xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision bebe2cf65d55febe21a5af8db2bd2e168caaa2e7)
1 
2 /*
3   Defines matrix-matrix product routines for pairs of MPIAIJ matrices
4           C = A * B
5 */
6 #include <../src/mat/impls/aij/seq/aij.h> /*I "petscmat.h" I*/
7 #include <../src/mat/utils/freespace.h>
8 #include <../src/mat/impls/aij/mpi/mpiaij.h>
9 #include <petscbt.h>
10 #include <../src/mat/impls/dense/mpi/mpidense.h>
11 #include <petsc/private/vecimpl.h>
12 
13 #undef __FUNCT__
14 #define __FUNCT__ "MatMatMult_MPIAIJ_MPIAIJ"
15 PetscErrorCode MatMatMult_MPIAIJ_MPIAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill, Mat *C)
16 {
17   PetscErrorCode ierr;
18   const char     *algTypes[2] = {"scalable","nonscalable"};
19   PetscInt       alg=0; /* set default algorithm */
20 
21   PetscFunctionBegin;
22   if (scall == MAT_INITIAL_MATRIX) {
23     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
24     ierr = PetscOptionsEList("-matmatmult_via","Algorithmic approach","MatMatMult",algTypes,2,algTypes[0],&alg,NULL);CHKERRQ(ierr);
25     ierr = PetscOptionsEnd();CHKERRQ(ierr);
26 
27     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
28     switch (alg) {
29     case 1:
30       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(A,B,fill,C);CHKERRQ(ierr);
31       break;
32     default:
33       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ(A,B,fill,C);CHKERRQ(ierr);
34       break;
35     }
36     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
37   }
38   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
39   ierr = (*(*C)->ops->matmultnumeric)(A,B,*C);CHKERRQ(ierr);
40   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
41   PetscFunctionReturn(0);
42 }
43 
44 #undef __FUNCT__
45 #define __FUNCT__ "MatDestroy_MPIAIJ_MatMatMult"
46 PetscErrorCode MatDestroy_MPIAIJ_MatMatMult(Mat A)
47 {
48   PetscErrorCode ierr;
49   Mat_MPIAIJ     *a    = (Mat_MPIAIJ*)A->data;
50   Mat_PtAPMPI    *ptap = a->ptap;
51 
52   PetscFunctionBegin;
53   ierr = PetscFree2(ptap->startsj_s,ptap->startsj_r);CHKERRQ(ierr);
54   ierr = PetscFree(ptap->bufa);CHKERRQ(ierr);
55   ierr = MatDestroy(&ptap->P_loc);CHKERRQ(ierr);
56   ierr = MatDestroy(&ptap->P_oth);CHKERRQ(ierr);
57   ierr = MatDestroy(&ptap->Pt);CHKERRQ(ierr);
58   ierr = PetscFree(ptap->api);CHKERRQ(ierr);
59   ierr = PetscFree(ptap->apj);CHKERRQ(ierr);
60   ierr = PetscFree(ptap->apa);CHKERRQ(ierr);
61   ierr = ptap->destroy(A);CHKERRQ(ierr);
62   ierr = PetscFree(ptap);CHKERRQ(ierr);
63   PetscFunctionReturn(0);
64 }
65 
66 #undef __FUNCT__
67 #define __FUNCT__ "MatDuplicate_MPIAIJ_MatMatMult"
68 PetscErrorCode MatDuplicate_MPIAIJ_MatMatMult(Mat A, MatDuplicateOption op, Mat *M)
69 {
70   PetscErrorCode ierr;
71   Mat_MPIAIJ     *a    = (Mat_MPIAIJ*)A->data;
72   Mat_PtAPMPI    *ptap = a->ptap;
73 
74   PetscFunctionBegin;
75   ierr = (*ptap->duplicate)(A,op,M);CHKERRQ(ierr);
76 
77   (*M)->ops->destroy   = ptap->destroy;   /* = MatDestroy_MPIAIJ, *M doesn't duplicate A's special structure! */
78   (*M)->ops->duplicate = ptap->duplicate; /* = MatDuplicate_MPIAIJ */
79   PetscFunctionReturn(0);
80 }
81 
82 #undef __FUNCT__
83 #define __FUNCT__ "MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable"
84 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,Mat C)
85 {
86   PetscErrorCode ierr;
87   Mat_MPIAIJ     *a  =(Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
88   Mat_SeqAIJ     *ad =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
89   Mat_SeqAIJ     *cd =(Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
90   PetscInt       *adi=ad->i,*adj,*aoi=ao->i,*aoj;
91   PetscScalar    *ada,*aoa,*cda=cd->a,*coa=co->a;
92   Mat_SeqAIJ     *p_loc,*p_oth;
93   PetscInt       *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
94   PetscScalar    *pa_loc,*pa_oth,*pa,*apa,valtmp,*ca;
95   PetscInt       cm   =C->rmap->n,anz,pnz;
96   Mat_PtAPMPI    *ptap=c->ptap;
97   PetscInt       *api,*apj,*apJ,i,j,k,row;
98   PetscInt       cstart=C->cmap->rstart;
99   PetscInt       cdnz,conz,k0,k1;
100   MPI_Comm       comm;
101   PetscMPIInt    size;
102 
103   PetscFunctionBegin;
104   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
105   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
106 
107   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
108   /*-----------------------------------------------------*/
109   /* update numerical values of P_oth and P_loc */
110   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
111   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
112 
113   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
114   /*----------------------------------------------------------*/
115   /* get data from symbolic products */
116   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
117   pi_loc=p_loc->i; pj_loc=p_loc->j; pa_loc=p_loc->a;
118   if (size >1) {
119     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
120     pi_oth=p_oth->i; pj_oth=p_oth->j; pa_oth=p_oth->a;
121   } else {
122     pi_oth=NULL; pj_oth=NULL; pa_oth=NULL;
123   }
124 
125   /* get apa for storing dense row A[i,:]*P */
126   apa = ptap->apa;
127 
128   api = ptap->api;
129   apj = ptap->apj;
130   for (i=0; i<cm; i++) {
131     /* diagonal portion of A */
132     anz = adi[i+1] - adi[i];
133     adj = ad->j + adi[i];
134     ada = ad->a + adi[i];
135     for (j=0; j<anz; j++) {
136       row = adj[j];
137       pnz = pi_loc[row+1] - pi_loc[row];
138       pj  = pj_loc + pi_loc[row];
139       pa  = pa_loc + pi_loc[row];
140 
141       /* perform dense axpy */
142       valtmp = ada[j];
143       for (k=0; k<pnz; k++) {
144         apa[pj[k]] += valtmp*pa[k];
145       }
146       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
147     }
148 
149     /* off-diagonal portion of A */
150     anz = aoi[i+1] - aoi[i];
151     aoj = ao->j + aoi[i];
152     aoa = ao->a + aoi[i];
153     for (j=0; j<anz; j++) {
154       row = aoj[j];
155       pnz = pi_oth[row+1] - pi_oth[row];
156       pj  = pj_oth + pi_oth[row];
157       pa  = pa_oth + pi_oth[row];
158 
159       /* perform dense axpy */
160       valtmp = aoa[j];
161       for (k=0; k<pnz; k++) {
162         apa[pj[k]] += valtmp*pa[k];
163       }
164       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
165     }
166 
167     /* set values in C */
168     apJ  = apj + api[i];
169     cdnz = cd->i[i+1] - cd->i[i];
170     conz = co->i[i+1] - co->i[i];
171 
172     /* 1st off-diagoanl part of C */
173     ca = coa + co->i[i];
174     k  = 0;
175     for (k0=0; k0<conz; k0++) {
176       if (apJ[k] >= cstart) break;
177       ca[k0]      = apa[apJ[k]];
178       apa[apJ[k]] = 0.0;
179       k++;
180     }
181 
182     /* diagonal part of C */
183     ca = cda + cd->i[i];
184     for (k1=0; k1<cdnz; k1++) {
185       ca[k1]      = apa[apJ[k]];
186       apa[apJ[k]] = 0.0;
187       k++;
188     }
189 
190     /* 2nd off-diagoanl part of C */
191     ca = coa + co->i[i];
192     for (; k0<conz; k0++) {
193       ca[k0]      = apa[apJ[k]];
194       apa[apJ[k]] = 0.0;
195       k++;
196     }
197   }
198   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
199   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
200   PetscFunctionReturn(0);
201 }
202 
203 #undef __FUNCT__
204 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable"
205 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,PetscReal fill,Mat *C)
206 {
207   PetscErrorCode     ierr;
208   MPI_Comm           comm;
209   PetscMPIInt        size;
210   Mat                Cmpi;
211   Mat_PtAPMPI        *ptap;
212   PetscFreeSpaceList free_space=NULL,current_space=NULL;
213   Mat_MPIAIJ         *a        =(Mat_MPIAIJ*)A->data,*c;
214   Mat_SeqAIJ         *ad       =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
215   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
216   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
217   PetscInt           *lnk,i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi;
218   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
219   PetscBT            lnkbt;
220   PetscScalar        *apa;
221   PetscReal          afill;
222   PetscInt           nlnk_max,armax,prmax;
223 
224   PetscFunctionBegin;
225   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
226   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
227 
228   if (A->cmap->rstart != P->rmap->rstart || A->cmap->rend != P->rmap->rend) {
229     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);
230   }
231 
232   /* create struct Mat_PtAPMPI and attached it to C later */
233   ierr = PetscNew(&ptap);CHKERRQ(ierr);
234 
235   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
236   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
237 
238   /* get P_loc by taking all local rows of P */
239   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
240 
241   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
242   pi_loc = p_loc->i; pj_loc = p_loc->j;
243   if (size > 1) {
244     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
245     pi_oth = p_oth->i; pj_oth = p_oth->j;
246   } else {
247     p_oth = NULL;
248     pi_oth = NULL; pj_oth = NULL;
249   }
250 
251   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
252   /*-------------------------------------------------------------------*/
253   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
254   ptap->api = api;
255   api[0]    = 0;
256 
257   /* create and initialize a linked list */
258   armax    = ad->rmax+ao->rmax;
259   if (size >1) {
260     prmax    = PetscMax(p_loc->rmax,p_oth->rmax);
261   } else {
262     prmax = p_loc->rmax;
263   }
264   nlnk_max = armax*prmax;
265   if (!nlnk_max || nlnk_max > pN) nlnk_max = pN;
266   ierr = PetscLLCondensedCreate(nlnk_max,pN,&lnk,&lnkbt);CHKERRQ(ierr);
267 
268   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
269   ierr = PetscFreeSpaceGet((PetscInt)(fill*(adi[am]+aoi[am]+pi_loc[pm])),&free_space);CHKERRQ(ierr);
270 
271   current_space = free_space;
272 
273   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
274   for (i=0; i<am; i++) {
275     /* diagonal portion of A */
276     nzi = adi[i+1] - adi[i];
277     for (j=0; j<nzi; j++) {
278       row  = *adj++;
279       pnz  = pi_loc[row+1] - pi_loc[row];
280       Jptr = pj_loc + pi_loc[row];
281       /* add non-zero cols of P into the sorted linked list lnk */
282       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
283     }
284     /* off-diagonal portion of A */
285     nzi = aoi[i+1] - aoi[i];
286     for (j=0; j<nzi; j++) {
287       row  = *aoj++;
288       pnz  = pi_oth[row+1] - pi_oth[row];
289       Jptr = pj_oth + pi_oth[row];
290       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
291     }
292 
293     apnz     = lnk[0];
294     api[i+1] = api[i] + apnz;
295 
296     /* if free space is not available, double the total space in the list */
297     if (current_space->local_remaining<apnz) {
298       ierr = PetscFreeSpaceGet(apnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
299       nspacedouble++;
300     }
301 
302     /* Copy data into free space, then initialize lnk */
303     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
304     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
305 
306     current_space->array           += apnz;
307     current_space->local_used      += apnz;
308     current_space->local_remaining -= apnz;
309   }
310 
311   /* Allocate space for apj, initialize apj, and */
312   /* destroy list of free space and other temporary array(s) */
313   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
314   apj  = ptap->apj;
315   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
316   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
317 
318   /* malloc apa to store dense row A[i,:]*P */
319   ierr = PetscCalloc1(pN,&apa);CHKERRQ(ierr);
320 
321   ptap->apa = apa;
322 
323   /* create and assemble symbolic parallel matrix Cmpi */
324   /*----------------------------------------------------*/
325   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
326   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
327   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
328 
329   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
330   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
331   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
332   for (i=0; i<am; i++) {
333     row  = i + rstart;
334     apnz = api[i+1] - api[i];
335     ierr = MatSetValues(Cmpi,1,&row,apnz,apj,apa,INSERT_VALUES);CHKERRQ(ierr);
336     apj += apnz;
337   }
338   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
339   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
340 
341   ptap->destroy        = Cmpi->ops->destroy;
342   ptap->duplicate      = Cmpi->ops->duplicate;
343   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
344   Cmpi->ops->destroy   = MatDestroy_MPIAIJ_MatMatMult;
345   Cmpi->ops->duplicate = MatDuplicate_MPIAIJ_MatMatMult;
346 
347   /* attach the supporting struct to Cmpi for reuse */
348   c       = (Mat_MPIAIJ*)Cmpi->data;
349   c->ptap = ptap;
350 
351   *C = Cmpi;
352 
353   /* set MatInfo */
354   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
355   if (afill < 1.0) afill = 1.0;
356   Cmpi->info.mallocs           = nspacedouble;
357   Cmpi->info.fill_ratio_given  = fill;
358   Cmpi->info.fill_ratio_needed = afill;
359 
360 #if defined(PETSC_USE_INFO)
361   if (api[am]) {
362     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
363     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
364   } else {
365     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
366   }
367 #endif
368   PetscFunctionReturn(0);
369 }
370 
371 #undef __FUNCT__
372 #define __FUNCT__ "MatMatMult_MPIAIJ_MPIDense"
373 PetscErrorCode MatMatMult_MPIAIJ_MPIDense(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
374 {
375   PetscErrorCode ierr;
376 
377   PetscFunctionBegin;
378   if (scall == MAT_INITIAL_MATRIX) {
379     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
380     ierr = MatMatMultSymbolic_MPIAIJ_MPIDense(A,B,fill,C);CHKERRQ(ierr);
381     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
382   }
383   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
384   ierr = MatMatMultNumeric_MPIAIJ_MPIDense(A,B,*C);CHKERRQ(ierr);
385   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
386   PetscFunctionReturn(0);
387 }
388 
389 typedef struct {
390   Mat         workB;
391   PetscScalar *rvalues,*svalues;
392   MPI_Request *rwaits,*swaits;
393 } MPIAIJ_MPIDense;
394 
395 #undef __FUNCT__
396 #define __FUNCT__ "MatMPIAIJ_MPIDenseDestroy"
397 PetscErrorCode MatMPIAIJ_MPIDenseDestroy(void *ctx)
398 {
399   MPIAIJ_MPIDense *contents = (MPIAIJ_MPIDense*) ctx;
400   PetscErrorCode  ierr;
401 
402   PetscFunctionBegin;
403   ierr = MatDestroy(&contents->workB);CHKERRQ(ierr);
404   ierr = PetscFree4(contents->rvalues,contents->svalues,contents->rwaits,contents->swaits);CHKERRQ(ierr);
405   ierr = PetscFree(contents);CHKERRQ(ierr);
406   PetscFunctionReturn(0);
407 }
408 
409 #undef __FUNCT__
410 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIDense"
411 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat A,Mat B,PetscReal fill,Mat *C)
412 {
413   PetscErrorCode         ierr;
414   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*) A->data;
415   PetscInt               nz   = aij->B->cmap->n;
416   PetscContainer         container;
417   MPIAIJ_MPIDense        *contents;
418   VecScatter             ctx   = aij->Mvctx;
419   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
420   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
421   PetscInt               m     = A->rmap->n,n=B->cmap->n;
422 
423   PetscFunctionBegin;
424   ierr = MatCreate(PetscObjectComm((PetscObject)B),C);CHKERRQ(ierr);
425   ierr = MatSetSizes(*C,m,n,A->rmap->N,B->cmap->N);CHKERRQ(ierr);
426   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
427   ierr = MatSetType(*C,MATMPIDENSE);CHKERRQ(ierr);
428   ierr = MatMPIDenseSetPreallocation(*C,NULL);CHKERRQ(ierr);
429   ierr = MatAssemblyBegin(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
430   ierr = MatAssemblyEnd(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
431 
432   (*C)->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
433 
434   ierr = PetscNew(&contents);CHKERRQ(ierr);
435   /* Create work matrix used to store off processor rows of B needed for local product */
436   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,B->cmap->N,NULL,&contents->workB);CHKERRQ(ierr);
437   /* Create work arrays needed */
438   ierr = PetscMalloc4(B->cmap->N*from->starts[from->n],&contents->rvalues,
439                       B->cmap->N*to->starts[to->n],&contents->svalues,
440                       from->n,&contents->rwaits,
441                       to->n,&contents->swaits);CHKERRQ(ierr);
442 
443   ierr = PetscContainerCreate(PetscObjectComm((PetscObject)A),&container);CHKERRQ(ierr);
444   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
445   ierr = PetscContainerSetUserDestroy(container,MatMPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
446   ierr = PetscObjectCompose((PetscObject)(*C),"workB",(PetscObject)container);CHKERRQ(ierr);
447   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
448   PetscFunctionReturn(0);
449 }
450 
451 #undef __FUNCT__
452 #define __FUNCT__ "MatMPIDenseScatter"
453 /*
454     Performs an efficient scatter on the rows of B needed by this process; this is
455     a modification of the VecScatterBegin_() routines.
456 */
457 PetscErrorCode MatMPIDenseScatter(Mat A,Mat B,Mat C,Mat *outworkB)
458 {
459   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*)A->data;
460   PetscErrorCode         ierr;
461   PetscScalar            *b,*w,*svalues,*rvalues;
462   VecScatter             ctx   = aij->Mvctx;
463   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
464   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
465   PetscInt               i,j,k;
466   PetscInt               *sindices,*sstarts,*rindices,*rstarts;
467   PetscMPIInt            *sprocs,*rprocs,nrecvs;
468   MPI_Request            *swaits,*rwaits;
469   MPI_Comm               comm;
470   PetscMPIInt            tag  = ((PetscObject)ctx)->tag,ncols = B->cmap->N, nrows = aij->B->cmap->n,imdex,nrowsB = B->rmap->n;
471   MPI_Status             status;
472   MPIAIJ_MPIDense        *contents;
473   PetscContainer         container;
474   Mat                    workB;
475 
476   PetscFunctionBegin;
477   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
478   ierr = PetscObjectQuery((PetscObject)C,"workB",(PetscObject*)&container);CHKERRQ(ierr);
479   if (!container) SETERRQ(comm,PETSC_ERR_PLIB,"Container does not exist");
480   ierr = PetscContainerGetPointer(container,(void**)&contents);CHKERRQ(ierr);
481 
482   workB = *outworkB = contents->workB;
483   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);
484   sindices = to->indices;
485   sstarts  = to->starts;
486   sprocs   = to->procs;
487   swaits   = contents->swaits;
488   svalues  = contents->svalues;
489 
490   rindices = from->indices;
491   rstarts  = from->starts;
492   rprocs   = from->procs;
493   rwaits   = contents->rwaits;
494   rvalues  = contents->rvalues;
495 
496   ierr = MatDenseGetArray(B,&b);CHKERRQ(ierr);
497   ierr = MatDenseGetArray(workB,&w);CHKERRQ(ierr);
498 
499   for (i=0; i<from->n; i++) {
500     ierr = MPI_Irecv(rvalues+ncols*rstarts[i],ncols*(rstarts[i+1]-rstarts[i]),MPIU_SCALAR,rprocs[i],tag,comm,rwaits+i);CHKERRQ(ierr);
501   }
502 
503   for (i=0; i<to->n; i++) {
504     /* pack a message at a time */
505     for (j=0; j<sstarts[i+1]-sstarts[i]; j++) {
506       for (k=0; k<ncols; k++) {
507         svalues[ncols*(sstarts[i] + j) + k] = b[sindices[sstarts[i]+j] + nrowsB*k];
508       }
509     }
510     ierr = MPI_Isend(svalues+ncols*sstarts[i],ncols*(sstarts[i+1]-sstarts[i]),MPIU_SCALAR,sprocs[i],tag,comm,swaits+i);CHKERRQ(ierr);
511   }
512 
513   nrecvs = from->n;
514   while (nrecvs) {
515     ierr = MPI_Waitany(from->n,rwaits,&imdex,&status);CHKERRQ(ierr);
516     nrecvs--;
517     /* unpack a message at a time */
518     for (j=0; j<rstarts[imdex+1]-rstarts[imdex]; j++) {
519       for (k=0; k<ncols; k++) {
520         w[rindices[rstarts[imdex]+j] + nrows*k] = rvalues[ncols*(rstarts[imdex] + j) + k];
521       }
522     }
523   }
524   if (to->n) {ierr = MPI_Waitall(to->n,swaits,to->sstatus);CHKERRQ(ierr);}
525 
526   ierr = MatDenseRestoreArray(B,&b);CHKERRQ(ierr);
527   ierr = MatDenseRestoreArray(workB,&w);CHKERRQ(ierr);
528   ierr = MatAssemblyBegin(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
529   ierr = MatAssemblyEnd(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
530   PetscFunctionReturn(0);
531 }
532 extern PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat,Mat,Mat);
533 
534 #undef __FUNCT__
535 #define __FUNCT__ "MatMatMultNumeric_MPIAIJ_MPIDense"
536 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat A,Mat B,Mat C)
537 {
538   PetscErrorCode ierr;
539   Mat_MPIAIJ     *aij    = (Mat_MPIAIJ*)A->data;
540   Mat_MPIDense   *bdense = (Mat_MPIDense*)B->data;
541   Mat_MPIDense   *cdense = (Mat_MPIDense*)C->data;
542   Mat            workB;
543 
544   PetscFunctionBegin;
545   /* diagonal block of A times all local rows of B*/
546   ierr = MatMatMultNumeric_SeqAIJ_SeqDense(aij->A,bdense->A,cdense->A);CHKERRQ(ierr);
547 
548   /* get off processor parts of B needed to complete the product */
549   ierr = MatMPIDenseScatter(A,B,C,&workB);CHKERRQ(ierr);
550 
551   /* off-diagonal block of A times nonlocal rows of B */
552   ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A);CHKERRQ(ierr);
553   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
554   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
555   PetscFunctionReturn(0);
556 }
557 
558 #undef __FUNCT__
559 #define __FUNCT__ "MatMatMultNumeric_MPIAIJ_MPIAIJ"
560 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
561 {
562   PetscErrorCode ierr;
563   Mat_MPIAIJ     *a   = (Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
564   Mat_SeqAIJ     *ad  = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
565   Mat_SeqAIJ     *cd  = (Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
566   PetscInt       *adi = ad->i,*adj,*aoi=ao->i,*aoj;
567   PetscScalar    *ada,*aoa,*cda=cd->a,*coa=co->a;
568   Mat_SeqAIJ     *p_loc,*p_oth;
569   PetscInt       *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
570   PetscScalar    *pa_loc,*pa_oth,*pa,valtmp,*ca;
571   PetscInt       cm          = C->rmap->n,anz,pnz;
572   Mat_PtAPMPI    *ptap       = c->ptap;
573   PetscScalar    *apa_sparse = ptap->apa;
574   PetscInt       *api,*apj,*apJ,i,j,k,row;
575   PetscInt       cstart = C->cmap->rstart;
576   PetscInt       cdnz,conz,k0,k1,nextp;
577   MPI_Comm       comm;
578   PetscMPIInt    size;
579 
580   PetscFunctionBegin;
581   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
582   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
583 
584   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
585   /*-----------------------------------------------------*/
586   /* update numerical values of P_oth and P_loc */
587   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
588   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
589 
590   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
591   /*----------------------------------------------------------*/
592   /* get data from symbolic products */
593   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
594   pi_loc = p_loc->i; pj_loc = p_loc->j; pa_loc = p_loc->a;
595   if (size >1) {
596     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
597     pi_oth = p_oth->i; pj_oth = p_oth->j; pa_oth = p_oth->a;
598   } else {
599     p_oth = NULL; pi_oth = NULL; pj_oth = NULL; pa_oth = NULL;
600   }
601 
602   api = ptap->api;
603   apj = ptap->apj;
604   for (i=0; i<cm; i++) {
605     apJ = apj + api[i];
606 
607     /* diagonal portion of A */
608     anz = adi[i+1] - adi[i];
609     adj = ad->j + adi[i];
610     ada = ad->a + adi[i];
611     for (j=0; j<anz; j++) {
612       row = adj[j];
613       pnz = pi_loc[row+1] - pi_loc[row];
614       pj  = pj_loc + pi_loc[row];
615       pa  = pa_loc + pi_loc[row];
616       /* perform sparse axpy */
617       valtmp = ada[j];
618       nextp  = 0;
619       for (k=0; nextp<pnz; k++) {
620         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
621           apa_sparse[k] += valtmp*pa[nextp++];
622         }
623       }
624       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
625     }
626 
627     /* off-diagonal portion of A */
628     anz = aoi[i+1] - aoi[i];
629     aoj = ao->j + aoi[i];
630     aoa = ao->a + aoi[i];
631     for (j=0; j<anz; j++) {
632       row = aoj[j];
633       pnz = pi_oth[row+1] - pi_oth[row];
634       pj  = pj_oth + pi_oth[row];
635       pa  = pa_oth + pi_oth[row];
636       /* perform sparse axpy */
637       valtmp = aoa[j];
638       nextp  = 0;
639       for (k=0; nextp<pnz; k++) {
640         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
641           apa_sparse[k] += valtmp*pa[nextp++];
642         }
643       }
644       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
645     }
646 
647     /* set values in C */
648     cdnz = cd->i[i+1] - cd->i[i];
649     conz = co->i[i+1] - co->i[i];
650 
651     /* 1st off-diagoanl part of C */
652     ca = coa + co->i[i];
653     k  = 0;
654     for (k0=0; k0<conz; k0++) {
655       if (apJ[k] >= cstart) break;
656       ca[k0]        = apa_sparse[k];
657       apa_sparse[k] = 0.0;
658       k++;
659     }
660 
661     /* diagonal part of C */
662     ca = cda + cd->i[i];
663     for (k1=0; k1<cdnz; k1++) {
664       ca[k1]        = apa_sparse[k];
665       apa_sparse[k] = 0.0;
666       k++;
667     }
668 
669     /* 2nd off-diagoanl part of C */
670     ca = coa + co->i[i];
671     for (; k0<conz; k0++) {
672       ca[k0]        = apa_sparse[k];
673       apa_sparse[k] = 0.0;
674       k++;
675     }
676   }
677   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
678   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
679   PetscFunctionReturn(0);
680 }
681 
682 /* same as MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(), except using LLCondensed to avoid O(BN) memory requirement */
683 #undef __FUNCT__
684 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIAIJ"
685 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat *C)
686 {
687   PetscErrorCode     ierr;
688   MPI_Comm           comm;
689   PetscMPIInt        size;
690   Mat                Cmpi;
691   Mat_PtAPMPI        *ptap;
692   PetscFreeSpaceList free_space = NULL,current_space=NULL;
693   Mat_MPIAIJ         *a         = (Mat_MPIAIJ*)A->data,*c;
694   Mat_SeqAIJ         *ad        = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
695   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
696   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
697   PetscInt           i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi,*lnk,apnz_max=0;
698   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
699   PetscInt           nlnk_max,armax,prmax;
700   PetscReal          afill;
701   PetscScalar        *apa;
702 
703   PetscFunctionBegin;
704   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
705   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
706 
707   /* create struct Mat_PtAPMPI and attached it to C later */
708   ierr = PetscNew(&ptap);CHKERRQ(ierr);
709 
710   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
711   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
712 
713   /* get P_loc by taking all local rows of P */
714   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
715 
716   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
717   pi_loc = p_loc->i; pj_loc = p_loc->j;
718   if (size > 1) {
719     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
720     pi_oth = p_oth->i; pj_oth = p_oth->j;
721   } else {
722     p_oth  = NULL;
723     pi_oth = NULL; pj_oth = NULL;
724   }
725 
726   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
727   /*-------------------------------------------------------------------*/
728   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
729   ptap->api = api;
730   api[0]    = 0;
731 
732   /* create and initialize a linked list */
733   armax = ad->rmax+ao->rmax;
734   if (size >1) {
735     prmax = PetscMax(p_loc->rmax,p_oth->rmax);
736   } else {
737     prmax = p_loc->rmax;
738   }
739   nlnk_max = armax*prmax;
740   if (!nlnk_max || nlnk_max > pN) nlnk_max = pN;
741   ierr = PetscLLCondensedCreate_Scalable(nlnk_max,&lnk);CHKERRQ(ierr);
742 
743   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
744   ierr = PetscFreeSpaceGet((PetscInt)(fill*(adi[am]+aoi[am]+pi_loc[pm])),&free_space);CHKERRQ(ierr);
745 
746   current_space = free_space;
747 
748   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
749   for (i=0; i<am; i++) {
750     /* diagonal portion of A */
751     nzi = adi[i+1] - adi[i];
752     for (j=0; j<nzi; j++) {
753       row  = *adj++;
754       pnz  = pi_loc[row+1] - pi_loc[row];
755       Jptr = pj_loc + pi_loc[row];
756       /* add non-zero cols of P into the sorted linked list lnk */
757       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
758     }
759     /* off-diagonal portion of A */
760     nzi = aoi[i+1] - aoi[i];
761     for (j=0; j<nzi; j++) {
762       row  = *aoj++;
763       pnz  = pi_oth[row+1] - pi_oth[row];
764       Jptr = pj_oth + pi_oth[row];
765       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
766     }
767 
768     apnz     = *lnk;
769     api[i+1] = api[i] + apnz;
770     if (apnz > apnz_max) apnz_max = apnz;
771 
772     /* if free space is not available, double the total space in the list */
773     if (current_space->local_remaining<apnz) {
774       ierr = PetscFreeSpaceGet(apnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
775       nspacedouble++;
776     }
777 
778     /* Copy data into free space, then initialize lnk */
779     ierr = PetscLLCondensedClean_Scalable(apnz,current_space->array,lnk);CHKERRQ(ierr);
780     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
781 
782     current_space->array           += apnz;
783     current_space->local_used      += apnz;
784     current_space->local_remaining -= apnz;
785   }
786 
787   /* Allocate space for apj, initialize apj, and */
788   /* destroy list of free space and other temporary array(s) */
789   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
790   apj  = ptap->apj;
791   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
792   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
793 
794   /* create and assemble symbolic parallel matrix Cmpi */
795   /*----------------------------------------------------*/
796   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
797   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
798   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
799   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
800   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
801   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
802 
803   /* malloc apa for assembly Cmpi */
804   ierr = PetscCalloc1(apnz_max,&apa);CHKERRQ(ierr);
805 
806   ptap->apa = apa;
807   for (i=0; i<am; i++) {
808     row  = i + rstart;
809     apnz = api[i+1] - api[i];
810     ierr = MatSetValues(Cmpi,1,&row,apnz,apj,apa,INSERT_VALUES);CHKERRQ(ierr);
811     apj += apnz;
812   }
813   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
814   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
815 
816   ptap->destroy             = Cmpi->ops->destroy;
817   ptap->duplicate           = Cmpi->ops->duplicate;
818   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ;
819   Cmpi->ops->destroy        = MatDestroy_MPIAIJ_MatMatMult;
820   Cmpi->ops->duplicate      = MatDuplicate_MPIAIJ_MatMatMult;
821 
822   /* attach the supporting struct to Cmpi for reuse */
823   c       = (Mat_MPIAIJ*)Cmpi->data;
824   c->ptap = ptap;
825 
826   *C = Cmpi;
827 
828   /* set MatInfo */
829   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
830   if (afill < 1.0) afill = 1.0;
831   Cmpi->info.mallocs           = nspacedouble;
832   Cmpi->info.fill_ratio_given  = fill;
833   Cmpi->info.fill_ratio_needed = afill;
834 
835 #if defined(PETSC_USE_INFO)
836   if (api[am]) {
837     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
838     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
839   } else {
840     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
841   }
842 #endif
843   PetscFunctionReturn(0);
844 }
845 
846 /*-------------------------------------------------------------------------*/
847 #undef __FUNCT__
848 #define __FUNCT__ "MatTransposeMatMult_MPIAIJ_MPIAIJ"
849 PetscErrorCode MatTransposeMatMult_MPIAIJ_MPIAIJ(Mat P,Mat A,MatReuse scall,PetscReal fill,Mat *C)
850 {
851   PetscErrorCode ierr;
852   const char     *algTypes[3] = {"scalable","nonscalable","matmatmult"};
853   PetscInt       alg=0; /* set default algorithm */
854 
855   PetscFunctionBegin;
856   if (scall == MAT_INITIAL_MATRIX) {
857     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
858     ierr = PetscOptionsEList("-mattransposematmult_via","Algorithmic approach","MatTransposeMatMult",algTypes,3,algTypes[0],&alg,NULL);CHKERRQ(ierr);
859     ierr = PetscOptionsEnd();CHKERRQ(ierr);
860 
861     ierr = PetscLogEventBegin(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
862     switch (alg) {
863     case 1:
864       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(P,A,fill,C);CHKERRQ(ierr);
865       break;
866     case 2:
867     {
868       Mat         Pt;
869       Mat_PtAPMPI *ptap;
870       Mat_MPIAIJ  *c;
871       ierr = MatTranspose(P,MAT_INITIAL_MATRIX,&Pt);CHKERRQ(ierr);
872       ierr = MatMatMult(Pt,A,MAT_INITIAL_MATRIX,fill,C);CHKERRQ(ierr);
873       c        = (Mat_MPIAIJ*)(*C)->data;
874       ptap     = c->ptap;
875       ptap->Pt = Pt;
876       (*C)->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult;
877       PetscFunctionReturn(0);
878     }
879       break;
880     default:
881       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(P,A,fill,C);CHKERRQ(ierr);
882       break;
883     }
884     ierr = PetscLogEventEnd(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
885   }
886   ierr = PetscLogEventBegin(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
887   ierr = (*(*C)->ops->mattransposemultnumeric)(P,A,*C);CHKERRQ(ierr);
888   ierr = PetscLogEventEnd(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
889   PetscFunctionReturn(0);
890 }
891 
892 /* This routine only works when scall=MAT_REUSE_MATRIX! */
893 #undef __FUNCT__
894 #define __FUNCT__ "MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult"
895 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult(Mat P,Mat A,Mat C)
896 {
897   PetscErrorCode ierr;
898   Mat_MPIAIJ     *c=(Mat_MPIAIJ*)C->data;
899   Mat_PtAPMPI    *ptap= c->ptap;
900   Mat            Pt=ptap->Pt;
901 
902   PetscFunctionBegin;
903   ierr = MatTranspose(P,MAT_REUSE_MATRIX,&Pt);CHKERRQ(ierr);
904   ierr = MatMatMultNumeric(Pt,A,C);CHKERRQ(ierr);
905   PetscFunctionReturn(0);
906 }
907 
908 /* Non-scalable version, use dense axpy */
909 #undef __FUNCT__
910 #define __FUNCT__ "MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable"
911 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,Mat C)
912 {
913   PetscErrorCode      ierr;
914   Mat_Merge_SeqsToMPI *merge;
915   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
916   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
917   Mat_PtAPMPI         *ptap;
918   PetscInt            *adj,*aJ;
919   PetscInt            i,j,k,anz,pnz,row,*cj;
920   MatScalar           *ada,*aval,*ca,valtmp;
921   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
922   MPI_Comm            comm;
923   PetscMPIInt         size,rank,taga,*len_s;
924   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
925   PetscInt            **buf_ri,**buf_rj;
926   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
927   MPI_Request         *s_waits,*r_waits;
928   MPI_Status          *status;
929   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
930   PetscInt            *ai,*aj,*coi,*coj;
931   PetscInt            *poJ,*pdJ;
932   Mat                 A_loc;
933   Mat_SeqAIJ          *a_loc;
934 
935   PetscFunctionBegin;
936   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
937   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
938   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
939 
940   ptap  = c->ptap;
941   merge = ptap->merge;
942 
943   /* 2) compute numeric C_seq = P_loc^T*A_loc*P - dominating part */
944   /*--------------------------------------------------------------*/
945   /* get data from symbolic products */
946   coi  = merge->coi; coj = merge->coj;
947   ierr = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
948 
949   bi     = merge->bi; bj = merge->bj;
950   owners = merge->rowmap->range;
951   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
952 
953   /* get A_loc by taking all local rows of A */
954   A_loc = ptap->A_loc;
955   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
956   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
957   ai    = a_loc->i;
958   aj    = a_loc->j;
959 
960   ierr = PetscCalloc1(A->cmap->N,&aval);CHKERRQ(ierr); /* non-scalable!!! */
961 
962   for (i=0; i<am; i++) {
963     /* 2-a) put A[i,:] to dense array aval */
964     anz = ai[i+1] - ai[i];
965     adj = aj + ai[i];
966     ada = a_loc->a + ai[i];
967     for (j=0; j<anz; j++) {
968       aval[adj[j]] = ada[j];
969     }
970 
971     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
972     /*--------------------------------------------------------------*/
973     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
974     pnz = po->i[i+1] - po->i[i];
975     poJ = po->j + po->i[i];
976     pA  = po->a + po->i[i];
977     for (j=0; j<pnz; j++) {
978       row = poJ[j];
979       cnz = coi[row+1] - coi[row];
980       cj  = coj + coi[row];
981       ca  = coa + coi[row];
982       /* perform dense axpy */
983       valtmp = pA[j];
984       for (k=0; k<cnz; k++) {
985         ca[k] += valtmp*aval[cj[k]];
986       }
987       ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
988     }
989 
990     /* put the value into Cd (diagonal part) */
991     pnz = pd->i[i+1] - pd->i[i];
992     pdJ = pd->j + pd->i[i];
993     pA  = pd->a + pd->i[i];
994     for (j=0; j<pnz; j++) {
995       row = pdJ[j];
996       cnz = bi[row+1] - bi[row];
997       cj  = bj + bi[row];
998       ca  = ba + bi[row];
999       /* perform dense axpy */
1000       valtmp = pA[j];
1001       for (k=0; k<cnz; k++) {
1002         ca[k] += valtmp*aval[cj[k]];
1003       }
1004       ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1005     }
1006 
1007     /* zero the current row of Pt*A */
1008     aJ = aj + ai[i];
1009     for (k=0; k<anz; k++) aval[aJ[k]] = 0.0;
1010   }
1011 
1012   /* 3) send and recv matrix values coa */
1013   /*------------------------------------*/
1014   buf_ri = merge->buf_ri;
1015   buf_rj = merge->buf_rj;
1016   len_s  = merge->len_s;
1017   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1018   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1019 
1020   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1021   for (proc=0,k=0; proc<size; proc++) {
1022     if (!len_s[proc]) continue;
1023     i    = merge->owners_co[proc];
1024     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1025     k++;
1026   }
1027   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1028   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1029 
1030   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1031   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1032   ierr = PetscFree(coa);CHKERRQ(ierr);
1033 
1034   /* 4) insert local Cseq and received values into Cmpi */
1035   /*----------------------------------------------------*/
1036   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1037   for (k=0; k<merge->nrecv; k++) {
1038     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1039     nrows       = *(buf_ri_k[k]);
1040     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1041     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1042   }
1043 
1044   for (i=0; i<cm; i++) {
1045     row  = owners[rank] + i; /* global row index of C_seq */
1046     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1047     ba_i = ba + bi[i];
1048     bnz  = bi[i+1] - bi[i];
1049     /* add received vals into ba */
1050     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1051       /* i-th row */
1052       if (i == *nextrow[k]) {
1053         cnz    = *(nextci[k]+1) - *nextci[k];
1054         cj     = buf_rj[k] + *(nextci[k]);
1055         ca     = abuf_r[k] + *(nextci[k]);
1056         nextcj = 0;
1057         for (j=0; nextcj<cnz; j++) {
1058           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1059             ba_i[j] += ca[nextcj++];
1060           }
1061         }
1062         nextrow[k]++; nextci[k]++;
1063         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1064       }
1065     }
1066     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1067   }
1068   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1069   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1070 
1071   ierr = PetscFree(ba);CHKERRQ(ierr);
1072   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1073   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1074   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1075   ierr = PetscFree(aval);CHKERRQ(ierr);
1076   PetscFunctionReturn(0);
1077 }
1078 
1079 PetscErrorCode MatDuplicate_MPIAIJ_MatPtAP(Mat, MatDuplicateOption,Mat*);
1080 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1081 #undef __FUNCT__
1082 #define __FUNCT__ "MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable"
1083 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,PetscReal fill,Mat *C)
1084 {
1085   PetscErrorCode      ierr;
1086   Mat                 Cmpi,A_loc,POt,PDt;
1087   Mat_PtAPMPI         *ptap;
1088   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1089   Mat_MPIAIJ          *p        =(Mat_MPIAIJ*)P->data,*c;
1090   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1091   PetscInt            nnz;
1092   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1093   PetscInt            am=A->rmap->n,pn=P->cmap->n;
1094   PetscBT             lnkbt;
1095   MPI_Comm            comm;
1096   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1097   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1098   PetscInt            len,proc,*dnz,*onz,*owners;
1099   PetscInt            nzi,*bi,*bj;
1100   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1101   MPI_Request         *swaits,*rwaits;
1102   MPI_Status          *sstatus,rstatus;
1103   Mat_Merge_SeqsToMPI *merge;
1104   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1105   PetscReal           afill  =1.0,afill_tmp;
1106   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Crmax;
1107   PetscScalar         *vals;
1108   Mat_SeqAIJ          *a_loc, *pdt,*pot;
1109 
1110   PetscFunctionBegin;
1111   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1112   /* check if matrix local sizes are compatible */
1113   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend) {
1114     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);
1115   }
1116 
1117   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1118   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1119 
1120   /* create struct Mat_PtAPMPI and attached it to C later */
1121   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1122 
1123   /* get A_loc by taking all local rows of A */
1124   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1125 
1126   ptap->A_loc = A_loc;
1127 
1128   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1129   ai    = a_loc->i;
1130   aj    = a_loc->j;
1131 
1132   /* determine symbolic Co=(p->B)^T*A - send to others */
1133   /*----------------------------------------------------*/
1134   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1135   pdt  = (Mat_SeqAIJ*)PDt->data;
1136   pdti = pdt->i; pdtj = pdt->j;
1137 
1138   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1139   pot  = (Mat_SeqAIJ*)POt->data;
1140   poti = pot->i; potj = pot->j;
1141 
1142   /* then, compute symbolic Co = (p->B)^T*A */
1143   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors >= (num of nonzero rows of C_seq) - pn */
1144   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
1145   coi[0] = 0;
1146 
1147   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1148   nnz           = fill*(poti[pon] + ai[am]);
1149   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1150   current_space = free_space;
1151 
1152   /* create and initialize a linked list */
1153   i     = PetscMax(pdt->rmax,pot->rmax);
1154   Crmax = i*a_loc->rmax*size;
1155   if (!Crmax || Crmax > aN) Crmax = aN;
1156   ierr = PetscLLCondensedCreate(Crmax,aN,&lnk,&lnkbt);CHKERRQ(ierr);
1157 
1158   for (i=0; i<pon; i++) {
1159     pnz = poti[i+1] - poti[i];
1160     ptJ = potj + poti[i];
1161     for (j=0; j<pnz; j++) {
1162       row  = ptJ[j]; /* row of A_loc == col of Pot */
1163       anz  = ai[row+1] - ai[row];
1164       Jptr = aj + ai[row];
1165       /* add non-zero cols of AP into the sorted linked list lnk */
1166       ierr = PetscLLCondensedAddSorted(anz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1167     }
1168     nnz = lnk[0];
1169 
1170     /* If free space is not available, double the total space in the list */
1171     if (current_space->local_remaining<nnz) {
1172       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1173       nspacedouble++;
1174     }
1175 
1176     /* Copy data into free space, and zero out denserows */
1177     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1178 
1179     current_space->array           += nnz;
1180     current_space->local_used      += nnz;
1181     current_space->local_remaining -= nnz;
1182 
1183     coi[i+1] = coi[i] + nnz;
1184   }
1185 
1186   ierr = PetscMalloc1(coi[pon]+1,&coj);CHKERRQ(ierr);
1187   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1188 
1189   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1190   if (afill_tmp > afill) afill = afill_tmp;
1191 
1192   /* send j-array (coj) of Co to other processors */
1193   /*----------------------------------------------*/
1194   /* determine row ownership */
1195   ierr = PetscNew(&merge);CHKERRQ(ierr);
1196   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1197 
1198   merge->rowmap->n  = pn;
1199   merge->rowmap->bs = 1;
1200 
1201   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1202   owners = merge->rowmap->range;
1203 
1204   /* determine the number of messages to send, their lengths */
1205   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
1206   ierr = PetscMalloc1(size,&merge->len_s);CHKERRQ(ierr);
1207 
1208   len_s        = merge->len_s;
1209   merge->nsend = 0;
1210 
1211   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
1212   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1213 
1214   proc = 0;
1215   for (i=0; i<pon; i++) {
1216     while (prmap[i] >= owners[proc+1]) proc++;
1217     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1218     len_s[proc] += coi[i+1] - coi[i];
1219   }
1220 
1221   len          = 0; /* max length of buf_si[] */
1222   owners_co[0] = 0;
1223   for (proc=0; proc<size; proc++) {
1224     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1225     if (len_si[proc]) {
1226       merge->nsend++;
1227       len_si[proc] = 2*(len_si[proc] + 1);
1228       len         += len_si[proc];
1229     }
1230   }
1231 
1232   /* determine the number and length of messages to receive for coi and coj  */
1233   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1234   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1235 
1236   /* post the Irecv and Isend of coj */
1237   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1238   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1239   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
1240   for (proc=0, k=0; proc<size; proc++) {
1241     if (!len_s[proc]) continue;
1242     i    = owners_co[proc];
1243     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1244     k++;
1245   }
1246 
1247   /* receives and sends of coj are complete */
1248   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
1249   for (i=0; i<merge->nrecv; i++) {
1250     PetscMPIInt icompleted;
1251     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1252   }
1253   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1254   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1255 
1256   /* send and recv coi */
1257   /*-------------------*/
1258   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1259   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1260   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1261   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1262   for (proc=0,k=0; proc<size; proc++) {
1263     if (!len_s[proc]) continue;
1264     /* form outgoing message for i-structure:
1265          buf_si[0]:                 nrows to be sent
1266                [1:nrows]:           row index (global)
1267                [nrows+1:2*nrows+1]: i-structure index
1268     */
1269     /*-------------------------------------------*/
1270     nrows       = len_si[proc]/2 - 1;
1271     buf_si_i    = buf_si + nrows+1;
1272     buf_si[0]   = nrows;
1273     buf_si_i[0] = 0;
1274     nrows       = 0;
1275     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1276       nzi               = coi[i+1] - coi[i];
1277       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi; /* i-structure */
1278       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1279       nrows++;
1280     }
1281     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1282     k++;
1283     buf_si += len_si[proc];
1284   }
1285   i = merge->nrecv;
1286   while (i--) {
1287     PetscMPIInt icompleted;
1288     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1289   }
1290   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1291   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1292   ierr = PetscFree(len_si);CHKERRQ(ierr);
1293   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1294   ierr = PetscFree(swaits);CHKERRQ(ierr);
1295   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1296   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1297 
1298   /* compute the local portion of C (mpi mat) */
1299   /*------------------------------------------*/
1300   /* allocate bi array and free space for accumulating nonzero column info */
1301   ierr  = PetscMalloc1(pn+1,&bi);CHKERRQ(ierr);
1302   bi[0] = 0;
1303 
1304   /* set initial free space to be fill*(nnz(P) + nnz(A)) */
1305   nnz           = fill*(pdti[pn] + poti[pon] + ai[am]);
1306   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1307   current_space = free_space;
1308 
1309   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1310   for (k=0; k<merge->nrecv; k++) {
1311     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1312     nrows       = *buf_ri_k[k];
1313     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1314     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1315   }
1316 
1317   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1318   rmax = 0;
1319   for (i=0; i<pn; i++) {
1320     /* add pdt[i,:]*AP into lnk */
1321     pnz = pdti[i+1] - pdti[i];
1322     ptJ = pdtj + pdti[i];
1323     for (j=0; j<pnz; j++) {
1324       row  = ptJ[j];  /* row of AP == col of Pt */
1325       anz  = ai[row+1] - ai[row];
1326       Jptr = aj + ai[row];
1327       /* add non-zero cols of AP into the sorted linked list lnk */
1328       ierr = PetscLLCondensedAddSorted(anz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1329     }
1330 
1331     /* add received col data into lnk */
1332     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1333       if (i == *nextrow[k]) { /* i-th row */
1334         nzi  = *(nextci[k]+1) - *nextci[k];
1335         Jptr = buf_rj[k] + *nextci[k];
1336         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1337         nextrow[k]++; nextci[k]++;
1338       }
1339     }
1340     nnz = lnk[0];
1341 
1342     /* if free space is not available, make more free space */
1343     if (current_space->local_remaining<nnz) {
1344       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1345       nspacedouble++;
1346     }
1347     /* copy data into free space, then initialize lnk */
1348     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1349     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1350 
1351     current_space->array           += nnz;
1352     current_space->local_used      += nnz;
1353     current_space->local_remaining -= nnz;
1354 
1355     bi[i+1] = bi[i] + nnz;
1356     if (nnz > rmax) rmax = nnz;
1357   }
1358   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1359 
1360   ierr = PetscMalloc1(bi[pn]+1,&bj);CHKERRQ(ierr);
1361   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1362 
1363   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
1364   if (afill_tmp > afill) afill = afill_tmp;
1365   ierr = PetscLLCondensedDestroy(lnk,lnkbt);CHKERRQ(ierr);
1366   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1367   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1368 
1369   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
1370   /*----------------------------------------------------------------------------------*/
1371   ierr = PetscCalloc1(rmax+1,&vals);CHKERRQ(ierr);
1372 
1373   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1374   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1375   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
1376   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1377   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1378   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1379   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
1380   for (i=0; i<pn; i++) {
1381     row  = i + rstart;
1382     nnz  = bi[i+1] - bi[i];
1383     Jptr = bj + bi[i];
1384     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
1385   }
1386   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1387   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1388   ierr = PetscFree(vals);CHKERRQ(ierr);
1389 
1390   merge->bi        = bi;
1391   merge->bj        = bj;
1392   merge->coi       = coi;
1393   merge->coj       = coj;
1394   merge->buf_ri    = buf_ri;
1395   merge->buf_rj    = buf_rj;
1396   merge->owners_co = owners_co;
1397   merge->destroy   = Cmpi->ops->destroy;
1398   merge->duplicate = Cmpi->ops->duplicate;
1399 
1400   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1401   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
1402   Cmpi->ops->duplicate               = MatDuplicate_MPIAIJ_MatPtAP;
1403 
1404   /* attach the supporting struct to Cmpi for reuse */
1405   c           = (Mat_MPIAIJ*)Cmpi->data;
1406   c->ptap     = ptap;
1407   ptap->api   = NULL;
1408   ptap->apj   = NULL;
1409   ptap->merge = merge;
1410   ptap->rmax  = rmax;
1411 
1412   *C = Cmpi;
1413 #if defined(PETSC_USE_INFO)
1414   if (bi[pn] != 0) {
1415     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1416     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
1417   } else {
1418     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1419   }
1420 #endif
1421   PetscFunctionReturn(0);
1422 }
1423 
1424 #undef __FUNCT__
1425 #define __FUNCT__ "MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ"
1426 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ(Mat P,Mat A,Mat C)
1427 {
1428   PetscErrorCode      ierr;
1429   Mat_Merge_SeqsToMPI *merge;
1430   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1431   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
1432   Mat_PtAPMPI         *ptap;
1433   PetscInt            *adj;
1434   PetscInt            i,j,k,anz,pnz,row,*cj,nexta;
1435   MatScalar           *ada,*ca,valtmp;
1436   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
1437   MPI_Comm            comm;
1438   PetscMPIInt         size,rank,taga,*len_s;
1439   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
1440   PetscInt            **buf_ri,**buf_rj;
1441   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
1442   MPI_Request         *s_waits,*r_waits;
1443   MPI_Status          *status;
1444   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
1445   PetscInt            *ai,*aj,*coi,*coj;
1446   PetscInt            *poJ,*pdJ;
1447   Mat                 A_loc;
1448   Mat_SeqAIJ          *a_loc;
1449 
1450   PetscFunctionBegin;
1451   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1452   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1453   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1454 
1455   ptap  = c->ptap;
1456   merge = ptap->merge;
1457 
1458   /* 2) compute numeric C_seq = P_loc^T*A_loc */
1459   /*------------------------------------------*/
1460   /* get data from symbolic products */
1461   coi    = merge->coi; coj = merge->coj;
1462   ierr   = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
1463   bi     = merge->bi; bj = merge->bj;
1464   owners = merge->rowmap->range;
1465   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
1466 
1467   /* get A_loc by taking all local rows of A */
1468   A_loc = ptap->A_loc;
1469   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
1470   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1471   ai    = a_loc->i;
1472   aj    = a_loc->j;
1473 
1474   for (i=0; i<am; i++) {
1475     anz = ai[i+1] - ai[i];
1476     adj = aj + ai[i];
1477     ada = a_loc->a + ai[i];
1478 
1479     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1480     /*-------------------------------------------------------------*/
1481     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1482     pnz = po->i[i+1] - po->i[i];
1483     poJ = po->j + po->i[i];
1484     pA  = po->a + po->i[i];
1485     for (j=0; j<pnz; j++) {
1486       row = poJ[j];
1487       cj  = coj + coi[row];
1488       ca  = coa + coi[row];
1489       /* perform sparse axpy */
1490       nexta  = 0;
1491       valtmp = pA[j];
1492       for (k=0; nexta<anz; k++) {
1493         if (cj[k] == adj[nexta]) {
1494           ca[k] += valtmp*ada[nexta];
1495           nexta++;
1496         }
1497       }
1498       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1499     }
1500 
1501     /* put the value into Cd (diagonal part) */
1502     pnz = pd->i[i+1] - pd->i[i];
1503     pdJ = pd->j + pd->i[i];
1504     pA  = pd->a + pd->i[i];
1505     for (j=0; j<pnz; j++) {
1506       row = pdJ[j];
1507       cj  = bj + bi[row];
1508       ca  = ba + bi[row];
1509       /* perform sparse axpy */
1510       nexta  = 0;
1511       valtmp = pA[j];
1512       for (k=0; nexta<anz; k++) {
1513         if (cj[k] == adj[nexta]) {
1514           ca[k] += valtmp*ada[nexta];
1515           nexta++;
1516         }
1517       }
1518       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1519     }
1520   }
1521 
1522   /* 3) send and recv matrix values coa */
1523   /*------------------------------------*/
1524   buf_ri = merge->buf_ri;
1525   buf_rj = merge->buf_rj;
1526   len_s  = merge->len_s;
1527   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1528   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1529 
1530   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1531   for (proc=0,k=0; proc<size; proc++) {
1532     if (!len_s[proc]) continue;
1533     i    = merge->owners_co[proc];
1534     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1535     k++;
1536   }
1537   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1538   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1539 
1540   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1541   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1542   ierr = PetscFree(coa);CHKERRQ(ierr);
1543 
1544   /* 4) insert local Cseq and received values into Cmpi */
1545   /*----------------------------------------------------*/
1546   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1547   for (k=0; k<merge->nrecv; k++) {
1548     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1549     nrows       = *(buf_ri_k[k]);
1550     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1551     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1552   }
1553 
1554   for (i=0; i<cm; i++) {
1555     row  = owners[rank] + i; /* global row index of C_seq */
1556     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1557     ba_i = ba + bi[i];
1558     bnz  = bi[i+1] - bi[i];
1559     /* add received vals into ba */
1560     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1561       /* i-th row */
1562       if (i == *nextrow[k]) {
1563         cnz    = *(nextci[k]+1) - *nextci[k];
1564         cj     = buf_rj[k] + *(nextci[k]);
1565         ca     = abuf_r[k] + *(nextci[k]);
1566         nextcj = 0;
1567         for (j=0; nextcj<cnz; j++) {
1568           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1569             ba_i[j] += ca[nextcj++];
1570           }
1571         }
1572         nextrow[k]++; nextci[k]++;
1573         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1574       }
1575     }
1576     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1577   }
1578   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1579   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1580 
1581   ierr = PetscFree(ba);CHKERRQ(ierr);
1582   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1583   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1584   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1585   PetscFunctionReturn(0);
1586 }
1587 
1588 PetscErrorCode MatDuplicate_MPIAIJ_MatPtAP(Mat, MatDuplicateOption,Mat*);
1589 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ();
1590    differ from MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable in using LLCondensedCreate_Scalable() */
1591 #undef __FUNCT__
1592 #define __FUNCT__ "MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ"
1593 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P,Mat A,PetscReal fill,Mat *C)
1594 {
1595   PetscErrorCode      ierr;
1596   Mat                 Cmpi,A_loc,POt,PDt;
1597   Mat_PtAPMPI         *ptap;
1598   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1599   Mat_MPIAIJ          *p        =(Mat_MPIAIJ*)P->data,*c;
1600   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1601   PetscInt            nnz;
1602   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1603   PetscInt            am  =A->rmap->n,pn=P->cmap->n;
1604   MPI_Comm            comm;
1605   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1606   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1607   PetscInt            len,proc,*dnz,*onz,*owners;
1608   PetscInt            nzi,*bi,*bj;
1609   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1610   MPI_Request         *swaits,*rwaits;
1611   MPI_Status          *sstatus,rstatus;
1612   Mat_Merge_SeqsToMPI *merge;
1613   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1614   PetscReal           afill  =1.0,afill_tmp;
1615   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Crmax;
1616   PetscScalar         *vals;
1617   Mat_SeqAIJ          *a_loc, *pdt,*pot;
1618 
1619   PetscFunctionBegin;
1620   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1621   /* check if matrix local sizes are compatible */
1622   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend) {
1623     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);
1624   }
1625 
1626   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1627   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1628 
1629   /* create struct Mat_PtAPMPI and attached it to C later */
1630   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1631 
1632   /* get A_loc by taking all local rows of A */
1633   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1634 
1635   ptap->A_loc = A_loc;
1636   a_loc       = (Mat_SeqAIJ*)(A_loc)->data;
1637   ai          = a_loc->i;
1638   aj          = a_loc->j;
1639 
1640   /* determine symbolic Co=(p->B)^T*A - send to others */
1641   /*----------------------------------------------------*/
1642   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1643   pdt  = (Mat_SeqAIJ*)PDt->data;
1644   pdti = pdt->i; pdtj = pdt->j;
1645 
1646   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1647   pot  = (Mat_SeqAIJ*)POt->data;
1648   poti = pot->i; potj = pot->j;
1649 
1650   /* then, compute symbolic Co = (p->B)^T*A */
1651   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1652                          >= (num of nonzero rows of C_seq) - pn */
1653   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
1654   coi[0] = 0;
1655 
1656   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1657   nnz           = fill*(poti[pon] + ai[am]);
1658   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1659   current_space = free_space;
1660 
1661   /* create and initialize a linked list */
1662   i     = PetscMax(pdt->rmax,pot->rmax);
1663   Crmax = i*a_loc->rmax*size; /* non-scalable! */
1664   if (!Crmax || Crmax > aN) Crmax = aN;
1665   ierr = PetscLLCondensedCreate_Scalable(Crmax,&lnk);CHKERRQ(ierr);
1666 
1667   for (i=0; i<pon; i++) {
1668     pnz = poti[i+1] - poti[i];
1669     ptJ = potj + poti[i];
1670     for (j=0; j<pnz; j++) {
1671       row  = ptJ[j]; /* row of A_loc == col of Pot */
1672       anz  = ai[row+1] - ai[row];
1673       Jptr = aj + ai[row];
1674       /* add non-zero cols of AP into the sorted linked list lnk */
1675       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1676     }
1677     nnz = lnk[0];
1678 
1679     /* If free space is not available, double the total space in the list */
1680     if (current_space->local_remaining<nnz) {
1681       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1682       nspacedouble++;
1683     }
1684 
1685     /* Copy data into free space, and zero out denserows */
1686     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1687 
1688     current_space->array           += nnz;
1689     current_space->local_used      += nnz;
1690     current_space->local_remaining -= nnz;
1691 
1692     coi[i+1] = coi[i] + nnz;
1693   }
1694 
1695   ierr = PetscMalloc1(coi[pon]+1,&coj);CHKERRQ(ierr);
1696   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1697 
1698   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1699   if (afill_tmp > afill) afill = afill_tmp;
1700 
1701   /* send j-array (coj) of Co to other processors */
1702   /*----------------------------------------------*/
1703   /* determine row ownership */
1704   ierr = PetscNew(&merge);CHKERRQ(ierr);
1705   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1706 
1707   merge->rowmap->n  = pn;
1708   merge->rowmap->bs = 1;
1709 
1710   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1711   owners = merge->rowmap->range;
1712 
1713   /* determine the number of messages to send, their lengths */
1714   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
1715   ierr = PetscMalloc1(size,&merge->len_s);CHKERRQ(ierr);
1716 
1717   len_s        = merge->len_s;
1718   merge->nsend = 0;
1719 
1720   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
1721   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1722 
1723   proc = 0;
1724   for (i=0; i<pon; i++) {
1725     while (prmap[i] >= owners[proc+1]) proc++;
1726     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1727     len_s[proc] += coi[i+1] - coi[i];
1728   }
1729 
1730   len          = 0; /* max length of buf_si[] */
1731   owners_co[0] = 0;
1732   for (proc=0; proc<size; proc++) {
1733     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1734     if (len_si[proc]) {
1735       merge->nsend++;
1736       len_si[proc] = 2*(len_si[proc] + 1);
1737       len         += len_si[proc];
1738     }
1739   }
1740 
1741   /* determine the number and length of messages to receive for coi and coj  */
1742   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1743   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1744 
1745   /* post the Irecv and Isend of coj */
1746   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1747   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1748   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
1749   for (proc=0, k=0; proc<size; proc++) {
1750     if (!len_s[proc]) continue;
1751     i    = owners_co[proc];
1752     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1753     k++;
1754   }
1755 
1756   /* receives and sends of coj are complete */
1757   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
1758   for (i=0; i<merge->nrecv; i++) {
1759     PetscMPIInt icompleted;
1760     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1761   }
1762   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1763   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1764 
1765   /* send and recv coi */
1766   /*-------------------*/
1767   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1768   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1769   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1770   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1771   for (proc=0,k=0; proc<size; proc++) {
1772     if (!len_s[proc]) continue;
1773     /* form outgoing message for i-structure:
1774          buf_si[0]:                 nrows to be sent
1775                [1:nrows]:           row index (global)
1776                [nrows+1:2*nrows+1]: i-structure index
1777     */
1778     /*-------------------------------------------*/
1779     nrows       = len_si[proc]/2 - 1;
1780     buf_si_i    = buf_si + nrows+1;
1781     buf_si[0]   = nrows;
1782     buf_si_i[0] = 0;
1783     nrows       = 0;
1784     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1785       nzi               = coi[i+1] - coi[i];
1786       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1787       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1788       nrows++;
1789     }
1790     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1791     k++;
1792     buf_si += len_si[proc];
1793   }
1794   i = merge->nrecv;
1795   while (i--) {
1796     PetscMPIInt icompleted;
1797     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1798   }
1799   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1800   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1801   ierr = PetscFree(len_si);CHKERRQ(ierr);
1802   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1803   ierr = PetscFree(swaits);CHKERRQ(ierr);
1804   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1805   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1806 
1807   /* compute the local portion of C (mpi mat) */
1808   /*------------------------------------------*/
1809   /* allocate bi array and free space for accumulating nonzero column info */
1810   ierr  = PetscMalloc1(pn+1,&bi);CHKERRQ(ierr);
1811   bi[0] = 0;
1812 
1813   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
1814   nnz           = fill*(pdti[pn] + poti[pon] + ai[am]);
1815   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1816   current_space = free_space;
1817 
1818   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1819   for (k=0; k<merge->nrecv; k++) {
1820     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1821     nrows       = *buf_ri_k[k];
1822     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1823     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th recieved i-structure  */
1824   }
1825 
1826   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1827   rmax = 0;
1828   for (i=0; i<pn; i++) {
1829     /* add pdt[i,:]*AP into lnk */
1830     pnz = pdti[i+1] - pdti[i];
1831     ptJ = pdtj + pdti[i];
1832     for (j=0; j<pnz; j++) {
1833       row  = ptJ[j];  /* row of AP == col of Pt */
1834       anz  = ai[row+1] - ai[row];
1835       Jptr = aj + ai[row];
1836       /* add non-zero cols of AP into the sorted linked list lnk */
1837       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1838     }
1839 
1840     /* add received col data into lnk */
1841     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1842       if (i == *nextrow[k]) { /* i-th row */
1843         nzi  = *(nextci[k]+1) - *nextci[k];
1844         Jptr = buf_rj[k] + *nextci[k];
1845         ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
1846         nextrow[k]++; nextci[k]++;
1847       }
1848     }
1849     nnz = lnk[0];
1850 
1851     /* if free space is not available, make more free space */
1852     if (current_space->local_remaining<nnz) {
1853       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
1854       nspacedouble++;
1855     }
1856     /* copy data into free space, then initialize lnk */
1857     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1858     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1859 
1860     current_space->array           += nnz;
1861     current_space->local_used      += nnz;
1862     current_space->local_remaining -= nnz;
1863 
1864     bi[i+1] = bi[i] + nnz;
1865     if (nnz > rmax) rmax = nnz;
1866   }
1867   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1868 
1869   ierr      = PetscMalloc1(bi[pn]+1,&bj);CHKERRQ(ierr);
1870   ierr      = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1871   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
1872   if (afill_tmp > afill) afill = afill_tmp;
1873   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
1874   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1875   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1876 
1877   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
1878   /*----------------------------------------------------------------------------------*/
1879   ierr = PetscCalloc1(rmax+1,&vals);CHKERRQ(ierr);
1880 
1881   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1882   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1883   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
1884   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1885   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1886   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1887   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
1888   for (i=0; i<pn; i++) {
1889     row  = i + rstart;
1890     nnz  = bi[i+1] - bi[i];
1891     Jptr = bj + bi[i];
1892     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
1893   }
1894   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1895   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1896   ierr = PetscFree(vals);CHKERRQ(ierr);
1897 
1898   merge->bi        = bi;
1899   merge->bj        = bj;
1900   merge->coi       = coi;
1901   merge->coj       = coj;
1902   merge->buf_ri    = buf_ri;
1903   merge->buf_rj    = buf_rj;
1904   merge->owners_co = owners_co;
1905   merge->destroy   = Cmpi->ops->destroy;
1906   merge->duplicate = Cmpi->ops->duplicate;
1907 
1908   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ;
1909   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
1910   Cmpi->ops->duplicate               = MatDuplicate_MPIAIJ_MatPtAP;
1911 
1912   /* attach the supporting struct to Cmpi for reuse */
1913   c = (Mat_MPIAIJ*)Cmpi->data;
1914 
1915   c->ptap     = ptap;
1916   ptap->api   = NULL;
1917   ptap->apj   = NULL;
1918   ptap->merge = merge;
1919   ptap->rmax  = rmax;
1920   ptap->apa   = NULL;
1921 
1922   *C = Cmpi;
1923 #if defined(PETSC_USE_INFO)
1924   if (bi[pn] != 0) {
1925     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1926     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
1927   } else {
1928     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1929   }
1930 #endif
1931   PetscFunctionReturn(0);
1932 }
1933