xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision 2bf68e3e0f2a61f71e7c65bee250bfa1c8ce0cdb)
1 
2 /*
3   Defines matrix-matrix product routines for pairs of MPIAIJ matrices
4           C = A * B
5 */
6 #include <../src/mat/impls/aij/seq/aij.h> /*I "petscmat.h" I*/
7 #include <../src/mat/utils/freespace.h>
8 #include <../src/mat/impls/aij/mpi/mpiaij.h>
9 #include <petscbt.h>
10 #include <../src/mat/impls/dense/mpi/mpidense.h>
11 #include <petsc/private/vecimpl.h>
12 
13 #if defined(PETSC_HAVE_HYPRE)
14 PETSC_INTERN PetscErrorCode MatMatMultSymbolic_AIJ_AIJ_wHYPRE(Mat,Mat,PetscReal,Mat*);
15 #endif
16 
17 #undef __FUNCT__
18 #define __FUNCT__ "MatMatMult_MPIAIJ_MPIAIJ"
19 PETSC_INTERN PetscErrorCode MatMatMult_MPIAIJ_MPIAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill, Mat *C)
20 {
21   PetscErrorCode ierr;
22 #if defined(PETSC_HAVE_HYPRE)
23   const char     *algTypes[3] = {"scalable","nonscalable","hypre"};
24   PetscInt       nalg = 3;
25 #else
26   const char     *algTypes[2] = {"scalable","nonscalable"};
27   PetscInt       nalg = 2;
28 #endif
29   PetscInt       alg = 1; /* set default algorithm */
30   MPI_Comm       comm;
31 
32   PetscFunctionBegin;
33   if (scall == MAT_INITIAL_MATRIX) {
34     ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
35     if (A->cmap->rstart != B->rmap->rstart || A->cmap->rend != B->rmap->rend) SETERRQ4(comm,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, (%D, %D) != (%D,%D)",A->cmap->rstart,A->cmap->rend,B->rmap->rstart,B->rmap->rend);
36 
37     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
38     ierr = PetscOptionsEList("-matmatmult_via","Algorithmic approach","MatMatMult",algTypes,nalg,algTypes[1],&alg,NULL);CHKERRQ(ierr);
39     ierr = PetscOptionsEnd();CHKERRQ(ierr);
40 
41     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
42     switch (alg) {
43     case 1:
44       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(A,B,fill,C);CHKERRQ(ierr);
45       break;
46 #if defined(PETSC_HAVE_HYPRE)
47     case 2:
48       ierr = MatMatMultSymbolic_AIJ_AIJ_wHYPRE(A,B,fill,C);CHKERRQ(ierr);
49       break;
50 #endif
51     default:
52       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ(A,B,fill,C);CHKERRQ(ierr);
53       break;
54     }
55     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
56   }
57   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
58   ierr = (*(*C)->ops->matmultnumeric)(A,B,*C);CHKERRQ(ierr);
59   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
60   PetscFunctionReturn(0);
61 }
62 
63 #undef __FUNCT__
64 #define __FUNCT__ "MatDestroy_MPIAIJ_MatMatMult"
65 PetscErrorCode MatDestroy_MPIAIJ_MatMatMult(Mat A)
66 {
67   PetscErrorCode ierr;
68   Mat_MPIAIJ     *a    = (Mat_MPIAIJ*)A->data;
69   Mat_PtAPMPI    *ptap = a->ptap;
70 
71   PetscFunctionBegin;
72   ierr = PetscFree2(ptap->startsj_s,ptap->startsj_r);CHKERRQ(ierr);
73   ierr = PetscFree(ptap->bufa);CHKERRQ(ierr);
74   ierr = MatDestroy(&ptap->P_loc);CHKERRQ(ierr);
75   ierr = MatDestroy(&ptap->P_oth);CHKERRQ(ierr);
76   ierr = MatDestroy(&ptap->Pt);CHKERRQ(ierr);
77   ierr = PetscFree(ptap->api);CHKERRQ(ierr);
78   ierr = PetscFree(ptap->apj);CHKERRQ(ierr);
79   ierr = PetscFree(ptap->apa);CHKERRQ(ierr);
80   ierr = ptap->destroy(A);CHKERRQ(ierr);
81   ierr = PetscFree(ptap);CHKERRQ(ierr);
82   PetscFunctionReturn(0);
83 }
84 
85 #undef __FUNCT__
86 #define __FUNCT__ "MatDuplicate_MPIAIJ_MatMatMult"
87 PetscErrorCode MatDuplicate_MPIAIJ_MatMatMult(Mat A, MatDuplicateOption op, Mat *M)
88 {
89   PetscErrorCode ierr;
90   Mat_MPIAIJ     *a    = (Mat_MPIAIJ*)A->data;
91   Mat_PtAPMPI    *ptap = a->ptap;
92 
93   PetscFunctionBegin;
94   ierr = (*ptap->duplicate)(A,op,M);CHKERRQ(ierr);
95 
96   (*M)->ops->destroy   = ptap->destroy;   /* = MatDestroy_MPIAIJ, *M doesn't duplicate A's special structure! */
97   (*M)->ops->duplicate = ptap->duplicate; /* = MatDuplicate_MPIAIJ */
98   PetscFunctionReturn(0);
99 }
100 
101 #undef __FUNCT__
102 #define __FUNCT__ "MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable"
103 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,Mat C)
104 {
105   PetscErrorCode ierr;
106   Mat_MPIAIJ     *a  =(Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
107   Mat_SeqAIJ     *ad =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
108   Mat_SeqAIJ     *cd =(Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
109   PetscScalar    *cda=cd->a,*coa=co->a;
110   Mat_SeqAIJ     *p_loc,*p_oth;
111   PetscScalar    *apa,*ca;
112   PetscInt       cm   =C->rmap->n;
113   Mat_PtAPMPI    *ptap=c->ptap;
114   PetscInt       *api,*apj,*apJ,i,k;
115   PetscInt       cstart=C->cmap->rstart;
116   PetscInt       cdnz,conz,k0,k1;
117   MPI_Comm       comm;
118   PetscMPIInt    size;
119 
120   PetscFunctionBegin;
121   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
122   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
123 
124   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
125   /*-----------------------------------------------------*/
126   /* update numerical values of P_oth and P_loc */
127   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
128   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
129 
130   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
131   /*----------------------------------------------------------*/
132   /* get data from symbolic products */
133   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
134   p_oth = NULL;
135   if (size >1) {
136     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
137   }
138 
139   /* get apa for storing dense row A[i,:]*P */
140   apa = ptap->apa;
141 
142   api = ptap->api;
143   apj = ptap->apj;
144   for (i=0; i<cm; i++) {
145     /* compute apa = A[i,:]*P */
146     AProw_nonscalable(i,ad,ao,p_loc,p_oth,apa);
147 
148     /* set values in C */
149     apJ  = apj + api[i];
150     cdnz = cd->i[i+1] - cd->i[i];
151     conz = co->i[i+1] - co->i[i];
152 
153     /* 1st off-diagoanl part of C */
154     ca = coa + co->i[i];
155     k  = 0;
156     for (k0=0; k0<conz; k0++) {
157       if (apJ[k] >= cstart) break;
158       ca[k0]      = apa[apJ[k]];
159       apa[apJ[k]] = 0.0;
160       k++;
161     }
162 
163     /* diagonal part of C */
164     ca = cda + cd->i[i];
165     for (k1=0; k1<cdnz; k1++) {
166       ca[k1]      = apa[apJ[k]];
167       apa[apJ[k]] = 0.0;
168       k++;
169     }
170 
171     /* 2nd off-diagoanl part of C */
172     ca = coa + co->i[i];
173     for (; k0<conz; k0++) {
174       ca[k0]      = apa[apJ[k]];
175       apa[apJ[k]] = 0.0;
176       k++;
177     }
178   }
179   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
180   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
181   PetscFunctionReturn(0);
182 }
183 
184 #undef __FUNCT__
185 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable"
186 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,PetscReal fill,Mat *C)
187 {
188   PetscErrorCode     ierr;
189   MPI_Comm           comm;
190   PetscMPIInt        size;
191   Mat                Cmpi;
192   Mat_PtAPMPI        *ptap;
193   PetscFreeSpaceList free_space=NULL,current_space=NULL;
194   Mat_MPIAIJ         *a        =(Mat_MPIAIJ*)A->data,*c;
195   Mat_SeqAIJ         *ad       =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
196   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
197   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
198   PetscInt           *lnk,i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi;
199   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n,Crmax;
200   PetscBT            lnkbt;
201   PetscScalar        *apa;
202   PetscReal          afill;
203   PetscTable         ta;
204 
205   PetscFunctionBegin;
206   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
207   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
208 
209   /* create struct Mat_PtAPMPI and attached it to C later */
210   ierr = PetscNew(&ptap);CHKERRQ(ierr);
211 
212   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
213   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
214 
215   /* get P_loc by taking all local rows of P */
216   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
217 
218   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
219   pi_loc = p_loc->i; pj_loc = p_loc->j;
220   if (size > 1) {
221     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
222     pi_oth = p_oth->i; pj_oth = p_oth->j;
223   } else {
224     p_oth = NULL;
225     pi_oth = NULL; pj_oth = NULL;
226   }
227 
228   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
229   /*-------------------------------------------------------------------*/
230   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
231   ptap->api = api;
232   api[0]    = 0;
233 
234   /* create and initialize a linked list */
235   ierr = PetscTableCreate(pN,pN,&ta);CHKERRQ(ierr);
236   MatRowMergeMax_SeqAIJ(p_loc,ptap->P_loc->rmap->N,ta);
237   MatRowMergeMax_SeqAIJ(p_oth,ptap->P_oth->rmap->N,ta);
238   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
239   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
240 
241   ierr = PetscLLCondensedCreate(Crmax,pN,&lnk,&lnkbt);CHKERRQ(ierr);
242 
243   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
244   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space);CHKERRQ(ierr);
245   current_space = free_space;
246 
247   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
248   for (i=0; i<am; i++) {
249     /* diagonal portion of A */
250     nzi = adi[i+1] - adi[i];
251     for (j=0; j<nzi; j++) {
252       row  = *adj++;
253       pnz  = pi_loc[row+1] - pi_loc[row];
254       Jptr = pj_loc + pi_loc[row];
255       /* add non-zero cols of P into the sorted linked list lnk */
256       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
257     }
258     /* off-diagonal portion of A */
259     nzi = aoi[i+1] - aoi[i];
260     for (j=0; j<nzi; j++) {
261       row  = *aoj++;
262       pnz  = pi_oth[row+1] - pi_oth[row];
263       Jptr = pj_oth + pi_oth[row];
264       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
265     }
266 
267     apnz     = lnk[0];
268     api[i+1] = api[i] + apnz;
269 
270     /* if free space is not available, double the total space in the list */
271     if (current_space->local_remaining<apnz) {
272       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
273       nspacedouble++;
274     }
275 
276     /* Copy data into free space, then initialize lnk */
277     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
278     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
279 
280     current_space->array           += apnz;
281     current_space->local_used      += apnz;
282     current_space->local_remaining -= apnz;
283   }
284 
285   /* Allocate space for apj, initialize apj, and */
286   /* destroy list of free space and other temporary array(s) */
287   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
288   apj  = ptap->apj;
289   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
290   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
291 
292   /* malloc apa to store dense row A[i,:]*P */
293   ierr = PetscCalloc1(pN,&apa);CHKERRQ(ierr);
294 
295   ptap->apa = apa;
296 
297   /* create and assemble symbolic parallel matrix Cmpi */
298   /*----------------------------------------------------*/
299   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
300   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
301   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
302 
303   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
304   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
305   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
306   for (i=0; i<am; i++) {
307     row  = i + rstart;
308     apnz = api[i+1] - api[i];
309     ierr = MatSetValues(Cmpi,1,&row,apnz,apj,apa,INSERT_VALUES);CHKERRQ(ierr);
310     apj += apnz;
311   }
312   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
313   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
314 
315   ptap->destroy        = Cmpi->ops->destroy;
316   ptap->duplicate      = Cmpi->ops->duplicate;
317   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
318   Cmpi->ops->destroy   = MatDestroy_MPIAIJ_MatMatMult;
319   Cmpi->ops->duplicate = MatDuplicate_MPIAIJ_MatMatMult;
320 
321   /* attach the supporting struct to Cmpi for reuse */
322   c       = (Mat_MPIAIJ*)Cmpi->data;
323   c->ptap = ptap;
324 
325   *C = Cmpi;
326 
327   /* set MatInfo */
328   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
329   if (afill < 1.0) afill = 1.0;
330   Cmpi->info.mallocs           = nspacedouble;
331   Cmpi->info.fill_ratio_given  = fill;
332   Cmpi->info.fill_ratio_needed = afill;
333 
334 #if defined(PETSC_USE_INFO)
335   if (api[am]) {
336     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
337     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
338   } else {
339     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
340   }
341 #endif
342   PetscFunctionReturn(0);
343 }
344 
345 #undef __FUNCT__
346 #define __FUNCT__ "MatMatMult_MPIAIJ_MPIDense"
347 PETSC_INTERN PetscErrorCode MatMatMult_MPIAIJ_MPIDense(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
348 {
349   PetscErrorCode ierr;
350 
351   PetscFunctionBegin;
352   if (scall == MAT_INITIAL_MATRIX) {
353     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
354     ierr = MatMatMultSymbolic_MPIAIJ_MPIDense(A,B,fill,C);CHKERRQ(ierr);
355     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
356   }
357   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
358   ierr = MatMatMultNumeric_MPIAIJ_MPIDense(A,B,*C);CHKERRQ(ierr);
359   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
360   PetscFunctionReturn(0);
361 }
362 
363 typedef struct {
364   Mat         workB;
365   PetscScalar *rvalues,*svalues;
366   MPI_Request *rwaits,*swaits;
367 } MPIAIJ_MPIDense;
368 
369 #undef __FUNCT__
370 #define __FUNCT__ "MatMPIAIJ_MPIDenseDestroy"
371 PetscErrorCode MatMPIAIJ_MPIDenseDestroy(void *ctx)
372 {
373   MPIAIJ_MPIDense *contents = (MPIAIJ_MPIDense*) ctx;
374   PetscErrorCode  ierr;
375 
376   PetscFunctionBegin;
377   ierr = MatDestroy(&contents->workB);CHKERRQ(ierr);
378   ierr = PetscFree4(contents->rvalues,contents->svalues,contents->rwaits,contents->swaits);CHKERRQ(ierr);
379   ierr = PetscFree(contents);CHKERRQ(ierr);
380   PetscFunctionReturn(0);
381 }
382 
383 #undef __FUNCT__
384 #define __FUNCT__ "MatMatMultNumeric_MPIDense"
385 /*
386     This is a "dummy function" that handles the case where matrix C was created as a dense matrix
387   directly by the user and passed to MatMatMult() with the MAT_REUSE_MATRIX option
388 
389   It is the same as MatMatMultSymbolic_MPIAIJ_MPIDense() except does not create C
390 */
391 PetscErrorCode MatMatMultNumeric_MPIDense(Mat A,Mat B,Mat C)
392 {
393   PetscErrorCode         ierr;
394   PetscBool              flg;
395   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*) A->data;
396   PetscInt               nz   = aij->B->cmap->n;
397   PetscContainer         container;
398   MPIAIJ_MPIDense        *contents;
399   VecScatter             ctx   = aij->Mvctx;
400   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
401   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
402 
403   PetscFunctionBegin;
404   ierr = PetscObjectTypeCompare((PetscObject)B,MATMPIDENSE,&flg);CHKERRQ(ierr);
405   if (!flg) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Second matrix must be mpidense");
406 
407   /* Handle case where where user provided the final C matrix rather than calling MatMatMult() with MAT_INITIAL_MATRIX*/
408   ierr = PetscObjectTypeCompare((PetscObject)A,MATMPIAIJ,&flg);CHKERRQ(ierr);
409   if (!flg) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"First matrix must be MPIAIJ");
410 
411   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
412 
413   ierr = PetscNew(&contents);CHKERRQ(ierr);
414   /* Create work matrix used to store off processor rows of B needed for local product */
415   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,B->cmap->N,NULL,&contents->workB);CHKERRQ(ierr);
416   /* Create work arrays needed */
417   ierr = PetscMalloc4(B->cmap->N*from->starts[from->n],&contents->rvalues,
418                       B->cmap->N*to->starts[to->n],&contents->svalues,
419                       from->n,&contents->rwaits,
420                       to->n,&contents->swaits);CHKERRQ(ierr);
421 
422   ierr = PetscContainerCreate(PetscObjectComm((PetscObject)A),&container);CHKERRQ(ierr);
423   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
424   ierr = PetscContainerSetUserDestroy(container,MatMPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
425   ierr = PetscObjectCompose((PetscObject)C,"workB",(PetscObject)container);CHKERRQ(ierr);
426   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
427 
428   ierr = (*C->ops->matmultnumeric)(A,B,C);CHKERRQ(ierr);
429   PetscFunctionReturn(0);
430 }
431 
432 #undef __FUNCT__
433 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIDense"
434 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat A,Mat B,PetscReal fill,Mat *C)
435 {
436   PetscErrorCode         ierr;
437   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*) A->data;
438   PetscInt               nz   = aij->B->cmap->n;
439   PetscContainer         container;
440   MPIAIJ_MPIDense        *contents;
441   VecScatter             ctx   = aij->Mvctx;
442   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
443   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
444   PetscInt               m     = A->rmap->n,n=B->cmap->n;
445 
446   PetscFunctionBegin;
447   ierr = MatCreate(PetscObjectComm((PetscObject)B),C);CHKERRQ(ierr);
448   ierr = MatSetSizes(*C,m,n,A->rmap->N,B->cmap->N);CHKERRQ(ierr);
449   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
450   ierr = MatSetType(*C,MATMPIDENSE);CHKERRQ(ierr);
451   ierr = MatMPIDenseSetPreallocation(*C,NULL);CHKERRQ(ierr);
452   ierr = MatAssemblyBegin(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
453   ierr = MatAssemblyEnd(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
454 
455   (*C)->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
456 
457   ierr = PetscNew(&contents);CHKERRQ(ierr);
458   /* Create work matrix used to store off processor rows of B needed for local product */
459   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,B->cmap->N,NULL,&contents->workB);CHKERRQ(ierr);
460   /* Create work arrays needed */
461   ierr = PetscMalloc4(B->cmap->N*from->starts[from->n],&contents->rvalues,
462                       B->cmap->N*to->starts[to->n],&contents->svalues,
463                       from->n,&contents->rwaits,
464                       to->n,&contents->swaits);CHKERRQ(ierr);
465 
466   ierr = PetscContainerCreate(PetscObjectComm((PetscObject)A),&container);CHKERRQ(ierr);
467   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
468   ierr = PetscContainerSetUserDestroy(container,MatMPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
469   ierr = PetscObjectCompose((PetscObject)(*C),"workB",(PetscObject)container);CHKERRQ(ierr);
470   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
471   PetscFunctionReturn(0);
472 }
473 
474 #undef __FUNCT__
475 #define __FUNCT__ "MatMPIDenseScatter"
476 /*
477     Performs an efficient scatter on the rows of B needed by this process; this is
478     a modification of the VecScatterBegin_() routines.
479 */
480 PetscErrorCode MatMPIDenseScatter(Mat A,Mat B,Mat C,Mat *outworkB)
481 {
482   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*)A->data;
483   PetscErrorCode         ierr;
484   PetscScalar            *b,*w,*svalues,*rvalues;
485   VecScatter             ctx   = aij->Mvctx;
486   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
487   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
488   PetscInt               i,j,k;
489   PetscInt               *sindices,*sstarts,*rindices,*rstarts;
490   PetscMPIInt            *sprocs,*rprocs,nrecvs;
491   MPI_Request            *swaits,*rwaits;
492   MPI_Comm               comm;
493   PetscMPIInt            tag  = ((PetscObject)ctx)->tag,ncols = B->cmap->N, nrows = aij->B->cmap->n,imdex,nrowsB = B->rmap->n;
494   MPI_Status             status;
495   MPIAIJ_MPIDense        *contents;
496   PetscContainer         container;
497   Mat                    workB;
498 
499   PetscFunctionBegin;
500   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
501   ierr = PetscObjectQuery((PetscObject)C,"workB",(PetscObject*)&container);CHKERRQ(ierr);
502   if (!container) SETERRQ(comm,PETSC_ERR_PLIB,"Container does not exist");
503   ierr = PetscContainerGetPointer(container,(void**)&contents);CHKERRQ(ierr);
504 
505   workB = *outworkB = contents->workB;
506   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);
507   sindices = to->indices;
508   sstarts  = to->starts;
509   sprocs   = to->procs;
510   swaits   = contents->swaits;
511   svalues  = contents->svalues;
512 
513   rindices = from->indices;
514   rstarts  = from->starts;
515   rprocs   = from->procs;
516   rwaits   = contents->rwaits;
517   rvalues  = contents->rvalues;
518 
519   ierr = MatDenseGetArray(B,&b);CHKERRQ(ierr);
520   ierr = MatDenseGetArray(workB,&w);CHKERRQ(ierr);
521 
522   for (i=0; i<from->n; i++) {
523     ierr = MPI_Irecv(rvalues+ncols*rstarts[i],ncols*(rstarts[i+1]-rstarts[i]),MPIU_SCALAR,rprocs[i],tag,comm,rwaits+i);CHKERRQ(ierr);
524   }
525 
526   for (i=0; i<to->n; i++) {
527     /* pack a message at a time */
528     for (j=0; j<sstarts[i+1]-sstarts[i]; j++) {
529       for (k=0; k<ncols; k++) {
530         svalues[ncols*(sstarts[i] + j) + k] = b[sindices[sstarts[i]+j] + nrowsB*k];
531       }
532     }
533     ierr = MPI_Isend(svalues+ncols*sstarts[i],ncols*(sstarts[i+1]-sstarts[i]),MPIU_SCALAR,sprocs[i],tag,comm,swaits+i);CHKERRQ(ierr);
534   }
535 
536   nrecvs = from->n;
537   while (nrecvs) {
538     ierr = MPI_Waitany(from->n,rwaits,&imdex,&status);CHKERRQ(ierr);
539     nrecvs--;
540     /* unpack a message at a time */
541     for (j=0; j<rstarts[imdex+1]-rstarts[imdex]; j++) {
542       for (k=0; k<ncols; k++) {
543         w[rindices[rstarts[imdex]+j] + nrows*k] = rvalues[ncols*(rstarts[imdex] + j) + k];
544       }
545     }
546   }
547   if (to->n) {ierr = MPI_Waitall(to->n,swaits,to->sstatus);CHKERRQ(ierr);}
548 
549   ierr = MatDenseRestoreArray(B,&b);CHKERRQ(ierr);
550   ierr = MatDenseRestoreArray(workB,&w);CHKERRQ(ierr);
551   ierr = MatAssemblyBegin(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
552   ierr = MatAssemblyEnd(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
553   PetscFunctionReturn(0);
554 }
555 extern PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat,Mat,Mat);
556 
557 #undef __FUNCT__
558 #define __FUNCT__ "MatMatMultNumeric_MPIAIJ_MPIDense"
559 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat A,Mat B,Mat C)
560 {
561   PetscErrorCode ierr;
562   Mat_MPIAIJ     *aij    = (Mat_MPIAIJ*)A->data;
563   Mat_MPIDense   *bdense = (Mat_MPIDense*)B->data;
564   Mat_MPIDense   *cdense = (Mat_MPIDense*)C->data;
565   Mat            workB;
566 
567   PetscFunctionBegin;
568   /* diagonal block of A times all local rows of B*/
569   ierr = MatMatMultNumeric_SeqAIJ_SeqDense(aij->A,bdense->A,cdense->A);CHKERRQ(ierr);
570 
571   /* get off processor parts of B needed to complete the product */
572   ierr = MatMPIDenseScatter(A,B,C,&workB);CHKERRQ(ierr);
573 
574   /* off-diagonal block of A times nonlocal rows of B */
575   ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A);CHKERRQ(ierr);
576   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
577   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
578   PetscFunctionReturn(0);
579 }
580 
581 #undef __FUNCT__
582 #define __FUNCT__ "MatMatMultNumeric_MPIAIJ_MPIAIJ"
583 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
584 {
585   PetscErrorCode ierr;
586   Mat_MPIAIJ     *a   = (Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
587   Mat_SeqAIJ     *ad  = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
588   Mat_SeqAIJ     *cd  = (Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
589   PetscInt       *adi = ad->i,*adj,*aoi=ao->i,*aoj;
590   PetscScalar    *ada,*aoa,*cda=cd->a,*coa=co->a;
591   Mat_SeqAIJ     *p_loc,*p_oth;
592   PetscInt       *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
593   PetscScalar    *pa_loc,*pa_oth,*pa,valtmp,*ca;
594   PetscInt       cm          = C->rmap->n,anz,pnz;
595   Mat_PtAPMPI    *ptap       = c->ptap;
596   PetscScalar    *apa_sparse = ptap->apa;
597   PetscInt       *api,*apj,*apJ,i,j,k,row;
598   PetscInt       cstart = C->cmap->rstart;
599   PetscInt       cdnz,conz,k0,k1,nextp;
600   MPI_Comm       comm;
601   PetscMPIInt    size;
602 
603   PetscFunctionBegin;
604   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
605   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
606 
607   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
608   /*-----------------------------------------------------*/
609   /* update numerical values of P_oth and P_loc */
610   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
611   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
612 
613   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
614   /*----------------------------------------------------------*/
615   /* get data from symbolic products */
616   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
617   pi_loc = p_loc->i; pj_loc = p_loc->j; pa_loc = p_loc->a;
618   if (size >1) {
619     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
620     pi_oth = p_oth->i; pj_oth = p_oth->j; pa_oth = p_oth->a;
621   } else {
622     p_oth = NULL; pi_oth = NULL; pj_oth = NULL; pa_oth = NULL;
623   }
624 
625   api = ptap->api;
626   apj = ptap->apj;
627   for (i=0; i<cm; i++) {
628     apJ = apj + api[i];
629 
630     /* diagonal portion of A */
631     anz = adi[i+1] - adi[i];
632     adj = ad->j + adi[i];
633     ada = ad->a + adi[i];
634     for (j=0; j<anz; j++) {
635       row = adj[j];
636       pnz = pi_loc[row+1] - pi_loc[row];
637       pj  = pj_loc + pi_loc[row];
638       pa  = pa_loc + pi_loc[row];
639       /* perform sparse axpy */
640       valtmp = ada[j];
641       nextp  = 0;
642       for (k=0; nextp<pnz; k++) {
643         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
644           apa_sparse[k] += valtmp*pa[nextp++];
645         }
646       }
647       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
648     }
649 
650     /* off-diagonal portion of A */
651     anz = aoi[i+1] - aoi[i];
652     aoj = ao->j + aoi[i];
653     aoa = ao->a + aoi[i];
654     for (j=0; j<anz; j++) {
655       row = aoj[j];
656       pnz = pi_oth[row+1] - pi_oth[row];
657       pj  = pj_oth + pi_oth[row];
658       pa  = pa_oth + pi_oth[row];
659       /* perform sparse axpy */
660       valtmp = aoa[j];
661       nextp  = 0;
662       for (k=0; nextp<pnz; k++) {
663         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
664           apa_sparse[k] += valtmp*pa[nextp++];
665         }
666       }
667       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
668     }
669 
670     /* set values in C */
671     cdnz = cd->i[i+1] - cd->i[i];
672     conz = co->i[i+1] - co->i[i];
673 
674     /* 1st off-diagoanl part of C */
675     ca = coa + co->i[i];
676     k  = 0;
677     for (k0=0; k0<conz; k0++) {
678       if (apJ[k] >= cstart) break;
679       ca[k0]        = apa_sparse[k];
680       apa_sparse[k] = 0.0;
681       k++;
682     }
683 
684     /* diagonal part of C */
685     ca = cda + cd->i[i];
686     for (k1=0; k1<cdnz; k1++) {
687       ca[k1]        = apa_sparse[k];
688       apa_sparse[k] = 0.0;
689       k++;
690     }
691 
692     /* 2nd off-diagoanl part of C */
693     ca = coa + co->i[i];
694     for (; k0<conz; k0++) {
695       ca[k0]        = apa_sparse[k];
696       apa_sparse[k] = 0.0;
697       k++;
698     }
699   }
700   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
701   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
702   PetscFunctionReturn(0);
703 }
704 
705 /* same as MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(), except using LLCondensed to avoid O(BN) memory requirement */
706 #undef __FUNCT__
707 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIAIJ"
708 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat *C)
709 {
710   PetscErrorCode     ierr;
711   MPI_Comm           comm;
712   PetscMPIInt        size;
713   Mat                Cmpi;
714   Mat_PtAPMPI        *ptap;
715   PetscFreeSpaceList free_space = NULL,current_space=NULL;
716   Mat_MPIAIJ         *a         = (Mat_MPIAIJ*)A->data,*c;
717   Mat_SeqAIJ         *ad        = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
718   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
719   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
720   PetscInt           i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi,*lnk,apnz_max;
721   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
722   PetscReal          afill;
723   PetscScalar        *apa;
724   PetscTable         ta;
725 
726   PetscFunctionBegin;
727   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
728   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
729 
730   /* create struct Mat_PtAPMPI and attached it to C later */
731   ierr = PetscNew(&ptap);CHKERRQ(ierr);
732 
733   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
734   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
735 
736   /* get P_loc by taking all local rows of P */
737   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
738 
739   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
740   pi_loc = p_loc->i; pj_loc = p_loc->j;
741   if (size > 1) {
742     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
743     pi_oth = p_oth->i; pj_oth = p_oth->j;
744   } else {
745     p_oth  = NULL;
746     pi_oth = NULL; pj_oth = NULL;
747   }
748 
749   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
750   /*-------------------------------------------------------------------*/
751   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
752   ptap->api = api;
753   api[0]    = 0;
754 
755   /* create and initialize a linked list */
756   apnz_max = 6*(p_loc->rmax + (PetscInt)(1.e-2*pN)); /* expected apnz_max */
757   if (apnz_max > pN) apnz_max = pN;
758   ierr = PetscTableCreate(apnz_max,pN,&ta);CHKERRQ(ierr);
759 
760   /* Calculate apnz_max */
761   apnz_max = 0;
762   for (i=0; i<am; i++) {
763     ierr = PetscTableRemoveAll(ta);CHKERRQ(ierr);
764     /* diagonal portion of A */
765     nzi  = adi[i+1] - adi[i];
766     Jptr = adj+adi[i];  /* cols of A_diag */
767     MatMergeRows_SeqAIJ(p_loc,nzi,Jptr,ta);
768     ierr = PetscTableGetCount(ta,&apnz);CHKERRQ(ierr);
769     if (apnz_max < apnz) apnz_max = apnz;
770 
771     /*  off-diagonal portion of A */
772     nzi = aoi[i+1] - aoi[i];
773     Jptr = aoj+aoi[i];  /* cols of A_off */
774     MatMergeRows_SeqAIJ(p_oth,nzi,Jptr,ta);
775     ierr = PetscTableGetCount(ta,&apnz);CHKERRQ(ierr);
776     if (apnz_max < apnz) apnz_max = apnz;
777   }
778   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
779 
780   ierr = PetscLLCondensedCreate_Scalable(apnz_max,&lnk);CHKERRQ(ierr);
781 
782   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
783   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space);CHKERRQ(ierr);
784   current_space = free_space;
785   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
786   for (i=0; i<am; i++) {
787     /* diagonal portion of A */
788     nzi = adi[i+1] - adi[i];
789     for (j=0; j<nzi; j++) {
790       row  = *adj++;
791       pnz  = pi_loc[row+1] - pi_loc[row];
792       Jptr = pj_loc + pi_loc[row];
793       /* add non-zero cols of P into the sorted linked list lnk */
794       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
795     }
796     /* off-diagonal portion of A */
797     nzi = aoi[i+1] - aoi[i];
798     for (j=0; j<nzi; j++) {
799       row  = *aoj++;
800       pnz  = pi_oth[row+1] - pi_oth[row];
801       Jptr = pj_oth + pi_oth[row];
802       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
803     }
804 
805     apnz     = *lnk;
806     api[i+1] = api[i] + apnz;
807 
808     /* if free space is not available, double the total space in the list */
809     if (current_space->local_remaining<apnz) {
810       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
811       nspacedouble++;
812     }
813 
814     /* Copy data into free space, then initialize lnk */
815     ierr = PetscLLCondensedClean_Scalable(apnz,current_space->array,lnk);CHKERRQ(ierr);
816     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
817 
818     current_space->array           += apnz;
819     current_space->local_used      += apnz;
820     current_space->local_remaining -= apnz;
821   }
822 
823   /* Allocate space for apj, initialize apj, and */
824   /* destroy list of free space and other temporary array(s) */
825   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
826   apj  = ptap->apj;
827   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
828   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
829 
830   /* create and assemble symbolic parallel matrix Cmpi */
831   /*----------------------------------------------------*/
832   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
833   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
834   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
835   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
836   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
837   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
838 
839   /* malloc apa for assembly Cmpi */
840   ierr = PetscCalloc1(apnz_max,&apa);CHKERRQ(ierr);
841 
842   ptap->apa = apa;
843   for (i=0; i<am; i++) {
844     row  = i + rstart;
845     apnz = api[i+1] - api[i];
846     ierr = MatSetValues(Cmpi,1,&row,apnz,apj,apa,INSERT_VALUES);CHKERRQ(ierr);
847     apj += apnz;
848   }
849   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
850   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
851 
852   ptap->destroy             = Cmpi->ops->destroy;
853   ptap->duplicate           = Cmpi->ops->duplicate;
854   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ;
855   Cmpi->ops->destroy        = MatDestroy_MPIAIJ_MatMatMult;
856   Cmpi->ops->duplicate      = MatDuplicate_MPIAIJ_MatMatMult;
857 
858   /* attach the supporting struct to Cmpi for reuse */
859   c       = (Mat_MPIAIJ*)Cmpi->data;
860   c->ptap = ptap;
861 
862   *C = Cmpi;
863 
864   /* set MatInfo */
865   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
866   if (afill < 1.0) afill = 1.0;
867   Cmpi->info.mallocs           = nspacedouble;
868   Cmpi->info.fill_ratio_given  = fill;
869   Cmpi->info.fill_ratio_needed = afill;
870 
871 #if defined(PETSC_USE_INFO)
872   if (api[am]) {
873     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
874     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
875   } else {
876     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
877   }
878 #endif
879   PetscFunctionReturn(0);
880 }
881 
882 /*-------------------------------------------------------------------------*/
883 #undef __FUNCT__
884 #define __FUNCT__ "MatTransposeMatMult_MPIAIJ_MPIAIJ"
885 PetscErrorCode MatTransposeMatMult_MPIAIJ_MPIAIJ(Mat P,Mat A,MatReuse scall,PetscReal fill,Mat *C)
886 {
887   PetscErrorCode ierr;
888   const char     *algTypes[3] = {"scalable","nonscalable","matmatmult"};
889   PetscInt       alg=0; /* set default algorithm */
890 
891   PetscFunctionBegin;
892   if (scall == MAT_INITIAL_MATRIX) {
893     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
894     ierr = PetscOptionsEList("-mattransposematmult_via","Algorithmic approach","MatTransposeMatMult",algTypes,3,algTypes[0],&alg,NULL);CHKERRQ(ierr);
895     ierr = PetscOptionsEnd();CHKERRQ(ierr);
896 
897     ierr = PetscLogEventBegin(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
898     switch (alg) {
899     case 1:
900       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(P,A,fill,C);CHKERRQ(ierr);
901       break;
902     case 2:
903     {
904       Mat         Pt;
905       Mat_PtAPMPI *ptap;
906       Mat_MPIAIJ  *c;
907       ierr = MatTranspose(P,MAT_INITIAL_MATRIX,&Pt);CHKERRQ(ierr);
908       ierr = MatMatMult(Pt,A,MAT_INITIAL_MATRIX,fill,C);CHKERRQ(ierr);
909       c        = (Mat_MPIAIJ*)(*C)->data;
910       ptap     = c->ptap;
911       ptap->Pt = Pt;
912       (*C)->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult;
913       PetscFunctionReturn(0);
914     }
915       break;
916     default:
917       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(P,A,fill,C);CHKERRQ(ierr);
918       break;
919     }
920     ierr = PetscLogEventEnd(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
921   }
922   ierr = PetscLogEventBegin(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
923   ierr = (*(*C)->ops->mattransposemultnumeric)(P,A,*C);CHKERRQ(ierr);
924   ierr = PetscLogEventEnd(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
925   PetscFunctionReturn(0);
926 }
927 
928 /* This routine only works when scall=MAT_REUSE_MATRIX! */
929 #undef __FUNCT__
930 #define __FUNCT__ "MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult"
931 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult(Mat P,Mat A,Mat C)
932 {
933   PetscErrorCode ierr;
934   Mat_MPIAIJ     *c=(Mat_MPIAIJ*)C->data;
935   Mat_PtAPMPI    *ptap= c->ptap;
936   Mat            Pt=ptap->Pt;
937 
938   PetscFunctionBegin;
939   ierr = MatTranspose(P,MAT_REUSE_MATRIX,&Pt);CHKERRQ(ierr);
940   ierr = MatMatMultNumeric(Pt,A,C);CHKERRQ(ierr);
941   PetscFunctionReturn(0);
942 }
943 
944 /* Non-scalable version, use dense axpy */
945 #undef __FUNCT__
946 #define __FUNCT__ "MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable"
947 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,Mat C)
948 {
949   PetscErrorCode      ierr;
950   Mat_Merge_SeqsToMPI *merge;
951   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
952   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
953   Mat_PtAPMPI         *ptap;
954   PetscInt            *adj,*aJ;
955   PetscInt            i,j,k,anz,pnz,row,*cj;
956   MatScalar           *ada,*aval,*ca,valtmp;
957   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
958   MPI_Comm            comm;
959   PetscMPIInt         size,rank,taga,*len_s;
960   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
961   PetscInt            **buf_ri,**buf_rj;
962   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
963   MPI_Request         *s_waits,*r_waits;
964   MPI_Status          *status;
965   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
966   PetscInt            *ai,*aj,*coi,*coj;
967   PetscInt            *poJ,*pdJ;
968   Mat                 A_loc;
969   Mat_SeqAIJ          *a_loc;
970 
971   PetscFunctionBegin;
972   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
973   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
974   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
975 
976   ptap  = c->ptap;
977   merge = ptap->merge;
978 
979   /* 2) compute numeric C_seq = P_loc^T*A_loc*P - dominating part */
980   /*--------------------------------------------------------------*/
981   /* get data from symbolic products */
982   coi  = merge->coi; coj = merge->coj;
983   ierr = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
984 
985   bi     = merge->bi; bj = merge->bj;
986   owners = merge->rowmap->range;
987   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
988 
989   /* get A_loc by taking all local rows of A */
990   A_loc = ptap->A_loc;
991   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
992   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
993   ai    = a_loc->i;
994   aj    = a_loc->j;
995 
996   ierr = PetscCalloc1(A->cmap->N,&aval);CHKERRQ(ierr); /* non-scalable!!! */
997 
998   for (i=0; i<am; i++) {
999     /* 2-a) put A[i,:] to dense array aval */
1000     anz = ai[i+1] - ai[i];
1001     adj = aj + ai[i];
1002     ada = a_loc->a + ai[i];
1003     for (j=0; j<anz; j++) {
1004       aval[adj[j]] = ada[j];
1005     }
1006 
1007     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1008     /*--------------------------------------------------------------*/
1009     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1010     pnz = po->i[i+1] - po->i[i];
1011     poJ = po->j + po->i[i];
1012     pA  = po->a + po->i[i];
1013     for (j=0; j<pnz; j++) {
1014       row = poJ[j];
1015       cnz = coi[row+1] - coi[row];
1016       cj  = coj + coi[row];
1017       ca  = coa + coi[row];
1018       /* perform dense axpy */
1019       valtmp = pA[j];
1020       for (k=0; k<cnz; k++) {
1021         ca[k] += valtmp*aval[cj[k]];
1022       }
1023       ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1024     }
1025 
1026     /* put the value into Cd (diagonal part) */
1027     pnz = pd->i[i+1] - pd->i[i];
1028     pdJ = pd->j + pd->i[i];
1029     pA  = pd->a + pd->i[i];
1030     for (j=0; j<pnz; j++) {
1031       row = pdJ[j];
1032       cnz = bi[row+1] - bi[row];
1033       cj  = bj + bi[row];
1034       ca  = ba + bi[row];
1035       /* perform dense axpy */
1036       valtmp = pA[j];
1037       for (k=0; k<cnz; k++) {
1038         ca[k] += valtmp*aval[cj[k]];
1039       }
1040       ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1041     }
1042 
1043     /* zero the current row of Pt*A */
1044     aJ = aj + ai[i];
1045     for (k=0; k<anz; k++) aval[aJ[k]] = 0.0;
1046   }
1047 
1048   /* 3) send and recv matrix values coa */
1049   /*------------------------------------*/
1050   buf_ri = merge->buf_ri;
1051   buf_rj = merge->buf_rj;
1052   len_s  = merge->len_s;
1053   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1054   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1055 
1056   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1057   for (proc=0,k=0; proc<size; proc++) {
1058     if (!len_s[proc]) continue;
1059     i    = merge->owners_co[proc];
1060     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1061     k++;
1062   }
1063   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1064   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1065 
1066   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1067   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1068   ierr = PetscFree(coa);CHKERRQ(ierr);
1069 
1070   /* 4) insert local Cseq and received values into Cmpi */
1071   /*----------------------------------------------------*/
1072   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1073   for (k=0; k<merge->nrecv; k++) {
1074     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1075     nrows       = *(buf_ri_k[k]);
1076     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1077     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1078   }
1079 
1080   for (i=0; i<cm; i++) {
1081     row  = owners[rank] + i; /* global row index of C_seq */
1082     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1083     ba_i = ba + bi[i];
1084     bnz  = bi[i+1] - bi[i];
1085     /* add received vals into ba */
1086     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1087       /* i-th row */
1088       if (i == *nextrow[k]) {
1089         cnz    = *(nextci[k]+1) - *nextci[k];
1090         cj     = buf_rj[k] + *(nextci[k]);
1091         ca     = abuf_r[k] + *(nextci[k]);
1092         nextcj = 0;
1093         for (j=0; nextcj<cnz; j++) {
1094           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1095             ba_i[j] += ca[nextcj++];
1096           }
1097         }
1098         nextrow[k]++; nextci[k]++;
1099         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1100       }
1101     }
1102     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1103   }
1104   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1105   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1106 
1107   ierr = PetscFree(ba);CHKERRQ(ierr);
1108   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1109   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1110   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1111   ierr = PetscFree(aval);CHKERRQ(ierr);
1112   PetscFunctionReturn(0);
1113 }
1114 
1115 PetscErrorCode MatDuplicate_MPIAIJ_MatPtAP(Mat, MatDuplicateOption,Mat*);
1116 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1117 #undef __FUNCT__
1118 #define __FUNCT__ "MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable"
1119 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,PetscReal fill,Mat *C)
1120 {
1121   PetscErrorCode      ierr;
1122   Mat                 Cmpi,A_loc,POt,PDt;
1123   Mat_PtAPMPI         *ptap;
1124   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1125   Mat_MPIAIJ          *p        =(Mat_MPIAIJ*)P->data,*c;
1126   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1127   PetscInt            nnz;
1128   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1129   PetscInt            am=A->rmap->n,pn=P->cmap->n;
1130   PetscBT             lnkbt;
1131   MPI_Comm            comm;
1132   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1133   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1134   PetscInt            len,proc,*dnz,*onz,*owners;
1135   PetscInt            nzi,*bi,*bj;
1136   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1137   MPI_Request         *swaits,*rwaits;
1138   MPI_Status          *sstatus,rstatus;
1139   Mat_Merge_SeqsToMPI *merge;
1140   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1141   PetscReal           afill  =1.0,afill_tmp;
1142   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N;
1143   PetscScalar         *vals;
1144   Mat_SeqAIJ          *a_loc, *pdt,*pot;
1145 
1146   PetscFunctionBegin;
1147   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1148   /* check if matrix local sizes are compatible */
1149   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend) SETERRQ4(comm,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, A (%D, %D) != P (%D,%D)",A->rmap->rstart,A->rmap->rend,P->rmap->rstart,P->rmap->rend);
1150 
1151   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1152   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1153 
1154   /* create struct Mat_PtAPMPI and attached it to C later */
1155   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1156 
1157   /* get A_loc by taking all local rows of A */
1158   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1159 
1160   ptap->A_loc = A_loc;
1161 
1162   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1163   ai    = a_loc->i;
1164   aj    = a_loc->j;
1165 
1166   /* determine symbolic Co=(p->B)^T*A - send to others */
1167   /*----------------------------------------------------*/
1168   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1169   pdt  = (Mat_SeqAIJ*)PDt->data;
1170   pdti = pdt->i; pdtj = pdt->j;
1171 
1172   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1173   pot  = (Mat_SeqAIJ*)POt->data;
1174   poti = pot->i; potj = pot->j;
1175 
1176   /* then, compute symbolic Co = (p->B)^T*A */
1177   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors >= (num of nonzero rows of C_seq) - pn */
1178   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
1179   coi[0] = 0;
1180 
1181   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1182   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(poti[pon],ai[am]));
1183   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1184   current_space = free_space;
1185 
1186   /* create and initialize a linked list */
1187   ierr = PetscLLCondensedCreate(aN,aN,&lnk,&lnkbt);CHKERRQ(ierr);
1188 
1189   for (i=0; i<pon; i++) {
1190     pnz = poti[i+1] - poti[i];
1191     ptJ = potj + poti[i];
1192     for (j=0; j<pnz; j++) {
1193       row  = ptJ[j]; /* row of A_loc == col of Pot */
1194       anz  = ai[row+1] - ai[row];
1195       Jptr = aj + ai[row];
1196       /* add non-zero cols of AP into the sorted linked list lnk */
1197       ierr = PetscLLCondensedAddSorted(anz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1198     }
1199     nnz = lnk[0];
1200 
1201     /* If free space is not available, double the total space in the list */
1202     if (current_space->local_remaining<nnz) {
1203       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1204       nspacedouble++;
1205     }
1206 
1207     /* Copy data into free space, and zero out denserows */
1208     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1209 
1210     current_space->array           += nnz;
1211     current_space->local_used      += nnz;
1212     current_space->local_remaining -= nnz;
1213 
1214     coi[i+1] = coi[i] + nnz;
1215   }
1216 
1217   ierr = PetscMalloc1(coi[pon]+1,&coj);CHKERRQ(ierr);
1218   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1219 
1220   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1221   if (afill_tmp > afill) afill = afill_tmp;
1222 
1223   /* send j-array (coj) of Co to other processors */
1224   /*----------------------------------------------*/
1225   /* determine row ownership */
1226   ierr = PetscNew(&merge);CHKERRQ(ierr);
1227   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1228 
1229   merge->rowmap->n  = pn;
1230   merge->rowmap->bs = 1;
1231 
1232   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1233   owners = merge->rowmap->range;
1234 
1235   /* determine the number of messages to send, their lengths */
1236   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
1237   ierr = PetscMalloc1(size,&merge->len_s);CHKERRQ(ierr);
1238 
1239   len_s        = merge->len_s;
1240   merge->nsend = 0;
1241 
1242   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
1243   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1244 
1245   proc = 0;
1246   for (i=0; i<pon; i++) {
1247     while (prmap[i] >= owners[proc+1]) proc++;
1248     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1249     len_s[proc] += coi[i+1] - coi[i];
1250   }
1251 
1252   len          = 0; /* max length of buf_si[] */
1253   owners_co[0] = 0;
1254   for (proc=0; proc<size; proc++) {
1255     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1256     if (len_si[proc]) {
1257       merge->nsend++;
1258       len_si[proc] = 2*(len_si[proc] + 1);
1259       len         += len_si[proc];
1260     }
1261   }
1262 
1263   /* determine the number and length of messages to receive for coi and coj  */
1264   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1265   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1266 
1267   /* post the Irecv and Isend of coj */
1268   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1269   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1270   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
1271   for (proc=0, k=0; proc<size; proc++) {
1272     if (!len_s[proc]) continue;
1273     i    = owners_co[proc];
1274     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1275     k++;
1276   }
1277 
1278   /* receives and sends of coj are complete */
1279   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
1280   for (i=0; i<merge->nrecv; i++) {
1281     PetscMPIInt icompleted;
1282     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1283   }
1284   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1285   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1286 
1287   /* send and recv coi */
1288   /*-------------------*/
1289   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1290   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1291   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1292   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1293   for (proc=0,k=0; proc<size; proc++) {
1294     if (!len_s[proc]) continue;
1295     /* form outgoing message for i-structure:
1296          buf_si[0]:                 nrows to be sent
1297                [1:nrows]:           row index (global)
1298                [nrows+1:2*nrows+1]: i-structure index
1299     */
1300     /*-------------------------------------------*/
1301     nrows       = len_si[proc]/2 - 1;
1302     buf_si_i    = buf_si + nrows+1;
1303     buf_si[0]   = nrows;
1304     buf_si_i[0] = 0;
1305     nrows       = 0;
1306     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1307       nzi               = coi[i+1] - coi[i];
1308       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi; /* i-structure */
1309       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1310       nrows++;
1311     }
1312     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1313     k++;
1314     buf_si += len_si[proc];
1315   }
1316   i = merge->nrecv;
1317   while (i--) {
1318     PetscMPIInt icompleted;
1319     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1320   }
1321   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1322   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1323   ierr = PetscFree(len_si);CHKERRQ(ierr);
1324   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1325   ierr = PetscFree(swaits);CHKERRQ(ierr);
1326   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1327   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1328 
1329   /* compute the local portion of C (mpi mat) */
1330   /*------------------------------------------*/
1331   /* allocate bi array and free space for accumulating nonzero column info */
1332   ierr  = PetscMalloc1(pn+1,&bi);CHKERRQ(ierr);
1333   bi[0] = 0;
1334 
1335   /* set initial free space to be fill*(nnz(P) + nnz(A)) */
1336   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(pdti[pn],PetscIntSumTruncate(poti[pon],ai[am])));
1337   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1338   current_space = free_space;
1339 
1340   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1341   for (k=0; k<merge->nrecv; k++) {
1342     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1343     nrows       = *buf_ri_k[k];
1344     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1345     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1346   }
1347 
1348   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1349   rmax = 0;
1350   for (i=0; i<pn; i++) {
1351     /* add pdt[i,:]*AP into lnk */
1352     pnz = pdti[i+1] - pdti[i];
1353     ptJ = pdtj + pdti[i];
1354     for (j=0; j<pnz; j++) {
1355       row  = ptJ[j];  /* row of AP == col of Pt */
1356       anz  = ai[row+1] - ai[row];
1357       Jptr = aj + ai[row];
1358       /* add non-zero cols of AP into the sorted linked list lnk */
1359       ierr = PetscLLCondensedAddSorted(anz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1360     }
1361 
1362     /* add received col data into lnk */
1363     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1364       if (i == *nextrow[k]) { /* i-th row */
1365         nzi  = *(nextci[k]+1) - *nextci[k];
1366         Jptr = buf_rj[k] + *nextci[k];
1367         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1368         nextrow[k]++; nextci[k]++;
1369       }
1370     }
1371     nnz = lnk[0];
1372 
1373     /* if free space is not available, make more free space */
1374     if (current_space->local_remaining<nnz) {
1375       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1376       nspacedouble++;
1377     }
1378     /* copy data into free space, then initialize lnk */
1379     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1380     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1381 
1382     current_space->array           += nnz;
1383     current_space->local_used      += nnz;
1384     current_space->local_remaining -= nnz;
1385 
1386     bi[i+1] = bi[i] + nnz;
1387     if (nnz > rmax) rmax = nnz;
1388   }
1389   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1390 
1391   ierr = PetscMalloc1(bi[pn]+1,&bj);CHKERRQ(ierr);
1392   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1393 
1394   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
1395   if (afill_tmp > afill) afill = afill_tmp;
1396   ierr = PetscLLCondensedDestroy(lnk,lnkbt);CHKERRQ(ierr);
1397   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1398   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1399 
1400   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
1401   /*----------------------------------------------------------------------------------*/
1402   ierr = PetscCalloc1(rmax+1,&vals);CHKERRQ(ierr);
1403 
1404   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1405   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1406   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
1407   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1408   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1409   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1410   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
1411   for (i=0; i<pn; i++) {
1412     row  = i + rstart;
1413     nnz  = bi[i+1] - bi[i];
1414     Jptr = bj + bi[i];
1415     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
1416   }
1417   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1418   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1419   ierr = PetscFree(vals);CHKERRQ(ierr);
1420 
1421   merge->bi        = bi;
1422   merge->bj        = bj;
1423   merge->coi       = coi;
1424   merge->coj       = coj;
1425   merge->buf_ri    = buf_ri;
1426   merge->buf_rj    = buf_rj;
1427   merge->owners_co = owners_co;
1428 
1429   /* attach the supporting struct to Cmpi for reuse */
1430   c           = (Mat_MPIAIJ*)Cmpi->data;
1431   c->ptap     = ptap;
1432   ptap->api   = NULL;
1433   ptap->apj   = NULL;
1434   ptap->merge = merge;
1435   ptap->destroy   = Cmpi->ops->destroy;
1436   ptap->duplicate = Cmpi->ops->duplicate;
1437 
1438   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1439   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
1440   Cmpi->ops->duplicate               = MatDuplicate_MPIAIJ_MatPtAP;
1441 
1442   *C = Cmpi;
1443 #if defined(PETSC_USE_INFO)
1444   if (bi[pn] != 0) {
1445     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1446     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
1447   } else {
1448     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1449   }
1450 #endif
1451   PetscFunctionReturn(0);
1452 }
1453 
1454 #undef __FUNCT__
1455 #define __FUNCT__ "MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ"
1456 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ(Mat P,Mat A,Mat C)
1457 {
1458   PetscErrorCode      ierr;
1459   Mat_Merge_SeqsToMPI *merge;
1460   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1461   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
1462   Mat_PtAPMPI         *ptap;
1463   PetscInt            *adj;
1464   PetscInt            i,j,k,anz,pnz,row,*cj,nexta;
1465   MatScalar           *ada,*ca,valtmp;
1466   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
1467   MPI_Comm            comm;
1468   PetscMPIInt         size,rank,taga,*len_s;
1469   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
1470   PetscInt            **buf_ri,**buf_rj;
1471   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
1472   MPI_Request         *s_waits,*r_waits;
1473   MPI_Status          *status;
1474   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
1475   PetscInt            *ai,*aj,*coi,*coj;
1476   PetscInt            *poJ,*pdJ;
1477   Mat                 A_loc;
1478   Mat_SeqAIJ          *a_loc;
1479 
1480   PetscFunctionBegin;
1481   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1482   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1483   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1484 
1485   ptap  = c->ptap;
1486   merge = ptap->merge;
1487 
1488   /* 2) compute numeric C_seq = P_loc^T*A_loc */
1489   /*------------------------------------------*/
1490   /* get data from symbolic products */
1491   coi    = merge->coi; coj = merge->coj;
1492   ierr   = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
1493   bi     = merge->bi; bj = merge->bj;
1494   owners = merge->rowmap->range;
1495   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
1496 
1497   /* get A_loc by taking all local rows of A */
1498   A_loc = ptap->A_loc;
1499   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
1500   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1501   ai    = a_loc->i;
1502   aj    = a_loc->j;
1503 
1504   for (i=0; i<am; i++) {
1505     anz = ai[i+1] - ai[i];
1506     adj = aj + ai[i];
1507     ada = a_loc->a + ai[i];
1508 
1509     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1510     /*-------------------------------------------------------------*/
1511     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1512     pnz = po->i[i+1] - po->i[i];
1513     poJ = po->j + po->i[i];
1514     pA  = po->a + po->i[i];
1515     for (j=0; j<pnz; j++) {
1516       row = poJ[j];
1517       cj  = coj + coi[row];
1518       ca  = coa + coi[row];
1519       /* perform sparse axpy */
1520       nexta  = 0;
1521       valtmp = pA[j];
1522       for (k=0; nexta<anz; k++) {
1523         if (cj[k] == adj[nexta]) {
1524           ca[k] += valtmp*ada[nexta];
1525           nexta++;
1526         }
1527       }
1528       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1529     }
1530 
1531     /* put the value into Cd (diagonal part) */
1532     pnz = pd->i[i+1] - pd->i[i];
1533     pdJ = pd->j + pd->i[i];
1534     pA  = pd->a + pd->i[i];
1535     for (j=0; j<pnz; j++) {
1536       row = pdJ[j];
1537       cj  = bj + bi[row];
1538       ca  = ba + bi[row];
1539       /* perform sparse axpy */
1540       nexta  = 0;
1541       valtmp = pA[j];
1542       for (k=0; nexta<anz; k++) {
1543         if (cj[k] == adj[nexta]) {
1544           ca[k] += valtmp*ada[nexta];
1545           nexta++;
1546         }
1547       }
1548       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1549     }
1550   }
1551 
1552   /* 3) send and recv matrix values coa */
1553   /*------------------------------------*/
1554   buf_ri = merge->buf_ri;
1555   buf_rj = merge->buf_rj;
1556   len_s  = merge->len_s;
1557   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1558   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1559 
1560   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1561   for (proc=0,k=0; proc<size; proc++) {
1562     if (!len_s[proc]) continue;
1563     i    = merge->owners_co[proc];
1564     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1565     k++;
1566   }
1567   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1568   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1569 
1570   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1571   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1572   ierr = PetscFree(coa);CHKERRQ(ierr);
1573 
1574   /* 4) insert local Cseq and received values into Cmpi */
1575   /*----------------------------------------------------*/
1576   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1577   for (k=0; k<merge->nrecv; k++) {
1578     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1579     nrows       = *(buf_ri_k[k]);
1580     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1581     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1582   }
1583 
1584   for (i=0; i<cm; i++) {
1585     row  = owners[rank] + i; /* global row index of C_seq */
1586     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1587     ba_i = ba + bi[i];
1588     bnz  = bi[i+1] - bi[i];
1589     /* add received vals into ba */
1590     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1591       /* i-th row */
1592       if (i == *nextrow[k]) {
1593         cnz    = *(nextci[k]+1) - *nextci[k];
1594         cj     = buf_rj[k] + *(nextci[k]);
1595         ca     = abuf_r[k] + *(nextci[k]);
1596         nextcj = 0;
1597         for (j=0; nextcj<cnz; j++) {
1598           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1599             ba_i[j] += ca[nextcj++];
1600           }
1601         }
1602         nextrow[k]++; nextci[k]++;
1603         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1604       }
1605     }
1606     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1607   }
1608   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1609   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1610 
1611   ierr = PetscFree(ba);CHKERRQ(ierr);
1612   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1613   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1614   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1615   PetscFunctionReturn(0);
1616 }
1617 
1618 PetscErrorCode MatDuplicate_MPIAIJ_MatPtAP(Mat, MatDuplicateOption,Mat*);
1619 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ();
1620    differ from MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable in using LLCondensedCreate_Scalable() */
1621 #undef __FUNCT__
1622 #define __FUNCT__ "MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ"
1623 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P,Mat A,PetscReal fill,Mat *C)
1624 {
1625   PetscErrorCode      ierr;
1626   Mat                 Cmpi,A_loc,POt,PDt;
1627   Mat_PtAPMPI         *ptap;
1628   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1629   Mat_MPIAIJ          *p        =(Mat_MPIAIJ*)P->data,*c;
1630   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1631   PetscInt            nnz;
1632   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1633   PetscInt            am  =A->rmap->n,pn=P->cmap->n;
1634   MPI_Comm            comm;
1635   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1636   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1637   PetscInt            len,proc,*dnz,*onz,*owners;
1638   PetscInt            nzi,*bi,*bj;
1639   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1640   MPI_Request         *swaits,*rwaits;
1641   MPI_Status          *sstatus,rstatus;
1642   Mat_Merge_SeqsToMPI *merge;
1643   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1644   PetscReal           afill  =1.0,afill_tmp;
1645   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Armax;
1646   PetscScalar         *vals;
1647   Mat_SeqAIJ          *a_loc,*pdt,*pot;
1648   PetscTable          ta;
1649 
1650   PetscFunctionBegin;
1651   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1652   /* check if matrix local sizes are compatible */
1653   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend) SETERRQ4(comm,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, A (%D, %D) != P (%D,%D)",A->rmap->rstart,A->rmap->rend,P->rmap->rstart,P->rmap->rend);
1654 
1655   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1656   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1657 
1658   /* create struct Mat_PtAPMPI and attached it to C later */
1659   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1660 
1661   /* get A_loc by taking all local rows of A */
1662   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1663 
1664   ptap->A_loc = A_loc;
1665   a_loc       = (Mat_SeqAIJ*)(A_loc)->data;
1666   ai          = a_loc->i;
1667   aj          = a_loc->j;
1668 
1669   /* determine symbolic Co=(p->B)^T*A - send to others */
1670   /*----------------------------------------------------*/
1671   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1672   pdt  = (Mat_SeqAIJ*)PDt->data;
1673   pdti = pdt->i; pdtj = pdt->j;
1674 
1675   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1676   pot  = (Mat_SeqAIJ*)POt->data;
1677   poti = pot->i; potj = pot->j;
1678 
1679   /* then, compute symbolic Co = (p->B)^T*A */
1680   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1681                          >= (num of nonzero rows of C_seq) - pn */
1682   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
1683   coi[0] = 0;
1684 
1685   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1686   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(poti[pon],ai[am]));
1687   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1688   current_space = free_space;
1689 
1690   /* create and initialize a linked list */
1691   ierr = PetscTableCreate(aN,aN,&ta);CHKERRQ(ierr);
1692   MatRowMergeMax_SeqAIJ(a_loc,am,ta);
1693   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
1694   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
1695 
1696   for (i=0; i<pon; i++) {
1697     pnz = poti[i+1] - poti[i];
1698     ptJ = potj + poti[i];
1699     for (j=0; j<pnz; j++) {
1700       row  = ptJ[j]; /* row of A_loc == col of Pot */
1701       anz  = ai[row+1] - ai[row];
1702       Jptr = aj + ai[row];
1703       /* add non-zero cols of AP into the sorted linked list lnk */
1704       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1705     }
1706     nnz = lnk[0];
1707 
1708     /* If free space is not available, double the total space in the list */
1709     if (current_space->local_remaining<nnz) {
1710       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1711       nspacedouble++;
1712     }
1713 
1714     /* Copy data into free space, and zero out denserows */
1715     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1716 
1717     current_space->array           += nnz;
1718     current_space->local_used      += nnz;
1719     current_space->local_remaining -= nnz;
1720 
1721     coi[i+1] = coi[i] + nnz;
1722   }
1723 
1724   ierr = PetscMalloc1(coi[pon]+1,&coj);CHKERRQ(ierr);
1725   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1726   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr); /* must destroy to get a new one for C */
1727 
1728   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1729   if (afill_tmp > afill) afill = afill_tmp;
1730 
1731   /* send j-array (coj) of Co to other processors */
1732   /*----------------------------------------------*/
1733   /* determine row ownership */
1734   ierr = PetscNew(&merge);CHKERRQ(ierr);
1735   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1736 
1737   merge->rowmap->n  = pn;
1738   merge->rowmap->bs = 1;
1739 
1740   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1741   owners = merge->rowmap->range;
1742 
1743   /* determine the number of messages to send, their lengths */
1744   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
1745   ierr = PetscMalloc1(size,&merge->len_s);CHKERRQ(ierr);
1746 
1747   len_s        = merge->len_s;
1748   merge->nsend = 0;
1749 
1750   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
1751   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1752 
1753   proc = 0;
1754   for (i=0; i<pon; i++) {
1755     while (prmap[i] >= owners[proc+1]) proc++;
1756     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1757     len_s[proc] += coi[i+1] - coi[i];
1758   }
1759 
1760   len          = 0; /* max length of buf_si[] */
1761   owners_co[0] = 0;
1762   for (proc=0; proc<size; proc++) {
1763     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1764     if (len_si[proc]) {
1765       merge->nsend++;
1766       len_si[proc] = 2*(len_si[proc] + 1);
1767       len         += len_si[proc];
1768     }
1769   }
1770 
1771   /* determine the number and length of messages to receive for coi and coj  */
1772   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1773   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1774 
1775   /* post the Irecv and Isend of coj */
1776   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1777   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1778   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
1779   for (proc=0, k=0; proc<size; proc++) {
1780     if (!len_s[proc]) continue;
1781     i    = owners_co[proc];
1782     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1783     k++;
1784   }
1785 
1786   /* receives and sends of coj are complete */
1787   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
1788   for (i=0; i<merge->nrecv; i++) {
1789     PetscMPIInt icompleted;
1790     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1791   }
1792   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1793   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1794 
1795   /* add received column indices into table to update Armax */
1796   for (k=0; k<merge->nrecv; k++) {/* k-th received message */
1797     Jptr = buf_rj[k];
1798     for (j=0; j<merge->len_r[k]; j++) {
1799       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
1800     }
1801   }
1802   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
1803 
1804   /* send and recv coi */
1805   /*-------------------*/
1806   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1807   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1808   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1809   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1810   for (proc=0,k=0; proc<size; proc++) {
1811     if (!len_s[proc]) continue;
1812     /* form outgoing message for i-structure:
1813          buf_si[0]:                 nrows to be sent
1814                [1:nrows]:           row index (global)
1815                [nrows+1:2*nrows+1]: i-structure index
1816     */
1817     /*-------------------------------------------*/
1818     nrows       = len_si[proc]/2 - 1;
1819     buf_si_i    = buf_si + nrows+1;
1820     buf_si[0]   = nrows;
1821     buf_si_i[0] = 0;
1822     nrows       = 0;
1823     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1824       nzi               = coi[i+1] - coi[i];
1825       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1826       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1827       nrows++;
1828     }
1829     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1830     k++;
1831     buf_si += len_si[proc];
1832   }
1833   i = merge->nrecv;
1834   while (i--) {
1835     PetscMPIInt icompleted;
1836     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1837   }
1838   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1839   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1840   ierr = PetscFree(len_si);CHKERRQ(ierr);
1841   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1842   ierr = PetscFree(swaits);CHKERRQ(ierr);
1843   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1844   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1845 
1846   /* compute the local portion of C (mpi mat) */
1847   /*------------------------------------------*/
1848   /* allocate bi array and free space for accumulating nonzero column info */
1849   ierr  = PetscMalloc1(pn+1,&bi);CHKERRQ(ierr);
1850   bi[0] = 0;
1851 
1852   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
1853   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(pdti[pn],PetscIntSumTruncate(poti[pon],ai[am])));
1854   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1855   current_space = free_space;
1856 
1857   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1858   for (k=0; k<merge->nrecv; k++) {
1859     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1860     nrows       = *buf_ri_k[k];
1861     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1862     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th recieved i-structure  */
1863   }
1864 
1865   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
1866   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1867   rmax = 0;
1868   for (i=0; i<pn; i++) {
1869     /* add pdt[i,:]*AP into lnk */
1870     pnz = pdti[i+1] - pdti[i];
1871     ptJ = pdtj + pdti[i];
1872     for (j=0; j<pnz; j++) {
1873       row  = ptJ[j];  /* row of AP == col of Pt */
1874       anz  = ai[row+1] - ai[row];
1875       Jptr = aj + ai[row];
1876       /* add non-zero cols of AP into the sorted linked list lnk */
1877       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1878     }
1879 
1880     /* add received col data into lnk */
1881     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1882       if (i == *nextrow[k]) { /* i-th row */
1883         nzi  = *(nextci[k]+1) - *nextci[k];
1884         Jptr = buf_rj[k] + *nextci[k];
1885         ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
1886         nextrow[k]++; nextci[k]++;
1887       }
1888     }
1889     nnz = lnk[0];
1890 
1891     /* if free space is not available, make more free space */
1892     if (current_space->local_remaining<nnz) {
1893       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1894       nspacedouble++;
1895     }
1896     /* copy data into free space, then initialize lnk */
1897     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1898     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1899 
1900     current_space->array           += nnz;
1901     current_space->local_used      += nnz;
1902     current_space->local_remaining -= nnz;
1903 
1904     bi[i+1] = bi[i] + nnz;
1905     if (nnz > rmax) rmax = nnz;
1906   }
1907   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1908 
1909   ierr      = PetscMalloc1(bi[pn]+1,&bj);CHKERRQ(ierr);
1910   ierr      = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1911   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
1912   if (afill_tmp > afill) afill = afill_tmp;
1913   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
1914   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
1915 
1916   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1917   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1918 
1919   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
1920   /*----------------------------------------------------------------------------------*/
1921   ierr = PetscCalloc1(rmax+1,&vals);CHKERRQ(ierr);
1922 
1923   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1924   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1925   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
1926   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1927   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1928   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1929   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
1930   for (i=0; i<pn; i++) {
1931     row  = i + rstart;
1932     nnz  = bi[i+1] - bi[i];
1933     Jptr = bj + bi[i];
1934     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
1935   }
1936   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1937   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1938   ierr = PetscFree(vals);CHKERRQ(ierr);
1939 
1940   merge->bi        = bi;
1941   merge->bj        = bj;
1942   merge->coi       = coi;
1943   merge->coj       = coj;
1944   merge->buf_ri    = buf_ri;
1945   merge->buf_rj    = buf_rj;
1946   merge->owners_co = owners_co;
1947 
1948   /* attach the supporting struct to Cmpi for reuse */
1949   c = (Mat_MPIAIJ*)Cmpi->data;
1950 
1951   c->ptap     = ptap;
1952   ptap->api   = NULL;
1953   ptap->apj   = NULL;
1954   ptap->merge = merge;
1955   ptap->apa   = NULL;
1956   ptap->destroy   = Cmpi->ops->destroy;
1957   ptap->duplicate = Cmpi->ops->duplicate;
1958 
1959   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ;
1960   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
1961   Cmpi->ops->duplicate               = MatDuplicate_MPIAIJ_MatPtAP;
1962 
1963   *C = Cmpi;
1964 #if defined(PETSC_USE_INFO)
1965   if (bi[pn] != 0) {
1966     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1967     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
1968   } else {
1969     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1970   }
1971 #endif
1972   PetscFunctionReturn(0);
1973 }
1974