xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision a44bcdf57b2bdc3efd71d40abec79a20d8d3f002)
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 = 6*(p_loc->rmax + (PetscInt)(1.e-2*pN));
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   PetscReal          afill;
712   PetscScalar        *apa;
713   PetscTable         ta;
714 
715   PetscFunctionBegin;
716   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
717   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
718 
719   /* create struct Mat_PtAPMPI and attached it to C later */
720   ierr = PetscNew(&ptap);CHKERRQ(ierr);
721 
722   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
723   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
724 
725   /* get P_loc by taking all local rows of P */
726   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
727 
728   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
729   pi_loc = p_loc->i; pj_loc = p_loc->j;
730   if (size > 1) {
731     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
732     pi_oth = p_oth->i; pj_oth = p_oth->j;
733   } else {
734     p_oth  = NULL;
735     pi_oth = NULL; pj_oth = NULL;
736   }
737 
738   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
739   /*-------------------------------------------------------------------*/
740   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
741   ptap->api = api;
742   api[0]    = 0;
743 
744   /* create and initialize a linked list */
745   Crmax = 6*(p_loc->rmax + (PetscInt)(1.e-2*pN)); /* expected Crmax */
746   if (Crmax > pN) Crmax = pN;
747   ierr = PetscTableCreate(Crmax,pN,&ta);CHKERRQ(ierr);
748   MatRowMergeMax_SeqAIJ(p_loc,ptap->P_loc->rmap->N,ta);
749   MatRowMergeMax_SeqAIJ(p_oth,ptap->P_oth->rmap->N,ta);
750   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
751   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
752 
753   ierr = PetscLLCondensedCreate_Scalable(Crmax,&lnk);CHKERRQ(ierr);
754 
755   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
756   ierr = PetscFreeSpaceGet((PetscInt)(fill*(adi[am]+aoi[am]+pi_loc[pm])),&free_space);CHKERRQ(ierr);
757 
758   current_space = free_space;
759 
760   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
761   for (i=0; i<am; i++) {
762     /* diagonal portion of A */
763     nzi = adi[i+1] - adi[i];
764     for (j=0; j<nzi; j++) {
765       row  = *adj++;
766       pnz  = pi_loc[row+1] - pi_loc[row];
767       Jptr = pj_loc + pi_loc[row];
768       /* add non-zero cols of P into the sorted linked list lnk */
769       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
770     }
771     /* off-diagonal portion of A */
772     nzi = aoi[i+1] - aoi[i];
773     for (j=0; j<nzi; j++) {
774       row  = *aoj++;
775       pnz  = pi_oth[row+1] - pi_oth[row];
776       Jptr = pj_oth + pi_oth[row];
777       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
778     }
779 
780     apnz     = *lnk;
781     api[i+1] = api[i] + apnz;
782     if (apnz > apnz_max) apnz_max = 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(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 #undef __FUNCT__
860 #define __FUNCT__ "MatTransposeMatMult_MPIAIJ_MPIAIJ"
861 PetscErrorCode MatTransposeMatMult_MPIAIJ_MPIAIJ(Mat P,Mat A,MatReuse scall,PetscReal fill,Mat *C)
862 {
863   PetscErrorCode ierr;
864   const char     *algTypes[3] = {"scalable","nonscalable","matmatmult"};
865   PetscInt       alg=0; /* set default algorithm */
866 
867   PetscFunctionBegin;
868   if (scall == MAT_INITIAL_MATRIX) {
869     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
870     ierr = PetscOptionsEList("-mattransposematmult_via","Algorithmic approach","MatTransposeMatMult",algTypes,3,algTypes[0],&alg,NULL);CHKERRQ(ierr);
871     ierr = PetscOptionsEnd();CHKERRQ(ierr);
872 
873     ierr = PetscLogEventBegin(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
874     switch (alg) {
875     case 1:
876       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(P,A,fill,C);CHKERRQ(ierr);
877       break;
878     case 2:
879     {
880       Mat         Pt;
881       Mat_PtAPMPI *ptap;
882       Mat_MPIAIJ  *c;
883       ierr = MatTranspose(P,MAT_INITIAL_MATRIX,&Pt);CHKERRQ(ierr);
884       ierr = MatMatMult(Pt,A,MAT_INITIAL_MATRIX,fill,C);CHKERRQ(ierr);
885       c        = (Mat_MPIAIJ*)(*C)->data;
886       ptap     = c->ptap;
887       ptap->Pt = Pt;
888       (*C)->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult;
889       PetscFunctionReturn(0);
890     }
891       break;
892     default:
893       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(P,A,fill,C);CHKERRQ(ierr);
894       break;
895     }
896     ierr = PetscLogEventEnd(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
897   }
898   ierr = PetscLogEventBegin(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
899   ierr = (*(*C)->ops->mattransposemultnumeric)(P,A,*C);CHKERRQ(ierr);
900   ierr = PetscLogEventEnd(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
901   PetscFunctionReturn(0);
902 }
903 
904 /* This routine only works when scall=MAT_REUSE_MATRIX! */
905 #undef __FUNCT__
906 #define __FUNCT__ "MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult"
907 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult(Mat P,Mat A,Mat C)
908 {
909   PetscErrorCode ierr;
910   Mat_MPIAIJ     *c=(Mat_MPIAIJ*)C->data;
911   Mat_PtAPMPI    *ptap= c->ptap;
912   Mat            Pt=ptap->Pt;
913 
914   PetscFunctionBegin;
915   ierr = MatTranspose(P,MAT_REUSE_MATRIX,&Pt);CHKERRQ(ierr);
916   ierr = MatMatMultNumeric(Pt,A,C);CHKERRQ(ierr);
917   PetscFunctionReturn(0);
918 }
919 
920 /* Non-scalable version, use dense axpy */
921 #undef __FUNCT__
922 #define __FUNCT__ "MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable"
923 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,Mat C)
924 {
925   PetscErrorCode      ierr;
926   Mat_Merge_SeqsToMPI *merge;
927   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
928   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
929   Mat_PtAPMPI         *ptap;
930   PetscInt            *adj,*aJ;
931   PetscInt            i,j,k,anz,pnz,row,*cj;
932   MatScalar           *ada,*aval,*ca,valtmp;
933   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
934   MPI_Comm            comm;
935   PetscMPIInt         size,rank,taga,*len_s;
936   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
937   PetscInt            **buf_ri,**buf_rj;
938   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
939   MPI_Request         *s_waits,*r_waits;
940   MPI_Status          *status;
941   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
942   PetscInt            *ai,*aj,*coi,*coj;
943   PetscInt            *poJ,*pdJ;
944   Mat                 A_loc;
945   Mat_SeqAIJ          *a_loc;
946 
947   PetscFunctionBegin;
948   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
949   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
950   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
951 
952   ptap  = c->ptap;
953   merge = ptap->merge;
954 
955   /* 2) compute numeric C_seq = P_loc^T*A_loc*P - dominating part */
956   /*--------------------------------------------------------------*/
957   /* get data from symbolic products */
958   coi  = merge->coi; coj = merge->coj;
959   ierr = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
960 
961   bi     = merge->bi; bj = merge->bj;
962   owners = merge->rowmap->range;
963   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
964 
965   /* get A_loc by taking all local rows of A */
966   A_loc = ptap->A_loc;
967   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
968   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
969   ai    = a_loc->i;
970   aj    = a_loc->j;
971 
972   ierr = PetscCalloc1(A->cmap->N,&aval);CHKERRQ(ierr); /* non-scalable!!! */
973 
974   for (i=0; i<am; i++) {
975     /* 2-a) put A[i,:] to dense array aval */
976     anz = ai[i+1] - ai[i];
977     adj = aj + ai[i];
978     ada = a_loc->a + ai[i];
979     for (j=0; j<anz; j++) {
980       aval[adj[j]] = ada[j];
981     }
982 
983     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
984     /*--------------------------------------------------------------*/
985     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
986     pnz = po->i[i+1] - po->i[i];
987     poJ = po->j + po->i[i];
988     pA  = po->a + po->i[i];
989     for (j=0; j<pnz; j++) {
990       row = poJ[j];
991       cnz = coi[row+1] - coi[row];
992       cj  = coj + coi[row];
993       ca  = coa + coi[row];
994       /* perform dense axpy */
995       valtmp = pA[j];
996       for (k=0; k<cnz; k++) {
997         ca[k] += valtmp*aval[cj[k]];
998       }
999       ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1000     }
1001 
1002     /* put the value into Cd (diagonal part) */
1003     pnz = pd->i[i+1] - pd->i[i];
1004     pdJ = pd->j + pd->i[i];
1005     pA  = pd->a + pd->i[i];
1006     for (j=0; j<pnz; j++) {
1007       row = pdJ[j];
1008       cnz = bi[row+1] - bi[row];
1009       cj  = bj + bi[row];
1010       ca  = ba + bi[row];
1011       /* perform dense axpy */
1012       valtmp = pA[j];
1013       for (k=0; k<cnz; k++) {
1014         ca[k] += valtmp*aval[cj[k]];
1015       }
1016       ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1017     }
1018 
1019     /* zero the current row of Pt*A */
1020     aJ = aj + ai[i];
1021     for (k=0; k<anz; k++) aval[aJ[k]] = 0.0;
1022   }
1023 
1024   /* 3) send and recv matrix values coa */
1025   /*------------------------------------*/
1026   buf_ri = merge->buf_ri;
1027   buf_rj = merge->buf_rj;
1028   len_s  = merge->len_s;
1029   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1030   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1031 
1032   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1033   for (proc=0,k=0; proc<size; proc++) {
1034     if (!len_s[proc]) continue;
1035     i    = merge->owners_co[proc];
1036     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1037     k++;
1038   }
1039   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1040   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1041 
1042   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1043   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1044   ierr = PetscFree(coa);CHKERRQ(ierr);
1045 
1046   /* 4) insert local Cseq and received values into Cmpi */
1047   /*----------------------------------------------------*/
1048   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1049   for (k=0; k<merge->nrecv; k++) {
1050     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1051     nrows       = *(buf_ri_k[k]);
1052     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1053     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1054   }
1055 
1056   for (i=0; i<cm; i++) {
1057     row  = owners[rank] + i; /* global row index of C_seq */
1058     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1059     ba_i = ba + bi[i];
1060     bnz  = bi[i+1] - bi[i];
1061     /* add received vals into ba */
1062     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1063       /* i-th row */
1064       if (i == *nextrow[k]) {
1065         cnz    = *(nextci[k]+1) - *nextci[k];
1066         cj     = buf_rj[k] + *(nextci[k]);
1067         ca     = abuf_r[k] + *(nextci[k]);
1068         nextcj = 0;
1069         for (j=0; nextcj<cnz; j++) {
1070           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1071             ba_i[j] += ca[nextcj++];
1072           }
1073         }
1074         nextrow[k]++; nextci[k]++;
1075         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1076       }
1077     }
1078     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1079   }
1080   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1081   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1082 
1083   ierr = PetscFree(ba);CHKERRQ(ierr);
1084   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1085   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1086   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1087   ierr = PetscFree(aval);CHKERRQ(ierr);
1088   PetscFunctionReturn(0);
1089 }
1090 
1091 PetscErrorCode MatDuplicate_MPIAIJ_MatPtAP(Mat, MatDuplicateOption,Mat*);
1092 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1093 #undef __FUNCT__
1094 #define __FUNCT__ "MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable"
1095 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,PetscReal fill,Mat *C)
1096 {
1097   PetscErrorCode      ierr;
1098   Mat                 Cmpi,A_loc,POt,PDt;
1099   Mat_PtAPMPI         *ptap;
1100   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1101   Mat_MPIAIJ          *p        =(Mat_MPIAIJ*)P->data,*c;
1102   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1103   PetscInt            nnz;
1104   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1105   PetscInt            am=A->rmap->n,pn=P->cmap->n;
1106   PetscBT             lnkbt;
1107   MPI_Comm            comm;
1108   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1109   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1110   PetscInt            len,proc,*dnz,*onz,*owners;
1111   PetscInt            nzi,*bi,*bj;
1112   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1113   MPI_Request         *swaits,*rwaits;
1114   MPI_Status          *sstatus,rstatus;
1115   Mat_Merge_SeqsToMPI *merge;
1116   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1117   PetscReal           afill  =1.0,afill_tmp;
1118   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N;
1119   PetscScalar         *vals;
1120   Mat_SeqAIJ          *a_loc, *pdt,*pot;
1121 
1122   PetscFunctionBegin;
1123   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1124   /* check if matrix local sizes are compatible */
1125   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend) {
1126     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);
1127   }
1128 
1129   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1130   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1131 
1132   /* create struct Mat_PtAPMPI and attached it to C later */
1133   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1134 
1135   /* get A_loc by taking all local rows of A */
1136   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1137 
1138   ptap->A_loc = A_loc;
1139 
1140   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1141   ai    = a_loc->i;
1142   aj    = a_loc->j;
1143 
1144   /* determine symbolic Co=(p->B)^T*A - send to others */
1145   /*----------------------------------------------------*/
1146   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1147   pdt  = (Mat_SeqAIJ*)PDt->data;
1148   pdti = pdt->i; pdtj = pdt->j;
1149 
1150   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1151   pot  = (Mat_SeqAIJ*)POt->data;
1152   poti = pot->i; potj = pot->j;
1153 
1154   /* then, compute symbolic Co = (p->B)^T*A */
1155   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors >= (num of nonzero rows of C_seq) - pn */
1156   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
1157   coi[0] = 0;
1158 
1159   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1160   nnz           = fill*(poti[pon] + ai[am]);
1161   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1162   current_space = free_space;
1163 
1164   /* create and initialize a linked list */
1165   ierr = PetscLLCondensedCreate(aN,aN,&lnk,&lnkbt);CHKERRQ(ierr);
1166 
1167   for (i=0; i<pon; i++) {
1168     pnz = poti[i+1] - poti[i];
1169     ptJ = potj + poti[i];
1170     for (j=0; j<pnz; j++) {
1171       row  = ptJ[j]; /* row of A_loc == col of Pot */
1172       anz  = ai[row+1] - ai[row];
1173       Jptr = aj + ai[row];
1174       /* add non-zero cols of AP into the sorted linked list lnk */
1175       ierr = PetscLLCondensedAddSorted(anz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1176     }
1177     nnz = lnk[0];
1178 
1179     /* If free space is not available, double the total space in the list */
1180     if (current_space->local_remaining<nnz) {
1181       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1182       nspacedouble++;
1183     }
1184 
1185     /* Copy data into free space, and zero out denserows */
1186     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1187 
1188     current_space->array           += nnz;
1189     current_space->local_used      += nnz;
1190     current_space->local_remaining -= nnz;
1191 
1192     coi[i+1] = coi[i] + nnz;
1193   }
1194 
1195   ierr = PetscMalloc1(coi[pon]+1,&coj);CHKERRQ(ierr);
1196   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1197 
1198   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1199   if (afill_tmp > afill) afill = afill_tmp;
1200 
1201   /* send j-array (coj) of Co to other processors */
1202   /*----------------------------------------------*/
1203   /* determine row ownership */
1204   ierr = PetscNew(&merge);CHKERRQ(ierr);
1205   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1206 
1207   merge->rowmap->n  = pn;
1208   merge->rowmap->bs = 1;
1209 
1210   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1211   owners = merge->rowmap->range;
1212 
1213   /* determine the number of messages to send, their lengths */
1214   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
1215   ierr = PetscMalloc1(size,&merge->len_s);CHKERRQ(ierr);
1216 
1217   len_s        = merge->len_s;
1218   merge->nsend = 0;
1219 
1220   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
1221   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1222 
1223   proc = 0;
1224   for (i=0; i<pon; i++) {
1225     while (prmap[i] >= owners[proc+1]) proc++;
1226     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1227     len_s[proc] += coi[i+1] - coi[i];
1228   }
1229 
1230   len          = 0; /* max length of buf_si[] */
1231   owners_co[0] = 0;
1232   for (proc=0; proc<size; proc++) {
1233     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1234     if (len_si[proc]) {
1235       merge->nsend++;
1236       len_si[proc] = 2*(len_si[proc] + 1);
1237       len         += len_si[proc];
1238     }
1239   }
1240 
1241   /* determine the number and length of messages to receive for coi and coj  */
1242   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1243   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1244 
1245   /* post the Irecv and Isend of coj */
1246   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1247   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1248   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
1249   for (proc=0, k=0; proc<size; proc++) {
1250     if (!len_s[proc]) continue;
1251     i    = owners_co[proc];
1252     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1253     k++;
1254   }
1255 
1256   /* receives and sends of coj are complete */
1257   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
1258   for (i=0; i<merge->nrecv; i++) {
1259     PetscMPIInt icompleted;
1260     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1261   }
1262   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1263   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1264 
1265   /* send and recv coi */
1266   /*-------------------*/
1267   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1268   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1269   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1270   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1271   for (proc=0,k=0; proc<size; proc++) {
1272     if (!len_s[proc]) continue;
1273     /* form outgoing message for i-structure:
1274          buf_si[0]:                 nrows to be sent
1275                [1:nrows]:           row index (global)
1276                [nrows+1:2*nrows+1]: i-structure index
1277     */
1278     /*-------------------------------------------*/
1279     nrows       = len_si[proc]/2 - 1;
1280     buf_si_i    = buf_si + nrows+1;
1281     buf_si[0]   = nrows;
1282     buf_si_i[0] = 0;
1283     nrows       = 0;
1284     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1285       nzi               = coi[i+1] - coi[i];
1286       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi; /* i-structure */
1287       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1288       nrows++;
1289     }
1290     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1291     k++;
1292     buf_si += len_si[proc];
1293   }
1294   i = merge->nrecv;
1295   while (i--) {
1296     PetscMPIInt icompleted;
1297     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1298   }
1299   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1300   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1301   ierr = PetscFree(len_si);CHKERRQ(ierr);
1302   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1303   ierr = PetscFree(swaits);CHKERRQ(ierr);
1304   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1305   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1306 
1307   /* compute the local portion of C (mpi mat) */
1308   /*------------------------------------------*/
1309   /* allocate bi array and free space for accumulating nonzero column info */
1310   ierr  = PetscMalloc1(pn+1,&bi);CHKERRQ(ierr);
1311   bi[0] = 0;
1312 
1313   /* set initial free space to be fill*(nnz(P) + nnz(A)) */
1314   nnz           = fill*(pdti[pn] + poti[pon] + ai[am]);
1315   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1316   current_space = free_space;
1317 
1318   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1319   for (k=0; k<merge->nrecv; k++) {
1320     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1321     nrows       = *buf_ri_k[k];
1322     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1323     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1324   }
1325 
1326   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1327   rmax = 0;
1328   for (i=0; i<pn; i++) {
1329     /* add pdt[i,:]*AP into lnk */
1330     pnz = pdti[i+1] - pdti[i];
1331     ptJ = pdtj + pdti[i];
1332     for (j=0; j<pnz; j++) {
1333       row  = ptJ[j];  /* row of AP == col of Pt */
1334       anz  = ai[row+1] - ai[row];
1335       Jptr = aj + ai[row];
1336       /* add non-zero cols of AP into the sorted linked list lnk */
1337       ierr = PetscLLCondensedAddSorted(anz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1338     }
1339 
1340     /* add received col data into lnk */
1341     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1342       if (i == *nextrow[k]) { /* i-th row */
1343         nzi  = *(nextci[k]+1) - *nextci[k];
1344         Jptr = buf_rj[k] + *nextci[k];
1345         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1346         nextrow[k]++; nextci[k]++;
1347       }
1348     }
1349     nnz = lnk[0];
1350 
1351     /* if free space is not available, make more free space */
1352     if (current_space->local_remaining<nnz) {
1353       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1354       nspacedouble++;
1355     }
1356     /* copy data into free space, then initialize lnk */
1357     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1358     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1359 
1360     current_space->array           += nnz;
1361     current_space->local_used      += nnz;
1362     current_space->local_remaining -= nnz;
1363 
1364     bi[i+1] = bi[i] + nnz;
1365     if (nnz > rmax) rmax = nnz;
1366   }
1367   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1368 
1369   ierr = PetscMalloc1(bi[pn]+1,&bj);CHKERRQ(ierr);
1370   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1371 
1372   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
1373   if (afill_tmp > afill) afill = afill_tmp;
1374   ierr = PetscLLCondensedDestroy(lnk,lnkbt);CHKERRQ(ierr);
1375   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1376   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1377 
1378   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
1379   /*----------------------------------------------------------------------------------*/
1380   ierr = PetscCalloc1(rmax+1,&vals);CHKERRQ(ierr);
1381 
1382   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1383   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1384   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
1385   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1386   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1387   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1388   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
1389   for (i=0; i<pn; i++) {
1390     row  = i + rstart;
1391     nnz  = bi[i+1] - bi[i];
1392     Jptr = bj + bi[i];
1393     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
1394   }
1395   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1396   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1397   ierr = PetscFree(vals);CHKERRQ(ierr);
1398 
1399   merge->bi        = bi;
1400   merge->bj        = bj;
1401   merge->coi       = coi;
1402   merge->coj       = coj;
1403   merge->buf_ri    = buf_ri;
1404   merge->buf_rj    = buf_rj;
1405   merge->owners_co = owners_co;
1406   merge->destroy   = Cmpi->ops->destroy;
1407   merge->duplicate = Cmpi->ops->duplicate;
1408 
1409   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1410   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
1411   Cmpi->ops->duplicate               = MatDuplicate_MPIAIJ_MatPtAP;
1412 
1413   /* attach the supporting struct to Cmpi for reuse */
1414   c           = (Mat_MPIAIJ*)Cmpi->data;
1415   c->ptap     = ptap;
1416   ptap->api   = NULL;
1417   ptap->apj   = NULL;
1418   ptap->merge = merge;
1419   ptap->rmax  = rmax;
1420 
1421   *C = Cmpi;
1422 #if defined(PETSC_USE_INFO)
1423   if (bi[pn] != 0) {
1424     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1425     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
1426   } else {
1427     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1428   }
1429 #endif
1430   PetscFunctionReturn(0);
1431 }
1432 
1433 #undef __FUNCT__
1434 #define __FUNCT__ "MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ"
1435 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ(Mat P,Mat A,Mat C)
1436 {
1437   PetscErrorCode      ierr;
1438   Mat_Merge_SeqsToMPI *merge;
1439   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1440   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
1441   Mat_PtAPMPI         *ptap;
1442   PetscInt            *adj;
1443   PetscInt            i,j,k,anz,pnz,row,*cj,nexta;
1444   MatScalar           *ada,*ca,valtmp;
1445   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
1446   MPI_Comm            comm;
1447   PetscMPIInt         size,rank,taga,*len_s;
1448   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
1449   PetscInt            **buf_ri,**buf_rj;
1450   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
1451   MPI_Request         *s_waits,*r_waits;
1452   MPI_Status          *status;
1453   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
1454   PetscInt            *ai,*aj,*coi,*coj;
1455   PetscInt            *poJ,*pdJ;
1456   Mat                 A_loc;
1457   Mat_SeqAIJ          *a_loc;
1458 
1459   PetscFunctionBegin;
1460   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1461   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1462   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1463 
1464   ptap  = c->ptap;
1465   merge = ptap->merge;
1466 
1467   /* 2) compute numeric C_seq = P_loc^T*A_loc */
1468   /*------------------------------------------*/
1469   /* get data from symbolic products */
1470   coi    = merge->coi; coj = merge->coj;
1471   ierr   = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
1472   bi     = merge->bi; bj = merge->bj;
1473   owners = merge->rowmap->range;
1474   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
1475 
1476   /* get A_loc by taking all local rows of A */
1477   A_loc = ptap->A_loc;
1478   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
1479   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1480   ai    = a_loc->i;
1481   aj    = a_loc->j;
1482 
1483   for (i=0; i<am; i++) {
1484     anz = ai[i+1] - ai[i];
1485     adj = aj + ai[i];
1486     ada = a_loc->a + ai[i];
1487 
1488     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1489     /*-------------------------------------------------------------*/
1490     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1491     pnz = po->i[i+1] - po->i[i];
1492     poJ = po->j + po->i[i];
1493     pA  = po->a + po->i[i];
1494     for (j=0; j<pnz; j++) {
1495       row = poJ[j];
1496       cj  = coj + coi[row];
1497       ca  = coa + coi[row];
1498       /* perform sparse axpy */
1499       nexta  = 0;
1500       valtmp = pA[j];
1501       for (k=0; nexta<anz; k++) {
1502         if (cj[k] == adj[nexta]) {
1503           ca[k] += valtmp*ada[nexta];
1504           nexta++;
1505         }
1506       }
1507       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1508     }
1509 
1510     /* put the value into Cd (diagonal part) */
1511     pnz = pd->i[i+1] - pd->i[i];
1512     pdJ = pd->j + pd->i[i];
1513     pA  = pd->a + pd->i[i];
1514     for (j=0; j<pnz; j++) {
1515       row = pdJ[j];
1516       cj  = bj + bi[row];
1517       ca  = ba + bi[row];
1518       /* perform sparse axpy */
1519       nexta  = 0;
1520       valtmp = pA[j];
1521       for (k=0; nexta<anz; k++) {
1522         if (cj[k] == adj[nexta]) {
1523           ca[k] += valtmp*ada[nexta];
1524           nexta++;
1525         }
1526       }
1527       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1528     }
1529   }
1530 
1531   /* 3) send and recv matrix values coa */
1532   /*------------------------------------*/
1533   buf_ri = merge->buf_ri;
1534   buf_rj = merge->buf_rj;
1535   len_s  = merge->len_s;
1536   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1537   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1538 
1539   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1540   for (proc=0,k=0; proc<size; proc++) {
1541     if (!len_s[proc]) continue;
1542     i    = merge->owners_co[proc];
1543     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1544     k++;
1545   }
1546   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1547   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1548 
1549   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1550   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1551   ierr = PetscFree(coa);CHKERRQ(ierr);
1552 
1553   /* 4) insert local Cseq and received values into Cmpi */
1554   /*----------------------------------------------------*/
1555   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1556   for (k=0; k<merge->nrecv; k++) {
1557     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1558     nrows       = *(buf_ri_k[k]);
1559     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1560     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1561   }
1562 
1563   for (i=0; i<cm; i++) {
1564     row  = owners[rank] + i; /* global row index of C_seq */
1565     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1566     ba_i = ba + bi[i];
1567     bnz  = bi[i+1] - bi[i];
1568     /* add received vals into ba */
1569     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1570       /* i-th row */
1571       if (i == *nextrow[k]) {
1572         cnz    = *(nextci[k]+1) - *nextci[k];
1573         cj     = buf_rj[k] + *(nextci[k]);
1574         ca     = abuf_r[k] + *(nextci[k]);
1575         nextcj = 0;
1576         for (j=0; nextcj<cnz; j++) {
1577           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1578             ba_i[j] += ca[nextcj++];
1579           }
1580         }
1581         nextrow[k]++; nextci[k]++;
1582         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1583       }
1584     }
1585     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1586   }
1587   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1588   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1589 
1590   ierr = PetscFree(ba);CHKERRQ(ierr);
1591   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1592   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1593   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1594   PetscFunctionReturn(0);
1595 }
1596 
1597 PetscErrorCode MatDuplicate_MPIAIJ_MatPtAP(Mat, MatDuplicateOption,Mat*);
1598 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ();
1599    differ from MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable in using LLCondensedCreate_Scalable() */
1600 #undef __FUNCT__
1601 #define __FUNCT__ "MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ"
1602 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P,Mat A,PetscReal fill,Mat *C)
1603 {
1604   PetscErrorCode      ierr;
1605   Mat                 Cmpi,A_loc,POt,PDt;
1606   Mat_PtAPMPI         *ptap;
1607   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1608   Mat_MPIAIJ          *p        =(Mat_MPIAIJ*)P->data,*c;
1609   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1610   PetscInt            nnz;
1611   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1612   PetscInt            am  =A->rmap->n,pn=P->cmap->n;
1613   MPI_Comm            comm;
1614   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1615   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1616   PetscInt            len,proc,*dnz,*onz,*owners;
1617   PetscInt            nzi,*bi,*bj;
1618   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1619   MPI_Request         *swaits,*rwaits;
1620   MPI_Status          *sstatus,rstatus;
1621   Mat_Merge_SeqsToMPI *merge;
1622   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1623   PetscReal           afill  =1.0,afill_tmp;
1624   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Armax;
1625   PetscScalar         *vals;
1626   Mat_SeqAIJ          *a_loc,*pdt,*pot;
1627   PetscTable          ta;
1628 
1629   PetscFunctionBegin;
1630   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1631   /* check if matrix local sizes are compatible */
1632   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend) {
1633     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);
1634   }
1635 
1636   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1637   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1638 
1639   /* create struct Mat_PtAPMPI and attached it to C later */
1640   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1641 
1642   /* get A_loc by taking all local rows of A */
1643   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1644 
1645   ptap->A_loc = A_loc;
1646   a_loc       = (Mat_SeqAIJ*)(A_loc)->data;
1647   ai          = a_loc->i;
1648   aj          = a_loc->j;
1649 
1650   /* determine symbolic Co=(p->B)^T*A - send to others */
1651   /*----------------------------------------------------*/
1652   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1653   pdt  = (Mat_SeqAIJ*)PDt->data;
1654   pdti = pdt->i; pdtj = pdt->j;
1655 
1656   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1657   pot  = (Mat_SeqAIJ*)POt->data;
1658   poti = pot->i; potj = pot->j;
1659 
1660   /* then, compute symbolic Co = (p->B)^T*A */
1661   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1662                          >= (num of nonzero rows of C_seq) - pn */
1663   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
1664   coi[0] = 0;
1665 
1666   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1667   nnz           = fill*(poti[pon] + ai[am]);
1668   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1669   current_space = free_space;
1670 
1671   /* create and initialize a linked list */
1672   ierr = PetscTableCreate(2*a_loc->rmax,aN,&ta);CHKERRQ(ierr);
1673   MatRowMergeMax_SeqAIJ(a_loc,am,ta);
1674   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
1675   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
1676 
1677   for (i=0; i<pon; i++) {
1678     pnz = poti[i+1] - poti[i];
1679     ptJ = potj + poti[i];
1680     for (j=0; j<pnz; j++) {
1681       row  = ptJ[j]; /* row of A_loc == col of Pot */
1682       anz  = ai[row+1] - ai[row];
1683       Jptr = aj + ai[row];
1684       /* add non-zero cols of AP into the sorted linked list lnk */
1685       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1686     }
1687     nnz = lnk[0];
1688 
1689     /* If free space is not available, double the total space in the list */
1690     if (current_space->local_remaining<nnz) {
1691       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1692       nspacedouble++;
1693     }
1694 
1695     /* Copy data into free space, and zero out denserows */
1696     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1697 
1698     current_space->array           += nnz;
1699     current_space->local_used      += nnz;
1700     current_space->local_remaining -= nnz;
1701 
1702     coi[i+1] = coi[i] + nnz;
1703   }
1704 
1705   ierr = PetscMalloc1(coi[pon]+1,&coj);CHKERRQ(ierr);
1706   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1707   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr); /* must destroy to get a new one for C */
1708 
1709   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1710   if (afill_tmp > afill) afill = afill_tmp;
1711 
1712   /* send j-array (coj) of Co to other processors */
1713   /*----------------------------------------------*/
1714   /* determine row ownership */
1715   ierr = PetscNew(&merge);CHKERRQ(ierr);
1716   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1717 
1718   merge->rowmap->n  = pn;
1719   merge->rowmap->bs = 1;
1720 
1721   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1722   owners = merge->rowmap->range;
1723 
1724   /* determine the number of messages to send, their lengths */
1725   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
1726   ierr = PetscMalloc1(size,&merge->len_s);CHKERRQ(ierr);
1727 
1728   len_s        = merge->len_s;
1729   merge->nsend = 0;
1730 
1731   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
1732   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1733 
1734   proc = 0;
1735   for (i=0; i<pon; i++) {
1736     while (prmap[i] >= owners[proc+1]) proc++;
1737     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1738     len_s[proc] += coi[i+1] - coi[i];
1739   }
1740 
1741   len          = 0; /* max length of buf_si[] */
1742   owners_co[0] = 0;
1743   for (proc=0; proc<size; proc++) {
1744     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1745     if (len_si[proc]) {
1746       merge->nsend++;
1747       len_si[proc] = 2*(len_si[proc] + 1);
1748       len         += len_si[proc];
1749     }
1750   }
1751 
1752   /* determine the number and length of messages to receive for coi and coj  */
1753   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1754   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1755 
1756   /* post the Irecv and Isend of coj */
1757   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1758   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1759   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
1760   for (proc=0, k=0; proc<size; proc++) {
1761     if (!len_s[proc]) continue;
1762     i    = owners_co[proc];
1763     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1764     k++;
1765   }
1766 
1767   /* receives and sends of coj are complete */
1768   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
1769   for (i=0; i<merge->nrecv; i++) {
1770     PetscMPIInt icompleted;
1771     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1772   }
1773   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1774   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1775 
1776   /* add received column indices into table to update Armax */
1777   for (k=0; k<merge->nrecv; k++) {/* k-th received message */
1778     Jptr = buf_rj[k];
1779     for (j=0; j<merge->len_r[k]; j++) {
1780       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
1781     }
1782   }
1783   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
1784 
1785   /* send and recv coi */
1786   /*-------------------*/
1787   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1788   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1789   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1790   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1791   for (proc=0,k=0; proc<size; proc++) {
1792     if (!len_s[proc]) continue;
1793     /* form outgoing message for i-structure:
1794          buf_si[0]:                 nrows to be sent
1795                [1:nrows]:           row index (global)
1796                [nrows+1:2*nrows+1]: i-structure index
1797     */
1798     /*-------------------------------------------*/
1799     nrows       = len_si[proc]/2 - 1;
1800     buf_si_i    = buf_si + nrows+1;
1801     buf_si[0]   = nrows;
1802     buf_si_i[0] = 0;
1803     nrows       = 0;
1804     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1805       nzi               = coi[i+1] - coi[i];
1806       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1807       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1808       nrows++;
1809     }
1810     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1811     k++;
1812     buf_si += len_si[proc];
1813   }
1814   i = merge->nrecv;
1815   while (i--) {
1816     PetscMPIInt icompleted;
1817     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1818   }
1819   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1820   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1821   ierr = PetscFree(len_si);CHKERRQ(ierr);
1822   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1823   ierr = PetscFree(swaits);CHKERRQ(ierr);
1824   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1825   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1826 
1827   /* compute the local portion of C (mpi mat) */
1828   /*------------------------------------------*/
1829   /* allocate bi array and free space for accumulating nonzero column info */
1830   ierr  = PetscMalloc1(pn+1,&bi);CHKERRQ(ierr);
1831   bi[0] = 0;
1832 
1833   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
1834   nnz           = fill*(pdti[pn] + poti[pon] + ai[am]);
1835   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1836   current_space = free_space;
1837 
1838   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1839   for (k=0; k<merge->nrecv; k++) {
1840     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1841     nrows       = *buf_ri_k[k];
1842     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1843     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th recieved i-structure  */
1844   }
1845 
1846   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
1847   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1848   rmax = 0;
1849   for (i=0; i<pn; i++) {
1850     /* add pdt[i,:]*AP into lnk */
1851     pnz = pdti[i+1] - pdti[i];
1852     ptJ = pdtj + pdti[i];
1853     for (j=0; j<pnz; j++) {
1854       row  = ptJ[j];  /* row of AP == col of Pt */
1855       anz  = ai[row+1] - ai[row];
1856       Jptr = aj + ai[row];
1857       /* add non-zero cols of AP into the sorted linked list lnk */
1858       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1859     }
1860 
1861     /* add received col data into lnk */
1862     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1863       if (i == *nextrow[k]) { /* i-th row */
1864         nzi  = *(nextci[k]+1) - *nextci[k];
1865         Jptr = buf_rj[k] + *nextci[k];
1866         ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
1867         nextrow[k]++; nextci[k]++;
1868       }
1869     }
1870     nnz = lnk[0];
1871 
1872     /* if free space is not available, make more free space */
1873     if (current_space->local_remaining<nnz) {
1874       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1875       nspacedouble++;
1876     }
1877     /* copy data into free space, then initialize lnk */
1878     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1879     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1880 
1881     current_space->array           += nnz;
1882     current_space->local_used      += nnz;
1883     current_space->local_remaining -= nnz;
1884 
1885     bi[i+1] = bi[i] + nnz;
1886     if (nnz > rmax) rmax = nnz;
1887   }
1888   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1889 
1890   ierr      = PetscMalloc1(bi[pn]+1,&bj);CHKERRQ(ierr);
1891   ierr      = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1892   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
1893   if (afill_tmp > afill) afill = afill_tmp;
1894   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
1895   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
1896 
1897   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1898   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1899 
1900   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
1901   /*----------------------------------------------------------------------------------*/
1902   ierr = PetscCalloc1(rmax+1,&vals);CHKERRQ(ierr);
1903 
1904   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1905   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1906   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
1907   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1908   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1909   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1910   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
1911   for (i=0; i<pn; i++) {
1912     row  = i + rstart;
1913     nnz  = bi[i+1] - bi[i];
1914     Jptr = bj + bi[i];
1915     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
1916   }
1917   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1918   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1919   ierr = PetscFree(vals);CHKERRQ(ierr);
1920 
1921   merge->bi        = bi;
1922   merge->bj        = bj;
1923   merge->coi       = coi;
1924   merge->coj       = coj;
1925   merge->buf_ri    = buf_ri;
1926   merge->buf_rj    = buf_rj;
1927   merge->owners_co = owners_co;
1928   merge->destroy   = Cmpi->ops->destroy;
1929   merge->duplicate = Cmpi->ops->duplicate;
1930 
1931   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ;
1932   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
1933   Cmpi->ops->duplicate               = MatDuplicate_MPIAIJ_MatPtAP;
1934 
1935   /* attach the supporting struct to Cmpi for reuse */
1936   c = (Mat_MPIAIJ*)Cmpi->data;
1937 
1938   c->ptap     = ptap;
1939   ptap->api   = NULL;
1940   ptap->apj   = NULL;
1941   ptap->merge = merge;
1942   ptap->rmax  = rmax;
1943   ptap->apa   = NULL;
1944 
1945   *C = Cmpi;
1946 #if defined(PETSC_USE_INFO)
1947   if (bi[pn] != 0) {
1948     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1949     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
1950   } else {
1951     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1952   }
1953 #endif
1954   PetscFunctionReturn(0);
1955 }
1956