP into the sorted linked list lnk */
1859       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1860     }
1861 
1862     /* add received col data into lnk */
1863     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1864       if (i == *nextrow[k]) { /* i-th row */
1865         nzi  = *(nextci[k]+1) - *nextci[k];
1866         Jptr = buf_rj[k] + *nextci[k];
1867         ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
1868         nextrow[k]++; nextci[k]++;
1869       }
1870     }
1871     nnz = lnk[0];
1872 
1873     /* if free space is not available, make more free space */
1874     if (current_space->local_remaining<nnz) {
1875       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1876       nspacedouble++;
1877     }
1878     /* copy data into free space, then initialize lnk */
1879     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1880     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1881 
1882     current_space->array           += nnz;
1883     current_space->local_used      += nnz;
1884     current_space->local_remaining -= nnz;
1885 
1886     bi[i+1] = bi[i] + nnz;
1887     if (nnz > rmax) rmax = nnz;
1888   }
1889   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1890 
1891   ierr      = PetscMalloc1(bi[pn]+1,&bj);CHKERRQ(ierr);
1892   ierr      = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1893   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
1894   if (afill_tmp > afill) afill = afill_tmp;
1895   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
1896   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
1897 
1898   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1899   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1900 
1901   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
1902   /*----------------------------------------------------------------------------------*/
1903   ierr = PetscCalloc1(rmax+1,&vals);CHKERRQ(ierr);
1904 
1905   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1906   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1907   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
1908   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1909   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1910   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1911   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
1912   for (i=0; i<pn; i++) {
1913     row  = i + rstart;
1914     nnz  = bi[i+1] - bi[i];
1915     Jptr = bj + bi[i];
1916     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
1917   }
1918   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1919   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1920   ierr = PetscFree(vals);CHKERRQ(ierr);
1921 
1922   merge->bi        = bi;
1923   merge->bj        = bj;
1924   merge->coi       = coi;
1925   merge->coj       = coj;
1926   merge->buf_ri    = buf_ri;
1927   merge->buf_rj    = buf_rj;
1928   merge->owners_co = owners_co;
1929   merge->destroy   = Cmpi->ops->destroy;
1930   merge->duplicate = Cmpi->ops->duplicate;
1931 
1932   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ;
1933   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
1934   Cmpi->ops->duplicate               = MatDuplicate_MPIAIJ_MatPtAP;
1935 
1936   /* attach the supporting struct to Cmpi for reuse */
1937   c = (Mat_MPIAIJ*)Cmpi->data;
1938 
1939   c->ptap     = ptap;
1940   ptap->api   = NULL;
1941   ptap->apj   = NULL;
1942   ptap->merge = merge;
1943   ptap->rmax  = rmax;
1944   ptap->apa   = NULL;
1945 
1946   *C = Cmpi;
1947 #if defined(PETSC_USE_INFO)
1948   if (bi[pn] != 0) {
1949     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1950     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
1951   } else {
1952     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1953   }
1954 #endif
1955   PetscFunctionReturn(0);
1956 }
1957