xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision f3f0eb19a322e1923f66aeb6cc7ffacd1eb54b2a)
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 #undef __FUNCT__
13 #define __FUNCT__ "MatMatMult_MPIAIJ_MPIAIJ"
14 PetscErrorCode MatMatMult_MPIAIJ_MPIAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill, Mat *C)
15 {
16   PetscErrorCode ierr;
17 
18   PetscFunctionBegin;
19   if (scall == MAT_INITIAL_MATRIX){
20     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
21     ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ(A,B,fill,C);CHKERRQ(ierr);
22     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
23   }
24 
25   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
26   ierr = (*(*C)->ops->matmultnumeric)(A,B,*C);CHKERRQ(ierr);
27   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
28   PetscFunctionReturn(0);
29 }
30 
31 #undef __FUNCT__
32 #define __FUNCT__ "MatDestroy_MPIAIJ_MatMatMult"
33 PetscErrorCode MatDestroy_MPIAIJ_MatMatMult(Mat A)
34 {
35   PetscErrorCode ierr;
36   Mat_MPIAIJ     *a=(Mat_MPIAIJ*)A->data;
37   Mat_PtAPMPI    *ptap=a->ptap;
38 
39   PetscFunctionBegin;
40   ierr = PetscFree2(ptap->startsj_s,ptap->startsj_r);CHKERRQ(ierr);
41   ierr = PetscFree(ptap->bufa);CHKERRQ(ierr);
42   ierr = MatDestroy(&ptap->P_loc);CHKERRQ(ierr);
43   ierr = MatDestroy(&ptap->P_oth);CHKERRQ(ierr);
44   ierr = MatDestroy(&ptap->Pt);CHKERRQ(ierr);
45   ierr = PetscFree(ptap->api);CHKERRQ(ierr);
46   ierr = PetscFree(ptap->apj);CHKERRQ(ierr);
47   ierr = PetscFree(ptap->apa);CHKERRQ(ierr);
48   ierr = ptap->destroy(A);CHKERRQ(ierr);
49   ierr = PetscFree(ptap);CHKERRQ(ierr);
50   PetscFunctionReturn(0);
51 }
52 
53 #undef __FUNCT__
54 #define __FUNCT__ "MatDuplicate_MPIAIJ_MatMatMult"
55 PetscErrorCode MatDuplicate_MPIAIJ_MatMatMult(Mat A, MatDuplicateOption op, Mat *M)
56 {
57   PetscErrorCode     ierr;
58   Mat_MPIAIJ         *a=(Mat_MPIAIJ*)A->data;
59   Mat_PtAPMPI        *ptap=a->ptap;
60 
61   PetscFunctionBegin;
62   ierr = (*ptap->duplicate)(A,op,M);CHKERRQ(ierr);
63   (*M)->ops->destroy   = ptap->destroy;   /* = MatDestroy_MPIAIJ, *M doesn't duplicate A's special structure! */
64   (*M)->ops->duplicate = ptap->duplicate; /* = MatDuplicate_MPIAIJ */
65   PetscFunctionReturn(0);
66 }
67 
68 #undef __FUNCT__
69 #define __FUNCT__ "MatMatMultNumeric_MPIAIJ_MPIAIJ"
70 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
71 {
72   PetscErrorCode     ierr;
73   Mat_MPIAIJ         *a=(Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
74   Mat_SeqAIJ         *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
75   Mat_SeqAIJ         *cd=(Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
76   PetscInt           *adi=ad->i,*adj,*aoi=ao->i,*aoj;
77   PetscScalar        *ada,*aoa,*cda=cd->a,*coa=co->a;
78   Mat_SeqAIJ         *p_loc,*p_oth;
79   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
80   PetscScalar        *pa_loc,*pa_oth,*pa,*apa,valtmp,*ca;
81   PetscInt           cm=C->rmap->n,anz,pnz;
82   Mat_PtAPMPI        *ptap=c->ptap;
83   PetscInt           *api,*apj,*apJ,i,j,k,row;
84   PetscInt           cstart=C->cmap->rstart;
85   PetscInt           cdnz,conz,k0,k1;
86 
87   PetscFunctionBegin;
88   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
89   /*-----------------------------------------------------*/
90   /* update numerical values of P_oth and P_loc */
91   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
92   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
93 
94   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
95   /*----------------------------------------------------------*/
96   /* get data from symbolic products */
97   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
98   p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
99   pi_loc=p_loc->i; pj_loc=p_loc->j; pa_loc=p_loc->a;
100   pi_oth=p_oth->i; pj_oth=p_oth->j; pa_oth=p_oth->a;
101 
102   /* get apa for storing dense row A[i,:]*P */
103   apa = ptap->apa;
104 
105   api = ptap->api;
106   apj = ptap->apj;
107   for (i=0; i<cm; i++) {
108     /* diagonal portion of A */
109     anz = adi[i+1] - adi[i];
110     adj = ad->j + adi[i];
111     ada = ad->a + adi[i];
112     for (j=0; j<anz; j++) {
113       row = adj[j];
114       pnz = pi_loc[row+1] - pi_loc[row];
115       pj  = pj_loc + pi_loc[row];
116       pa  = pa_loc + pi_loc[row];
117 
118       /* perform dense axpy */
119       valtmp = ada[j];
120       for (k=0; k<pnz; k++){
121         apa[pj[k]] += valtmp*pa[k];
122       }
123       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
124     }
125 
126     /* off-diagonal portion of A */
127     anz = aoi[i+1] - aoi[i];
128     aoj = ao->j + aoi[i];
129     aoa = ao->a + aoi[i];
130     for (j=0; j<anz; j++) {
131       row = aoj[j];
132       pnz = pi_oth[row+1] - pi_oth[row];
133       pj  = pj_oth + pi_oth[row];
134       pa  = pa_oth + pi_oth[row];
135 
136       /* perform dense axpy */
137       valtmp = aoa[j];
138       for (k=0; k<pnz; k++){
139         apa[pj[k]] += valtmp*pa[k];
140       }
141       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
142     }
143 
144     /* set values in C */
145     apJ = apj + api[i];
146     cdnz = cd->i[i+1] - cd->i[i];
147     conz = co->i[i+1] - co->i[i];
148 
149     /* 1st off-diagoanl part of C */
150     ca = coa + co->i[i];
151     k  = 0;
152     for (k0=0; k0<conz; k0++){
153       if (apJ[k] >= cstart) break;
154       ca[k0]      = apa[apJ[k]];
155       apa[apJ[k]] = 0.0;
156       k++;
157     }
158 
159     /* diagonal part of C */
160     ca = cda + cd->i[i];
161     for (k1=0; k1<cdnz; k1++){
162       ca[k1]      = apa[apJ[k]];
163       apa[apJ[k]] = 0.0;
164       k++;
165     }
166 
167     /* 2nd off-diagoanl part of C */
168     ca = coa + co->i[i];
169     for (; k0<conz; k0++){
170       ca[k0]      = apa[apJ[k]];
171       apa[apJ[k]] = 0.0;
172       k++;
173     }
174   }
175   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
176   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
177   PetscFunctionReturn(0);
178 }
179 
180 #undef __FUNCT__
181 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIAIJ"
182 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat *C)
183 {
184   PetscErrorCode       ierr;
185   MPI_Comm             comm=((PetscObject)A)->comm;
186   Mat                  Cmpi;
187   Mat_PtAPMPI          *ptap;
188   PetscFreeSpaceList   free_space=PETSC_NULL,current_space=PETSC_NULL;
189   Mat_MPIAIJ           *a=(Mat_MPIAIJ*)A->data,*c;
190   Mat_SeqAIJ           *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
191   PetscInt             *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
192   PetscInt             *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
193   PetscInt             *lnk,i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi;
194   PetscInt             am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
195   PetscBT              lnkbt;
196   PetscScalar          *apa;
197   PetscReal            afill;
198   PetscBool            scalable=PETSC_TRUE;
199   PetscInt             nlnk_max,armax,prmax;
200 
201   PetscFunctionBegin;
202   if (A->cmap->rstart != P->rmap->rstart || A->cmap->rend != P->rmap->rend){
203     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);
204   }
205 
206   ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
207     ierr = PetscOptionsBool("-matmatmult_scalable","Use a scalable but slower C=A*B","",scalable,&scalable,PETSC_NULL);CHKERRQ(ierr);
208     if (scalable){
209       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable(A,P,fill,C);;CHKERRQ(ierr);
210       PetscFunctionReturn(0);
211     }
212   ierr = PetscOptionsEnd();CHKERRQ(ierr);
213 
214   /* create struct Mat_PtAPMPI and attached it to C later */
215   ierr = PetscNew(Mat_PtAPMPI,&ptap);CHKERRQ(ierr);
216 
217   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
218   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
219 
220   /* get P_loc by taking all local rows of P */
221   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
222 
223   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
224   p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
225   pi_loc = p_loc->i; pj_loc = p_loc->j;
226   pi_oth = p_oth->i; pj_oth = p_oth->j;
227 
228   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
229   /*-------------------------------------------------------------------*/
230   ierr  = PetscMalloc((am+2)*sizeof(PetscInt),&api);CHKERRQ(ierr);
231   ptap->api = api;
232   api[0]    = 0;
233 
234   /* create and initialize a linked list */
235   armax = ad->rmax+ao->rmax;
236   prmax = PetscMax(p_loc->rmax,p_oth->rmax);
237   nlnk_max = armax*prmax;
238   if (!nlnk_max || nlnk_max > pN) nlnk_max = pN;
239   ierr = PetscLLCondensedCreate(nlnk_max,pN,&lnk,&lnkbt);CHKERRQ(ierr);
240 
241   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
242   ierr = PetscFreeSpaceGet((PetscInt)(fill*(adi[am]+aoi[am]+pi_loc[pm])),&free_space);CHKERRQ(ierr);
243   current_space = free_space;
244 
245   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
246   for (i=0; i<am; i++) {
247     apnz = 0;
248     /* diagonal portion of A */
249     nzi = adi[i+1] - adi[i];
250     for (j=0; j<nzi; j++){
251       row = *adj++;
252       pnz = pi_loc[row+1] - pi_loc[row];
253       Jptr  = pj_loc + pi_loc[row];
254       /* add non-zero cols of P into the sorted linked list lnk */
255       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
256     }
257     /* off-diagonal portion of A */
258     nzi = aoi[i+1] - aoi[i];
259     for (j=0; j<nzi; j++){
260       row = *aoj++;
261       pnz = pi_oth[row+1] - pi_oth[row];
262       Jptr  = pj_oth + pi_oth[row];
263       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
264     }
265 
266     apnz     = lnk[0];
267     api[i+1] = api[i] + apnz;
268 
269     /* if free space is not available, double the total space in the list */
270     if (current_space->local_remaining<apnz) {
271       ierr = PetscFreeSpaceGet(apnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
272       nspacedouble++;
273     }
274 
275     /* Copy data into free space, then initialize lnk */
276     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
277     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
278     current_space->array           += apnz;
279     current_space->local_used      += apnz;
280     current_space->local_remaining -= apnz;
281   }
282 
283   /* Allocate space for apj, initialize apj, and */
284   /* destroy list of free space and other temporary array(s) */
285   ierr = PetscMalloc((api[am]+1)*sizeof(PetscInt),&ptap->apj);CHKERRQ(ierr);
286   apj  = ptap->apj;
287   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
288   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
289 
290   /* malloc apa to store dense row A[i,:]*P */
291   ierr = PetscMalloc(pN*sizeof(PetscScalar),&apa);CHKERRQ(ierr);
292   ierr = PetscMemzero(apa,pN*sizeof(PetscScalar));CHKERRQ(ierr);
293   ptap->apa = apa;
294 
295   /* create and assemble symbolic parallel matrix Cmpi */
296   /*----------------------------------------------------*/
297   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
298   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
299   ierr = MatSetBlockSizes(Cmpi,A->rmap->bs,P->cmap->bs);CHKERRQ(ierr);
300 
301   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
302   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
303   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
304   for (i=0; i<am; i++){
305     row  = i + rstart;
306     apnz = api[i+1] - api[i];
307     ierr = MatSetValues(Cmpi,1,&row,apnz,apj,apa,INSERT_VALUES);CHKERRQ(ierr);
308     apj += apnz;
309   }
310   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
311   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
312 
313   ptap->destroy             = Cmpi->ops->destroy;
314   ptap->duplicate           = Cmpi->ops->duplicate;
315   Cmpi->ops->destroy        = MatDestroy_MPIAIJ_MatMatMult;
316   Cmpi->ops->duplicate      = MatDuplicate_MPIAIJ_MatMatMult;
317 
318   /* attach the supporting struct to Cmpi for reuse */
319   c = (Mat_MPIAIJ*)Cmpi->data;
320   c->ptap  = ptap;
321 
322   *C = Cmpi;
323 
324   /* set MatInfo */
325   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
326   if (afill < 1.0) afill = 1.0;
327   Cmpi->info.mallocs           = nspacedouble;
328   Cmpi->info.fill_ratio_given  = fill;
329   Cmpi->info.fill_ratio_needed = afill;
330 
331 #if defined(PETSC_USE_INFO)
332   if (api[am]) {
333     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %G needed %G.\n",nspacedouble,fill,afill);CHKERRQ(ierr);
334     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%G,&C) for best performance.;\n",afill);CHKERRQ(ierr);
335   } else {
336     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
337   }
338 #endif
339   PetscFunctionReturn(0);
340 }
341 
342 #undef __FUNCT__
343 #define __FUNCT__ "MatMatMult_MPIAIJ_MPIDense"
344 PetscErrorCode MatMatMult_MPIAIJ_MPIDense(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
345 {
346   PetscErrorCode ierr;
347 
348   PetscFunctionBegin;
349   if (scall == MAT_INITIAL_MATRIX){
350     ierr = MatMatMultSymbolic_MPIAIJ_MPIDense(A,B,fill,C);CHKERRQ(ierr);
351   }
352   ierr = MatMatMultNumeric_MPIAIJ_MPIDense(A,B,*C);CHKERRQ(ierr);
353   PetscFunctionReturn(0);
354 }
355 
356 typedef struct {
357   Mat         workB;
358   PetscScalar *rvalues,*svalues;
359   MPI_Request *rwaits,*swaits;
360 } MPIAIJ_MPIDense;
361 
362 #undef __FUNCT__
363 #define __FUNCT__ "MatMPIAIJ_MPIDenseDestroy"
364 PetscErrorCode MatMPIAIJ_MPIDenseDestroy(void *ctx)
365 {
366   MPIAIJ_MPIDense *contents = (MPIAIJ_MPIDense*) ctx;
367   PetscErrorCode  ierr;
368 
369   PetscFunctionBegin;
370   ierr = MatDestroy(&contents->workB);CHKERRQ(ierr);
371   ierr = PetscFree4(contents->rvalues,contents->svalues,contents->rwaits,contents->swaits);CHKERRQ(ierr);
372   ierr = PetscFree(contents);CHKERRQ(ierr);
373   PetscFunctionReturn(0);
374 }
375 
376 #undef __FUNCT__
377 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIDense"
378 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat A,Mat B,PetscReal fill,Mat *C)
379 {
380   PetscErrorCode         ierr;
381   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*) A->data;
382   PetscInt               nz = aij->B->cmap->n;
383   PetscContainer         container;
384   MPIAIJ_MPIDense        *contents;
385   VecScatter             ctx = aij->Mvctx;
386   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
387   VecScatter_MPI_General *to   = ( VecScatter_MPI_General*) ctx->todata;
388   PetscInt               m=A->rmap->n,n=B->cmap->n;
389 
390   PetscFunctionBegin;
391   ierr = MatCreate(((PetscObject)B)->comm,C);CHKERRQ(ierr);
392   ierr = MatSetSizes(*C,m,n,A->rmap->N,B->cmap->N);CHKERRQ(ierr);
393   ierr = MatSetBlockSizes(*C,A->rmap->bs,B->cmap->bs);CHKERRQ(ierr);
394   ierr = MatSetType(*C,MATMPIDENSE);CHKERRQ(ierr);
395   ierr = MatMPIDenseSetPreallocation(*C,PETSC_NULL);CHKERRQ(ierr);
396   ierr = MatAssemblyBegin(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
397   ierr = MatAssemblyEnd(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
398   (*C)->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
399 
400   ierr = PetscNew(MPIAIJ_MPIDense,&contents);CHKERRQ(ierr);
401   /* Create work matrix used to store off processor rows of B needed for local product */
402   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,B->cmap->N,PETSC_NULL,&contents->workB);CHKERRQ(ierr);
403   /* Create work arrays needed */
404   ierr = PetscMalloc4(B->cmap->N*from->starts[from->n],PetscScalar,&contents->rvalues,
405                       B->cmap->N*to->starts[to->n],PetscScalar,&contents->svalues,
406                       from->n,MPI_Request,&contents->rwaits,
407                       to->n,MPI_Request,&contents->swaits);CHKERRQ(ierr);
408 
409   ierr = PetscContainerCreate(((PetscObject)A)->comm,&container);CHKERRQ(ierr);
410   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
411   ierr = PetscContainerSetUserDestroy(container,MatMPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
412   ierr = PetscObjectCompose((PetscObject)(*C),"workB",(PetscObject)container);CHKERRQ(ierr);
413   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
414   PetscFunctionReturn(0);
415 }
416 
417 #undef __FUNCT__
418 #define __FUNCT__ "MatMPIDenseScatter"
419 /*
420     Performs an efficient scatter on the rows of B needed by this process; this is
421     a modification of the VecScatterBegin_() routines.
422 */
423 PetscErrorCode MatMPIDenseScatter(Mat A,Mat B,Mat C,Mat *outworkB)
424 {
425   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*)A->data;
426   PetscErrorCode         ierr;
427   PetscScalar            *b,*w,*svalues,*rvalues;
428   VecScatter             ctx = aij->Mvctx;
429   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
430   VecScatter_MPI_General *to   = ( VecScatter_MPI_General*) ctx->todata;
431   PetscInt               i,j,k;
432   PetscInt               *sindices,*sstarts,*rindices,*rstarts;
433   PetscMPIInt            *sprocs,*rprocs,nrecvs;
434   MPI_Request            *swaits,*rwaits;
435   MPI_Comm               comm = ((PetscObject)A)->comm;
436   PetscMPIInt            tag = ((PetscObject)ctx)->tag,ncols = B->cmap->N, nrows = aij->B->cmap->n,imdex,nrowsB = B->rmap->n;
437   MPI_Status             status;
438   MPIAIJ_MPIDense        *contents;
439   PetscContainer         container;
440   Mat                    workB;
441 
442   PetscFunctionBegin;
443   ierr = PetscObjectQuery((PetscObject)C,"workB",(PetscObject*)&container);CHKERRQ(ierr);
444   if (!container) SETERRQ(comm,PETSC_ERR_PLIB,"Container does not exist");
445   ierr = PetscContainerGetPointer(container,(void**)&contents);CHKERRQ(ierr);
446 
447   workB = *outworkB = contents->workB;
448   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);
449   sindices  = to->indices;
450   sstarts   = to->starts;
451   sprocs    = to->procs;
452   swaits    = contents->swaits;
453   svalues   = contents->svalues;
454 
455   rindices  = from->indices;
456   rstarts   = from->starts;
457   rprocs    = from->procs;
458   rwaits    = contents->rwaits;
459   rvalues   = contents->rvalues;
460 
461   ierr = MatDenseGetArray(B,&b);CHKERRQ(ierr);
462   ierr = MatDenseGetArray(workB,&w);CHKERRQ(ierr);
463 
464   for (i=0; i<from->n; i++) {
465     ierr = MPI_Irecv(rvalues+ncols*rstarts[i],ncols*(rstarts[i+1]-rstarts[i]),MPIU_SCALAR,rprocs[i],tag,comm,rwaits+i);CHKERRQ(ierr);
466   }
467 
468   for (i=0; i<to->n; i++) {
469     /* pack a message at a time */
470     CHKMEMQ;
471     for (j=0; j<sstarts[i+1]-sstarts[i]; j++){
472       for (k=0; k<ncols; k++) {
473         svalues[ncols*(sstarts[i] + j) + k] = b[sindices[sstarts[i]+j] + nrowsB*k];
474       }
475     }
476     CHKMEMQ;
477     ierr = MPI_Isend(svalues+ncols*sstarts[i],ncols*(sstarts[i+1]-sstarts[i]),MPIU_SCALAR,sprocs[i],tag,comm,swaits+i);CHKERRQ(ierr);
478   }
479 
480   nrecvs = from->n;
481   while (nrecvs) {
482     ierr = MPI_Waitany(from->n,rwaits,&imdex,&status);CHKERRQ(ierr);
483     nrecvs--;
484     /* unpack a message at a time */
485     CHKMEMQ;
486     for (j=0; j<rstarts[imdex+1]-rstarts[imdex]; j++){
487       for (k=0; k<ncols; k++) {
488         w[rindices[rstarts[imdex]+j] + nrows*k] = rvalues[ncols*(rstarts[imdex] + j) + k];
489       }
490     }
491     CHKMEMQ;
492   }
493   if (to->n) {ierr = MPI_Waitall(to->n,swaits,to->sstatus);CHKERRQ(ierr);}
494 
495   ierr = MatDenseRestoreArray(B,&b);CHKERRQ(ierr);
496   ierr = MatDenseRestoreArray(workB,&w);CHKERRQ(ierr);
497   ierr = MatAssemblyBegin(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
498   ierr = MatAssemblyEnd(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
499   PetscFunctionReturn(0);
500 }
501 extern PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat,Mat,Mat);
502 
503 #undef __FUNCT__
504 #define __FUNCT__ "MatMatMultNumeric_MPIAIJ_MPIDense"
505 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat A,Mat B,Mat C)
506 {
507   PetscErrorCode       ierr;
508   Mat_MPIAIJ           *aij = (Mat_MPIAIJ*)A->data;
509   Mat_MPIDense         *bdense = (Mat_MPIDense*)B->data;
510   Mat_MPIDense         *cdense = (Mat_MPIDense*)C->data;
511   Mat                  workB;
512 
513   PetscFunctionBegin;
514 
515   /* diagonal block of A times all local rows of B*/
516   ierr = MatMatMultNumeric_SeqAIJ_SeqDense(aij->A,bdense->A,cdense->A);CHKERRQ(ierr);
517 
518   /* get off processor parts of B needed to complete the product */
519   ierr = MatMPIDenseScatter(A,B,C,&workB);CHKERRQ(ierr);
520 
521   /* off-diagonal block of A times nonlocal rows of B */
522   ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A);CHKERRQ(ierr);
523   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
524   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
525   PetscFunctionReturn(0);
526 }
527 
528 #undef __FUNCT__
529 #define __FUNCT__ "MatMatMultNumeric_MPIAIJ_MPIAIJ_Scalable"
530 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ_Scalable(Mat A,Mat P,Mat C)
531 {
532   PetscErrorCode     ierr;
533   Mat_MPIAIJ         *a=(Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
534   Mat_SeqAIJ         *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
535   Mat_SeqAIJ         *cd=(Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
536   PetscInt           *adi=ad->i,*adj,*aoi=ao->i,*aoj;
537   PetscScalar        *ada,*aoa,*cda=cd->a,*coa=co->a;
538   Mat_SeqAIJ         *p_loc,*p_oth;
539   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
540   PetscScalar        *pa_loc,*pa_oth,*pa,valtmp,*ca;
541   PetscInt           cm=C->rmap->n,anz,pnz;
542   Mat_PtAPMPI        *ptap=c->ptap;
543   PetscScalar        *apa_sparse=ptap->apa;
544   PetscInt           *api,*apj,*apJ,i,j,k,row;
545   PetscInt           cstart=C->cmap->rstart;
546   PetscInt           cdnz,conz,k0,k1,nextp;
547 
548   PetscFunctionBegin;
549   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
550   /*-----------------------------------------------------*/
551   /* update numerical values of P_oth and P_loc */
552   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
553   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
554 
555   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
556   /*----------------------------------------------------------*/
557   /* get data from symbolic products */
558   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
559   p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
560   pi_loc=p_loc->i; pj_loc=p_loc->j; pa_loc=p_loc->a;
561   pi_oth=p_oth->i; pj_oth=p_oth->j; pa_oth=p_oth->a;
562 
563   api = ptap->api;
564   apj = ptap->apj;
565   for (i=0; i<cm; i++) {
566     apJ = apj + api[i];
567 
568     /* diagonal portion of A */
569     anz = adi[i+1] - adi[i];
570     adj = ad->j + adi[i];
571     ada = ad->a + adi[i];
572     for (j=0; j<anz; j++) {
573       row = adj[j];
574       pnz = pi_loc[row+1] - pi_loc[row];
575       pj  = pj_loc + pi_loc[row];
576       pa  = pa_loc + pi_loc[row];
577       /* perform sparse axpy */
578       valtmp = ada[j];
579       nextp  = 0;
580       for (k=0; nextp<pnz; k++) {
581         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
582           apa_sparse[k] += valtmp*pa[nextp++];
583         }
584       }
585       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
586     }
587 
588     /* off-diagonal portion of A */
589     anz = aoi[i+1] - aoi[i];
590     aoj = ao->j + aoi[i];
591     aoa = ao->a + aoi[i];
592     for (j=0; j<anz; j++) {
593       row = aoj[j];
594       pnz = pi_oth[row+1] - pi_oth[row];
595       pj  = pj_oth + pi_oth[row];
596       pa  = pa_oth + pi_oth[row];
597       /* perform sparse axpy */
598       valtmp = aoa[j];
599       nextp  = 0;
600       for (k=0; nextp<pnz; k++) {
601         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
602           apa_sparse[k] += valtmp*pa[nextp++];
603         }
604       }
605       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
606     }
607 
608     /* set values in C */
609     cdnz = cd->i[i+1] - cd->i[i];
610     conz = co->i[i+1] - co->i[i];
611 
612     /* 1st off-diagoanl part of C */
613     ca = coa + co->i[i];
614     k  = 0;
615     for (k0=0; k0<conz; k0++){
616       if (apJ[k] >= cstart) break;
617       ca[k0]      = apa_sparse[k];
618       apa_sparse[k] = 0.0;
619       k++;
620     }
621 
622     /* diagonal part of C */
623     ca = cda + cd->i[i];
624     for (k1=0; k1<cdnz; k1++){
625       ca[k1]      = apa_sparse[k];
626       apa_sparse[k] = 0.0;
627       k++;
628     }
629 
630     /* 2nd off-diagoanl part of C */
631     ca = coa + co->i[i];
632     for (; k0<conz; k0++){
633       ca[k0]      = apa_sparse[k];
634       apa_sparse[k] = 0.0;
635       k++;
636     }
637   }
638   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
639   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
640   PetscFunctionReturn(0);
641 }
642 
643 /* same as MatMatMultSymbolic_MPIAIJ_MPIAIJ(), except using LLCondensed to avoid O(BN) memory requirement */
644 #undef __FUNCT__
645 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable"
646 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable(Mat A,Mat P,PetscReal fill,Mat *C)
647 {
648   PetscErrorCode       ierr;
649   MPI_Comm             comm=((PetscObject)A)->comm;
650   Mat                  Cmpi;
651   Mat_PtAPMPI          *ptap;
652   PetscFreeSpaceList   free_space=PETSC_NULL,current_space=PETSC_NULL;
653   Mat_MPIAIJ           *a=(Mat_MPIAIJ*)A->data,*c;
654   Mat_SeqAIJ           *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
655   PetscInt             *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
656   PetscInt             *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
657   PetscInt             i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi,*lnk,apnz_max=0;
658   PetscInt             am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
659   PetscInt             nlnk_max,armax,prmax;
660   PetscReal            afill;
661   PetscScalar          *apa;
662 
663   PetscFunctionBegin;
664   /* create struct Mat_PtAPMPI and attached it to C later */
665   ierr = PetscNew(Mat_PtAPMPI,&ptap);CHKERRQ(ierr);
666 
667   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
668   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
669 
670   /* get P_loc by taking all local rows of P */
671   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
672 
673   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
674   p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
675   pi_loc = p_loc->i; pj_loc = p_loc->j;
676   pi_oth = p_oth->i; pj_oth = p_oth->j;
677 
678   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
679   /*-------------------------------------------------------------------*/
680   ierr  = PetscMalloc((am+2)*sizeof(PetscInt),&api);CHKERRQ(ierr);
681   ptap->api = api;
682   api[0]    = 0;
683 
684   /* create and initialize a linked list */
685   armax = ad->rmax+ao->rmax;
686   prmax = PetscMax(p_loc->rmax,p_oth->rmax);
687   nlnk_max = armax*prmax;
688   if (!nlnk_max || nlnk_max > pN) nlnk_max = pN;
689   ierr = PetscLLCondensedCreate_Scalable(nlnk_max,&lnk);CHKERRQ(ierr);
690 
691   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
692   ierr = PetscFreeSpaceGet((PetscInt)(fill*(adi[am]+aoi[am]+pi_loc[pm])),&free_space);CHKERRQ(ierr);
693   current_space = free_space;
694 
695   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
696   for (i=0; i<am; i++) {
697     apnz = 0;
698     /* diagonal portion of A */
699     nzi = adi[i+1] - adi[i];
700     for (j=0; j<nzi; j++){
701       row = *adj++;
702       pnz = pi_loc[row+1] - pi_loc[row];
703       Jptr  = pj_loc + pi_loc[row];
704       /* add non-zero cols of P into the sorted linked list lnk */
705       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
706     }
707     /* off-diagonal portion of A */
708     nzi = aoi[i+1] - aoi[i];
709     for (j=0; j<nzi; j++){
710       row = *aoj++;
711       pnz = pi_oth[row+1] - pi_oth[row];
712       Jptr  = pj_oth + pi_oth[row];
713       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
714     }
715 
716     apnz     = *lnk;
717     api[i+1] = api[i] + apnz;
718     if (apnz > apnz_max) apnz_max = apnz;
719 
720     /* if free space is not available, double the total space in the list */
721     if (current_space->local_remaining<apnz) {
722       ierr = PetscFreeSpaceGet(apnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
723       nspacedouble++;
724     }
725 
726     /* Copy data into free space, then initialize lnk */
727     ierr = PetscLLCondensedClean_Scalable(apnz,current_space->array,lnk);CHKERRQ(ierr);
728     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
729     current_space->array           += apnz;
730     current_space->local_used      += apnz;
731     current_space->local_remaining -= apnz;
732   }
733 
734   /* Allocate space for apj, initialize apj, and */
735   /* destroy list of free space and other temporary array(s) */
736   ierr = PetscMalloc((api[am]+1)*sizeof(PetscInt),&ptap->apj);CHKERRQ(ierr);
737   apj  = ptap->apj;
738   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
739   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
740 
741   /* create and assemble symbolic parallel matrix Cmpi */
742   /*----------------------------------------------------*/
743   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
744   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
745   ierr = MatSetBlockSizes(Cmpi,A->rmap->bs,P->cmap->bs);CHKERRQ(ierr);
746   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
747   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
748   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
749 
750   /* malloc apa for assembly Cmpi */
751   ierr = PetscMalloc(apnz_max*sizeof(PetscScalar),&apa);CHKERRQ(ierr);
752   ierr = PetscMemzero(apa,apnz_max*sizeof(PetscScalar));CHKERRQ(ierr);
753   ptap->apa = apa;
754   for (i=0; i<am; i++){
755     row  = i + rstart;
756     apnz = api[i+1] - api[i];
757     ierr = MatSetValues(Cmpi,1,&row,apnz,apj,apa,INSERT_VALUES);CHKERRQ(ierr);
758     apj += apnz;
759   }
760   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
761   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
762 
763   ptap->destroy             = Cmpi->ops->destroy;
764   ptap->duplicate           = Cmpi->ops->duplicate;
765   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_Scalable;
766   Cmpi->ops->destroy        = MatDestroy_MPIAIJ_MatMatMult;
767   Cmpi->ops->duplicate      = MatDuplicate_MPIAIJ_MatMatMult;
768 
769   /* attach the supporting struct to Cmpi for reuse */
770   c = (Mat_MPIAIJ*)Cmpi->data;
771   c->ptap  = ptap;
772 
773   *C = Cmpi;
774 
775   /* set MatInfo */
776   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
777   if (afill < 1.0) afill = 1.0;
778   Cmpi->info.mallocs           = nspacedouble;
779   Cmpi->info.fill_ratio_given  = fill;
780   Cmpi->info.fill_ratio_needed = afill;
781 
782 #if defined(PETSC_USE_INFO)
783   if (api[am]) {
784     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %G needed %G.\n",nspacedouble,fill,afill);CHKERRQ(ierr);
785     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%G,&C) for best performance.;\n",afill);CHKERRQ(ierr);
786   } else {
787     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
788   }
789 #endif
790   PetscFunctionReturn(0);
791 }
792 
793 /*-------------------------------------------------------------------------*/
794 #undef __FUNCT__
795 #define __FUNCT__ "MatTransposeMatMult_MPIAIJ_MPIAIJ"
796 PetscErrorCode MatTransposeMatMult_MPIAIJ_MPIAIJ(Mat P,Mat A,MatReuse scall,PetscReal fill,Mat *C)
797 {
798   PetscErrorCode ierr;
799   PetscBool      scalable=PETSC_TRUE,viamatmatmult=PETSC_FALSE;
800 
801   PetscFunctionBegin;
802   ierr = PetscOptionsBool("-mattransposematmult_viamatmatmult","Use R=Pt and C=R*A","",viamatmatmult,&viamatmatmult,PETSC_NULL);CHKERRQ(ierr);
803   if (viamatmatmult){
804     Mat         Pt;
805     Mat_PtAPMPI *ptap;
806     Mat_MPIAIJ  *c;
807     if (scall == MAT_INITIAL_MATRIX){
808       ierr = MatTranspose(P,MAT_INITIAL_MATRIX,&Pt);CHKERRQ(ierr);
809       ierr = MatMatMult(Pt,A,MAT_INITIAL_MATRIX,fill,C);CHKERRQ(ierr);
810 
811       c        = (Mat_MPIAIJ*)(*C)->data;
812       ptap     = c->ptap;
813       ptap->Pt = Pt;
814     } else if (scall == MAT_REUSE_MATRIX){
815       c    = (Mat_MPIAIJ*)(*C)->data;
816       ptap = c->ptap;
817       Pt   = ptap->Pt;
818       ierr = MatTranspose(P,scall,&Pt);CHKERRQ(ierr);
819       ierr = MatMatMult(Pt,A,scall,fill,C);CHKERRQ(ierr);
820     } else {
821       SETERRQ(((PetscObject)A)->comm,PETSC_ERR_ARG_WRONGSTATE,"Not supported");
822     }
823     PetscFunctionReturn(0);
824   }
825 
826   if (scall == MAT_INITIAL_MATRIX){
827     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
828       ierr = PetscOptionsBool("-mattransposematmult_scalable","Use a scalable but slower C=Pt*A","",scalable,&scalable,PETSC_NULL);CHKERRQ(ierr);
829       if  (scalable){
830         ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable(P,A,fill,C);CHKERRQ(ierr);
831       } else {
832         ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(P,A,fill,C);CHKERRQ(ierr);
833       }
834     ierr = PetscOptionsEnd();CHKERRQ(ierr);
835   }
836   ierr = (*(*C)->ops->mattransposemultnumeric)(P,A,*C);CHKERRQ(ierr);
837   PetscFunctionReturn(0);
838 }
839 
840 #undef __FUNCT__
841 #define __FUNCT__ "MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ"
842 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ(Mat P,Mat A,Mat C)
843 {
844   PetscErrorCode       ierr;
845   Mat_Merge_SeqsToMPI  *merge;
846   Mat_MPIAIJ           *p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
847   Mat_SeqAIJ           *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
848   Mat_PtAPMPI          *ptap;
849   PetscInt             *adj,*aJ;
850   PetscInt             i,j,k,anz,pnz,row,*cj;
851   MatScalar            *ada,*aval,*ca,valtmp;
852   PetscInt             am=A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
853   MPI_Comm             comm=((PetscObject)C)->comm;
854   PetscMPIInt          size,rank,taga,*len_s;
855   PetscInt             *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
856   PetscInt             **buf_ri,**buf_rj;
857   PetscInt             cnz=0,*bj_i,*bi,*bj,bnz,nextcj; /* bi,bj,ba: local array of C(mpi mat) */
858   MPI_Request          *s_waits,*r_waits;
859   MPI_Status           *status;
860   MatScalar            **abuf_r,*ba_i,*pA,*coa,*ba;
861   PetscInt             *ai,*aj,*coi,*coj;
862   PetscInt             *poJ=po->j,*pdJ=pd->j;
863   Mat                  A_loc;
864   Mat_SeqAIJ           *a_loc;
865 
866   PetscFunctionBegin;
867   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
868   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
869 
870   ptap  = c->ptap;
871   merge = ptap->merge;
872 
873   /* 2) compute numeric C_seq = P_loc^T*A_loc*P - dominating part */
874   /*--------------------------------------------------------------*/
875   /* get data from symbolic products */
876   coi = merge->coi; coj = merge->coj;
877   ierr = PetscMalloc((coi[pon]+1)*sizeof(MatScalar),&coa);CHKERRQ(ierr);
878   ierr = PetscMemzero(coa,coi[pon]*sizeof(MatScalar));CHKERRQ(ierr);
879 
880   bi     = merge->bi; bj = merge->bj;
881   owners = merge->rowmap->range;
882   ierr   = PetscMalloc((bi[cm]+1)*sizeof(MatScalar),&ba);CHKERRQ(ierr);
883   ierr   = PetscMemzero(ba,bi[cm]*sizeof(MatScalar));CHKERRQ(ierr);
884 
885   /* get A_loc by taking all local rows of A */
886   A_loc = ptap->A_loc;
887   ierr = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
888   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
889   ai   = a_loc->i;
890   aj   = a_loc->j;
891 
892   ierr = PetscMalloc((A->cmap->N)*sizeof(PetscScalar),&aval);CHKERRQ(ierr); /* non-scalable!!! */
893   ierr = PetscMemzero(aval,A->cmap->N*sizeof(PetscScalar));CHKERRQ(ierr);
894 
895     for (i=0; i<am; i++) {
896       /* 2-a) put A[i,:] to dense array aval */
897       anz = ai[i+1] - ai[i];
898       adj = aj + ai[i];
899       ada = a_loc->a + ai[i];
900       for (j=0; j<anz; j++){
901         aval[adj[j]] = ada[j];
902       }
903 
904       /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
905       /*--------------------------------------------------------------*/
906       /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
907       pnz = po->i[i+1] - po->i[i];
908       poJ = po->j + po->i[i];
909       pA  = po->a + po->i[i];
910       for (j=0; j<pnz; j++){
911         row = poJ[j];
912         cnz = coi[row+1] - coi[row];
913         cj  = coj + coi[row];
914         ca  = coa + coi[row];
915         /* perform dense axpy */
916         valtmp = pA[j];
917         for (k=0; k<cnz; k++) {
918           ca[k] += valtmp*aval[cj[k]];
919         }
920         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
921       }
922 
923       /* put the value into Cd (diagonal part) */
924       pnz = pd->i[i+1] - pd->i[i];
925       pdJ = pd->j + pd->i[i];
926       pA  = pd->a + pd->i[i];
927       for (j=0; j<pnz; j++){
928         row = pdJ[j];
929         cnz = bi[row+1] - bi[row];
930         cj  = bj + bi[row];
931         ca  = ba + bi[row];
932         /* perform dense axpy */
933         valtmp = pA[j];
934         for (k=0; k<cnz; k++) {
935           ca[k] += valtmp*aval[cj[k]];
936         }
937         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
938       }
939 
940       /* zero the current row of Pt*A */
941       aJ = aj + ai[i];
942       for (k=0; k<anz; k++) aval[aJ[k]] = 0.0;
943     }
944 
945   /* 3) send and recv matrix values coa */
946   /*------------------------------------*/
947   buf_ri = merge->buf_ri;
948   buf_rj = merge->buf_rj;
949   len_s  = merge->len_s;
950   ierr = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
951   ierr = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
952 
953   ierr = PetscMalloc2(merge->nsend+1,MPI_Request,&s_waits,size,MPI_Status,&status);CHKERRQ(ierr);
954   for (proc=0,k=0; proc<size; proc++){
955     if (!len_s[proc]) continue;
956     i = merge->owners_co[proc];
957     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
958     k++;
959   }
960   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
961   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
962 
963   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
964   ierr = PetscFree(r_waits);CHKERRQ(ierr);
965   ierr = PetscFree(coa);CHKERRQ(ierr);
966 
967   /* 4) insert local Cseq and received values into Cmpi */
968   /*----------------------------------------------------*/
969   ierr = PetscMalloc3(merge->nrecv,PetscInt**,&buf_ri_k,merge->nrecv,PetscInt*,&nextrow,merge->nrecv,PetscInt*,&nextci);CHKERRQ(ierr);
970   for (k=0; k<merge->nrecv; k++){
971     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
972     nrows       = *(buf_ri_k[k]);
973     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
974     nextci[k]   = buf_ri_k[k] + (nrows + 1);/* poins to the next i-structure of k-th recved i-structure  */
975   }
976 
977   for (i=0; i<cm; i++) {
978     row = owners[rank] + i; /* global row index of C_seq */
979     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
980     ba_i = ba + bi[i];
981     bnz  = bi[i+1] - bi[i];
982     /* add received vals into ba */
983     for (k=0; k<merge->nrecv; k++){ /* k-th received message */
984       /* i-th row */
985       if (i == *nextrow[k]) {
986         cnz = *(nextci[k]+1) - *nextci[k];
987         cj  = buf_rj[k] + *(nextci[k]);
988         ca  = abuf_r[k] + *(nextci[k]);
989         nextcj = 0;
990         for (j=0; nextcj<cnz; j++){
991           if (bj_i[j] == cj[nextcj]){ /* bcol == ccol */
992             ba_i[j] += ca[nextcj++];
993           }
994         }
995         nextrow[k]++; nextci[k]++;
996         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
997       }
998     }
999     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1000   }
1001   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1002   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1003 
1004   ierr = PetscFree(ba);CHKERRQ(ierr);
1005   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1006   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1007   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1008   ierr = PetscFree(aval);CHKERRQ(ierr);
1009   PetscFunctionReturn(0);
1010 }
1011 
1012 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1013 #undef __FUNCT__
1014 #define __FUNCT__ "MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ"
1015 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P,Mat A,PetscReal fill,Mat *C)
1016 {
1017   PetscErrorCode       ierr;
1018   Mat                  Cmpi,A_loc,POt,PDt;
1019   Mat_PtAPMPI          *ptap;
1020   PetscFreeSpaceList   free_space=PETSC_NULL,current_space=PETSC_NULL;
1021   Mat_MPIAIJ           *p=(Mat_MPIAIJ*)P->data,*c;
1022   PetscInt             *pdti,*pdtj,*poti,*potj,*ptJ;
1023   PetscInt             nnz;
1024   PetscInt             *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1025   PetscInt             am=A->rmap->n,pn=P->cmap->n;
1026   PetscBT              lnkbt;
1027   MPI_Comm             comm=((PetscObject)A)->comm;
1028   PetscMPIInt          size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1029   PetscInt             **buf_rj,**buf_ri,**buf_ri_k;
1030   PetscInt             len,proc,*dnz,*onz,*owners;
1031   PetscInt             nzi,*bi,*bj;
1032   PetscInt             nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1033   MPI_Request          *swaits,*rwaits;
1034   MPI_Status           *sstatus,rstatus;
1035   Mat_Merge_SeqsToMPI  *merge;
1036   PetscInt             *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1037   PetscReal            afill=1.0,afill_tmp;
1038   PetscInt             rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Crmax;
1039   PetscScalar          *vals;
1040   Mat_SeqAIJ           *a_loc, *pdt,*pot;
1041 
1042   PetscFunctionBegin;
1043   /* check if matrix local sizes are compatible */
1044   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend){
1045     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);
1046   }
1047 
1048   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1049   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1050 
1051   /* create struct Mat_PtAPMPI and attached it to C later */
1052   ierr = PetscNew(Mat_PtAPMPI,&ptap);CHKERRQ(ierr);
1053 
1054   /* get A_loc by taking all local rows of A */
1055   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1056   ptap->A_loc = A_loc;
1057   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1058   ai   = a_loc->i;
1059   aj   = a_loc->j;
1060 
1061   /* determine symbolic Co=(p->B)^T*A - send to others */
1062   /*----------------------------------------------------*/
1063   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1064   pdt = (Mat_SeqAIJ*)PDt->data;
1065   pdti = pdt->i; pdtj = pdt->j;
1066 
1067   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1068   pot = (Mat_SeqAIJ*)POt->data;
1069   poti = pot->i; potj = pot->j;
1070 
1071   /* then, compute symbolic Co = (p->B)^T*A */
1072   pon = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1073                          >= (num of nonzero rows of C_seq) - pn */
1074   ierr = PetscMalloc((pon+1)*sizeof(PetscInt),&coi);CHKERRQ(ierr);
1075   coi[0] = 0;
1076 
1077   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1078   nnz           = fill*(poti[pon] + ai[am]);
1079   ierr          = PetscFreeSpaceGet(nnz,&free_space);
1080   current_space = free_space;
1081 
1082   /* create and initialize a linked list */
1083   i = PetscMax(pdt->rmax,pot->rmax);
1084   Crmax = i*a_loc->rmax*size;
1085   if (!Crmax || Crmax > aN) Crmax = aN;
1086   ierr = PetscLLCondensedCreate(Crmax,aN,&lnk,&lnkbt);CHKERRQ(ierr);
1087 
1088   for (i=0; i<pon; i++) {
1089     pnz = poti[i+1] - poti[i];
1090     ptJ = potj + poti[i];
1091     for (j=0; j<pnz; j++){
1092       row  = ptJ[j]; /* row of A_loc == col of Pot */
1093       anz  = ai[row+1] - ai[row];
1094       Jptr = aj + ai[row];
1095       /* add non-zero cols of AP into the sorted linked list lnk */
1096       ierr = PetscLLCondensedAddSorted(anz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1097     }
1098     nnz = lnk[0];
1099 
1100     /* If free space is not available, double the total space in the list */
1101     if (current_space->local_remaining<nnz) {
1102       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1103       nspacedouble++;
1104     }
1105 
1106     /* Copy data into free space, and zero out denserows */
1107     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1108     current_space->array           += nnz;
1109     current_space->local_used      += nnz;
1110     current_space->local_remaining -= nnz;
1111     coi[i+1] = coi[i] + nnz;
1112   }
1113 
1114   ierr = PetscMalloc((coi[pon]+1)*sizeof(PetscInt),&coj);CHKERRQ(ierr);
1115   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1116   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1117   if (afill_tmp > afill) afill = afill_tmp;
1118 
1119   /* send j-array (coj) of Co to other processors */
1120   /*----------------------------------------------*/
1121   /* determine row ownership */
1122   ierr = PetscNew(Mat_Merge_SeqsToMPI,&merge);CHKERRQ(ierr);
1123   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1124   merge->rowmap->n = pn;
1125   merge->rowmap->bs = 1;
1126   ierr = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1127   owners = merge->rowmap->range;
1128 
1129   /* determine the number of messages to send, their lengths */
1130   ierr = PetscMalloc(size*sizeof(PetscMPIInt),&len_si);CHKERRQ(ierr);
1131   ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1132   ierr = PetscMalloc(size*sizeof(PetscMPIInt),&merge->len_s);CHKERRQ(ierr);
1133   len_s = merge->len_s;
1134   merge->nsend = 0;
1135 
1136   ierr = PetscMalloc((size+2)*sizeof(PetscInt),&owners_co);CHKERRQ(ierr);
1137   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1138 
1139   proc = 0;
1140   for (i=0; i<pon; i++){
1141     while (prmap[i] >= owners[proc+1]) proc++;
1142     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1143     len_s[proc] += coi[i+1] - coi[i];
1144   }
1145 
1146   len   = 0;  /* max length of buf_si[] */
1147   owners_co[0] = 0;
1148   for (proc=0; proc<size; proc++){
1149     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1150     if (len_si[proc]){
1151       merge->nsend++;
1152       len_si[proc] = 2*(len_si[proc] + 1);
1153       len += len_si[proc];
1154     }
1155   }
1156 
1157   /* determine the number and length of messages to receive for coi and coj  */
1158   ierr = PetscGatherNumberOfMessages(comm,PETSC_NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1159   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1160 
1161   /* post the Irecv and Isend of coj */
1162   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1163   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1164   ierr = PetscMalloc((merge->nsend+1)*sizeof(MPI_Request),&swaits);CHKERRQ(ierr);
1165   for (proc=0, k=0; proc<size; proc++){
1166     if (!len_s[proc]) continue;
1167     i = owners_co[proc];
1168     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1169     k++;
1170   }
1171 
1172   /* receives and sends of coj are complete */
1173   ierr = PetscMalloc(size*sizeof(MPI_Status),&sstatus);CHKERRQ(ierr);
1174   for (i=0; i<merge->nrecv; i++){
1175     PetscMPIInt icompleted;
1176     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1177   }
1178   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1179   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1180 
1181   /* send and recv coi */
1182   /*-------------------*/
1183   ierr = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1184   ierr = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1185   ierr = PetscMalloc((len+1)*sizeof(PetscInt),&buf_s);CHKERRQ(ierr);
1186   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1187   for (proc=0,k=0; proc<size; proc++){
1188     if (!len_s[proc]) continue;
1189     /* form outgoing message for i-structure:
1190          buf_si[0]:                 nrows to be sent
1191                [1:nrows]:           row index (global)
1192                [nrows+1:2*nrows+1]: i-structure index
1193     */
1194     /*-------------------------------------------*/
1195     nrows = len_si[proc]/2 - 1;
1196     buf_si_i    = buf_si + nrows+1;
1197     buf_si[0]   = nrows;
1198     buf_si_i[0] = 0;
1199     nrows = 0;
1200     for (i=owners_co[proc]; i<owners_co[proc+1]; i++){
1201       nzi = coi[i+1] - coi[i];
1202       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi; /* i-structure */
1203       buf_si[nrows+1] =prmap[i] -owners[proc]; /* local row index */
1204       nrows++;
1205     }
1206     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1207     k++;
1208     buf_si += len_si[proc];
1209   }
1210   i = merge->nrecv;
1211   while (i--) {
1212     PetscMPIInt icompleted;
1213     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1214   }
1215   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1216   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1217   ierr = PetscFree(len_si);CHKERRQ(ierr);
1218   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1219   ierr = PetscFree(swaits);CHKERRQ(ierr);
1220   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1221   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1222 
1223   /* compute the local portion of C (mpi mat) */
1224   /*------------------------------------------*/
1225   /* allocate bi array and free space for accumulating nonzero column info */
1226   ierr = PetscMalloc((pn+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
1227   bi[0] = 0;
1228 
1229   /* set initial free space to be fill*(nnz(P) + nnz(A)) */
1230   nnz           = fill*(pdti[pn] + poti[pon] + ai[am]);
1231   ierr          = PetscFreeSpaceGet(nnz,&free_space);
1232   current_space = free_space;
1233 
1234   ierr = PetscMalloc3(merge->nrecv,PetscInt**,&buf_ri_k,merge->nrecv,PetscInt*,&nextrow,merge->nrecv,PetscInt*,&nextci);CHKERRQ(ierr);
1235   for (k=0; k<merge->nrecv; k++){
1236     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1237     nrows       = *buf_ri_k[k];
1238     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1239     nextci[k]   = buf_ri_k[k] + (nrows + 1);/* poins to the next i-structure of k-th recved i-structure  */
1240   }
1241 
1242   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1243   rmax = 0;
1244   for (i=0; i<pn; i++) {
1245     /* add pdt[i,:]*AP into lnk */
1246     pnz = pdti[i+1] - pdti[i];
1247     ptJ = pdtj + pdti[i];
1248     for (j=0; j<pnz; j++){
1249       row  = ptJ[j];  /* row of AP == col of Pt */
1250       anz  = ai[row+1] - ai[row];
1251       Jptr = aj + ai[row];
1252       /* add non-zero cols of AP into the sorted linked list lnk */
1253       ierr = PetscLLCondensedAddSorted(anz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1254     }
1255 
1256     /* add received col data into lnk */
1257     for (k=0; k<merge->nrecv; k++){ /* k-th received message */
1258       if (i == *nextrow[k]) { /* i-th row */
1259         nzi = *(nextci[k]+1) - *nextci[k];
1260         Jptr  = buf_rj[k] + *nextci[k];
1261         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1262         nextrow[k]++; nextci[k]++;
1263       }
1264     }
1265     nnz = lnk[0];
1266 
1267     /* if free space is not available, make more free space */
1268     if (current_space->local_remaining<nnz) {
1269       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1270       nspacedouble++;
1271     }
1272     /* copy data into free space, then initialize lnk */
1273     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1274     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1275     current_space->array           += nnz;
1276     current_space->local_used      += nnz;
1277     current_space->local_remaining -= nnz;
1278     bi[i+1] = bi[i] + nnz;
1279     if (nnz > rmax) rmax = nnz;
1280   }
1281   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1282 
1283   ierr = PetscMalloc((bi[pn]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
1284   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1285   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
1286   if (afill_tmp > afill) afill = afill_tmp;
1287   ierr = PetscLLCondensedDestroy(lnk,lnkbt);CHKERRQ(ierr);
1288   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1289   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1290 
1291   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
1292   /*----------------------------------------------------------------------------------*/
1293   ierr = PetscMalloc((rmax+1)*sizeof(PetscScalar),&vals);CHKERRQ(ierr);
1294   ierr = PetscMemzero(vals,rmax*sizeof(PetscScalar));CHKERRQ(ierr);
1295 
1296   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1297   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1298   ierr = MatSetBlockSizes(Cmpi,P->cmap->bs,A->cmap->bs);CHKERRQ(ierr);
1299   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1300   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1301   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1302   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
1303   for (i=0; i<pn; i++){
1304     row = i + rstart;
1305     nnz = bi[i+1] - bi[i];
1306     Jptr = bj + bi[i];
1307     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
1308   }
1309   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1310   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1311   ierr = PetscFree(vals);CHKERRQ(ierr);
1312 
1313   merge->bi            = bi;
1314   merge->bj            = bj;
1315   merge->coi           = coi;
1316   merge->coj           = coj;
1317   merge->buf_ri        = buf_ri;
1318   merge->buf_rj        = buf_rj;
1319   merge->owners_co     = owners_co;
1320   merge->destroy       = Cmpi->ops->destroy;
1321   merge->duplicate     = Cmpi->ops->duplicate;
1322 
1323   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ;
1324   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
1325 
1326   /* attach the supporting struct to Cmpi for reuse */
1327   c = (Mat_MPIAIJ*)Cmpi->data;
1328   c->ptap        = ptap;
1329   ptap->api      = PETSC_NULL;
1330   ptap->apj      = PETSC_NULL;
1331   ptap->merge    = merge;
1332   ptap->rmax     = rmax;
1333 
1334   *C = Cmpi;
1335 #if defined(PETSC_USE_INFO)
1336   if (bi[pn] != 0) {
1337     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %G needed %G.\n",nspacedouble,fill,afill);CHKERRQ(ierr);
1338     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%G,&C) for best performance.\n",afill);CHKERRQ(ierr);
1339   } else {
1340     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1341   }
1342 #endif
1343   PetscFunctionReturn(0);
1344 }
1345 
1346 #undef __FUNCT__
1347 #define __FUNCT__ "MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_Scalable"
1348 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_Scalable(Mat P,Mat A,Mat C)
1349 {
1350   PetscErrorCode       ierr;
1351   Mat_Merge_SeqsToMPI  *merge;
1352   Mat_MPIAIJ           *p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1353   Mat_SeqAIJ           *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
1354   Mat_PtAPMPI          *ptap;
1355   PetscInt             *adj;
1356   PetscInt             i,j,k,anz,pnz,row,*cj,nexta;
1357   MatScalar            *ada,*ca,valtmp;
1358   PetscInt             am=A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
1359   MPI_Comm             comm=((PetscObject)C)->comm;
1360   PetscMPIInt          size,rank,taga,*len_s;
1361   PetscInt             *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
1362   PetscInt             **buf_ri,**buf_rj;
1363   PetscInt             cnz=0,*bj_i,*bi,*bj,bnz,nextcj; /* bi,bj,ba: local array of C(mpi mat) */
1364   MPI_Request          *s_waits,*r_waits;
1365   MPI_Status           *status;
1366   MatScalar            **abuf_r,*ba_i,*pA,*coa,*ba;
1367   PetscInt             *ai,*aj,*coi,*coj;
1368   PetscInt             *poJ=po->j,*pdJ=pd->j;
1369   Mat                  A_loc;
1370   Mat_SeqAIJ           *a_loc;
1371 
1372   PetscFunctionBegin;
1373   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1374   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1375 
1376   ptap  = c->ptap;
1377   merge = ptap->merge;
1378 
1379   /* 2) compute numeric C_seq = P_loc^T*A_loc */
1380   /*------------------------------------------*/
1381   /* get data from symbolic products */
1382   coi = merge->coi; coj = merge->coj;
1383   ierr = PetscMalloc((coi[pon]+1)*sizeof(MatScalar),&coa);CHKERRQ(ierr);
1384   ierr = PetscMemzero(coa,coi[pon]*sizeof(MatScalar));CHKERRQ(ierr);
1385   bi     = merge->bi; bj = merge->bj;
1386   owners = merge->rowmap->range;
1387   ierr   = PetscMalloc((bi[cm]+1)*sizeof(MatScalar),&ba);CHKERRQ(ierr);
1388   ierr   = PetscMemzero(ba,bi[cm]*sizeof(MatScalar));CHKERRQ(ierr);
1389 
1390   /* get A_loc by taking all local rows of A */
1391   A_loc = ptap->A_loc;
1392   ierr = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
1393   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1394   ai   = a_loc->i;
1395   aj   = a_loc->j;
1396 
1397   for (i=0; i<am; i++) {
1398     anz = ai[i+1] - ai[i];
1399     adj = aj + ai[i];
1400     ada = a_loc->a + ai[i];
1401 
1402     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1403     /*-------------------------------------------------------------*/
1404     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1405     pnz = po->i[i+1] - po->i[i];
1406     poJ = po->j + po->i[i];
1407     pA  = po->a + po->i[i];
1408     for (j=0; j<pnz; j++){
1409       row = poJ[j];
1410       cnz = coi[row+1] - coi[row];
1411       cj  = coj + coi[row];
1412       ca  = coa + coi[row];
1413       /* perform sparse axpy */
1414       nexta  = 0;
1415       valtmp = pA[j];
1416       for (k=0; nexta<anz; k++) {
1417         if (cj[k] == adj[nexta]){
1418           ca[k] += valtmp*ada[nexta];
1419           nexta++;
1420         }
1421       }
1422       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1423     }
1424 
1425     /* put the value into Cd (diagonal part) */
1426     pnz = pd->i[i+1] - pd->i[i];
1427     pdJ = pd->j + pd->i[i];
1428     pA  = pd->a + pd->i[i];
1429     for (j=0; j<pnz; j++){
1430       row = pdJ[j];
1431       cnz = bi[row+1] - bi[row];
1432       cj  = bj + bi[row];
1433       ca  = ba + bi[row];
1434       /* perform sparse axpy */
1435       nexta  = 0;
1436       valtmp = pA[j];
1437       for (k=0; nexta<anz; k++) {
1438         if (cj[k] == adj[nexta]){
1439           ca[k] += valtmp*ada[nexta];
1440           nexta++;
1441         }
1442       }
1443       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1444     }
1445   }
1446 
1447   /* 3) send and recv matrix values coa */
1448   /*------------------------------------*/
1449   buf_ri = merge->buf_ri;
1450   buf_rj = merge->buf_rj;
1451   len_s  = merge->len_s;
1452   ierr = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1453   ierr = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1454 
1455   ierr = PetscMalloc2(merge->nsend+1,MPI_Request,&s_waits,size,MPI_Status,&status);CHKERRQ(ierr);
1456   for (proc=0,k=0; proc<size; proc++){
1457     if (!len_s[proc]) continue;
1458     i = merge->owners_co[proc];
1459     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1460     k++;
1461   }
1462   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1463   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1464 
1465   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1466   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1467   ierr = PetscFree(coa);CHKERRQ(ierr);
1468 
1469   /* 4) insert local Cseq and received values into Cmpi */
1470   /*----------------------------------------------------*/
1471   ierr = PetscMalloc3(merge->nrecv,PetscInt**,&buf_ri_k,merge->nrecv,PetscInt*,&nextrow,merge->nrecv,PetscInt*,&nextci);CHKERRQ(ierr);
1472   for (k=0; k<merge->nrecv; k++){
1473     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1474     nrows       = *(buf_ri_k[k]);
1475     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1476     nextci[k]   = buf_ri_k[k] + (nrows + 1);/* poins to the next i-structure of k-th recved i-structure  */
1477   }
1478 
1479   for (i=0; i<cm; i++) {
1480     row = owners[rank] + i; /* global row index of C_seq */
1481     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1482     ba_i = ba + bi[i];
1483     bnz  = bi[i+1] - bi[i];
1484     /* add received vals into ba */
1485     for (k=0; k<merge->nrecv; k++){ /* k-th received message */
1486       /* i-th row */
1487       if (i == *nextrow[k]) {
1488         cnz = *(nextci[k]+1) - *nextci[k];
1489         cj  = buf_rj[k] + *(nextci[k]);
1490         ca  = abuf_r[k] + *(nextci[k]);
1491         nextcj = 0;
1492         for (j=0; nextcj<cnz; j++){
1493           if (bj_i[j] == cj[nextcj]){ /* bcol == ccol */
1494             ba_i[j] += ca[nextcj++];
1495           }
1496         }
1497         nextrow[k]++; nextci[k]++;
1498         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1499       }
1500     }
1501     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1502   }
1503   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1504   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1505 
1506   ierr = PetscFree(ba);CHKERRQ(ierr);
1507   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1508   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1509   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1510   PetscFunctionReturn(0);
1511 }
1512 
1513 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1514 #undef __FUNCT__
1515 #define __FUNCT__ "MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable"
1516 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable(Mat P,Mat A,PetscReal fill,Mat *C)
1517 {
1518   PetscErrorCode       ierr;
1519   Mat                  Cmpi,A_loc,POt,PDt;
1520   Mat_PtAPMPI          *ptap;
1521   PetscFreeSpaceList   free_space=PETSC_NULL,current_space=PETSC_NULL;
1522   Mat_MPIAIJ           *p=(Mat_MPIAIJ*)P->data,*c;
1523   PetscInt             *pdti,*pdtj,*poti,*potj,*ptJ;
1524   PetscInt             nnz;
1525   PetscInt             *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1526   PetscInt             am=A->rmap->n,pn=P->cmap->n;
1527   MPI_Comm             comm=((PetscObject)A)->comm;
1528   PetscMPIInt          size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1529   PetscInt             **buf_rj,**buf_ri,**buf_ri_k;
1530   PetscInt             len,proc,*dnz,*onz,*owners;
1531   PetscInt             nzi,*bi,*bj;
1532   PetscInt             nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1533   MPI_Request          *swaits,*rwaits;
1534   MPI_Status           *sstatus,rstatus;
1535   Mat_Merge_SeqsToMPI  *merge;
1536   PetscInt             *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1537   PetscReal            afill=1.0,afill_tmp;
1538   PetscInt             rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Crmax;
1539   PetscScalar          *vals;
1540   Mat_SeqAIJ           *a_loc, *pdt,*pot;
1541 
1542   PetscFunctionBegin;
1543   /* check if matrix local sizes are compatible */
1544   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend){
1545     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);
1546   }
1547 
1548   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1549   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1550 
1551   /* create struct Mat_PtAPMPI and attached it to C later */
1552   ierr = PetscNew(Mat_PtAPMPI,&ptap);CHKERRQ(ierr);
1553 
1554   /* get A_loc by taking all local rows of A */
1555   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1556   ptap->A_loc = A_loc;
1557   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1558   ai   = a_loc->i;
1559   aj   = a_loc->j;
1560 
1561   /* determine symbolic Co=(p->B)^T*A - send to others */
1562   /*----------------------------------------------------*/
1563   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1564   pdt = (Mat_SeqAIJ*)PDt->data;
1565   pdti = pdt->i; pdtj = pdt->j;
1566 
1567   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1568   pot = (Mat_SeqAIJ*)POt->data;
1569   poti = pot->i; potj = pot->j;
1570 
1571   /* then, compute symbolic Co = (p->B)^T*A */
1572   pon = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1573                          >= (num of nonzero rows of C_seq) - pn */
1574   ierr = PetscMalloc((pon+1)*sizeof(PetscInt),&coi);CHKERRQ(ierr);
1575   coi[0] = 0;
1576 
1577   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1578   nnz           = fill*(poti[pon] + ai[am]);
1579   ierr          = PetscFreeSpaceGet(nnz,&free_space);
1580   current_space = free_space;
1581 
1582   /* create and initialize a linked list */
1583   i = PetscMax(pdt->rmax,pot->rmax);
1584   Crmax = i*a_loc->rmax*size; /* non-scalable! */
1585   if (!Crmax || Crmax > aN) Crmax = aN;
1586   ierr = PetscLLCondensedCreate_Scalable(Crmax,&lnk);CHKERRQ(ierr);
1587 
1588   for (i=0; i<pon; i++) {
1589     nnz = 0;
1590     pnz = poti[i+1] - poti[i];
1591     ptJ = potj + poti[i];
1592     for (j=0; j<pnz; j++){
1593       row  = ptJ[j]; /* row of A_loc == col of Pot */
1594       anz  = ai[row+1] - ai[row];
1595       Jptr = aj + ai[row];
1596       /* add non-zero cols of AP into the sorted linked list lnk */
1597       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1598     }
1599     nnz = lnk[0];
1600 
1601     /* If free space is not available, double the total space in the list */
1602     if (current_space->local_remaining<nnz) {
1603       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1604       nspacedouble++;
1605     }
1606 
1607     /* Copy data into free space, and zero out denserows */
1608     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1609     current_space->array           += nnz;
1610     current_space->local_used      += nnz;
1611     current_space->local_remaining -= nnz;
1612     coi[i+1] = coi[i] + nnz;
1613   }
1614 
1615   ierr = PetscMalloc((coi[pon]+1)*sizeof(PetscInt),&coj);CHKERRQ(ierr);
1616   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1617   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1618   if (afill_tmp > afill) afill = afill_tmp;
1619 
1620   /* send j-array (coj) of Co to other processors */
1621   /*----------------------------------------------*/
1622   /* determine row ownership */
1623   ierr = PetscNew(Mat_Merge_SeqsToMPI,&merge);CHKERRQ(ierr);
1624   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1625   merge->rowmap->n = pn;
1626   merge->rowmap->bs = 1;
1627   ierr = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1628   owners = merge->rowmap->range;
1629 
1630   /* determine the number of messages to send, their lengths */
1631   ierr = PetscMalloc(size*sizeof(PetscMPIInt),&len_si);CHKERRQ(ierr);
1632   ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1633   ierr = PetscMalloc(size*sizeof(PetscMPIInt),&merge->len_s);CHKERRQ(ierr);
1634   len_s = merge->len_s;
1635   merge->nsend = 0;
1636 
1637   ierr = PetscMalloc((size+2)*sizeof(PetscInt),&owners_co);CHKERRQ(ierr);
1638   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1639 
1640   proc = 0;
1641   for (i=0; i<pon; i++){
1642     while (prmap[i] >= owners[proc+1]) proc++;
1643     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1644     len_s[proc] += coi[i+1] - coi[i];
1645   }
1646 
1647   len   = 0;  /* max length of buf_si[] */
1648   owners_co[0] = 0;
1649   for (proc=0; proc<size; proc++){
1650     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1651     if (len_si[proc]){
1652       merge->nsend++;
1653       len_si[proc] = 2*(len_si[proc] + 1);
1654       len += len_si[proc];
1655     }
1656   }
1657 
1658   /* determine the number and length of messages to receive for coi and coj  */
1659   ierr = PetscGatherNumberOfMessages(comm,PETSC_NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1660   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1661 
1662   /* post the Irecv and Isend of coj */
1663   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1664   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1665   ierr = PetscMalloc((merge->nsend+1)*sizeof(MPI_Request),&swaits);CHKERRQ(ierr);
1666   for (proc=0, k=0; proc<size; proc++){
1667     if (!len_s[proc]) continue;
1668     i = owners_co[proc];
1669     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1670     k++;
1671   }
1672 
1673   /* receives and sends of coj are complete */
1674   ierr = PetscMalloc(size*sizeof(MPI_Status),&sstatus);CHKERRQ(ierr);
1675   for (i=0; i<merge->nrecv; i++){
1676     PetscMPIInt icompleted;
1677     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1678   }
1679   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1680   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1681 
1682   /* send and recv coi */
1683   /*-------------------*/
1684   ierr = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1685   ierr = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1686   ierr = PetscMalloc((len+1)*sizeof(PetscInt),&buf_s);CHKERRQ(ierr);
1687   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1688   for (proc=0,k=0; proc<size; proc++){
1689     if (!len_s[proc]) continue;
1690     /* form outgoing message for i-structure:
1691          buf_si[0]:                 nrows to be sent
1692                [1:nrows]:           row index (global)
1693                [nrows+1:2*nrows+1]: i-structure index
1694     */
1695     /*-------------------------------------------*/
1696     nrows = len_si[proc]/2 - 1;
1697     buf_si_i    = buf_si + nrows+1;
1698     buf_si[0]   = nrows;
1699     buf_si_i[0] = 0;
1700     nrows = 0;
1701     for (i=owners_co[proc]; i<owners_co[proc+1]; i++){
1702       nzi = coi[i+1] - coi[i];
1703       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi; /* i-structure */
1704       buf_si[nrows+1] =prmap[i] -owners[proc]; /* local row index */
1705       nrows++;
1706     }
1707     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1708     k++;
1709     buf_si += len_si[proc];
1710   }
1711   i = merge->nrecv;
1712   while (i--) {
1713     PetscMPIInt icompleted;
1714     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1715   }
1716   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1717   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1718   ierr = PetscFree(len_si);CHKERRQ(ierr);
1719   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1720   ierr = PetscFree(swaits);CHKERRQ(ierr);
1721   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1722   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1723 
1724   /* compute the local portion of C (mpi mat) */
1725   /*------------------------------------------*/
1726   /* allocate bi array and free space for accumulating nonzero column info */
1727   ierr = PetscMalloc((pn+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
1728   bi[0] = 0;
1729 
1730   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
1731   nnz           = fill*(pdti[pn] + poti[pon] + ai[am]);
1732   ierr          = PetscFreeSpaceGet(nnz,&free_space);
1733   current_space = free_space;
1734 
1735   ierr = PetscMalloc3(merge->nrecv,PetscInt**,&buf_ri_k,merge->nrecv,PetscInt*,&nextrow,merge->nrecv,PetscInt*,&nextci);CHKERRQ(ierr);
1736   for (k=0; k<merge->nrecv; k++){
1737     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1738     nrows       = *buf_ri_k[k];
1739     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1740     nextci[k]   = buf_ri_k[k] + (nrows + 1);/* poins to the next i-structure of k-th recved i-structure  */
1741   }
1742 
1743   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1744   rmax = 0;
1745   for (i=0; i<pn; i++) {
1746     /* add pdt[i,:]*AP into lnk */
1747     pnz = pdti[i+1] - pdti[i];
1748     ptJ = pdtj + pdti[i];
1749     for (j=0; j<pnz; j++){
1750       row  = ptJ[j];  /* row of AP == col of Pt */
1751       anz  = ai[row+1] - ai[row];
1752       Jptr = aj + ai[row];
1753       /* add non-zero cols of AP into the sorted linked list lnk */
1754       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1755     }
1756 
1757     /* add received col data into lnk */
1758     for (k=0; k<merge->nrecv; k++){ /* k-th received message */
1759       if (i == *nextrow[k]) { /* i-th row */
1760         nzi = *(nextci[k]+1) - *nextci[k];
1761         Jptr  = buf_rj[k] + *nextci[k];
1762         ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
1763         nextrow[k]++; nextci[k]++;
1764       }
1765     }
1766     nnz = lnk[0];
1767 
1768     /* if free space is not available, make more free space */
1769     if (current_space->local_remaining<nnz) {
1770       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1771       nspacedouble++;
1772     }
1773     /* copy data into free space, then initialize lnk */
1774     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1775     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1776     current_space->array           += nnz;
1777     current_space->local_used      += nnz;
1778     current_space->local_remaining -= nnz;
1779     bi[i+1] = bi[i] + nnz;
1780     if (nnz > rmax) rmax = nnz;
1781   }
1782   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1783 
1784   ierr = PetscMalloc((bi[pn]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
1785   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1786   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
1787   if (afill_tmp > afill) afill = afill_tmp;
1788   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
1789   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1790   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1791 
1792   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
1793   /*----------------------------------------------------------------------------------*/
1794   ierr = PetscMalloc((rmax+1)*sizeof(PetscScalar),&vals);CHKERRQ(ierr);
1795   ierr = PetscMemzero(vals,rmax*sizeof(PetscScalar));CHKERRQ(ierr);
1796 
1797   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1798   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1799   ierr = MatSetBlockSizes(Cmpi,P->cmap->bs,A->cmap->bs); CHKERRQ(ierr);
1800   ierr = MatSetType(Cmpi,MATMPIAIJ); CHKERRQ(ierr);
1801   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1802   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1803   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
1804   for (i=0; i<pn; i++){
1805     row = i + rstart;
1806     nnz = bi[i+1] - bi[i];
1807     Jptr = bj + bi[i];
1808     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
1809   }
1810   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1811   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1812   ierr = PetscFree(vals);CHKERRQ(ierr);
1813 
1814   merge->bi            = bi;
1815   merge->bj            = bj;
1816   merge->coi           = coi;
1817   merge->coj           = coj;
1818   merge->buf_ri        = buf_ri;
1819   merge->buf_rj        = buf_rj;
1820   merge->owners_co     = owners_co;
1821   merge->destroy       = Cmpi->ops->destroy;
1822   merge->duplicate     = Cmpi->ops->duplicate;
1823 
1824   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_Scalable;
1825   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
1826 
1827   /* attach the supporting struct to Cmpi for reuse */
1828   c = (Mat_MPIAIJ*)Cmpi->data;
1829   c->ptap        = ptap;
1830   ptap->api      = PETSC_NULL;
1831   ptap->apj      = PETSC_NULL;
1832   ptap->merge    = merge;
1833   ptap->rmax     = rmax;
1834   ptap->apa      = PETSC_NULL;
1835 
1836   *C = Cmpi;
1837 #if defined(PETSC_USE_INFO)
1838   if (bi[pn] != 0) {
1839     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %G needed %G.\n",nspacedouble,fill,afill);CHKERRQ(ierr);
1840     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%G,&C) for best performance.\n",afill);CHKERRQ(ierr);
1841   } else {
1842     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1843   }
1844 #endif
1845   PetscFunctionReturn(0);
1846 }
1847