xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision e856ceec863ef703e49eb5c04ffcf8aa49e2215c)
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 /*
12 #define DEBUG_MATMATMULT
13  */
14 /*
15 #define DEBUG_MATTrMatMult
16  */
17 
18 #undef __FUNCT__
19 #define __FUNCT__ "MatMatMult_MPIAIJ_MPIAIJ"
20 PetscErrorCode MatMatMult_MPIAIJ_MPIAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill, Mat *C)
21 {
22   PetscErrorCode ierr;
23 
24   PetscFunctionBegin;
25   if (scall == MAT_INITIAL_MATRIX){
26     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
27     ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ(A,B,fill,C);CHKERRQ(ierr);
28     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
29   }
30   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
31   ierr = (*(*C)->ops->matmultnumeric)(A,B,*C);CHKERRQ(ierr);
32   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
33   PetscFunctionReturn(0);
34 }
35 
36 #undef __FUNCT__
37 #define __FUNCT__ "PetscContainerDestroy_Mat_MatMatMultMPI"
38 PetscErrorCode PetscContainerDestroy_Mat_MatMatMultMPI(void *ptr)
39 {
40   PetscErrorCode       ierr;
41   Mat_MatMatMultMPI    *mult=(Mat_MatMatMultMPI*)ptr;
42 
43   PetscFunctionBegin;
44   ierr = ISDestroy(&mult->isrowa);CHKERRQ(ierr);
45   ierr = ISDestroy(&mult->isrowb);CHKERRQ(ierr);
46   ierr = ISDestroy(&mult->iscolb);CHKERRQ(ierr);
47   ierr = MatDestroy(&mult->C_seq);CHKERRQ(ierr);
48   ierr = MatDestroy(&mult->A_loc);CHKERRQ(ierr);
49   ierr = MatDestroy(&mult->B_seq);CHKERRQ(ierr);
50   ierr = PetscFree(mult);CHKERRQ(ierr);
51   PetscFunctionReturn(0);
52 }
53 
54 #undef __FUNCT__
55 #define __FUNCT__ "MatDestroy_MPIAIJ_MatMatMult"
56 PetscErrorCode MatDestroy_MPIAIJ_MatMatMult(Mat A)
57 {
58   PetscErrorCode ierr;
59   Mat_MPIAIJ     *a=(Mat_MPIAIJ*)A->data;
60   Mat_PtAPMPI    *ptap=a->ptap;
61 
62   PetscFunctionBegin;
63   ierr = PetscFree2(ptap->startsj_s,ptap->startsj_r);CHKERRQ(ierr);
64   ierr = PetscFree(ptap->bufa);CHKERRQ(ierr);
65   ierr = MatDestroy(&ptap->P_loc);CHKERRQ(ierr);
66   ierr = MatDestroy(&ptap->P_oth);CHKERRQ(ierr);
67   ierr = PetscFree(ptap->api);CHKERRQ(ierr);
68   ierr = PetscFree(ptap->apj);CHKERRQ(ierr);
69   ierr = PetscFree(ptap->apa);CHKERRQ(ierr);
70   ierr = ptap->destroy(A);CHKERRQ(ierr);
71   ierr = PetscFree(ptap);CHKERRQ(ierr);
72   PetscFunctionReturn(0);
73 }
74 
75 #undef __FUNCT__
76 #define __FUNCT__ "MatDestroy_MPIAIJ_MatMatMult_32"
77 PetscErrorCode MatDestroy_MPIAIJ_MatMatMult_32(Mat A)
78 {
79   PetscErrorCode     ierr;
80   PetscContainer     container;
81   Mat_MatMatMultMPI  *mult=PETSC_NULL;
82 
83   PetscFunctionBegin;
84   ierr = PetscObjectQuery((PetscObject)A,"Mat_MatMatMultMPI",(PetscObject *)&container);CHKERRQ(ierr);
85   if (!container) SETERRQ(((PetscObject)A)->comm,PETSC_ERR_PLIB,"Container does not exist");
86   ierr = PetscContainerGetPointer(container,(void **)&mult);CHKERRQ(ierr);
87   A->ops->destroy   = mult->destroy;
88   A->ops->duplicate = mult->duplicate;
89   if (A->ops->destroy) {
90     ierr = (*A->ops->destroy)(A);CHKERRQ(ierr);
91   }
92   ierr = PetscObjectCompose((PetscObject)A,"Mat_MatMatMultMPI",0);CHKERRQ(ierr);
93   PetscFunctionReturn(0);
94 }
95 
96 #undef __FUNCT__
97 #define __FUNCT__ "MatDuplicate_MPIAIJ_MatMatMult_32"
98 PetscErrorCode MatDuplicate_MPIAIJ_MatMatMult_32(Mat A, MatDuplicateOption op, Mat *M)
99 {
100   PetscErrorCode     ierr;
101   Mat_MatMatMultMPI  *mult;
102   PetscContainer     container;
103 
104   PetscFunctionBegin;
105   ierr = PetscObjectQuery((PetscObject)A,"Mat_MatMatMultMPI",(PetscObject *)&container);CHKERRQ(ierr);
106   if (!container) SETERRQ(((PetscObject)A)->comm,PETSC_ERR_PLIB,"Container does not exist");
107   ierr  = PetscContainerGetPointer(container,(void **)&mult);CHKERRQ(ierr);
108   /* Note: the container is not duplicated, because it requires deep copying of
109      several large data sets (see PetscContainerDestroy_Mat_MatMatMultMPI()).
110      These data sets are only used for repeated calling of MatMatMultNumeric().
111      *M is unlikely being used in this way. Thus we create *M with pure mpiaij format */
112   ierr = (*mult->duplicate)(A,op,M);CHKERRQ(ierr);
113   (*M)->ops->destroy   = mult->destroy;   /* = MatDestroy_MPIAIJ, *M doesn't duplicate A's container! */
114   (*M)->ops->duplicate = mult->duplicate; /* = MatDuplicate_MPIAIJ */
115   PetscFunctionReturn(0);
116 }
117 
118 #undef __FUNCT__
119 #define __FUNCT__ "MatDuplicate_MPIAIJ_MatMatMult"
120 PetscErrorCode MatDuplicate_MPIAIJ_MatMatMult(Mat A, MatDuplicateOption op, Mat *M)
121 {
122   PetscErrorCode     ierr;
123   Mat_MPIAIJ         *a=(Mat_MPIAIJ*)A->data;
124   Mat_PtAPMPI        *ptap=a->ptap;
125 
126   PetscFunctionBegin;
127   ierr = (*ptap->duplicate)(A,op,M);CHKERRQ(ierr);
128   (*M)->ops->destroy   = ptap->destroy;   /* = MatDestroy_MPIAIJ, *M doesn't duplicate A's special structure! */
129   (*M)->ops->duplicate = ptap->duplicate; /* = MatDuplicate_MPIAIJ */
130   PetscFunctionReturn(0);
131 }
132 
133 #undef __FUNCT__
134 #define __FUNCT__ "MatMatMultNumeric_MPIAIJ_MPIAIJ"
135 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
136 {
137   PetscErrorCode     ierr;
138   Mat_MPIAIJ         *a=(Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
139   Mat_SeqAIJ         *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
140   Mat_SeqAIJ         *cd=(Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
141   PetscInt           *adi=ad->i,*adj,*aoi=ao->i,*aoj;
142   PetscScalar        *ada,*aoa,*cda=cd->a,*coa=co->a;
143   Mat_SeqAIJ         *p_loc,*p_oth;
144   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
145   PetscScalar        *pa_loc,*pa_oth,*pa,*apa,valtmp,*ca;
146   PetscInt           cm=C->rmap->n,anz,pnz;
147   Mat_PtAPMPI        *ptap=c->ptap;
148   PetscInt           *api,*apj,*apJ,i,j,k,row;
149   PetscInt           cstart=C->cmap->rstart;
150   PetscInt           cdnz,conz,k0,k1;
151 #if defined(DEBUG_MATMATMULT)
152   PetscMPIInt        rank=a->rank;
153 #endif
154 
155   PetscFunctionBegin;
156 #if defined(DEBUG_MATMATMULT)
157   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] call MatMatMultNumeric_MPIAIJ_MPIAIJ()...\n",rank);
158 #endif
159 
160   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
161   /*-----------------------------------------------------*/
162   /* update numerical values of P_oth and P_loc */
163   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
164   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
165 #if defined(DEBUG_MATMATMULT)
166   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] got P_oth and P_loc...\n",rank);
167 #endif
168 
169   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
170   /*----------------------------------------------------------*/
171   /* get data from symbolic products */
172   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
173   p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
174   pi_loc=p_loc->i; pj_loc=p_loc->j; pa_loc=p_loc->a;
175   pi_oth=p_oth->i; pj_oth=p_oth->j; pa_oth=p_oth->a;
176 
177   /* get apa for storing dense row A[i,:]*P */
178   apa = ptap->apa;
179 
180   api = ptap->api;
181   apj = ptap->apj;
182   for (i=0; i<cm; i++) {
183     /* diagonal portion of A */
184     anz = adi[i+1] - adi[i];
185     adj = ad->j + adi[i];
186     ada = ad->a + adi[i];
187     for (j=0; j<anz; j++) {
188       row = adj[j];
189       pnz = pi_loc[row+1] - pi_loc[row];
190       pj  = pj_loc + pi_loc[row];
191       pa  = pa_loc + pi_loc[row];
192 
193       /* perform dense axpy */
194       valtmp = ada[j];
195       for (k=0; k<pnz; k++){
196         apa[pj[k]] += valtmp*pa[k];
197       }
198       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
199     }
200 
201     /* off-diagonal portion of A */
202     anz = aoi[i+1] - aoi[i];
203     aoj = ao->j + aoi[i];
204     aoa = ao->a + aoi[i];
205     for (j=0; j<anz; j++) {
206       row = aoj[j];
207       pnz = pi_oth[row+1] - pi_oth[row];
208       pj  = pj_oth + pi_oth[row];
209       pa  = pa_oth + pi_oth[row];
210 
211       /* perform dense axpy */
212       valtmp = aoa[j];
213       for (k=0; k<pnz; k++){
214         apa[pj[k]] += valtmp*pa[k];
215       }
216       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
217     }
218 
219     /* set values in C */
220     apJ = apj + api[i];
221     cdnz = cd->i[i+1] - cd->i[i];
222     conz = co->i[i+1] - co->i[i];
223 
224     /* 1st off-diagoanl part of C */
225     ca = coa + co->i[i];
226     k  = 0;
227     for (k0=0; k0<conz; k0++){
228       if (apJ[k] >= cstart) break;
229       ca[k0]      = apa[apJ[k]];
230       apa[apJ[k]] = 0.0;
231       k++;
232     }
233 
234     /* diagonal part of C */
235     ca = cda + cd->i[i];
236     for (k1=0; k1<cdnz; k1++){
237       ca[k1]      = apa[apJ[k]];
238       apa[apJ[k]] = 0.0;
239       k++;
240     }
241 
242     /* 2nd off-diagoanl part of C */
243     ca = coa + co->i[i];
244     for (; k0<conz; k0++){
245       ca[k0]      = apa[apJ[k]];
246       apa[apJ[k]] = 0.0;
247       k++;
248     }
249   }
250   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
251   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
252   PetscFunctionReturn(0);
253 }
254 
255 #undef __FUNCT__
256 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIAIJ"
257 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat *C)
258 {
259   PetscErrorCode       ierr;
260   MPI_Comm             comm=((PetscObject)A)->comm;
261   Mat                  Cmpi;
262   Mat_PtAPMPI          *ptap;
263   PetscFreeSpaceList   free_space=PETSC_NULL,current_space=PETSC_NULL;
264   Mat_MPIAIJ           *a=(Mat_MPIAIJ*)A->data,*c;
265   Mat_SeqAIJ           *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
266   PetscInt             *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
267   PetscInt             *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
268   PetscInt             *lnk,i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi;
269   PetscInt             am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
270   PetscBT              lnkbt;
271   PetscScalar          *apa;
272   PetscReal            afill;
273   PetscBool            scalable=PETSC_FALSE;
274   PetscInt             nlnk_max,armax,prmax;
275 #if defined(DEBUG_MATMATMULT)
276   PetscMPIInt          rank=a->rank;
277 #endif
278 
279   PetscFunctionBegin;
280   if (A->cmap->rstart != P->rmap->rstart || A->cmap->rend != P->rmap->rend){
281     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);
282   }
283 
284   ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
285 
286     ierr = PetscOptionsBool("-matmatmult_32","Use a scalable but slower C=A*B","",scalable,&scalable,PETSC_NULL);CHKERRQ(ierr);
287     if (scalable){
288       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable_32(A,P,fill,C);;CHKERRQ(ierr);
289       PetscFunctionReturn(0);
290     }
291     ierr = PetscOptionsBool("-matmatmult_scalable","Use a scalable but slower C=A*B","",scalable,&scalable,PETSC_NULL);CHKERRQ(ierr);
292     if (scalable){
293       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable(A,P,fill,C);;CHKERRQ(ierr);
294       PetscFunctionReturn(0);
295     }
296   ierr = PetscOptionsEnd();CHKERRQ(ierr);
297 
298 #if defined(DEBUG_MATMATMULT)
299   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] call MatMatMultSymbolic_MPIAIJ_MPIAIJ()...\n",rank);
300 #endif
301 
302   /* create struct Mat_PtAPMPI and attached it to C later */
303   ierr = PetscNew(Mat_PtAPMPI,&ptap);CHKERRQ(ierr);
304 
305   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
306   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
307 #if defined(DEBUG_MATMATMULT)
308   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] P_oth is done...\n",rank);
309 #endif
310   /* get P_loc by taking all local rows of P */
311   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
312 
313   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
314   p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
315   pi_loc = p_loc->i; pj_loc = p_loc->j;
316   pi_oth = p_oth->i; pj_oth = p_oth->j;
317 
318 #if defined(DEBUG_MATMATMULT)
319   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] P_loc is done, Annz %D * P_locnnz %D = %D...\n",rank,ad->i[am]+ao->i[am],p_loc->rmax,(ad->i[am]+ao->i[am])*p_loc->rmax);
320 #endif
321 
322   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
323   /*-------------------------------------------------------------------*/
324   ierr  = PetscMalloc((am+2)*sizeof(PetscInt),&api);CHKERRQ(ierr);
325   ptap->api = api;
326   api[0]    = 0;
327 
328   /* create and initialize a linked list */
329   armax = ad->rmax+ao->rmax;
330   prmax = PetscMax(p_loc->rmax,p_oth->rmax);
331   nlnk_max = armax*prmax;
332   if (!nlnk_max || nlnk_max > pN) nlnk_max = pN;
333 #if defined(DEBUG_MATMATMULT)
334   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] pN %d; nlnk_max %d; Armax %d+%d=%d; Prmax Max(%d,%d)=%d\n",rank,pN,nlnk_max,ad->rmax,ao->rmax,armax,p_loc->rmax,p_oth->rmax,prmax);
335 #endif
336   ierr = PetscLLCondensedCreate(nlnk_max,pN,&lnk,&lnkbt);CHKERRQ(ierr);
337 
338   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
339   ierr = PetscFreeSpaceGet((PetscInt)(fill*(adi[am]+aoi[am]+pi_loc[pm])),&free_space);CHKERRQ(ierr);
340   current_space = free_space;
341 
342   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
343   for (i=0; i<am; i++) {
344     apnz = 0;
345     /* diagonal portion of A */
346     nzi = adi[i+1] - adi[i];
347     for (j=0; j<nzi; j++){
348       row = *adj++;
349       pnz = pi_loc[row+1] - pi_loc[row];
350       Jptr  = pj_loc + pi_loc[row];
351       /* add non-zero cols of P into the sorted linked list lnk */
352       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
353     }
354     /* off-diagonal portion of A */
355     nzi = aoi[i+1] - aoi[i];
356     for (j=0; j<nzi; j++){
357       row = *aoj++;
358       pnz = pi_oth[row+1] - pi_oth[row];
359       Jptr  = pj_oth + pi_oth[row];
360       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
361     }
362 
363     apnz     = lnk[0];
364     api[i+1] = api[i] + apnz;
365 
366     /* if free space is not available, double the total space in the list */
367     if (current_space->local_remaining<apnz) {
368       ierr = PetscFreeSpaceGet(apnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
369       nspacedouble++;
370     }
371 
372     /* Copy data into free space, then initialize lnk */
373     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
374     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
375     current_space->array           += apnz;
376     current_space->local_used      += apnz;
377     current_space->local_remaining -= apnz;
378   }
379 
380   /* Allocate space for apj, initialize apj, and */
381   /* destroy list of free space and other temporary array(s) */
382   ierr = PetscMalloc((api[am]+1)*sizeof(PetscInt),&ptap->apj);CHKERRQ(ierr);
383   apj  = ptap->apj;
384   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
385   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
386 #if defined(DEBUG_MATMATMULT)
387   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] AP is done...\n",rank);
388 #endif
389 
390   /* malloc apa to store dense row A[i,:]*P */
391   ierr = PetscMalloc(pN*sizeof(PetscScalar),&apa);CHKERRQ(ierr);
392   ierr = PetscMemzero(apa,pN*sizeof(PetscScalar));CHKERRQ(ierr);
393   ptap->apa = apa;
394 #if defined(DEBUG_MATMATMULT)
395   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] Malloc apa pN %D is done...\n",rank,pN);
396 #endif
397 
398   /* create and assemble symbolic parallel matrix Cmpi */
399   /*----------------------------------------------------*/
400   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
401   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
402   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
403   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
404   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
405   for (i=0; i<am; i++){
406     row  = i + rstart;
407     apnz = api[i+1] - api[i];
408     ierr = MatSetValues(Cmpi,1,&row,apnz,apj,apa,INSERT_VALUES);CHKERRQ(ierr);
409     apj += apnz;
410   }
411   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
412   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
413 
414   ptap->destroy             = Cmpi->ops->destroy;
415   ptap->duplicate           = Cmpi->ops->duplicate;
416   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ;
417   Cmpi->ops->destroy        = MatDestroy_MPIAIJ_MatMatMult;
418   Cmpi->ops->duplicate      = MatDuplicate_MPIAIJ_MatMatMult;
419 
420   /* attach the supporting struct to Cmpi for reuse */
421   c = (Mat_MPIAIJ*)Cmpi->data;
422   c->ptap  = ptap;
423 
424   *C = Cmpi;
425 
426   /* set MatInfo */
427   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]) + 1.e-5;
428   if (afill < 1.0) afill = 1.0;
429   Cmpi->info.mallocs           = nspacedouble;
430   Cmpi->info.fill_ratio_given  = fill;
431   Cmpi->info.fill_ratio_needed = afill;
432 
433 #if defined(PETSC_USE_INFO)
434   if (api[am]) {
435     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %G needed %G.\n",nspacedouble,fill,afill);CHKERRQ(ierr);
436     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%G,&C) for best performance.;\n",afill);CHKERRQ(ierr);
437   } else {
438     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
439   }
440 #endif
441   PetscFunctionReturn(0);
442 }
443 
444 /* implementation used in PETSc-3.2 */
445 /* This routine is called ONLY in the case of reusing previously computed symbolic C */
446 #undef __FUNCT__
447 #define __FUNCT__ "MatMatMultNumeric_MPIAIJ_MPIAIJ_Scalable_32"
448 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ_Scalable_32(Mat A,Mat B,Mat C)
449 {
450   PetscErrorCode     ierr;
451   Mat                *seq;
452   Mat_MatMatMultMPI  *mult;
453   PetscContainer     container;
454 
455   PetscFunctionBegin;
456   ierr = PetscObjectQuery((PetscObject)C,"Mat_MatMatMultMPI",(PetscObject *)&container);CHKERRQ(ierr);
457   if (!container) SETERRQ(((PetscObject)A)->comm,PETSC_ERR_PLIB,"Container does not exist");
458   ierr  = PetscContainerGetPointer(container,(void **)&mult);CHKERRQ(ierr);
459 
460   if (mult->skipNumeric){ /* first numeric product is done during symbolic product */
461     mult->skipNumeric = PETSC_FALSE;
462     PetscFunctionReturn(0);
463   }
464 #if defined(DEBUG_MATMATMULT)
465   PetscMPIInt rank;
466   MPI_Comm    comm = ((PetscObject)C)->comm;
467   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
468   ierr = MPI_Barrier(comm);CHKERRQ(ierr);
469   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] call MatMatMultNumeric_MPIAIJ_MPIAIJ_Scalable_32()...\n",rank);
470 #endif
471 
472   seq = &mult->B_seq;
473   ierr = MatGetSubMatrices(B,1,&mult->isrowb,&mult->iscolb,MAT_REUSE_MATRIX,&seq);CHKERRQ(ierr);
474   mult->B_seq = *seq;
475 
476   seq = &mult->A_loc;
477   ierr = MatGetSubMatrices(A,1,&mult->isrowa,&mult->isrowb,MAT_REUSE_MATRIX,&seq);CHKERRQ(ierr);
478   mult->A_loc = *seq;
479 
480   ierr = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable(mult->A_loc,mult->B_seq,mult->C_seq);CHKERRQ(ierr);
481 
482   ierr = PetscObjectReference((PetscObject)mult->C_seq);CHKERRQ(ierr);
483   ierr = MatMerge(((PetscObject)A)->comm,mult->C_seq,B->cmap->n,MAT_REUSE_MATRIX,&C);CHKERRQ(ierr);
484   PetscFunctionReturn(0);
485 }
486 
487 /* numeric product is computed as well */
488 #undef __FUNCT__
489 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable_32"
490 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable_32(Mat A,Mat B,PetscReal fill,Mat *C)
491 {
492   PetscErrorCode     ierr;
493   Mat_MatMatMultMPI  *mult;
494   PetscContainer     container;
495   Mat                AB,*seq;
496   Mat_MPIAIJ         *a=(Mat_MPIAIJ*)A->data;
497   PetscInt           *idx,i,start,ncols,nzA,nzB,*cmap,imark;
498 #if defined(DEBUG_MATMATMULT)
499   MPI_Comm             comm = ((PetscObject)A)->comm;
500   PetscMPIInt          rank=a->rank;
501 #endif
502 
503   PetscFunctionBegin;
504 #if defined(DEBUG_MATMATMULT)
505   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] call MatMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable_32()...\n",rank);
506 #endif
507   if (A->cmap->rstart != B->rmap->rstart || A->cmap->rend != B->rmap->rend){
508     SETERRQ4(((PetscObject)A)->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);
509   }
510 
511   ierr = PetscNew(Mat_MatMatMultMPI,&mult);CHKERRQ(ierr);
512 
513   /* get isrowb: nonzero col of A */
514   start = A->cmap->rstart;
515   cmap  = a->garray;
516   nzA   = a->A->cmap->n;
517   nzB   = a->B->cmap->n;
518   ierr  = PetscMalloc((nzA+nzB)*sizeof(PetscInt), &idx);CHKERRQ(ierr);
519   ncols = 0;
520   for (i=0; i<nzB; i++) {  /* row < local row index */
521     if (cmap[i] < start) idx[ncols++] = cmap[i];
522     else break;
523   }
524   imark = i;
525   for (i=0; i<nzA; i++) idx[ncols++] = start + i;  /* local rows */
526   for (i=imark; i<nzB; i++) idx[ncols++] = cmap[i]; /* row > local row index */
527   ierr = ISCreateGeneral(PETSC_COMM_SELF,ncols,idx,PETSC_OWN_POINTER,&mult->isrowb);CHKERRQ(ierr);
528   ierr = ISCreateStride(PETSC_COMM_SELF,B->cmap->N,0,1,&mult->iscolb);CHKERRQ(ierr);
529 
530   /*  get isrowa: all local rows of A */
531   ierr = ISCreateStride(PETSC_COMM_SELF,A->rmap->n,A->rmap->rstart,1,&mult->isrowa);CHKERRQ(ierr);
532 
533   /* Below should go to MatMatMultNumeric_MPIAIJ_MPIAIJ() - How to generate C there? */
534   /* create a seq matrix B_seq = submatrix of B by taking rows of B that equal to nonzero col of A */
535   ierr = MatGetSubMatrices(B,1,&mult->isrowb,&mult->iscolb,MAT_INITIAL_MATRIX,&seq);CHKERRQ(ierr);
536   mult->B_seq = *seq;
537   ierr = PetscFree(seq);CHKERRQ(ierr);
538 #if defined(DEBUG_MATMATMULT)
539   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] B_seq is done...\n",rank);
540 #endif
541 
542   /*  create a seq matrix A_seq = submatrix of A by taking all local rows of A */
543   ierr = MatGetSubMatrices(A,1,&mult->isrowa,&mult->isrowb,MAT_INITIAL_MATRIX,&seq);CHKERRQ(ierr);
544   mult->A_loc = *seq;
545   ierr = PetscFree(seq);CHKERRQ(ierr);
546 #if defined(DEBUG_MATMATMULT)
547   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] A_loc is done...\n",rank);
548 #endif
549 
550   /* compute C_seq = A_seq * B_seq */
551   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable(mult->A_loc,mult->B_seq,fill,&mult->C_seq);CHKERRQ(ierr);
552 #if defined(DEBUG_MATMATMULT)
553   ierr = MPI_Barrier(comm);CHKERRQ(ierr);
554   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] C_seq Symbolic is done...\n",rank);
555 #endif
556   ierr = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable(mult->A_loc,mult->B_seq,mult->C_seq);CHKERRQ(ierr);
557 #if defined(DEBUG_MATMATMULT)
558   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] C_seq Numeric is done...\n",rank);
559 #endif
560 
561   /* create mpi matrix C by concatinating C_seq */
562   ierr = PetscObjectReference((PetscObject)mult->C_seq);CHKERRQ(ierr); /* prevent C_seq being destroyed by MatMerge() */
563   ierr = MatMergeSymbolic(((PetscObject)A)->comm,mult->C_seq,B->cmap->n,&AB);CHKERRQ(ierr);
564   ierr = MatMergeNumeric(((PetscObject)A)->comm,mult->C_seq,B->cmap->n,AB);CHKERRQ(ierr);
565 #if defined(DEBUG_MATMATMULT)
566   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] Merge is done...\n",rank);
567 #endif
568 
569   /* attach the supporting struct to C for reuse of symbolic C */
570   ierr = PetscContainerCreate(PETSC_COMM_SELF,&container);CHKERRQ(ierr);
571   ierr = PetscContainerSetPointer(container,mult);CHKERRQ(ierr);
572   ierr = PetscContainerSetUserDestroy(container,PetscContainerDestroy_Mat_MatMatMultMPI);CHKERRQ(ierr);
573   ierr = PetscObjectCompose((PetscObject)AB,"Mat_MatMatMultMPI",(PetscObject)container);CHKERRQ(ierr);
574   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
575   mult->skipNumeric  = PETSC_TRUE; /* a numeric product is done here */
576   mult->destroy      = AB->ops->destroy;
577   mult->duplicate    = AB->ops->duplicate;
578   AB->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_Scalable_32;
579   AB->ops->destroy        = MatDestroy_MPIAIJ_MatMatMult_32;
580   AB->ops->duplicate      = MatDuplicate_MPIAIJ_MatMatMult_32;
581   AB->ops->matmult        = MatMatMult_MPIAIJ_MPIAIJ;
582   *C                      = AB;
583   PetscFunctionReturn(0);
584 }
585 
586 #undef __FUNCT__
587 #define __FUNCT__ "MatMatMult_MPIAIJ_MPIDense"
588 PetscErrorCode MatMatMult_MPIAIJ_MPIDense(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
589 {
590   PetscErrorCode ierr;
591 
592   PetscFunctionBegin;
593   if (scall == MAT_INITIAL_MATRIX){
594     ierr = MatMatMultSymbolic_MPIAIJ_MPIDense(A,B,fill,C);CHKERRQ(ierr);
595   }
596   ierr = MatMatMultNumeric_MPIAIJ_MPIDense(A,B,*C);CHKERRQ(ierr);
597   PetscFunctionReturn(0);
598 }
599 
600 typedef struct {
601   Mat         workB;
602   PetscScalar *rvalues,*svalues;
603   MPI_Request *rwaits,*swaits;
604 } MPIAIJ_MPIDense;
605 
606 #undef __FUNCT__
607 #define __FUNCT__ "MPIAIJ_MPIDenseDestroy"
608 PetscErrorCode MPIAIJ_MPIDenseDestroy(void *ctx)
609 {
610   MPIAIJ_MPIDense *contents = (MPIAIJ_MPIDense*) ctx;
611   PetscErrorCode  ierr;
612 
613   PetscFunctionBegin;
614   ierr = MatDestroy(&contents->workB);CHKERRQ(ierr);
615   ierr = PetscFree4(contents->rvalues,contents->svalues,contents->rwaits,contents->swaits);CHKERRQ(ierr);
616   ierr = PetscFree(contents);CHKERRQ(ierr);
617   PetscFunctionReturn(0);
618 }
619 
620 #undef __FUNCT__
621 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIDense"
622 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat A,Mat B,PetscReal fill,Mat *C)
623 {
624   PetscErrorCode         ierr;
625   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*) A->data;
626   PetscInt               nz = aij->B->cmap->n;
627   PetscContainer         container;
628   MPIAIJ_MPIDense        *contents;
629   VecScatter             ctx = aij->Mvctx;
630   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
631   VecScatter_MPI_General *to   = ( VecScatter_MPI_General*) ctx->todata;
632   PetscInt               m=A->rmap->n,n=B->cmap->n;
633 
634   PetscFunctionBegin;
635   ierr = MatCreate(((PetscObject)B)->comm,C);CHKERRQ(ierr);
636   ierr = MatSetSizes(*C,m,n,A->rmap->N,B->cmap->N);CHKERRQ(ierr);
637   ierr = MatSetType(*C,MATMPIDENSE);CHKERRQ(ierr);
638   ierr = MatAssemblyBegin(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
639   ierr = MatAssemblyEnd(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
640   (*C)->ops->matmult = MatMatMult_MPIAIJ_MPIDense;
641 
642   ierr = PetscNew(MPIAIJ_MPIDense,&contents);CHKERRQ(ierr);
643   /* Create work matrix used to store off processor rows of B needed for local product */
644   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,B->cmap->N,PETSC_NULL,&contents->workB);CHKERRQ(ierr);
645   /* Create work arrays needed */
646   ierr = PetscMalloc4(B->cmap->N*from->starts[from->n],PetscScalar,&contents->rvalues,
647                       B->cmap->N*to->starts[to->n],PetscScalar,&contents->svalues,
648                       from->n,MPI_Request,&contents->rwaits,
649                       to->n,MPI_Request,&contents->swaits);CHKERRQ(ierr);
650 
651   ierr = PetscContainerCreate(((PetscObject)A)->comm,&container);CHKERRQ(ierr);
652   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
653   ierr = PetscContainerSetUserDestroy(container,MPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
654   ierr = PetscObjectCompose((PetscObject)(*C),"workB",(PetscObject)container);CHKERRQ(ierr);
655   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
656   PetscFunctionReturn(0);
657 }
658 
659 #undef __FUNCT__
660 #define __FUNCT__ "MatMPIDenseScatter"
661 /*
662     Performs an efficient scatter on the rows of B needed by this process; this is
663     a modification of the VecScatterBegin_() routines.
664 */
665 PetscErrorCode MatMPIDenseScatter(Mat A,Mat B,Mat C,Mat *outworkB)
666 {
667   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*)A->data;
668   PetscErrorCode         ierr;
669   PetscScalar            *b,*w,*svalues,*rvalues;
670   VecScatter             ctx = aij->Mvctx;
671   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
672   VecScatter_MPI_General *to   = ( VecScatter_MPI_General*) ctx->todata;
673   PetscInt               i,j,k;
674   PetscInt               *sindices,*sstarts,*rindices,*rstarts;
675   PetscMPIInt            *sprocs,*rprocs,nrecvs;
676   MPI_Request            *swaits,*rwaits;
677   MPI_Comm               comm = ((PetscObject)A)->comm;
678   PetscMPIInt            tag = ((PetscObject)ctx)->tag,ncols = B->cmap->N, nrows = aij->B->cmap->n,imdex,nrowsB = B->rmap->n;
679   MPI_Status             status;
680   MPIAIJ_MPIDense        *contents;
681   PetscContainer         container;
682   Mat                    workB;
683 
684   PetscFunctionBegin;
685   ierr = PetscObjectQuery((PetscObject)C,"workB",(PetscObject*)&container);CHKERRQ(ierr);
686   if (!container) SETERRQ(comm,PETSC_ERR_PLIB,"Container does not exist");
687   ierr = PetscContainerGetPointer(container,(void**)&contents);CHKERRQ(ierr);
688 
689   workB = *outworkB = contents->workB;
690   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);
691   sindices  = to->indices;
692   sstarts   = to->starts;
693   sprocs    = to->procs;
694   swaits    = contents->swaits;
695   svalues   = contents->svalues;
696 
697   rindices  = from->indices;
698   rstarts   = from->starts;
699   rprocs    = from->procs;
700   rwaits    = contents->rwaits;
701   rvalues   = contents->rvalues;
702 
703   ierr = MatGetArray(B,&b);CHKERRQ(ierr);
704   ierr = MatGetArray(workB,&w);CHKERRQ(ierr);
705 
706   for (i=0; i<from->n; i++) {
707     ierr = MPI_Irecv(rvalues+ncols*rstarts[i],ncols*(rstarts[i+1]-rstarts[i]),MPIU_SCALAR,rprocs[i],tag,comm,rwaits+i);CHKERRQ(ierr);
708   }
709 
710   for (i=0; i<to->n; i++) {
711     /* pack a message at a time */
712     CHKMEMQ;
713     for (j=0; j<sstarts[i+1]-sstarts[i]; j++){
714       for (k=0; k<ncols; k++) {
715         svalues[ncols*(sstarts[i] + j) + k] = b[sindices[sstarts[i]+j] + nrowsB*k];
716       }
717     }
718     CHKMEMQ;
719     ierr = MPI_Isend(svalues+ncols*sstarts[i],ncols*(sstarts[i+1]-sstarts[i]),MPIU_SCALAR,sprocs[i],tag,comm,swaits+i);CHKERRQ(ierr);
720   }
721 
722   nrecvs = from->n;
723   while (nrecvs) {
724     ierr = MPI_Waitany(from->n,rwaits,&imdex,&status);CHKERRQ(ierr);
725     nrecvs--;
726     /* unpack a message at a time */
727     CHKMEMQ;
728     for (j=0; j<rstarts[imdex+1]-rstarts[imdex]; j++){
729       for (k=0; k<ncols; k++) {
730         w[rindices[rstarts[imdex]+j] + nrows*k] = rvalues[ncols*(rstarts[imdex] + j) + k];
731       }
732     }
733     CHKMEMQ;
734   }
735   if (to->n) {ierr = MPI_Waitall(to->n,swaits,to->sstatus);CHKERRQ(ierr);}
736 
737   ierr = MatRestoreArray(B,&b);CHKERRQ(ierr);
738   ierr = MatRestoreArray(workB,&w);CHKERRQ(ierr);
739   ierr = MatAssemblyBegin(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
740   ierr = MatAssemblyEnd(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
741   PetscFunctionReturn(0);
742 }
743 extern PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat,Mat,Mat);
744 
745 #undef __FUNCT__
746 #define __FUNCT__ "MatMatMultNumeric_MPIAIJ_MPIDense"
747 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat A,Mat B,Mat C)
748 {
749   PetscErrorCode       ierr;
750   Mat_MPIAIJ           *aij = (Mat_MPIAIJ*)A->data;
751   Mat_MPIDense         *bdense = (Mat_MPIDense*)B->data;
752   Mat_MPIDense         *cdense = (Mat_MPIDense*)C->data;
753   Mat                  workB;
754 
755   PetscFunctionBegin;
756 
757   /* diagonal block of A times all local rows of B*/
758   ierr = MatMatMultNumeric_SeqAIJ_SeqDense(aij->A,bdense->A,cdense->A);CHKERRQ(ierr);
759 
760   /* get off processor parts of B needed to complete the product */
761   ierr = MatMPIDenseScatter(A,B,C,&workB);CHKERRQ(ierr);
762 
763   /* off-diagonal block of A times nonlocal rows of B */
764   ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A);CHKERRQ(ierr);
765   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
766   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
767   PetscFunctionReturn(0);
768 }
769 
770 #undef __FUNCT__
771 #define __FUNCT__ "MatMatMultNumeric_MPIAIJ_MPIAIJ_Scalable"
772 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ_Scalable(Mat A,Mat P,Mat C)
773 {
774   PetscErrorCode     ierr;
775   Mat_MPIAIJ         *a=(Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
776   Mat_SeqAIJ         *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
777   Mat_SeqAIJ         *cd=(Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
778   PetscInt           *adi=ad->i,*adj,*aoi=ao->i,*aoj;
779   PetscScalar        *ada,*aoa,*cda=cd->a,*coa=co->a;
780   Mat_SeqAIJ         *p_loc,*p_oth;
781   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
782   PetscScalar        *pa_loc,*pa_oth,*pa,valtmp,*ca;
783   PetscInt           cm=C->rmap->n,anz,pnz;
784   Mat_PtAPMPI        *ptap=c->ptap;
785   PetscScalar        *apa_sparse=ptap->apa;
786   PetscInt           *api,*apj,*apJ,i,j,k,row;
787   PetscInt           cstart=C->cmap->rstart;
788   PetscInt           cdnz,conz,k0,k1,nextp;
789 #if defined(DEBUG_MATMATMULT)
790   PetscMPIInt        rank=a->rank;
791 #endif
792 
793   PetscFunctionBegin;
794 #if defined(DEBUG_MATMATMULT)
795   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] call MatMatMultNumeric_MPIAIJ_MPIAIJ_Scalable()...\n",rank);
796 #endif
797 
798   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
799   /*-----------------------------------------------------*/
800   /* update numerical values of P_oth and P_loc */
801   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
802   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
803 #if defined(DEBUG_MATMATMULT)
804   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] got P_oth and P_loc...\n",rank);
805 #endif
806 
807   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
808   /*----------------------------------------------------------*/
809   /* get data from symbolic products */
810   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
811   p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
812   pi_loc=p_loc->i; pj_loc=p_loc->j; pa_loc=p_loc->a;
813   pi_oth=p_oth->i; pj_oth=p_oth->j; pa_oth=p_oth->a;
814 
815   api = ptap->api;
816   apj = ptap->apj;
817   for (i=0; i<cm; i++) {
818     apJ = apj + api[i];
819 
820     /* diagonal portion of A */
821     anz = adi[i+1] - adi[i];
822     adj = ad->j + adi[i];
823     ada = ad->a + adi[i];
824     for (j=0; j<anz; j++) {
825       row = adj[j];
826       pnz = pi_loc[row+1] - pi_loc[row];
827       pj  = pj_loc + pi_loc[row];
828       pa  = pa_loc + pi_loc[row];
829       /* perform sparse axpy */
830       valtmp = ada[j];
831       nextp  = 0;
832       for (k=0; nextp<pnz; k++) {
833         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
834           apa_sparse[k] += valtmp*pa[nextp++];
835         }
836       }
837       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
838     }
839 
840     /* off-diagonal portion of A */
841     anz = aoi[i+1] - aoi[i];
842     aoj = ao->j + aoi[i];
843     aoa = ao->a + aoi[i];
844     for (j=0; j<anz; j++) {
845       row = aoj[j];
846       pnz = pi_oth[row+1] - pi_oth[row];
847       pj  = pj_oth + pi_oth[row];
848       pa  = pa_oth + pi_oth[row];
849       /* perform sparse axpy */
850       valtmp = aoa[j];
851       nextp  = 0;
852       for (k=0; nextp<pnz; k++) {
853         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
854           apa_sparse[k] += valtmp*pa[nextp++];
855         }
856       }
857       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
858     }
859 
860     /* set values in C */
861     cdnz = cd->i[i+1] - cd->i[i];
862     conz = co->i[i+1] - co->i[i];
863 
864     /* 1st off-diagoanl part of C */
865     ca = coa + co->i[i];
866     k  = 0;
867     for (k0=0; k0<conz; k0++){
868       if (apJ[k] >= cstart) break;
869       ca[k0]      = apa_sparse[k];
870       apa_sparse[k] = 0.0;
871       k++;
872     }
873 
874     /* diagonal part of C */
875     ca = cda + cd->i[i];
876     for (k1=0; k1<cdnz; k1++){
877       ca[k1]      = apa_sparse[k];
878       apa_sparse[k] = 0.0;
879       k++;
880     }
881 
882     /* 2nd off-diagoanl part of C */
883     ca = coa + co->i[i];
884     for (; k0<conz; k0++){
885       ca[k0]      = apa_sparse[k];
886       apa_sparse[k] = 0.0;
887       k++;
888     }
889   }
890   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
891   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
892   PetscFunctionReturn(0);
893 }
894 
895 /* same as MatMatMultSymbolic_MPIAIJ_MPIAIJ(), except using LLCondensed to avoid O(BN) memory requirement */
896 #undef __FUNCT__
897 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable"
898 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable(Mat A,Mat P,PetscReal fill,Mat *C)
899 {
900   PetscErrorCode       ierr;
901   MPI_Comm             comm=((PetscObject)A)->comm;
902   Mat                  Cmpi;
903   Mat_PtAPMPI          *ptap;
904   PetscFreeSpaceList   free_space=PETSC_NULL,current_space=PETSC_NULL;
905   Mat_MPIAIJ           *a=(Mat_MPIAIJ*)A->data,*c;
906   Mat_SeqAIJ           *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
907   PetscInt             *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
908   PetscInt             *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
909   PetscInt             i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi,*lnk,apnz_max=0;
910   PetscInt             am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
911   PetscInt             nlnk_max,armax,prmax;
912   PetscReal            afill;
913   PetscScalar          *apa;
914 #if defined(DEBUG_MATMATMULT)
915   PetscMPIInt          rank=a->rank;
916 #endif
917 
918   PetscFunctionBegin;
919 #if defined(DEBUG_MATMATMULT)
920   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
921   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] call MatMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable()...\n",rank);
922 #endif
923 
924   /* create struct Mat_PtAPMPI and attached it to C later */
925   ierr = PetscNew(Mat_PtAPMPI,&ptap);CHKERRQ(ierr);
926 
927   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
928   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
929 #if defined(DEBUG_MATMATMULT)
930   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] P_oth is done...\n",rank);
931 #endif
932   /* get P_loc by taking all local rows of P */
933   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
934 
935   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
936   p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
937   pi_loc = p_loc->i; pj_loc = p_loc->j;
938   pi_oth = p_oth->i; pj_oth = p_oth->j;
939 
940 #if defined(DEBUG_MATMATMULT)
941   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] P_loc is done, Annz %D * P_locnnz %D = %D...\n",rank,ad->i[am]+ao->i[am],p_loc->rmax,(ad->i[am]+ao->i[am])*p_loc->rmax);
942 #endif
943 
944   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
945   /*-------------------------------------------------------------------*/
946   ierr  = PetscMalloc((am+2)*sizeof(PetscInt),&api);CHKERRQ(ierr);
947   ptap->api = api;
948   api[0]    = 0;
949 
950   /* create and initialize a linked list */
951   armax = ad->rmax+ao->rmax;
952   prmax = PetscMax(p_loc->rmax,p_oth->rmax);
953   nlnk_max = armax*prmax;
954   if (!nlnk_max || nlnk_max > pN) nlnk_max = pN;
955 #if defined(DEBUG_MATMATMULT)
956   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] pN %d; nlnk_max %d; Armax %d+%d=%d; Prmax Max(%d,%d)=%d\n",rank,pN,nlnk_max,ad->rmax,ao->rmax,armax,p_loc->rmax,p_oth->rmax,prmax);
957 #endif
958   ierr = PetscLLCondensedCreate_Scalable(nlnk_max,&lnk);CHKERRQ(ierr);
959 
960   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
961   ierr = PetscFreeSpaceGet((PetscInt)(fill*(adi[am]+aoi[am]+pi_loc[pm])),&free_space);CHKERRQ(ierr);
962   current_space = free_space;
963 
964   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
965   for (i=0; i<am; i++) {
966     apnz = 0;
967     /* diagonal portion of A */
968     nzi = adi[i+1] - adi[i];
969     for (j=0; j<nzi; j++){
970       row = *adj++;
971       pnz = pi_loc[row+1] - pi_loc[row];
972       Jptr  = pj_loc + pi_loc[row];
973       /* add non-zero cols of P into the sorted linked list lnk */
974       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
975     }
976     /* off-diagonal portion of A */
977     nzi = aoi[i+1] - aoi[i];
978     for (j=0; j<nzi; j++){
979       row = *aoj++;
980       pnz = pi_oth[row+1] - pi_oth[row];
981       Jptr  = pj_oth + pi_oth[row];
982       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
983     }
984 
985     apnz     = *lnk;
986     api[i+1] = api[i] + apnz;
987     if (apnz > apnz_max) apnz_max = apnz;
988 
989     /* if free space is not available, double the total space in the list */
990     if (current_space->local_remaining<apnz) {
991       ierr = PetscFreeSpaceGet(apnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
992       nspacedouble++;
993     }
994 
995     /* Copy data into free space, then initialize lnk */
996     ierr = PetscLLCondensedClean_Scalable(apnz,current_space->array,lnk);CHKERRQ(ierr);
997     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
998     current_space->array           += apnz;
999     current_space->local_used      += apnz;
1000     current_space->local_remaining -= apnz;
1001   }
1002 
1003   /* Allocate space for apj, initialize apj, and */
1004   /* destroy list of free space and other temporary array(s) */
1005   ierr = PetscMalloc((api[am]+1)*sizeof(PetscInt),&ptap->apj);CHKERRQ(ierr);
1006   apj  = ptap->apj;
1007   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
1008   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
1009 #if defined(DEBUG_MATMATMULT)
1010   if (!rank) ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] AP is done..., apnz_max %d\n",rank,apnz_max);
1011 #endif
1012 
1013   /* create and assemble symbolic parallel matrix Cmpi */
1014   /*----------------------------------------------------*/
1015   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1016   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1017   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1018   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1019   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1020   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
1021 
1022   /* malloc apa for assembly Cmpi */
1023   ierr = PetscMalloc(apnz_max*sizeof(PetscScalar),&apa);CHKERRQ(ierr);
1024   ierr = PetscMemzero(apa,apnz_max*sizeof(PetscScalar));CHKERRQ(ierr);
1025   ptap->apa = apa;
1026   for (i=0; i<am; i++){
1027     row  = i + rstart;
1028     apnz = api[i+1] - api[i];
1029     ierr = MatSetValues(Cmpi,1,&row,apnz,apj,apa,INSERT_VALUES);CHKERRQ(ierr);
1030     apj += apnz;
1031   }
1032   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1033   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1034 
1035   ptap->destroy             = Cmpi->ops->destroy;
1036   ptap->duplicate           = Cmpi->ops->duplicate;
1037   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_Scalable;
1038   Cmpi->ops->destroy        = MatDestroy_MPIAIJ_MatMatMult;
1039   Cmpi->ops->duplicate      = MatDuplicate_MPIAIJ_MatMatMult;
1040 
1041   /* attach the supporting struct to Cmpi for reuse */
1042   c = (Mat_MPIAIJ*)Cmpi->data;
1043   c->ptap  = ptap;
1044 
1045   *C = Cmpi;
1046 
1047   /* set MatInfo */
1048   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]) + 1.e-5;
1049   if (afill < 1.0) afill = 1.0;
1050   Cmpi->info.mallocs           = nspacedouble;
1051   Cmpi->info.fill_ratio_given  = fill;
1052   Cmpi->info.fill_ratio_needed = afill;
1053 
1054 #if defined(PETSC_USE_INFO)
1055   if (api[am]) {
1056     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %G needed %G.\n",nspacedouble,fill,afill);CHKERRQ(ierr);
1057     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%G,&C) for best performance.;\n",afill);CHKERRQ(ierr);
1058   } else {
1059     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1060   }
1061 #endif
1062   PetscFunctionReturn(0);
1063 }
1064 
1065 /*-------------------------------------------------------------------------*/
1066 #undef __FUNCT__
1067 #define __FUNCT__ "MatTransposeMatMult_MPIAIJ_MPIAIJ"
1068 PetscErrorCode MatTransposeMatMult_MPIAIJ_MPIAIJ(Mat P,Mat A,MatReuse scall,PetscReal fill,Mat *C)
1069 {
1070   PetscErrorCode ierr;
1071   PetscBool      scalable=PETSC_FALSE;
1072 
1073   PetscFunctionBegin;
1074   if (scall == MAT_INITIAL_MATRIX){
1075     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
1076       ierr = PetscOptionsBool("-mattransposematmult_scalable","Use a scalable but slower C=Pt*A","",scalable,&scalable,PETSC_NULL);CHKERRQ(ierr);
1077       if  (scalable){
1078         ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable(P,A,fill,C);CHKERRQ(ierr);
1079       } else {
1080         ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(P,A,fill,C);CHKERRQ(ierr);
1081       }
1082     ierr = PetscOptionsEnd();CHKERRQ(ierr);
1083   }
1084   ierr = (*(*C)->ops->mattransposemultnumeric)(P,A,*C);CHKERRQ(ierr);
1085   PetscFunctionReturn(0);
1086 }
1087 
1088 #undef __FUNCT__
1089 #define __FUNCT__ "MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ"
1090 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ(Mat P,Mat A,Mat C)
1091 {
1092   PetscErrorCode       ierr;
1093   Mat_Merge_SeqsToMPI  *merge;
1094   Mat_MPIAIJ           *p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1095   Mat_SeqAIJ           *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
1096   Mat_PtAPMPI          *ptap;
1097   PetscInt             *adj,*aJ;
1098   PetscInt             i,j,k,anz,pnz,row,*cj;
1099   MatScalar            *ada,*aval,*ca,valtmp;
1100   PetscInt             am=A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
1101   MPI_Comm             comm=((PetscObject)C)->comm;
1102   PetscMPIInt          size,rank,taga,*len_s;
1103   PetscInt             *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
1104   PetscInt             **buf_ri,**buf_rj;
1105   PetscInt             cnz=0,*bj_i,*bi,*bj,bnz,nextcj; /* bi,bj,ba: local array of C(mpi mat) */
1106   MPI_Request          *s_waits,*r_waits;
1107   MPI_Status           *status;
1108   MatScalar            **abuf_r,*ba_i,*pA,*coa,*ba;
1109   PetscInt             *ai,*aj,*coi,*coj;
1110   PetscInt             *poJ=po->j,*pdJ=pd->j;
1111   Mat                  A_loc;
1112   Mat_SeqAIJ           *a_loc;
1113 
1114   PetscFunctionBegin;
1115   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1116   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1117 
1118   ptap  = c->ptap;
1119   merge = ptap->merge;
1120 
1121  #if defined(DEBUG_MATTrMatMult)
1122   ierr = PetscSynchronizedPrintf(comm,"[%d] TransposeMatMultNumeric: Crmax %d \n",rank,ptap->rmax);
1123   ierr = PetscSynchronizedFlush(comm);CHKERRQ(ierr);
1124 #endif
1125 
1126   /* 2) compute numeric C_seq = P_loc^T*A_loc*P - dominating part */
1127   /*--------------------------------------------------------------*/
1128   /* get data from symbolic products */
1129   coi = merge->coi; coj = merge->coj;
1130   ierr = PetscMalloc((coi[pon]+1)*sizeof(MatScalar),&coa);CHKERRQ(ierr);
1131   ierr = PetscMemzero(coa,coi[pon]*sizeof(MatScalar));CHKERRQ(ierr);
1132 
1133   bi     = merge->bi; bj = merge->bj;
1134   owners = merge->rowmap->range;
1135   ierr   = PetscMalloc((bi[cm]+1)*sizeof(MatScalar),&ba);CHKERRQ(ierr);
1136   ierr   = PetscMemzero(ba,bi[cm]*sizeof(MatScalar));CHKERRQ(ierr);
1137 
1138   /* get A_loc by taking all local rows of A */
1139   A_loc = ptap->A_loc;
1140   ierr = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
1141   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1142   ai   = a_loc->i;
1143   aj   = a_loc->j;
1144 
1145   ierr = PetscMalloc((A->cmap->N)*sizeof(PetscScalar),&aval);CHKERRQ(ierr); /* non-scalable!!! */
1146   ierr = PetscMemzero(aval,A->cmap->N*sizeof(PetscScalar));CHKERRQ(ierr);
1147 
1148     for (i=0; i<am; i++) {
1149       /* 2-a) put A[i,:] to dense array aval */
1150       anz = ai[i+1] - ai[i];
1151       adj = aj + ai[i];
1152       ada = a_loc->a + ai[i];
1153       for (j=0; j<anz; j++){
1154         aval[adj[j]] = ada[j];
1155       }
1156 
1157       /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1158       /*--------------------------------------------------------------*/
1159       /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1160       pnz = po->i[i+1] - po->i[i];
1161       poJ = po->j + po->i[i];
1162       pA  = po->a + po->i[i];
1163       for (j=0; j<pnz; j++){
1164         row = poJ[j];
1165         cnz = coi[row+1] - coi[row];
1166         cj  = coj + coi[row];
1167         ca  = coa + coi[row];
1168         /* perform dense axpy */
1169         valtmp = pA[j];
1170         for (k=0; k<cnz; k++) {
1171           ca[k] += valtmp*aval[cj[k]];
1172         }
1173         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1174       }
1175 
1176       /* put the value into Cd (diagonal part) */
1177       pnz = pd->i[i+1] - pd->i[i];
1178       pdJ = pd->j + pd->i[i];
1179       pA  = pd->a + pd->i[i];
1180       for (j=0; j<pnz; j++){
1181         row = pdJ[j];
1182         cnz = bi[row+1] - bi[row];
1183         cj  = bj + bi[row];
1184         ca  = ba + bi[row];
1185         /* perform dense axpy */
1186         valtmp = pA[j];
1187         for (k=0; k<cnz; k++) {
1188           ca[k] += valtmp*aval[cj[k]];
1189         }
1190         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1191       }
1192 
1193       /* zero the current row of Pt*A */
1194       aJ = aj + ai[i];
1195       for (k=0; k<anz; k++) aval[aJ[k]] = 0.0;
1196     }
1197 
1198   /* 3) send and recv matrix values coa */
1199   /*------------------------------------*/
1200   buf_ri = merge->buf_ri;
1201   buf_rj = merge->buf_rj;
1202   len_s  = merge->len_s;
1203   ierr = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1204   ierr = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1205 
1206   ierr = PetscMalloc2(merge->nsend+1,MPI_Request,&s_waits,size,MPI_Status,&status);CHKERRQ(ierr);
1207   for (proc=0,k=0; proc<size; proc++){
1208     if (!len_s[proc]) continue;
1209     i = merge->owners_co[proc];
1210     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1211     k++;
1212   }
1213   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1214   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1215 
1216   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1217   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1218   ierr = PetscFree(coa);CHKERRQ(ierr);
1219 
1220   /* 4) insert local Cseq and received values into Cmpi */
1221   /*----------------------------------------------------*/
1222   ierr = PetscMalloc3(merge->nrecv,PetscInt**,&buf_ri_k,merge->nrecv,PetscInt*,&nextrow,merge->nrecv,PetscInt*,&nextci);CHKERRQ(ierr);
1223   for (k=0; k<merge->nrecv; k++){
1224     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1225     nrows       = *(buf_ri_k[k]);
1226     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1227     nextci[k]   = buf_ri_k[k] + (nrows + 1);/* poins to the next i-structure of k-th recved i-structure  */
1228   }
1229 
1230   for (i=0; i<cm; i++) {
1231     row = owners[rank] + i; /* global row index of C_seq */
1232     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1233     ba_i = ba + bi[i];
1234     bnz  = bi[i+1] - bi[i];
1235     /* add received vals into ba */
1236     for (k=0; k<merge->nrecv; k++){ /* k-th received message */
1237       /* i-th row */
1238       if (i == *nextrow[k]) {
1239         cnz = *(nextci[k]+1) - *nextci[k];
1240         cj  = buf_rj[k] + *(nextci[k]);
1241         ca  = abuf_r[k] + *(nextci[k]);
1242         nextcj = 0;
1243         for (j=0; nextcj<cnz; j++){
1244           if (bj_i[j] == cj[nextcj]){ /* bcol == ccol */
1245             ba_i[j] += ca[nextcj++];
1246           }
1247         }
1248         nextrow[k]++; nextci[k]++;
1249         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1250       }
1251     }
1252     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1253   }
1254   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1255   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1256 
1257   ierr = PetscFree(ba);CHKERRQ(ierr);
1258   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1259   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1260   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1261   ierr = PetscFree(aval);CHKERRQ(ierr);
1262   PetscFunctionReturn(0);
1263 }
1264 
1265 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1266 #undef __FUNCT__
1267 #define __FUNCT__ "MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ"
1268 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P,Mat A,PetscReal fill,Mat *C)
1269 {
1270   PetscErrorCode       ierr;
1271   Mat                  Cmpi,A_loc,POt,PDt;
1272   Mat_PtAPMPI          *ptap;
1273   PetscFreeSpaceList   free_space=PETSC_NULL,current_space=PETSC_NULL;
1274   Mat_MPIAIJ           *p=(Mat_MPIAIJ*)P->data,*c;
1275   PetscInt             *pdti,*pdtj,*poti,*potj,*ptJ;
1276   PetscInt             nnz;
1277   PetscInt             *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1278   PetscInt             am=A->rmap->n,pn=P->cmap->n;
1279   PetscBT              lnkbt;
1280   MPI_Comm             comm=((PetscObject)A)->comm;
1281   PetscMPIInt          size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1282   PetscInt             **buf_rj,**buf_ri,**buf_ri_k;
1283   PetscInt             len,proc,*dnz,*onz,*owners;
1284   PetscInt             nzi,*bi,*bj;
1285   PetscInt             nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1286   MPI_Request          *swaits,*rwaits;
1287   MPI_Status           *sstatus,rstatus;
1288   Mat_Merge_SeqsToMPI  *merge;
1289   PetscInt             *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1290   PetscReal            afill=1.0,afill_tmp;
1291   PetscInt             rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Crmax;
1292   PetscScalar          *vals;
1293   Mat_SeqAIJ           *a_loc, *pdt,*pot;
1294 
1295   PetscFunctionBegin;
1296   /* check if matrix local sizes are compatible */
1297   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend){
1298     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);
1299   }
1300 
1301   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1302   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1303 #if defined(DEBUG_MATTrMatMult)
1304   ierr = PetscSynchronizedPrintf(comm,"[%d] TransposeMatMultSymbolic P: %d %d, %d %d; A %d %d, %d %d\n",rank,P->rmap->N,P->cmap->N,P->rmap->n,P->cmap->n,A->rmap->N,aN,A->rmap->n,A->cmap->n);
1305   ierr = PetscSynchronizedFlush(comm);CHKERRQ(ierr);
1306 #endif
1307 
1308   /* create struct Mat_PtAPMPI and attached it to C later */
1309   ierr = PetscNew(Mat_PtAPMPI,&ptap);CHKERRQ(ierr);
1310 
1311   /* get A_loc by taking all local rows of A */
1312   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1313   ptap->A_loc = A_loc;
1314   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1315   ai   = a_loc->i;
1316   aj   = a_loc->j;
1317 
1318   /* determine symbolic Co=(p->B)^T*A - send to others */
1319   /*----------------------------------------------------*/
1320   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1321   pdt = (Mat_SeqAIJ*)PDt->data;
1322   pdti = pdt->i; pdtj = pdt->j;
1323 
1324   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1325   pot = (Mat_SeqAIJ*)POt->data;
1326   poti = pot->i; potj = pot->j;
1327 
1328   /* then, compute symbolic Co = (p->B)^T*A */
1329   pon = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1330                          >= (num of nonzero rows of C_seq) - pn */
1331   ierr = PetscMalloc((pon+1)*sizeof(PetscInt),&coi);CHKERRQ(ierr);
1332   coi[0] = 0;
1333 
1334   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1335   nnz           = fill*(poti[pon] + ai[am]);
1336   ierr          = PetscFreeSpaceGet(nnz,&free_space);
1337   current_space = free_space;
1338 #if defined(DEBUG_MATTrMatMult)
1339   ierr = PetscSynchronizedPrintf(comm, "[%d] nnz = fill %g *(%d + %d)\n",rank,fill,poti[pon],ai[am]);CHKERRQ(ierr);
1340   ierr = PetscSynchronizedFlush(comm);CHKERRQ(ierr);
1341 #endif
1342   /* create and initialize a linked list */
1343   i = PetscMax(pdt->rmax,pot->rmax);
1344   Crmax = i*a_loc->rmax*size;
1345   if (!Crmax || Crmax > aN) Crmax = aN;
1346 #if defined(DEBUG_MATTrMatMult)
1347   printf("[%d] rmax A_loc %d * max(PD %d, PO %d)=%d, Crmax %d\n",rank,a_loc->rmax,pdt->rmax,pot->rmax,i*a_loc->rmax,Crmax);
1348 #endif
1349   ierr = PetscLLCondensedCreate(Crmax,aN,&lnk,&lnkbt);CHKERRQ(ierr);
1350 
1351   for (i=0; i<pon; i++) {
1352     pnz = poti[i+1] - poti[i];
1353     ptJ = potj + poti[i];
1354     for (j=0; j<pnz; j++){
1355       row  = ptJ[j]; /* row of A_loc == col of Pot */
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     nnz = lnk[0];
1362 
1363     /* If free space is not available, double the total space in the list */
1364     if (current_space->local_remaining<nnz) {
1365       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1366       nspacedouble++;
1367     }
1368 
1369     /* Copy data into free space, and zero out denserows */
1370     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1371     current_space->array           += nnz;
1372     current_space->local_used      += nnz;
1373     current_space->local_remaining -= nnz;
1374     coi[i+1] = coi[i] + nnz;
1375   }
1376 
1377   ierr = PetscMalloc((coi[pon]+1)*sizeof(PetscInt),&coj);CHKERRQ(ierr);
1378   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1379   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]);
1380   if (afill_tmp > afill) afill = afill_tmp;
1381 
1382   /* send j-array (coj) of Co to other processors */
1383   /*----------------------------------------------*/
1384   /* determine row ownership */
1385   ierr = PetscNew(Mat_Merge_SeqsToMPI,&merge);CHKERRQ(ierr);
1386   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1387   merge->rowmap->n = pn;
1388   merge->rowmap->bs = 1;
1389   ierr = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1390   owners = merge->rowmap->range;
1391 
1392   /* determine the number of messages to send, their lengths */
1393   ierr = PetscMalloc(size*sizeof(PetscMPIInt),&len_si);CHKERRQ(ierr);
1394   ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1395   ierr = PetscMalloc(size*sizeof(PetscMPIInt),&merge->len_s);CHKERRQ(ierr);
1396   len_s = merge->len_s;
1397   merge->nsend = 0;
1398 
1399   ierr = PetscMalloc((size+2)*sizeof(PetscInt),&owners_co);CHKERRQ(ierr);
1400   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1401 
1402   proc = 0;
1403   for (i=0; i<pon; i++){
1404     while (prmap[i] >= owners[proc+1]) proc++;
1405     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1406     len_s[proc] += coi[i+1] - coi[i];
1407   }
1408 
1409   len   = 0;  /* max length of buf_si[] */
1410   owners_co[0] = 0;
1411   for (proc=0; proc<size; proc++){
1412     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1413     if (len_si[proc]){
1414       merge->nsend++;
1415       len_si[proc] = 2*(len_si[proc] + 1);
1416       len += len_si[proc];
1417     }
1418   }
1419 
1420   /* determine the number and length of messages to receive for coi and coj  */
1421   ierr = PetscGatherNumberOfMessages(comm,PETSC_NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1422   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1423 
1424   /* post the Irecv and Isend of coj */
1425   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1426   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1427   ierr = PetscMalloc((merge->nsend+1)*sizeof(MPI_Request),&swaits);CHKERRQ(ierr);
1428   for (proc=0, k=0; proc<size; proc++){
1429     if (!len_s[proc]) continue;
1430     i = owners_co[proc];
1431     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1432     k++;
1433   }
1434 
1435   /* receives and sends of coj are complete */
1436   ierr = PetscMalloc(size*sizeof(MPI_Status),&sstatus);CHKERRQ(ierr);
1437   for (i=0; i<merge->nrecv; i++){
1438     ierr = MPI_Waitany(merge->nrecv,rwaits,&j,&rstatus);CHKERRQ(ierr);
1439   }
1440   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1441   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1442 
1443   /* send and recv coi */
1444   /*-------------------*/
1445   ierr = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1446   ierr = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1447   ierr = PetscMalloc((len+1)*sizeof(PetscInt),&buf_s);CHKERRQ(ierr);
1448   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1449   for (proc=0,k=0; proc<size; proc++){
1450     if (!len_s[proc]) continue;
1451     /* form outgoing message for i-structure:
1452          buf_si[0]:                 nrows to be sent
1453                [1:nrows]:           row index (global)
1454                [nrows+1:2*nrows+1]: i-structure index
1455     */
1456     /*-------------------------------------------*/
1457     nrows = len_si[proc]/2 - 1;
1458     buf_si_i    = buf_si + nrows+1;
1459     buf_si[0]   = nrows;
1460     buf_si_i[0] = 0;
1461     nrows = 0;
1462     for (i=owners_co[proc]; i<owners_co[proc+1]; i++){
1463       nzi = coi[i+1] - coi[i];
1464       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi; /* i-structure */
1465       buf_si[nrows+1] =prmap[i] -owners[proc]; /* local row index */
1466       nrows++;
1467     }
1468     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1469     k++;
1470     buf_si += len_si[proc];
1471   }
1472   i = merge->nrecv;
1473   while (i--) {
1474     ierr = MPI_Waitany(merge->nrecv,rwaits,&j,&rstatus);CHKERRQ(ierr);
1475   }
1476   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1477   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1478   ierr = PetscFree(len_si);CHKERRQ(ierr);
1479   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1480   ierr = PetscFree(swaits);CHKERRQ(ierr);
1481   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1482   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1483 
1484   /* compute the local portion of C (mpi mat) */
1485   /*------------------------------------------*/
1486   /* allocate bi array and free space for accumulating nonzero column info */
1487   ierr = PetscMalloc((pn+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
1488   bi[0] = 0;
1489 
1490   /* set initial free space to be fill*(nnz(P) + nnz(A)) */
1491   nnz           = fill*(pdti[pn] + poti[pon] + ai[am]);
1492   ierr          = PetscFreeSpaceGet(nnz,&free_space);
1493   current_space = free_space;
1494 
1495 #if defined(DEBUG_MATTrMatMult)
1496   ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] nnz=fill %g*(%d + %d + %d)\n",rank,fill,pdti[pn],poti[pon],ai[am]);
1497 #endif
1498 
1499   ierr = PetscMalloc3(merge->nrecv,PetscInt**,&buf_ri_k,merge->nrecv,PetscInt*,&nextrow,merge->nrecv,PetscInt*,&nextci);CHKERRQ(ierr);
1500   for (k=0; k<merge->nrecv; k++){
1501     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1502     nrows       = *buf_ri_k[k];
1503     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1504     nextci[k]   = buf_ri_k[k] + (nrows + 1);/* poins to the next i-structure of k-th recved i-structure  */
1505   }
1506 
1507   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1508   rmax = 0;
1509   for (i=0; i<pn; i++) {
1510     /* add pdt[i,:]*AP into lnk */
1511     pnz = pdti[i+1] - pdti[i];
1512     ptJ = pdtj + pdti[i];
1513     for (j=0; j<pnz; j++){
1514       row  = ptJ[j];  /* row of AP == col of Pt */
1515       anz  = ai[row+1] - ai[row];
1516       Jptr = aj + ai[row];
1517       /* add non-zero cols of AP into the sorted linked list lnk */
1518       ierr = PetscLLCondensedAddSorted(anz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1519     }
1520 
1521     /* add received col data into lnk */
1522     for (k=0; k<merge->nrecv; k++){ /* k-th received message */
1523       if (i == *nextrow[k]) { /* i-th row */
1524         nzi = *(nextci[k]+1) - *nextci[k];
1525         Jptr  = buf_rj[k] + *nextci[k];
1526         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1527         nextrow[k]++; nextci[k]++;
1528       }
1529     }
1530     nnz = lnk[0];
1531 
1532     /* if free space is not available, make more free space */
1533     if (current_space->local_remaining<nnz) {
1534       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1535       nspacedouble++;
1536     }
1537     /* copy data into free space, then initialize lnk */
1538     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1539     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1540     current_space->array           += nnz;
1541     current_space->local_used      += nnz;
1542     current_space->local_remaining -= nnz;
1543     bi[i+1] = bi[i] + nnz;
1544     if (nnz > rmax) rmax = nnz;
1545   }
1546   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1547 
1548   ierr = PetscMalloc((bi[pn]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
1549   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1550   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]);
1551   if (afill_tmp > afill) afill = afill_tmp;
1552   ierr = PetscLLCondensedDestroy(lnk,lnkbt);CHKERRQ(ierr);
1553   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1554   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1555 
1556   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
1557   /*----------------------------------------------------------------------------------*/
1558   ierr = PetscMalloc((rmax+1)*sizeof(PetscScalar),&vals);CHKERRQ(ierr);
1559   ierr = PetscMemzero(vals,rmax*sizeof(PetscScalar));CHKERRQ(ierr);
1560 
1561   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1562   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1563   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1564   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1565   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1566   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
1567   for (i=0; i<pn; i++){
1568     row = i + rstart;
1569     nnz = bi[i+1] - bi[i];
1570     Jptr = bj + bi[i];
1571     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
1572   }
1573   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1574   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1575   ierr = PetscFree(vals);CHKERRQ(ierr);
1576 
1577   merge->bi            = bi;
1578   merge->bj            = bj;
1579   merge->coi           = coi;
1580   merge->coj           = coj;
1581   merge->buf_ri        = buf_ri;
1582   merge->buf_rj        = buf_rj;
1583   merge->owners_co     = owners_co;
1584   merge->destroy       = Cmpi->ops->destroy;
1585   merge->duplicate     = Cmpi->ops->duplicate;
1586 
1587   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ;
1588   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
1589 
1590   /* attach the supporting struct to Cmpi for reuse */
1591   c = (Mat_MPIAIJ*)Cmpi->data;
1592   c->ptap        = ptap;
1593   ptap->api      = PETSC_NULL;
1594   ptap->apj      = PETSC_NULL;
1595   ptap->merge    = merge;
1596   ptap->rmax     = rmax;
1597 
1598   *C = Cmpi;
1599 #if defined(PETSC_USE_INFO)
1600   if (bi[pn] != 0) {
1601     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %G needed %G.\n",nspacedouble,fill,afill);CHKERRQ(ierr);
1602     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%G,&C) for best performance.\n",afill);CHKERRQ(ierr);
1603   } else {
1604     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1605   }
1606 #endif
1607   PetscFunctionReturn(0);
1608 }
1609 
1610 #undef __FUNCT__
1611 #define __FUNCT__ "MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_Scalable"
1612 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_Scalable(Mat P,Mat A,Mat C)
1613 {
1614   PetscErrorCode       ierr;
1615   Mat_Merge_SeqsToMPI  *merge;
1616   Mat_MPIAIJ           *p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1617   Mat_SeqAIJ           *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
1618   Mat_PtAPMPI          *ptap;
1619   PetscInt             *adj;
1620   PetscInt             i,j,k,anz,pnz,row,*cj,nexta;
1621   MatScalar            *ada,*ca,valtmp;
1622   PetscInt             am=A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
1623   MPI_Comm             comm=((PetscObject)C)->comm;
1624   PetscMPIInt          size,rank,taga,*len_s;
1625   PetscInt             *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
1626   PetscInt             **buf_ri,**buf_rj;
1627   PetscInt             cnz=0,*bj_i,*bi,*bj,bnz,nextcj; /* bi,bj,ba: local array of C(mpi mat) */
1628   MPI_Request          *s_waits,*r_waits;
1629   MPI_Status           *status;
1630   MatScalar            **abuf_r,*ba_i,*pA,*coa,*ba;
1631   PetscInt             *ai,*aj,*coi,*coj;
1632   PetscInt             *poJ=po->j,*pdJ=pd->j;
1633   Mat                  A_loc;
1634   Mat_SeqAIJ           *a_loc;
1635 
1636   PetscFunctionBegin;
1637   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1638   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1639 
1640   ptap  = c->ptap;
1641   merge = ptap->merge;
1642 
1643  #if defined(DEBUG_MATTrMatMult)
1644   ierr = PetscSynchronizedPrintf(comm,"[%d] TransposeMatMultNumeric_Scalable: Crmax %d \n",rank,ptap->rmax);
1645   ierr = PetscSynchronizedFlush(comm);CHKERRQ(ierr);
1646 #endif
1647 
1648   /* 2) compute numeric C_seq = P_loc^T*A_loc */
1649   /*------------------------------------------*/
1650   /* get data from symbolic products */
1651   coi = merge->coi; coj = merge->coj;
1652   ierr = PetscMalloc((coi[pon]+1)*sizeof(MatScalar),&coa);CHKERRQ(ierr);
1653   ierr = PetscMemzero(coa,coi[pon]*sizeof(MatScalar));CHKERRQ(ierr);
1654 
1655   bi     = merge->bi; bj = merge->bj;
1656   owners = merge->rowmap->range;
1657   ierr   = PetscMalloc((bi[cm]+1)*sizeof(MatScalar),&ba);CHKERRQ(ierr);
1658   ierr   = PetscMemzero(ba,bi[cm]*sizeof(MatScalar));CHKERRQ(ierr);
1659 
1660   /* get A_loc by taking all local rows of A */
1661   A_loc = ptap->A_loc;
1662   ierr = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
1663   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1664   ai   = a_loc->i;
1665   aj   = a_loc->j;
1666 
1667   for (i=0; i<am; i++) {
1668     /* 2-a) put A[i,:] to dense array aval */
1669     anz = ai[i+1] - ai[i];
1670     adj = aj + ai[i];
1671     ada = a_loc->a + ai[i];
1672 
1673     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1674     /*-------------------------------------------------------------*/
1675     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1676     pnz = po->i[i+1] - po->i[i];
1677     poJ = po->j + po->i[i];
1678     pA  = po->a + po->i[i];
1679     for (j=0; j<pnz; j++){
1680       row = poJ[j];
1681       cnz = coi[row+1] - coi[row];
1682       cj  = coj + coi[row];
1683       ca  = coa + coi[row];
1684       /* perform sparse axpy */
1685       nexta  = 0;
1686       valtmp = pA[j];
1687       for (k=0; nexta<anz; k++) {
1688         if (cj[k] == adj[nexta]){
1689           ca[k] += valtmp*ada[nexta];
1690           nexta++;
1691         }
1692       }
1693       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1694     }
1695 
1696     /* put the value into Cd (diagonal part) */
1697     pnz = pd->i[i+1] - pd->i[i];
1698     pdJ = pd->j + pd->i[i];
1699     pA  = pd->a + pd->i[i];
1700     for (j=0; j<pnz; j++){
1701       row = pdJ[j];
1702       cnz = bi[row+1] - bi[row];
1703       cj  = bj + bi[row];
1704       ca  = ba + bi[row];
1705       /* perform sparse axpy */
1706       nexta  = 0;
1707       valtmp = pA[j];
1708       for (k=0; nexta<anz; k++) {
1709         if (cj[k] == adj[nexta]){
1710           ca[k] += valtmp*ada[nexta];
1711           nexta++;
1712         }
1713       }
1714       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1715     }
1716 
1717   }
1718 
1719   /* 3) send and recv matrix values coa */
1720   /*------------------------------------*/
1721   buf_ri = merge->buf_ri;
1722   buf_rj = merge->buf_rj;
1723   len_s  = merge->len_s;
1724   ierr = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1725   ierr = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1726 
1727   ierr = PetscMalloc2(merge->nsend+1,MPI_Request,&s_waits,size,MPI_Status,&status);CHKERRQ(ierr);
1728   for (proc=0,k=0; proc<size; proc++){
1729     if (!len_s[proc]) continue;
1730     i = merge->owners_co[proc];
1731     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1732     k++;
1733   }
1734   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1735   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1736 
1737   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1738   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1739   ierr = PetscFree(coa);CHKERRQ(ierr);
1740 
1741   /* 4) insert local Cseq and received values into Cmpi */
1742   /*----------------------------------------------------*/
1743   ierr = PetscMalloc3(merge->nrecv,PetscInt**,&buf_ri_k,merge->nrecv,PetscInt*,&nextrow,merge->nrecv,PetscInt*,&nextci);CHKERRQ(ierr);
1744   for (k=0; k<merge->nrecv; k++){
1745     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1746     nrows       = *(buf_ri_k[k]);
1747     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1748     nextci[k]   = buf_ri_k[k] + (nrows + 1);/* poins to the next i-structure of k-th recved i-structure  */
1749   }
1750 
1751   for (i=0; i<cm; i++) {
1752     row = owners[rank] + i; /* global row index of C_seq */
1753     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1754     ba_i = ba + bi[i];
1755     bnz  = bi[i+1] - bi[i];
1756     /* add received vals into ba */
1757     for (k=0; k<merge->nrecv; k++){ /* k-th received message */
1758       /* i-th row */
1759       if (i == *nextrow[k]) {
1760         cnz = *(nextci[k]+1) - *nextci[k];
1761         cj  = buf_rj[k] + *(nextci[k]);
1762         ca  = abuf_r[k] + *(nextci[k]);
1763         nextcj = 0;
1764         for (j=0; nextcj<cnz; j++){
1765           if (bj_i[j] == cj[nextcj]){ /* bcol == ccol */
1766             ba_i[j] += ca[nextcj++];
1767           }
1768         }
1769         nextrow[k]++; nextci[k]++;
1770         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1771       }
1772     }
1773     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1774   }
1775   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1776   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1777 
1778   ierr = PetscFree(ba);CHKERRQ(ierr);
1779   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1780   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1781   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1782   PetscFunctionReturn(0);
1783 }
1784 
1785 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1786 #undef __FUNCT__
1787 #define __FUNCT__ "MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable"
1788 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable(Mat P,Mat A,PetscReal fill,Mat *C)
1789 {
1790   PetscErrorCode       ierr;
1791   Mat                  Cmpi,A_loc,POt,PDt;
1792   Mat_PtAPMPI          *ptap;
1793   PetscFreeSpaceList   free_space=PETSC_NULL,current_space=PETSC_NULL;
1794   Mat_MPIAIJ           *p=(Mat_MPIAIJ*)P->data,*c;
1795   PetscInt             *pdti,*pdtj,*poti,*potj,*ptJ;
1796   PetscInt             nnz;
1797   PetscInt             *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1798   PetscInt             am=A->rmap->n,pn=P->cmap->n;
1799   MPI_Comm             comm=((PetscObject)A)->comm;
1800   PetscMPIInt          size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1801   PetscInt             **buf_rj,**buf_ri,**buf_ri_k;
1802   PetscInt             len,proc,*dnz,*onz,*owners;
1803   PetscInt             nzi,*bi,*bj;
1804   PetscInt             nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1805   MPI_Request          *swaits,*rwaits;
1806   MPI_Status           *sstatus,rstatus;
1807   Mat_Merge_SeqsToMPI  *merge;
1808   PetscInt             *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1809   PetscReal            afill=1.0,afill_tmp;
1810   PetscInt             rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Crmax;
1811   PetscScalar          *vals;
1812   Mat_SeqAIJ           *a_loc, *pdt,*pot;
1813 
1814   PetscFunctionBegin;
1815   /* check if matrix local sizes are compatible */
1816   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend){
1817     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);
1818   }
1819 
1820   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1821   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1822 #if defined(DEBUG_MATTrMatMult)
1823   ierr = PetscSynchronizedPrintf(comm,"[%d] TransposeMatMultSymbolic_Scalable P: %d %d, %d %d; A %d %d, %d %d\n",rank,P->rmap->N,P->cmap->N,P->rmap->n,P->cmap->n,A->rmap->N,aN,A->rmap->n,A->cmap->n);
1824   ierr = PetscSynchronizedFlush(comm);CHKERRQ(ierr);
1825 #endif
1826 
1827   /* create struct Mat_PtAPMPI and attached it to C later */
1828   ierr = PetscNew(Mat_PtAPMPI,&ptap);CHKERRQ(ierr);
1829 
1830   /* get A_loc by taking all local rows of A */
1831   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1832   ptap->A_loc = A_loc;
1833   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1834   ai   = a_loc->i;
1835   aj   = a_loc->j;
1836 
1837   /* determine symbolic Co=(p->B)^T*A - send to others */
1838   /*----------------------------------------------------*/
1839   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1840   pdt = (Mat_SeqAIJ*)PDt->data;
1841   pdti = pdt->i; pdtj = pdt->j;
1842 
1843   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1844   pot = (Mat_SeqAIJ*)POt->data;
1845   poti = pot->i; potj = pot->j;
1846 
1847   /* then, compute symbolic Co = (p->B)^T*A */
1848   pon = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1849                          >= (num of nonzero rows of C_seq) - pn */
1850   ierr = PetscMalloc((pon+1)*sizeof(PetscInt),&coi);CHKERRQ(ierr);
1851   coi[0] = 0;
1852 
1853   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1854   nnz           = fill*(poti[pon] + ai[am]);
1855   ierr          = PetscFreeSpaceGet(nnz,&free_space);
1856   current_space = free_space;
1857 #if defined(DEBUG_MATTrMatMult)
1858   ierr = PetscSynchronizedPrintf(comm, "[%d] nnz = fill %g *(%d + %d)\n",rank,fill,poti[pon],ai[am]);CHKERRQ(ierr);
1859   ierr = PetscSynchronizedFlush(comm);CHKERRQ(ierr);
1860 #endif
1861   /* create and initialize a linked list */
1862   i = PetscMax(pdt->rmax,pot->rmax);
1863   Crmax = i*a_loc->rmax*size; /* non-scalable! */
1864   if (!Crmax || Crmax > aN) Crmax = aN;
1865 #if defined(DEBUG_MATTrMatMult)
1866   printf("[%d] rmax A_loc %d * max(PD %d, PO %d)=%d, Crmax %d\n",rank,a_loc->rmax,pdt->rmax,pot->rmax,i*a_loc->rmax,Crmax);
1867 #endif
1868   ierr = PetscLLCondensedCreate_Scalable(Crmax,&lnk);CHKERRQ(ierr);
1869 
1870   for (i=0; i<pon; i++) {
1871     nnz = 0;
1872     pnz = poti[i+1] - poti[i];
1873     ptJ = potj + poti[i];
1874     for (j=0; j<pnz; j++){
1875       row  = ptJ[j]; /* row of A_loc == col of Pot */
1876       anz  = ai[row+1] - ai[row];
1877       Jptr = aj + ai[row];
1878       /* add non-zero cols of AP into the sorted linked list lnk */
1879       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1880     }
1881     nnz = lnk[0];
1882 
1883     /* If free space is not available, double the total space in the list */
1884     if (current_space->local_remaining<nnz) {
1885       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1886       nspacedouble++;
1887     }
1888 
1889     /* Copy data into free space, and zero out denserows */
1890     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1891     current_space->array           += nnz;
1892     current_space->local_used      += nnz;
1893     current_space->local_remaining -= nnz;
1894     coi[i+1] = coi[i] + nnz;
1895   }
1896 
1897   ierr = PetscMalloc((coi[pon]+1)*sizeof(PetscInt),&coj);CHKERRQ(ierr);
1898   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1899   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]);
1900   if (afill_tmp > afill) afill = afill_tmp;
1901 
1902   /* send j-array (coj) of Co to other processors */
1903   /*----------------------------------------------*/
1904   /* determine row ownership */
1905   ierr = PetscNew(Mat_Merge_SeqsToMPI,&merge);CHKERRQ(ierr);
1906   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1907   merge->rowmap->n = pn;
1908   merge->rowmap->bs = 1;
1909   ierr = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1910   owners = merge->rowmap->range;
1911 
1912   /* determine the number of messages to send, their lengths */
1913   ierr = PetscMalloc(size*sizeof(PetscMPIInt),&len_si);CHKERRQ(ierr);
1914   ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1915   ierr = PetscMalloc(size*sizeof(PetscMPIInt),&merge->len_s);CHKERRQ(ierr);
1916   len_s = merge->len_s;
1917   merge->nsend = 0;
1918 
1919   ierr = PetscMalloc((size+2)*sizeof(PetscInt),&owners_co);CHKERRQ(ierr);
1920   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1921 
1922   proc = 0;
1923   for (i=0; i<pon; i++){
1924     while (prmap[i] >= owners[proc+1]) proc++;
1925     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1926     len_s[proc] += coi[i+1] - coi[i];
1927   }
1928 
1929   len   = 0;  /* max length of buf_si[] */
1930   owners_co[0] = 0;
1931   for (proc=0; proc<size; proc++){
1932     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1933     if (len_si[proc]){
1934       merge->nsend++;
1935       len_si[proc] = 2*(len_si[proc] + 1);
1936       len += len_si[proc];
1937     }
1938   }
1939 
1940   /* determine the number and length of messages to receive for coi and coj  */
1941   ierr = PetscGatherNumberOfMessages(comm,PETSC_NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1942   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1943 
1944   /* post the Irecv and Isend of coj */
1945   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1946   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1947   ierr = PetscMalloc((merge->nsend+1)*sizeof(MPI_Request),&swaits);CHKERRQ(ierr);
1948   for (proc=0, k=0; proc<size; proc++){
1949     if (!len_s[proc]) continue;
1950     i = owners_co[proc];
1951     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1952     k++;
1953   }
1954 
1955   /* receives and sends of coj are complete */
1956   ierr = PetscMalloc(size*sizeof(MPI_Status),&sstatus);CHKERRQ(ierr);
1957   for (i=0; i<merge->nrecv; i++){
1958     ierr = MPI_Waitany(merge->nrecv,rwaits,&j,&rstatus);CHKERRQ(ierr);
1959   }
1960   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1961   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1962 
1963   /* send and recv coi */
1964   /*-------------------*/
1965   ierr = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1966   ierr = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1967   ierr = PetscMalloc((len+1)*sizeof(PetscInt),&buf_s);CHKERRQ(ierr);
1968   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1969   for (proc=0,k=0; proc<size; proc++){
1970     if (!len_s[proc]) continue;
1971     /* form outgoing message for i-structure:
1972          buf_si[0]:                 nrows to be sent
1973                [1:nrows]:           row index (global)
1974                [nrows+1:2*nrows+1]: i-structure index
1975     */
1976     /*-------------------------------------------*/
1977     nrows = len_si[proc]/2 - 1;
1978     buf_si_i    = buf_si + nrows+1;
1979     buf_si[0]   = nrows;
1980     buf_si_i[0] = 0;
1981     nrows = 0;
1982     for (i=owners_co[proc]; i<owners_co[proc+1]; i++){
1983       nzi = coi[i+1] - coi[i];
1984       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi; /* i-structure */
1985       buf_si[nrows+1] =prmap[i] -owners[proc]; /* local row index */
1986       nrows++;
1987     }
1988     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1989     k++;
1990     buf_si += len_si[proc];
1991   }
1992   i = merge->nrecv;
1993   while (i--) {
1994     ierr = MPI_Waitany(merge->nrecv,rwaits,&j,&rstatus);CHKERRQ(ierr);
1995   }
1996   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1997   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1998   ierr = PetscFree(len_si);CHKERRQ(ierr);
1999   ierr = PetscFree(len_ri);CHKERRQ(ierr);
2000   ierr = PetscFree(swaits);CHKERRQ(ierr);
2001   ierr = PetscFree(sstatus);CHKERRQ(ierr);
2002   ierr = PetscFree(buf_s);CHKERRQ(ierr);
2003 
2004   /* compute the local portion of C (mpi mat) */
2005   /*------------------------------------------*/
2006   /* allocate bi array and free space for accumulating nonzero column info */
2007   ierr = PetscMalloc((pn+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
2008   bi[0] = 0;
2009 
2010   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
2011   nnz           = fill*(pdti[pn] + poti[pon] + ai[am]);
2012   ierr          = PetscFreeSpaceGet(nnz,&free_space);
2013   current_space = free_space;
2014 
2015 #if defined(DEBUG_MATTrMatMult)
2016   ierr = PetscPrintf(PETSC_COMM_SELF,"[%d] nnz=%d + %d + %d\n",rank,pdti[pn],poti[pon],ai[am]);
2017 #endif
2018 
2019   ierr = PetscMalloc3(merge->nrecv,PetscInt**,&buf_ri_k,merge->nrecv,PetscInt*,&nextrow,merge->nrecv,PetscInt*,&nextci);CHKERRQ(ierr);
2020   for (k=0; k<merge->nrecv; k++){
2021     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
2022     nrows       = *buf_ri_k[k];
2023     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
2024     nextci[k]   = buf_ri_k[k] + (nrows + 1);/* poins to the next i-structure of k-th recved i-structure  */
2025   }
2026 
2027   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
2028   rmax = 0;
2029   for (i=0; i<pn; i++) {
2030     /* add pdt[i,:]*AP into lnk */
2031     pnz = pdti[i+1] - pdti[i];
2032     ptJ = pdtj + pdti[i];
2033     for (j=0; j<pnz; j++){
2034       row  = ptJ[j];  /* row of AP == col of Pt */
2035       anz  = ai[row+1] - ai[row];
2036       Jptr = aj + ai[row];
2037       /* add non-zero cols of AP into the sorted linked list lnk */
2038       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
2039     }
2040 
2041     /* add received col data into lnk */
2042     for (k=0; k<merge->nrecv; k++){ /* k-th received message */
2043       if (i == *nextrow[k]) { /* i-th row */
2044         nzi = *(nextci[k]+1) - *nextci[k];
2045         Jptr  = buf_rj[k] + *nextci[k];
2046         ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
2047         nextrow[k]++; nextci[k]++;
2048       }
2049     }
2050     nnz = lnk[0];
2051 
2052     /* if free space is not available, make more free space */
2053     if (current_space->local_remaining<nnz) {
2054       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
2055       nspacedouble++;
2056     }
2057     /* copy data into free space, then initialize lnk */
2058     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
2059     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
2060     current_space->array           += nnz;
2061     current_space->local_used      += nnz;
2062     current_space->local_remaining -= nnz;
2063     bi[i+1] = bi[i] + nnz;
2064     if (nnz > rmax) rmax = nnz;
2065   }
2066   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
2067 
2068   ierr = PetscMalloc((bi[pn]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
2069   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
2070   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]);
2071   if (afill_tmp > afill) afill = afill_tmp;
2072   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
2073   ierr = MatDestroy(&POt);CHKERRQ(ierr);
2074   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
2075 
2076   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
2077   /*----------------------------------------------------------------------------------*/
2078   ierr = PetscMalloc((rmax+1)*sizeof(PetscScalar),&vals);CHKERRQ(ierr);
2079   ierr = PetscMemzero(vals,rmax*sizeof(PetscScalar));CHKERRQ(ierr);
2080 
2081   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
2082   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
2083   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
2084   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
2085   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
2086   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
2087   for (i=0; i<pn; i++){
2088     row = i + rstart;
2089     nnz = bi[i+1] - bi[i];
2090     Jptr = bj + bi[i];
2091     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
2092   }
2093   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2094   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2095   ierr = PetscFree(vals);CHKERRQ(ierr);
2096 
2097   merge->bi            = bi;
2098   merge->bj            = bj;
2099   merge->coi           = coi;
2100   merge->coj           = coj;
2101   merge->buf_ri        = buf_ri;
2102   merge->buf_rj        = buf_rj;
2103   merge->owners_co     = owners_co;
2104   merge->destroy       = Cmpi->ops->destroy;
2105   merge->duplicate     = Cmpi->ops->duplicate;
2106 
2107   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_Scalable;
2108   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
2109 
2110   /* attach the supporting struct to Cmpi for reuse */
2111   c = (Mat_MPIAIJ*)Cmpi->data;
2112   c->ptap        = ptap;
2113   ptap->api      = PETSC_NULL;
2114   ptap->apj      = PETSC_NULL;
2115   ptap->merge    = merge;
2116   ptap->rmax     = rmax;
2117   ptap->apa      = PETSC_NULL;
2118 
2119   *C = Cmpi;
2120 #if defined(PETSC_USE_INFO)
2121   if (bi[pn] != 0) {
2122     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %G needed %G.\n",nspacedouble,fill,afill);CHKERRQ(ierr);
2123     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%G,&C) for best performance.\n",afill);CHKERRQ(ierr);
2124   } else {
2125     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
2126   }
2127 #endif
2128   PetscFunctionReturn(0);
2129 }
2130