xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision 9467f45fc2f06212a563b445f167c8c6a3337d2e)
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=0; /* set default algorithm */
20 
21   PetscFunctionBegin;
22   if (scall == MAT_INITIAL_MATRIX) {
23     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
24     ierr = PetscOptionsEList("-matmatmult_via","Algorithmic approach","MatMatMult",algTypes,2,algTypes[0],&alg,NULL);CHKERRQ(ierr);
25     ierr = PetscOptionsEnd();CHKERRQ(ierr);
26 
27     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
28     switch (alg) {
29     case 1:
30       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(A,B,fill,C);CHKERRQ(ierr);
31       break;
32     default:
33       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ(A,B,fill,C);CHKERRQ(ierr);
34       break;
35     }
36     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
37   }
38   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
39   ierr = (*(*C)->ops->matmultnumeric)(A,B,*C);CHKERRQ(ierr);
40   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
41   PetscFunctionReturn(0);
42 }
43 
44 #undef __FUNCT__
45 #define __FUNCT__ "MatDestroy_MPIAIJ_MatMatMult"
46 PetscErrorCode MatDestroy_MPIAIJ_MatMatMult(Mat A)
47 {
48   PetscErrorCode ierr;
49   Mat_MPIAIJ     *a    = (Mat_MPIAIJ*)A->data;
50   Mat_PtAPMPI    *ptap = a->ptap;
51 
52   PetscFunctionBegin;
53   ierr = PetscFree2(ptap->startsj_s,ptap->startsj_r);CHKERRQ(ierr);
54   ierr = PetscFree(ptap->bufa);CHKERRQ(ierr);
55   ierr = MatDestroy(&ptap->P_loc);CHKERRQ(ierr);
56   ierr = MatDestroy(&ptap->P_oth);CHKERRQ(ierr);
57   ierr = MatDestroy(&ptap->Pt);CHKERRQ(ierr);
58   ierr = PetscFree(ptap->api);CHKERRQ(ierr);
59   ierr = PetscFree(ptap->apj);CHKERRQ(ierr);
60   ierr = PetscFree(ptap->apa);CHKERRQ(ierr);
61   ierr = ptap->destroy(A);CHKERRQ(ierr);
62   ierr = PetscFree(ptap);CHKERRQ(ierr);
63   PetscFunctionReturn(0);
64 }
65 
66 #undef __FUNCT__
67 #define __FUNCT__ "MatDuplicate_MPIAIJ_MatMatMult"
68 PetscErrorCode MatDuplicate_MPIAIJ_MatMatMult(Mat A, MatDuplicateOption op, Mat *M)
69 {
70   PetscErrorCode ierr;
71   Mat_MPIAIJ     *a    = (Mat_MPIAIJ*)A->data;
72   Mat_PtAPMPI    *ptap = a->ptap;
73 
74   PetscFunctionBegin;
75   ierr = (*ptap->duplicate)(A,op,M);CHKERRQ(ierr);
76 
77   (*M)->ops->destroy   = ptap->destroy;   /* = MatDestroy_MPIAIJ, *M doesn't duplicate A's special structure! */
78   (*M)->ops->duplicate = ptap->duplicate; /* = MatDuplicate_MPIAIJ */
79   PetscFunctionReturn(0);
80 }
81 
82 #undef __FUNCT__
83 #define __FUNCT__ "MatMatMultNumeric_MPIAIJ_MPIAIJ"
84 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
85 {
86   PetscErrorCode ierr;
87   Mat_MPIAIJ     *a  =(Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
88   Mat_SeqAIJ     *ad =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
89   Mat_SeqAIJ     *cd =(Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
90   PetscInt       *adi=ad->i,*adj,*aoi=ao->i,*aoj;
91   PetscScalar    *ada,*aoa,*cda=cd->a,*coa=co->a;
92   Mat_SeqAIJ     *p_loc,*p_oth;
93   PetscInt       *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
94   PetscScalar    *pa_loc,*pa_oth,*pa,*apa,valtmp,*ca;
95   PetscInt       cm   =C->rmap->n,anz,pnz;
96   Mat_PtAPMPI    *ptap=c->ptap;
97   PetscInt       *api,*apj,*apJ,i,j,k,row;
98   PetscInt       cstart=C->cmap->rstart;
99   PetscInt       cdnz,conz,k0,k1;
100   MPI_Comm       comm;
101   PetscMPIInt    size;
102 
103   PetscFunctionBegin;
104   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
105   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
106 
107   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
108   /*-----------------------------------------------------*/
109   /* update numerical values of P_oth and P_loc */
110   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
111   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
112 
113   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
114   /*----------------------------------------------------------*/
115   /* get data from symbolic products */
116   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
117   pi_loc=p_loc->i; pj_loc=p_loc->j; pa_loc=p_loc->a;
118   if (size >1) {
119     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
120     pi_oth=p_oth->i; pj_oth=p_oth->j; pa_oth=p_oth->a;
121   }
122 
123   /* get apa for storing dense row A[i,:]*P */
124   apa = ptap->apa;
125 
126   api = ptap->api;
127   apj = ptap->apj;
128   for (i=0; i<cm; i++) {
129     /* diagonal portion of A */
130     anz = adi[i+1] - adi[i];
131     adj = ad->j + adi[i];
132     ada = ad->a + adi[i];
133     for (j=0; j<anz; j++) {
134       row = adj[j];
135       pnz = pi_loc[row+1] - pi_loc[row];
136       pj  = pj_loc + pi_loc[row];
137       pa  = pa_loc + pi_loc[row];
138 
139       /* perform dense axpy */
140       valtmp = ada[j];
141       for (k=0; k<pnz; k++) {
142         apa[pj[k]] += valtmp*pa[k];
143       }
144       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
145     }
146 
147     /* off-diagonal portion of A */
148     anz = aoi[i+1] - aoi[i];
149     aoj = ao->j + aoi[i];
150     aoa = ao->a + aoi[i];
151     for (j=0; j<anz; j++) {
152       row = aoj[j];
153       pnz = pi_oth[row+1] - pi_oth[row];
154       pj  = pj_oth + pi_oth[row];
155       pa  = pa_oth + pi_oth[row];
156 
157       /* perform dense axpy */
158       valtmp = aoa[j];
159       for (k=0; k<pnz; k++) {
160         apa[pj[k]] += valtmp*pa[k];
161       }
162       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
163     }
164 
165     /* set values in C */
166     apJ  = apj + api[i];
167     cdnz = cd->i[i+1] - cd->i[i];
168     conz = co->i[i+1] - co->i[i];
169 
170     /* 1st off-diagoanl part of C */
171     ca = coa + co->i[i];
172     k  = 0;
173     for (k0=0; k0<conz; k0++) {
174       if (apJ[k] >= cstart) break;
175       ca[k0]      = apa[apJ[k]];
176       apa[apJ[k]] = 0.0;
177       k++;
178     }
179 
180     /* diagonal part of C */
181     ca = cda + cd->i[i];
182     for (k1=0; k1<cdnz; k1++) {
183       ca[k1]      = apa[apJ[k]];
184       apa[apJ[k]] = 0.0;
185       k++;
186     }
187 
188     /* 2nd off-diagoanl part of C */
189     ca = coa + co->i[i];
190     for (; k0<conz; k0++) {
191       ca[k0]      = apa[apJ[k]];
192       apa[apJ[k]] = 0.0;
193       k++;
194     }
195   }
196   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
197   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
198   PetscFunctionReturn(0);
199 }
200 
201 #undef __FUNCT__
202 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable"
203 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,PetscReal fill,Mat *C)
204 {
205   PetscErrorCode     ierr;
206   MPI_Comm           comm;
207   PetscMPIInt        size;
208   Mat                Cmpi;
209   Mat_PtAPMPI        *ptap;
210   PetscFreeSpaceList free_space=NULL,current_space=NULL;
211   Mat_MPIAIJ         *a        =(Mat_MPIAIJ*)A->data,*c;
212   Mat_SeqAIJ         *ad       =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
213   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
214   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
215   PetscInt           *lnk,i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi;
216   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
217   PetscBT            lnkbt;
218   PetscScalar        *apa;
219   PetscReal          afill;
220   PetscInt           nlnk_max,armax,prmax;
221 
222   PetscFunctionBegin;
223   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
224   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
225 
226   if (A->cmap->rstart != P->rmap->rstart || A->cmap->rend != P->rmap->rend) {
227     SETERRQ4(comm,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, (%D, %D) != (%D,%D)",A->cmap->rstart,A->cmap->rend,P->rmap->rstart,P->rmap->rend);
228   }
229 
230   /* create struct Mat_PtAPMPI and attached it to C later */
231   ierr = PetscNew(&ptap);CHKERRQ(ierr);
232 
233   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
234   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
235 
236   /* get P_loc by taking all local rows of P */
237   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
238 
239   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
240   pi_loc = p_loc->i; pj_loc = p_loc->j;
241   if (size > 1) {
242     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
243     pi_oth = p_oth->i; pj_oth = p_oth->j;
244   }
245 
246   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
247   /*-------------------------------------------------------------------*/
248   ierr      = PetscMalloc1((am+2),&api);CHKERRQ(ierr);
249   ptap->api = api;
250   api[0]    = 0;
251 
252   /* create and initialize a linked list */
253   armax    = ad->rmax+ao->rmax;
254   if (size >1) {
255     prmax    = PetscMax(p_loc->rmax,p_oth->rmax);
256   } else {
257     prmax = p_loc->rmax;
258   }
259   nlnk_max = armax*prmax;
260   if (!nlnk_max || nlnk_max > pN) nlnk_max = pN;
261   ierr = PetscLLCondensedCreate(nlnk_max,pN,&lnk,&lnkbt);CHKERRQ(ierr);
262 
263   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
264   ierr = PetscFreeSpaceGet((PetscInt)(fill*(adi[am]+aoi[am]+pi_loc[pm])),&free_space);CHKERRQ(ierr);
265 
266   current_space = free_space;
267 
268   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
269   for (i=0; i<am; i++) {
270     /* diagonal portion of A */
271     nzi = adi[i+1] - adi[i];
272     for (j=0; j<nzi; j++) {
273       row  = *adj++;
274       pnz  = pi_loc[row+1] - pi_loc[row];
275       Jptr = pj_loc + pi_loc[row];
276       /* add non-zero cols of P into the sorted linked list lnk */
277       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
278     }
279     /* off-diagonal portion of A */
280     nzi = aoi[i+1] - aoi[i];
281     for (j=0; j<nzi; j++) {
282       row  = *aoj++;
283       pnz  = pi_oth[row+1] - pi_oth[row];
284       Jptr = pj_oth + pi_oth[row];
285       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
286     }
287 
288     apnz     = lnk[0];
289     api[i+1] = api[i] + apnz;
290 
291     /* if free space is not available, double the total space in the list */
292     if (current_space->local_remaining<apnz) {
293       ierr = PetscFreeSpaceGet(apnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
294       nspacedouble++;
295     }
296 
297     /* Copy data into free space, then initialize lnk */
298     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
299     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
300 
301     current_space->array           += apnz;
302     current_space->local_used      += apnz;
303     current_space->local_remaining -= apnz;
304   }
305 
306   /* Allocate space for apj, initialize apj, and */
307   /* destroy list of free space and other temporary array(s) */
308   ierr = PetscMalloc1((api[am]+1),&ptap->apj);CHKERRQ(ierr);
309   apj  = ptap->apj;
310   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
311   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
312 
313   /* malloc apa to store dense row A[i,:]*P */
314   ierr = PetscCalloc1(pN,&apa);CHKERRQ(ierr);
315 
316   ptap->apa = apa;
317 
318   /* create and assemble symbolic parallel matrix Cmpi */
319   /*----------------------------------------------------*/
320   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
321   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
322   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
323 
324   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
325   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
326   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
327   for (i=0; i<am; i++) {
328     row  = i + rstart;
329     apnz = api[i+1] - api[i];
330     ierr = MatSetValues(Cmpi,1,&row,apnz,apj,apa,INSERT_VALUES);CHKERRQ(ierr);
331     apj += apnz;
332   }
333   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
334   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
335 
336   ptap->destroy        = Cmpi->ops->destroy;
337   ptap->duplicate      = Cmpi->ops->duplicate;
338   Cmpi->ops->destroy   = MatDestroy_MPIAIJ_MatMatMult;
339   Cmpi->ops->duplicate = MatDuplicate_MPIAIJ_MatMatMult;
340 
341   /* attach the supporting struct to Cmpi for reuse */
342   c       = (Mat_MPIAIJ*)Cmpi->data;
343   c->ptap = ptap;
344 
345   *C = Cmpi;
346 
347   /* set MatInfo */
348   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
349   if (afill < 1.0) afill = 1.0;
350   Cmpi->info.mallocs           = nspacedouble;
351   Cmpi->info.fill_ratio_given  = fill;
352   Cmpi->info.fill_ratio_needed = afill;
353 
354 #if defined(PETSC_USE_INFO)
355   if (api[am]) {
356     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
357     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
358   } else {
359     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
360   }
361 #endif
362   PetscFunctionReturn(0);
363 }
364 
365 #undef __FUNCT__
366 #define __FUNCT__ "MatMatMult_MPIAIJ_MPIDense"
367 PetscErrorCode MatMatMult_MPIAIJ_MPIDense(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
368 {
369   PetscErrorCode ierr;
370 
371   PetscFunctionBegin;
372   if (scall == MAT_INITIAL_MATRIX) {
373     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
374     ierr = MatMatMultSymbolic_MPIAIJ_MPIDense(A,B,fill,C);CHKERRQ(ierr);
375     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
376   }
377   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
378   ierr = MatMatMultNumeric_MPIAIJ_MPIDense(A,B,*C);CHKERRQ(ierr);
379   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
380   PetscFunctionReturn(0);
381 }
382 
383 typedef struct {
384   Mat         workB;
385   PetscScalar *rvalues,*svalues;
386   MPI_Request *rwaits,*swaits;
387 } MPIAIJ_MPIDense;
388 
389 #undef __FUNCT__
390 #define __FUNCT__ "MatMPIAIJ_MPIDenseDestroy"
391 PetscErrorCode MatMPIAIJ_MPIDenseDestroy(void *ctx)
392 {
393   MPIAIJ_MPIDense *contents = (MPIAIJ_MPIDense*) ctx;
394   PetscErrorCode  ierr;
395 
396   PetscFunctionBegin;
397   ierr = MatDestroy(&contents->workB);CHKERRQ(ierr);
398   ierr = PetscFree4(contents->rvalues,contents->svalues,contents->rwaits,contents->swaits);CHKERRQ(ierr);
399   ierr = PetscFree(contents);CHKERRQ(ierr);
400   PetscFunctionReturn(0);
401 }
402 
403 #undef __FUNCT__
404 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIDense"
405 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat A,Mat B,PetscReal fill,Mat *C)
406 {
407   PetscErrorCode         ierr;
408   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*) A->data;
409   PetscInt               nz   = aij->B->cmap->n;
410   PetscContainer         container;
411   MPIAIJ_MPIDense        *contents;
412   VecScatter             ctx   = aij->Mvctx;
413   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
414   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
415   PetscInt               m     = A->rmap->n,n=B->cmap->n;
416 
417   PetscFunctionBegin;
418   ierr = MatCreate(PetscObjectComm((PetscObject)B),C);CHKERRQ(ierr);
419   ierr = MatSetSizes(*C,m,n,A->rmap->N,B->cmap->N);CHKERRQ(ierr);
420   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
421   ierr = MatSetType(*C,MATMPIDENSE);CHKERRQ(ierr);
422   ierr = MatMPIDenseSetPreallocation(*C,NULL);CHKERRQ(ierr);
423   ierr = MatAssemblyBegin(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
424   ierr = MatAssemblyEnd(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
425 
426   (*C)->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
427 
428   ierr = PetscNew(&contents);CHKERRQ(ierr);
429   /* Create work matrix used to store off processor rows of B needed for local product */
430   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,B->cmap->N,NULL,&contents->workB);CHKERRQ(ierr);
431   /* Create work arrays needed */
432   ierr = PetscMalloc4(B->cmap->N*from->starts[from->n],&contents->rvalues,
433                       B->cmap->N*to->starts[to->n],&contents->svalues,
434                       from->n,&contents->rwaits,
435                       to->n,&contents->swaits);CHKERRQ(ierr);
436 
437   ierr = PetscContainerCreate(PetscObjectComm((PetscObject)A),&container);CHKERRQ(ierr);
438   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
439   ierr = PetscContainerSetUserDestroy(container,MatMPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
440   ierr = PetscObjectCompose((PetscObject)(*C),"workB",(PetscObject)container);CHKERRQ(ierr);
441   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
442   PetscFunctionReturn(0);
443 }
444 
445 #undef __FUNCT__
446 #define __FUNCT__ "MatMPIDenseScatter"
447 /*
448     Performs an efficient scatter on the rows of B needed by this process; this is
449     a modification of the VecScatterBegin_() routines.
450 */
451 PetscErrorCode MatMPIDenseScatter(Mat A,Mat B,Mat C,Mat *outworkB)
452 {
453   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*)A->data;
454   PetscErrorCode         ierr;
455   PetscScalar            *b,*w,*svalues,*rvalues;
456   VecScatter             ctx   = aij->Mvctx;
457   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
458   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
459   PetscInt               i,j,k;
460   PetscInt               *sindices,*sstarts,*rindices,*rstarts;
461   PetscMPIInt            *sprocs,*rprocs,nrecvs;
462   MPI_Request            *swaits,*rwaits;
463   MPI_Comm               comm;
464   PetscMPIInt            tag  = ((PetscObject)ctx)->tag,ncols = B->cmap->N, nrows = aij->B->cmap->n,imdex,nrowsB = B->rmap->n;
465   MPI_Status             status;
466   MPIAIJ_MPIDense        *contents;
467   PetscContainer         container;
468   Mat                    workB;
469 
470   PetscFunctionBegin;
471   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
472   ierr = PetscObjectQuery((PetscObject)C,"workB",(PetscObject*)&container);CHKERRQ(ierr);
473   if (!container) SETERRQ(comm,PETSC_ERR_PLIB,"Container does not exist");
474   ierr = PetscContainerGetPointer(container,(void**)&contents);CHKERRQ(ierr);
475 
476   workB = *outworkB = contents->workB;
477   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);
478   sindices = to->indices;
479   sstarts  = to->starts;
480   sprocs   = to->procs;
481   swaits   = contents->swaits;
482   svalues  = contents->svalues;
483 
484   rindices = from->indices;
485   rstarts  = from->starts;
486   rprocs   = from->procs;
487   rwaits   = contents->rwaits;
488   rvalues  = contents->rvalues;
489 
490   ierr = MatDenseGetArray(B,&b);CHKERRQ(ierr);
491   ierr = MatDenseGetArray(workB,&w);CHKERRQ(ierr);
492 
493   for (i=0; i<from->n; i++) {
494     ierr = MPI_Irecv(rvalues+ncols*rstarts[i],ncols*(rstarts[i+1]-rstarts[i]),MPIU_SCALAR,rprocs[i],tag,comm,rwaits+i);CHKERRQ(ierr);
495   }
496 
497   for (i=0; i<to->n; i++) {
498     /* pack a message at a time */
499     for (j=0; j<sstarts[i+1]-sstarts[i]; j++) {
500       for (k=0; k<ncols; k++) {
501         svalues[ncols*(sstarts[i] + j) + k] = b[sindices[sstarts[i]+j] + nrowsB*k];
502       }
503     }
504     ierr = MPI_Isend(svalues+ncols*sstarts[i],ncols*(sstarts[i+1]-sstarts[i]),MPIU_SCALAR,sprocs[i],tag,comm,swaits+i);CHKERRQ(ierr);
505   }
506 
507   nrecvs = from->n;
508   while (nrecvs) {
509     ierr = MPI_Waitany(from->n,rwaits,&imdex,&status);CHKERRQ(ierr);
510     nrecvs--;
511     /* unpack a message at a time */
512     for (j=0; j<rstarts[imdex+1]-rstarts[imdex]; j++) {
513       for (k=0; k<ncols; k++) {
514         w[rindices[rstarts[imdex]+j] + nrows*k] = rvalues[ncols*(rstarts[imdex] + j) + k];
515       }
516     }
517   }
518   if (to->n) {ierr = MPI_Waitall(to->n,swaits,to->sstatus);CHKERRQ(ierr);}
519 
520   ierr = MatDenseRestoreArray(B,&b);CHKERRQ(ierr);
521   ierr = MatDenseRestoreArray(workB,&w);CHKERRQ(ierr);
522   ierr = MatAssemblyBegin(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
523   ierr = MatAssemblyEnd(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
524   PetscFunctionReturn(0);
525 }
526 extern PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat,Mat,Mat);
527 
528 #undef __FUNCT__
529 #define __FUNCT__ "MatMatMultNumeric_MPIAIJ_MPIDense"
530 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat A,Mat B,Mat C)
531 {
532   PetscErrorCode ierr;
533   Mat_MPIAIJ     *aij    = (Mat_MPIAIJ*)A->data;
534   Mat_MPIDense   *bdense = (Mat_MPIDense*)B->data;
535   Mat_MPIDense   *cdense = (Mat_MPIDense*)C->data;
536   Mat            workB;
537 
538   PetscFunctionBegin;
539   /* diagonal block of A times all local rows of B*/
540   ierr = MatMatMultNumeric_SeqAIJ_SeqDense(aij->A,bdense->A,cdense->A);CHKERRQ(ierr);
541 
542   /* get off processor parts of B needed to complete the product */
543   ierr = MatMPIDenseScatter(A,B,C,&workB);CHKERRQ(ierr);
544 
545   /* off-diagonal block of A times nonlocal rows of B */
546   ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A);CHKERRQ(ierr);
547   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
548   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
549   PetscFunctionReturn(0);
550 }
551 
552 #undef __FUNCT__
553 #define __FUNCT__ "MatMatMultNumeric_MPIAIJ_MPIAIJ_Scalable"
554 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ_Scalable(Mat A,Mat P,Mat C)
555 {
556   PetscErrorCode ierr;
557   Mat_MPIAIJ     *a   = (Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
558   Mat_SeqAIJ     *ad  = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
559   Mat_SeqAIJ     *cd  = (Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
560   PetscInt       *adi = ad->i,*adj,*aoi=ao->i,*aoj;
561   PetscScalar    *ada,*aoa,*cda=cd->a,*coa=co->a;
562   Mat_SeqAIJ     *p_loc,*p_oth;
563   PetscInt       *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
564   PetscScalar    *pa_loc,*pa_oth,*pa,valtmp,*ca;
565   PetscInt       cm          = C->rmap->n,anz,pnz;
566   Mat_PtAPMPI    *ptap       = c->ptap;
567   PetscScalar    *apa_sparse = ptap->apa;
568   PetscInt       *api,*apj,*apJ,i,j,k,row;
569   PetscInt       cstart = C->cmap->rstart;
570   PetscInt       cdnz,conz,k0,k1,nextp;
571   MPI_Comm       comm;
572   PetscMPIInt    size;
573 
574   PetscFunctionBegin;
575   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
576   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
577 
578   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
579   /*-----------------------------------------------------*/
580   /* update numerical values of P_oth and P_loc */
581   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
582   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
583 
584   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
585   /*----------------------------------------------------------*/
586   /* get data from symbolic products */
587   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
588   pi_loc=p_loc->i; pj_loc=p_loc->j; pa_loc=p_loc->a;
589   if (size >1) {
590     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
591     pi_oth=p_oth->i; pj_oth=p_oth->j; pa_oth=p_oth->a;
592   }
593 
594   api = ptap->api;
595   apj = ptap->apj;
596   for (i=0; i<cm; i++) {
597     apJ = apj + api[i];
598 
599     /* diagonal portion of A */
600     anz = adi[i+1] - adi[i];
601     adj = ad->j + adi[i];
602     ada = ad->a + adi[i];
603     for (j=0; j<anz; j++) {
604       row = adj[j];
605       pnz = pi_loc[row+1] - pi_loc[row];
606       pj  = pj_loc + pi_loc[row];
607       pa  = pa_loc + pi_loc[row];
608       /* perform sparse axpy */
609       valtmp = ada[j];
610       nextp  = 0;
611       for (k=0; nextp<pnz; k++) {
612         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
613           apa_sparse[k] += valtmp*pa[nextp++];
614         }
615       }
616       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
617     }
618 
619     /* off-diagonal portion of A */
620     anz = aoi[i+1] - aoi[i];
621     aoj = ao->j + aoi[i];
622     aoa = ao->a + aoi[i];
623     for (j=0; j<anz; j++) {
624       row = aoj[j];
625       pnz = pi_oth[row+1] - pi_oth[row];
626       pj  = pj_oth + pi_oth[row];
627       pa  = pa_oth + pi_oth[row];
628       /* perform sparse axpy */
629       valtmp = aoa[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     /* set values in C */
640     cdnz = cd->i[i+1] - cd->i[i];
641     conz = co->i[i+1] - co->i[i];
642 
643     /* 1st off-diagoanl part of C */
644     ca = coa + co->i[i];
645     k  = 0;
646     for (k0=0; k0<conz; k0++) {
647       if (apJ[k] >= cstart) break;
648       ca[k0]        = apa_sparse[k];
649       apa_sparse[k] = 0.0;
650       k++;
651     }
652 
653     /* diagonal part of C */
654     ca = cda + cd->i[i];
655     for (k1=0; k1<cdnz; k1++) {
656       ca[k1]        = apa_sparse[k];
657       apa_sparse[k] = 0.0;
658       k++;
659     }
660 
661     /* 2nd off-diagoanl part of C */
662     ca = coa + co->i[i];
663     for (; k0<conz; k0++) {
664       ca[k0]        = apa_sparse[k];
665       apa_sparse[k] = 0.0;
666       k++;
667     }
668   }
669   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
670   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
671   PetscFunctionReturn(0);
672 }
673 
674 /* same as MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(), except using LLCondensed to avoid O(BN) memory requirement */
675 #undef __FUNCT__
676 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIAIJ"
677 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat *C)
678 {
679   PetscErrorCode     ierr;
680   MPI_Comm           comm;
681   PetscMPIInt        size;
682   Mat                Cmpi;
683   Mat_PtAPMPI        *ptap;
684   PetscFreeSpaceList free_space = NULL,current_space=NULL;
685   Mat_MPIAIJ         *a         = (Mat_MPIAIJ*)A->data,*c;
686   Mat_SeqAIJ         *ad        = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
687   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
688   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
689   PetscInt           i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi,*lnk,apnz_max=0;
690   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
691   PetscInt           nlnk_max,armax,prmax;
692   PetscReal          afill;
693   PetscScalar        *apa;
694 
695   PetscFunctionBegin;
696   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
697   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
698 
699   /* create struct Mat_PtAPMPI and attached it to C later */
700   ierr = PetscNew(&ptap);CHKERRQ(ierr);
701 
702   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
703   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
704 
705   /* get P_loc by taking all local rows of P */
706   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
707 
708   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
709   pi_loc = p_loc->i; pj_loc = p_loc->j;
710   if (size > 1) {
711     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
712     pi_oth = p_oth->i;
713     pj_oth = p_oth->j;
714   }
715 
716   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
717   /*-------------------------------------------------------------------*/
718   ierr      = PetscMalloc1((am+2),&api);CHKERRQ(ierr);
719   ptap->api = api;
720   api[0]    = 0;
721 
722   /* create and initialize a linked list */
723   armax = ad->rmax+ao->rmax;
724   if (size >1) {
725     prmax = PetscMax(p_loc->rmax,p_oth->rmax);
726   } else {
727     prmax = p_loc->rmax;
728   }
729   nlnk_max = armax*prmax;
730   if (!nlnk_max || nlnk_max > pN) nlnk_max = pN;
731   ierr = PetscLLCondensedCreate_Scalable(nlnk_max,&lnk);CHKERRQ(ierr);
732 
733   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
734   ierr = PetscFreeSpaceGet((PetscInt)(fill*(adi[am]+aoi[am]+pi_loc[pm])),&free_space);CHKERRQ(ierr);
735 
736   current_space = free_space;
737 
738   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
739   for (i=0; i<am; i++) {
740     /* diagonal portion of A */
741     nzi = adi[i+1] - adi[i];
742     for (j=0; j<nzi; j++) {
743       row  = *adj++;
744       pnz  = pi_loc[row+1] - pi_loc[row];
745       Jptr = pj_loc + pi_loc[row];
746       /* add non-zero cols of P into the sorted linked list lnk */
747       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
748     }
749     /* off-diagonal portion of A */
750     nzi = aoi[i+1] - aoi[i];
751     for (j=0; j<nzi; j++) {
752       row  = *aoj++;
753       pnz  = pi_oth[row+1] - pi_oth[row];
754       Jptr = pj_oth + pi_oth[row];
755       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
756     }
757 
758     apnz     = *lnk;
759     api[i+1] = api[i] + apnz;
760     if (apnz > apnz_max) apnz_max = apnz;
761 
762     /* if free space is not available, double the total space in the list */
763     if (current_space->local_remaining<apnz) {
764       ierr = PetscFreeSpaceGet(apnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
765       nspacedouble++;
766     }
767 
768     /* Copy data into free space, then initialize lnk */
769     ierr = PetscLLCondensedClean_Scalable(apnz,current_space->array,lnk);CHKERRQ(ierr);
770     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
771 
772     current_space->array           += apnz;
773     current_space->local_used      += apnz;
774     current_space->local_remaining -= apnz;
775   }
776 
777   /* Allocate space for apj, initialize apj, and */
778   /* destroy list of free space and other temporary array(s) */
779   ierr = PetscMalloc1((api[am]+1),&ptap->apj);CHKERRQ(ierr);
780   apj  = ptap->apj;
781   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
782   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
783 
784   /* create and assemble symbolic parallel matrix Cmpi */
785   /*----------------------------------------------------*/
786   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
787   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
788   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
789   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
790   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
791   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
792 
793   /* malloc apa for assembly Cmpi */
794   ierr = PetscCalloc1(apnz_max,&apa);CHKERRQ(ierr);
795 
796   ptap->apa = apa;
797   for (i=0; i<am; i++) {
798     row  = i + rstart;
799     apnz = api[i+1] - api[i];
800     ierr = MatSetValues(Cmpi,1,&row,apnz,apj,apa,INSERT_VALUES);CHKERRQ(ierr);
801     apj += apnz;
802   }
803   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
804   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
805 
806   ptap->destroy             = Cmpi->ops->destroy;
807   ptap->duplicate           = Cmpi->ops->duplicate;
808   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_Scalable;
809   Cmpi->ops->destroy        = MatDestroy_MPIAIJ_MatMatMult;
810   Cmpi->ops->duplicate      = MatDuplicate_MPIAIJ_MatMatMult;
811 
812   /* attach the supporting struct to Cmpi for reuse */
813   c       = (Mat_MPIAIJ*)Cmpi->data;
814   c->ptap = ptap;
815 
816   *C = Cmpi;
817 
818   /* set MatInfo */
819   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
820   if (afill < 1.0) afill = 1.0;
821   Cmpi->info.mallocs           = nspacedouble;
822   Cmpi->info.fill_ratio_given  = fill;
823   Cmpi->info.fill_ratio_needed = afill;
824 
825 #if defined(PETSC_USE_INFO)
826   if (api[am]) {
827     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
828     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
829   } else {
830     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
831   }
832 #endif
833   PetscFunctionReturn(0);
834 }
835 
836 /*-------------------------------------------------------------------------*/
837 #undef __FUNCT__
838 #define __FUNCT__ "MatTransposeMatMult_MPIAIJ_MPIAIJ"
839 PetscErrorCode MatTransposeMatMult_MPIAIJ_MPIAIJ(Mat P,Mat A,MatReuse scall,PetscReal fill,Mat *C)
840 {
841   PetscErrorCode ierr;
842   const char     *algTypes[3] = {"scalable","nonscalable","matmatmult"};
843   PetscInt       alg=0; /* set default algorithm */
844 
845   PetscFunctionBegin;
846   if (scall == MAT_INITIAL_MATRIX) {
847     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
848     ierr = PetscOptionsEList("-mattransposematmult_via","Algorithmic approach","MatTransposeMatMult",algTypes,3,algTypes[0],&alg,NULL);CHKERRQ(ierr);
849     ierr = PetscOptionsEnd();CHKERRQ(ierr);
850 
851     ierr = PetscLogEventBegin(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
852     switch (alg) {
853     case 1:
854       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(P,A,fill,C);CHKERRQ(ierr);
855       break;
856     case 2:
857     {
858       Mat         Pt;
859       Mat_PtAPMPI *ptap;
860       Mat_MPIAIJ  *c;
861       ierr = MatTranspose(P,MAT_INITIAL_MATRIX,&Pt);CHKERRQ(ierr);
862       ierr = MatMatMult(Pt,A,MAT_INITIAL_MATRIX,fill,C);CHKERRQ(ierr);
863       c        = (Mat_MPIAIJ*)(*C)->data;
864       ptap     = c->ptap;
865       ptap->Pt = Pt;
866       (*C)->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult;
867       PetscFunctionReturn(0);
868     }
869       break;
870     default:
871       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(P,A,fill,C);CHKERRQ(ierr);
872       break;
873     }
874     ierr = PetscLogEventEnd(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
875   }
876   ierr = PetscLogEventBegin(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
877   ierr = (*(*C)->ops->mattransposemultnumeric)(P,A,*C);CHKERRQ(ierr);
878   ierr = PetscLogEventEnd(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
879   PetscFunctionReturn(0);
880 }
881 
882 /* This routine only works when scall=MAT_REUSE_MATRIX! */
883 #undef __FUNCT__
884 #define __FUNCT__ "MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult"
885 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult(Mat P,Mat A,Mat C)
886 {
887   PetscErrorCode ierr;
888   Mat_MPIAIJ     *c=(Mat_MPIAIJ*)C->data;
889   Mat_PtAPMPI    *ptap= c->ptap;
890   Mat            Pt=ptap->Pt;
891 
892   PetscFunctionBegin;
893   ierr = MatTranspose(P,MAT_REUSE_MATRIX,&Pt);CHKERRQ(ierr);
894   ierr = MatMatMultNumeric(Pt,A,C);CHKERRQ(ierr);
895   PetscFunctionReturn(0);
896 }
897 
898 /* Non-scalable version, use dense axpy */
899 #undef __FUNCT__
900 #define __FUNCT__ "MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable"
901 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,Mat C)
902 {
903   PetscErrorCode      ierr;
904   Mat_Merge_SeqsToMPI *merge;
905   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
906   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
907   Mat_PtAPMPI         *ptap;
908   PetscInt            *adj,*aJ;
909   PetscInt            i,j,k,anz,pnz,row,*cj;
910   MatScalar           *ada,*aval,*ca,valtmp;
911   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
912   MPI_Comm            comm;
913   PetscMPIInt         size,rank,taga,*len_s;
914   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
915   PetscInt            **buf_ri,**buf_rj;
916   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
917   MPI_Request         *s_waits,*r_waits;
918   MPI_Status          *status;
919   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
920   PetscInt            *ai,*aj,*coi,*coj;
921   PetscInt            *poJ,*pdJ;
922   Mat                 A_loc;
923   Mat_SeqAIJ          *a_loc;
924 
925   PetscFunctionBegin;
926   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
927   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
928   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
929 
930   ptap  = c->ptap;
931   merge = ptap->merge;
932 
933   /* 2) compute numeric C_seq = P_loc^T*A_loc*P - dominating part */
934   /*--------------------------------------------------------------*/
935   /* get data from symbolic products */
936   coi  = merge->coi; coj = merge->coj;
937   ierr = PetscCalloc1((coi[pon]+1),&coa);CHKERRQ(ierr);
938 
939   bi     = merge->bi; bj = merge->bj;
940   owners = merge->rowmap->range;
941   ierr   = PetscCalloc1((bi[cm]+1),&ba);CHKERRQ(ierr);
942 
943   /* get A_loc by taking all local rows of A */
944   A_loc = ptap->A_loc;
945   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
946   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
947   ai    = a_loc->i;
948   aj    = a_loc->j;
949 
950   ierr = PetscCalloc1((A->cmap->N),&aval);CHKERRQ(ierr); /* non-scalable!!! */
951 
952   for (i=0; i<am; i++) {
953     /* 2-a) put A[i,:] to dense array aval */
954     anz = ai[i+1] - ai[i];
955     adj = aj + ai[i];
956     ada = a_loc->a + ai[i];
957     for (j=0; j<anz; j++) {
958       aval[adj[j]] = ada[j];
959     }
960 
961     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
962     /*--------------------------------------------------------------*/
963     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
964     pnz = po->i[i+1] - po->i[i];
965     poJ = po->j + po->i[i];
966     pA  = po->a + po->i[i];
967     for (j=0; j<pnz; j++) {
968       row = poJ[j];
969       cnz = coi[row+1] - coi[row];
970       cj  = coj + coi[row];
971       ca  = coa + coi[row];
972       /* perform dense axpy */
973       valtmp = pA[j];
974       for (k=0; k<cnz; k++) {
975         ca[k] += valtmp*aval[cj[k]];
976       }
977       ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
978     }
979 
980     /* put the value into Cd (diagonal part) */
981     pnz = pd->i[i+1] - pd->i[i];
982     pdJ = pd->j + pd->i[i];
983     pA  = pd->a + pd->i[i];
984     for (j=0; j<pnz; j++) {
985       row = pdJ[j];
986       cnz = bi[row+1] - bi[row];
987       cj  = bj + bi[row];
988       ca  = ba + bi[row];
989       /* perform dense axpy */
990       valtmp = pA[j];
991       for (k=0; k<cnz; k++) {
992         ca[k] += valtmp*aval[cj[k]];
993       }
994       ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
995     }
996 
997     /* zero the current row of Pt*A */
998     aJ = aj + ai[i];
999     for (k=0; k<anz; k++) aval[aJ[k]] = 0.0;
1000   }
1001 
1002   /* 3) send and recv matrix values coa */
1003   /*------------------------------------*/
1004   buf_ri = merge->buf_ri;
1005   buf_rj = merge->buf_rj;
1006   len_s  = merge->len_s;
1007   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1008   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1009 
1010   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1011   for (proc=0,k=0; proc<size; proc++) {
1012     if (!len_s[proc]) continue;
1013     i    = merge->owners_co[proc];
1014     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1015     k++;
1016   }
1017   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1018   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1019 
1020   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1021   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1022   ierr = PetscFree(coa);CHKERRQ(ierr);
1023 
1024   /* 4) insert local Cseq and received values into Cmpi */
1025   /*----------------------------------------------------*/
1026   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1027   for (k=0; k<merge->nrecv; k++) {
1028     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1029     nrows       = *(buf_ri_k[k]);
1030     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1031     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1032   }
1033 
1034   for (i=0; i<cm; i++) {
1035     row  = owners[rank] + i; /* global row index of C_seq */
1036     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1037     ba_i = ba + bi[i];
1038     bnz  = bi[i+1] - bi[i];
1039     /* add received vals into ba */
1040     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1041       /* i-th row */
1042       if (i == *nextrow[k]) {
1043         cnz    = *(nextci[k]+1) - *nextci[k];
1044         cj     = buf_rj[k] + *(nextci[k]);
1045         ca     = abuf_r[k] + *(nextci[k]);
1046         nextcj = 0;
1047         for (j=0; nextcj<cnz; j++) {
1048           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1049             ba_i[j] += ca[nextcj++];
1050           }
1051         }
1052         nextrow[k]++; nextci[k]++;
1053         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1054       }
1055     }
1056     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1057   }
1058   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1059   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1060 
1061   ierr = PetscFree(ba);CHKERRQ(ierr);
1062   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1063   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1064   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1065   ierr = PetscFree(aval);CHKERRQ(ierr);
1066   PetscFunctionReturn(0);
1067 }
1068 
1069 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1070 #undef __FUNCT__
1071 #define __FUNCT__ "MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable"
1072 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,PetscReal fill,Mat *C)
1073 {
1074   PetscErrorCode      ierr;
1075   Mat                 Cmpi,A_loc,POt,PDt;
1076   Mat_PtAPMPI         *ptap;
1077   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1078   Mat_MPIAIJ          *p        =(Mat_MPIAIJ*)P->data,*c;
1079   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1080   PetscInt            nnz;
1081   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1082   PetscInt            am=A->rmap->n,pn=P->cmap->n;
1083   PetscBT             lnkbt;
1084   MPI_Comm            comm;
1085   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1086   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1087   PetscInt            len,proc,*dnz,*onz,*owners;
1088   PetscInt            nzi,*bi,*bj;
1089   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1090   MPI_Request         *swaits,*rwaits;
1091   MPI_Status          *sstatus,rstatus;
1092   Mat_Merge_SeqsToMPI *merge;
1093   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1094   PetscReal           afill  =1.0,afill_tmp;
1095   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Crmax;
1096   PetscScalar         *vals;
1097   Mat_SeqAIJ          *a_loc, *pdt,*pot;
1098 
1099   PetscFunctionBegin;
1100   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1101   /* check if matrix local sizes are compatible */
1102   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend) {
1103     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);
1104   }
1105 
1106   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1107   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1108 
1109   /* create struct Mat_PtAPMPI and attached it to C later */
1110   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1111 
1112   /* get A_loc by taking all local rows of A */
1113   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1114 
1115   ptap->A_loc = A_loc;
1116 
1117   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1118   ai    = a_loc->i;
1119   aj    = a_loc->j;
1120 
1121   /* determine symbolic Co=(p->B)^T*A - send to others */
1122   /*----------------------------------------------------*/
1123   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1124   pdt  = (Mat_SeqAIJ*)PDt->data;
1125   pdti = pdt->i; pdtj = pdt->j;
1126 
1127   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1128   pot  = (Mat_SeqAIJ*)POt->data;
1129   poti = pot->i; potj = pot->j;
1130 
1131   /* then, compute symbolic Co = (p->B)^T*A */
1132   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors >= (num of nonzero rows of C_seq) - pn */
1133   ierr   = PetscMalloc1((pon+1),&coi);CHKERRQ(ierr);
1134   coi[0] = 0;
1135 
1136   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1137   nnz           = fill*(poti[pon] + ai[am]);
1138   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1139   current_space = free_space;
1140 
1141   /* create and initialize a linked list */
1142   i     = PetscMax(pdt->rmax,pot->rmax);
1143   Crmax = i*a_loc->rmax*size;
1144   if (!Crmax || Crmax > aN) Crmax = aN;
1145   ierr = PetscLLCondensedCreate(Crmax,aN,&lnk,&lnkbt);CHKERRQ(ierr);
1146 
1147   for (i=0; i<pon; i++) {
1148     pnz = poti[i+1] - poti[i];
1149     ptJ = potj + poti[i];
1150     for (j=0; j<pnz; j++) {
1151       row  = ptJ[j]; /* row of A_loc == col of Pot */
1152       anz  = ai[row+1] - ai[row];
1153       Jptr = aj + ai[row];
1154       /* add non-zero cols of AP into the sorted linked list lnk */
1155       ierr = PetscLLCondensedAddSorted(anz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1156     }
1157     nnz = lnk[0];
1158 
1159     /* If free space is not available, double the total space in the list */
1160     if (current_space->local_remaining<nnz) {
1161       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1162       nspacedouble++;
1163     }
1164 
1165     /* Copy data into free space, and zero out denserows */
1166     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1167 
1168     current_space->array           += nnz;
1169     current_space->local_used      += nnz;
1170     current_space->local_remaining -= nnz;
1171 
1172     coi[i+1] = coi[i] + nnz;
1173   }
1174 
1175   ierr = PetscMalloc1((coi[pon]+1),&coj);CHKERRQ(ierr);
1176   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1177 
1178   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1179   if (afill_tmp > afill) afill = afill_tmp;
1180 
1181   /* send j-array (coj) of Co to other processors */
1182   /*----------------------------------------------*/
1183   /* determine row ownership */
1184   ierr = PetscNew(&merge);CHKERRQ(ierr);
1185   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1186 
1187   merge->rowmap->n  = pn;
1188   merge->rowmap->bs = 1;
1189 
1190   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1191   owners = merge->rowmap->range;
1192 
1193   /* determine the number of messages to send, their lengths */
1194   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
1195   ierr = PetscMalloc1(size,&merge->len_s);CHKERRQ(ierr);
1196 
1197   len_s        = merge->len_s;
1198   merge->nsend = 0;
1199 
1200   ierr = PetscMalloc1((size+2),&owners_co);CHKERRQ(ierr);
1201   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1202 
1203   proc = 0;
1204   for (i=0; i<pon; i++) {
1205     while (prmap[i] >= owners[proc+1]) proc++;
1206     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1207     len_s[proc] += coi[i+1] - coi[i];
1208   }
1209 
1210   len          = 0; /* max length of buf_si[] */
1211   owners_co[0] = 0;
1212   for (proc=0; proc<size; proc++) {
1213     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1214     if (len_si[proc]) {
1215       merge->nsend++;
1216       len_si[proc] = 2*(len_si[proc] + 1);
1217       len         += len_si[proc];
1218     }
1219   }
1220 
1221   /* determine the number and length of messages to receive for coi and coj  */
1222   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1223   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1224 
1225   /* post the Irecv and Isend of coj */
1226   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1227   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1228   ierr = PetscMalloc1((merge->nsend+1),&swaits);CHKERRQ(ierr);
1229   for (proc=0, k=0; proc<size; proc++) {
1230     if (!len_s[proc]) continue;
1231     i    = owners_co[proc];
1232     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1233     k++;
1234   }
1235 
1236   /* receives and sends of coj are complete */
1237   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
1238   for (i=0; i<merge->nrecv; i++) {
1239     PetscMPIInt icompleted;
1240     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1241   }
1242   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1243   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1244 
1245   /* send and recv coi */
1246   /*-------------------*/
1247   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1248   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1249   ierr   = PetscMalloc1((len+1),&buf_s);CHKERRQ(ierr);
1250   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1251   for (proc=0,k=0; proc<size; proc++) {
1252     if (!len_s[proc]) continue;
1253     /* form outgoing message for i-structure:
1254          buf_si[0]:                 nrows to be sent
1255                [1:nrows]:           row index (global)
1256                [nrows+1:2*nrows+1]: i-structure index
1257     */
1258     /*-------------------------------------------*/
1259     nrows       = len_si[proc]/2 - 1;
1260     buf_si_i    = buf_si + nrows+1;
1261     buf_si[0]   = nrows;
1262     buf_si_i[0] = 0;
1263     nrows       = 0;
1264     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1265       nzi               = coi[i+1] - coi[i];
1266       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi; /* i-structure */
1267       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1268       nrows++;
1269     }
1270     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1271     k++;
1272     buf_si += len_si[proc];
1273   }
1274   i = merge->nrecv;
1275   while (i--) {
1276     PetscMPIInt icompleted;
1277     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1278   }
1279   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1280   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1281   ierr = PetscFree(len_si);CHKERRQ(ierr);
1282   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1283   ierr = PetscFree(swaits);CHKERRQ(ierr);
1284   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1285   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1286 
1287   /* compute the local portion of C (mpi mat) */
1288   /*------------------------------------------*/
1289   /* allocate bi array and free space for accumulating nonzero column info */
1290   ierr  = PetscMalloc1((pn+1),&bi);CHKERRQ(ierr);
1291   bi[0] = 0;
1292 
1293   /* set initial free space to be fill*(nnz(P) + nnz(A)) */
1294   nnz           = fill*(pdti[pn] + poti[pon] + ai[am]);
1295   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1296   current_space = free_space;
1297 
1298   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1299   for (k=0; k<merge->nrecv; k++) {
1300     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1301     nrows       = *buf_ri_k[k];
1302     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1303     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1304   }
1305 
1306   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1307   rmax = 0;
1308   for (i=0; i<pn; i++) {
1309     /* add pdt[i,:]*AP into lnk */
1310     pnz = pdti[i+1] - pdti[i];
1311     ptJ = pdtj + pdti[i];
1312     for (j=0; j<pnz; j++) {
1313       row  = ptJ[j];  /* row of AP == col of Pt */
1314       anz  = ai[row+1] - ai[row];
1315       Jptr = aj + ai[row];
1316       /* add non-zero cols of AP into the sorted linked list lnk */
1317       ierr = PetscLLCondensedAddSorted(anz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1318     }
1319 
1320     /* add received col data into lnk */
1321     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1322       if (i == *nextrow[k]) { /* i-th row */
1323         nzi  = *(nextci[k]+1) - *nextci[k];
1324         Jptr = buf_rj[k] + *nextci[k];
1325         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1326         nextrow[k]++; nextci[k]++;
1327       }
1328     }
1329     nnz = lnk[0];
1330 
1331     /* if free space is not available, make more free space */
1332     if (current_space->local_remaining<nnz) {
1333       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1334       nspacedouble++;
1335     }
1336     /* copy data into free space, then initialize lnk */
1337     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1338     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1339 
1340     current_space->array           += nnz;
1341     current_space->local_used      += nnz;
1342     current_space->local_remaining -= nnz;
1343 
1344     bi[i+1] = bi[i] + nnz;
1345     if (nnz > rmax) rmax = nnz;
1346   }
1347   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1348 
1349   ierr = PetscMalloc1((bi[pn]+1),&bj);CHKERRQ(ierr);
1350   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1351 
1352   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
1353   if (afill_tmp > afill) afill = afill_tmp;
1354   ierr = PetscLLCondensedDestroy(lnk,lnkbt);CHKERRQ(ierr);
1355   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1356   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1357 
1358   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
1359   /*----------------------------------------------------------------------------------*/
1360   ierr = PetscCalloc1(rmax+1,&vals);CHKERRQ(ierr);
1361 
1362   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1363   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1364   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
1365   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1366   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1367   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1368   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
1369   for (i=0; i<pn; i++) {
1370     row  = i + rstart;
1371     nnz  = bi[i+1] - bi[i];
1372     Jptr = bj + bi[i];
1373     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
1374   }
1375   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1376   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1377   ierr = PetscFree(vals);CHKERRQ(ierr);
1378 
1379   merge->bi        = bi;
1380   merge->bj        = bj;
1381   merge->coi       = coi;
1382   merge->coj       = coj;
1383   merge->buf_ri    = buf_ri;
1384   merge->buf_rj    = buf_rj;
1385   merge->owners_co = owners_co;
1386   merge->destroy   = Cmpi->ops->destroy;
1387   merge->duplicate = Cmpi->ops->duplicate;
1388 
1389   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1390   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
1391 
1392   /* attach the supporting struct to Cmpi for reuse */
1393   c           = (Mat_MPIAIJ*)Cmpi->data;
1394   c->ptap     = ptap;
1395   ptap->api   = NULL;
1396   ptap->apj   = NULL;
1397   ptap->merge = merge;
1398   ptap->rmax  = rmax;
1399 
1400   *C = Cmpi;
1401 #if defined(PETSC_USE_INFO)
1402   if (bi[pn] != 0) {
1403     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1404     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
1405   } else {
1406     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1407   }
1408 #endif
1409   PetscFunctionReturn(0);
1410 }
1411 
1412 #undef __FUNCT__
1413 #define __FUNCT__ "MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ"
1414 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ(Mat P,Mat A,Mat C)
1415 {
1416   PetscErrorCode      ierr;
1417   Mat_Merge_SeqsToMPI *merge;
1418   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1419   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
1420   Mat_PtAPMPI         *ptap;
1421   PetscInt            *adj;
1422   PetscInt            i,j,k,anz,pnz,row,*cj,nexta;
1423   MatScalar           *ada,*ca,valtmp;
1424   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
1425   MPI_Comm            comm;
1426   PetscMPIInt         size,rank,taga,*len_s;
1427   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
1428   PetscInt            **buf_ri,**buf_rj;
1429   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
1430   MPI_Request         *s_waits,*r_waits;
1431   MPI_Status          *status;
1432   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
1433   PetscInt            *ai,*aj,*coi,*coj;
1434   PetscInt            *poJ,*pdJ;
1435   Mat                 A_loc;
1436   Mat_SeqAIJ          *a_loc;
1437 
1438   PetscFunctionBegin;
1439   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1440   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1441   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1442 
1443   ptap  = c->ptap;
1444   merge = ptap->merge;
1445 
1446   /* 2) compute numeric C_seq = P_loc^T*A_loc */
1447   /*------------------------------------------*/
1448   /* get data from symbolic products */
1449   coi    = merge->coi; coj = merge->coj;
1450   ierr   = PetscCalloc1((coi[pon]+1),&coa);CHKERRQ(ierr);
1451   bi     = merge->bi; bj = merge->bj;
1452   owners = merge->rowmap->range;
1453   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
1454 
1455   /* get A_loc by taking all local rows of A */
1456   A_loc = ptap->A_loc;
1457   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
1458   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1459   ai    = a_loc->i;
1460   aj    = a_loc->j;
1461 
1462   for (i=0; i<am; i++) {
1463     anz = ai[i+1] - ai[i];
1464     adj = aj + ai[i];
1465     ada = a_loc->a + ai[i];
1466 
1467     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1468     /*-------------------------------------------------------------*/
1469     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1470     pnz = po->i[i+1] - po->i[i];
1471     poJ = po->j + po->i[i];
1472     pA  = po->a + po->i[i];
1473     for (j=0; j<pnz; j++) {
1474       row = poJ[j];
1475       cj  = coj + coi[row];
1476       ca  = coa + coi[row];
1477       /* perform sparse axpy */
1478       nexta  = 0;
1479       valtmp = pA[j];
1480       for (k=0; nexta<anz; k++) {
1481         if (cj[k] == adj[nexta]) {
1482           ca[k] += valtmp*ada[nexta];
1483           nexta++;
1484         }
1485       }
1486       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1487     }
1488 
1489     /* put the value into Cd (diagonal part) */
1490     pnz = pd->i[i+1] - pd->i[i];
1491     pdJ = pd->j + pd->i[i];
1492     pA  = pd->a + pd->i[i];
1493     for (j=0; j<pnz; j++) {
1494       row = pdJ[j];
1495       cj  = bj + bi[row];
1496       ca  = ba + bi[row];
1497       /* perform sparse axpy */
1498       nexta  = 0;
1499       valtmp = pA[j];
1500       for (k=0; nexta<anz; k++) {
1501         if (cj[k] == adj[nexta]) {
1502           ca[k] += valtmp*ada[nexta];
1503           nexta++;
1504         }
1505       }
1506       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1507     }
1508   }
1509 
1510   /* 3) send and recv matrix values coa */
1511   /*------------------------------------*/
1512   buf_ri = merge->buf_ri;
1513   buf_rj = merge->buf_rj;
1514   len_s  = merge->len_s;
1515   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1516   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1517 
1518   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1519   for (proc=0,k=0; proc<size; proc++) {
1520     if (!len_s[proc]) continue;
1521     i    = merge->owners_co[proc];
1522     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1523     k++;
1524   }
1525   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1526   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1527 
1528   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1529   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1530   ierr = PetscFree(coa);CHKERRQ(ierr);
1531 
1532   /* 4) insert local Cseq and received values into Cmpi */
1533   /*----------------------------------------------------*/
1534   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1535   for (k=0; k<merge->nrecv; k++) {
1536     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1537     nrows       = *(buf_ri_k[k]);
1538     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1539     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1540   }
1541 
1542   for (i=0; i<cm; i++) {
1543     row  = owners[rank] + i; /* global row index of C_seq */
1544     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1545     ba_i = ba + bi[i];
1546     bnz  = bi[i+1] - bi[i];
1547     /* add received vals into ba */
1548     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1549       /* i-th row */
1550       if (i == *nextrow[k]) {
1551         cnz    = *(nextci[k]+1) - *nextci[k];
1552         cj     = buf_rj[k] + *(nextci[k]);
1553         ca     = abuf_r[k] + *(nextci[k]);
1554         nextcj = 0;
1555         for (j=0; nextcj<cnz; j++) {
1556           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1557             ba_i[j] += ca[nextcj++];
1558           }
1559         }
1560         nextrow[k]++; nextci[k]++;
1561         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1562       }
1563     }
1564     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1565   }
1566   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1567   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1568 
1569   ierr = PetscFree(ba);CHKERRQ(ierr);
1570   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1571   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1572   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1573   PetscFunctionReturn(0);
1574 }
1575 
1576 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ();
1577    differ from MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable in using LLCondensedCreate_Scalable() */
1578 #undef __FUNCT__
1579 #define __FUNCT__ "MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ"
1580 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P,Mat A,PetscReal fill,Mat *C)
1581 {
1582   PetscErrorCode      ierr;
1583   Mat                 Cmpi,A_loc,POt,PDt;
1584   Mat_PtAPMPI         *ptap;
1585   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1586   Mat_MPIAIJ          *p        =(Mat_MPIAIJ*)P->data,*c;
1587   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1588   PetscInt            nnz;
1589   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1590   PetscInt            am  =A->rmap->n,pn=P->cmap->n;
1591   MPI_Comm            comm;
1592   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1593   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1594   PetscInt            len,proc,*dnz,*onz,*owners;
1595   PetscInt            nzi,*bi,*bj;
1596   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1597   MPI_Request         *swaits,*rwaits;
1598   MPI_Status          *sstatus,rstatus;
1599   Mat_Merge_SeqsToMPI *merge;
1600   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1601   PetscReal           afill  =1.0,afill_tmp;
1602   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Crmax;
1603   PetscScalar         *vals;
1604   Mat_SeqAIJ          *a_loc, *pdt,*pot;
1605 
1606   PetscFunctionBegin;
1607   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1608   /* check if matrix local sizes are compatible */
1609   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend) {
1610     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);
1611   }
1612 
1613   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1614   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1615 
1616   /* create struct Mat_PtAPMPI and attached it to C later */
1617   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1618 
1619   /* get A_loc by taking all local rows of A */
1620   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1621 
1622   ptap->A_loc = A_loc;
1623   a_loc       = (Mat_SeqAIJ*)(A_loc)->data;
1624   ai          = a_loc->i;
1625   aj          = a_loc->j;
1626 
1627   /* determine symbolic Co=(p->B)^T*A - send to others */
1628   /*----------------------------------------------------*/
1629   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1630   pdt  = (Mat_SeqAIJ*)PDt->data;
1631   pdti = pdt->i; pdtj = pdt->j;
1632 
1633   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1634   pot  = (Mat_SeqAIJ*)POt->data;
1635   poti = pot->i; potj = pot->j;
1636 
1637   /* then, compute symbolic Co = (p->B)^T*A */
1638   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1639                          >= (num of nonzero rows of C_seq) - pn */
1640   ierr   = PetscMalloc1((pon+1),&coi);CHKERRQ(ierr);
1641   coi[0] = 0;
1642 
1643   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1644   nnz           = fill*(poti[pon] + ai[am]);
1645   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1646   current_space = free_space;
1647 
1648   /* create and initialize a linked list */
1649   i     = PetscMax(pdt->rmax,pot->rmax);
1650   Crmax = i*a_loc->rmax*size; /* non-scalable! */
1651   if (!Crmax || Crmax > aN) Crmax = aN;
1652   ierr = PetscLLCondensedCreate_Scalable(Crmax,&lnk);CHKERRQ(ierr);
1653 
1654   for (i=0; i<pon; i++) {
1655     pnz = poti[i+1] - poti[i];
1656     ptJ = potj + poti[i];
1657     for (j=0; j<pnz; j++) {
1658       row  = ptJ[j]; /* row of A_loc == col of Pot */
1659       anz  = ai[row+1] - ai[row];
1660       Jptr = aj + ai[row];
1661       /* add non-zero cols of AP into the sorted linked list lnk */
1662       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1663     }
1664     nnz = lnk[0];
1665 
1666     /* If free space is not available, double the total space in the list */
1667     if (current_space->local_remaining<nnz) {
1668       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1669       nspacedouble++;
1670     }
1671 
1672     /* Copy data into free space, and zero out denserows */
1673     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1674 
1675     current_space->array           += nnz;
1676     current_space->local_used      += nnz;
1677     current_space->local_remaining -= nnz;
1678 
1679     coi[i+1] = coi[i] + nnz;
1680   }
1681 
1682   ierr = PetscMalloc1((coi[pon]+1),&coj);CHKERRQ(ierr);
1683   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1684 
1685   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1686   if (afill_tmp > afill) afill = afill_tmp;
1687 
1688   /* send j-array (coj) of Co to other processors */
1689   /*----------------------------------------------*/
1690   /* determine row ownership */
1691   ierr = PetscNew(&merge);CHKERRQ(ierr);
1692   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1693 
1694   merge->rowmap->n  = pn;
1695   merge->rowmap->bs = 1;
1696 
1697   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1698   owners = merge->rowmap->range;
1699 
1700   /* determine the number of messages to send, their lengths */
1701   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
1702   ierr = PetscMalloc1(size,&merge->len_s);CHKERRQ(ierr);
1703 
1704   len_s        = merge->len_s;
1705   merge->nsend = 0;
1706 
1707   ierr = PetscMalloc1((size+2),&owners_co);CHKERRQ(ierr);
1708   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1709 
1710   proc = 0;
1711   for (i=0; i<pon; i++) {
1712     while (prmap[i] >= owners[proc+1]) proc++;
1713     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1714     len_s[proc] += coi[i+1] - coi[i];
1715   }
1716 
1717   len          = 0; /* max length of buf_si[] */
1718   owners_co[0] = 0;
1719   for (proc=0; proc<size; proc++) {
1720     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1721     if (len_si[proc]) {
1722       merge->nsend++;
1723       len_si[proc] = 2*(len_si[proc] + 1);
1724       len         += len_si[proc];
1725     }
1726   }
1727 
1728   /* determine the number and length of messages to receive for coi and coj  */
1729   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1730   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1731 
1732   /* post the Irecv and Isend of coj */
1733   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1734   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1735   ierr = PetscMalloc1((merge->nsend+1),&swaits);CHKERRQ(ierr);
1736   for (proc=0, k=0; proc<size; proc++) {
1737     if (!len_s[proc]) continue;
1738     i    = owners_co[proc];
1739     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1740     k++;
1741   }
1742 
1743   /* receives and sends of coj are complete */
1744   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
1745   for (i=0; i<merge->nrecv; i++) {
1746     PetscMPIInt icompleted;
1747     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1748   }
1749   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1750   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1751 
1752   /* send and recv coi */
1753   /*-------------------*/
1754   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1755   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1756   ierr   = PetscMalloc1((len+1),&buf_s);CHKERRQ(ierr);
1757   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1758   for (proc=0,k=0; proc<size; proc++) {
1759     if (!len_s[proc]) continue;
1760     /* form outgoing message for i-structure:
1761          buf_si[0]:                 nrows to be sent
1762                [1:nrows]:           row index (global)
1763                [nrows+1:2*nrows+1]: i-structure index
1764     */
1765     /*-------------------------------------------*/
1766     nrows       = len_si[proc]/2 - 1;
1767     buf_si_i    = buf_si + nrows+1;
1768     buf_si[0]   = nrows;
1769     buf_si_i[0] = 0;
1770     nrows       = 0;
1771     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1772       nzi               = coi[i+1] - coi[i];
1773       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1774       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1775       nrows++;
1776     }
1777     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1778     k++;
1779     buf_si += len_si[proc];
1780   }
1781   i = merge->nrecv;
1782   while (i--) {
1783     PetscMPIInt icompleted;
1784     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1785   }
1786   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1787   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1788   ierr = PetscFree(len_si);CHKERRQ(ierr);
1789   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1790   ierr = PetscFree(swaits);CHKERRQ(ierr);
1791   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1792   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1793 
1794   /* compute the local portion of C (mpi mat) */
1795   /*------------------------------------------*/
1796   /* allocate bi array and free space for accumulating nonzero column info */
1797   ierr  = PetscMalloc1((pn+1),&bi);CHKERRQ(ierr);
1798   bi[0] = 0;
1799 
1800   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
1801   nnz           = fill*(pdti[pn] + poti[pon] + ai[am]);
1802   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1803   current_space = free_space;
1804 
1805   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1806   for (k=0; k<merge->nrecv; k++) {
1807     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1808     nrows       = *buf_ri_k[k];
1809     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1810     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th recieved i-structure  */
1811   }
1812 
1813   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1814   rmax = 0;
1815   for (i=0; i<pn; i++) {
1816     /* add pdt[i,:]*AP into lnk */
1817     pnz = pdti[i+1] - pdti[i];
1818     ptJ = pdtj + pdti[i];
1819     for (j=0; j<pnz; j++) {
1820       row  = ptJ[j];  /* row of AP == col of Pt */
1821       anz  = ai[row+1] - ai[row];
1822       Jptr = aj + ai[row];
1823       /* add non-zero cols of AP into the sorted linked list lnk */
1824       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1825     }
1826 
1827     /* add received col data into lnk */
1828     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1829       if (i == *nextrow[k]) { /* i-th row */
1830         nzi  = *(nextci[k]+1) - *nextci[k];
1831         Jptr = buf_rj[k] + *nextci[k];
1832         ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
1833         nextrow[k]++; nextci[k]++;
1834       }
1835     }
1836     nnz = lnk[0];
1837 
1838     /* if free space is not available, make more free space */
1839     if (current_space->local_remaining<nnz) {
1840       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1841       nspacedouble++;
1842     }
1843     /* copy data into free space, then initialize lnk */
1844     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1845     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1846 
1847     current_space->array           += nnz;
1848     current_space->local_used      += nnz;
1849     current_space->local_remaining -= nnz;
1850 
1851     bi[i+1] = bi[i] + nnz;
1852     if (nnz > rmax) rmax = nnz;
1853   }
1854   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1855 
1856   ierr      = PetscMalloc1((bi[pn]+1),&bj);CHKERRQ(ierr);
1857   ierr      = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1858   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
1859   if (afill_tmp > afill) afill = afill_tmp;
1860   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
1861   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1862   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1863 
1864   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
1865   /*----------------------------------------------------------------------------------*/
1866   ierr = PetscCalloc1(rmax+1,&vals);CHKERRQ(ierr);
1867 
1868   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1869   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1870   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
1871   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1872   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1873   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1874   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
1875   for (i=0; i<pn; i++) {
1876     row  = i + rstart;
1877     nnz  = bi[i+1] - bi[i];
1878     Jptr = bj + bi[i];
1879     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
1880   }
1881   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1882   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1883   ierr = PetscFree(vals);CHKERRQ(ierr);
1884 
1885   merge->bi        = bi;
1886   merge->bj        = bj;
1887   merge->coi       = coi;
1888   merge->coj       = coj;
1889   merge->buf_ri    = buf_ri;
1890   merge->buf_rj    = buf_rj;
1891   merge->owners_co = owners_co;
1892   merge->destroy   = Cmpi->ops->destroy;
1893   merge->duplicate = Cmpi->ops->duplicate;
1894 
1895   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ;
1896   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
1897 
1898   /* attach the supporting struct to Cmpi for reuse */
1899   c = (Mat_MPIAIJ*)Cmpi->data;
1900 
1901   c->ptap     = ptap;
1902   ptap->api   = NULL;
1903   ptap->apj   = NULL;
1904   ptap->merge = merge;
1905   ptap->rmax  = rmax;
1906   ptap->apa   = NULL;
1907 
1908   *C = Cmpi;
1909 #if defined(PETSC_USE_INFO)
1910   if (bi[pn] != 0) {
1911     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1912     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
1913   } else {
1914     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1915   }
1916 #endif
1917   PetscFunctionReturn(0);
1918 }
1919