xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision e383bbd07f1c175fdb04eea9fdcaf8d9f9e92c0f)
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     /* diagonal portion of A */
248     nzi = adi[i+1] - adi[i];
249     for (j=0; j<nzi; j++){
250       row = *adj++;
251       pnz = pi_loc[row+1] - pi_loc[row];
252       Jptr  = pj_loc + pi_loc[row];
253       /* add non-zero cols of P into the sorted linked list lnk */
254       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
255     }
256     /* off-diagonal portion of A */
257     nzi = aoi[i+1] - aoi[i];
258     for (j=0; j<nzi; j++){
259       row = *aoj++;
260       pnz = pi_oth[row+1] - pi_oth[row];
261       Jptr  = pj_oth + pi_oth[row];
262       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
263     }
264 
265     apnz     = lnk[0];
266     api[i+1] = api[i] + apnz;
267 
268     /* if free space is not available, double the total space in the list */
269     if (current_space->local_remaining<apnz) {
270       ierr = PetscFreeSpaceGet(apnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
271       nspacedouble++;
272     }
273 
274     /* Copy data into free space, then initialize lnk */
275     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
276     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
277     current_space->array           += apnz;
278     current_space->local_used      += apnz;
279     current_space->local_remaining -= apnz;
280   }
281 
282   /* Allocate space for apj, initialize apj, and */
283   /* destroy list of free space and other temporary array(s) */
284   ierr = PetscMalloc((api[am]+1)*sizeof(PetscInt),&ptap->apj);CHKERRQ(ierr);
285   apj  = ptap->apj;
286   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
287   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
288 
289   /* malloc apa to store dense row A[i,:]*P */
290   ierr = PetscMalloc(pN*sizeof(PetscScalar),&apa);CHKERRQ(ierr);
291   ierr = PetscMemzero(apa,pN*sizeof(PetscScalar));CHKERRQ(ierr);
292   ptap->apa = apa;
293 
294   /* create and assemble symbolic parallel matrix Cmpi */
295   /*----------------------------------------------------*/
296   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
297   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
298   ierr = MatSetBlockSizes(Cmpi,A->rmap->bs,P->cmap->bs);CHKERRQ(ierr);
299 
300   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
301   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
302   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
303   for (i=0; i<am; i++){
304     row  = i + rstart;
305     apnz = api[i+1] - api[i];
306     ierr = MatSetValues(Cmpi,1,&row,apnz,apj,apa,INSERT_VALUES);CHKERRQ(ierr);
307     apj += apnz;
308   }
309   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
310   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
311 
312   ptap->destroy             = Cmpi->ops->destroy;
313   ptap->duplicate           = Cmpi->ops->duplicate;
314   Cmpi->ops->destroy        = MatDestroy_MPIAIJ_MatMatMult;
315   Cmpi->ops->duplicate      = MatDuplicate_MPIAIJ_MatMatMult;
316 
317   /* attach the supporting struct to Cmpi for reuse */
318   c = (Mat_MPIAIJ*)Cmpi->data;
319   c->ptap  = ptap;
320 
321   *C = Cmpi;
322 
323   /* set MatInfo */
324   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
325   if (afill < 1.0) afill = 1.0;
326   Cmpi->info.mallocs           = nspacedouble;
327   Cmpi->info.fill_ratio_given  = fill;
328   Cmpi->info.fill_ratio_needed = afill;
329 
330 #if defined(PETSC_USE_INFO)
331   if (api[am]) {
332     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %G needed %G.\n",nspacedouble,fill,afill);CHKERRQ(ierr);
333     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%G,&C) for best performance.;\n",afill);CHKERRQ(ierr);
334   } else {
335     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
336   }
337 #endif
338   PetscFunctionReturn(0);
339 }
340 
341 #undef __FUNCT__
342 #define __FUNCT__ "MatMatMult_MPIAIJ_MPIDense"
343 PetscErrorCode MatMatMult_MPIAIJ_MPIDense(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
344 {
345   PetscErrorCode ierr;
346 
347   PetscFunctionBegin;
348   if (scall == MAT_INITIAL_MATRIX){
349     ierr = MatMatMultSymbolic_MPIAIJ_MPIDense(A,B,fill,C);CHKERRQ(ierr);
350   }
351   ierr = MatMatMultNumeric_MPIAIJ_MPIDense(A,B,*C);CHKERRQ(ierr);
352   PetscFunctionReturn(0);
353 }
354 
355 typedef struct {
356   Mat         workB;
357   PetscScalar *rvalues,*svalues;
358   MPI_Request *rwaits,*swaits;
359 } MPIAIJ_MPIDense;
360 
361 #undef __FUNCT__
362 #define __FUNCT__ "MatMPIAIJ_MPIDenseDestroy"
363 PetscErrorCode MatMPIAIJ_MPIDenseDestroy(void *ctx)
364 {
365   MPIAIJ_MPIDense *contents = (MPIAIJ_MPIDense*) ctx;
366   PetscErrorCode  ierr;
367 
368   PetscFunctionBegin;
369   ierr = MatDestroy(&contents->workB);CHKERRQ(ierr);
370   ierr = PetscFree4(contents->rvalues,contents->svalues,contents->rwaits,contents->swaits);CHKERRQ(ierr);
371   ierr = PetscFree(contents);CHKERRQ(ierr);
372   PetscFunctionReturn(0);
373 }
374 
375 #undef __FUNCT__
376 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIDense"
377 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat A,Mat B,PetscReal fill,Mat *C)
378 {
379   PetscErrorCode         ierr;
380   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*) A->data;
381   PetscInt               nz = aij->B->cmap->n;
382   PetscContainer         container;
383   MPIAIJ_MPIDense        *contents;
384   VecScatter             ctx = aij->Mvctx;
385   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
386   VecScatter_MPI_General *to   = ( VecScatter_MPI_General*) ctx->todata;
387   PetscInt               m=A->rmap->n,n=B->cmap->n;
388 
389   PetscFunctionBegin;
390   ierr = MatCreate(((PetscObject)B)->comm,C);CHKERRQ(ierr);
391   ierr = MatSetSizes(*C,m,n,A->rmap->N,B->cmap->N);CHKERRQ(ierr);
392   ierr = MatSetBlockSizes(*C,A->rmap->bs,B->cmap->bs);CHKERRQ(ierr);
393   ierr = MatSetType(*C,MATMPIDENSE);CHKERRQ(ierr);
394   ierr = MatMPIDenseSetPreallocation(*C,PETSC_NULL);CHKERRQ(ierr);
395   ierr = MatAssemblyBegin(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
396   ierr = MatAssemblyEnd(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
397   (*C)->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
398 
399   ierr = PetscNew(MPIAIJ_MPIDense,&contents);CHKERRQ(ierr);
400   /* Create work matrix used to store off processor rows of B needed for local product */
401   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,B->cmap->N,PETSC_NULL,&contents->workB);CHKERRQ(ierr);
402   /* Create work arrays needed */
403   ierr = PetscMalloc4(B->cmap->N*from->starts[from->n],PetscScalar,&contents->rvalues,
404                       B->cmap->N*to->starts[to->n],PetscScalar,&contents->svalues,
405                       from->n,MPI_Request,&contents->rwaits,
406                       to->n,MPI_Request,&contents->swaits);CHKERRQ(ierr);
407 
408   ierr = PetscContainerCreate(((PetscObject)A)->comm,&container);CHKERRQ(ierr);
409   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
410   ierr = PetscContainerSetUserDestroy(container,MatMPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
411   ierr = PetscObjectCompose((PetscObject)(*C),"workB",(PetscObject)container);CHKERRQ(ierr);
412   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
413   PetscFunctionReturn(0);
414 }
415 
416 #undef __FUNCT__
417 #define __FUNCT__ "MatMPIDenseScatter"
418 /*
419     Performs an efficient scatter on the rows of B needed by this process; this is
420     a modification of the VecScatterBegin_() routines.
421 */
422 PetscErrorCode MatMPIDenseScatter(Mat A,Mat B,Mat C,Mat *outworkB)
423 {
424   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*)A->data;
425   PetscErrorCode         ierr;
426   PetscScalar            *b,*w,*svalues,*rvalues;
427   VecScatter             ctx = aij->Mvctx;
428   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
429   VecScatter_MPI_General *to   = ( VecScatter_MPI_General*) ctx->todata;
430   PetscInt               i,j,k;
431   PetscInt               *sindices,*sstarts,*rindices,*rstarts;
432   PetscMPIInt            *sprocs,*rprocs,nrecvs;
433   MPI_Request            *swaits,*rwaits;
434   MPI_Comm               comm = ((PetscObject)A)->comm;
435   PetscMPIInt            tag = ((PetscObject)ctx)->tag,ncols = B->cmap->N, nrows = aij->B->cmap->n,imdex,nrowsB = B->rmap->n;
436   MPI_Status             status;
437   MPIAIJ_MPIDense        *contents;
438   PetscContainer         container;
439   Mat                    workB;
440 
441   PetscFunctionBegin;
442   ierr = PetscObjectQuery((PetscObject)C,"workB",(PetscObject*)&container);CHKERRQ(ierr);
443   if (!container) SETERRQ(comm,PETSC_ERR_PLIB,"Container does not exist");
444   ierr = PetscContainerGetPointer(container,(void**)&contents);CHKERRQ(ierr);
445 
446   workB = *outworkB = contents->workB;
447   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);
448   sindices  = to->indices;
449   sstarts   = to->starts;
450   sprocs    = to->procs;
451   swaits    = contents->swaits;
452   svalues   = contents->svalues;
453 
454   rindices  = from->indices;
455   rstarts   = from->starts;
456   rprocs    = from->procs;
457   rwaits    = contents->rwaits;
458   rvalues   = contents->rvalues;
459 
460   ierr = MatDenseGetArray(B,&b);CHKERRQ(ierr);
461   ierr = MatDenseGetArray(workB,&w);CHKERRQ(ierr);
462 
463   for (i=0; i<from->n; i++) {
464     ierr = MPI_Irecv(rvalues+ncols*rstarts[i],ncols*(rstarts[i+1]-rstarts[i]),MPIU_SCALAR,rprocs[i],tag,comm,rwaits+i);CHKERRQ(ierr);
465   }
466 
467   for (i=0; i<to->n; i++) {
468     /* pack a message at a time */
469     CHKMEMQ;
470     for (j=0; j<sstarts[i+1]-sstarts[i]; j++){
471       for (k=0; k<ncols; k++) {
472         svalues[ncols*(sstarts[i] + j) + k] = b[sindices[sstarts[i]+j] + nrowsB*k];
473       }
474     }
475     CHKMEMQ;
476     ierr = MPI_Isend(svalues+ncols*sstarts[i],ncols*(sstarts[i+1]-sstarts[i]),MPIU_SCALAR,sprocs[i],tag,comm,swaits+i);CHKERRQ(ierr);
477   }
478 
479   nrecvs = from->n;
480   while (nrecvs) {
481     ierr = MPI_Waitany(from->n,rwaits,&imdex,&status);CHKERRQ(ierr);
482     nrecvs--;
483     /* unpack a message at a time */
484     CHKMEMQ;
485     for (j=0; j<rstarts[imdex+1]-rstarts[imdex]; j++){
486       for (k=0; k<ncols; k++) {
487         w[rindices[rstarts[imdex]+j] + nrows*k] = rvalues[ncols*(rstarts[imdex] + j) + k];
488       }
489     }
490     CHKMEMQ;
491   }
492   if (to->n) {ierr = MPI_Waitall(to->n,swaits,to->sstatus);CHKERRQ(ierr);}
493 
494   ierr = MatDenseRestoreArray(B,&b);CHKERRQ(ierr);
495   ierr = MatDenseRestoreArray(workB,&w);CHKERRQ(ierr);
496   ierr = MatAssemblyBegin(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
497   ierr = MatAssemblyEnd(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
498   PetscFunctionReturn(0);
499 }
500 extern PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat,Mat,Mat);
501 
502 #undef __FUNCT__
503 #define __FUNCT__ "MatMatMultNumeric_MPIAIJ_MPIDense"
504 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat A,Mat B,Mat C)
505 {
506   PetscErrorCode       ierr;
507   Mat_MPIAIJ           *aij = (Mat_MPIAIJ*)A->data;
508   Mat_MPIDense         *bdense = (Mat_MPIDense*)B->data;
509   Mat_MPIDense         *cdense = (Mat_MPIDense*)C->data;
510   Mat                  workB;
511 
512   PetscFunctionBegin;
513 
514   /* diagonal block of A times all local rows of B*/
515   ierr = MatMatMultNumeric_SeqAIJ_SeqDense(aij->A,bdense->A,cdense->A);CHKERRQ(ierr);
516 
517   /* get off processor parts of B needed to complete the product */
518   ierr = MatMPIDenseScatter(A,B,C,&workB);CHKERRQ(ierr);
519 
520   /* off-diagonal block of A times nonlocal rows of B */
521   ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A);CHKERRQ(ierr);
522   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
523   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
524   PetscFunctionReturn(0);
525 }
526 
527 #undef __FUNCT__
528 #define __FUNCT__ "MatMatMultNumeric_MPIAIJ_MPIAIJ_Scalable"
529 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ_Scalable(Mat A,Mat P,Mat C)
530 {
531   PetscErrorCode     ierr;
532   Mat_MPIAIJ         *a=(Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
533   Mat_SeqAIJ         *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
534   Mat_SeqAIJ         *cd=(Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
535   PetscInt           *adi=ad->i,*adj,*aoi=ao->i,*aoj;
536   PetscScalar        *ada,*aoa,*cda=cd->a,*coa=co->a;
537   Mat_SeqAIJ         *p_loc,*p_oth;
538   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
539   PetscScalar        *pa_loc,*pa_oth,*pa,valtmp,*ca;
540   PetscInt           cm=C->rmap->n,anz,pnz;
541   Mat_PtAPMPI        *ptap=c->ptap;
542   PetscScalar        *apa_sparse=ptap->apa;
543   PetscInt           *api,*apj,*apJ,i,j,k,row;
544   PetscInt           cstart=C->cmap->rstart;
545   PetscInt           cdnz,conz,k0,k1,nextp;
546 
547   PetscFunctionBegin;
548   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
549   /*-----------------------------------------------------*/
550   /* update numerical values of P_oth and P_loc */
551   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
552   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
553 
554   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
555   /*----------------------------------------------------------*/
556   /* get data from symbolic products */
557   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
558   p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
559   pi_loc=p_loc->i; pj_loc=p_loc->j; pa_loc=p_loc->a;
560   pi_oth=p_oth->i; pj_oth=p_oth->j; pa_oth=p_oth->a;
561 
562   api = ptap->api;
563   apj = ptap->apj;
564   for (i=0; i<cm; i++) {
565     apJ = apj + api[i];
566 
567     /* diagonal portion of A */
568     anz = adi[i+1] - adi[i];
569     adj = ad->j + adi[i];
570     ada = ad->a + adi[i];
571     for (j=0; j<anz; j++) {
572       row = adj[j];
573       pnz = pi_loc[row+1] - pi_loc[row];
574       pj  = pj_loc + pi_loc[row];
575       pa  = pa_loc + pi_loc[row];
576       /* perform sparse axpy */
577       valtmp = ada[j];
578       nextp  = 0;
579       for (k=0; nextp<pnz; k++) {
580         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
581           apa_sparse[k] += valtmp*pa[nextp++];
582         }
583       }
584       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
585     }
586 
587     /* off-diagonal portion of A */
588     anz = aoi[i+1] - aoi[i];
589     aoj = ao->j + aoi[i];
590     aoa = ao->a + aoi[i];
591     for (j=0; j<anz; j++) {
592       row = aoj[j];
593       pnz = pi_oth[row+1] - pi_oth[row];
594       pj  = pj_oth + pi_oth[row];
595       pa  = pa_oth + pi_oth[row];
596       /* perform sparse axpy */
597       valtmp = aoa[j];
598       nextp  = 0;
599       for (k=0; nextp<pnz; k++) {
600         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
601           apa_sparse[k] += valtmp*pa[nextp++];
602         }
603       }
604       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
605     }
606 
607     /* set values in C */
608     cdnz = cd->i[i+1] - cd->i[i];
609     conz = co->i[i+1] - co->i[i];
610 
611     /* 1st off-diagoanl part of C */
612     ca = coa + co->i[i];
613     k  = 0;
614     for (k0=0; k0<conz; k0++){
615       if (apJ[k] >= cstart) break;
616       ca[k0]      = apa_sparse[k];
617       apa_sparse[k] = 0.0;
618       k++;
619     }
620 
621     /* diagonal part of C */
622     ca = cda + cd->i[i];
623     for (k1=0; k1<cdnz; k1++){
624       ca[k1]      = apa_sparse[k];
625       apa_sparse[k] = 0.0;
626       k++;
627     }
628 
629     /* 2nd off-diagoanl part of C */
630     ca = coa + co->i[i];
631     for (; k0<conz; k0++){
632       ca[k0]      = apa_sparse[k];
633       apa_sparse[k] = 0.0;
634       k++;
635     }
636   }
637   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
638   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
639   PetscFunctionReturn(0);
640 }
641 
642 /* same as MatMatMultSymbolic_MPIAIJ_MPIAIJ(), except using LLCondensed to avoid O(BN) memory requirement */
643 #undef __FUNCT__
644 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable"
645 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable(Mat A,Mat P,PetscReal fill,Mat *C)
646 {
647   PetscErrorCode       ierr;
648   MPI_Comm             comm=((PetscObject)A)->comm;
649   Mat                  Cmpi;
650   Mat_PtAPMPI          *ptap;
651   PetscFreeSpaceList   free_space=PETSC_NULL,current_space=PETSC_NULL;
652   Mat_MPIAIJ           *a=(Mat_MPIAIJ*)A->data,*c;
653   Mat_SeqAIJ           *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
654   PetscInt             *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
655   PetscInt             *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
656   PetscInt             i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi,*lnk,apnz_max=0;
657   PetscInt             am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
658   PetscInt             nlnk_max,armax,prmax;
659   PetscReal            afill;
660   PetscScalar          *apa;
661 
662   PetscFunctionBegin;
663   /* create struct Mat_PtAPMPI and attached it to C later */
664   ierr = PetscNew(Mat_PtAPMPI,&ptap);CHKERRQ(ierr);
665 
666   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
667   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
668 
669   /* get P_loc by taking all local rows of P */
670   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
671 
672   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
673   p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
674   pi_loc = p_loc->i; pj_loc = p_loc->j;
675   pi_oth = p_oth->i; pj_oth = p_oth->j;
676 
677   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
678   /*-------------------------------------------------------------------*/
679   ierr  = PetscMalloc((am+2)*sizeof(PetscInt),&api);CHKERRQ(ierr);
680   ptap->api = api;
681   api[0]    = 0;
682 
683   /* create and initialize a linked list */
684   armax = ad->rmax+ao->rmax;
685   prmax = PetscMax(p_loc->rmax,p_oth->rmax);
686   nlnk_max = armax*prmax;
687   if (!nlnk_max || nlnk_max > pN) nlnk_max = pN;
688   ierr = PetscLLCondensedCreate_Scalable(nlnk_max,&lnk);CHKERRQ(ierr);
689 
690   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
691   ierr = PetscFreeSpaceGet((PetscInt)(fill*(adi[am]+aoi[am]+pi_loc[pm])),&free_space);CHKERRQ(ierr);
692   current_space = free_space;
693 
694   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
695   for (i=0; i<am; i++) {
696     /* diagonal portion of A */
697     nzi = adi[i+1] - adi[i];
698     for (j=0; j<nzi; j++){
699       row = *adj++;
700       pnz = pi_loc[row+1] - pi_loc[row];
701       Jptr  = pj_loc + pi_loc[row];
702       /* add non-zero cols of P into the sorted linked list lnk */
703       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
704     }
705     /* off-diagonal portion of A */
706     nzi = aoi[i+1] - aoi[i];
707     for (j=0; j<nzi; j++){
708       row = *aoj++;
709       pnz = pi_oth[row+1] - pi_oth[row];
710       Jptr  = pj_oth + pi_oth[row];
711       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
712     }
713 
714     apnz     = *lnk;
715     api[i+1] = api[i] + apnz;
716     if (apnz > apnz_max) apnz_max = apnz;
717 
718     /* if free space is not available, double the total space in the list */
719     if (current_space->local_remaining<apnz) {
720       ierr = PetscFreeSpaceGet(apnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
721       nspacedouble++;
722     }
723 
724     /* Copy data into free space, then initialize lnk */
725     ierr = PetscLLCondensedClean_Scalable(apnz,current_space->array,lnk);CHKERRQ(ierr);
726     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
727     current_space->array           += apnz;
728     current_space->local_used      += apnz;
729     current_space->local_remaining -= apnz;
730   }
731 
732   /* Allocate space for apj, initialize apj, and */
733   /* destroy list of free space and other temporary array(s) */
734   ierr = PetscMalloc((api[am]+1)*sizeof(PetscInt),&ptap->apj);CHKERRQ(ierr);
735   apj  = ptap->apj;
736   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
737   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
738 
739   /* create and assemble symbolic parallel matrix Cmpi */
740   /*----------------------------------------------------*/
741   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
742   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
743   ierr = MatSetBlockSizes(Cmpi,A->rmap->bs,P->cmap->bs);CHKERRQ(ierr);
744   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
745   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
746   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
747 
748   /* malloc apa for assembly Cmpi */
749   ierr = PetscMalloc(apnz_max*sizeof(PetscScalar),&apa);CHKERRQ(ierr);
750   ierr = PetscMemzero(apa,apnz_max*sizeof(PetscScalar));CHKERRQ(ierr);
751   ptap->apa = apa;
752   for (i=0; i<am; i++){
753     row  = i + rstart;
754     apnz = api[i+1] - api[i];
755     ierr = MatSetValues(Cmpi,1,&row,apnz,apj,apa,INSERT_VALUES);CHKERRQ(ierr);
756     apj += apnz;
757   }
758   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
759   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
760 
761   ptap->destroy             = Cmpi->ops->destroy;
762   ptap->duplicate           = Cmpi->ops->duplicate;
763   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_Scalable;
764   Cmpi->ops->destroy        = MatDestroy_MPIAIJ_MatMatMult;
765   Cmpi->ops->duplicate      = MatDuplicate_MPIAIJ_MatMatMult;
766 
767   /* attach the supporting struct to Cmpi for reuse */
768   c = (Mat_MPIAIJ*)Cmpi->data;
769   c->ptap  = ptap;
770 
771   *C = Cmpi;
772 
773   /* set MatInfo */
774   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
775   if (afill < 1.0) afill = 1.0;
776   Cmpi->info.mallocs           = nspacedouble;
777   Cmpi->info.fill_ratio_given  = fill;
778   Cmpi->info.fill_ratio_needed = afill;
779 
780 #if defined(PETSC_USE_INFO)
781   if (api[am]) {
782     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %G needed %G.\n",nspacedouble,fill,afill);CHKERRQ(ierr);
783     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%G,&C) for best performance.;\n",afill);CHKERRQ(ierr);
784   } else {
785     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
786   }
787 #endif
788   PetscFunctionReturn(0);
789 }
790 
791 /*-------------------------------------------------------------------------*/
792 #undef __FUNCT__
793 #define __FUNCT__ "MatTransposeMatMult_MPIAIJ_MPIAIJ"
794 PetscErrorCode MatTransposeMatMult_MPIAIJ_MPIAIJ(Mat P,Mat A,MatReuse scall,PetscReal fill,Mat *C)
795 {
796   PetscErrorCode ierr;
797   PetscBool      scalable=PETSC_TRUE,viamatmatmult=PETSC_FALSE;
798 
799   PetscFunctionBegin;
800   ierr = PetscOptionsBool("-mattransposematmult_viamatmatmult","Use R=Pt and C=R*A","",viamatmatmult,&viamatmatmult,PETSC_NULL);CHKERRQ(ierr);
801   if (viamatmatmult){
802     Mat         Pt;
803     Mat_PtAPMPI *ptap;
804     Mat_MPIAIJ  *c;
805     if (scall == MAT_INITIAL_MATRIX){
806       ierr = MatTranspose(P,MAT_INITIAL_MATRIX,&Pt);CHKERRQ(ierr);
807       ierr = MatMatMult(Pt,A,MAT_INITIAL_MATRIX,fill,C);CHKERRQ(ierr);
808 
809       c        = (Mat_MPIAIJ*)(*C)->data;
810       ptap     = c->ptap;
811       ptap->Pt = Pt;
812     } else if (scall == MAT_REUSE_MATRIX){
813       c    = (Mat_MPIAIJ*)(*C)->data;
814       ptap = c->ptap;
815       Pt   = ptap->Pt;
816       ierr = MatTranspose(P,scall,&Pt);CHKERRQ(ierr);
817       ierr = MatMatMult(Pt,A,scall,fill,C);CHKERRQ(ierr);
818     } else {
819       SETERRQ(((PetscObject)A)->comm,PETSC_ERR_ARG_WRONGSTATE,"Not supported");
820     }
821     PetscFunctionReturn(0);
822   }
823 
824   if (scall == MAT_INITIAL_MATRIX){
825     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
826       ierr = PetscOptionsBool("-mattransposematmult_scalable","Use a scalable but slower C=Pt*A","",scalable,&scalable,PETSC_NULL);CHKERRQ(ierr);
827       if  (scalable){
828         ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable(P,A,fill,C);CHKERRQ(ierr);
829       } else {
830         ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(P,A,fill,C);CHKERRQ(ierr);
831       }
832     ierr = PetscOptionsEnd();CHKERRQ(ierr);
833   }
834   ierr = (*(*C)->ops->mattransposemultnumeric)(P,A,*C);CHKERRQ(ierr);
835   PetscFunctionReturn(0);
836 }
837 
838 #undef __FUNCT__
839 #define __FUNCT__ "MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ"
840 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ(Mat P,Mat A,Mat C)
841 {
842   PetscErrorCode       ierr;
843   Mat_Merge_SeqsToMPI  *merge;
844   Mat_MPIAIJ           *p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
845   Mat_SeqAIJ           *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
846   Mat_PtAPMPI          *ptap;
847   PetscInt             *adj,*aJ;
848   PetscInt             i,j,k,anz,pnz,row,*cj;
849   MatScalar            *ada,*aval,*ca,valtmp;
850   PetscInt             am=A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
851   MPI_Comm             comm=((PetscObject)C)->comm;
852   PetscMPIInt          size,rank,taga,*len_s;
853   PetscInt             *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
854   PetscInt             **buf_ri,**buf_rj;
855   PetscInt             cnz=0,*bj_i,*bi,*bj,bnz,nextcj; /* bi,bj,ba: local array of C(mpi mat) */
856   MPI_Request          *s_waits,*r_waits;
857   MPI_Status           *status;
858   MatScalar            **abuf_r,*ba_i,*pA,*coa,*ba;
859   PetscInt             *ai,*aj,*coi,*coj;
860   PetscInt             *poJ,*pdJ;
861   Mat                  A_loc;
862   Mat_SeqAIJ           *a_loc;
863 
864   PetscFunctionBegin;
865   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
866   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
867 
868   ptap  = c->ptap;
869   merge = ptap->merge;
870 
871   /* 2) compute numeric C_seq = P_loc^T*A_loc*P - dominating part */
872   /*--------------------------------------------------------------*/
873   /* get data from symbolic products */
874   coi = merge->coi; coj = merge->coj;
875   ierr = PetscMalloc((coi[pon]+1)*sizeof(MatScalar),&coa);CHKERRQ(ierr);
876   ierr = PetscMemzero(coa,coi[pon]*sizeof(MatScalar));CHKERRQ(ierr);
877 
878   bi     = merge->bi; bj = merge->bj;
879   owners = merge->rowmap->range;
880   ierr   = PetscMalloc((bi[cm]+1)*sizeof(MatScalar),&ba);CHKERRQ(ierr);
881   ierr   = PetscMemzero(ba,bi[cm]*sizeof(MatScalar));CHKERRQ(ierr);
882 
883   /* get A_loc by taking all local rows of A */
884   A_loc = ptap->A_loc;
885   ierr = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
886   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
887   ai   = a_loc->i;
888   aj   = a_loc->j;
889 
890   ierr = PetscMalloc((A->cmap->N)*sizeof(PetscScalar),&aval);CHKERRQ(ierr); /* non-scalable!!! */
891   ierr = PetscMemzero(aval,A->cmap->N*sizeof(PetscScalar));CHKERRQ(ierr);
892 
893     for (i=0; i<am; i++) {
894       /* 2-a) put A[i,:] to dense array aval */
895       anz = ai[i+1] - ai[i];
896       adj = aj + ai[i];
897       ada = a_loc->a + ai[i];
898       for (j=0; j<anz; j++){
899         aval[adj[j]] = ada[j];
900       }
901 
902       /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
903       /*--------------------------------------------------------------*/
904       /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
905       pnz = po->i[i+1] - po->i[i];
906       poJ = po->j + po->i[i];
907       pA  = po->a + po->i[i];
908       for (j=0; j<pnz; j++){
909         row = poJ[j];
910         cnz = coi[row+1] - coi[row];
911         cj  = coj + coi[row];
912         ca  = coa + coi[row];
913         /* perform dense axpy */
914         valtmp = pA[j];
915         for (k=0; k<cnz; k++) {
916           ca[k] += valtmp*aval[cj[k]];
917         }
918         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
919       }
920 
921       /* put the value into Cd (diagonal part) */
922       pnz = pd->i[i+1] - pd->i[i];
923       pdJ = pd->j + pd->i[i];
924       pA  = pd->a + pd->i[i];
925       for (j=0; j<pnz; j++){
926         row = pdJ[j];
927         cnz = bi[row+1] - bi[row];
928         cj  = bj + bi[row];
929         ca  = ba + bi[row];
930         /* perform dense axpy */
931         valtmp = pA[j];
932         for (k=0; k<cnz; k++) {
933           ca[k] += valtmp*aval[cj[k]];
934         }
935         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
936       }
937 
938       /* zero the current row of Pt*A */
939       aJ = aj + ai[i];
940       for (k=0; k<anz; k++) aval[aJ[k]] = 0.0;
941     }
942 
943   /* 3) send and recv matrix values coa */
944   /*------------------------------------*/
945   buf_ri = merge->buf_ri;
946   buf_rj = merge->buf_rj;
947   len_s  = merge->len_s;
948   ierr = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
949   ierr = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
950 
951   ierr = PetscMalloc2(merge->nsend+1,MPI_Request,&s_waits,size,MPI_Status,&status);CHKERRQ(ierr);
952   for (proc=0,k=0; proc<size; proc++){
953     if (!len_s[proc]) continue;
954     i = merge->owners_co[proc];
955     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
956     k++;
957   }
958   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
959   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
960 
961   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
962   ierr = PetscFree(r_waits);CHKERRQ(ierr);
963   ierr = PetscFree(coa);CHKERRQ(ierr);
964 
965   /* 4) insert local Cseq and received values into Cmpi */
966   /*----------------------------------------------------*/
967   ierr = PetscMalloc3(merge->nrecv,PetscInt**,&buf_ri_k,merge->nrecv,PetscInt*,&nextrow,merge->nrecv,PetscInt*,&nextci);CHKERRQ(ierr);
968   for (k=0; k<merge->nrecv; k++){
969     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
970     nrows       = *(buf_ri_k[k]);
971     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
972     nextci[k]   = buf_ri_k[k] + (nrows + 1);/* poins to the next i-structure of k-th recved i-structure  */
973   }
974 
975   for (i=0; i<cm; i++) {
976     row = owners[rank] + i; /* global row index of C_seq */
977     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
978     ba_i = ba + bi[i];
979     bnz  = bi[i+1] - bi[i];
980     /* add received vals into ba */
981     for (k=0; k<merge->nrecv; k++){ /* k-th received message */
982       /* i-th row */
983       if (i == *nextrow[k]) {
984         cnz = *(nextci[k]+1) - *nextci[k];
985         cj  = buf_rj[k] + *(nextci[k]);
986         ca  = abuf_r[k] + *(nextci[k]);
987         nextcj = 0;
988         for (j=0; nextcj<cnz; j++){
989           if (bj_i[j] == cj[nextcj]){ /* bcol == ccol */
990             ba_i[j] += ca[nextcj++];
991           }
992         }
993         nextrow[k]++; nextci[k]++;
994         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
995       }
996     }
997     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
998   }
999   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1000   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1001 
1002   ierr = PetscFree(ba);CHKERRQ(ierr);
1003   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1004   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1005   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1006   ierr = PetscFree(aval);CHKERRQ(ierr);
1007   PetscFunctionReturn(0);
1008 }
1009 
1010 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1011 #undef __FUNCT__
1012 #define __FUNCT__ "MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ"
1013 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P,Mat A,PetscReal fill,Mat *C)
1014 {
1015   PetscErrorCode       ierr;
1016   Mat                  Cmpi,A_loc,POt,PDt;
1017   Mat_PtAPMPI          *ptap;
1018   PetscFreeSpaceList   free_space=PETSC_NULL,current_space=PETSC_NULL;
1019   Mat_MPIAIJ           *p=(Mat_MPIAIJ*)P->data,*c;
1020   PetscInt             *pdti,*pdtj,*poti,*potj,*ptJ;
1021   PetscInt             nnz;
1022   PetscInt             *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1023   PetscInt             am=A->rmap->n,pn=P->cmap->n;
1024   PetscBT              lnkbt;
1025   MPI_Comm             comm=((PetscObject)A)->comm;
1026   PetscMPIInt          size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1027   PetscInt             **buf_rj,**buf_ri,**buf_ri_k;
1028   PetscInt             len,proc,*dnz,*onz,*owners;
1029   PetscInt             nzi,*bi,*bj;
1030   PetscInt             nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1031   MPI_Request          *swaits,*rwaits;
1032   MPI_Status           *sstatus,rstatus;
1033   Mat_Merge_SeqsToMPI  *merge;
1034   PetscInt             *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1035   PetscReal            afill=1.0,afill_tmp;
1036   PetscInt             rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Crmax;
1037   PetscScalar          *vals;
1038   Mat_SeqAIJ           *a_loc, *pdt,*pot;
1039 
1040   PetscFunctionBegin;
1041   /* check if matrix local sizes are compatible */
1042   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend){
1043     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);
1044   }
1045 
1046   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1047   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1048 
1049   /* create struct Mat_PtAPMPI and attached it to C later */
1050   ierr = PetscNew(Mat_PtAPMPI,&ptap);CHKERRQ(ierr);
1051 
1052   /* get A_loc by taking all local rows of A */
1053   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1054   ptap->A_loc = A_loc;
1055   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1056   ai   = a_loc->i;
1057   aj   = a_loc->j;
1058 
1059   /* determine symbolic Co=(p->B)^T*A - send to others */
1060   /*----------------------------------------------------*/
1061   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1062   pdt = (Mat_SeqAIJ*)PDt->data;
1063   pdti = pdt->i; pdtj = pdt->j;
1064 
1065   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1066   pot = (Mat_SeqAIJ*)POt->data;
1067   poti = pot->i; potj = pot->j;
1068 
1069   /* then, compute symbolic Co = (p->B)^T*A */
1070   pon = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1071                          >= (num of nonzero rows of C_seq) - pn */
1072   ierr = PetscMalloc((pon+1)*sizeof(PetscInt),&coi);CHKERRQ(ierr);
1073   coi[0] = 0;
1074 
1075   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1076   nnz           = fill*(poti[pon] + ai[am]);
1077   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1078   current_space = free_space;
1079 
1080   /* create and initialize a linked list */
1081   i = PetscMax(pdt->rmax,pot->rmax);
1082   Crmax = i*a_loc->rmax*size;
1083   if (!Crmax || Crmax > aN) Crmax = aN;
1084   ierr = PetscLLCondensedCreate(Crmax,aN,&lnk,&lnkbt);CHKERRQ(ierr);
1085 
1086   for (i=0; i<pon; i++) {
1087     pnz = poti[i+1] - poti[i];
1088     ptJ = potj + poti[i];
1089     for (j=0; j<pnz; j++){
1090       row  = ptJ[j]; /* row of A_loc == col of Pot */
1091       anz  = ai[row+1] - ai[row];
1092       Jptr = aj + ai[row];
1093       /* add non-zero cols of AP into the sorted linked list lnk */
1094       ierr = PetscLLCondensedAddSorted(anz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1095     }
1096     nnz = lnk[0];
1097 
1098     /* If free space is not available, double the total space in the list */
1099     if (current_space->local_remaining<nnz) {
1100       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1101       nspacedouble++;
1102     }
1103 
1104     /* Copy data into free space, and zero out denserows */
1105     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1106     current_space->array           += nnz;
1107     current_space->local_used      += nnz;
1108     current_space->local_remaining -= nnz;
1109     coi[i+1] = coi[i] + nnz;
1110   }
1111 
1112   ierr = PetscMalloc((coi[pon]+1)*sizeof(PetscInt),&coj);CHKERRQ(ierr);
1113   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1114   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1115   if (afill_tmp > afill) afill = afill_tmp;
1116 
1117   /* send j-array (coj) of Co to other processors */
1118   /*----------------------------------------------*/
1119   /* determine row ownership */
1120   ierr = PetscNew(Mat_Merge_SeqsToMPI,&merge);CHKERRQ(ierr);
1121   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1122   merge->rowmap->n = pn;
1123   merge->rowmap->bs = 1;
1124   ierr = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1125   owners = merge->rowmap->range;
1126 
1127   /* determine the number of messages to send, their lengths */
1128   ierr = PetscMalloc(size*sizeof(PetscMPIInt),&len_si);CHKERRQ(ierr);
1129   ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1130   ierr = PetscMalloc(size*sizeof(PetscMPIInt),&merge->len_s);CHKERRQ(ierr);
1131   len_s = merge->len_s;
1132   merge->nsend = 0;
1133 
1134   ierr = PetscMalloc((size+2)*sizeof(PetscInt),&owners_co);CHKERRQ(ierr);
1135   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1136 
1137   proc = 0;
1138   for (i=0; i<pon; i++){
1139     while (prmap[i] >= owners[proc+1]) proc++;
1140     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1141     len_s[proc] += coi[i+1] - coi[i];
1142   }
1143 
1144   len   = 0;  /* max length of buf_si[] */
1145   owners_co[0] = 0;
1146   for (proc=0; proc<size; proc++){
1147     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1148     if (len_si[proc]){
1149       merge->nsend++;
1150       len_si[proc] = 2*(len_si[proc] + 1);
1151       len += len_si[proc];
1152     }
1153   }
1154 
1155   /* determine the number and length of messages to receive for coi and coj  */
1156   ierr = PetscGatherNumberOfMessages(comm,PETSC_NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1157   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1158 
1159   /* post the Irecv and Isend of coj */
1160   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1161   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1162   ierr = PetscMalloc((merge->nsend+1)*sizeof(MPI_Request),&swaits);CHKERRQ(ierr);
1163   for (proc=0, k=0; proc<size; proc++){
1164     if (!len_s[proc]) continue;
1165     i = owners_co[proc];
1166     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1167     k++;
1168   }
1169 
1170   /* receives and sends of coj are complete */
1171   ierr = PetscMalloc(size*sizeof(MPI_Status),&sstatus);CHKERRQ(ierr);
1172   for (i=0; i<merge->nrecv; i++){
1173     PetscMPIInt icompleted;
1174     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1175   }
1176   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1177   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1178 
1179   /* send and recv coi */
1180   /*-------------------*/
1181   ierr = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1182   ierr = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1183   ierr = PetscMalloc((len+1)*sizeof(PetscInt),&buf_s);CHKERRQ(ierr);
1184   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1185   for (proc=0,k=0; proc<size; proc++){
1186     if (!len_s[proc]) continue;
1187     /* form outgoing message for i-structure:
1188          buf_si[0]:                 nrows to be sent
1189                [1:nrows]:           row index (global)
1190                [nrows+1:2*nrows+1]: i-structure index
1191     */
1192     /*-------------------------------------------*/
1193     nrows = len_si[proc]/2 - 1;
1194     buf_si_i    = buf_si + nrows+1;
1195     buf_si[0]   = nrows;
1196     buf_si_i[0] = 0;
1197     nrows = 0;
1198     for (i=owners_co[proc]; i<owners_co[proc+1]; i++){
1199       nzi = coi[i+1] - coi[i];
1200       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi; /* i-structure */
1201       buf_si[nrows+1] =prmap[i] -owners[proc]; /* local row index */
1202       nrows++;
1203     }
1204     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1205     k++;
1206     buf_si += len_si[proc];
1207   }
1208   i = merge->nrecv;
1209   while (i--) {
1210     PetscMPIInt icompleted;
1211     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1212   }
1213   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1214   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1215   ierr = PetscFree(len_si);CHKERRQ(ierr);
1216   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1217   ierr = PetscFree(swaits);CHKERRQ(ierr);
1218   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1219   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1220 
1221   /* compute the local portion of C (mpi mat) */
1222   /*------------------------------------------*/
1223   /* allocate bi array and free space for accumulating nonzero column info */
1224   ierr = PetscMalloc((pn+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
1225   bi[0] = 0;
1226 
1227   /* set initial free space to be fill*(nnz(P) + nnz(A)) */
1228   nnz           = fill*(pdti[pn] + poti[pon] + ai[am]);
1229   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1230   current_space = free_space;
1231 
1232   ierr = PetscMalloc3(merge->nrecv,PetscInt**,&buf_ri_k,merge->nrecv,PetscInt*,&nextrow,merge->nrecv,PetscInt*,&nextci);CHKERRQ(ierr);
1233   for (k=0; k<merge->nrecv; k++){
1234     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1235     nrows       = *buf_ri_k[k];
1236     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1237     nextci[k]   = buf_ri_k[k] + (nrows + 1);/* poins to the next i-structure of k-th recved i-structure  */
1238   }
1239 
1240   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1241   rmax = 0;
1242   for (i=0; i<pn; i++) {
1243     /* add pdt[i,:]*AP into lnk */
1244     pnz = pdti[i+1] - pdti[i];
1245     ptJ = pdtj + pdti[i];
1246     for (j=0; j<pnz; j++){
1247       row  = ptJ[j];  /* row of AP == col of Pt */
1248       anz  = ai[row+1] - ai[row];
1249       Jptr = aj + ai[row];
1250       /* add non-zero cols of AP into the sorted linked list lnk */
1251       ierr = PetscLLCondensedAddSorted(anz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1252     }
1253 
1254     /* add received col data into lnk */
1255     for (k=0; k<merge->nrecv; k++){ /* k-th received message */
1256       if (i == *nextrow[k]) { /* i-th row */
1257         nzi = *(nextci[k]+1) - *nextci[k];
1258         Jptr  = buf_rj[k] + *nextci[k];
1259         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1260         nextrow[k]++; nextci[k]++;
1261       }
1262     }
1263     nnz = lnk[0];
1264 
1265     /* if free space is not available, make more free space */
1266     if (current_space->local_remaining<nnz) {
1267       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1268       nspacedouble++;
1269     }
1270     /* copy data into free space, then initialize lnk */
1271     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1272     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1273     current_space->array           += nnz;
1274     current_space->local_used      += nnz;
1275     current_space->local_remaining -= nnz;
1276     bi[i+1] = bi[i] + nnz;
1277     if (nnz > rmax) rmax = nnz;
1278   }
1279   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1280 
1281   ierr = PetscMalloc((bi[pn]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
1282   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1283   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
1284   if (afill_tmp > afill) afill = afill_tmp;
1285   ierr = PetscLLCondensedDestroy(lnk,lnkbt);CHKERRQ(ierr);
1286   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1287   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1288 
1289   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
1290   /*----------------------------------------------------------------------------------*/
1291   ierr = PetscMalloc((rmax+1)*sizeof(PetscScalar),&vals);CHKERRQ(ierr);
1292   ierr = PetscMemzero(vals,rmax*sizeof(PetscScalar));CHKERRQ(ierr);
1293 
1294   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1295   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1296   ierr = MatSetBlockSizes(Cmpi,P->cmap->bs,A->cmap->bs);CHKERRQ(ierr);
1297   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1298   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1299   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1300   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
1301   for (i=0; i<pn; i++){
1302     row = i + rstart;
1303     nnz = bi[i+1] - bi[i];
1304     Jptr = bj + bi[i];
1305     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
1306   }
1307   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1308   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1309   ierr = PetscFree(vals);CHKERRQ(ierr);
1310 
1311   merge->bi            = bi;
1312   merge->bj            = bj;
1313   merge->coi           = coi;
1314   merge->coj           = coj;
1315   merge->buf_ri        = buf_ri;
1316   merge->buf_rj        = buf_rj;
1317   merge->owners_co     = owners_co;
1318   merge->destroy       = Cmpi->ops->destroy;
1319   merge->duplicate     = Cmpi->ops->duplicate;
1320 
1321   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ;
1322   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
1323 
1324   /* attach the supporting struct to Cmpi for reuse */
1325   c = (Mat_MPIAIJ*)Cmpi->data;
1326   c->ptap        = ptap;
1327   ptap->api      = PETSC_NULL;
1328   ptap->apj      = PETSC_NULL;
1329   ptap->merge    = merge;
1330   ptap->rmax     = rmax;
1331 
1332   *C = Cmpi;
1333 #if defined(PETSC_USE_INFO)
1334   if (bi[pn] != 0) {
1335     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %G needed %G.\n",nspacedouble,fill,afill);CHKERRQ(ierr);
1336     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%G,&C) for best performance.\n",afill);CHKERRQ(ierr);
1337   } else {
1338     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1339   }
1340 #endif
1341   PetscFunctionReturn(0);
1342 }
1343 
1344 #undef __FUNCT__
1345 #define __FUNCT__ "MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_Scalable"
1346 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_Scalable(Mat P,Mat A,Mat C)
1347 {
1348   PetscErrorCode       ierr;
1349   Mat_Merge_SeqsToMPI  *merge;
1350   Mat_MPIAIJ           *p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1351   Mat_SeqAIJ           *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
1352   Mat_PtAPMPI          *ptap;
1353   PetscInt             *adj;
1354   PetscInt             i,j,k,anz,pnz,row,*cj,nexta;
1355   MatScalar            *ada,*ca,valtmp;
1356   PetscInt             am=A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
1357   MPI_Comm             comm=((PetscObject)C)->comm;
1358   PetscMPIInt          size,rank,taga,*len_s;
1359   PetscInt             *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
1360   PetscInt             **buf_ri,**buf_rj;
1361   PetscInt             cnz=0,*bj_i,*bi,*bj,bnz,nextcj; /* bi,bj,ba: local array of C(mpi mat) */
1362   MPI_Request          *s_waits,*r_waits;
1363   MPI_Status           *status;
1364   MatScalar            **abuf_r,*ba_i,*pA,*coa,*ba;
1365   PetscInt             *ai,*aj,*coi,*coj;
1366   PetscInt             *poJ,*pdJ;
1367   Mat                  A_loc;
1368   Mat_SeqAIJ           *a_loc;
1369 
1370   PetscFunctionBegin;
1371   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1372   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1373 
1374   ptap  = c->ptap;
1375   merge = ptap->merge;
1376 
1377   /* 2) compute numeric C_seq = P_loc^T*A_loc */
1378   /*------------------------------------------*/
1379   /* get data from symbolic products */
1380   coi = merge->coi; coj = merge->coj;
1381   ierr = PetscMalloc((coi[pon]+1)*sizeof(MatScalar),&coa);CHKERRQ(ierr);
1382   ierr = PetscMemzero(coa,coi[pon]*sizeof(MatScalar));CHKERRQ(ierr);
1383   bi     = merge->bi; bj = merge->bj;
1384   owners = merge->rowmap->range;
1385   ierr   = PetscMalloc((bi[cm]+1)*sizeof(MatScalar),&ba);CHKERRQ(ierr);
1386   ierr   = PetscMemzero(ba,bi[cm]*sizeof(MatScalar));CHKERRQ(ierr);
1387 
1388   /* get A_loc by taking all local rows of A */
1389   A_loc = ptap->A_loc;
1390   ierr = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
1391   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1392   ai   = a_loc->i;
1393   aj   = a_loc->j;
1394 
1395   for (i=0; i<am; i++) {
1396     anz = ai[i+1] - ai[i];
1397     adj = aj + ai[i];
1398     ada = a_loc->a + ai[i];
1399 
1400     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1401     /*-------------------------------------------------------------*/
1402     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1403     pnz = po->i[i+1] - po->i[i];
1404     poJ = po->j + po->i[i];
1405     pA  = po->a + po->i[i];
1406     for (j=0; j<pnz; j++){
1407       row = poJ[j];
1408       cj  = coj + coi[row];
1409       ca  = coa + coi[row];
1410       /* perform sparse axpy */
1411       nexta  = 0;
1412       valtmp = pA[j];
1413       for (k=0; nexta<anz; k++) {
1414         if (cj[k] == adj[nexta]){
1415           ca[k] += valtmp*ada[nexta];
1416           nexta++;
1417         }
1418       }
1419       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1420     }
1421 
1422     /* put the value into Cd (diagonal part) */
1423     pnz = pd->i[i+1] - pd->i[i];
1424     pdJ = pd->j + pd->i[i];
1425     pA  = pd->a + pd->i[i];
1426     for (j=0; j<pnz; j++){
1427       row = pdJ[j];
1428       cj  = bj + bi[row];
1429       ca  = ba + bi[row];
1430       /* perform sparse axpy */
1431       nexta  = 0;
1432       valtmp = pA[j];
1433       for (k=0; nexta<anz; k++) {
1434         if (cj[k] == adj[nexta]){
1435           ca[k] += valtmp*ada[nexta];
1436           nexta++;
1437         }
1438       }
1439       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1440     }
1441   }
1442 
1443   /* 3) send and recv matrix values coa */
1444   /*------------------------------------*/
1445   buf_ri = merge->buf_ri;
1446   buf_rj = merge->buf_rj;
1447   len_s  = merge->len_s;
1448   ierr = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1449   ierr = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1450 
1451   ierr = PetscMalloc2(merge->nsend+1,MPI_Request,&s_waits,size,MPI_Status,&status);CHKERRQ(ierr);
1452   for (proc=0,k=0; proc<size; proc++){
1453     if (!len_s[proc]) continue;
1454     i = merge->owners_co[proc];
1455     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1456     k++;
1457   }
1458   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1459   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1460 
1461   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1462   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1463   ierr = PetscFree(coa);CHKERRQ(ierr);
1464 
1465   /* 4) insert local Cseq and received values into Cmpi */
1466   /*----------------------------------------------------*/
1467   ierr = PetscMalloc3(merge->nrecv,PetscInt**,&buf_ri_k,merge->nrecv,PetscInt*,&nextrow,merge->nrecv,PetscInt*,&nextci);CHKERRQ(ierr);
1468   for (k=0; k<merge->nrecv; k++){
1469     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1470     nrows       = *(buf_ri_k[k]);
1471     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1472     nextci[k]   = buf_ri_k[k] + (nrows + 1);/* poins to the next i-structure of k-th recved i-structure  */
1473   }
1474 
1475   for (i=0; i<cm; i++) {
1476     row = owners[rank] + i; /* global row index of C_seq */
1477     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1478     ba_i = ba + bi[i];
1479     bnz  = bi[i+1] - bi[i];
1480     /* add received vals into ba */
1481     for (k=0; k<merge->nrecv; k++){ /* k-th received message */
1482       /* i-th row */
1483       if (i == *nextrow[k]) {
1484         cnz = *(nextci[k]+1) - *nextci[k];
1485         cj  = buf_rj[k] + *(nextci[k]);
1486         ca  = abuf_r[k] + *(nextci[k]);
1487         nextcj = 0;
1488         for (j=0; nextcj<cnz; j++){
1489           if (bj_i[j] == cj[nextcj]){ /* bcol == ccol */
1490             ba_i[j] += ca[nextcj++];
1491           }
1492         }
1493         nextrow[k]++; nextci[k]++;
1494         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1495       }
1496     }
1497     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1498   }
1499   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1500   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1501 
1502   ierr = PetscFree(ba);CHKERRQ(ierr);
1503   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1504   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1505   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1506   PetscFunctionReturn(0);
1507 }
1508 
1509 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1510 #undef __FUNCT__
1511 #define __FUNCT__ "MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable"
1512 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_Scalable(Mat P,Mat A,PetscReal fill,Mat *C)
1513 {
1514   PetscErrorCode       ierr;
1515   Mat                  Cmpi,A_loc,POt,PDt;
1516   Mat_PtAPMPI          *ptap;
1517   PetscFreeSpaceList   free_space=PETSC_NULL,current_space=PETSC_NULL;
1518   Mat_MPIAIJ           *p=(Mat_MPIAIJ*)P->data,*c;
1519   PetscInt             *pdti,*pdtj,*poti,*potj,*ptJ;
1520   PetscInt             nnz;
1521   PetscInt             *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1522   PetscInt             am=A->rmap->n,pn=P->cmap->n;
1523   MPI_Comm             comm=((PetscObject)A)->comm;
1524   PetscMPIInt          size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1525   PetscInt             **buf_rj,**buf_ri,**buf_ri_k;
1526   PetscInt             len,proc,*dnz,*onz,*owners;
1527   PetscInt             nzi,*bi,*bj;
1528   PetscInt             nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1529   MPI_Request          *swaits,*rwaits;
1530   MPI_Status           *sstatus,rstatus;
1531   Mat_Merge_SeqsToMPI  *merge;
1532   PetscInt             *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1533   PetscReal            afill=1.0,afill_tmp;
1534   PetscInt             rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Crmax;
1535   PetscScalar          *vals;
1536   Mat_SeqAIJ           *a_loc, *pdt,*pot;
1537 
1538   PetscFunctionBegin;
1539   /* check if matrix local sizes are compatible */
1540   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend){
1541     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);
1542   }
1543 
1544   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1545   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1546 
1547   /* create struct Mat_PtAPMPI and attached it to C later */
1548   ierr = PetscNew(Mat_PtAPMPI,&ptap);CHKERRQ(ierr);
1549 
1550   /* get A_loc by taking all local rows of A */
1551   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1552   ptap->A_loc = A_loc;
1553   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1554   ai   = a_loc->i;
1555   aj   = a_loc->j;
1556 
1557   /* determine symbolic Co=(p->B)^T*A - send to others */
1558   /*----------------------------------------------------*/
1559   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1560   pdt = (Mat_SeqAIJ*)PDt->data;
1561   pdti = pdt->i; pdtj = pdt->j;
1562 
1563   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1564   pot = (Mat_SeqAIJ*)POt->data;
1565   poti = pot->i; potj = pot->j;
1566 
1567   /* then, compute symbolic Co = (p->B)^T*A */
1568   pon = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1569                          >= (num of nonzero rows of C_seq) - pn */
1570   ierr = PetscMalloc((pon+1)*sizeof(PetscInt),&coi);CHKERRQ(ierr);
1571   coi[0] = 0;
1572 
1573   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1574   nnz           = fill*(poti[pon] + ai[am]);
1575   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1576   current_space = free_space;
1577 
1578   /* create and initialize a linked list */
1579   i = PetscMax(pdt->rmax,pot->rmax);
1580   Crmax = i*a_loc->rmax*size; /* non-scalable! */
1581   if (!Crmax || Crmax > aN) Crmax = aN;
1582   ierr = PetscLLCondensedCreate_Scalable(Crmax,&lnk);CHKERRQ(ierr);
1583 
1584   for (i=0; i<pon; i++) {
1585     pnz = poti[i+1] - poti[i];
1586     ptJ = potj + poti[i];
1587     for (j=0; j<pnz; j++){
1588       row  = ptJ[j]; /* row of A_loc == col of Pot */
1589       anz  = ai[row+1] - ai[row];
1590       Jptr = aj + ai[row];
1591       /* add non-zero cols of AP into the sorted linked list lnk */
1592       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1593     }
1594     nnz = lnk[0];
1595 
1596     /* If free space is not available, double the total space in the list */
1597     if (current_space->local_remaining<nnz) {
1598       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1599       nspacedouble++;
1600     }
1601 
1602     /* Copy data into free space, and zero out denserows */
1603     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1604     current_space->array           += nnz;
1605     current_space->local_used      += nnz;
1606     current_space->local_remaining -= nnz;
1607     coi[i+1] = coi[i] + nnz;
1608   }
1609 
1610   ierr = PetscMalloc((coi[pon]+1)*sizeof(PetscInt),&coj);CHKERRQ(ierr);
1611   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1612   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1613   if (afill_tmp > afill) afill = afill_tmp;
1614 
1615   /* send j-array (coj) of Co to other processors */
1616   /*----------------------------------------------*/
1617   /* determine row ownership */
1618   ierr = PetscNew(Mat_Merge_SeqsToMPI,&merge);CHKERRQ(ierr);
1619   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1620   merge->rowmap->n = pn;
1621   merge->rowmap->bs = 1;
1622   ierr = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1623   owners = merge->rowmap->range;
1624 
1625   /* determine the number of messages to send, their lengths */
1626   ierr = PetscMalloc(size*sizeof(PetscMPIInt),&len_si);CHKERRQ(ierr);
1627   ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1628   ierr = PetscMalloc(size*sizeof(PetscMPIInt),&merge->len_s);CHKERRQ(ierr);
1629   len_s = merge->len_s;
1630   merge->nsend = 0;
1631 
1632   ierr = PetscMalloc((size+2)*sizeof(PetscInt),&owners_co);CHKERRQ(ierr);
1633   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1634 
1635   proc = 0;
1636   for (i=0; i<pon; i++){
1637     while (prmap[i] >= owners[proc+1]) proc++;
1638     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1639     len_s[proc] += coi[i+1] - coi[i];
1640   }
1641 
1642   len   = 0;  /* max length of buf_si[] */
1643   owners_co[0] = 0;
1644   for (proc=0; proc<size; proc++){
1645     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1646     if (len_si[proc]){
1647       merge->nsend++;
1648       len_si[proc] = 2*(len_si[proc] + 1);
1649       len += len_si[proc];
1650     }
1651   }
1652 
1653   /* determine the number and length of messages to receive for coi and coj  */
1654   ierr = PetscGatherNumberOfMessages(comm,PETSC_NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1655   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1656 
1657   /* post the Irecv and Isend of coj */
1658   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1659   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1660   ierr = PetscMalloc((merge->nsend+1)*sizeof(MPI_Request),&swaits);CHKERRQ(ierr);
1661   for (proc=0, k=0; proc<size; proc++){
1662     if (!len_s[proc]) continue;
1663     i = owners_co[proc];
1664     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1665     k++;
1666   }
1667 
1668   /* receives and sends of coj are complete */
1669   ierr = PetscMalloc(size*sizeof(MPI_Status),&sstatus);CHKERRQ(ierr);
1670   for (i=0; i<merge->nrecv; i++){
1671     PetscMPIInt icompleted;
1672     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1673   }
1674   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1675   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1676 
1677   /* send and recv coi */
1678   /*-------------------*/
1679   ierr = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1680   ierr = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1681   ierr = PetscMalloc((len+1)*sizeof(PetscInt),&buf_s);CHKERRQ(ierr);
1682   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1683   for (proc=0,k=0; proc<size; proc++){
1684     if (!len_s[proc]) continue;
1685     /* form outgoing message for i-structure:
1686          buf_si[0]:                 nrows to be sent
1687                [1:nrows]:           row index (global)
1688                [nrows+1:2*nrows+1]: i-structure index
1689     */
1690     /*-------------------------------------------*/
1691     nrows = len_si[proc]/2 - 1;
1692     buf_si_i    = buf_si + nrows+1;
1693     buf_si[0]   = nrows;
1694     buf_si_i[0] = 0;
1695     nrows = 0;
1696     for (i=owners_co[proc]; i<owners_co[proc+1]; i++){
1697       nzi = coi[i+1] - coi[i];
1698       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi; /* i-structure */
1699       buf_si[nrows+1] =prmap[i] -owners[proc]; /* local row index */
1700       nrows++;
1701     }
1702     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1703     k++;
1704     buf_si += len_si[proc];
1705   }
1706   i = merge->nrecv;
1707   while (i--) {
1708     PetscMPIInt icompleted;
1709     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1710   }
1711   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1712   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1713   ierr = PetscFree(len_si);CHKERRQ(ierr);
1714   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1715   ierr = PetscFree(swaits);CHKERRQ(ierr);
1716   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1717   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1718 
1719   /* compute the local portion of C (mpi mat) */
1720   /*------------------------------------------*/
1721   /* allocate bi array and free space for accumulating nonzero column info */
1722   ierr = PetscMalloc((pn+1)*sizeof(PetscInt),&bi);CHKERRQ(ierr);
1723   bi[0] = 0;
1724 
1725   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
1726   nnz           = fill*(pdti[pn] + poti[pon] + ai[am]);
1727   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1728   current_space = free_space;
1729 
1730   ierr = PetscMalloc3(merge->nrecv,PetscInt**,&buf_ri_k,merge->nrecv,PetscInt*,&nextrow,merge->nrecv,PetscInt*,&nextci);CHKERRQ(ierr);
1731   for (k=0; k<merge->nrecv; k++){
1732     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1733     nrows       = *buf_ri_k[k];
1734     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1735     nextci[k]   = buf_ri_k[k] + (nrows + 1);/* poins to the next i-structure of k-th recved i-structure  */
1736   }
1737 
1738   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1739   rmax = 0;
1740   for (i=0; i<pn; i++) {
1741     /* add pdt[i,:]*AP into lnk */
1742     pnz = pdti[i+1] - pdti[i];
1743     ptJ = pdtj + pdti[i];
1744     for (j=0; j<pnz; j++){
1745       row  = ptJ[j];  /* row of AP == col of Pt */
1746       anz  = ai[row+1] - ai[row];
1747       Jptr = aj + ai[row];
1748       /* add non-zero cols of AP into the sorted linked list lnk */
1749       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1750     }
1751 
1752     /* add received col data into lnk */
1753     for (k=0; k<merge->nrecv; k++){ /* k-th received message */
1754       if (i == *nextrow[k]) { /* i-th row */
1755         nzi = *(nextci[k]+1) - *nextci[k];
1756         Jptr  = buf_rj[k] + *nextci[k];
1757         ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
1758         nextrow[k]++; nextci[k]++;
1759       }
1760     }
1761     nnz = lnk[0];
1762 
1763     /* if free space is not available, make more free space */
1764     if (current_space->local_remaining<nnz) {
1765       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1766       nspacedouble++;
1767     }
1768     /* copy data into free space, then initialize lnk */
1769     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1770     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1771     current_space->array           += nnz;
1772     current_space->local_used      += nnz;
1773     current_space->local_remaining -= nnz;
1774     bi[i+1] = bi[i] + nnz;
1775     if (nnz > rmax) rmax = nnz;
1776   }
1777   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1778 
1779   ierr = PetscMalloc((bi[pn]+1)*sizeof(PetscInt),&bj);CHKERRQ(ierr);
1780   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1781   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
1782   if (afill_tmp > afill) afill = afill_tmp;
1783   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
1784   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1785   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1786 
1787   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
1788   /*----------------------------------------------------------------------------------*/
1789   ierr = PetscMalloc((rmax+1)*sizeof(PetscScalar),&vals);CHKERRQ(ierr);
1790   ierr = PetscMemzero(vals,rmax*sizeof(PetscScalar));CHKERRQ(ierr);
1791 
1792   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1793   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1794   ierr = MatSetBlockSizes(Cmpi,P->cmap->bs,A->cmap->bs);CHKERRQ(ierr);
1795   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1796   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1797   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1798   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
1799   for (i=0; i<pn; i++){
1800     row = i + rstart;
1801     nnz = bi[i+1] - bi[i];
1802     Jptr = bj + bi[i];
1803     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
1804   }
1805   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1806   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1807   ierr = PetscFree(vals);CHKERRQ(ierr);
1808 
1809   merge->bi            = bi;
1810   merge->bj            = bj;
1811   merge->coi           = coi;
1812   merge->coj           = coj;
1813   merge->buf_ri        = buf_ri;
1814   merge->buf_rj        = buf_rj;
1815   merge->owners_co     = owners_co;
1816   merge->destroy       = Cmpi->ops->destroy;
1817   merge->duplicate     = Cmpi->ops->duplicate;
1818 
1819   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_Scalable;
1820   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
1821 
1822   /* attach the supporting struct to Cmpi for reuse */
1823   c = (Mat_MPIAIJ*)Cmpi->data;
1824   c->ptap        = ptap;
1825   ptap->api      = PETSC_NULL;
1826   ptap->apj      = PETSC_NULL;
1827   ptap->merge    = merge;
1828   ptap->rmax     = rmax;
1829   ptap->apa      = PETSC_NULL;
1830 
1831   *C = Cmpi;
1832 #if defined(PETSC_USE_INFO)
1833   if (bi[pn] != 0) {
1834     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %G needed %G.\n",nspacedouble,fill,afill);CHKERRQ(ierr);
1835     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%G,&C) for best performance.\n",afill);CHKERRQ(ierr);
1836   } else {
1837     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1838   }
1839 #endif
1840   PetscFunctionReturn(0);
1841 }
1842