xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision 8a210135f4b6c0580b34d5c022516dba42e89106)
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[4] = {"scalable","nonscalable","seqmpi","hypre"};
22   PetscInt       nalg = 4;
23 #else
24   const char     *algTypes[3] = {"scalable","nonscalable","seqmpi"};
25   PetscInt       nalg = 3;
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     case 2:
65       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ_seqMPI(A,B,fill,C);CHKERRQ(ierr);
66       break;
67 #if defined(PETSC_HAVE_HYPRE)
68     case 3:
69       ierr = MatMatMultSymbolic_AIJ_AIJ_wHYPRE(A,B,fill,C);CHKERRQ(ierr);
70       break;
71 #endif
72     default:
73       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ(A,B,fill,C);CHKERRQ(ierr);
74       break;
75     }
76     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
77   }
78   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
79   ierr = (*(*C)->ops->matmultnumeric)(A,B,*C);CHKERRQ(ierr);
80   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
81   PetscFunctionReturn(0);
82 }
83 
84 PetscErrorCode MatDestroy_MPIAIJ_MatMatMult(Mat A)
85 {
86   PetscErrorCode ierr;
87   Mat_MPIAIJ     *a    = (Mat_MPIAIJ*)A->data;
88   Mat_PtAPMPI    *ptap = a->ptap;
89 
90   PetscFunctionBegin;
91   ierr = PetscFree2(ptap->startsj_s,ptap->startsj_r);CHKERRQ(ierr);
92   ierr = PetscFree(ptap->bufa);CHKERRQ(ierr);
93   ierr = MatDestroy(&ptap->P_loc);CHKERRQ(ierr);
94   ierr = MatDestroy(&ptap->P_oth);CHKERRQ(ierr);
95   ierr = MatDestroy(&ptap->Pt);CHKERRQ(ierr);
96   ierr = PetscFree(ptap->api);CHKERRQ(ierr);
97   ierr = PetscFree(ptap->apj);CHKERRQ(ierr);
98   ierr = PetscFree(ptap->apa);CHKERRQ(ierr);
99   ierr = ptap->destroy(A);CHKERRQ(ierr);
100   ierr = PetscFree(ptap);CHKERRQ(ierr);
101   PetscFunctionReturn(0);
102 }
103 
104 PetscErrorCode MatDuplicate_MPIAIJ_MatMatMult(Mat A, MatDuplicateOption op, Mat *M)
105 {
106   PetscErrorCode ierr;
107   Mat_MPIAIJ     *a    = (Mat_MPIAIJ*)A->data;
108   Mat_PtAPMPI    *ptap = a->ptap;
109 
110   PetscFunctionBegin;
111   ierr = (*ptap->duplicate)(A,op,M);CHKERRQ(ierr);
112 
113   (*M)->ops->destroy   = ptap->destroy;   /* = MatDestroy_MPIAIJ, *M doesn't duplicate A's special structure! */
114   (*M)->ops->duplicate = ptap->duplicate; /* = MatDuplicate_MPIAIJ */
115   PetscFunctionReturn(0);
116 }
117 
118 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,Mat C)
119 {
120   PetscErrorCode ierr;
121   Mat_MPIAIJ     *a  =(Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
122   Mat_SeqAIJ     *ad =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
123   Mat_SeqAIJ     *cd =(Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
124   PetscScalar    *cda=cd->a,*coa=co->a;
125   Mat_SeqAIJ     *p_loc,*p_oth;
126   PetscScalar    *apa,*ca;
127   PetscInt       cm   =C->rmap->n;
128   Mat_PtAPMPI    *ptap=c->ptap;
129   PetscInt       *api,*apj,*apJ,i,k;
130   PetscInt       cstart=C->cmap->rstart;
131   PetscInt       cdnz,conz,k0,k1;
132   MPI_Comm       comm;
133   PetscMPIInt    size;
134 
135   PetscFunctionBegin;
136   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
137   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
138 
139   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
140   /*-----------------------------------------------------*/
141   /* update numerical values of P_oth and P_loc */
142   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
143   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
144 
145   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
146   /*----------------------------------------------------------*/
147   /* get data from symbolic products */
148   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
149   p_oth = NULL;
150   if (size >1) {
151     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
152   }
153 
154   /* get apa for storing dense row A[i,:]*P */
155   apa = ptap->apa;
156 
157   api = ptap->api;
158   apj = ptap->apj;
159   for (i=0; i<cm; i++) {
160     /* compute apa = A[i,:]*P */
161     AProw_nonscalable(i,ad,ao,p_loc,p_oth,apa);
162 
163     /* set values in C */
164     apJ  = apj + api[i];
165     cdnz = cd->i[i+1] - cd->i[i];
166     conz = co->i[i+1] - co->i[i];
167 
168     /* 1st off-diagoanl part of C */
169     ca = coa + co->i[i];
170     k  = 0;
171     for (k0=0; k0<conz; k0++) {
172       if (apJ[k] >= cstart) break;
173       ca[k0]      = apa[apJ[k]];
174       apa[apJ[k++]] = 0.0;
175     }
176 
177     /* diagonal part of C */
178     ca = cda + cd->i[i];
179     for (k1=0; k1<cdnz; k1++) {
180       ca[k1]      = apa[apJ[k]];
181       apa[apJ[k++]] = 0.0;
182     }
183 
184     /* 2nd off-diagoanl part of C */
185     ca = coa + co->i[i];
186     for (; k0<conz; k0++) {
187       ca[k0]      = apa[apJ[k]];
188       apa[apJ[k++]] = 0.0;
189     }
190   }
191   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
192   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
193   PetscFunctionReturn(0);
194 }
195 
196 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,PetscReal fill,Mat *C)
197 {
198   PetscErrorCode     ierr;
199   MPI_Comm           comm;
200   PetscMPIInt        size;
201   Mat                Cmpi;
202   Mat_PtAPMPI        *ptap;
203   PetscFreeSpaceList free_space=NULL,current_space=NULL;
204   Mat_MPIAIJ         *a        =(Mat_MPIAIJ*)A->data,*c;
205   Mat_SeqAIJ         *ad       =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
206   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
207   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
208   PetscInt           *lnk,i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi;
209   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
210   PetscBT            lnkbt;
211   PetscScalar        *apa;
212   PetscReal          afill;
213 
214   PetscFunctionBegin;
215   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
216   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
217 
218   /* create struct Mat_PtAPMPI and attached it to C later */
219   ierr = PetscNew(&ptap);CHKERRQ(ierr);
220 
221   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
222   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
223 
224   /* get P_loc by taking all local rows of P */
225   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
226 
227   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
228   pi_loc = p_loc->i; pj_loc = p_loc->j;
229   if (size > 1) {
230     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
231     pi_oth = p_oth->i; pj_oth = p_oth->j;
232   } else {
233     p_oth = NULL;
234     pi_oth = NULL; pj_oth = NULL;
235   }
236 
237   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
238   /*-------------------------------------------------------------------*/
239   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
240   ptap->api = api;
241   api[0]    = 0;
242 
243   /* create and initialize a linked list */
244   ierr = PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt);CHKERRQ(ierr);
245 
246   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
247   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space);CHKERRQ(ierr);
248   current_space = free_space;
249 
250   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
251   for (i=0; i<am; i++) {
252     /* diagonal portion of A */
253     nzi = adi[i+1] - adi[i];
254     for (j=0; j<nzi; j++) {
255       row  = *adj++;
256       pnz  = pi_loc[row+1] - pi_loc[row];
257       Jptr = pj_loc + pi_loc[row];
258       /* add non-zero cols of P into the sorted linked list lnk */
259       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
260     }
261     /* off-diagonal portion of A */
262     nzi = aoi[i+1] - aoi[i];
263     for (j=0; j<nzi; j++) {
264       row  = *aoj++;
265       pnz  = pi_oth[row+1] - pi_oth[row];
266       Jptr = pj_oth + pi_oth[row];
267       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
268     }
269 
270     apnz     = lnk[0];
271     api[i+1] = api[i] + apnz;
272 
273     /* if free space is not available, double the total space in the list */
274     if (current_space->local_remaining<apnz) {
275       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
276       nspacedouble++;
277     }
278 
279     /* Copy data into free space, then initialize lnk */
280     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
281     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
282 
283     current_space->array           += apnz;
284     current_space->local_used      += apnz;
285     current_space->local_remaining -= apnz;
286   }
287 
288   /* Allocate space for apj, initialize apj, and */
289   /* destroy list of free space and other temporary array(s) */
290   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
291   apj  = ptap->apj;
292   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
293   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
294 
295   /* malloc apa to store dense row A[i,:]*P */
296   ierr = PetscCalloc1(pN,&apa);CHKERRQ(ierr);
297 
298   ptap->apa = apa;
299 
300   /* create and assemble symbolic parallel matrix Cmpi */
301   /*----------------------------------------------------*/
302   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
303   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
304   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
305 
306   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
307   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
308 
309   ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(Cmpi, apj, api, dnz, onz);CHKERRQ(ierr);
310   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
311   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
312   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
313 
314   ptap->destroy        = Cmpi->ops->destroy;
315   ptap->duplicate      = Cmpi->ops->duplicate;
316   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
317   Cmpi->ops->destroy   = MatDestroy_MPIAIJ_MatMatMult;
318   Cmpi->ops->duplicate = MatDuplicate_MPIAIJ_MatMatMult;
319 
320   /* attach the supporting struct to Cmpi for reuse */
321   c       = (Mat_MPIAIJ*)Cmpi->data;
322   c->ptap = ptap;
323 
324   *C = Cmpi;
325 
326   /* set MatInfo */
327   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
328   if (afill < 1.0) afill = 1.0;
329   Cmpi->info.mallocs           = nspacedouble;
330   Cmpi->info.fill_ratio_given  = fill;
331   Cmpi->info.fill_ratio_needed = afill;
332 
333 #if defined(PETSC_USE_INFO)
334   if (api[am]) {
335     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
336     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
337   } else {
338     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
339   }
340 #endif
341   PetscFunctionReturn(0);
342 }
343 
344 PETSC_INTERN PetscErrorCode MatMatMult_MPIAIJ_MPIDense(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
345 {
346   PetscErrorCode ierr;
347 
348   PetscFunctionBegin;
349   if (scall == MAT_INITIAL_MATRIX) {
350     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
351     ierr = MatMatMultSymbolic_MPIAIJ_MPIDense(A,B,fill,C);CHKERRQ(ierr);
352     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
353   }
354   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
355   ierr = MatMatMultNumeric_MPIAIJ_MPIDense(A,B,*C);CHKERRQ(ierr);
356   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
357   PetscFunctionReturn(0);
358 }
359 
360 typedef struct {
361   Mat         workB;
362   PetscScalar *rvalues,*svalues;
363   MPI_Request *rwaits,*swaits;
364 } MPIAIJ_MPIDense;
365 
366 PetscErrorCode MatMPIAIJ_MPIDenseDestroy(void *ctx)
367 {
368   MPIAIJ_MPIDense *contents = (MPIAIJ_MPIDense*) ctx;
369   PetscErrorCode  ierr;
370 
371   PetscFunctionBegin;
372   ierr = MatDestroy(&contents->workB);CHKERRQ(ierr);
373   ierr = PetscFree4(contents->rvalues,contents->svalues,contents->rwaits,contents->swaits);CHKERRQ(ierr);
374   ierr = PetscFree(contents);CHKERRQ(ierr);
375   PetscFunctionReturn(0);
376 }
377 
378 /*
379     This is a "dummy function" that handles the case where matrix C was created as a dense matrix
380   directly by the user and passed to MatMatMult() with the MAT_REUSE_MATRIX option
381 
382   It is the same as MatMatMultSymbolic_MPIAIJ_MPIDense() except does not create C
383 */
384 PetscErrorCode MatMatMultNumeric_MPIDense(Mat A,Mat B,Mat C)
385 {
386   PetscErrorCode         ierr;
387   PetscBool              flg;
388   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*) A->data;
389   PetscInt               nz   = aij->B->cmap->n;
390   PetscContainer         container;
391   MPIAIJ_MPIDense        *contents;
392   VecScatter             ctx   = aij->Mvctx;
393   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
394   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
395 
396   PetscFunctionBegin;
397   ierr = PetscObjectTypeCompare((PetscObject)B,MATMPIDENSE,&flg);CHKERRQ(ierr);
398   if (!flg) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Second matrix must be mpidense");
399 
400   /* Handle case where where user provided the final C matrix rather than calling MatMatMult() with MAT_INITIAL_MATRIX*/
401   ierr = PetscObjectTypeCompare((PetscObject)A,MATMPIAIJ,&flg);CHKERRQ(ierr);
402   if (!flg) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"First matrix must be MPIAIJ");
403 
404   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
405 
406   ierr = PetscNew(&contents);CHKERRQ(ierr);
407   /* Create work matrix used to store off processor rows of B needed for local product */
408   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,B->cmap->N,NULL,&contents->workB);CHKERRQ(ierr);
409   /* Create work arrays needed */
410   ierr = PetscMalloc4(B->cmap->N*from->starts[from->n],&contents->rvalues,
411                       B->cmap->N*to->starts[to->n],&contents->svalues,
412                       from->n,&contents->rwaits,
413                       to->n,&contents->swaits);CHKERRQ(ierr);
414 
415   ierr = PetscContainerCreate(PetscObjectComm((PetscObject)A),&container);CHKERRQ(ierr);
416   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
417   ierr = PetscContainerSetUserDestroy(container,MatMPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
418   ierr = PetscObjectCompose((PetscObject)C,"workB",(PetscObject)container);CHKERRQ(ierr);
419   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
420 
421   ierr = (*C->ops->matmultnumeric)(A,B,C);CHKERRQ(ierr);
422   PetscFunctionReturn(0);
423 }
424 
425 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat A,Mat B,PetscReal fill,Mat *C)
426 {
427   PetscErrorCode         ierr;
428   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*) A->data;
429   PetscInt               nz   = aij->B->cmap->n;
430   PetscContainer         container;
431   MPIAIJ_MPIDense        *contents;
432   VecScatter             ctx   = aij->Mvctx;
433   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
434   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
435   PetscInt               m     = A->rmap->n,n=B->cmap->n;
436 
437   PetscFunctionBegin;
438   ierr = MatCreate(PetscObjectComm((PetscObject)B),C);CHKERRQ(ierr);
439   ierr = MatSetSizes(*C,m,n,A->rmap->N,B->cmap->N);CHKERRQ(ierr);
440   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
441   ierr = MatSetType(*C,MATMPIDENSE);CHKERRQ(ierr);
442   ierr = MatMPIDenseSetPreallocation(*C,NULL);CHKERRQ(ierr);
443   ierr = MatAssemblyBegin(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
444   ierr = MatAssemblyEnd(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
445 
446   (*C)->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
447 
448   ierr = PetscNew(&contents);CHKERRQ(ierr);
449   /* Create work matrix used to store off processor rows of B needed for local product */
450   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,B->cmap->N,NULL,&contents->workB);CHKERRQ(ierr);
451   /* Create work arrays needed */
452   ierr = PetscMalloc4(B->cmap->N*from->starts[from->n],&contents->rvalues,
453                       B->cmap->N*to->starts[to->n],&contents->svalues,
454                       from->n,&contents->rwaits,
455                       to->n,&contents->swaits);CHKERRQ(ierr);
456 
457   ierr = PetscContainerCreate(PetscObjectComm((PetscObject)A),&container);CHKERRQ(ierr);
458   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
459   ierr = PetscContainerSetUserDestroy(container,MatMPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
460   ierr = PetscObjectCompose((PetscObject)(*C),"workB",(PetscObject)container);CHKERRQ(ierr);
461   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
462   PetscFunctionReturn(0);
463 }
464 
465 /*
466     Performs an efficient scatter on the rows of B needed by this process; this is
467     a modification of the VecScatterBegin_() routines.
468 */
469 PetscErrorCode MatMPIDenseScatter(Mat A,Mat B,Mat C,Mat *outworkB)
470 {
471   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*)A->data;
472   PetscErrorCode         ierr;
473   PetscScalar            *b,*w,*svalues,*rvalues;
474   VecScatter             ctx   = aij->Mvctx;
475   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
476   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
477   PetscInt               i,j,k;
478   PetscInt               *sindices,*sstarts,*rindices,*rstarts;
479   PetscMPIInt            *sprocs,*rprocs,nrecvs;
480   MPI_Request            *swaits,*rwaits;
481   MPI_Comm               comm;
482   PetscMPIInt            tag  = ((PetscObject)ctx)->tag,ncols = B->cmap->N, nrows = aij->B->cmap->n,imdex,nrowsB = B->rmap->n;
483   MPI_Status             status;
484   MPIAIJ_MPIDense        *contents;
485   PetscContainer         container;
486   Mat                    workB;
487 
488   PetscFunctionBegin;
489   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
490   ierr = PetscObjectQuery((PetscObject)C,"workB",(PetscObject*)&container);CHKERRQ(ierr);
491   if (!container) SETERRQ(comm,PETSC_ERR_PLIB,"Container does not exist");
492   ierr = PetscContainerGetPointer(container,(void**)&contents);CHKERRQ(ierr);
493 
494   workB = *outworkB = contents->workB;
495   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);
496   sindices = to->indices;
497   sstarts  = to->starts;
498   sprocs   = to->procs;
499   swaits   = contents->swaits;
500   svalues  = contents->svalues;
501 
502   rindices = from->indices;
503   rstarts  = from->starts;
504   rprocs   = from->procs;
505   rwaits   = contents->rwaits;
506   rvalues  = contents->rvalues;
507 
508   ierr = MatDenseGetArray(B,&b);CHKERRQ(ierr);
509   ierr = MatDenseGetArray(workB,&w);CHKERRQ(ierr);
510 
511   for (i=0; i<from->n; i++) {
512     ierr = MPI_Irecv(rvalues+ncols*rstarts[i],ncols*(rstarts[i+1]-rstarts[i]),MPIU_SCALAR,rprocs[i],tag,comm,rwaits+i);CHKERRQ(ierr);
513   }
514 
515   for (i=0; i<to->n; i++) {
516     /* pack a message at a time */
517     for (j=0; j<sstarts[i+1]-sstarts[i]; j++) {
518       for (k=0; k<ncols; k++) {
519         svalues[ncols*(sstarts[i] + j) + k] = b[sindices[sstarts[i]+j] + nrowsB*k];
520       }
521     }
522     ierr = MPI_Isend(svalues+ncols*sstarts[i],ncols*(sstarts[i+1]-sstarts[i]),MPIU_SCALAR,sprocs[i],tag,comm,swaits+i);CHKERRQ(ierr);
523   }
524 
525   nrecvs = from->n;
526   while (nrecvs) {
527     ierr = MPI_Waitany(from->n,rwaits,&imdex,&status);CHKERRQ(ierr);
528     nrecvs--;
529     /* unpack a message at a time */
530     for (j=0; j<rstarts[imdex+1]-rstarts[imdex]; j++) {
531       for (k=0; k<ncols; k++) {
532         w[rindices[rstarts[imdex]+j] + nrows*k] = rvalues[ncols*(rstarts[imdex] + j) + k];
533       }
534     }
535   }
536   if (to->n) {ierr = MPI_Waitall(to->n,swaits,to->sstatus);CHKERRQ(ierr);}
537 
538   ierr = MatDenseRestoreArray(B,&b);CHKERRQ(ierr);
539   ierr = MatDenseRestoreArray(workB,&w);CHKERRQ(ierr);
540   ierr = MatAssemblyBegin(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
541   ierr = MatAssemblyEnd(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
542   PetscFunctionReturn(0);
543 }
544 extern PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat,Mat,Mat);
545 
546 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat A,Mat B,Mat C)
547 {
548   PetscErrorCode ierr;
549   Mat_MPIAIJ     *aij    = (Mat_MPIAIJ*)A->data;
550   Mat_MPIDense   *bdense = (Mat_MPIDense*)B->data;
551   Mat_MPIDense   *cdense = (Mat_MPIDense*)C->data;
552   Mat            workB;
553 
554   PetscFunctionBegin;
555   /* diagonal block of A times all local rows of B*/
556   ierr = MatMatMultNumeric_SeqAIJ_SeqDense(aij->A,bdense->A,cdense->A);CHKERRQ(ierr);
557 
558   /* get off processor parts of B needed to complete the product */
559   ierr = MatMPIDenseScatter(A,B,C,&workB);CHKERRQ(ierr);
560 
561   /* off-diagonal block of A times nonlocal rows of B */
562   ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A);CHKERRQ(ierr);
563   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
564   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
565   PetscFunctionReturn(0);
566 }
567 
568 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
569 {
570   PetscErrorCode ierr;
571   Mat_MPIAIJ     *a   = (Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
572   Mat_SeqAIJ     *ad  = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
573   Mat_SeqAIJ     *cd  = (Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
574   PetscInt       *adi = ad->i,*adj,*aoi=ao->i,*aoj;
575   PetscScalar    *ada,*aoa,*cda=cd->a,*coa=co->a;
576   Mat_SeqAIJ     *p_loc,*p_oth;
577   PetscInt       *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
578   PetscScalar    *pa_loc,*pa_oth,*pa,valtmp,*ca;
579   PetscInt       cm          = C->rmap->n,anz,pnz;
580   Mat_PtAPMPI    *ptap       = c->ptap;
581   PetscScalar    *apa_sparse = ptap->apa;
582   PetscInt       *api,*apj,*apJ,i,j,k,row;
583   PetscInt       cstart = C->cmap->rstart;
584   PetscInt       cdnz,conz,k0,k1,nextp;
585   MPI_Comm       comm;
586   PetscMPIInt    size;
587 
588   PetscFunctionBegin;
589   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
590   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
591 
592   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
593   /*-----------------------------------------------------*/
594   /* update numerical values of P_oth and P_loc */
595   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
596   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
597 
598   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
599   /*----------------------------------------------------------*/
600   /* get data from symbolic products */
601   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
602   pi_loc = p_loc->i; pj_loc = p_loc->j; pa_loc = p_loc->a;
603   if (size >1) {
604     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
605     pi_oth = p_oth->i; pj_oth = p_oth->j; pa_oth = p_oth->a;
606   } else {
607     p_oth = NULL; pi_oth = NULL; pj_oth = NULL; pa_oth = NULL;
608   }
609 
610   api = ptap->api;
611   apj = ptap->apj;
612   for (i=0; i<cm; i++) {
613     apJ = apj + api[i];
614 
615     /* diagonal portion of A */
616     anz = adi[i+1] - adi[i];
617     adj = ad->j + adi[i];
618     ada = ad->a + adi[i];
619     for (j=0; j<anz; j++) {
620       row = adj[j];
621       pnz = pi_loc[row+1] - pi_loc[row];
622       pj  = pj_loc + pi_loc[row];
623       pa  = pa_loc + pi_loc[row];
624       /* perform sparse axpy */
625       valtmp = ada[j];
626       nextp  = 0;
627       for (k=0; nextp<pnz; k++) {
628         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
629           apa_sparse[k] += valtmp*pa[nextp++];
630         }
631       }
632       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
633     }
634 
635     /* off-diagonal portion of A */
636     anz = aoi[i+1] - aoi[i];
637     aoj = ao->j + aoi[i];
638     aoa = ao->a + aoi[i];
639     for (j=0; j<anz; j++) {
640       row = aoj[j];
641       pnz = pi_oth[row+1] - pi_oth[row];
642       pj  = pj_oth + pi_oth[row];
643       pa  = pa_oth + pi_oth[row];
644       /* perform sparse axpy */
645       valtmp = aoa[j];
646       nextp  = 0;
647       for (k=0; nextp<pnz; k++) {
648         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
649           apa_sparse[k] += valtmp*pa[nextp++];
650         }
651       }
652       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
653     }
654 
655     /* set values in C */
656     cdnz = cd->i[i+1] - cd->i[i];
657     conz = co->i[i+1] - co->i[i];
658 
659     /* 1st off-diagoanl part of C */
660     ca = coa + co->i[i];
661     k  = 0;
662     for (k0=0; k0<conz; k0++) {
663       if (apJ[k] >= cstart) break;
664       ca[k0]        = apa_sparse[k];
665       apa_sparse[k] = 0.0;
666       k++;
667     }
668 
669     /* diagonal part of C */
670     ca = cda + cd->i[i];
671     for (k1=0; k1<cdnz; k1++) {
672       ca[k1]        = apa_sparse[k];
673       apa_sparse[k] = 0.0;
674       k++;
675     }
676 
677     /* 2nd off-diagoanl part of C */
678     ca = coa + co->i[i];
679     for (; k0<conz; k0++) {
680       ca[k0]        = apa_sparse[k];
681       apa_sparse[k] = 0.0;
682       k++;
683     }
684   }
685   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
686   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
687   PetscFunctionReturn(0);
688 }
689 
690 /* same as MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(), except using LLCondensed to avoid O(BN) memory requirement */
691 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat *C)
692 {
693   PetscErrorCode     ierr;
694   MPI_Comm           comm;
695   PetscMPIInt        size;
696   Mat                Cmpi;
697   Mat_PtAPMPI        *ptap;
698   PetscFreeSpaceList free_space = NULL,current_space=NULL;
699   Mat_MPIAIJ         *a         = (Mat_MPIAIJ*)A->data,*c;
700   Mat_SeqAIJ         *ad        = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
701   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
702   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
703   PetscInt           i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi,*lnk,apnz_max=0;
704   PetscInt           am=A->rmap->n,pn=P->cmap->n,pm=P->rmap->n,lsize=20;
705   PetscReal          afill;
706   PetscScalar        *apa;
707 
708   PetscFunctionBegin;
709   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
710   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
711 
712   /* create struct Mat_PtAPMPI and attached it to C later */
713   ierr = PetscNew(&ptap);CHKERRQ(ierr);
714 
715   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
716   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
717 
718   /* get P_loc by taking all local rows of P */
719   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
720 
721   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
722   pi_loc = p_loc->i; pj_loc = p_loc->j;
723   if (size > 1) {
724     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
725     pi_oth = p_oth->i; pj_oth = p_oth->j;
726   } else {
727     p_oth  = NULL;
728     pi_oth = NULL; pj_oth = NULL;
729   }
730 
731   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
732   /*-------------------------------------------------------------------*/
733   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
734   ptap->api = api;
735   api[0]    = 0;
736 
737   ierr = PetscLLCondensedCreate_Scalable(lsize,&lnk);CHKERRQ(ierr);
738 
739   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
740   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space);CHKERRQ(ierr);
741   current_space = free_space;
742   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
743   for (i=0; i<am; i++) {
744     /* diagonal portion of A */
745     nzi = adi[i+1] - adi[i];
746     for (j=0; j<nzi; j++) {
747       row  = *adj++;
748       pnz  = pi_loc[row+1] - pi_loc[row];
749       Jptr = pj_loc + pi_loc[row];
750       /* Expand list if it is not long enough */
751       while (pnz+apnz_max > lsize) {
752         /* Expand list by 30%. This expansion typically occurs only rarely and the list */
753         /* is typically quite short, so this factor does not have a big impact on performance and memory usage. */
754         lsize *= 1.3;
755         ierr = PetscLLCondensedExpand_Scalable(lsize, &lnk);
756       }
757       /* add non-zero cols of P into the sorted linked list lnk */
758       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
759       apnz     = *lnk; /* The first element in the list is the number of items in the list */
760       api[i+1] = api[i] + apnz;
761       if (apnz > apnz_max) apnz_max = apnz;
762     }
763     /* off-diagonal portion of A */
764     nzi = aoi[i+1] - aoi[i];
765     for (j=0; j<nzi; j++) {
766       row  = *aoj++;
767       pnz  = pi_oth[row+1] - pi_oth[row];
768       Jptr = pj_oth + pi_oth[row];
769       /* Expand list if it is not long enough */
770       while (pnz+apnz_max > lsize) {
771         lsize *= 1.3;
772         ierr = PetscLLCondensedExpand_Scalable(lsize, &lnk);
773       }
774       /* add non-zero cols of P into the sorted linked list lnk */
775       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
776       apnz     = *lnk;  /* The first element in the list is the number of items in the list */
777       api[i+1] = api[i] + apnz;
778       if (apnz > apnz_max) apnz_max = apnz;
779     }
780     apnz     = *lnk;
781     api[i+1] = api[i] + apnz;
782     if (apnz > apnz_max) apnz_max = apnz;
783 
784     /* if free space is not available, double the total space in the list */
785     if (current_space->local_remaining<apnz) {
786       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
787       nspacedouble++;
788     }
789 
790     /* Copy data into free space, then initialize lnk */
791     ierr = PetscLLCondensedClean_Scalable(apnz,current_space->array,lnk);CHKERRQ(ierr);
792     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
793 
794     current_space->array           += apnz;
795     current_space->local_used      += apnz;
796     current_space->local_remaining -= apnz;
797   }
798 
799   /* Allocate space for apj, initialize apj, and */
800   /* destroy list of free space and other temporary array(s) */
801   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
802   apj  = ptap->apj;
803   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
804   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
805 
806   /* create and assemble symbolic parallel matrix Cmpi */
807   /*----------------------------------------------------*/
808   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
809   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
810   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
811   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
812   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
813 
814   /* malloc apa for assembly Cmpi */
815   ierr = PetscCalloc1(apnz_max,&apa);CHKERRQ(ierr);
816   ptap->apa = apa;
817 
818   ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(Cmpi, apj, api, dnz, onz);CHKERRQ(ierr);
819   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
820   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
821   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
822 
823   ptap->destroy             = Cmpi->ops->destroy;
824   ptap->duplicate           = Cmpi->ops->duplicate;
825   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ;
826   Cmpi->ops->destroy        = MatDestroy_MPIAIJ_MatMatMult;
827   Cmpi->ops->duplicate      = MatDuplicate_MPIAIJ_MatMatMult;
828 
829   /* attach the supporting struct to Cmpi for reuse */
830   c       = (Mat_MPIAIJ*)Cmpi->data;
831   c->ptap = ptap;
832   *C = Cmpi;
833 
834   /* set MatInfo */
835   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
836   if (afill < 1.0) afill = 1.0;
837   Cmpi->info.mallocs           = nspacedouble;
838   Cmpi->info.fill_ratio_given  = fill;
839   Cmpi->info.fill_ratio_needed = afill;
840 
841 #if defined(PETSC_USE_INFO)
842   if (api[am]) {
843     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
844     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
845   } else {
846     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
847   }
848 #endif
849   PetscFunctionReturn(0);
850 }
851 
852 /* This function is needed for the seqMPI matrix-matrix multiplication.  */
853 /* Three input arrays are merged to one output array. The size of the    */
854 /* output array is also output. Duplicate entries only show up once.     */
855 static void Merge3SortedArrays(PetscInt  size1, PetscInt *in1,
856                                PetscInt  size2, PetscInt *in2,
857                                PetscInt  size3, PetscInt *in3,
858                                PetscInt *size4, PetscInt *out)
859 {
860   int i = 0, j = 0, k = 0, l = 0;
861 
862   /* Traverse all three arrays */
863   while (i<size1 && j<size2 && k<size3) {
864     if (in1[i] < in2[j] && in1[i] < in3[k]) {
865       out[l++] = in1[i++];
866     }
867     else if(in2[j] < in1[i] && in2[j] < in3[k]) {
868       out[l++] = in2[j++];
869     }
870     else if(in3[k] < in1[i] && in3[k] < in2[j]) {
871       out[l++] = in3[k++];
872     }
873     else if(in1[i] == in2[j] && in1[i] < in3[k]) {
874       out[l++] = in1[i];
875       i++, j++;
876     }
877     else if(in1[i] == in3[k] && in1[i] < in2[j]) {
878       out[l++] = in1[i];
879       i++, k++;
880     }
881     else if(in3[k] == in2[j] && in2[j] < in1[i])  {
882       out[l++] = in2[j];
883       k++, j++;
884     }
885     else if(in1[i] == in2[j] && in1[i] == in3[k]) {
886       out[l++] = in1[i];
887       i++, j++, k++;
888     }
889   }
890 
891   /* Traverse two remaining arrays */
892   while (i<size1 && j<size2) {
893     if (in1[i] < in2[j]) {
894       out[l++] = in1[i++];
895     }
896     else if(in1[i] > in2[j]) {
897       out[l++] = in2[j++];
898     }
899     else {
900       out[l++] = in1[i];
901       i++, j++;
902     }
903   }
904 
905   while (i<size1 && k<size3) {
906     if (in1[i] < in3[k]) {
907       out[l++] = in1[i++];
908     }
909     else if(in1[i] > in3[k]) {
910       out[l++] = in3[k++];
911     }
912     else {
913       out[l++] = in1[i];
914       i++, k++;
915     }
916   }
917 
918   while (k<size3 && j<size2)  {
919     if (in3[k] < in2[j]) {
920       out[l++] = in3[k++];
921     }
922     else if(in3[k] > in2[j]) {
923       out[l++] = in2[j++];
924     }
925     else {
926       out[l++] = in3[k];
927       k++, j++;
928     }
929   }
930 
931   /* Traverse one remaining array */
932   while (i<size1) out[l++] = in1[i++];
933   while (j<size2) out[l++] = in2[j++];
934   while (k<size3) out[l++] = in3[k++];
935 
936   *size4 = l;
937 }
938 
939 /* This matrix-matrix multiplication algorithm divides the multiplication into three multiplications and  */
940 /* adds up the products. Two of these three multiplications are performed with existing (sequential)      */
941 /* matrix-matrix multiplications.  */
942 #undef __FUNCT__
943 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIAIJ_seqMPI"
944 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_seqMPI(Mat A, Mat P, PetscReal fill, Mat *C)
945 {
946   PetscErrorCode     ierr;
947   MPI_Comm           comm;
948   PetscMPIInt        size;
949   Mat                Cmpi;
950   Mat_PtAPMPI        *ptap;
951   PetscFreeSpaceList free_space_diag=NULL, current_space=NULL;
952   Mat_MPIAIJ         *a        =(Mat_MPIAIJ*)A->data;
953   Mat_SeqAIJ         *ad       =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc;
954   Mat_MPIAIJ         *p        =(Mat_MPIAIJ*)P->data;
955   Mat_MPIAIJ         *c;
956   Mat_SeqAIJ         *adpd_seq, *p_off, *aopoth_seq;
957   PetscInt           adponz, adpdnz;
958   PetscInt           *pi_loc,*dnz,*onz;
959   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,rstart=A->rmap->rstart;
960   PetscInt           *lnk,i, i1=0,pnz,row,*adpoi,*adpoj, *api, *adpoJ, *aopJ, *apJ,*Jptr, aopnz, nspacedouble=0,j,nzi,
961                      *apj,apnz, *adpdi, *adpdj, *adpdJ, *poff_i, *poff_j, *j_temp, *aopothi, *aopothj;
962   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n, p_colstart, p_colend;
963   PetscBT            lnkbt;
964   PetscScalar        *apa;
965   PetscReal          afill;
966   PetscMPIInt        rank;
967   Mat                adpd, aopoth;
968 
969   PetscFunctionBegin;
970   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
971   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
972   ierr = MPI_Comm_rank(comm, &rank);CHKERRQ(ierr);
973   ierr = MatGetOwnershipRangeColumn(P, &p_colstart, &p_colend); CHKERRQ(ierr);
974 
975   /* create struct Mat_PtAPMPI and attached it to C later */
976   ierr = PetscNew(&ptap);CHKERRQ(ierr);
977 
978   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
979   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
980 
981   /* get P_loc by taking all local rows of P */
982   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
983 
984 
985   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
986   pi_loc = p_loc->i;
987 
988   /* Allocate memory for the i arrays of the matrices A*P, A_diag*P_off and A_offd * P */
989   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
990   ierr      = PetscMalloc1(am+2,&adpoi);CHKERRQ(ierr);
991 
992   adpoi[0]    = 0;
993   ptap->api = api;
994   api[0] = 0;
995 
996   /* create and initialize a linked list, will be used for both A_diag * P_loc_off and A_offd * P_oth */
997   ierr = PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt);CHKERRQ(ierr);
998   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
999 
1000   /* Symbolic calc of A_loc_diag * P_loc_diag */
1001   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(a->A, p->A, fill, &adpd);CHKERRQ(ierr);
1002   adpd_seq = (Mat_SeqAIJ*)((adpd)->data);
1003   adpdi = adpd_seq->i; adpdj = adpd_seq->j;
1004   p_off = (Mat_SeqAIJ*)((p->B)->data);
1005   poff_i = p_off->i; poff_j = p_off->j;
1006 
1007   /* j_temp stores indices of a result row before they are added to the linked list */
1008   ierr = PetscMalloc1(pN+2,&j_temp);CHKERRQ(ierr);
1009 
1010 
1011   /* Symbolic calc of the A_diag * p_loc_off */
1012   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
1013   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space_diag);CHKERRQ(ierr);
1014   current_space = free_space_diag;
1015 
1016   for (i=0; i<am; i++) {
1017     /* A_diag * P_loc_off */
1018     nzi = adi[i+1] - adi[i];
1019     for (j=0; j<nzi; j++) {
1020       row  = *adj++;
1021       pnz  = poff_i[row+1] - poff_i[row];
1022       Jptr = poff_j + poff_i[row];
1023       for(i1 = 0; i1 < pnz; i1++) {
1024         j_temp[i1] = p->garray[Jptr[i1]];
1025       }
1026       /* add non-zero cols of P into the sorted linked list lnk */
1027       ierr = PetscLLCondensedAddSorted(pnz,j_temp,lnk,lnkbt);CHKERRQ(ierr);
1028     }
1029 
1030     adponz     = lnk[0];
1031     adpoi[i+1] = adpoi[i] + adponz;
1032 
1033     /* if free space is not available, double the total space in the list */
1034     if (current_space->local_remaining<adponz) {
1035       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(adponz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1036       nspacedouble++;
1037     }
1038 
1039     /* Copy data into free space, then initialize lnk */
1040     ierr = PetscLLCondensedClean(pN,adponz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1041 
1042     current_space->array           += adponz;
1043     current_space->local_used      += adponz;
1044     current_space->local_remaining -= adponz;
1045   }
1046 
1047   /* Symbolic calc of A_off * P_oth */
1048   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(a->B, ptap->P_oth, fill, &aopoth);CHKERRQ(ierr);
1049   aopoth_seq = (Mat_SeqAIJ*)((aopoth)->data);
1050   aopothi = aopoth_seq->i; aopothj = aopoth_seq->j;
1051 
1052   /* Allocate space for apj, adpj, aopj, ... */
1053   /* destroy lists of free space and other temporary array(s) */
1054 
1055   ierr = PetscMalloc1(aopothi[am] + adpoi[am] + adpdi[am]+2, &ptap->apj);CHKERRQ(ierr);
1056   ierr = PetscMalloc1(adpoi[am]+2, &adpoj);CHKERRQ(ierr);
1057 
1058   /* Copy from linked list to j-array */
1059   ierr = PetscFreeSpaceContiguous(&free_space_diag,adpoj);CHKERRQ(ierr);
1060   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1061 
1062   adpoJ = adpoj;
1063   adpdJ = adpdj;
1064   aopJ = aopothj;
1065   apj  = ptap->apj;
1066   apJ = apj; /* still empty */
1067 
1068   /* Merge j-arrays of A_off * P, A_diag * P_loc_off, and */
1069   /* A_diag * P_loc_diag to get A*P */
1070   for (i = 0; i < am; i++) {
1071     aopnz  =  aopothi[i+1] -  aopothi[i];
1072     adponz = adpoi[i+1] - adpoi[i];
1073     adpdnz = adpdi[i+1] - adpdi[i];
1074 
1075     /* Correct indices from A_diag*P_diag */
1076     for(i1 = 0; i1 < adpdnz; i1++) {
1077       adpdJ[i1] += p_colstart;
1078     }
1079     /* Merge j-arrays of A_diag * P_loc_off and A_diag * P_loc_diag and A_off * P_oth */
1080     Merge3SortedArrays(adponz, adpoJ, adpdnz, adpdJ, aopnz, aopJ, &apnz, apJ);
1081     ierr = MatPreallocateSet(i+rstart, apnz, apJ, dnz, onz); CHKERRQ(ierr);
1082 
1083     aopJ += aopnz;
1084     adpoJ += adponz;
1085     adpdJ += adpdnz;
1086     apJ += apnz;
1087     api[i+1] = api[i] + apnz;
1088   }
1089 
1090   /* malloc apa to store dense row A[i,:]*P */
1091   ierr = PetscCalloc1(pN+2,&apa);CHKERRQ(ierr);
1092 
1093   ptap->apa = apa;
1094   /* create and assemble symbolic parallel matrix Cmpi */
1095   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1096   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1097   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
1098 
1099   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1100   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1101 
1102 
1103   ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(Cmpi, apj, api, dnz, onz);CHKERRQ(ierr);
1104   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1105   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1106   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1107 
1108 
1109   ptap->destroy        = Cmpi->ops->destroy;
1110   ptap->duplicate      = Cmpi->ops->duplicate;
1111   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1112   Cmpi->ops->destroy   = MatDestroy_MPIAIJ_MatMatMult;
1113   Cmpi->ops->duplicate = MatDuplicate_MPIAIJ_MatMatMult;
1114 
1115   /* attach the supporting struct to Cmpi for reuse */
1116   c       = (Mat_MPIAIJ*)Cmpi->data;
1117   c->ptap = ptap;
1118   *C = Cmpi;
1119 
1120   /* set MatInfo */
1121   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
1122   if (afill < 1.0) afill = 1.0;
1123   Cmpi->info.mallocs           = nspacedouble;
1124   Cmpi->info.fill_ratio_given  = fill;
1125   Cmpi->info.fill_ratio_needed = afill;
1126 
1127 #if defined(PETSC_USE_INFO)
1128   if (api[am]) {
1129     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1130     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
1131   } else {
1132     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1133   }
1134 #endif
1135 
1136   ierr = MatDestroy(&aopoth);CHKERRQ(ierr);
1137   ierr = MatDestroy(&adpd);CHKERRQ(ierr);
1138   ierr = PetscFree(j_temp);CHKERRQ(ierr);
1139   ierr = PetscFree(adpoj);CHKERRQ(ierr);
1140   ierr = PetscFree(adpoi);CHKERRQ(ierr);
1141   PetscFunctionReturn(0);
1142 }
1143 
1144 
1145 /*-------------------------------------------------------------------------*/
1146 PetscErrorCode MatTransposeMatMult_MPIAIJ_MPIAIJ(Mat P,Mat A,MatReuse scall,PetscReal fill,Mat *C)
1147 {
1148   PetscErrorCode ierr;
1149   const char     *algTypes[3] = {"scalable","nonscalable","matmatmult"};
1150   PetscInt       aN=A->cmap->N,alg=1; /* set default algorithm */
1151   PetscBool      flg;
1152 
1153   PetscFunctionBegin;
1154   if (scall == MAT_INITIAL_MATRIX) {
1155     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
1156     PetscOptionsObject->alreadyprinted = PETSC_FALSE; /* a hack to ensure the option shows in '-help' */
1157     ierr = PetscOptionsEList("-mattransposematmult_via","Algorithmic approach","MatTransposeMatMult",algTypes,3,algTypes[1],&alg,&flg);CHKERRQ(ierr);
1158     ierr = PetscOptionsEnd();CHKERRQ(ierr);
1159 
1160     ierr = PetscLogEventBegin(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
1161     switch (alg) {
1162     case 1:
1163       if (!flg && aN > 100000) { /* may switch to scalable algorithm as default */
1164         MatInfo     Ainfo,Pinfo;
1165         PetscInt    nz_local;
1166         PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
1167         MPI_Comm    comm;
1168 
1169         ierr = MatGetInfo(A,MAT_LOCAL,&Ainfo);CHKERRQ(ierr);
1170         ierr = MatGetInfo(P,MAT_LOCAL,&Pinfo);CHKERRQ(ierr);
1171         nz_local = (PetscInt)(Ainfo.nz_allocated + Pinfo.nz_allocated); /* estimated local nonzero entries */
1172 
1173         if (aN > fill*nz_local) alg_scalable_loc = PETSC_TRUE;
1174         ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1175         ierr = MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm);CHKERRQ(ierr);
1176 
1177         if (alg_scalable) {
1178           alg  = 0; /* scalable algorithm would slower than nonscalable algorithm */
1179           ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(P,A,fill,C);CHKERRQ(ierr);
1180           break;
1181         }
1182       }
1183       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(P,A,fill,C);CHKERRQ(ierr);
1184       break;
1185     case 2:
1186     {
1187       Mat         Pt;
1188       Mat_PtAPMPI *ptap;
1189       Mat_MPIAIJ  *c;
1190       ierr = MatTranspose(P,MAT_INITIAL_MATRIX,&Pt);CHKERRQ(ierr);
1191       ierr = MatMatMult(Pt,A,MAT_INITIAL_MATRIX,fill,C);CHKERRQ(ierr);
1192       c        = (Mat_MPIAIJ*)(*C)->data;
1193       ptap     = c->ptap;
1194       ptap->Pt = Pt;
1195       (*C)->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult;
1196       PetscFunctionReturn(0);
1197     }
1198       break;
1199     default: /* scalable algorithm */
1200       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(P,A,fill,C);CHKERRQ(ierr);
1201       break;
1202     }
1203     ierr = PetscLogEventEnd(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
1204   }
1205   ierr = PetscLogEventBegin(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
1206   ierr = (*(*C)->ops->mattransposemultnumeric)(P,A,*C);CHKERRQ(ierr);
1207   ierr = PetscLogEventEnd(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
1208   PetscFunctionReturn(0);
1209 }
1210 
1211 /* This routine only works when scall=MAT_REUSE_MATRIX! */
1212 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult(Mat P,Mat A,Mat C)
1213 {
1214   PetscErrorCode ierr;
1215   Mat_MPIAIJ     *c=(Mat_MPIAIJ*)C->data;
1216   Mat_PtAPMPI    *ptap= c->ptap;
1217   Mat            Pt=ptap->Pt;
1218 
1219   PetscFunctionBegin;
1220   ierr = MatTranspose(P,MAT_REUSE_MATRIX,&Pt);CHKERRQ(ierr);
1221   ierr = MatMatMultNumeric(Pt,A,C);CHKERRQ(ierr);
1222   PetscFunctionReturn(0);
1223 }
1224 
1225 PetscErrorCode MatDuplicate_MPIAIJ_MatPtAP(Mat,MatDuplicateOption,Mat*);
1226 
1227 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1228 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,PetscReal fill,Mat *C)
1229 {
1230   PetscErrorCode      ierr;
1231   Mat_PtAPMPI         *ptap;
1232   Mat_MPIAIJ          *p=(Mat_MPIAIJ*)P->data,*c;
1233   MPI_Comm            comm;
1234   PetscMPIInt         size,rank;
1235   Mat                 Cmpi;
1236   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1237   PetscInt            pn=P->cmap->n,aN=A->cmap->N,an=A->cmap->n;
1238   PetscInt            *lnk,i,k,nsend;
1239   PetscBT             lnkbt;
1240   PetscMPIInt         tagi,tagj,*len_si,*len_s,*len_ri,icompleted=0,nrecv;
1241   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1242   PetscInt            len,proc,*dnz,*onz,*owners,nzi;
1243   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1244   MPI_Request         *swaits,*rwaits;
1245   MPI_Status          *sstatus,rstatus;
1246   PetscLayout         rowmap;
1247   PetscInt            *owners_co,*coi,*coj;    /* i and j array of (p->B)^T*A*P - used in the communication */
1248   PetscMPIInt         *len_r,*id_r;    /* array of length of comm->size, store send/recv matrix values */
1249   PetscInt            *Jptr,*prmap=p->garray,con,j,Crmax;
1250   Mat_SeqAIJ          *a_loc,*c_loc,*c_oth;
1251   PetscTable          ta;
1252 
1253   PetscFunctionBegin;
1254   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1255   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1256   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1257 
1258   /* create symbolic parallel matrix Cmpi */
1259   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1260   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1261 
1262   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1263 
1264   /* create struct Mat_PtAPMPI and attached it to C later */
1265   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1266   ptap->reuse = MAT_INITIAL_MATRIX;
1267 
1268   /* (0) compute Rd = Pd^T, Ro = Po^T  */
1269   /* --------------------------------- */
1270   ierr = MatTranspose_SeqAIJ(p->A,MAT_INITIAL_MATRIX,&ptap->Rd);CHKERRQ(ierr);
1271   ierr = MatTranspose_SeqAIJ(p->B,MAT_INITIAL_MATRIX,&ptap->Ro);CHKERRQ(ierr);
1272 
1273   /* (1) compute symbolic A_loc */
1274   /* ---------------------------*/
1275   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&ptap->A_loc);CHKERRQ(ierr);
1276 
1277   /* (2-1) compute symbolic C_oth = Ro*A_loc  */
1278   /* ------------------------------------ */
1279   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Ro,ptap->A_loc,fill,&ptap->C_oth);CHKERRQ(ierr);
1280 
1281   /* (3) send coj of C_oth to other processors  */
1282   /* ------------------------------------------ */
1283   /* determine row ownership */
1284   ierr = PetscLayoutCreate(comm,&rowmap);CHKERRQ(ierr);
1285   rowmap->n  = pn;
1286   rowmap->bs = 1;
1287   ierr   = PetscLayoutSetUp(rowmap);CHKERRQ(ierr);
1288   owners = rowmap->range;
1289 
1290   /* determine the number of messages to send, their lengths */
1291   ierr = PetscMalloc4(size,&len_s,size,&len_si,size,&sstatus,size+2,&owners_co);CHKERRQ(ierr);
1292   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1293   ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1294 
1295   c_oth = (Mat_SeqAIJ*)ptap->C_oth->data;
1296   coi   = c_oth->i; coj = c_oth->j;
1297   con   = ptap->C_oth->rmap->n;
1298   proc  = 0;
1299   for (i=0; i<con; i++) {
1300     while (prmap[i] >= owners[proc+1]) proc++;
1301     len_si[proc]++;               /* num of rows in Co(=Pt*A) to be sent to [proc] */
1302     len_s[proc] += coi[i+1] - coi[i]; /* num of nonzeros in Co to be sent to [proc] */
1303   }
1304 
1305   len          = 0; /* max length of buf_si[], see (4) */
1306   owners_co[0] = 0;
1307   nsend        = 0;
1308   for (proc=0; proc<size; proc++) {
1309     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1310     if (len_s[proc]) {
1311       nsend++;
1312       len_si[proc] = 2*(len_si[proc] + 1); /* length of buf_si to be sent to [proc] */
1313       len         += len_si[proc];
1314     }
1315   }
1316 
1317   /* determine the number and length of messages to receive for coi and coj  */
1318   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&nrecv);CHKERRQ(ierr);
1319   ierr = PetscGatherMessageLengths2(comm,nsend,nrecv,len_s,len_si,&id_r,&len_r,&len_ri);CHKERRQ(ierr);
1320 
1321   /* post the Irecv and Isend of coj */
1322   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1323   ierr = PetscPostIrecvInt(comm,tagj,nrecv,id_r,len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1324   ierr = PetscMalloc1(nsend+1,&swaits);CHKERRQ(ierr);
1325   for (proc=0, k=0; proc<size; proc++) {
1326     if (!len_s[proc]) continue;
1327     i    = owners_co[proc];
1328     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1329     k++;
1330   }
1331 
1332   /* (2-2) compute symbolic C_loc = Rd*A_loc */
1333   /* ---------------------------------------- */
1334   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Rd,ptap->A_loc,fill,&ptap->C_loc);CHKERRQ(ierr);
1335   c_loc = (Mat_SeqAIJ*)ptap->C_loc->data;
1336 
1337   /* receives coj are complete */
1338   for (i=0; i<nrecv; i++) {
1339     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1340   }
1341   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1342   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
1343 
1344   /* add received column indices into ta to update Crmax */
1345   a_loc = (Mat_SeqAIJ*)(ptap->A_loc)->data;
1346 
1347   /* create and initialize a linked list */
1348   ierr = PetscTableCreate(an,aN,&ta);CHKERRQ(ierr); /* for compute Crmax */
1349   MatRowMergeMax_SeqAIJ(a_loc,ptap->A_loc->rmap->N,ta);
1350 
1351   for (k=0; k<nrecv; k++) {/* k-th received message */
1352     Jptr = buf_rj[k];
1353     for (j=0; j<len_r[k]; j++) {
1354       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
1355     }
1356   }
1357   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
1358   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
1359 
1360   /* (4) send and recv coi */
1361   /*-----------------------*/
1362   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1363   ierr   = PetscPostIrecvInt(comm,tagi,nrecv,id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1364   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1365   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1366   for (proc=0,k=0; proc<size; proc++) {
1367     if (!len_s[proc]) continue;
1368     /* form outgoing message for i-structure:
1369          buf_si[0]:                 nrows to be sent
1370                [1:nrows]:           row index (global)
1371                [nrows+1:2*nrows+1]: i-structure index
1372     */
1373     /*-------------------------------------------*/
1374     nrows       = len_si[proc]/2 - 1; /* num of rows in Co to be sent to [proc] */
1375     buf_si_i    = buf_si + nrows+1;
1376     buf_si[0]   = nrows;
1377     buf_si_i[0] = 0;
1378     nrows       = 0;
1379     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1380       nzi = coi[i+1] - coi[i];
1381       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1382       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1383       nrows++;
1384     }
1385     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1386     k++;
1387     buf_si += len_si[proc];
1388   }
1389   for (i=0; i<nrecv; i++) {
1390     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1391   }
1392   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1393   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
1394 
1395   ierr = PetscFree4(len_s,len_si,sstatus,owners_co);CHKERRQ(ierr);
1396   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1397   ierr = PetscFree(swaits);CHKERRQ(ierr);
1398   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1399 
1400   /* (5) compute the local portion of Cmpi      */
1401   /* ------------------------------------------ */
1402   /* set initial free space to be Crmax, sufficient for holding nozeros in each row of Cmpi */
1403   ierr          = PetscFreeSpaceGet(Crmax,&free_space);CHKERRQ(ierr);
1404   current_space = free_space;
1405 
1406   ierr = PetscMalloc3(nrecv,&buf_ri_k,nrecv,&nextrow,nrecv,&nextci);CHKERRQ(ierr);
1407   for (k=0; k<nrecv; k++) {
1408     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1409     nrows       = *buf_ri_k[k];
1410     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1411     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1412   }
1413 
1414   ierr = MatPreallocateInitialize(comm,pn,an,dnz,onz);CHKERRQ(ierr);
1415   ierr = PetscLLCondensedCreate(Crmax,aN,&lnk,&lnkbt);CHKERRQ(ierr);
1416   for (i=0; i<pn; i++) {
1417     /* add C_loc into Cmpi */
1418     nzi  = c_loc->i[i+1] - c_loc->i[i];
1419     Jptr = c_loc->j + c_loc->i[i];
1420     ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1421 
1422     /* add received col data into lnk */
1423     for (k=0; k<nrecv; k++) { /* k-th received message */
1424       if (i == *nextrow[k]) { /* i-th row */
1425         nzi  = *(nextci[k]+1) - *nextci[k];
1426         Jptr = buf_rj[k] + *nextci[k];
1427         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1428         nextrow[k]++; nextci[k]++;
1429       }
1430     }
1431     nzi = lnk[0];
1432 
1433     /* copy data into free space, then initialize lnk */
1434     ierr = PetscLLCondensedClean(aN,nzi,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1435     ierr = MatPreallocateSet(i+owners[rank],nzi,current_space->array,dnz,onz);CHKERRQ(ierr);
1436   }
1437   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1438   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1439   ierr = PetscFreeSpaceDestroy(free_space);CHKERRQ(ierr);
1440 
1441   /* local sizes and preallocation */
1442   ierr = MatSetSizes(Cmpi,pn,an,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1443   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(P->cmap->bs));CHKERRQ(ierr);
1444   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1445   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1446 
1447   /* members in merge */
1448   ierr = PetscFree(id_r);CHKERRQ(ierr);
1449   ierr = PetscFree(len_r);CHKERRQ(ierr);
1450   ierr = PetscFree(buf_ri[0]);CHKERRQ(ierr);
1451   ierr = PetscFree(buf_ri);CHKERRQ(ierr);
1452   ierr = PetscFree(buf_rj[0]);CHKERRQ(ierr);
1453   ierr = PetscFree(buf_rj);CHKERRQ(ierr);
1454   ierr = PetscLayoutDestroy(&rowmap);CHKERRQ(ierr);
1455 
1456   /* attach the supporting struct to Cmpi for reuse */
1457   c = (Mat_MPIAIJ*)Cmpi->data;
1458   c->ptap         = ptap;
1459   ptap->duplicate = Cmpi->ops->duplicate;
1460   ptap->destroy   = Cmpi->ops->destroy;
1461 
1462   /* Cmpi is not ready for use - assembly will be done by MatPtAPNumeric() */
1463   Cmpi->assembled        = PETSC_FALSE;
1464   Cmpi->ops->destroy     = MatDestroy_MPIAIJ_PtAP;
1465   Cmpi->ops->duplicate   = MatDuplicate_MPIAIJ_MatPtAP;
1466   *C                     = Cmpi;
1467   PetscFunctionReturn(0);
1468 }
1469 
1470 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,Mat C)
1471 {
1472   PetscErrorCode    ierr;
1473   Mat_MPIAIJ        *p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1474   Mat_SeqAIJ        *c_seq;
1475   Mat_PtAPMPI       *ptap = c->ptap;
1476   Mat               A_loc,C_loc,C_oth;
1477   PetscInt          i,rstart,rend,cm,ncols,row;
1478   const PetscInt    *cols;
1479   const PetscScalar *vals;
1480 
1481   PetscFunctionBegin;
1482   ierr = MatZeroEntries(C);CHKERRQ(ierr);
1483 
1484   if (ptap->reuse == MAT_REUSE_MATRIX) {
1485     /* These matrices are obtained in MatTransposeMatMultSymbolic() */
1486     /* 1) get R = Pd^T, Ro = Po^T */
1487     /*----------------------------*/
1488     ierr = MatTranspose_SeqAIJ(p->A,MAT_REUSE_MATRIX,&ptap->Rd);CHKERRQ(ierr);
1489     ierr = MatTranspose_SeqAIJ(p->B,MAT_REUSE_MATRIX,&ptap->Ro);CHKERRQ(ierr);
1490 
1491     /* 2) compute numeric A_loc */
1492     /*--------------------------*/
1493     ierr = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&ptap->A_loc);CHKERRQ(ierr);
1494   }
1495 
1496   /* 3) C_loc = Rd*A_loc, C_oth = Ro*A_loc */
1497   A_loc = ptap->A_loc;
1498   ierr = ((ptap->C_loc)->ops->matmultnumeric)(ptap->Rd,A_loc,ptap->C_loc);CHKERRQ(ierr);
1499   ierr = ((ptap->C_oth)->ops->matmultnumeric)(ptap->Ro,A_loc,ptap->C_oth);CHKERRQ(ierr);
1500   C_loc = ptap->C_loc;
1501   C_oth = ptap->C_oth;
1502 
1503   /* add C_loc and Co to to C */
1504   ierr = MatGetOwnershipRange(C,&rstart,&rend);CHKERRQ(ierr);
1505 
1506   /* C_loc -> C */
1507   cm    = C_loc->rmap->N;
1508   c_seq = (Mat_SeqAIJ*)C_loc->data;
1509   cols = c_seq->j;
1510   vals = c_seq->a;
1511   for (i=0; i<cm; i++) {
1512     ncols = c_seq->i[i+1] - c_seq->i[i];
1513     row = rstart + i;
1514     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1515     cols += ncols; vals += ncols;
1516   }
1517 
1518   /* Co -> C, off-processor part */
1519   cm    = C_oth->rmap->N;
1520   c_seq = (Mat_SeqAIJ*)C_oth->data;
1521   cols  = c_seq->j;
1522   vals  = c_seq->a;
1523   for (i=0; i<cm; i++) {
1524     ncols = c_seq->i[i+1] - c_seq->i[i];
1525     row = p->garray[i];
1526     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1527     cols += ncols; vals += ncols;
1528   }
1529   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1530   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1531 
1532   ptap->reuse = MAT_REUSE_MATRIX;
1533   PetscFunctionReturn(0);
1534 }
1535 
1536 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ(Mat P,Mat A,Mat C)
1537 {
1538   PetscErrorCode      ierr;
1539   Mat_Merge_SeqsToMPI *merge;
1540   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1541   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
1542   Mat_PtAPMPI         *ptap;
1543   PetscInt            *adj;
1544   PetscInt            i,j,k,anz,pnz,row,*cj,nexta;
1545   MatScalar           *ada,*ca,valtmp;
1546   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
1547   MPI_Comm            comm;
1548   PetscMPIInt         size,rank,taga,*len_s;
1549   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
1550   PetscInt            **buf_ri,**buf_rj;
1551   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
1552   MPI_Request         *s_waits,*r_waits;
1553   MPI_Status          *status;
1554   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
1555   PetscInt            *ai,*aj,*coi,*coj,*poJ,*pdJ;
1556   Mat                 A_loc;
1557   Mat_SeqAIJ          *a_loc;
1558 
1559   PetscFunctionBegin;
1560   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1561   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1562   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1563 
1564   ptap  = c->ptap;
1565   merge = ptap->merge;
1566 
1567   /* 2) compute numeric C_seq = P_loc^T*A_loc */
1568   /*------------------------------------------*/
1569   /* get data from symbolic products */
1570   coi    = merge->coi; coj = merge->coj;
1571   ierr   = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
1572   bi     = merge->bi; bj = merge->bj;
1573   owners = merge->rowmap->range;
1574   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
1575 
1576   /* get A_loc by taking all local rows of A */
1577   A_loc = ptap->A_loc;
1578   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
1579   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1580   ai    = a_loc->i;
1581   aj    = a_loc->j;
1582 
1583   for (i=0; i<am; i++) {
1584     anz = ai[i+1] - ai[i];
1585     adj = aj + ai[i];
1586     ada = a_loc->a + ai[i];
1587 
1588     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1589     /*-------------------------------------------------------------*/
1590     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1591     pnz = po->i[i+1] - po->i[i];
1592     poJ = po->j + po->i[i];
1593     pA  = po->a + po->i[i];
1594     for (j=0; j<pnz; j++) {
1595       row = poJ[j];
1596       cj  = coj + coi[row];
1597       ca  = coa + coi[row];
1598       /* perform sparse axpy */
1599       nexta  = 0;
1600       valtmp = pA[j];
1601       for (k=0; nexta<anz; k++) {
1602         if (cj[k] == adj[nexta]) {
1603           ca[k] += valtmp*ada[nexta];
1604           nexta++;
1605         }
1606       }
1607       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1608     }
1609 
1610     /* put the value into Cd (diagonal part) */
1611     pnz = pd->i[i+1] - pd->i[i];
1612     pdJ = pd->j + pd->i[i];
1613     pA  = pd->a + pd->i[i];
1614     for (j=0; j<pnz; j++) {
1615       row = pdJ[j];
1616       cj  = bj + bi[row];
1617       ca  = ba + bi[row];
1618       /* perform sparse axpy */
1619       nexta  = 0;
1620       valtmp = pA[j];
1621       for (k=0; nexta<anz; k++) {
1622         if (cj[k] == adj[nexta]) {
1623           ca[k] += valtmp*ada[nexta];
1624           nexta++;
1625         }
1626       }
1627       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1628     }
1629   }
1630 
1631   /* 3) send and recv matrix values coa */
1632   /*------------------------------------*/
1633   buf_ri = merge->buf_ri;
1634   buf_rj = merge->buf_rj;
1635   len_s  = merge->len_s;
1636   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1637   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1638 
1639   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1640   for (proc=0,k=0; proc<size; proc++) {
1641     if (!len_s[proc]) continue;
1642     i    = merge->owners_co[proc];
1643     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1644     k++;
1645   }
1646   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1647   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1648 
1649   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1650   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1651   ierr = PetscFree(coa);CHKERRQ(ierr);
1652 
1653   /* 4) insert local Cseq and received values into Cmpi */
1654   /*----------------------------------------------------*/
1655   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1656   for (k=0; k<merge->nrecv; k++) {
1657     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1658     nrows       = *(buf_ri_k[k]);
1659     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1660     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1661   }
1662 
1663   for (i=0; i<cm; i++) {
1664     row  = owners[rank] + i; /* global row index of C_seq */
1665     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1666     ba_i = ba + bi[i];
1667     bnz  = bi[i+1] - bi[i];
1668     /* add received vals into ba */
1669     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1670       /* i-th row */
1671       if (i == *nextrow[k]) {
1672         cnz    = *(nextci[k]+1) - *nextci[k];
1673         cj     = buf_rj[k] + *(nextci[k]);
1674         ca     = abuf_r[k] + *(nextci[k]);
1675         nextcj = 0;
1676         for (j=0; nextcj<cnz; j++) {
1677           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1678             ba_i[j] += ca[nextcj++];
1679           }
1680         }
1681         nextrow[k]++; nextci[k]++;
1682         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1683       }
1684     }
1685     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1686   }
1687   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1688   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1689 
1690   ierr = PetscFree(ba);CHKERRQ(ierr);
1691   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1692   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1693   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1694   PetscFunctionReturn(0);
1695 }
1696 
1697 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P,Mat A,PetscReal fill,Mat *C)
1698 {
1699   PetscErrorCode      ierr;
1700   Mat                 Cmpi,A_loc,POt,PDt;
1701   Mat_PtAPMPI         *ptap;
1702   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1703   Mat_MPIAIJ          *p=(Mat_MPIAIJ*)P->data,*a=(Mat_MPIAIJ*)A->data,*c;
1704   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1705   PetscInt            nnz;
1706   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1707   PetscInt            am  =A->rmap->n,pn=P->cmap->n;
1708   MPI_Comm            comm;
1709   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1710   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1711   PetscInt            len,proc,*dnz,*onz,*owners;
1712   PetscInt            nzi,*bi,*bj;
1713   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1714   MPI_Request         *swaits,*rwaits;
1715   MPI_Status          *sstatus,rstatus;
1716   Mat_Merge_SeqsToMPI *merge;
1717   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1718   PetscReal           afill  =1.0,afill_tmp;
1719   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Armax;
1720   PetscScalar         *vals;
1721   Mat_SeqAIJ          *a_loc,*pdt,*pot;
1722   PetscTable          ta;
1723 
1724   PetscFunctionBegin;
1725   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1726   /* check if matrix local sizes are compatible */
1727   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);
1728 
1729   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1730   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1731 
1732   /* create struct Mat_PtAPMPI and attached it to C later */
1733   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1734 
1735   /* get A_loc by taking all local rows of A */
1736   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1737 
1738   ptap->A_loc = A_loc;
1739   a_loc       = (Mat_SeqAIJ*)(A_loc)->data;
1740   ai          = a_loc->i;
1741   aj          = a_loc->j;
1742 
1743   /* determine symbolic Co=(p->B)^T*A - send to others */
1744   /*----------------------------------------------------*/
1745   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1746   pdt  = (Mat_SeqAIJ*)PDt->data;
1747   pdti = pdt->i; pdtj = pdt->j;
1748 
1749   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1750   pot  = (Mat_SeqAIJ*)POt->data;
1751   poti = pot->i; potj = pot->j;
1752 
1753   /* then, compute symbolic Co = (p->B)^T*A */
1754   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1755                          >= (num of nonzero rows of C_seq) - pn */
1756   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
1757   coi[0] = 0;
1758 
1759   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1760   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(poti[pon],ai[am]));
1761   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1762   current_space = free_space;
1763 
1764   /* create and initialize a linked list */
1765   ierr = PetscTableCreate(A->cmap->n + a->B->cmap->N,aN,&ta);CHKERRQ(ierr);
1766   MatRowMergeMax_SeqAIJ(a_loc,am,ta);
1767   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
1768 
1769   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
1770 
1771   for (i=0; i<pon; i++) {
1772     pnz = poti[i+1] - poti[i];
1773     ptJ = potj + poti[i];
1774     for (j=0; j<pnz; j++) {
1775       row  = ptJ[j]; /* row of A_loc == col of Pot */
1776       anz  = ai[row+1] - ai[row];
1777       Jptr = aj + ai[row];
1778       /* add non-zero cols of AP into the sorted linked list lnk */
1779       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1780     }
1781     nnz = lnk[0];
1782 
1783     /* If free space is not available, double the total space in the list */
1784     if (current_space->local_remaining<nnz) {
1785       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1786       nspacedouble++;
1787     }
1788 
1789     /* Copy data into free space, and zero out denserows */
1790     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1791 
1792     current_space->array           += nnz;
1793     current_space->local_used      += nnz;
1794     current_space->local_remaining -= nnz;
1795 
1796     coi[i+1] = coi[i] + nnz;
1797   }
1798 
1799   ierr = PetscMalloc1(coi[pon]+1,&coj);CHKERRQ(ierr);
1800   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1801   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr); /* must destroy to get a new one for C */
1802 
1803   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1804   if (afill_tmp > afill) afill = afill_tmp;
1805 
1806   /* send j-array (coj) of Co to other processors */
1807   /*----------------------------------------------*/
1808   /* determine row ownership */
1809   ierr = PetscNew(&merge);CHKERRQ(ierr);
1810   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1811 
1812   merge->rowmap->n  = pn;
1813   merge->rowmap->bs = 1;
1814 
1815   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1816   owners = merge->rowmap->range;
1817 
1818   /* determine the number of messages to send, their lengths */
1819   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
1820   ierr = PetscMalloc1(size,&merge->len_s);CHKERRQ(ierr);
1821 
1822   len_s        = merge->len_s;
1823   merge->nsend = 0;
1824 
1825   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
1826   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1827 
1828   proc = 0;
1829   for (i=0; i<pon; i++) {
1830     while (prmap[i] >= owners[proc+1]) proc++;
1831     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1832     len_s[proc] += coi[i+1] - coi[i];
1833   }
1834 
1835   len          = 0; /* max length of buf_si[] */
1836   owners_co[0] = 0;
1837   for (proc=0; proc<size; proc++) {
1838     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1839     if (len_si[proc]) {
1840       merge->nsend++;
1841       len_si[proc] = 2*(len_si[proc] + 1);
1842       len         += len_si[proc];
1843     }
1844   }
1845 
1846   /* determine the number and length of messages to receive for coi and coj  */
1847   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1848   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1849 
1850   /* post the Irecv and Isend of coj */
1851   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1852   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1853   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
1854   for (proc=0, k=0; proc<size; proc++) {
1855     if (!len_s[proc]) continue;
1856     i    = owners_co[proc];
1857     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1858     k++;
1859   }
1860 
1861   /* receives and sends of coj are complete */
1862   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
1863   for (i=0; i<merge->nrecv; i++) {
1864     PetscMPIInt icompleted;
1865     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1866   }
1867   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1868   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1869 
1870   /* add received column indices into table to update Armax */
1871   /* Armax can be as large as aN if a P[row,:] is dense, see src/ksp/ksp/examples/tutorials/ex56.c! */
1872   for (k=0; k<merge->nrecv; k++) {/* k-th received message */
1873     Jptr = buf_rj[k];
1874     for (j=0; j<merge->len_r[k]; j++) {
1875       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
1876     }
1877   }
1878   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
1879   /* 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); */
1880 
1881   /* send and recv coi */
1882   /*-------------------*/
1883   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1884   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1885   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1886   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1887   for (proc=0,k=0; proc<size; proc++) {
1888     if (!len_s[proc]) continue;
1889     /* form outgoing message for i-structure:
1890          buf_si[0]:                 nrows to be sent
1891                [1:nrows]:           row index (global)
1892                [nrows+1:2*nrows+1]: i-structure index
1893     */
1894     /*-------------------------------------------*/
1895     nrows       = len_si[proc]/2 - 1;
1896     buf_si_i    = buf_si + nrows+1;
1897     buf_si[0]   = nrows;
1898     buf_si_i[0] = 0;
1899     nrows       = 0;
1900     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1901       nzi               = coi[i+1] - coi[i];
1902       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1903       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1904       nrows++;
1905     }
1906     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1907     k++;
1908     buf_si += len_si[proc];
1909   }
1910   i = merge->nrecv;
1911   while (i--) {
1912     PetscMPIInt icompleted;
1913     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1914   }
1915   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1916   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1917   ierr = PetscFree(len_si);CHKERRQ(ierr);
1918   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1919   ierr = PetscFree(swaits);CHKERRQ(ierr);
1920   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1921   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1922 
1923   /* compute the local portion of C (mpi mat) */
1924   /*------------------------------------------*/
1925   /* allocate bi array and free space for accumulating nonzero column info */
1926   ierr  = PetscMalloc1(pn+1,&bi);CHKERRQ(ierr);
1927   bi[0] = 0;
1928 
1929   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
1930   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(pdti[pn],PetscIntSumTruncate(poti[pon],ai[am])));
1931   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1932   current_space = free_space;
1933 
1934   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1935   for (k=0; k<merge->nrecv; k++) {
1936     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1937     nrows       = *buf_ri_k[k];
1938     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1939     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th recieved i-structure  */
1940   }
1941 
1942   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
1943   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1944   rmax = 0;
1945   for (i=0; i<pn; i++) {
1946     /* add pdt[i,:]*AP into lnk */
1947     pnz = pdti[i+1] - pdti[i];
1948     ptJ = pdtj + pdti[i];
1949     for (j=0; j<pnz; j++) {
1950       row  = ptJ[j];  /* row of AP == col of Pt */
1951       anz  = ai[row+1] - ai[row];
1952       Jptr = aj + ai[row];
1953       /* add non-zero cols of AP into the sorted linked list lnk */
1954       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1955     }
1956 
1957     /* add received col data into lnk */
1958     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1959       if (i == *nextrow[k]) { /* i-th row */
1960         nzi  = *(nextci[k]+1) - *nextci[k];
1961         Jptr = buf_rj[k] + *nextci[k];
1962         ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
1963         nextrow[k]++; nextci[k]++;
1964       }
1965     }
1966     nnz = lnk[0];
1967 
1968     /* if free space is not available, make more free space */
1969     if (current_space->local_remaining<nnz) {
1970       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1971       nspacedouble++;
1972     }
1973     /* copy data into free space, then initialize lnk */
1974     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1975     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1976 
1977     current_space->array           += nnz;
1978     current_space->local_used      += nnz;
1979     current_space->local_remaining -= nnz;
1980 
1981     bi[i+1] = bi[i] + nnz;
1982     if (nnz > rmax) rmax = nnz;
1983   }
1984   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1985 
1986   ierr      = PetscMalloc1(bi[pn]+1,&bj);CHKERRQ(ierr);
1987   ierr      = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1988   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
1989   if (afill_tmp > afill) afill = afill_tmp;
1990   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
1991   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
1992 
1993   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1994   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1995 
1996   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
1997   /*----------------------------------------------------------------------------------*/
1998   ierr = PetscCalloc1(rmax+1,&vals);CHKERRQ(ierr);
1999 
2000   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
2001   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
2002   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
2003   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
2004   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
2005   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
2006   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
2007   for (i=0; i<pn; i++) {
2008     row  = i + rstart;
2009     nnz  = bi[i+1] - bi[i];
2010     Jptr = bj + bi[i];
2011     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
2012   }
2013   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2014   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2015   ierr = PetscFree(vals);CHKERRQ(ierr);
2016 
2017   merge->bi        = bi;
2018   merge->bj        = bj;
2019   merge->coi       = coi;
2020   merge->coj       = coj;
2021   merge->buf_ri    = buf_ri;
2022   merge->buf_rj    = buf_rj;
2023   merge->owners_co = owners_co;
2024 
2025   /* attach the supporting struct to Cmpi for reuse */
2026   c = (Mat_MPIAIJ*)Cmpi->data;
2027 
2028   c->ptap     = ptap;
2029   ptap->api   = NULL;
2030   ptap->apj   = NULL;
2031   ptap->merge = merge;
2032   ptap->apa   = NULL;
2033   ptap->destroy   = Cmpi->ops->destroy;
2034   ptap->duplicate = Cmpi->ops->duplicate;
2035 
2036   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ;
2037   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
2038   Cmpi->ops->duplicate               = MatDuplicate_MPIAIJ_MatPtAP;
2039 
2040   *C = Cmpi;
2041 #if defined(PETSC_USE_INFO)
2042   if (bi[pn] != 0) {
2043     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
2044     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
2045   } else {
2046     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
2047   }
2048 #endif
2049   PetscFunctionReturn(0);
2050 }
2051