xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision 4ede91e33c37eef5dcb7d7897beb5856941149d0)
1 
2 /*
3   Defines matrix-matrix product routines for pairs of MPIAIJ matrices
4           C = A * B
5 */
6 #include <../src/mat/impls/aij/seq/aij.h> /*I "petscmat.h" I*/
7 #include <../src/mat/utils/freespace.h>
8 #include <../src/mat/impls/aij/mpi/mpiaij.h>
9 #include <petscbt.h>
10 #include <../src/mat/impls/dense/mpi/mpidense.h>
11 #include <petsc/private/vecimpl.h>
12 
13 #if defined(PETSC_HAVE_HYPRE)
14 PETSC_INTERN PetscErrorCode MatMatMultSymbolic_AIJ_AIJ_wHYPRE(Mat,Mat,PetscReal,Mat*);
15 #endif
16 
17 PETSC_INTERN PetscErrorCode MatMatMult_MPIAIJ_MPIAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill, Mat *C)
18 {
19   PetscErrorCode ierr;
20 #if defined(PETSC_HAVE_HYPRE)
21   const char     *algTypes[3] = {"scalable","nonscalable","hypre"};
22   PetscInt       nalg = 3;
23 #else
24   const char     *algTypes[2] = {"scalable","nonscalable"};
25   PetscInt       nalg = 2;
26 #endif
27   PetscInt       alg = 1; /* set nonscalable algorithm as default */
28   MPI_Comm       comm;
29   PetscBool      flg;
30 
31   PetscFunctionBegin;
32   if (scall == MAT_INITIAL_MATRIX) {
33     ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
34     if (A->cmap->rstart != B->rmap->rstart || A->cmap->rend != B->rmap->rend) SETERRQ4(comm,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, (%D, %D) != (%D,%D)",A->cmap->rstart,A->cmap->rend,B->rmap->rstart,B->rmap->rend);
35 
36     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
37     PetscOptionsObject->alreadyprinted = PETSC_FALSE; /* a hack to ensure the option shows in '-help' */
38     ierr = PetscOptionsEList("-matmatmult_via","Algorithmic approach","MatMatMult",algTypes,nalg,algTypes[1],&alg,&flg);CHKERRQ(ierr);
39     ierr = PetscOptionsEnd();CHKERRQ(ierr);
40 
41     if (!flg && B->cmap->N > 100000) { /* may switch to scalable algorithm as default */
42       MatInfo     Ainfo,Binfo;
43       PetscInt    nz_local;
44       PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
45 
46       ierr = MatGetInfo(A,MAT_LOCAL,&Ainfo);CHKERRQ(ierr);
47       ierr = MatGetInfo(B,MAT_LOCAL,&Binfo);CHKERRQ(ierr);
48       nz_local = (PetscInt)(Ainfo.nz_allocated + Binfo.nz_allocated);
49 
50       if (B->cmap->N > fill*nz_local) alg_scalable_loc = PETSC_TRUE;
51       ierr = MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm);CHKERRQ(ierr);
52 
53       if (alg_scalable) {
54         alg  = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
55         ierr = PetscInfo2(B,"Use scalable algorithm, BN %D, fill*nz_allocated %g\n",B->cmap->N,fill*nz_local);CHKERRQ(ierr);
56       }
57     }
58 
59     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
60     switch (alg) {
61     case 1:
62       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(A,B,fill,C);CHKERRQ(ierr);
63       break;
64 #if defined(PETSC_HAVE_HYPRE)
65     case 2:
66       ierr = MatMatMultSymbolic_AIJ_AIJ_wHYPRE(A,B,fill,C);CHKERRQ(ierr);
67       break;
68 #endif
69     default:
70       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ(A,B,fill,C);CHKERRQ(ierr);
71       break;
72     }
73     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
74   }
75   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
76   ierr = (*(*C)->ops->matmultnumeric)(A,B,*C);CHKERRQ(ierr);
77   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
78   PetscFunctionReturn(0);
79 }
80 
81 PetscErrorCode MatDestroy_MPIAIJ_MatMatMult(Mat A)
82 {
83   PetscErrorCode ierr;
84   Mat_MPIAIJ     *a    = (Mat_MPIAIJ*)A->data;
85   Mat_PtAPMPI    *ptap = a->ptap;
86 
87   PetscFunctionBegin;
88   ierr = PetscFree2(ptap->startsj_s,ptap->startsj_r);CHKERRQ(ierr);
89   ierr = PetscFree(ptap->bufa);CHKERRQ(ierr);
90   ierr = MatDestroy(&ptap->P_loc);CHKERRQ(ierr);
91   ierr = MatDestroy(&ptap->P_oth);CHKERRQ(ierr);
92   ierr = MatDestroy(&ptap->Pt);CHKERRQ(ierr);
93   ierr = PetscFree(ptap->api);CHKERRQ(ierr);
94   ierr = PetscFree(ptap->apj);CHKERRQ(ierr);
95   ierr = PetscFree(ptap->apa);CHKERRQ(ierr);
96   ierr = ptap->destroy(A);CHKERRQ(ierr);
97   ierr = PetscFree(ptap);CHKERRQ(ierr);
98   PetscFunctionReturn(0);
99 }
100 
101 PetscErrorCode MatDuplicate_MPIAIJ_MatMatMult(Mat A, MatDuplicateOption op, Mat *M)
102 {
103   PetscErrorCode ierr;
104   Mat_MPIAIJ     *a    = (Mat_MPIAIJ*)A->data;
105   Mat_PtAPMPI    *ptap = a->ptap;
106 
107   PetscFunctionBegin;
108   ierr = (*ptap->duplicate)(A,op,M);CHKERRQ(ierr);
109 
110   (*M)->ops->destroy   = ptap->destroy;   /* = MatDestroy_MPIAIJ, *M doesn't duplicate A's special structure! */
111   (*M)->ops->duplicate = ptap->duplicate; /* = MatDuplicate_MPIAIJ */
112   PetscFunctionReturn(0);
113 }
114 
115 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,Mat C)
116 {
117   PetscErrorCode ierr;
118   Mat_MPIAIJ     *a  =(Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
119   Mat_SeqAIJ     *ad =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
120   Mat_SeqAIJ     *cd =(Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
121   PetscScalar    *cda=cd->a,*coa=co->a;
122   Mat_SeqAIJ     *p_loc,*p_oth;
123   PetscScalar    *apa,*ca;
124   PetscInt       cm   =C->rmap->n;
125   Mat_PtAPMPI    *ptap=c->ptap;
126   PetscInt       *api,*apj,*apJ,i,k;
127   PetscInt       cstart=C->cmap->rstart;
128   PetscInt       cdnz,conz,k0,k1;
129   MPI_Comm       comm;
130   PetscMPIInt    size;
131 
132   PetscFunctionBegin;
133   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
134   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
135 
136   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
137   /*-----------------------------------------------------*/
138   /* update numerical values of P_oth and P_loc */
139   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
140   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
141 
142   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
143   /*----------------------------------------------------------*/
144   /* get data from symbolic products */
145   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
146   p_oth = NULL;
147   if (size >1) {
148     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
149   }
150 
151   /* get apa for storing dense row A[i,:]*P */
152   apa = ptap->apa;
153 
154   api = ptap->api;
155   apj = ptap->apj;
156   for (i=0; i<cm; i++) {
157     /* compute apa = A[i,:]*P */
158     AProw_nonscalable(i,ad,ao,p_loc,p_oth,apa);
159 
160     /* set values in C */
161     apJ  = apj + api[i];
162     cdnz = cd->i[i+1] - cd->i[i];
163     conz = co->i[i+1] - co->i[i];
164 
165     /* 1st off-diagoanl part of C */
166     ca = coa + co->i[i];
167     k  = 0;
168     for (k0=0; k0<conz; k0++) {
169       if (apJ[k] >= cstart) break;
170       ca[k0]      = apa[apJ[k]];
171       apa[apJ[k++]] = 0.0;
172     }
173 
174     /* diagonal part of C */
175     ca = cda + cd->i[i];
176     for (k1=0; k1<cdnz; k1++) {
177       ca[k1]      = apa[apJ[k]];
178       apa[apJ[k++]] = 0.0;
179     }
180 
181     /* 2nd off-diagoanl part of C */
182     ca = coa + co->i[i];
183     for (; k0<conz; k0++) {
184       ca[k0]      = apa[apJ[k]];
185       apa[apJ[k++]] = 0.0;
186     }
187   }
188   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
189   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
190   PetscFunctionReturn(0);
191 }
192 
193 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,PetscReal fill,Mat *C)
194 {
195   PetscErrorCode     ierr;
196   MPI_Comm           comm;
197   PetscMPIInt        size;
198   Mat                Cmpi;
199   Mat_PtAPMPI        *ptap;
200   PetscFreeSpaceList free_space=NULL,current_space=NULL;
201   Mat_MPIAIJ         *a        =(Mat_MPIAIJ*)A->data,*c;
202   Mat_SeqAIJ         *ad       =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
203   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
204   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
205   PetscInt           *lnk,i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi;
206   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
207   PetscBT            lnkbt;
208   PetscScalar        *apa;
209   PetscReal          afill;
210 
211   PetscFunctionBegin;
212   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
213   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
214 
215   /* create struct Mat_PtAPMPI and attached it to C later */
216   ierr = PetscNew(&ptap);CHKERRQ(ierr);
217 
218   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
219   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
220 
221   /* get P_loc by taking all local rows of P */
222   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
223 
224   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
225   pi_loc = p_loc->i; pj_loc = p_loc->j;
226   if (size > 1) {
227     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
228     pi_oth = p_oth->i; pj_oth = p_oth->j;
229   } else {
230     p_oth = NULL;
231     pi_oth = NULL; pj_oth = NULL;
232   }
233 
234   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
235   /*-------------------------------------------------------------------*/
236   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
237   ptap->api = api;
238   api[0]    = 0;
239 
240   /* create and initialize a linked list */
241   ierr = PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt);CHKERRQ(ierr);
242 
243   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
244   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space);CHKERRQ(ierr);
245   current_space = free_space;
246 
247   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
248   for (i=0; i<am; i++) {
249     /* diagonal portion of A */
250     nzi = adi[i+1] - adi[i];
251     for (j=0; j<nzi; j++) {
252       row  = *adj++;
253       pnz  = pi_loc[row+1] - pi_loc[row];
254       Jptr = pj_loc + pi_loc[row];
255       /* add non-zero cols of P into the sorted linked list lnk */
256       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
257     }
258     /* off-diagonal portion of A */
259     nzi = aoi[i+1] - aoi[i];
260     for (j=0; j<nzi; j++) {
261       row  = *aoj++;
262       pnz  = pi_oth[row+1] - pi_oth[row];
263       Jptr = pj_oth + pi_oth[row];
264       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
265     }
266 
267     apnz     = lnk[0];
268     api[i+1] = api[i] + apnz;
269 
270     /* if free space is not available, double the total space in the list */
271     if (current_space->local_remaining<apnz) {
272       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
273       nspacedouble++;
274     }
275 
276     /* Copy data into free space, then initialize lnk */
277     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
278     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
279 
280     current_space->array           += apnz;
281     current_space->local_used      += apnz;
282     current_space->local_remaining -= apnz;
283   }
284 
285   /* Allocate space for apj, initialize apj, and */
286   /* destroy list of free space and other temporary array(s) */
287   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
288   apj  = ptap->apj;
289   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
290   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
291 
292   /* malloc apa to store dense row A[i,:]*P */
293   ierr = PetscCalloc1(pN,&apa);CHKERRQ(ierr);
294 
295   ptap->apa = apa;
296 
297   /* create and assemble symbolic parallel matrix Cmpi */
298   /*----------------------------------------------------*/
299   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
300   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
301   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
302 
303   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
304   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
305   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
306   for (i=0; i<am; i++) {
307     row  = i + rstart;
308     apnz = api[i+1] - api[i];
309     ierr = MatSetValues(Cmpi,1,&row,apnz,apj,apa,INSERT_VALUES);CHKERRQ(ierr);
310     apj += apnz;
311   }
312   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
313   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
314 
315   ptap->destroy        = Cmpi->ops->destroy;
316   ptap->duplicate      = Cmpi->ops->duplicate;
317   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
318   Cmpi->ops->destroy   = MatDestroy_MPIAIJ_MatMatMult;
319   Cmpi->ops->duplicate = MatDuplicate_MPIAIJ_MatMatMult;
320 
321   /* attach the supporting struct to Cmpi for reuse */
322   c       = (Mat_MPIAIJ*)Cmpi->data;
323   c->ptap = ptap;
324 
325   *C = Cmpi;
326 
327   /* set MatInfo */
328   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
329   if (afill < 1.0) afill = 1.0;
330   Cmpi->info.mallocs           = nspacedouble;
331   Cmpi->info.fill_ratio_given  = fill;
332   Cmpi->info.fill_ratio_needed = afill;
333 
334 #if defined(PETSC_USE_INFO)
335   if (api[am]) {
336     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
337     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
338   } else {
339     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
340   }
341 #endif
342   PetscFunctionReturn(0);
343 }
344 
345 PETSC_INTERN PetscErrorCode MatMatMult_MPIAIJ_MPIDense(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
346 {
347   PetscErrorCode ierr;
348 
349   PetscFunctionBegin;
350   if (scall == MAT_INITIAL_MATRIX) {
351     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
352     ierr = MatMatMultSymbolic_MPIAIJ_MPIDense(A,B,fill,C);CHKERRQ(ierr);
353     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
354   }
355   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
356   ierr = MatMatMultNumeric_MPIAIJ_MPIDense(A,B,*C);CHKERRQ(ierr);
357   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
358   PetscFunctionReturn(0);
359 }
360 
361 typedef struct {
362   Mat         workB;
363   PetscScalar *rvalues,*svalues;
364   MPI_Request *rwaits,*swaits;
365 } MPIAIJ_MPIDense;
366 
367 PetscErrorCode MatMPIAIJ_MPIDenseDestroy(void *ctx)
368 {
369   MPIAIJ_MPIDense *contents = (MPIAIJ_MPIDense*) ctx;
370   PetscErrorCode  ierr;
371 
372   PetscFunctionBegin;
373   ierr = MatDestroy(&contents->workB);CHKERRQ(ierr);
374   ierr = PetscFree4(contents->rvalues,contents->svalues,contents->rwaits,contents->swaits);CHKERRQ(ierr);
375   ierr = PetscFree(contents);CHKERRQ(ierr);
376   PetscFunctionReturn(0);
377 }
378 
379 /*
380     This is a "dummy function" that handles the case where matrix C was created as a dense matrix
381   directly by the user and passed to MatMatMult() with the MAT_REUSE_MATRIX option
382 
383   It is the same as MatMatMultSymbolic_MPIAIJ_MPIDense() except does not create C
384 */
385 PetscErrorCode MatMatMultNumeric_MPIDense(Mat A,Mat B,Mat C)
386 {
387   PetscErrorCode         ierr;
388   PetscBool              flg;
389   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*) A->data;
390   PetscInt               nz   = aij->B->cmap->n;
391   PetscContainer         container;
392   MPIAIJ_MPIDense        *contents;
393   VecScatter             ctx   = aij->Mvctx;
394   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
395   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
396 
397   PetscFunctionBegin;
398   ierr = PetscObjectTypeCompare((PetscObject)B,MATMPIDENSE,&flg);CHKERRQ(ierr);
399   if (!flg) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Second matrix must be mpidense");
400 
401   /* Handle case where where user provided the final C matrix rather than calling MatMatMult() with MAT_INITIAL_MATRIX*/
402   ierr = PetscObjectTypeCompare((PetscObject)A,MATMPIAIJ,&flg);CHKERRQ(ierr);
403   if (!flg) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"First matrix must be MPIAIJ");
404 
405   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
406 
407   ierr = PetscNew(&contents);CHKERRQ(ierr);
408   /* Create work matrix used to store off processor rows of B needed for local product */
409   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,B->cmap->N,NULL,&contents->workB);CHKERRQ(ierr);
410   /* Create work arrays needed */
411   ierr = PetscMalloc4(B->cmap->N*from->starts[from->n],&contents->rvalues,
412                       B->cmap->N*to->starts[to->n],&contents->svalues,
413                       from->n,&contents->rwaits,
414                       to->n,&contents->swaits);CHKERRQ(ierr);
415 
416   ierr = PetscContainerCreate(PetscObjectComm((PetscObject)A),&container);CHKERRQ(ierr);
417   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
418   ierr = PetscContainerSetUserDestroy(container,MatMPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
419   ierr = PetscObjectCompose((PetscObject)C,"workB",(PetscObject)container);CHKERRQ(ierr);
420   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
421 
422   ierr = (*C->ops->matmultnumeric)(A,B,C);CHKERRQ(ierr);
423   PetscFunctionReturn(0);
424 }
425 
426 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat A,Mat B,PetscReal fill,Mat *C)
427 {
428   PetscErrorCode         ierr;
429   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*) A->data;
430   PetscInt               nz   = aij->B->cmap->n;
431   PetscContainer         container;
432   MPIAIJ_MPIDense        *contents;
433   VecScatter             ctx   = aij->Mvctx;
434   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
435   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
436   PetscInt               m     = A->rmap->n,n=B->cmap->n;
437 
438   PetscFunctionBegin;
439   ierr = MatCreate(PetscObjectComm((PetscObject)B),C);CHKERRQ(ierr);
440   ierr = MatSetSizes(*C,m,n,A->rmap->N,B->cmap->N);CHKERRQ(ierr);
441   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
442   ierr = MatSetType(*C,MATMPIDENSE);CHKERRQ(ierr);
443   ierr = MatMPIDenseSetPreallocation(*C,NULL);CHKERRQ(ierr);
444   ierr = MatAssemblyBegin(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
445   ierr = MatAssemblyEnd(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
446 
447   (*C)->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
448 
449   ierr = PetscNew(&contents);CHKERRQ(ierr);
450   /* Create work matrix used to store off processor rows of B needed for local product */
451   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,B->cmap->N,NULL,&contents->workB);CHKERRQ(ierr);
452   /* Create work arrays needed */
453   ierr = PetscMalloc4(B->cmap->N*from->starts[from->n],&contents->rvalues,
454                       B->cmap->N*to->starts[to->n],&contents->svalues,
455                       from->n,&contents->rwaits,
456                       to->n,&contents->swaits);CHKERRQ(ierr);
457 
458   ierr = PetscContainerCreate(PetscObjectComm((PetscObject)A),&container);CHKERRQ(ierr);
459   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
460   ierr = PetscContainerSetUserDestroy(container,MatMPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
461   ierr = PetscObjectCompose((PetscObject)(*C),"workB",(PetscObject)container);CHKERRQ(ierr);
462   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
463   PetscFunctionReturn(0);
464 }
465 
466 /*
467     Performs an efficient scatter on the rows of B needed by this process; this is
468     a modification of the VecScatterBegin_() routines.
469 */
470 PetscErrorCode MatMPIDenseScatter(Mat A,Mat B,Mat C,Mat *outworkB)
471 {
472   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*)A->data;
473   PetscErrorCode         ierr;
474   PetscScalar            *b,*w,*svalues,*rvalues;
475   VecScatter             ctx   = aij->Mvctx;
476   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
477   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
478   PetscInt               i,j,k;
479   PetscInt               *sindices,*sstarts,*rindices,*rstarts;
480   PetscMPIInt            *sprocs,*rprocs,nrecvs;
481   MPI_Request            *swaits,*rwaits;
482   MPI_Comm               comm;
483   PetscMPIInt            tag  = ((PetscObject)ctx)->tag,ncols = B->cmap->N, nrows = aij->B->cmap->n,imdex,nrowsB = B->rmap->n;
484   MPI_Status             status;
485   MPIAIJ_MPIDense        *contents;
486   PetscContainer         container;
487   Mat                    workB;
488 
489   PetscFunctionBegin;
490   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
491   ierr = PetscObjectQuery((PetscObject)C,"workB",(PetscObject*)&container);CHKERRQ(ierr);
492   if (!container) SETERRQ(comm,PETSC_ERR_PLIB,"Container does not exist");
493   ierr = PetscContainerGetPointer(container,(void**)&contents);CHKERRQ(ierr);
494 
495   workB = *outworkB = contents->workB;
496   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);
497   sindices = to->indices;
498   sstarts  = to->starts;
499   sprocs   = to->procs;
500   swaits   = contents->swaits;
501   svalues  = contents->svalues;
502 
503   rindices = from->indices;
504   rstarts  = from->starts;
505   rprocs   = from->procs;
506   rwaits   = contents->rwaits;
507   rvalues  = contents->rvalues;
508 
509   ierr = MatDenseGetArray(B,&b);CHKERRQ(ierr);
510   ierr = MatDenseGetArray(workB,&w);CHKERRQ(ierr);
511 
512   for (i=0; i<from->n; i++) {
513     ierr = MPI_Irecv(rvalues+ncols*rstarts[i],ncols*(rstarts[i+1]-rstarts[i]),MPIU_SCALAR,rprocs[i],tag,comm,rwaits+i);CHKERRQ(ierr);
514   }
515 
516   for (i=0; i<to->n; i++) {
517     /* pack a message at a time */
518     for (j=0; j<sstarts[i+1]-sstarts[i]; j++) {
519       for (k=0; k<ncols; k++) {
520         svalues[ncols*(sstarts[i] + j) + k] = b[sindices[sstarts[i]+j] + nrowsB*k];
521       }
522     }
523     ierr = MPI_Isend(svalues+ncols*sstarts[i],ncols*(sstarts[i+1]-sstarts[i]),MPIU_SCALAR,sprocs[i],tag,comm,swaits+i);CHKERRQ(ierr);
524   }
525 
526   nrecvs = from->n;
527   while (nrecvs) {
528     ierr = MPI_Waitany(from->n,rwaits,&imdex,&status);CHKERRQ(ierr);
529     nrecvs--;
530     /* unpack a message at a time */
531     for (j=0; j<rstarts[imdex+1]-rstarts[imdex]; j++) {
532       for (k=0; k<ncols; k++) {
533         w[rindices[rstarts[imdex]+j] + nrows*k] = rvalues[ncols*(rstarts[imdex] + j) + k];
534       }
535     }
536   }
537   if (to->n) {ierr = MPI_Waitall(to->n,swaits,to->sstatus);CHKERRQ(ierr);}
538 
539   ierr = MatDenseRestoreArray(B,&b);CHKERRQ(ierr);
540   ierr = MatDenseRestoreArray(workB,&w);CHKERRQ(ierr);
541   ierr = MatAssemblyBegin(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
542   ierr = MatAssemblyEnd(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
543   PetscFunctionReturn(0);
544 }
545 extern PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat,Mat,Mat);
546 
547 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat A,Mat B,Mat C)
548 {
549   PetscErrorCode ierr;
550   Mat_MPIAIJ     *aij    = (Mat_MPIAIJ*)A->data;
551   Mat_MPIDense   *bdense = (Mat_MPIDense*)B->data;
552   Mat_MPIDense   *cdense = (Mat_MPIDense*)C->data;
553   Mat            workB;
554 
555   PetscFunctionBegin;
556   /* diagonal block of A times all local rows of B*/
557   ierr = MatMatMultNumeric_SeqAIJ_SeqDense(aij->A,bdense->A,cdense->A);CHKERRQ(ierr);
558 
559   /* get off processor parts of B needed to complete the product */
560   ierr = MatMPIDenseScatter(A,B,C,&workB);CHKERRQ(ierr);
561 
562   /* off-diagonal block of A times nonlocal rows of B */
563   ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A);CHKERRQ(ierr);
564   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
565   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
566   PetscFunctionReturn(0);
567 }
568 
569 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
570 {
571   PetscErrorCode ierr;
572   Mat_MPIAIJ     *a   = (Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
573   Mat_SeqAIJ     *ad  = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
574   Mat_SeqAIJ     *cd  = (Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
575   PetscInt       *adi = ad->i,*adj,*aoi=ao->i,*aoj;
576   PetscScalar    *ada,*aoa,*cda=cd->a,*coa=co->a;
577   Mat_SeqAIJ     *p_loc,*p_oth;
578   PetscInt       *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
579   PetscScalar    *pa_loc,*pa_oth,*pa,valtmp,*ca;
580   PetscInt       cm          = C->rmap->n,anz,pnz;
581   Mat_PtAPMPI    *ptap       = c->ptap;
582   PetscScalar    *apa_sparse = ptap->apa;
583   PetscInt       *api,*apj,*apJ,i,j,k,row;
584   PetscInt       cstart = C->cmap->rstart;
585   PetscInt       cdnz,conz,k0,k1,nextp;
586   MPI_Comm       comm;
587   PetscMPIInt    size;
588 
589   PetscFunctionBegin;
590   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
591   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
592 
593   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
594   /*-----------------------------------------------------*/
595   /* update numerical values of P_oth and P_loc */
596   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
597   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
598 
599   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
600   /*----------------------------------------------------------*/
601   /* get data from symbolic products */
602   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
603   pi_loc = p_loc->i; pj_loc = p_loc->j; pa_loc = p_loc->a;
604   if (size >1) {
605     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
606     pi_oth = p_oth->i; pj_oth = p_oth->j; pa_oth = p_oth->a;
607   } else {
608     p_oth = NULL; pi_oth = NULL; pj_oth = NULL; pa_oth = NULL;
609   }
610 
611   api = ptap->api;
612   apj = ptap->apj;
613   for (i=0; i<cm; i++) {
614     apJ = apj + api[i];
615 
616     /* diagonal portion of A */
617     anz = adi[i+1] - adi[i];
618     adj = ad->j + adi[i];
619     ada = ad->a + adi[i];
620     for (j=0; j<anz; j++) {
621       row = adj[j];
622       pnz = pi_loc[row+1] - pi_loc[row];
623       pj  = pj_loc + pi_loc[row];
624       pa  = pa_loc + pi_loc[row];
625       /* perform sparse axpy */
626       valtmp = ada[j];
627       nextp  = 0;
628       for (k=0; nextp<pnz; k++) {
629         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
630           apa_sparse[k] += valtmp*pa[nextp++];
631         }
632       }
633       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
634     }
635 
636     /* off-diagonal portion of A */
637     anz = aoi[i+1] - aoi[i];
638     aoj = ao->j + aoi[i];
639     aoa = ao->a + aoi[i];
640     for (j=0; j<anz; j++) {
641       row = aoj[j];
642       pnz = pi_oth[row+1] - pi_oth[row];
643       pj  = pj_oth + pi_oth[row];
644       pa  = pa_oth + pi_oth[row];
645       /* perform sparse axpy */
646       valtmp = aoa[j];
647       nextp  = 0;
648       for (k=0; nextp<pnz; k++) {
649         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
650           apa_sparse[k] += valtmp*pa[nextp++];
651         }
652       }
653       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
654     }
655 
656     /* set values in C */
657     cdnz = cd->i[i+1] - cd->i[i];
658     conz = co->i[i+1] - co->i[i];
659 
660     /* 1st off-diagoanl part of C */
661     ca = coa + co->i[i];
662     k  = 0;
663     for (k0=0; k0<conz; k0++) {
664       if (apJ[k] >= cstart) break;
665       ca[k0]        = apa_sparse[k];
666       apa_sparse[k] = 0.0;
667       k++;
668     }
669 
670     /* diagonal part of C */
671     ca = cda + cd->i[i];
672     for (k1=0; k1<cdnz; k1++) {
673       ca[k1]        = apa_sparse[k];
674       apa_sparse[k] = 0.0;
675       k++;
676     }
677 
678     /* 2nd off-diagoanl part of C */
679     ca = coa + co->i[i];
680     for (; k0<conz; k0++) {
681       ca[k0]        = apa_sparse[k];
682       apa_sparse[k] = 0.0;
683       k++;
684     }
685   }
686   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
687   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
688   PetscFunctionReturn(0);
689 }
690 
691 /* same as MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(), except using LLCondensed to avoid O(BN) memory requirement */
692 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat *C)
693 {
694   PetscErrorCode     ierr;
695   MPI_Comm           comm;
696   PetscMPIInt        size;
697   Mat                Cmpi;
698   Mat_PtAPMPI        *ptap;
699   PetscFreeSpaceList free_space = NULL,current_space=NULL;
700   Mat_MPIAIJ         *a         = (Mat_MPIAIJ*)A->data,*c;
701   Mat_SeqAIJ         *ad        = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
702   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
703   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
704   PetscInt           i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi,*lnk,apnz_max;
705   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
706   PetscReal          afill;
707   PetscScalar        *apa;
708   PetscTable         ta;
709 
710   PetscFunctionBegin;
711   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
712   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
713 
714   /* create struct Mat_PtAPMPI and attached it to C later */
715   ierr = PetscNew(&ptap);CHKERRQ(ierr);
716 
717   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
718   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
719 
720   /* get P_loc by taking all local rows of P */
721   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
722 
723   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
724   pi_loc = p_loc->i; pj_loc = p_loc->j;
725   if (size > 1) {
726     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
727     pi_oth = p_oth->i; pj_oth = p_oth->j;
728   } else {
729     p_oth  = NULL;
730     pi_oth = NULL; pj_oth = NULL;
731   }
732 
733   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
734   /*-------------------------------------------------------------------*/
735   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
736   ptap->api = api;
737   api[0]    = 0;
738 
739   /* create and initialize a linked list */
740   ierr = PetscTableCreate(pn,pN,&ta);CHKERRQ(ierr);
741 
742   /* Calculate apnz_max */
743   apnz_max = 0;
744   for (i=0; i<am; i++) {
745     ierr = PetscTableRemoveAll(ta);CHKERRQ(ierr);
746     /* diagonal portion of A */
747     nzi  = adi[i+1] - adi[i];
748     Jptr = adj+adi[i];  /* cols of A_diag */
749     MatMergeRows_SeqAIJ(p_loc,nzi,Jptr,ta);
750     ierr = PetscTableGetCount(ta,&apnz);CHKERRQ(ierr);
751     if (apnz_max < apnz) apnz_max = apnz;
752 
753     /*  off-diagonal portion of A */
754     nzi = aoi[i+1] - aoi[i];
755     Jptr = aoj+aoi[i];  /* cols of A_off */
756     MatMergeRows_SeqAIJ(p_oth,nzi,Jptr,ta);
757     ierr = PetscTableGetCount(ta,&apnz);CHKERRQ(ierr);
758     if (apnz_max < apnz) apnz_max = apnz;
759   }
760   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
761 
762   ierr = PetscLLCondensedCreate_Scalable(apnz_max,&lnk);CHKERRQ(ierr);
763 
764   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
765   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space);CHKERRQ(ierr);
766   current_space = free_space;
767   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
768   for (i=0; i<am; i++) {
769     /* diagonal portion of A */
770     nzi = adi[i+1] - adi[i];
771     for (j=0; j<nzi; j++) {
772       row  = *adj++;
773       pnz  = pi_loc[row+1] - pi_loc[row];
774       Jptr = pj_loc + pi_loc[row];
775       /* add non-zero cols of P into the sorted linked list lnk */
776       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
777     }
778     /* off-diagonal portion of A */
779     nzi = aoi[i+1] - aoi[i];
780     for (j=0; j<nzi; j++) {
781       row  = *aoj++;
782       pnz  = pi_oth[row+1] - pi_oth[row];
783       Jptr = pj_oth + pi_oth[row];
784       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
785     }
786 
787     apnz     = *lnk;
788     api[i+1] = api[i] + apnz;
789 
790     /* if free space is not available, double the total space in the list */
791     if (current_space->local_remaining<apnz) {
792       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
793       nspacedouble++;
794     }
795 
796     /* Copy data into free space, then initialize lnk */
797     ierr = PetscLLCondensedClean_Scalable(apnz,current_space->array,lnk);CHKERRQ(ierr);
798     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
799 
800     current_space->array           += apnz;
801     current_space->local_used      += apnz;
802     current_space->local_remaining -= apnz;
803   }
804 
805   /* Allocate space for apj, initialize apj, and */
806   /* destroy list of free space and other temporary array(s) */
807   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
808   apj  = ptap->apj;
809   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
810   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
811 
812   /* create and assemble symbolic parallel matrix Cmpi */
813   /*----------------------------------------------------*/
814   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
815   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
816   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
817   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
818   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
819   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
820 
821   /* malloc apa for assembly Cmpi */
822   ierr = PetscCalloc1(apnz_max,&apa);CHKERRQ(ierr);
823 
824   ptap->apa = apa;
825   for (i=0; i<am; i++) {
826     row  = i + rstart;
827     apnz = api[i+1] - api[i];
828     ierr = MatSetValues(Cmpi,1,&row,apnz,apj,apa,INSERT_VALUES);CHKERRQ(ierr);
829     apj += apnz;
830   }
831   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
832   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
833 
834   ptap->destroy             = Cmpi->ops->destroy;
835   ptap->duplicate           = Cmpi->ops->duplicate;
836   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ;
837   Cmpi->ops->destroy        = MatDestroy_MPIAIJ_MatMatMult;
838   Cmpi->ops->duplicate      = MatDuplicate_MPIAIJ_MatMatMult;
839 
840   /* attach the supporting struct to Cmpi for reuse */
841   c       = (Mat_MPIAIJ*)Cmpi->data;
842   c->ptap = ptap;
843 
844   *C = Cmpi;
845 
846   /* set MatInfo */
847   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
848   if (afill < 1.0) afill = 1.0;
849   Cmpi->info.mallocs           = nspacedouble;
850   Cmpi->info.fill_ratio_given  = fill;
851   Cmpi->info.fill_ratio_needed = afill;
852 
853 #if defined(PETSC_USE_INFO)
854   if (api[am]) {
855     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
856     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
857   } else {
858     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
859   }
860 #endif
861   PetscFunctionReturn(0);
862 }
863 
864 /*-------------------------------------------------------------------------*/
865 PetscErrorCode MatTransposeMatMult_MPIAIJ_MPIAIJ(Mat P,Mat A,MatReuse scall,PetscReal fill,Mat *C)
866 {
867   PetscErrorCode ierr;
868   const char     *algTypes[3] = {"scalable","nonscalable","matmatmult"};
869   PetscInt       aN=A->cmap->N,alg=1; /* set default algorithm */
870   PetscBool      flg;
871 
872   PetscFunctionBegin;
873   if (scall == MAT_INITIAL_MATRIX) {
874     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
875     PetscOptionsObject->alreadyprinted = PETSC_FALSE; /* a hack to ensure the option shows in '-help' */
876     ierr = PetscOptionsEList("-mattransposematmult_via","Algorithmic approach","MatTransposeMatMult",algTypes,3,algTypes[1],&alg,&flg);CHKERRQ(ierr);
877     ierr = PetscOptionsEnd();CHKERRQ(ierr);
878 
879     ierr = PetscLogEventBegin(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
880     switch (alg) {
881     case 1:
882       if (!flg && aN > 100000) { /* may switch to scalable algorithm as default */
883         MatInfo     Ainfo,Pinfo;
884         PetscInt    nz_local;
885         PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
886         MPI_Comm    comm;
887 
888         ierr = MatGetInfo(A,MAT_LOCAL,&Ainfo);CHKERRQ(ierr);
889         ierr = MatGetInfo(P,MAT_LOCAL,&Pinfo);CHKERRQ(ierr);
890         nz_local = (PetscInt)(Ainfo.nz_allocated + Pinfo.nz_allocated); /* estimated local nonzero entries */
891 
892         if (aN > fill*nz_local) alg_scalable_loc = PETSC_TRUE;
893         ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
894         ierr = MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm);CHKERRQ(ierr);
895 
896         if (alg_scalable) {
897           alg  = 0; /* scalable algorithm would slower than nonscalable algorithm */
898           ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(P,A,fill,C);CHKERRQ(ierr);
899           break;
900         }
901       }
902       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(P,A,fill,C);CHKERRQ(ierr);
903       break;
904     case 2:
905     {
906       Mat         Pt;
907       Mat_PtAPMPI *ptap;
908       Mat_MPIAIJ  *c;
909       ierr = MatTranspose(P,MAT_INITIAL_MATRIX,&Pt);CHKERRQ(ierr);
910       ierr = MatMatMult(Pt,A,MAT_INITIAL_MATRIX,fill,C);CHKERRQ(ierr);
911       c        = (Mat_MPIAIJ*)(*C)->data;
912       ptap     = c->ptap;
913       ptap->Pt = Pt;
914       (*C)->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult;
915       PetscFunctionReturn(0);
916     }
917       break;
918     default: /* scalable algorithm */
919       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(P,A,fill,C);CHKERRQ(ierr);
920       break;
921     }
922     ierr = PetscLogEventEnd(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
923   }
924   ierr = PetscLogEventBegin(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
925   ierr = (*(*C)->ops->mattransposemultnumeric)(P,A,*C);CHKERRQ(ierr);
926   ierr = PetscLogEventEnd(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
927   PetscFunctionReturn(0);
928 }
929 
930 /* This routine only works when scall=MAT_REUSE_MATRIX! */
931 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult(Mat P,Mat A,Mat C)
932 {
933   PetscErrorCode ierr;
934   Mat_MPIAIJ     *c=(Mat_MPIAIJ*)C->data;
935   Mat_PtAPMPI    *ptap= c->ptap;
936   Mat            Pt=ptap->Pt;
937 
938   PetscFunctionBegin;
939   ierr = MatTranspose(P,MAT_REUSE_MATRIX,&Pt);CHKERRQ(ierr);
940   ierr = MatMatMultNumeric(Pt,A,C);CHKERRQ(ierr);
941   PetscFunctionReturn(0);
942 }
943 
944 /* Non-scalable version, use dense axpy */
945 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable_old(Mat P,Mat A,Mat C)
946 {
947   PetscErrorCode      ierr;
948   Mat_Merge_SeqsToMPI *merge;
949   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
950   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
951   Mat_PtAPMPI         *ptap;
952   PetscInt            *adj,*aJ;
953   PetscInt            i,j,k,anz,pnz,row,*cj;
954   MatScalar           *ada,*aval,*ca,valtmp;
955   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
956   MPI_Comm            comm;
957   PetscMPIInt         size,rank,taga,*len_s;
958   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
959   PetscInt            **buf_ri,**buf_rj;
960   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
961   MPI_Request         *s_waits,*r_waits;
962   MPI_Status          *status;
963   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
964   PetscInt            *ai,*aj,*coi,*coj;
965   PetscInt            *poJ,*pdJ;
966   Mat                 A_loc;
967   Mat_SeqAIJ          *a_loc;
968 
969   PetscFunctionBegin;
970   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
971   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
972   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
973 
974   ptap  = c->ptap;
975   merge = ptap->merge;
976 
977   /* 2) compute numeric C_seq = P_loc^T*A_loc - dominating part */
978   /*------------------------------------------------------------*/
979   /* get data from symbolic products */
980   coi  = merge->coi; coj = merge->coj;
981   ierr = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
982 
983   bi     = merge->bi; bj = merge->bj;
984   owners = merge->rowmap->range;
985   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
986 
987   /* get A_loc by taking all local rows of A */
988   A_loc = ptap->A_loc;
989   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
990   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
991   ai    = a_loc->i;
992   aj    = a_loc->j;
993 
994   ierr = PetscCalloc1(A->cmap->N,&aval);CHKERRQ(ierr); /* non-scalable!!! */
995 
996   for (i=0; i<am; i++) {
997     /* 2-a) put A[i,:] to dense array aval */
998     anz = ai[i+1] - ai[i];
999     adj = aj + ai[i];
1000     ada = a_loc->a + ai[i];
1001     for (j=0; j<anz; j++) {
1002       aval[adj[j]] = ada[j];
1003     }
1004 
1005     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1006     /*--------------------------------------------------------------*/
1007     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1008     pnz = po->i[i+1] - po->i[i];
1009     poJ = po->j + po->i[i];
1010     pA  = po->a + po->i[i];
1011     for (j=0; j<pnz; j++) {
1012       row = poJ[j];
1013       cnz = coi[row+1] - coi[row];
1014       cj  = coj + coi[row];
1015       ca  = coa + coi[row];
1016       /* perform dense axpy */
1017       valtmp = pA[j];
1018       for (k=0; k<cnz; k++) {
1019         ca[k] += valtmp*aval[cj[k]];
1020       }
1021       ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1022     }
1023 
1024     /* put the value into Cd (diagonal part) */
1025     pnz = pd->i[i+1] - pd->i[i];
1026     pdJ = pd->j + pd->i[i];
1027     pA  = pd->a + pd->i[i];
1028     for (j=0; j<pnz; j++) {
1029       row = pdJ[j];
1030       cnz = bi[row+1] - bi[row];
1031       cj  = bj + bi[row];
1032       ca  = ba + bi[row];
1033       /* perform dense axpy */
1034       valtmp = pA[j];
1035       for (k=0; k<cnz; k++) {
1036         ca[k] += valtmp*aval[cj[k]];
1037       }
1038       ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1039     }
1040 
1041     /* zero the current row of Pt*A */
1042     aJ = aj + ai[i];
1043     for (k=0; k<anz; k++) aval[aJ[k]] = 0.0;
1044   }
1045 
1046   /* 3) send and recv matrix values coa */
1047   /*------------------------------------*/
1048   buf_ri = merge->buf_ri;
1049   buf_rj = merge->buf_rj;
1050   len_s  = merge->len_s;
1051   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1052   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1053 
1054   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1055   for (proc=0,k=0; proc<size; proc++) {
1056     if (!len_s[proc]) continue;
1057     i    = merge->owners_co[proc];
1058     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1059     k++;
1060   }
1061   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1062   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1063 
1064   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1065   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1066   ierr = PetscFree(coa);CHKERRQ(ierr);
1067 
1068   /* 4) insert local Cseq and received values into Cmpi */
1069   /*----------------------------------------------------*/
1070   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1071   for (k=0; k<merge->nrecv; k++) {
1072     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1073     nrows       = *(buf_ri_k[k]);
1074     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1075     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1076   }
1077 
1078   for (i=0; i<cm; i++) {
1079     row  = owners[rank] + i; /* global row index of C_seq */
1080     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1081     ba_i = ba + bi[i];
1082     bnz  = bi[i+1] - bi[i];
1083     /* add received vals into ba */
1084     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1085       /* i-th row */
1086       if (i == *nextrow[k]) {
1087         cnz    = *(nextci[k]+1) - *nextci[k];
1088         cj     = buf_rj[k] + *(nextci[k]);
1089         ca     = abuf_r[k] + *(nextci[k]);
1090         nextcj = 0;
1091         for (j=0; nextcj<cnz; j++) {
1092           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1093             ba_i[j] += ca[nextcj++];
1094           }
1095         }
1096         nextrow[k]++; nextci[k]++;
1097         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1098       }
1099     }
1100     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1101   }
1102   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1103   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1104 
1105   ierr = PetscFree(ba);CHKERRQ(ierr);
1106   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1107   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1108   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1109   ierr = PetscFree(aval);CHKERRQ(ierr);
1110   PetscFunctionReturn(0);
1111 }
1112 
1113 PetscErrorCode MatDuplicate_MPIAIJ_MatPtAP(Mat,MatDuplicateOption,Mat*);
1114 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1115 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable_old(Mat P,Mat A,PetscReal fill,Mat *C)
1116 {
1117   PetscErrorCode      ierr;
1118   Mat                 Cmpi,A_loc,POt,PDt;
1119   Mat_PtAPMPI         *ptap;
1120   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1121   Mat_MPIAIJ          *p        =(Mat_MPIAIJ*)P->data,*c;
1122   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1123   PetscInt            nnz;
1124   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1125   PetscInt            am=A->rmap->n,pn=P->cmap->n;
1126   PetscBT             lnkbt;
1127   MPI_Comm            comm;
1128   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1129   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1130   PetscInt            len,proc,*dnz,*onz,*owners;
1131   PetscInt            nzi,*bi,*bj;
1132   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1133   MPI_Request         *swaits,*rwaits;
1134   MPI_Status          *sstatus,rstatus;
1135   Mat_Merge_SeqsToMPI *merge;
1136   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1137   PetscReal           afill  =1.0,afill_tmp;
1138   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N;
1139   PetscScalar         *vals;
1140   Mat_SeqAIJ          *a_loc, *pdt,*pot;
1141 
1142   PetscFunctionBegin;
1143   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1144   /* check if matrix local sizes are compatible */
1145   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend) SETERRQ4(comm,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, A (%D, %D) != P (%D,%D)",A->rmap->rstart,A->rmap->rend,P->rmap->rstart,P->rmap->rend);
1146 
1147   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1148   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1149 
1150   /* create struct Mat_PtAPMPI and attached it to C later */
1151   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1152 
1153   /* get A_loc by taking all local rows of A */
1154   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1155 
1156   ptap->A_loc = A_loc;
1157 
1158   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1159   ai    = a_loc->i;
1160   aj    = a_loc->j;
1161 
1162   /* determine symbolic Co=(p->B)^T*A - send to others */
1163   /*----------------------------------------------------*/
1164   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1165   pdt  = (Mat_SeqAIJ*)PDt->data;
1166   pdti = pdt->i; pdtj = pdt->j;
1167 
1168   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1169   pot  = (Mat_SeqAIJ*)POt->data;
1170   poti = pot->i; potj = pot->j;
1171 
1172   /* then, compute symbolic Co = (p->B)^T*A */
1173   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors >= (num of nonzero rows of C_seq) - pn */
1174   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
1175   coi[0] = 0;
1176 
1177   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1178   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(poti[pon],ai[am]));
1179   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1180   current_space = free_space;
1181 
1182   /* create and initialize a linked list */
1183   ierr = PetscLLCondensedCreate(aN,aN,&lnk,&lnkbt);CHKERRQ(ierr);
1184 
1185   for (i=0; i<pon; i++) {
1186     pnz = poti[i+1] - poti[i];
1187     ptJ = potj + poti[i];
1188     for (j=0; j<pnz; j++) {
1189       row  = ptJ[j]; /* row of A_loc == col of Pot */
1190       anz  = ai[row+1] - ai[row];
1191       Jptr = aj + ai[row];
1192       /* add non-zero cols of AP into the sorted linked list lnk */
1193       ierr = PetscLLCondensedAddSorted(anz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1194     }
1195     nnz = lnk[0];
1196 
1197     /* If free space is not available, double the total space in the list */
1198     if (current_space->local_remaining<nnz) {
1199       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1200       nspacedouble++;
1201     }
1202 
1203     /* Copy data into free space, and zero out denserows */
1204     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1205 
1206     current_space->array           += nnz;
1207     current_space->local_used      += nnz;
1208     current_space->local_remaining -= nnz;
1209 
1210     coi[i+1] = coi[i] + nnz;
1211   }
1212 
1213   ierr = PetscMalloc1(coi[pon]+1,&coj);CHKERRQ(ierr);
1214   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1215 
1216   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1217   if (afill_tmp > afill) afill = afill_tmp;
1218 
1219   /* send j-array (coj) of Co to other processors */
1220   /*----------------------------------------------*/
1221   /* determine row ownership */
1222   ierr = PetscNew(&merge);CHKERRQ(ierr);
1223   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1224 
1225   merge->rowmap->n  = pn;
1226   merge->rowmap->bs = 1;
1227 
1228   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1229   owners = merge->rowmap->range;
1230 
1231   /* determine the number of messages to send, their lengths */
1232   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
1233   ierr = PetscMalloc1(size,&merge->len_s);CHKERRQ(ierr);
1234 
1235   len_s        = merge->len_s;
1236   merge->nsend = 0;
1237 
1238   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
1239   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1240 
1241   proc = 0;
1242   for (i=0; i<pon; i++) {
1243     while (prmap[i] >= owners[proc+1]) proc++;
1244     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1245     len_s[proc] += coi[i+1] - coi[i];
1246   }
1247 
1248   len          = 0; /* max length of buf_si[] */
1249   owners_co[0] = 0;
1250   for (proc=0; proc<size; proc++) {
1251     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1252     if (len_si[proc]) {
1253       merge->nsend++;
1254       len_si[proc] = 2*(len_si[proc] + 1);
1255       len         += len_si[proc];
1256     }
1257   }
1258 
1259   /* determine the number and length of messages to receive for coi and coj  */
1260   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1261   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1262 
1263   /* post the Irecv and Isend of coj */
1264   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1265   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1266   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
1267   for (proc=0, k=0; proc<size; proc++) {
1268     if (!len_s[proc]) continue;
1269     i    = owners_co[proc];
1270     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1271     k++;
1272   }
1273 
1274   /* receives and sends of coj are complete */
1275   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
1276   for (i=0; i<merge->nrecv; i++) {
1277     PetscMPIInt icompleted;
1278     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1279   }
1280   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1281   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1282 
1283   /* send and recv coi */
1284   /*-------------------*/
1285   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1286   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1287   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1288   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1289   for (proc=0,k=0; proc<size; proc++) {
1290     if (!len_s[proc]) continue;
1291     /* form outgoing message for i-structure:
1292          buf_si[0]:                 nrows to be sent
1293                [1:nrows]:           row index (global)
1294                [nrows+1:2*nrows+1]: i-structure index
1295     */
1296     /*-------------------------------------------*/
1297     nrows       = len_si[proc]/2 - 1;
1298     buf_si_i    = buf_si + nrows+1;
1299     buf_si[0]   = nrows;
1300     buf_si_i[0] = 0;
1301     nrows       = 0;
1302     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1303       nzi               = coi[i+1] - coi[i];
1304       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi; /* i-structure */
1305       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1306       nrows++;
1307     }
1308     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1309     k++;
1310     buf_si += len_si[proc];
1311   }
1312   i = merge->nrecv;
1313   while (i--) {
1314     PetscMPIInt icompleted;
1315     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1316   }
1317   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1318   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1319   ierr = PetscFree(len_si);CHKERRQ(ierr);
1320   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1321   ierr = PetscFree(swaits);CHKERRQ(ierr);
1322   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1323   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1324 
1325   /* compute the local portion of C (mpi mat) */
1326   /*------------------------------------------*/
1327   /* allocate bi array and free space for accumulating nonzero column info */
1328   ierr  = PetscMalloc1(pn+1,&bi);CHKERRQ(ierr);
1329   bi[0] = 0;
1330 
1331   /* set initial free space to be fill*(nnz(P) + nnz(A)) */
1332   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(pdti[pn],PetscIntSumTruncate(poti[pon],ai[am])));
1333   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1334   current_space = free_space;
1335 
1336   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1337   for (k=0; k<merge->nrecv; k++) {
1338     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1339     nrows       = *buf_ri_k[k];
1340     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1341     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1342   }
1343 
1344   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1345   rmax = 0;
1346   for (i=0; i<pn; i++) {
1347     /* add pdt[i,:]*AP into lnk */
1348     pnz = pdti[i+1] - pdti[i];
1349     ptJ = pdtj + pdti[i];
1350     for (j=0; j<pnz; j++) {
1351       row  = ptJ[j];  /* row of AP == col of Pt */
1352       anz  = ai[row+1] - ai[row];
1353       Jptr = aj + ai[row];
1354       /* add non-zero cols of AP into the sorted linked list lnk */
1355       ierr = PetscLLCondensedAddSorted(anz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1356     }
1357 
1358     /* add received col data into lnk */
1359     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1360       if (i == *nextrow[k]) { /* i-th row */
1361         nzi  = *(nextci[k]+1) - *nextci[k];
1362         Jptr = buf_rj[k] + *nextci[k];
1363         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1364         nextrow[k]++; nextci[k]++;
1365       }
1366     }
1367     nnz = lnk[0];
1368 
1369     /* if free space is not available, make more free space */
1370     if (current_space->local_remaining<nnz) {
1371       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1372       nspacedouble++;
1373     }
1374     /* copy data into free space, then initialize lnk */
1375     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1376     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1377 
1378     current_space->array           += nnz;
1379     current_space->local_used      += nnz;
1380     current_space->local_remaining -= nnz;
1381 
1382     bi[i+1] = bi[i] + nnz;
1383     if (nnz > rmax) rmax = nnz;
1384   }
1385   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1386 
1387   ierr = PetscMalloc1(bi[pn]+1,&bj);CHKERRQ(ierr);
1388   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1389 
1390   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
1391   if (afill_tmp > afill) afill = afill_tmp;
1392   ierr = PetscLLCondensedDestroy(lnk,lnkbt);CHKERRQ(ierr);
1393   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1394   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1395 
1396   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
1397   /*----------------------------------------------------------------------------------*/
1398   ierr = PetscCalloc1(rmax+1,&vals);CHKERRQ(ierr);
1399 
1400   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1401   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1402   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
1403   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1404   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1405   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1406   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
1407   for (i=0; i<pn; i++) {
1408     row  = i + rstart;
1409     nnz  = bi[i+1] - bi[i];
1410     Jptr = bj + bi[i];
1411     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
1412   }
1413   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1414   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1415   ierr = PetscFree(vals);CHKERRQ(ierr);
1416 
1417   merge->bi        = bi;
1418   merge->bj        = bj;
1419   merge->coi       = coi;
1420   merge->coj       = coj;
1421   merge->buf_ri    = buf_ri;
1422   merge->buf_rj    = buf_rj;
1423   merge->owners_co = owners_co;
1424 
1425   /* attach the supporting struct to Cmpi for reuse */
1426   c           = (Mat_MPIAIJ*)Cmpi->data;
1427   c->ptap     = ptap;
1428   ptap->api   = NULL;
1429   ptap->apj   = NULL;
1430   ptap->merge = merge;
1431   ptap->destroy   = Cmpi->ops->destroy;
1432   ptap->duplicate = Cmpi->ops->duplicate;
1433 
1434   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable_old;
1435   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
1436   Cmpi->ops->duplicate               = MatDuplicate_MPIAIJ_MatPtAP;
1437 
1438   *C = Cmpi;
1439 #if defined(PETSC_USE_INFO)
1440   if (bi[pn] != 0) {
1441     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1442     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
1443   } else {
1444     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1445   }
1446 #endif
1447   PetscFunctionReturn(0);
1448 }
1449 
1450 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,PetscReal fill,Mat *C)
1451 {
1452   PetscErrorCode      ierr;
1453   Mat_PtAPMPI         *ptap;
1454   Mat_MPIAIJ          *p=(Mat_MPIAIJ*)P->data,*c;
1455   MPI_Comm            comm;
1456   PetscMPIInt         size,rank;
1457   Mat                 Cmpi;
1458   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1459   PetscInt            pn=P->cmap->n,aN=A->cmap->N,an=A->cmap->n;
1460   PetscInt            *lnk,i,k,nsend;
1461   PetscBT             lnkbt;
1462   PetscMPIInt         tagi,tagj,*len_si,*len_s,*len_ri,icompleted=0,nrecv;
1463   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1464   PetscInt            len,proc,*dnz,*onz,*owners,nzi;
1465   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1466   MPI_Request         *swaits,*rwaits;
1467   MPI_Status          *sstatus,rstatus;
1468   PetscLayout         rowmap;
1469   PetscInt            *owners_co,*coi,*coj;    /* i and j array of (p->B)^T*A*P - used in the communication */
1470   PetscMPIInt         *len_r,*id_r;    /* array of length of comm->size, store send/recv matrix values */
1471   PetscInt            *Jptr,*prmap=p->garray,con,j,Crmax;
1472   Mat_SeqAIJ          *a_loc,*c_loc,*c_oth;
1473   PetscTable          ta;
1474 
1475   PetscFunctionBegin;
1476   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1477   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1478   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1479 
1480   /* create symbolic parallel matrix Cmpi */
1481   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1482   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1483 
1484   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1485 
1486   /* create struct Mat_PtAPMPI and attached it to C later */
1487   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1488   ptap->reuse = MAT_INITIAL_MATRIX;
1489 
1490   /* (0) compute Rd = Pd^T, Ro = Po^T  */
1491   /* --------------------------------- */
1492   ierr = MatTranspose_SeqAIJ(p->A,MAT_INITIAL_MATRIX,&ptap->Rd);CHKERRQ(ierr);
1493   ierr = MatTranspose_SeqAIJ(p->B,MAT_INITIAL_MATRIX,&ptap->Ro);CHKERRQ(ierr);
1494 
1495   /* (1) compute symbolic A_loc */
1496   /* ---------------------------*/
1497   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&ptap->A_loc);CHKERRQ(ierr);
1498 
1499   /* (2-1) compute symbolic C_oth = Ro*A_loc  */
1500   /* ------------------------------------ */
1501   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Ro,ptap->A_loc,fill,&ptap->C_oth);CHKERRQ(ierr);
1502 
1503   /* (3) send coj of C_oth to other processors  */
1504   /* ------------------------------------------ */
1505   /* determine row ownership */
1506   ierr = PetscLayoutCreate(comm,&rowmap);CHKERRQ(ierr);
1507   rowmap->n  = pn;
1508   rowmap->bs = 1;
1509   ierr   = PetscLayoutSetUp(rowmap);CHKERRQ(ierr);
1510   owners = rowmap->range;
1511 
1512   /* determine the number of messages to send, their lengths */
1513   ierr = PetscMalloc4(size,&len_s,size,&len_si,size,&sstatus,size+2,&owners_co);CHKERRQ(ierr);
1514   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1515   ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1516 
1517   c_oth = (Mat_SeqAIJ*)ptap->C_oth->data;
1518   coi   = c_oth->i; coj = c_oth->j;
1519   con   = ptap->C_oth->rmap->n;
1520   proc  = 0;
1521   for (i=0; i<con; i++) {
1522     while (prmap[i] >= owners[proc+1]) proc++;
1523     len_si[proc]++;               /* num of rows in Co(=Pt*A) to be sent to [proc] */
1524     len_s[proc] += coi[i+1] - coi[i]; /* num of nonzeros in Co to be sent to [proc] */
1525   }
1526 
1527   len          = 0; /* max length of buf_si[], see (4) */
1528   owners_co[0] = 0;
1529   nsend        = 0;
1530   for (proc=0; proc<size; proc++) {
1531     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1532     if (len_s[proc]) {
1533       nsend++;
1534       len_si[proc] = 2*(len_si[proc] + 1); /* length of buf_si to be sent to [proc] */
1535       len         += len_si[proc];
1536     }
1537   }
1538 
1539   /* determine the number and length of messages to receive for coi and coj  */
1540   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&nrecv);CHKERRQ(ierr);
1541   ierr = PetscGatherMessageLengths2(comm,nsend,nrecv,len_s,len_si,&id_r,&len_r,&len_ri);CHKERRQ(ierr);
1542 
1543   /* post the Irecv and Isend of coj */
1544   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1545   ierr = PetscPostIrecvInt(comm,tagj,nrecv,id_r,len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1546   ierr = PetscMalloc1(nsend+1,&swaits);CHKERRQ(ierr);
1547   for (proc=0, k=0; proc<size; proc++) {
1548     if (!len_s[proc]) continue;
1549     i    = owners_co[proc];
1550     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1551     k++;
1552   }
1553 
1554   /* (2-2) compute symbolic C_loc = Rd*A_loc */
1555   /* ---------------------------------------- */
1556   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Rd,ptap->A_loc,fill,&ptap->C_loc);CHKERRQ(ierr);
1557   c_loc = (Mat_SeqAIJ*)ptap->C_loc->data;
1558 
1559   /* receives coj are complete */
1560   for (i=0; i<nrecv; i++) {
1561     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1562   }
1563   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1564   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
1565 
1566   /* add received column indices into ta to update Crmax */
1567   a_loc = (Mat_SeqAIJ*)(ptap->A_loc)->data;
1568 
1569   /* create and initialize a linked list */
1570   ierr = PetscTableCreate(an,aN,&ta);CHKERRQ(ierr); /* for compute Crmax */
1571   MatRowMergeMax_SeqAIJ(a_loc,ptap->A_loc->rmap->N,ta);
1572 
1573   for (k=0; k<nrecv; k++) {/* k-th received message */
1574     Jptr = buf_rj[k];
1575     for (j=0; j<len_r[k]; j++) {
1576       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
1577     }
1578   }
1579   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
1580   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
1581 
1582   /* (4) send and recv coi */
1583   /*-----------------------*/
1584   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1585   ierr   = PetscPostIrecvInt(comm,tagi,nrecv,id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1586   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1587   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1588   for (proc=0,k=0; proc<size; proc++) {
1589     if (!len_s[proc]) continue;
1590     /* form outgoing message for i-structure:
1591          buf_si[0]:                 nrows to be sent
1592                [1:nrows]:           row index (global)
1593                [nrows+1:2*nrows+1]: i-structure index
1594     */
1595     /*-------------------------------------------*/
1596     nrows       = len_si[proc]/2 - 1; /* num of rows in Co to be sent to [proc] */
1597     buf_si_i    = buf_si + nrows+1;
1598     buf_si[0]   = nrows;
1599     buf_si_i[0] = 0;
1600     nrows       = 0;
1601     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1602       nzi = coi[i+1] - coi[i];
1603       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1604       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1605       nrows++;
1606     }
1607     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1608     k++;
1609     buf_si += len_si[proc];
1610   }
1611   for (i=0; i<nrecv; i++) {
1612     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1613   }
1614   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1615   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
1616 
1617   ierr = PetscFree4(len_s,len_si,sstatus,owners_co);CHKERRQ(ierr);
1618   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1619   ierr = PetscFree(swaits);CHKERRQ(ierr);
1620   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1621 
1622   /* (5) compute the local portion of Cmpi      */
1623   /* ------------------------------------------ */
1624   /* set initial free space to be Crmax, sufficient for holding nozeros in each row of Cmpi */
1625   ierr          = PetscFreeSpaceGet(Crmax,&free_space);CHKERRQ(ierr);
1626   current_space = free_space;
1627 
1628   ierr = PetscMalloc3(nrecv,&buf_ri_k,nrecv,&nextrow,nrecv,&nextci);CHKERRQ(ierr);
1629   for (k=0; k<nrecv; k++) {
1630     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1631     nrows       = *buf_ri_k[k];
1632     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1633     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1634   }
1635 
1636   ierr = MatPreallocateInitialize(comm,pn,an,dnz,onz);CHKERRQ(ierr);
1637   ierr = PetscLLCondensedCreate(Crmax,aN,&lnk,&lnkbt);CHKERRQ(ierr);
1638   for (i=0; i<pn; i++) {
1639     /* add C_loc into Cmpi */
1640     nzi  = c_loc->i[i+1] - c_loc->i[i];
1641     Jptr = c_loc->j + c_loc->i[i];
1642     ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1643 
1644     /* add received col data into lnk */
1645     for (k=0; k<nrecv; k++) { /* k-th received message */
1646       if (i == *nextrow[k]) { /* i-th row */
1647         nzi  = *(nextci[k]+1) - *nextci[k];
1648         Jptr = buf_rj[k] + *nextci[k];
1649         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1650         nextrow[k]++; nextci[k]++;
1651       }
1652     }
1653     nzi = lnk[0];
1654 
1655     /* copy data into free space, then initialize lnk */
1656     ierr = PetscLLCondensedClean(aN,nzi,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1657     ierr = MatPreallocateSet(i+owners[rank],nzi,current_space->array,dnz,onz);CHKERRQ(ierr);
1658   }
1659   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1660   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1661   ierr = PetscFreeSpaceDestroy(free_space);CHKERRQ(ierr);
1662 
1663   /* local sizes and preallocation */
1664   ierr = MatSetSizes(Cmpi,pn,an,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1665   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(P->cmap->bs));CHKERRQ(ierr);
1666   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1667   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1668 
1669   /* members in merge */
1670   ierr = PetscFree(id_r);CHKERRQ(ierr);
1671   ierr = PetscFree(len_r);CHKERRQ(ierr);
1672   ierr = PetscFree(buf_ri[0]);CHKERRQ(ierr);
1673   ierr = PetscFree(buf_ri);CHKERRQ(ierr);
1674   ierr = PetscFree(buf_rj[0]);CHKERRQ(ierr);
1675   ierr = PetscFree(buf_rj);CHKERRQ(ierr);
1676   ierr = PetscLayoutDestroy(&rowmap);CHKERRQ(ierr);
1677 
1678   /* attach the supporting struct to Cmpi for reuse */
1679   c = (Mat_MPIAIJ*)Cmpi->data;
1680   c->ptap         = ptap;
1681   ptap->duplicate = Cmpi->ops->duplicate;
1682   ptap->destroy   = Cmpi->ops->destroy;
1683 
1684   /* Cmpi is not ready for use - assembly will be done by MatPtAPNumeric() */
1685   Cmpi->assembled        = PETSC_FALSE;
1686   Cmpi->ops->destroy     = MatDestroy_MPIAIJ_PtAP;
1687   Cmpi->ops->duplicate   = MatDuplicate_MPIAIJ_MatPtAP;
1688   *C                     = Cmpi;
1689   PetscFunctionReturn(0);
1690 }
1691 
1692 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,Mat C)
1693 {
1694   PetscErrorCode    ierr;
1695   Mat_MPIAIJ        *p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1696   Mat_SeqAIJ        *c_seq;
1697   Mat_PtAPMPI       *ptap = c->ptap;
1698   Mat               A_loc,C_loc,C_oth;
1699   PetscInt          i,rstart,rend,cm,ncols,row;
1700   const PetscInt    *cols;
1701   const PetscScalar *vals;
1702 
1703   PetscFunctionBegin;
1704   ierr = MatZeroEntries(C);CHKERRQ(ierr);
1705 
1706   if (ptap->reuse == MAT_REUSE_MATRIX) {
1707     /* These matrices are obtained in MatTransposeMatMultSymbolic() */
1708     /* 1) get R = Pd^T, Ro = Po^T */
1709     /*----------------------------*/
1710     ierr = MatTranspose_SeqAIJ(p->A,MAT_REUSE_MATRIX,&ptap->Rd);CHKERRQ(ierr);
1711     ierr = MatTranspose_SeqAIJ(p->B,MAT_REUSE_MATRIX,&ptap->Ro);CHKERRQ(ierr);
1712 
1713     /* 2) compute numeric A_loc */
1714     /*--------------------------*/
1715     ierr = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&ptap->A_loc);CHKERRQ(ierr);
1716   }
1717 
1718   /* 3) C_loc = Rd*A_loc, C_oth = Ro*A_loc */
1719   A_loc = ptap->A_loc;
1720   ierr = ((ptap->C_loc)->ops->matmultnumeric)(ptap->Rd,A_loc,ptap->C_loc);CHKERRQ(ierr);
1721   ierr = ((ptap->C_oth)->ops->matmultnumeric)(ptap->Ro,A_loc,ptap->C_oth);CHKERRQ(ierr);
1722   C_loc = ptap->C_loc;
1723   C_oth = ptap->C_oth;
1724 
1725   /* add C_loc and Co to to C */
1726   ierr = MatGetOwnershipRange(C,&rstart,&rend);CHKERRQ(ierr);
1727 
1728   /* C_loc -> C */
1729   cm    = C_loc->rmap->N;
1730   c_seq = (Mat_SeqAIJ*)C_loc->data;
1731   cols = c_seq->j;
1732   vals = c_seq->a;
1733   for (i=0; i<cm; i++) {
1734     ncols = c_seq->i[i+1] - c_seq->i[i];
1735     row = rstart + i;
1736     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1737     cols += ncols; vals += ncols;
1738   }
1739 
1740   /* Co -> C, off-processor part */
1741   cm    = C_oth->rmap->N;
1742   c_seq = (Mat_SeqAIJ*)C_oth->data;
1743   cols  = c_seq->j;
1744   vals  = c_seq->a;
1745   for (i=0; i<cm; i++) {
1746     ncols = c_seq->i[i+1] - c_seq->i[i];
1747     row = p->garray[i];
1748     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1749     cols += ncols; vals += ncols;
1750   }
1751   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1752   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1753 
1754   ptap->reuse = MAT_REUSE_MATRIX;
1755   PetscFunctionReturn(0);
1756 }
1757 
1758 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ(Mat P,Mat A,Mat C)
1759 {
1760   PetscErrorCode      ierr;
1761   Mat_Merge_SeqsToMPI *merge;
1762   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1763   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
1764   Mat_PtAPMPI         *ptap;
1765   PetscInt            *adj;
1766   PetscInt            i,j,k,anz,pnz,row,*cj,nexta;
1767   MatScalar           *ada,*ca,valtmp;
1768   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
1769   MPI_Comm            comm;
1770   PetscMPIInt         size,rank,taga,*len_s;
1771   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
1772   PetscInt            **buf_ri,**buf_rj;
1773   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
1774   MPI_Request         *s_waits,*r_waits;
1775   MPI_Status          *status;
1776   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
1777   PetscInt            *ai,*aj,*coi,*coj;
1778   PetscInt            *poJ,*pdJ;
1779   Mat                 A_loc;
1780   Mat_SeqAIJ          *a_loc;
1781 
1782   PetscFunctionBegin;
1783   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1784   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1785   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1786 
1787   ptap  = c->ptap;
1788   merge = ptap->merge;
1789 
1790   /* 2) compute numeric C_seq = P_loc^T*A_loc */
1791   /*------------------------------------------*/
1792   /* get data from symbolic products */
1793   coi    = merge->coi; coj = merge->coj;
1794   ierr   = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
1795   bi     = merge->bi; bj = merge->bj;
1796   owners = merge->rowmap->range;
1797   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
1798 
1799   /* get A_loc by taking all local rows of A */
1800   A_loc = ptap->A_loc;
1801   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
1802   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1803   ai    = a_loc->i;
1804   aj    = a_loc->j;
1805 
1806   for (i=0; i<am; i++) {
1807     anz = ai[i+1] - ai[i];
1808     adj = aj + ai[i];
1809     ada = a_loc->a + ai[i];
1810 
1811     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1812     /*-------------------------------------------------------------*/
1813     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1814     pnz = po->i[i+1] - po->i[i];
1815     poJ = po->j + po->i[i];
1816     pA  = po->a + po->i[i];
1817     for (j=0; j<pnz; j++) {
1818       row = poJ[j];
1819       cj  = coj + coi[row];
1820       ca  = coa + coi[row];
1821       /* perform sparse axpy */
1822       nexta  = 0;
1823       valtmp = pA[j];
1824       for (k=0; nexta<anz; k++) {
1825         if (cj[k] == adj[nexta]) {
1826           ca[k] += valtmp*ada[nexta];
1827           nexta++;
1828         }
1829       }
1830       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1831     }
1832 
1833     /* put the value into Cd (diagonal part) */
1834     pnz = pd->i[i+1] - pd->i[i];
1835     pdJ = pd->j + pd->i[i];
1836     pA  = pd->a + pd->i[i];
1837     for (j=0; j<pnz; j++) {
1838       row = pdJ[j];
1839       cj  = bj + bi[row];
1840       ca  = ba + bi[row];
1841       /* perform sparse axpy */
1842       nexta  = 0;
1843       valtmp = pA[j];
1844       for (k=0; nexta<anz; k++) {
1845         if (cj[k] == adj[nexta]) {
1846           ca[k] += valtmp*ada[nexta];
1847           nexta++;
1848         }
1849       }
1850       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1851     }
1852   }
1853 
1854   /* 3) send and recv matrix values coa */
1855   /*------------------------------------*/
1856   buf_ri = merge->buf_ri;
1857   buf_rj = merge->buf_rj;
1858   len_s  = merge->len_s;
1859   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1860   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1861 
1862   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1863   for (proc=0,k=0; proc<size; proc++) {
1864     if (!len_s[proc]) continue;
1865     i    = merge->owners_co[proc];
1866     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1867     k++;
1868   }
1869   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1870   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1871 
1872   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1873   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1874   ierr = PetscFree(coa);CHKERRQ(ierr);
1875 
1876   /* 4) insert local Cseq and received values into Cmpi */
1877   /*----------------------------------------------------*/
1878   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1879   for (k=0; k<merge->nrecv; k++) {
1880     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1881     nrows       = *(buf_ri_k[k]);
1882     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1883     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1884   }
1885 
1886   for (i=0; i<cm; i++) {
1887     row  = owners[rank] + i; /* global row index of C_seq */
1888     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1889     ba_i = ba + bi[i];
1890     bnz  = bi[i+1] - bi[i];
1891     /* add received vals into ba */
1892     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1893       /* i-th row */
1894       if (i == *nextrow[k]) {
1895         cnz    = *(nextci[k]+1) - *nextci[k];
1896         cj     = buf_rj[k] + *(nextci[k]);
1897         ca     = abuf_r[k] + *(nextci[k]);
1898         nextcj = 0;
1899         for (j=0; nextcj<cnz; j++) {
1900           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1901             ba_i[j] += ca[nextcj++];
1902           }
1903         }
1904         nextrow[k]++; nextci[k]++;
1905         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1906       }
1907     }
1908     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1909   }
1910   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1911   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1912 
1913   ierr = PetscFree(ba);CHKERRQ(ierr);
1914   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1915   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1916   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1917   PetscFunctionReturn(0);
1918 }
1919 
1920 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P,Mat A,PetscReal fill,Mat *C)
1921 {
1922   PetscErrorCode      ierr;
1923   Mat                 Cmpi,A_loc,POt,PDt;
1924   Mat_PtAPMPI         *ptap;
1925   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1926   Mat_MPIAIJ          *p=(Mat_MPIAIJ*)P->data,*a=(Mat_MPIAIJ*)A->data,*c;
1927   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1928   PetscInt            nnz;
1929   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1930   PetscInt            am  =A->rmap->n,pn=P->cmap->n;
1931   MPI_Comm            comm;
1932   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1933   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1934   PetscInt            len,proc,*dnz,*onz,*owners;
1935   PetscInt            nzi,*bi,*bj;
1936   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1937   MPI_Request         *swaits,*rwaits;
1938   MPI_Status          *sstatus,rstatus;
1939   Mat_Merge_SeqsToMPI *merge;
1940   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1941   PetscReal           afill  =1.0,afill_tmp;
1942   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Armax;
1943   PetscScalar         *vals;
1944   Mat_SeqAIJ          *a_loc,*pdt,*pot;
1945   PetscTable          ta;
1946 
1947   PetscFunctionBegin;
1948   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1949   /* check if matrix local sizes are compatible */
1950   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend) SETERRQ4(comm,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, A (%D, %D) != P (%D,%D)",A->rmap->rstart,A->rmap->rend,P->rmap->rstart,P->rmap->rend);
1951 
1952   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1953   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1954 
1955   /* create struct Mat_PtAPMPI and attached it to C later */
1956   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1957 
1958   /* get A_loc by taking all local rows of A */
1959   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1960 
1961   ptap->A_loc = A_loc;
1962   a_loc       = (Mat_SeqAIJ*)(A_loc)->data;
1963   ai          = a_loc->i;
1964   aj          = a_loc->j;
1965 
1966   /* determine symbolic Co=(p->B)^T*A - send to others */
1967   /*----------------------------------------------------*/
1968   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1969   pdt  = (Mat_SeqAIJ*)PDt->data;
1970   pdti = pdt->i; pdtj = pdt->j;
1971 
1972   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1973   pot  = (Mat_SeqAIJ*)POt->data;
1974   poti = pot->i; potj = pot->j;
1975 
1976   /* then, compute symbolic Co = (p->B)^T*A */
1977   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1978                          >= (num of nonzero rows of C_seq) - pn */
1979   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
1980   coi[0] = 0;
1981 
1982   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1983   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(poti[pon],ai[am]));
1984   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1985   current_space = free_space;
1986 
1987   /* create and initialize a linked list */
1988   ierr = PetscTableCreate(A->cmap->n + a->B->cmap->N,aN,&ta);CHKERRQ(ierr);
1989   MatRowMergeMax_SeqAIJ(a_loc,am,ta);
1990   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
1991 
1992   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
1993 
1994   for (i=0; i<pon; i++) {
1995     pnz = poti[i+1] - poti[i];
1996     ptJ = potj + poti[i];
1997     for (j=0; j<pnz; j++) {
1998       row  = ptJ[j]; /* row of A_loc == col of Pot */
1999       anz  = ai[row+1] - ai[row];
2000       Jptr = aj + ai[row];
2001       /* add non-zero cols of AP into the sorted linked list lnk */
2002       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
2003     }
2004     nnz = lnk[0];
2005 
2006     /* If free space is not available, double the total space in the list */
2007     if (current_space->local_remaining<nnz) {
2008       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
2009       nspacedouble++;
2010     }
2011 
2012     /* Copy data into free space, and zero out denserows */
2013     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
2014 
2015     current_space->array           += nnz;
2016     current_space->local_used      += nnz;
2017     current_space->local_remaining -= nnz;
2018 
2019     coi[i+1] = coi[i] + nnz;
2020   }
2021 
2022   ierr = PetscMalloc1(coi[pon]+1,&coj);CHKERRQ(ierr);
2023   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
2024   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr); /* must destroy to get a new one for C */
2025 
2026   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
2027   if (afill_tmp > afill) afill = afill_tmp;
2028 
2029   /* send j-array (coj) of Co to other processors */
2030   /*----------------------------------------------*/
2031   /* determine row ownership */
2032   ierr = PetscNew(&merge);CHKERRQ(ierr);
2033   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
2034 
2035   merge->rowmap->n  = pn;
2036   merge->rowmap->bs = 1;
2037 
2038   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
2039   owners = merge->rowmap->range;
2040 
2041   /* determine the number of messages to send, their lengths */
2042   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
2043   ierr = PetscMalloc1(size,&merge->len_s);CHKERRQ(ierr);
2044 
2045   len_s        = merge->len_s;
2046   merge->nsend = 0;
2047 
2048   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
2049   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
2050 
2051   proc = 0;
2052   for (i=0; i<pon; i++) {
2053     while (prmap[i] >= owners[proc+1]) proc++;
2054     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
2055     len_s[proc] += coi[i+1] - coi[i];
2056   }
2057 
2058   len          = 0; /* max length of buf_si[] */
2059   owners_co[0] = 0;
2060   for (proc=0; proc<size; proc++) {
2061     owners_co[proc+1] = owners_co[proc] + len_si[proc];
2062     if (len_si[proc]) {
2063       merge->nsend++;
2064       len_si[proc] = 2*(len_si[proc] + 1);
2065       len         += len_si[proc];
2066     }
2067   }
2068 
2069   /* determine the number and length of messages to receive for coi and coj  */
2070   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
2071   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
2072 
2073   /* post the Irecv and Isend of coj */
2074   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
2075   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
2076   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
2077   for (proc=0, k=0; proc<size; proc++) {
2078     if (!len_s[proc]) continue;
2079     i    = owners_co[proc];
2080     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
2081     k++;
2082   }
2083 
2084   /* receives and sends of coj are complete */
2085   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
2086   for (i=0; i<merge->nrecv; i++) {
2087     PetscMPIInt icompleted;
2088     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
2089   }
2090   ierr = PetscFree(rwaits);CHKERRQ(ierr);
2091   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
2092 
2093   /* add received column indices into table to update Armax */
2094   /* Armax can be as large as aN if a P[row,:] is dense, see src/ksp/ksp/examples/tutorials/ex56.c! */
2095   for (k=0; k<merge->nrecv; k++) {/* k-th received message */
2096     Jptr = buf_rj[k];
2097     for (j=0; j<merge->len_r[k]; j++) {
2098       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
2099     }
2100   }
2101   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
2102   /* printf("Armax %d, an %d + Bn %d = %d, aN %d\n",Armax,A->cmap->n,a->B->cmap->N,A->cmap->n+a->B->cmap->N,aN); */
2103 
2104   /* send and recv coi */
2105   /*-------------------*/
2106   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
2107   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
2108   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
2109   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
2110   for (proc=0,k=0; proc<size; proc++) {
2111     if (!len_s[proc]) continue;
2112     /* form outgoing message for i-structure:
2113          buf_si[0]:                 nrows to be sent
2114                [1:nrows]:           row index (global)
2115                [nrows+1:2*nrows+1]: i-structure index
2116     */
2117     /*-------------------------------------------*/
2118     nrows       = len_si[proc]/2 - 1;
2119     buf_si_i    = buf_si + nrows+1;
2120     buf_si[0]   = nrows;
2121     buf_si_i[0] = 0;
2122     nrows       = 0;
2123     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
2124       nzi               = coi[i+1] - coi[i];
2125       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
2126       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
2127       nrows++;
2128     }
2129     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
2130     k++;
2131     buf_si += len_si[proc];
2132   }
2133   i = merge->nrecv;
2134   while (i--) {
2135     PetscMPIInt icompleted;
2136     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
2137   }
2138   ierr = PetscFree(rwaits);CHKERRQ(ierr);
2139   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
2140   ierr = PetscFree(len_si);CHKERRQ(ierr);
2141   ierr = PetscFree(len_ri);CHKERRQ(ierr);
2142   ierr = PetscFree(swaits);CHKERRQ(ierr);
2143   ierr = PetscFree(sstatus);CHKERRQ(ierr);
2144   ierr = PetscFree(buf_s);CHKERRQ(ierr);
2145 
2146   /* compute the local portion of C (mpi mat) */
2147   /*------------------------------------------*/
2148   /* allocate bi array and free space for accumulating nonzero column info */
2149   ierr  = PetscMalloc1(pn+1,&bi);CHKERRQ(ierr);
2150   bi[0] = 0;
2151 
2152   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
2153   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(pdti[pn],PetscIntSumTruncate(poti[pon],ai[am])));
2154   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
2155   current_space = free_space;
2156 
2157   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
2158   for (k=0; k<merge->nrecv; k++) {
2159     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
2160     nrows       = *buf_ri_k[k];
2161     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
2162     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th recieved i-structure  */
2163   }
2164 
2165   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
2166   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
2167   rmax = 0;
2168   for (i=0; i<pn; i++) {
2169     /* add pdt[i,:]*AP into lnk */
2170     pnz = pdti[i+1] - pdti[i];
2171     ptJ = pdtj + pdti[i];
2172     for (j=0; j<pnz; j++) {
2173       row  = ptJ[j];  /* row of AP == col of Pt */
2174       anz  = ai[row+1] - ai[row];
2175       Jptr = aj + ai[row];
2176       /* add non-zero cols of AP into the sorted linked list lnk */
2177       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
2178     }
2179 
2180     /* add received col data into lnk */
2181     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
2182       if (i == *nextrow[k]) { /* i-th row */
2183         nzi  = *(nextci[k]+1) - *nextci[k];
2184         Jptr = buf_rj[k] + *nextci[k];
2185         ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
2186         nextrow[k]++; nextci[k]++;
2187       }
2188     }
2189     nnz = lnk[0];
2190 
2191     /* if free space is not available, make more free space */
2192     if (current_space->local_remaining<nnz) {
2193       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
2194       nspacedouble++;
2195     }
2196     /* copy data into free space, then initialize lnk */
2197     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
2198     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
2199 
2200     current_space->array           += nnz;
2201     current_space->local_used      += nnz;
2202     current_space->local_remaining -= nnz;
2203 
2204     bi[i+1] = bi[i] + nnz;
2205     if (nnz > rmax) rmax = nnz;
2206   }
2207   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
2208 
2209   ierr      = PetscMalloc1(bi[pn]+1,&bj);CHKERRQ(ierr);
2210   ierr      = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
2211   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
2212   if (afill_tmp > afill) afill = afill_tmp;
2213   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
2214   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
2215 
2216   ierr = MatDestroy(&POt);CHKERRQ(ierr);
2217   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
2218 
2219   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
2220   /*----------------------------------------------------------------------------------*/
2221   ierr = PetscCalloc1(rmax+1,&vals);CHKERRQ(ierr);
2222 
2223   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
2224   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
2225   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
2226   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
2227   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
2228   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
2229   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
2230   for (i=0; i<pn; i++) {
2231     row  = i + rstart;
2232     nnz  = bi[i+1] - bi[i];
2233     Jptr = bj + bi[i];
2234     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
2235   }
2236   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2237   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2238   ierr = PetscFree(vals);CHKERRQ(ierr);
2239 
2240   merge->bi        = bi;
2241   merge->bj        = bj;
2242   merge->coi       = coi;
2243   merge->coj       = coj;
2244   merge->buf_ri    = buf_ri;
2245   merge->buf_rj    = buf_rj;
2246   merge->owners_co = owners_co;
2247 
2248   /* attach the supporting struct to Cmpi for reuse */
2249   c = (Mat_MPIAIJ*)Cmpi->data;
2250 
2251   c->ptap     = ptap;
2252   ptap->api   = NULL;
2253   ptap->apj   = NULL;
2254   ptap->merge = merge;
2255   ptap->apa   = NULL;
2256   ptap->destroy   = Cmpi->ops->destroy;
2257   ptap->duplicate = Cmpi->ops->duplicate;
2258 
2259   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ;
2260   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
2261   Cmpi->ops->duplicate               = MatDuplicate_MPIAIJ_MatPtAP;
2262 
2263   *C = Cmpi;
2264 #if defined(PETSC_USE_INFO)
2265   if (bi[pn] != 0) {
2266     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
2267     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
2268   } else {
2269     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
2270   }
2271 #endif
2272   PetscFunctionReturn(0);
2273 }
2274