xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision 38cfc46e4752cf65941cec0be9bac4d1d6832cc8)
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 #include <petsc/private/vecscatterimpl.h>
13 
14 #if defined(PETSC_HAVE_HYPRE)
15 PETSC_INTERN PetscErrorCode MatMatMultSymbolic_AIJ_AIJ_wHYPRE(Mat,Mat,PetscReal,Mat*);
16 #endif
17 
18 PETSC_INTERN PetscErrorCode MatMatMult_MPIAIJ_MPIAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill, Mat *C)
19 {
20   PetscErrorCode ierr;
21 #if defined(PETSC_HAVE_HYPRE)
22   const char     *algTypes[4] = {"scalable","nonscalable","seqmpi","hypre"};
23   PetscInt       nalg = 4;
24 #else
25   const char     *algTypes[3] = {"scalable","nonscalable","seqmpi"};
26   PetscInt       nalg = 3;
27 #endif
28   PetscInt       alg = 1; /* set nonscalable algorithm as default */
29   MPI_Comm       comm;
30   PetscBool      flg;
31 
32   PetscFunctionBegin;
33   if (scall == MAT_INITIAL_MATRIX) {
34     ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
35     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);
36 
37     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)A),((PetscObject)A)->prefix,"MatMatMult","Mat");CHKERRQ(ierr);
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     if (alg == 0 || alg == 1) {
79       Mat_MPIAIJ *c  = (Mat_MPIAIJ*)(*C)->data;
80       Mat_APMPI  *ap = c->ap;
81       ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)(*C)),((PetscObject)(*C))->prefix,"MatFreeIntermediateDataStructures","Mat");CHKERRQ(ierr);
82       ap->freestruct = PETSC_FALSE;
83       ierr = PetscOptionsBool("-mat_freeintermediatedatastructures","Free intermediate data structures", "MatFreeIntermediateDataStructures",ap->freestruct,&ap->freestruct, NULL);CHKERRQ(ierr);
84       ierr = PetscOptionsEnd();CHKERRQ(ierr);
85     }
86   }
87 
88   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
89   ierr = (*(*C)->ops->matmultnumeric)(A,B,*C);CHKERRQ(ierr);
90   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
91   PetscFunctionReturn(0);
92 }
93 
94 PetscErrorCode MatDestroy_MPIAIJ_MatMatMult(Mat A)
95 {
96   PetscErrorCode ierr;
97   Mat_MPIAIJ     *a    = (Mat_MPIAIJ*)A->data;
98   Mat_APMPI      *ptap = a->ap;
99 
100   PetscFunctionBegin;
101   ierr = PetscFree2(ptap->startsj_s,ptap->startsj_r);CHKERRQ(ierr);
102   ierr = PetscFree(ptap->bufa);CHKERRQ(ierr);
103   ierr = MatDestroy(&ptap->P_loc);CHKERRQ(ierr);
104   ierr = MatDestroy(&ptap->P_oth);CHKERRQ(ierr);
105   ierr = MatDestroy(&ptap->Pt);CHKERRQ(ierr);
106   ierr = PetscFree(ptap->api);CHKERRQ(ierr);
107   ierr = PetscFree(ptap->apj);CHKERRQ(ierr);
108   ierr = PetscFree(ptap->apa);CHKERRQ(ierr);
109   ierr = ptap->destroy(A);CHKERRQ(ierr);
110   ierr = PetscFree(ptap);CHKERRQ(ierr);
111   PetscFunctionReturn(0);
112 }
113 
114 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,Mat C)
115 {
116   PetscErrorCode ierr;
117   Mat_MPIAIJ     *a  =(Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
118   Mat_SeqAIJ     *ad =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
119   Mat_SeqAIJ     *cd =(Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
120   PetscScalar    *cda=cd->a,*coa=co->a;
121   Mat_SeqAIJ     *p_loc,*p_oth;
122   PetscScalar    *apa,*ca;
123   PetscInt       cm   =C->rmap->n;
124   Mat_APMPI      *ptap=c->ap;
125   PetscInt       *api,*apj,*apJ,i,k;
126   PetscInt       cstart=C->cmap->rstart;
127   PetscInt       cdnz,conz,k0,k1;
128   MPI_Comm       comm;
129   PetscMPIInt    size;
130 
131   PetscFunctionBegin;
132   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
133   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
134 
135   if (!ptap->P_oth && size>1) SETERRQ(comm,PETSC_ERR_ARG_WRONGSTATE,"AP cannot be reused. Do not call MatFreeIntermediateDataStructures() or use '-mat_freeintermediatedatastructures'");
136 
137   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
138   /*-----------------------------------------------------*/
139   /* update numerical values of P_oth and P_loc */
140   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
141   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
142 
143   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
144   /*----------------------------------------------------------*/
145   /* get data from symbolic products */
146   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
147   p_oth = NULL;
148   if (size >1) {
149     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
150   }
151 
152   /* get apa for storing dense row A[i,:]*P */
153   apa = ptap->apa;
154 
155   api = ptap->api;
156   apj = ptap->apj;
157   for (i=0; i<cm; i++) {
158     /* compute apa = A[i,:]*P */
159     AProw_nonscalable(i,ad,ao,p_loc,p_oth,apa);
160 
161     /* set values in C */
162     apJ  = apj + api[i];
163     cdnz = cd->i[i+1] - cd->i[i];
164     conz = co->i[i+1] - co->i[i];
165 
166     /* 1st off-diagoanl part of C */
167     ca = coa + co->i[i];
168     k  = 0;
169     for (k0=0; k0<conz; k0++) {
170       if (apJ[k] >= cstart) break;
171       ca[k0]      = apa[apJ[k]];
172       apa[apJ[k++]] = 0.0;
173     }
174 
175     /* diagonal part of C */
176     ca = cda + cd->i[i];
177     for (k1=0; k1<cdnz; k1++) {
178       ca[k1]      = apa[apJ[k]];
179       apa[apJ[k++]] = 0.0;
180     }
181 
182     /* 2nd off-diagoanl part of C */
183     ca = coa + co->i[i];
184     for (; k0<conz; k0++) {
185       ca[k0]      = apa[apJ[k]];
186       apa[apJ[k++]] = 0.0;
187     }
188   }
189   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
190   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
191 
192   if (ptap->freestruct) {
193     ierr = MatFreeIntermediateDataStructures(C);CHKERRQ(ierr);
194   }
195   PetscFunctionReturn(0);
196 }
197 
198 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,PetscReal fill,Mat *C)
199 {
200   PetscErrorCode     ierr;
201   MPI_Comm           comm;
202   PetscMPIInt        size;
203   Mat                Cmpi;
204   Mat_APMPI          *ptap;
205   PetscFreeSpaceList free_space=NULL,current_space=NULL;
206   Mat_MPIAIJ         *a        =(Mat_MPIAIJ*)A->data,*c;
207   Mat_SeqAIJ         *ad       =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
208   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
209   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
210   PetscInt           *lnk,i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi;
211   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
212   PetscBT            lnkbt;
213   PetscReal          afill;
214   MatType            mtype;
215 
216   PetscFunctionBegin;
217   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
218   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
219 
220   /* create struct Mat_APMPI and attached it to C later */
221   ierr = PetscNew(&ptap);CHKERRQ(ierr);
222 
223   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
224   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
225 
226   /* get P_loc by taking all local rows of P */
227   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
228 
229   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
230   pi_loc = p_loc->i; pj_loc = p_loc->j;
231   if (size > 1) {
232     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
233     pi_oth = p_oth->i; pj_oth = p_oth->j;
234   } else {
235     p_oth = NULL;
236     pi_oth = NULL; pj_oth = NULL;
237   }
238 
239   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
240   /*-------------------------------------------------------------------*/
241   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
242   ptap->api = api;
243   api[0]    = 0;
244 
245   /* create and initialize a linked list */
246   ierr = PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt);CHKERRQ(ierr);
247 
248   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
249   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space);CHKERRQ(ierr);
250   current_space = free_space;
251 
252   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
253   for (i=0; i<am; i++) {
254     /* diagonal portion of A */
255     nzi = adi[i+1] - adi[i];
256     for (j=0; j<nzi; j++) {
257       row  = *adj++;
258       pnz  = pi_loc[row+1] - pi_loc[row];
259       Jptr = pj_loc + pi_loc[row];
260       /* add non-zero cols of P into the sorted linked list lnk */
261       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
262     }
263     /* off-diagonal portion of A */
264     nzi = aoi[i+1] - aoi[i];
265     for (j=0; j<nzi; j++) {
266       row  = *aoj++;
267       pnz  = pi_oth[row+1] - pi_oth[row];
268       Jptr = pj_oth + pi_oth[row];
269       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
270     }
271 
272     apnz     = lnk[0];
273     api[i+1] = api[i] + apnz;
274 
275     /* if free space is not available, double the total space in the list */
276     if (current_space->local_remaining<apnz) {
277       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
278       nspacedouble++;
279     }
280 
281     /* Copy data into free space, then initialize lnk */
282     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
283     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
284 
285     current_space->array           += apnz;
286     current_space->local_used      += apnz;
287     current_space->local_remaining -= apnz;
288   }
289 
290   /* Allocate space for apj, initialize apj, and */
291   /* destroy list of free space and other temporary array(s) */
292   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
293   apj  = ptap->apj;
294   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
295   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
296 
297   /* malloc apa to store dense row A[i,:]*P */
298   ierr = PetscCalloc1(pN,&ptap->apa);CHKERRQ(ierr);
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 = MatGetType(A,&mtype);CHKERRQ(ierr);
307   ierr = MatSetType(Cmpi,mtype);CHKERRQ(ierr);
308   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
309 
310   ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(Cmpi, apj, api);CHKERRQ(ierr);
311   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
312   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
313   ierr = MatPreallocateFinalize(dnz,onz);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->freeintermediatedatastructures = MatFreeIntermediateDataStructures_MPIAIJ_AP;
320 
321   /* attach the supporting struct to Cmpi for reuse */
322   c       = (Mat_MPIAIJ*)Cmpi->data;
323   c->ap = 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,Bb,Cb,workB1,Bb1,Cb1;
363   MPI_Request  *rwaits,*swaits;
364   PetscInt     numBb;  /* num of Bb matrices */
365   PetscInt     nsends,nrecvs;
366   MPI_Datatype *stype,*rtype;
367 } MPIAIJ_MPIDense;
368 
369 PetscErrorCode MatMPIAIJ_MPIDenseDestroy(void *ctx)
370 {
371   MPIAIJ_MPIDense *contents = (MPIAIJ_MPIDense*)ctx;
372   PetscErrorCode  ierr;
373   PetscInt        i;
374 
375   PetscFunctionBegin;
376   ierr = MatDestroy(&contents->workB);CHKERRQ(ierr);
377 
378   if (contents->numBb) {
379     ierr = MatDestroy(&contents->Bb);CHKERRQ(ierr);
380     ierr = MatDestroy(&contents->Cb);CHKERRQ(ierr);
381 
382     ierr = MatDestroy(&contents->workB1);CHKERRQ(ierr);
383     ierr = MatDestroy(&contents->Bb1);CHKERRQ(ierr);
384     ierr = MatDestroy(&contents->Cb1);CHKERRQ(ierr);
385   }
386   for (i=0; i<contents->nsends; i++) {
387     ierr = MPI_Type_free(&contents->stype[i]);CHKERRQ(ierr);
388   }
389   for (i=0; i<contents->nrecvs; i++) {
390     ierr = MPI_Type_free(&contents->rtype[i]);CHKERRQ(ierr);
391   }
392   ierr = PetscFree4(contents->stype,contents->rtype,contents->rwaits,contents->swaits);CHKERRQ(ierr);
393   ierr = PetscFree(contents);CHKERRQ(ierr);
394   PetscFunctionReturn(0);
395 }
396 
397 /*
398     This is a "dummy function" that handles the case where matrix C was created as a dense matrix
399   directly by the user and passed to MatMatMult() with the MAT_REUSE_MATRIX option
400 
401   It is the same as MatMatMultSymbolic_MPIAIJ_MPIDense() except does not create C
402 */
403 PetscErrorCode MatMatMultNumeric_MPIDense(Mat A,Mat B,Mat C)
404 {
405   PetscErrorCode         ierr;
406   PetscBool              flg;
407   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*) A->data;
408   PetscInt               nz = aij->B->cmap->n,to_n,to_entries,from_n,from_entries;
409   PetscContainer         container;
410   MPIAIJ_MPIDense        *contents;
411   VecScatter             ctx = aij->Mvctx;
412 
413   PetscFunctionBegin;
414   ierr = PetscObjectTypeCompare((PetscObject)B,MATMPIDENSE,&flg);CHKERRQ(ierr);
415   if (!flg) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Second matrix must be mpidense");
416 
417   /* Handle case where where user provided the final C matrix rather than calling MatMatMult() with MAT_INITIAL_MATRIX*/
418   ierr = PetscObjectTypeCompare((PetscObject)A,MATMPIAIJ,&flg);CHKERRQ(ierr);
419   if (!flg) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"First matrix must be MPIAIJ");
420 
421   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
422 
423   ierr = PetscNew(&contents);CHKERRQ(ierr);
424   /* Create work matrix used to store off processor rows of B needed for local product */
425   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,B->cmap->N,NULL,&contents->workB);CHKERRQ(ierr);
426   /* Create work arrays needed */
427   ierr = VecScatterGetRemoteCount_Private(ctx,PETSC_TRUE/*send*/,&to_n,&to_entries);CHKERRQ(ierr);
428   ierr = VecScatterGetRemoteCount_Private(ctx,PETSC_FALSE/*recv*/,&from_n,&from_entries);CHKERRQ(ierr);
429   ierr = PetscMalloc2(from_n,&contents->rwaits,to_n,&contents->swaits);CHKERRQ(ierr);
430 
431   ierr = PetscContainerCreate(PetscObjectComm((PetscObject)A),&container);CHKERRQ(ierr);
432   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
433   ierr = PetscContainerSetUserDestroy(container,MatMPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
434   ierr = PetscObjectCompose((PetscObject)C,"workB",(PetscObject)container);CHKERRQ(ierr);
435   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
436 
437   ierr = (*C->ops->matmultnumeric)(A,B,C);CHKERRQ(ierr);
438   PetscFunctionReturn(0);
439 }
440 
441 /*
442   Create Bb, Cb, Bb1 and Cb1 matrices to be used by MatMatMult_MPIAIJ_MPIDense().
443   These matrices are used as wrappers for sub-columns of B and C, thus their own matrix operations are not used.
444   Modified from MatCreateDense().
445 */
446 PETSC_STATIC_INLINE PetscErrorCode MatCreateSubMPIDense_private(MPI_Comm comm,PetscInt m,PetscInt n,PetscInt M,PetscInt N,PetscInt rbs,PetscInt cbs,PetscScalar *data,Mat *A)
447 {
448   PetscErrorCode ierr;
449 
450   PetscFunctionBegin;
451   ierr = MatCreate(comm,A);CHKERRQ(ierr);
452   ierr = MatSetSizes(*A,m,n,M,N);CHKERRQ(ierr);
453   ierr = MatSetBlockSizes(*A,rbs,cbs);CHKERRQ(ierr);
454   ierr = MatSetType(*A,MATMPIDENSE);CHKERRQ(ierr);
455   ierr = MatMPIDenseSetPreallocation(*A,data);CHKERRQ(ierr);
456   (*A)->assembled = PETSC_TRUE;
457   PetscFunctionReturn(0);
458 }
459 
460 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat A,Mat B,PetscReal fill,Mat *C)
461 {
462   PetscErrorCode  ierr;
463   Mat_MPIAIJ      *aij=(Mat_MPIAIJ*)A->data;
464   PetscInt        nz=aij->B->cmap->n,nsends,nrecvs,i,nrows_to,j;
465   PetscContainer  container;
466   MPIAIJ_MPIDense *contents;
467   VecScatter      ctx=aij->Mvctx;
468   PetscInt        Am=A->rmap->n,Bm=B->rmap->n,Bn=B->cmap->n,BN=B->cmap->N,Bbn,Bbn1,bs,nrows_from;
469   MPI_Comm        comm;
470   MPI_Datatype    type1,*stype,*rtype;
471   const PetscInt  *sindices,*sstarts,*rstarts;
472   PetscMPIInt     *disp;
473 
474   PetscFunctionBegin;
475   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
476 
477   ierr = MatCreate(comm,C);CHKERRQ(ierr);
478   ierr = MatSetSizes(*C,Am,Bn,A->rmap->N,BN);CHKERRQ(ierr);
479   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
480   ierr = MatSetType(*C,MATMPIDENSE);CHKERRQ(ierr);
481   ierr = MatMPIDenseSetPreallocation(*C,NULL);CHKERRQ(ierr);
482   ierr = MatAssemblyBegin(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
483   ierr = MatAssemblyEnd(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
484 
485   (*C)->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
486 
487   ierr = PetscNew(&contents);CHKERRQ(ierr);
488   contents->numBb = 0;
489 
490   ierr = VecScatterGetRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,NULL,NULL);CHKERRQ(ierr);
491   ierr = VecScatterGetRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,NULL,NULL);CHKERRQ(ierr);
492 
493   /* Create column block of B and C for memory scalability when BN is too large */
494   /* Estimate Bbn, column size of Bb */
495   if (nz) {
496     Bbn1 = 2*Am*BN/nz;
497   } else Bbn1 = BN;
498 
499   bs = PetscAbs(B->cmap->bs);
500   Bbn1 = Bbn1/bs *bs; /* Bbn1 is a multiple of bs */
501   if (Bbn1 > BN) Bbn1 = BN;
502   ierr = MPI_Allreduce(&Bbn1,&Bbn,1,MPIU_INT,MPI_MAX,comm);CHKERRQ(ierr);
503 
504   /* Enable runtime option for Bbn */
505   ierr = PetscOptionsBegin(comm,((PetscObject)*C)->prefix,"MatMatMult","Mat");CHKERRQ(ierr);
506   ierr = PetscOptionsInt("-matmatmult_Bbn","Number of columns in Bb","MatMatMult",Bbn,&Bbn,NULL);CHKERRQ(ierr);
507   if (Bbn > BN) SETERRQ2(comm,PETSC_ERR_ARG_SIZ,"Bbn=%D cannot be larger than %D, column size of B",Bbn,BN);
508   ierr = PetscOptionsEnd();CHKERRQ(ierr);
509 
510   if (Bbn < BN) {
511     contents->numBb = BN/Bbn;
512     Bbn1 = BN - contents->numBb*Bbn;
513   }
514 
515   if (contents->numBb) {
516     PetscScalar data[1]; /* fake array for Bb and Cb */
517     ierr = PetscInfo3(*C,"use Bb, BN=%D, Bbn=%D; numBb=%D\n",BN,Bbn,contents->numBb);CHKERRQ(ierr);
518     ierr = MatCreateSubMPIDense_private(comm,B->rmap->n,PETSC_DECIDE,A->rmap->N,Bbn,B->rmap->bs,B->cmap->bs,data,&contents->Bb);CHKERRQ(ierr);
519     ierr = MatCreateSubMPIDense_private(comm,Am,PETSC_DECIDE,A->rmap->N,Bbn,(*C)->rmap->bs,(*C)->cmap->bs,data,&contents->Cb);CHKERRQ(ierr);
520 
521     if (Bbn1) { /* Create Bb1 and Cb1 for the remaining columns */
522       ierr = PetscInfo2(*C,"use Bb1, BN=%D, Bbn1=%D\n",BN,Bbn1);CHKERRQ(ierr);
523       ierr = MatCreateSubMPIDense_private(comm,B->rmap->n,PETSC_DECIDE,A->rmap->N,Bbn1,B->rmap->bs,B->cmap->bs,data,&contents->Bb1);CHKERRQ(ierr);
524       ierr = MatCreateSubMPIDense_private(comm,Am,PETSC_DECIDE,A->rmap->N,Bbn1,(*C)->rmap->bs,(*C)->cmap->bs,data,&contents->Cb1);CHKERRQ(ierr);
525 
526       /* Create work matrix used to store off processor rows of B needed for local product */
527       ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,Bbn1,NULL,&contents->workB1);CHKERRQ(ierr);
528     }
529   }
530 
531   /* Create work matrix used to store off processor rows of B needed for local product */
532   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,Bbn,NULL,&contents->workB);CHKERRQ(ierr);
533 
534   /* Use MPI derived data type to reduce memory required by the send/recv buffers */
535   ierr = PetscMalloc4(nsends,&stype,nrecvs,&rtype,nrecvs,&contents->rwaits,nsends,&contents->swaits);CHKERRQ(ierr);
536   contents->stype  = stype;
537   contents->nsends = nsends;
538 
539   contents->rtype  = rtype;
540   contents->nrecvs = nrecvs;
541 
542   ierr = PetscMalloc1(Bm+1,&disp);CHKERRQ(ierr);
543   for (i=0; i<nsends; i++) {
544     nrows_to = sstarts[i+1]-sstarts[i];
545     for (j=0; j<nrows_to; j++){
546       disp[j] = sindices[sstarts[i]+j]; /* rowB to be sent */
547     }
548     ierr = MPI_Type_create_indexed_block(nrows_to,1,(const PetscMPIInt *)disp,MPIU_SCALAR,&type1);CHKERRQ(ierr);
549     ierr = MPI_Type_create_resized(type1,0,Bm*sizeof(PetscScalar),&stype[i]);CHKERRQ(ierr);
550     ierr = MPI_Type_commit(&stype[i]);CHKERRQ(ierr);
551     ierr = MPI_Type_free(&type1);CHKERRQ(ierr);
552   }
553 
554   for (i=0; i<nrecvs; i++) {
555     /* received values from a process form a (nrows_from x Bbn) row block in workB (column-wise) */
556     nrows_from = rstarts[i+1]-rstarts[i];
557     disp[0] = 0;
558     ierr = MPI_Type_create_indexed_block(1, nrows_from, (const PetscMPIInt *)disp, MPIU_SCALAR, &type1);CHKERRQ(ierr);
559     ierr = MPI_Type_create_resized(type1, 0, nz*sizeof(PetscScalar), &rtype[i]);CHKERRQ(ierr);
560     ierr = MPI_Type_commit(&rtype[i]);CHKERRQ(ierr);
561     ierr = MPI_Type_free(&type1);CHKERRQ(ierr);
562   }
563 
564   ierr = PetscFree(disp);CHKERRQ(ierr);
565   ierr = VecScatterRestoreRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,NULL,NULL);CHKERRQ(ierr);
566   ierr = VecScatterRestoreRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,NULL,NULL);CHKERRQ(ierr);
567 
568   ierr = PetscContainerCreate(comm,&container);CHKERRQ(ierr);
569   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
570   ierr = PetscContainerSetUserDestroy(container,MatMPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
571   ierr = PetscObjectCompose((PetscObject)(*C),"workB",(PetscObject)container);CHKERRQ(ierr);
572   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
573   PetscFunctionReturn(0);
574 }
575 
576 extern PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat,Mat,Mat);
577 /*
578     Performs an efficient scatter on the rows of B needed by this process; this is
579     a modification of the VecScatterBegin_() routines.
580 
581     Input: Bbidx = 0: B = Bb
582                  = 1: B = Bb1, see MatMatMultSymbolic_MPIAIJ_MPIDense()
583 */
584 PetscErrorCode MatMPIDenseScatter(Mat A,Mat B,PetscInt Bbidx,Mat C,Mat *outworkB)
585 {
586   Mat_MPIAIJ        *aij = (Mat_MPIAIJ*)A->data;
587   PetscErrorCode    ierr;
588   const PetscScalar *b;
589   PetscScalar       *rvalues;
590   VecScatter        ctx = aij->Mvctx;
591   const PetscInt    *sindices,*sstarts,*rstarts;
592   const PetscMPIInt *sprocs,*rprocs;
593   PetscInt          i,nsends,nrecvs,nrecvs2;
594   MPI_Request       *swaits,*rwaits;
595   MPI_Comm          comm;
596   PetscMPIInt       tag=((PetscObject)ctx)->tag,ncols=B->cmap->N,nrows=aij->B->cmap->n,imdex,nsends_mpi,nrecvs_mpi;
597   MPI_Status        status;
598   MPIAIJ_MPIDense   *contents;
599   PetscContainer    container;
600   Mat               workB;
601   MPI_Datatype      *stype,*rtype;
602 
603   PetscFunctionBegin;
604   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
605   ierr = PetscObjectQuery((PetscObject)C,"workB",(PetscObject*)&container);CHKERRQ(ierr);
606   if (!container) SETERRQ(comm,PETSC_ERR_PLIB,"Container does not exist");
607   ierr = PetscContainerGetPointer(container,(void**)&contents);CHKERRQ(ierr);
608 
609   ierr = VecScatterGetRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,&sprocs,NULL/*bs*/);CHKERRQ(ierr);
610   ierr = VecScatterGetRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,&rprocs,NULL/*bs*/);CHKERRQ(ierr);
611   ierr = PetscMPIIntCast(nsends,&nsends_mpi);CHKERRQ(ierr);
612   ierr = PetscMPIIntCast(nrecvs,&nrecvs_mpi);CHKERRQ(ierr);
613   if (Bbidx == 0) {
614     workB = *outworkB = contents->workB;
615   } else {
616     workB = *outworkB = contents->workB1;
617   }
618   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);
619   swaits  = contents->swaits;
620   rwaits  = contents->rwaits;
621 
622   ierr = MatDenseGetArrayRead(B,&b);CHKERRQ(ierr);
623   ierr = MatDenseGetArray(workB,&rvalues);CHKERRQ(ierr);
624 
625   /* Post recv, use MPI derived data type to save memory */
626   rtype = contents->rtype;
627   for (i=0; i<nrecvs; i++) {
628     ierr = MPI_Irecv(rvalues+(rstarts[i]-rstarts[0]),ncols,rtype[i],rprocs[i],tag,comm,rwaits+i);CHKERRQ(ierr);
629   }
630 
631   stype = contents->stype;
632   for (i=0; i<nsends; i++) {
633     ierr = MPI_Isend(b,ncols,stype[i],sprocs[i],tag,comm,swaits+i);CHKERRQ(ierr);
634   }
635 
636   nrecvs2 = nrecvs;
637   while (nrecvs2) {
638     ierr = MPI_Waitany(nrecvs_mpi,rwaits,&imdex,&status);CHKERRQ(ierr);
639     nrecvs2--;
640   }
641   if (nsends) {ierr = MPI_Waitall(nsends_mpi,swaits,MPI_STATUSES_IGNORE);CHKERRQ(ierr);}
642 
643   ierr = VecScatterRestoreRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,&sprocs,NULL);CHKERRQ(ierr);
644   ierr = VecScatterRestoreRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,&rprocs,NULL);CHKERRQ(ierr);
645   ierr = MatDenseRestoreArrayRead(B,&b);CHKERRQ(ierr);
646   ierr = MatDenseRestoreArray(workB,&rvalues);CHKERRQ(ierr);
647   ierr = MatAssemblyBegin(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
648   ierr = MatAssemblyEnd(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
649   PetscFunctionReturn(0);
650 }
651 
652 /*
653   Compute Cb = A*Bb
654 */
655 PETSC_STATIC_INLINE PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense_private(Mat A,Mat Bb,PetscInt Bbidx,PetscInt start,Mat C,const PetscScalar *barray,PetscScalar *carray,Mat Cb)
656 {
657   PetscErrorCode  ierr;
658   PetscInt        start1;
659   Mat             workB;
660   Mat_MPIAIJ      *aij = (Mat_MPIAIJ*)A->data;
661   Mat_MPIDense    *cbdense = (Mat_MPIDense*)Cb->data;
662 
663   PetscFunctionBegin;
664   /* Place barray to Bb */
665   start1 = start*Bb->rmap->n;
666   ierr = MatDensePlaceArray(Bb,barray+start1);CHKERRQ(ierr);
667 
668   /* get off processor parts of Bb needed to complete Cb=A*Bb */
669   ierr = MatMPIDenseScatter(A,Bb,Bbidx,C,&workB);CHKERRQ(ierr);
670   ierr = MatDenseResetArray(Bb);CHKERRQ(ierr);
671 
672   /* off-diagonal block of A times nonlocal rows of Bb */
673   /* Place carray to Cb */
674   start1 = start*Cb->rmap->n;
675   ierr = MatDensePlaceArray(Cb,carray+start1);CHKERRQ(ierr);
676   ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cbdense->A);CHKERRQ(ierr);
677   ierr = MatDenseResetArray(Cb);CHKERRQ(ierr);
678   PetscFunctionReturn(0);
679 }
680 
681 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat A,Mat B,Mat C)
682 {
683   PetscErrorCode  ierr;
684   Mat_MPIAIJ      *aij    = (Mat_MPIAIJ*)A->data;
685   Mat_MPIDense    *bdense = (Mat_MPIDense*)B->data;
686   Mat_MPIDense    *cdense = (Mat_MPIDense*)C->data;
687   Mat             workB;
688   MPIAIJ_MPIDense *contents;
689   PetscContainer  container;
690   MPI_Comm        comm;
691   PetscInt        numBb;
692 
693   PetscFunctionBegin;
694   /* diagonal block of A times all local rows of B*/
695   ierr = MatMatMultNumeric_SeqAIJ_SeqDense(aij->A,bdense->A,cdense->A);CHKERRQ(ierr);
696 
697   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
698   ierr = PetscObjectQuery((PetscObject)C,"workB",(PetscObject*)&container);CHKERRQ(ierr);
699   if (!container) SETERRQ(comm,PETSC_ERR_PLIB,"Container does not exist");
700   ierr = PetscContainerGetPointer(container,(void**)&contents);CHKERRQ(ierr);
701   numBb = contents->numBb;
702 
703   if (!numBb) {
704     /* get off processor parts of B needed to complete C=A*B */
705     ierr = MatMPIDenseScatter(A,B,0,C,&workB);CHKERRQ(ierr);
706 
707     /* off-diagonal block of A times nonlocal rows of B */
708     ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A);CHKERRQ(ierr);
709 
710   } else {
711     const PetscScalar *barray;
712     PetscScalar       *carray;
713     Mat               Bb=contents->Bb,Cb=contents->Cb;
714     PetscInt          BbN=Bb->cmap->N,start,i;
715 
716     ierr = MatDenseGetArrayRead(B,&barray);CHKERRQ(ierr);
717     ierr = MatDenseGetArray(C,&carray);CHKERRQ(ierr);
718     for (i=0; i<numBb; i++) {
719       start = i*BbN;
720       ierr = MatMatMultNumeric_MPIAIJ_MPIDense_private(A,Bb,0,start,C,barray,carray,Cb);CHKERRQ(ierr);
721     }
722 
723     if (contents->Bb1) {
724       Bb = contents->Bb1; Cb = contents->Cb1;
725       start = i*BbN;
726       ierr = MatMatMultNumeric_MPIAIJ_MPIDense_private(A,Bb,1,start,C,barray,carray,Cb);CHKERRQ(ierr);
727     }
728     ierr = MatDenseRestoreArrayRead(B,&barray);CHKERRQ(ierr);
729     ierr = MatDenseRestoreArray(C,&carray);CHKERRQ(ierr);
730   }
731   PetscFunctionReturn(0);
732 }
733 
734 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
735 {
736   PetscErrorCode ierr;
737   Mat_MPIAIJ     *a   = (Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
738   Mat_SeqAIJ     *ad  = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
739   Mat_SeqAIJ     *cd  = (Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
740   PetscInt       *adi = ad->i,*adj,*aoi=ao->i,*aoj;
741   PetscScalar    *ada,*aoa,*cda=cd->a,*coa=co->a;
742   Mat_SeqAIJ     *p_loc,*p_oth;
743   PetscInt       *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
744   PetscScalar    *pa_loc,*pa_oth,*pa,valtmp,*ca;
745   PetscInt       cm    = C->rmap->n,anz,pnz;
746   Mat_APMPI      *ptap = c->ap;
747   PetscScalar    *apa_sparse;
748   PetscInt       *api,*apj,*apJ,i,j,k,row;
749   PetscInt       cstart = C->cmap->rstart;
750   PetscInt       cdnz,conz,k0,k1,nextp;
751   MPI_Comm       comm;
752   PetscMPIInt    size;
753 
754   PetscFunctionBegin;
755   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
756   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
757 
758   if (!ptap->P_oth && size>1) {
759     SETERRQ(comm,PETSC_ERR_ARG_WRONGSTATE,"AP cannot be reused. Do not call MatFreeIntermediateDataStructures() or use '-mat_freeintermediatedatastructures'");
760   }
761   apa_sparse = ptap->apa;
762 
763   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
764   /*-----------------------------------------------------*/
765   /* update numerical values of P_oth and P_loc */
766   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
767   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
768 
769   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
770   /*----------------------------------------------------------*/
771   /* get data from symbolic products */
772   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
773   pi_loc = p_loc->i; pj_loc = p_loc->j; pa_loc = p_loc->a;
774   if (size >1) {
775     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
776     pi_oth = p_oth->i; pj_oth = p_oth->j; pa_oth = p_oth->a;
777   } else {
778     p_oth = NULL; pi_oth = NULL; pj_oth = NULL; pa_oth = NULL;
779   }
780 
781   api = ptap->api;
782   apj = ptap->apj;
783   for (i=0; i<cm; i++) {
784     apJ = apj + api[i];
785 
786     /* diagonal portion of A */
787     anz = adi[i+1] - adi[i];
788     adj = ad->j + adi[i];
789     ada = ad->a + adi[i];
790     for (j=0; j<anz; j++) {
791       row = adj[j];
792       pnz = pi_loc[row+1] - pi_loc[row];
793       pj  = pj_loc + pi_loc[row];
794       pa  = pa_loc + pi_loc[row];
795       /* perform sparse axpy */
796       valtmp = ada[j];
797       nextp  = 0;
798       for (k=0; nextp<pnz; k++) {
799         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
800           apa_sparse[k] += valtmp*pa[nextp++];
801         }
802       }
803       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
804     }
805 
806     /* off-diagonal portion of A */
807     anz = aoi[i+1] - aoi[i];
808     aoj = ao->j + aoi[i];
809     aoa = ao->a + aoi[i];
810     for (j=0; j<anz; j++) {
811       row = aoj[j];
812       pnz = pi_oth[row+1] - pi_oth[row];
813       pj  = pj_oth + pi_oth[row];
814       pa  = pa_oth + pi_oth[row];
815       /* perform sparse axpy */
816       valtmp = aoa[j];
817       nextp  = 0;
818       for (k=0; nextp<pnz; k++) {
819         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
820           apa_sparse[k] += valtmp*pa[nextp++];
821         }
822       }
823       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
824     }
825 
826     /* set values in C */
827     cdnz = cd->i[i+1] - cd->i[i];
828     conz = co->i[i+1] - co->i[i];
829 
830     /* 1st off-diagoanl part of C */
831     ca = coa + co->i[i];
832     k  = 0;
833     for (k0=0; k0<conz; k0++) {
834       if (apJ[k] >= cstart) break;
835       ca[k0]        = apa_sparse[k];
836       apa_sparse[k] = 0.0;
837       k++;
838     }
839 
840     /* diagonal part of C */
841     ca = cda + cd->i[i];
842     for (k1=0; k1<cdnz; k1++) {
843       ca[k1]        = apa_sparse[k];
844       apa_sparse[k] = 0.0;
845       k++;
846     }
847 
848     /* 2nd off-diagoanl part of C */
849     ca = coa + co->i[i];
850     for (; k0<conz; k0++) {
851       ca[k0]        = apa_sparse[k];
852       apa_sparse[k] = 0.0;
853       k++;
854     }
855   }
856   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
857   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
858 
859   if (ptap->freestruct) {
860     ierr = MatFreeIntermediateDataStructures(C);CHKERRQ(ierr);
861   }
862   PetscFunctionReturn(0);
863 }
864 
865 /* same as MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(), except using LLCondensed to avoid O(BN) memory requirement */
866 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat *C)
867 {
868   PetscErrorCode     ierr;
869   MPI_Comm           comm;
870   PetscMPIInt        size;
871   Mat                Cmpi;
872   Mat_APMPI          *ptap;
873   PetscFreeSpaceList free_space = NULL,current_space=NULL;
874   Mat_MPIAIJ         *a         = (Mat_MPIAIJ*)A->data,*c;
875   Mat_SeqAIJ         *ad        = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
876   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
877   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
878   PetscInt           i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi,*lnk,apnz_max=0;
879   PetscInt           am=A->rmap->n,pn=P->cmap->n,pm=P->rmap->n,lsize=pn+20;
880   PetscReal          afill;
881   MatType            mtype;
882 
883   PetscFunctionBegin;
884   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
885   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
886 
887   /* create struct Mat_APMPI and attached it to C later */
888   ierr = PetscNew(&ptap);CHKERRQ(ierr);
889 
890   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
891   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
892 
893   /* get P_loc by taking all local rows of P */
894   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
895 
896   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
897   pi_loc = p_loc->i; pj_loc = p_loc->j;
898   if (size > 1) {
899     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
900     pi_oth = p_oth->i; pj_oth = p_oth->j;
901   } else {
902     p_oth  = NULL;
903     pi_oth = NULL; pj_oth = NULL;
904   }
905 
906   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
907   /*-------------------------------------------------------------------*/
908   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
909   ptap->api = api;
910   api[0]    = 0;
911 
912   ierr = PetscLLCondensedCreate_Scalable(lsize,&lnk);CHKERRQ(ierr);
913 
914   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
915   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space);CHKERRQ(ierr);
916   current_space = free_space;
917   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
918   for (i=0; i<am; i++) {
919     /* diagonal portion of A */
920     nzi = adi[i+1] - adi[i];
921     for (j=0; j<nzi; j++) {
922       row  = *adj++;
923       pnz  = pi_loc[row+1] - pi_loc[row];
924       Jptr = pj_loc + pi_loc[row];
925       /* Expand list if it is not long enough */
926       if (pnz+apnz_max > lsize) {
927         lsize = pnz+apnz_max;
928         ierr = PetscLLCondensedExpand_Scalable(lsize, &lnk);CHKERRQ(ierr);
929       }
930       /* add non-zero cols of P into the sorted linked list lnk */
931       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
932       apnz     = *lnk; /* The first element in the list is the number of items in the list */
933       api[i+1] = api[i] + apnz;
934       if (apnz > apnz_max) apnz_max = apnz;
935     }
936     /* off-diagonal portion of A */
937     nzi = aoi[i+1] - aoi[i];
938     for (j=0; j<nzi; j++) {
939       row  = *aoj++;
940       pnz  = pi_oth[row+1] - pi_oth[row];
941       Jptr = pj_oth + pi_oth[row];
942       /* Expand list if it is not long enough */
943       if (pnz+apnz_max > lsize) {
944         lsize = pnz + apnz_max;
945         ierr = PetscLLCondensedExpand_Scalable(lsize, &lnk);CHKERRQ(ierr);
946       }
947       /* add non-zero cols of P into the sorted linked list lnk */
948       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
949       apnz     = *lnk;  /* The first element in the list is the number of items in the list */
950       api[i+1] = api[i] + apnz;
951       if (apnz > apnz_max) apnz_max = apnz;
952     }
953     apnz     = *lnk;
954     api[i+1] = api[i] + apnz;
955     if (apnz > apnz_max) apnz_max = apnz;
956 
957     /* if free space is not available, double the total space in the list */
958     if (current_space->local_remaining<apnz) {
959       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
960       nspacedouble++;
961     }
962 
963     /* Copy data into free space, then initialize lnk */
964     ierr = PetscLLCondensedClean_Scalable(apnz,current_space->array,lnk);CHKERRQ(ierr);
965     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
966 
967     current_space->array           += apnz;
968     current_space->local_used      += apnz;
969     current_space->local_remaining -= apnz;
970   }
971 
972   /* Allocate space for apj, initialize apj, and */
973   /* destroy list of free space and other temporary array(s) */
974   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
975   apj  = ptap->apj;
976   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
977   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
978 
979   /* create and assemble symbolic parallel matrix Cmpi */
980   /*----------------------------------------------------*/
981   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
982   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
983   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
984   ierr = MatGetType(A,&mtype);CHKERRQ(ierr);
985   ierr = MatSetType(Cmpi,mtype);CHKERRQ(ierr);
986   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
987 
988   /* malloc apa for assembly Cmpi */
989   ierr = PetscCalloc1(apnz_max,&ptap->apa);CHKERRQ(ierr);
990 
991   ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(Cmpi, apj, api);CHKERRQ(ierr);
992   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
993   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
994   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
995 
996   ptap->destroy             = Cmpi->ops->destroy;
997   ptap->duplicate           = Cmpi->ops->duplicate;
998   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ;
999   Cmpi->ops->destroy        = MatDestroy_MPIAIJ_MatMatMult;
1000   Cmpi->ops->freeintermediatedatastructures = MatFreeIntermediateDataStructures_MPIAIJ_AP;
1001 
1002   /* attach the supporting struct to Cmpi for reuse */
1003   c       = (Mat_MPIAIJ*)Cmpi->data;
1004   c->ap = ptap;
1005   *C = Cmpi;
1006 
1007   /* set MatInfo */
1008   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
1009   if (afill < 1.0) afill = 1.0;
1010   Cmpi->info.mallocs           = nspacedouble;
1011   Cmpi->info.fill_ratio_given  = fill;
1012   Cmpi->info.fill_ratio_needed = afill;
1013 
1014 #if defined(PETSC_USE_INFO)
1015   if (api[am]) {
1016     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1017     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
1018   } else {
1019     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1020   }
1021 #endif
1022   PetscFunctionReturn(0);
1023 }
1024 
1025 /* This function is needed for the seqMPI matrix-matrix multiplication.  */
1026 /* Three input arrays are merged to one output array. The size of the    */
1027 /* output array is also output. Duplicate entries only show up once.     */
1028 static void Merge3SortedArrays(PetscInt  size1, PetscInt *in1,
1029                                PetscInt  size2, PetscInt *in2,
1030                                PetscInt  size3, PetscInt *in3,
1031                                PetscInt *size4, PetscInt *out)
1032 {
1033   int i = 0, j = 0, k = 0, l = 0;
1034 
1035   /* Traverse all three arrays */
1036   while (i<size1 && j<size2 && k<size3) {
1037     if (in1[i] < in2[j] && in1[i] < in3[k]) {
1038       out[l++] = in1[i++];
1039     }
1040     else if(in2[j] < in1[i] && in2[j] < in3[k]) {
1041       out[l++] = in2[j++];
1042     }
1043     else if(in3[k] < in1[i] && in3[k] < in2[j]) {
1044       out[l++] = in3[k++];
1045     }
1046     else if(in1[i] == in2[j] && in1[i] < in3[k]) {
1047       out[l++] = in1[i];
1048       i++, j++;
1049     }
1050     else if(in1[i] == in3[k] && in1[i] < in2[j]) {
1051       out[l++] = in1[i];
1052       i++, k++;
1053     }
1054     else if(in3[k] == in2[j] && in2[j] < in1[i])  {
1055       out[l++] = in2[j];
1056       k++, j++;
1057     }
1058     else if(in1[i] == in2[j] && in1[i] == in3[k]) {
1059       out[l++] = in1[i];
1060       i++, j++, k++;
1061     }
1062   }
1063 
1064   /* Traverse two remaining arrays */
1065   while (i<size1 && j<size2) {
1066     if (in1[i] < in2[j]) {
1067       out[l++] = in1[i++];
1068     }
1069     else if(in1[i] > in2[j]) {
1070       out[l++] = in2[j++];
1071     }
1072     else {
1073       out[l++] = in1[i];
1074       i++, j++;
1075     }
1076   }
1077 
1078   while (i<size1 && k<size3) {
1079     if (in1[i] < in3[k]) {
1080       out[l++] = in1[i++];
1081     }
1082     else if(in1[i] > in3[k]) {
1083       out[l++] = in3[k++];
1084     }
1085     else {
1086       out[l++] = in1[i];
1087       i++, k++;
1088     }
1089   }
1090 
1091   while (k<size3 && j<size2)  {
1092     if (in3[k] < in2[j]) {
1093       out[l++] = in3[k++];
1094     }
1095     else if(in3[k] > in2[j]) {
1096       out[l++] = in2[j++];
1097     }
1098     else {
1099       out[l++] = in3[k];
1100       k++, j++;
1101     }
1102   }
1103 
1104   /* Traverse one remaining array */
1105   while (i<size1) out[l++] = in1[i++];
1106   while (j<size2) out[l++] = in2[j++];
1107   while (k<size3) out[l++] = in3[k++];
1108 
1109   *size4 = l;
1110 }
1111 
1112 /* This matrix-matrix multiplication algorithm divides the multiplication into three multiplications and  */
1113 /* adds up the products. Two of these three multiplications are performed with existing (sequential)      */
1114 /* matrix-matrix multiplications.  */
1115 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_seqMPI(Mat A, Mat P, PetscReal fill, Mat *C)
1116 {
1117   PetscErrorCode     ierr;
1118   MPI_Comm           comm;
1119   PetscMPIInt        size;
1120   Mat                Cmpi;
1121   Mat_APMPI          *ptap;
1122   PetscFreeSpaceList free_space_diag=NULL, current_space=NULL;
1123   Mat_MPIAIJ         *a        =(Mat_MPIAIJ*)A->data;
1124   Mat_SeqAIJ         *ad       =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc;
1125   Mat_MPIAIJ         *p        =(Mat_MPIAIJ*)P->data;
1126   Mat_MPIAIJ         *c;
1127   Mat_SeqAIJ         *adpd_seq, *p_off, *aopoth_seq;
1128   PetscInt           adponz, adpdnz;
1129   PetscInt           *pi_loc,*dnz,*onz;
1130   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,rstart=A->rmap->rstart;
1131   PetscInt           *lnk,i, i1=0,pnz,row,*adpoi,*adpoj, *api, *adpoJ, *aopJ, *apJ,*Jptr, aopnz, nspacedouble=0,j,nzi,
1132                      *apj,apnz, *adpdi, *adpdj, *adpdJ, *poff_i, *poff_j, *j_temp, *aopothi, *aopothj;
1133   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n, p_colstart, p_colend;
1134   PetscBT            lnkbt;
1135   PetscReal          afill;
1136   PetscMPIInt        rank;
1137   Mat                adpd, aopoth;
1138   MatType            mtype;
1139   const char         *prefix;
1140 
1141   PetscFunctionBegin;
1142   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1143   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1144   ierr = MPI_Comm_rank(comm, &rank);CHKERRQ(ierr);
1145   ierr = MatGetOwnershipRangeColumn(P, &p_colstart, &p_colend); CHKERRQ(ierr);
1146 
1147   /* create struct Mat_APMPI and attached it to C later */
1148   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1149 
1150   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
1151   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
1152 
1153   /* get P_loc by taking all local rows of P */
1154   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
1155 
1156 
1157   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
1158   pi_loc = p_loc->i;
1159 
1160   /* Allocate memory for the i arrays of the matrices A*P, A_diag*P_off and A_offd * P */
1161   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
1162   ierr      = PetscMalloc1(am+2,&adpoi);CHKERRQ(ierr);
1163 
1164   adpoi[0]    = 0;
1165   ptap->api = api;
1166   api[0] = 0;
1167 
1168   /* create and initialize a linked list, will be used for both A_diag * P_loc_off and A_offd * P_oth */
1169   ierr = PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt);CHKERRQ(ierr);
1170   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
1171 
1172   /* Symbolic calc of A_loc_diag * P_loc_diag */
1173   ierr = MatGetOptionsPrefix(A,&prefix);CHKERRQ(ierr);
1174   ierr = MatSetOptionsPrefix(a->A,prefix);CHKERRQ(ierr);
1175   ierr = MatAppendOptionsPrefix(a->A,"inner_diag_");CHKERRQ(ierr);
1176   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(a->A, p->A, fill, &adpd);CHKERRQ(ierr);
1177   adpd_seq = (Mat_SeqAIJ*)((adpd)->data);
1178   adpdi = adpd_seq->i; adpdj = adpd_seq->j;
1179   p_off = (Mat_SeqAIJ*)((p->B)->data);
1180   poff_i = p_off->i; poff_j = p_off->j;
1181 
1182   /* j_temp stores indices of a result row before they are added to the linked list */
1183   ierr = PetscMalloc1(pN+2,&j_temp);CHKERRQ(ierr);
1184 
1185 
1186   /* Symbolic calc of the A_diag * p_loc_off */
1187   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
1188   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space_diag);CHKERRQ(ierr);
1189   current_space = free_space_diag;
1190 
1191   for (i=0; i<am; i++) {
1192     /* A_diag * P_loc_off */
1193     nzi = adi[i+1] - adi[i];
1194     for (j=0; j<nzi; j++) {
1195       row  = *adj++;
1196       pnz  = poff_i[row+1] - poff_i[row];
1197       Jptr = poff_j + poff_i[row];
1198       for(i1 = 0; i1 < pnz; i1++) {
1199         j_temp[i1] = p->garray[Jptr[i1]];
1200       }
1201       /* add non-zero cols of P into the sorted linked list lnk */
1202       ierr = PetscLLCondensedAddSorted(pnz,j_temp,lnk,lnkbt);CHKERRQ(ierr);
1203     }
1204 
1205     adponz     = lnk[0];
1206     adpoi[i+1] = adpoi[i] + adponz;
1207 
1208     /* if free space is not available, double the total space in the list */
1209     if (current_space->local_remaining<adponz) {
1210       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(adponz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1211       nspacedouble++;
1212     }
1213 
1214     /* Copy data into free space, then initialize lnk */
1215     ierr = PetscLLCondensedClean(pN,adponz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1216 
1217     current_space->array           += adponz;
1218     current_space->local_used      += adponz;
1219     current_space->local_remaining -= adponz;
1220   }
1221 
1222   /* Symbolic calc of A_off * P_oth */
1223   ierr = MatSetOptionsPrefix(a->B,prefix);CHKERRQ(ierr);
1224   ierr = MatAppendOptionsPrefix(a->B,"inner_offdiag_");CHKERRQ(ierr);
1225   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(a->B, ptap->P_oth, fill, &aopoth);CHKERRQ(ierr);
1226   aopoth_seq = (Mat_SeqAIJ*)((aopoth)->data);
1227   aopothi = aopoth_seq->i; aopothj = aopoth_seq->j;
1228 
1229   /* Allocate space for apj, adpj, aopj, ... */
1230   /* destroy lists of free space and other temporary array(s) */
1231 
1232   ierr = PetscMalloc1(aopothi[am] + adpoi[am] + adpdi[am]+2, &ptap->apj);CHKERRQ(ierr);
1233   ierr = PetscMalloc1(adpoi[am]+2, &adpoj);CHKERRQ(ierr);
1234 
1235   /* Copy from linked list to j-array */
1236   ierr = PetscFreeSpaceContiguous(&free_space_diag,adpoj);CHKERRQ(ierr);
1237   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1238 
1239   adpoJ = adpoj;
1240   adpdJ = adpdj;
1241   aopJ = aopothj;
1242   apj  = ptap->apj;
1243   apJ = apj; /* still empty */
1244 
1245   /* Merge j-arrays of A_off * P, A_diag * P_loc_off, and */
1246   /* A_diag * P_loc_diag to get A*P */
1247   for (i = 0; i < am; i++) {
1248     aopnz  =  aopothi[i+1] -  aopothi[i];
1249     adponz = adpoi[i+1] - adpoi[i];
1250     adpdnz = adpdi[i+1] - adpdi[i];
1251 
1252     /* Correct indices from A_diag*P_diag */
1253     for(i1 = 0; i1 < adpdnz; i1++) {
1254       adpdJ[i1] += p_colstart;
1255     }
1256     /* Merge j-arrays of A_diag * P_loc_off and A_diag * P_loc_diag and A_off * P_oth */
1257     Merge3SortedArrays(adponz, adpoJ, adpdnz, adpdJ, aopnz, aopJ, &apnz, apJ);
1258     ierr = MatPreallocateSet(i+rstart, apnz, apJ, dnz, onz); CHKERRQ(ierr);
1259 
1260     aopJ += aopnz;
1261     adpoJ += adponz;
1262     adpdJ += adpdnz;
1263     apJ += apnz;
1264     api[i+1] = api[i] + apnz;
1265   }
1266 
1267   /* malloc apa to store dense row A[i,:]*P */
1268   ierr = PetscCalloc1(pN+2,&ptap->apa);CHKERRQ(ierr);
1269 
1270   /* create and assemble symbolic parallel matrix Cmpi */
1271   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1272   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1273   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
1274   ierr = MatGetType(A,&mtype);CHKERRQ(ierr);
1275   ierr = MatSetType(Cmpi,mtype);CHKERRQ(ierr);
1276   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1277 
1278 
1279   ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(Cmpi, apj, api);CHKERRQ(ierr);
1280   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1281   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1282   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1283 
1284 
1285   ptap->destroy        = Cmpi->ops->destroy;
1286   ptap->duplicate      = Cmpi->ops->duplicate;
1287   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1288   Cmpi->ops->destroy   = MatDestroy_MPIAIJ_MatMatMult;
1289 
1290   /* attach the supporting struct to Cmpi for reuse */
1291   c       = (Mat_MPIAIJ*)Cmpi->data;
1292   c->ap = ptap;
1293   *C = Cmpi;
1294 
1295   /* set MatInfo */
1296   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
1297   if (afill < 1.0) afill = 1.0;
1298   Cmpi->info.mallocs           = nspacedouble;
1299   Cmpi->info.fill_ratio_given  = fill;
1300   Cmpi->info.fill_ratio_needed = afill;
1301 
1302 #if defined(PETSC_USE_INFO)
1303   if (api[am]) {
1304     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1305     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
1306   } else {
1307     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1308   }
1309 #endif
1310 
1311   ierr = MatDestroy(&aopoth);CHKERRQ(ierr);
1312   ierr = MatDestroy(&adpd);CHKERRQ(ierr);
1313   ierr = PetscFree(j_temp);CHKERRQ(ierr);
1314   ierr = PetscFree(adpoj);CHKERRQ(ierr);
1315   ierr = PetscFree(adpoi);CHKERRQ(ierr);
1316   PetscFunctionReturn(0);
1317 }
1318 
1319 
1320 /*-------------------------------------------------------------------------*/
1321 PetscErrorCode MatTransposeMatMult_MPIAIJ_MPIAIJ(Mat P,Mat A,MatReuse scall,PetscReal fill,Mat *C)
1322 {
1323   PetscErrorCode ierr;
1324   const char     *algTypes[3] = {"scalable","nonscalable","matmatmult"};
1325   PetscInt       aN=A->cmap->N,alg=1; /* set default algorithm */
1326   PetscBool      flg;
1327 
1328   PetscFunctionBegin;
1329   if (scall == MAT_INITIAL_MATRIX) {
1330     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)A),((PetscObject)A)->prefix,"MatTransposeMatMult","Mat");CHKERRQ(ierr);
1331     ierr = PetscOptionsEList("-mattransposematmult_via","Algorithmic approach","MatTransposeMatMult",algTypes,3,algTypes[1],&alg,&flg);CHKERRQ(ierr);
1332     ierr = PetscOptionsEnd();CHKERRQ(ierr);
1333 
1334     ierr = PetscLogEventBegin(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
1335     switch (alg) {
1336     case 1:
1337       if (!flg && aN > 100000) { /* may switch to scalable algorithm as default */
1338         MatInfo     Ainfo,Pinfo;
1339         PetscInt    nz_local;
1340         PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
1341         MPI_Comm    comm;
1342 
1343         ierr = MatGetInfo(A,MAT_LOCAL,&Ainfo);CHKERRQ(ierr);
1344         ierr = MatGetInfo(P,MAT_LOCAL,&Pinfo);CHKERRQ(ierr);
1345         nz_local = (PetscInt)(Ainfo.nz_allocated + Pinfo.nz_allocated); /* estimated local nonzero entries */
1346 
1347         if (aN > fill*nz_local) alg_scalable_loc = PETSC_TRUE;
1348         ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1349         ierr = MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm);CHKERRQ(ierr);
1350 
1351         if (alg_scalable) {
1352           alg  = 0; /* scalable algorithm would slower than nonscalable algorithm */
1353           ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(P,A,fill,C);CHKERRQ(ierr);
1354           break;
1355         }
1356       }
1357       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(P,A,fill,C);CHKERRQ(ierr);
1358       break;
1359     case 2:
1360     {
1361       Mat         Pt;
1362       Mat_APMPI   *ptap;
1363       Mat_MPIAIJ  *c;
1364       ierr = MatTranspose(P,MAT_INITIAL_MATRIX,&Pt);CHKERRQ(ierr);
1365       ierr = MatMatMult(Pt,A,MAT_INITIAL_MATRIX,fill,C);CHKERRQ(ierr);
1366       c        = (Mat_MPIAIJ*)(*C)->data;
1367       ptap     = c->ap;
1368       if (ptap) {
1369        ptap->Pt = Pt;
1370        (*C)->ops->freeintermediatedatastructures = MatFreeIntermediateDataStructures_MPIAIJ_AP;
1371       }
1372       (*C)->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult;
1373       PetscFunctionReturn(0);
1374     }
1375       break;
1376     default: /* scalable algorithm */
1377       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(P,A,fill,C);CHKERRQ(ierr);
1378       break;
1379     }
1380     ierr = PetscLogEventEnd(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
1381 
1382     {
1383       Mat_MPIAIJ *c  = (Mat_MPIAIJ*)(*C)->data;
1384       Mat_APMPI  *ap = c->ap;
1385       ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)(*C)),((PetscObject)(*C))->prefix,"MatFreeIntermediateDataStructures","Mat");CHKERRQ(ierr);
1386       ap->freestruct = PETSC_FALSE;
1387       ierr = PetscOptionsBool("-mat_freeintermediatedatastructures","Free intermediate data structures", "MatFreeIntermediateDataStructures",ap->freestruct,&ap->freestruct, NULL);CHKERRQ(ierr);
1388       ierr = PetscOptionsEnd();CHKERRQ(ierr);
1389     }
1390   }
1391 
1392   ierr = PetscLogEventBegin(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
1393   ierr = (*(*C)->ops->mattransposemultnumeric)(P,A,*C);CHKERRQ(ierr);
1394   ierr = PetscLogEventEnd(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
1395   PetscFunctionReturn(0);
1396 }
1397 
1398 /* This routine only works when scall=MAT_REUSE_MATRIX! */
1399 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult(Mat P,Mat A,Mat C)
1400 {
1401   PetscErrorCode ierr;
1402   Mat_MPIAIJ     *c=(Mat_MPIAIJ*)C->data;
1403   Mat_APMPI      *ptap= c->ap;
1404   Mat            Pt;
1405 
1406   PetscFunctionBegin;
1407   if (!ptap->Pt) {
1408     MPI_Comm comm;
1409     ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1410     SETERRQ(comm,PETSC_ERR_ARG_WRONGSTATE,"PtA cannot be reused. Do not call MatFreeIntermediateDataStructures() or use '-mat_freeintermediatedatastructures'");
1411   }
1412 
1413   Pt=ptap->Pt;
1414   ierr = MatTranspose(P,MAT_REUSE_MATRIX,&Pt);CHKERRQ(ierr);
1415   ierr = MatMatMultNumeric(Pt,A,C);CHKERRQ(ierr);
1416 
1417   /* supporting struct ptap consumes almost same amount of memory as C=PtAP, release it if C will not be updated by A and P */
1418   if (ptap->freestruct) {
1419     ierr = MatFreeIntermediateDataStructures(C);CHKERRQ(ierr);
1420   }
1421   PetscFunctionReturn(0);
1422 }
1423 
1424 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1425 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,PetscReal fill,Mat *C)
1426 {
1427   PetscErrorCode      ierr;
1428   Mat_APMPI           *ptap;
1429   Mat_MPIAIJ          *p=(Mat_MPIAIJ*)P->data,*c;
1430   MPI_Comm            comm;
1431   PetscMPIInt         size,rank;
1432   Mat                 Cmpi;
1433   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1434   PetscInt            pn=P->cmap->n,aN=A->cmap->N,an=A->cmap->n;
1435   PetscInt            *lnk,i,k,nsend;
1436   PetscBT             lnkbt;
1437   PetscMPIInt         tagi,tagj,*len_si,*len_s,*len_ri,icompleted=0,nrecv;
1438   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1439   PetscInt            len,proc,*dnz,*onz,*owners,nzi;
1440   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1441   MPI_Request         *swaits,*rwaits;
1442   MPI_Status          *sstatus,rstatus;
1443   PetscLayout         rowmap;
1444   PetscInt            *owners_co,*coi,*coj;    /* i and j array of (p->B)^T*A*P - used in the communication */
1445   PetscMPIInt         *len_r,*id_r;    /* array of length of comm->size, store send/recv matrix values */
1446   PetscInt            *Jptr,*prmap=p->garray,con,j,Crmax;
1447   Mat_SeqAIJ          *a_loc,*c_loc,*c_oth;
1448   PetscTable          ta;
1449   MatType             mtype;
1450   const char          *prefix;
1451 
1452   PetscFunctionBegin;
1453   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1454   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1455   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1456 
1457   /* create symbolic parallel matrix Cmpi */
1458   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1459   ierr = MatGetType(A,&mtype);CHKERRQ(ierr);
1460   ierr = MatSetType(Cmpi,mtype);CHKERRQ(ierr);
1461 
1462   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1463 
1464   /* create struct Mat_APMPI and attached it to C later */
1465   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1466   ptap->reuse = MAT_INITIAL_MATRIX;
1467 
1468   /* (0) compute Rd = Pd^T, Ro = Po^T  */
1469   /* --------------------------------- */
1470   ierr = MatTranspose_SeqAIJ(p->A,MAT_INITIAL_MATRIX,&ptap->Rd);CHKERRQ(ierr);
1471   ierr = MatTranspose_SeqAIJ(p->B,MAT_INITIAL_MATRIX,&ptap->Ro);CHKERRQ(ierr);
1472 
1473   /* (1) compute symbolic A_loc */
1474   /* ---------------------------*/
1475   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&ptap->A_loc);CHKERRQ(ierr);
1476 
1477   /* (2-1) compute symbolic C_oth = Ro*A_loc  */
1478   /* ------------------------------------ */
1479   ierr = MatGetOptionsPrefix(A,&prefix);CHKERRQ(ierr);
1480   ierr = MatSetOptionsPrefix(ptap->Ro,prefix);CHKERRQ(ierr);
1481   ierr = MatAppendOptionsPrefix(ptap->Ro,"inner_offdiag_");CHKERRQ(ierr);
1482   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Ro,ptap->A_loc,fill,&ptap->C_oth);CHKERRQ(ierr);
1483 
1484   /* (3) send coj of C_oth to other processors  */
1485   /* ------------------------------------------ */
1486   /* determine row ownership */
1487   ierr = PetscLayoutCreate(comm,&rowmap);CHKERRQ(ierr);
1488   rowmap->n  = pn;
1489   rowmap->bs = 1;
1490   ierr   = PetscLayoutSetUp(rowmap);CHKERRQ(ierr);
1491   owners = rowmap->range;
1492 
1493   /* determine the number of messages to send, their lengths */
1494   ierr = PetscMalloc4(size,&len_s,size,&len_si,size,&sstatus,size+2,&owners_co);CHKERRQ(ierr);
1495   ierr = PetscArrayzero(len_s,size);CHKERRQ(ierr);
1496   ierr = PetscArrayzero(len_si,size);CHKERRQ(ierr);
1497 
1498   c_oth = (Mat_SeqAIJ*)ptap->C_oth->data;
1499   coi   = c_oth->i; coj = c_oth->j;
1500   con   = ptap->C_oth->rmap->n;
1501   proc  = 0;
1502   for (i=0; i<con; i++) {
1503     while (prmap[i] >= owners[proc+1]) proc++;
1504     len_si[proc]++;               /* num of rows in Co(=Pt*A) to be sent to [proc] */
1505     len_s[proc] += coi[i+1] - coi[i]; /* num of nonzeros in Co to be sent to [proc] */
1506   }
1507 
1508   len          = 0; /* max length of buf_si[], see (4) */
1509   owners_co[0] = 0;
1510   nsend        = 0;
1511   for (proc=0; proc<size; proc++) {
1512     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1513     if (len_s[proc]) {
1514       nsend++;
1515       len_si[proc] = 2*(len_si[proc] + 1); /* length of buf_si to be sent to [proc] */
1516       len         += len_si[proc];
1517     }
1518   }
1519 
1520   /* determine the number and length of messages to receive for coi and coj  */
1521   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&nrecv);CHKERRQ(ierr);
1522   ierr = PetscGatherMessageLengths2(comm,nsend,nrecv,len_s,len_si,&id_r,&len_r,&len_ri);CHKERRQ(ierr);
1523 
1524   /* post the Irecv and Isend of coj */
1525   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1526   ierr = PetscPostIrecvInt(comm,tagj,nrecv,id_r,len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1527   ierr = PetscMalloc1(nsend+1,&swaits);CHKERRQ(ierr);
1528   for (proc=0, k=0; proc<size; proc++) {
1529     if (!len_s[proc]) continue;
1530     i    = owners_co[proc];
1531     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1532     k++;
1533   }
1534 
1535   /* (2-2) compute symbolic C_loc = Rd*A_loc */
1536   /* ---------------------------------------- */
1537   ierr = MatSetOptionsPrefix(ptap->Rd,prefix);CHKERRQ(ierr);
1538   ierr = MatAppendOptionsPrefix(ptap->Rd,"inner_diag_");CHKERRQ(ierr);
1539   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Rd,ptap->A_loc,fill,&ptap->C_loc);CHKERRQ(ierr);
1540   c_loc = (Mat_SeqAIJ*)ptap->C_loc->data;
1541 
1542   /* receives coj are complete */
1543   for (i=0; i<nrecv; i++) {
1544     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1545   }
1546   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1547   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
1548 
1549   /* add received column indices into ta to update Crmax */
1550   a_loc = (Mat_SeqAIJ*)(ptap->A_loc)->data;
1551 
1552   /* create and initialize a linked list */
1553   ierr = PetscTableCreate(an,aN,&ta);CHKERRQ(ierr); /* for compute Crmax */
1554   MatRowMergeMax_SeqAIJ(a_loc,ptap->A_loc->rmap->N,ta);
1555 
1556   for (k=0; k<nrecv; k++) {/* k-th received message */
1557     Jptr = buf_rj[k];
1558     for (j=0; j<len_r[k]; j++) {
1559       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
1560     }
1561   }
1562   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
1563   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
1564 
1565   /* (4) send and recv coi */
1566   /*-----------------------*/
1567   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1568   ierr   = PetscPostIrecvInt(comm,tagi,nrecv,id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1569   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1570   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1571   for (proc=0,k=0; proc<size; proc++) {
1572     if (!len_s[proc]) continue;
1573     /* form outgoing message for i-structure:
1574          buf_si[0]:                 nrows to be sent
1575                [1:nrows]:           row index (global)
1576                [nrows+1:2*nrows+1]: i-structure index
1577     */
1578     /*-------------------------------------------*/
1579     nrows       = len_si[proc]/2 - 1; /* num of rows in Co to be sent to [proc] */
1580     buf_si_i    = buf_si + nrows+1;
1581     buf_si[0]   = nrows;
1582     buf_si_i[0] = 0;
1583     nrows       = 0;
1584     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1585       nzi = coi[i+1] - coi[i];
1586       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1587       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1588       nrows++;
1589     }
1590     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1591     k++;
1592     buf_si += len_si[proc];
1593   }
1594   for (i=0; i<nrecv; i++) {
1595     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1596   }
1597   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1598   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
1599 
1600   ierr = PetscFree4(len_s,len_si,sstatus,owners_co);CHKERRQ(ierr);
1601   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1602   ierr = PetscFree(swaits);CHKERRQ(ierr);
1603   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1604 
1605   /* (5) compute the local portion of Cmpi      */
1606   /* ------------------------------------------ */
1607   /* set initial free space to be Crmax, sufficient for holding nozeros in each row of Cmpi */
1608   ierr          = PetscFreeSpaceGet(Crmax,&free_space);CHKERRQ(ierr);
1609   current_space = free_space;
1610 
1611   ierr = PetscMalloc3(nrecv,&buf_ri_k,nrecv,&nextrow,nrecv,&nextci);CHKERRQ(ierr);
1612   for (k=0; k<nrecv; k++) {
1613     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1614     nrows       = *buf_ri_k[k];
1615     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1616     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1617   }
1618 
1619   ierr = MatPreallocateInitialize(comm,pn,an,dnz,onz);CHKERRQ(ierr);
1620   ierr = PetscLLCondensedCreate(Crmax,aN,&lnk,&lnkbt);CHKERRQ(ierr);
1621   for (i=0; i<pn; i++) {
1622     /* add C_loc into Cmpi */
1623     nzi  = c_loc->i[i+1] - c_loc->i[i];
1624     Jptr = c_loc->j + c_loc->i[i];
1625     ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1626 
1627     /* add received col data into lnk */
1628     for (k=0; k<nrecv; k++) { /* k-th received message */
1629       if (i == *nextrow[k]) { /* i-th row */
1630         nzi  = *(nextci[k]+1) - *nextci[k];
1631         Jptr = buf_rj[k] + *nextci[k];
1632         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1633         nextrow[k]++; nextci[k]++;
1634       }
1635     }
1636     nzi = lnk[0];
1637 
1638     /* copy data into free space, then initialize lnk */
1639     ierr = PetscLLCondensedClean(aN,nzi,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1640     ierr = MatPreallocateSet(i+owners[rank],nzi,current_space->array,dnz,onz);CHKERRQ(ierr);
1641   }
1642   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1643   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1644   ierr = PetscFreeSpaceDestroy(free_space);CHKERRQ(ierr);
1645 
1646   /* local sizes and preallocation */
1647   ierr = MatSetSizes(Cmpi,pn,an,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1648   if (P->cmap->bs > 0) {ierr = PetscLayoutSetBlockSize(Cmpi->rmap,P->cmap->bs);CHKERRQ(ierr);}
1649   if (A->cmap->bs > 0) {ierr = PetscLayoutSetBlockSize(Cmpi->cmap,A->cmap->bs);CHKERRQ(ierr);}
1650   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1651   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1652 
1653   /* members in merge */
1654   ierr = PetscFree(id_r);CHKERRQ(ierr);
1655   ierr = PetscFree(len_r);CHKERRQ(ierr);
1656   ierr = PetscFree(buf_ri[0]);CHKERRQ(ierr);
1657   ierr = PetscFree(buf_ri);CHKERRQ(ierr);
1658   ierr = PetscFree(buf_rj[0]);CHKERRQ(ierr);
1659   ierr = PetscFree(buf_rj);CHKERRQ(ierr);
1660   ierr = PetscLayoutDestroy(&rowmap);CHKERRQ(ierr);
1661 
1662   /* attach the supporting struct to Cmpi for reuse */
1663   c = (Mat_MPIAIJ*)Cmpi->data;
1664   c->ap         = ptap;
1665   ptap->destroy = Cmpi->ops->destroy;
1666 
1667   /* Cmpi is not ready for use - assembly will be done by MatPtAPNumeric() */
1668   Cmpi->assembled        = PETSC_FALSE;
1669   Cmpi->ops->destroy     = MatDestroy_MPIAIJ_PtAP;
1670   Cmpi->ops->freeintermediatedatastructures = MatFreeIntermediateDataStructures_MPIAIJ_AP;
1671 
1672   *C                     = Cmpi;
1673   PetscFunctionReturn(0);
1674 }
1675 
1676 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,Mat C)
1677 {
1678   PetscErrorCode    ierr;
1679   Mat_MPIAIJ        *p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1680   Mat_SeqAIJ        *c_seq;
1681   Mat_APMPI         *ptap = c->ap;
1682   Mat               A_loc,C_loc,C_oth;
1683   PetscInt          i,rstart,rend,cm,ncols,row;
1684   const PetscInt    *cols;
1685   const PetscScalar *vals;
1686 
1687   PetscFunctionBegin;
1688   if (!ptap->A_loc) {
1689     MPI_Comm comm;
1690     ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1691     SETERRQ(comm,PETSC_ERR_ARG_WRONGSTATE,"PtA cannot be reused. Do not call MatFreeIntermediateDataStructures() or use '-mat_freeintermediatedatastructures'");
1692   }
1693 
1694   ierr = MatZeroEntries(C);CHKERRQ(ierr);
1695 
1696   if (ptap->reuse == MAT_REUSE_MATRIX) {
1697     /* These matrices are obtained in MatTransposeMatMultSymbolic() */
1698     /* 1) get R = Pd^T, Ro = Po^T */
1699     /*----------------------------*/
1700     ierr = MatTranspose_SeqAIJ(p->A,MAT_REUSE_MATRIX,&ptap->Rd);CHKERRQ(ierr);
1701     ierr = MatTranspose_SeqAIJ(p->B,MAT_REUSE_MATRIX,&ptap->Ro);CHKERRQ(ierr);
1702 
1703     /* 2) compute numeric A_loc */
1704     /*--------------------------*/
1705     ierr = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&ptap->A_loc);CHKERRQ(ierr);
1706   }
1707 
1708   /* 3) C_loc = Rd*A_loc, C_oth = Ro*A_loc */
1709   A_loc = ptap->A_loc;
1710   ierr = ((ptap->C_loc)->ops->matmultnumeric)(ptap->Rd,A_loc,ptap->C_loc);CHKERRQ(ierr);
1711   ierr = ((ptap->C_oth)->ops->matmultnumeric)(ptap->Ro,A_loc,ptap->C_oth);CHKERRQ(ierr);
1712   C_loc = ptap->C_loc;
1713   C_oth = ptap->C_oth;
1714 
1715   /* add C_loc and Co to to C */
1716   ierr = MatGetOwnershipRange(C,&rstart,&rend);CHKERRQ(ierr);
1717 
1718   /* C_loc -> C */
1719   cm    = C_loc->rmap->N;
1720   c_seq = (Mat_SeqAIJ*)C_loc->data;
1721   cols = c_seq->j;
1722   vals = c_seq->a;
1723   for (i=0; i<cm; i++) {
1724     ncols = c_seq->i[i+1] - c_seq->i[i];
1725     row = rstart + i;
1726     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1727     cols += ncols; vals += ncols;
1728   }
1729 
1730   /* Co -> C, off-processor part */
1731   cm    = C_oth->rmap->N;
1732   c_seq = (Mat_SeqAIJ*)C_oth->data;
1733   cols  = c_seq->j;
1734   vals  = c_seq->a;
1735   for (i=0; i<cm; i++) {
1736     ncols = c_seq->i[i+1] - c_seq->i[i];
1737     row = p->garray[i];
1738     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1739     cols += ncols; vals += ncols;
1740   }
1741   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1742   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1743 
1744   ptap->reuse = MAT_REUSE_MATRIX;
1745 
1746   /* supporting struct ptap consumes almost same amount of memory as C=PtAP, release it if C will not be updated by A and P */
1747   if (ptap->freestruct) {
1748     ierr = MatFreeIntermediateDataStructures(C);CHKERRQ(ierr);
1749   }
1750   PetscFunctionReturn(0);
1751 }
1752 
1753 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ(Mat P,Mat A,Mat C)
1754 {
1755   PetscErrorCode      ierr;
1756   Mat_Merge_SeqsToMPI *merge;
1757   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1758   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
1759   Mat_APMPI           *ptap;
1760   PetscInt            *adj;
1761   PetscInt            i,j,k,anz,pnz,row,*cj,nexta;
1762   MatScalar           *ada,*ca,valtmp;
1763   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
1764   MPI_Comm            comm;
1765   PetscMPIInt         size,rank,taga,*len_s;
1766   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
1767   PetscInt            **buf_ri,**buf_rj;
1768   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
1769   MPI_Request         *s_waits,*r_waits;
1770   MPI_Status          *status;
1771   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
1772   PetscInt            *ai,*aj,*coi,*coj,*poJ,*pdJ;
1773   Mat                 A_loc;
1774   Mat_SeqAIJ          *a_loc;
1775 
1776   PetscFunctionBegin;
1777   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1778   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1779   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1780 
1781   ptap  = c->ap;
1782   if (!ptap->A_loc) SETERRQ(comm,PETSC_ERR_ARG_WRONGSTATE,"PtA cannot be reused. Do not call MatFreeIntermediateDataStructures() or use '-mat_freeintermediatedatastructures'");
1783   merge = ptap->merge;
1784 
1785   /* 2) compute numeric C_seq = P_loc^T*A_loc */
1786   /*------------------------------------------*/
1787   /* get data from symbolic products */
1788   coi    = merge->coi; coj = merge->coj;
1789   ierr   = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
1790   bi     = merge->bi; bj = merge->bj;
1791   owners = merge->rowmap->range;
1792   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
1793 
1794   /* get A_loc by taking all local rows of A */
1795   A_loc = ptap->A_loc;
1796   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
1797   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1798   ai    = a_loc->i;
1799   aj    = a_loc->j;
1800 
1801   for (i=0; i<am; i++) {
1802     anz = ai[i+1] - ai[i];
1803     adj = aj + ai[i];
1804     ada = a_loc->a + ai[i];
1805 
1806     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1807     /*-------------------------------------------------------------*/
1808     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1809     pnz = po->i[i+1] - po->i[i];
1810     poJ = po->j + po->i[i];
1811     pA  = po->a + po->i[i];
1812     for (j=0; j<pnz; j++) {
1813       row = poJ[j];
1814       cj  = coj + coi[row];
1815       ca  = coa + coi[row];
1816       /* perform sparse axpy */
1817       nexta  = 0;
1818       valtmp = pA[j];
1819       for (k=0; nexta<anz; k++) {
1820         if (cj[k] == adj[nexta]) {
1821           ca[k] += valtmp*ada[nexta];
1822           nexta++;
1823         }
1824       }
1825       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1826     }
1827 
1828     /* put the value into Cd (diagonal part) */
1829     pnz = pd->i[i+1] - pd->i[i];
1830     pdJ = pd->j + pd->i[i];
1831     pA  = pd->a + pd->i[i];
1832     for (j=0; j<pnz; j++) {
1833       row = pdJ[j];
1834       cj  = bj + bi[row];
1835       ca  = ba + bi[row];
1836       /* perform sparse axpy */
1837       nexta  = 0;
1838       valtmp = pA[j];
1839       for (k=0; nexta<anz; k++) {
1840         if (cj[k] == adj[nexta]) {
1841           ca[k] += valtmp*ada[nexta];
1842           nexta++;
1843         }
1844       }
1845       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1846     }
1847   }
1848 
1849   /* 3) send and recv matrix values coa */
1850   /*------------------------------------*/
1851   buf_ri = merge->buf_ri;
1852   buf_rj = merge->buf_rj;
1853   len_s  = merge->len_s;
1854   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1855   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1856 
1857   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1858   for (proc=0,k=0; proc<size; proc++) {
1859     if (!len_s[proc]) continue;
1860     i    = merge->owners_co[proc];
1861     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1862     k++;
1863   }
1864   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1865   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1866 
1867   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1868   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1869   ierr = PetscFree(coa);CHKERRQ(ierr);
1870 
1871   /* 4) insert local Cseq and received values into Cmpi */
1872   /*----------------------------------------------------*/
1873   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1874   for (k=0; k<merge->nrecv; k++) {
1875     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1876     nrows       = *(buf_ri_k[k]);
1877     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1878     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1879   }
1880 
1881   for (i=0; i<cm; i++) {
1882     row  = owners[rank] + i; /* global row index of C_seq */
1883     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1884     ba_i = ba + bi[i];
1885     bnz  = bi[i+1] - bi[i];
1886     /* add received vals into ba */
1887     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1888       /* i-th row */
1889       if (i == *nextrow[k]) {
1890         cnz    = *(nextci[k]+1) - *nextci[k];
1891         cj     = buf_rj[k] + *(nextci[k]);
1892         ca     = abuf_r[k] + *(nextci[k]);
1893         nextcj = 0;
1894         for (j=0; nextcj<cnz; j++) {
1895           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1896             ba_i[j] += ca[nextcj++];
1897           }
1898         }
1899         nextrow[k]++; nextci[k]++;
1900         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1901       }
1902     }
1903     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1904   }
1905   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1906   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1907 
1908   ierr = PetscFree(ba);CHKERRQ(ierr);
1909   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1910   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1911   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1912 
1913   if (ptap->freestruct) {
1914     ierr = MatFreeIntermediateDataStructures(C);CHKERRQ(ierr);
1915   }
1916   PetscFunctionReturn(0);
1917 }
1918 
1919 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P,Mat A,PetscReal fill,Mat *C)
1920 {
1921   PetscErrorCode      ierr;
1922   Mat                 Cmpi,A_loc,POt,PDt;
1923   Mat_APMPI           *ptap;
1924   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1925   Mat_MPIAIJ          *p=(Mat_MPIAIJ*)P->data,*a=(Mat_MPIAIJ*)A->data,*c;
1926   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1927   PetscInt            nnz;
1928   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1929   PetscInt            am  =A->rmap->n,pn=P->cmap->n;
1930   MPI_Comm            comm;
1931   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1932   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1933   PetscInt            len,proc,*dnz,*onz,*owners;
1934   PetscInt            nzi,*bi,*bj;
1935   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1936   MPI_Request         *swaits,*rwaits;
1937   MPI_Status          *sstatus,rstatus;
1938   Mat_Merge_SeqsToMPI *merge;
1939   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1940   PetscReal           afill  =1.0,afill_tmp;
1941   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Armax;
1942   Mat_SeqAIJ          *a_loc,*pdt,*pot;
1943   PetscTable          ta;
1944   MatType             mtype;
1945 
1946   PetscFunctionBegin;
1947   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1948   /* check if matrix local sizes are compatible */
1949   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);
1950 
1951   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1952   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1953 
1954   /* create struct Mat_APMPI and attached it to C later */
1955   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1956 
1957   /* get A_loc by taking all local rows of A */
1958   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1959 
1960   ptap->A_loc = A_loc;
1961   a_loc       = (Mat_SeqAIJ*)(A_loc)->data;
1962   ai          = a_loc->i;
1963   aj          = a_loc->j;
1964 
1965   /* determine symbolic Co=(p->B)^T*A - send to others */
1966   /*----------------------------------------------------*/
1967   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1968   pdt  = (Mat_SeqAIJ*)PDt->data;
1969   pdti = pdt->i; pdtj = pdt->j;
1970 
1971   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1972   pot  = (Mat_SeqAIJ*)POt->data;
1973   poti = pot->i; potj = pot->j;
1974 
1975   /* then, compute symbolic Co = (p->B)^T*A */
1976   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1977                          >= (num of nonzero rows of C_seq) - pn */
1978   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
1979   coi[0] = 0;
1980 
1981   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1982   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(poti[pon],ai[am]));
1983   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1984   current_space = free_space;
1985 
1986   /* create and initialize a linked list */
1987   ierr = PetscTableCreate(A->cmap->n + a->B->cmap->N,aN,&ta);CHKERRQ(ierr);
1988   MatRowMergeMax_SeqAIJ(a_loc,am,ta);
1989   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
1990 
1991   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
1992 
1993   for (i=0; i<pon; i++) {
1994     pnz = poti[i+1] - poti[i];
1995     ptJ = potj + poti[i];
1996     for (j=0; j<pnz; j++) {
1997       row  = ptJ[j]; /* row of A_loc == col of Pot */
1998       anz  = ai[row+1] - ai[row];
1999       Jptr = aj + ai[row];
2000       /* add non-zero cols of AP into the sorted linked list lnk */
2001       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
2002     }
2003     nnz = lnk[0];
2004 
2005     /* If free space is not available, double the total space in the list */
2006     if (current_space->local_remaining<nnz) {
2007       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
2008       nspacedouble++;
2009     }
2010 
2011     /* Copy data into free space, and zero out denserows */
2012     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
2013 
2014     current_space->array           += nnz;
2015     current_space->local_used      += nnz;
2016     current_space->local_remaining -= nnz;
2017 
2018     coi[i+1] = coi[i] + nnz;
2019   }
2020 
2021   ierr = PetscMalloc1(coi[pon]+1,&coj);CHKERRQ(ierr);
2022   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
2023   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr); /* must destroy to get a new one for C */
2024 
2025   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
2026   if (afill_tmp > afill) afill = afill_tmp;
2027 
2028   /* send j-array (coj) of Co to other processors */
2029   /*----------------------------------------------*/
2030   /* determine row ownership */
2031   ierr = PetscNew(&merge);CHKERRQ(ierr);
2032   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
2033 
2034   merge->rowmap->n  = pn;
2035   merge->rowmap->bs = 1;
2036 
2037   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
2038   owners = merge->rowmap->range;
2039 
2040   /* determine the number of messages to send, their lengths */
2041   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
2042   ierr = PetscCalloc1(size,&merge->len_s);CHKERRQ(ierr);
2043 
2044   len_s        = merge->len_s;
2045   merge->nsend = 0;
2046 
2047   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
2048 
2049   proc = 0;
2050   for (i=0; i<pon; i++) {
2051     while (prmap[i] >= owners[proc+1]) proc++;
2052     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
2053     len_s[proc] += coi[i+1] - coi[i];
2054   }
2055 
2056   len          = 0; /* max length of buf_si[] */
2057   owners_co[0] = 0;
2058   for (proc=0; proc<size; proc++) {
2059     owners_co[proc+1] = owners_co[proc] + len_si[proc];
2060     if (len_si[proc]) {
2061       merge->nsend++;
2062       len_si[proc] = 2*(len_si[proc] + 1);
2063       len         += len_si[proc];
2064     }
2065   }
2066 
2067   /* determine the number and length of messages to receive for coi and coj  */
2068   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
2069   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
2070 
2071   /* post the Irecv and Isend of coj */
2072   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
2073   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
2074   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
2075   for (proc=0, k=0; proc<size; proc++) {
2076     if (!len_s[proc]) continue;
2077     i    = owners_co[proc];
2078     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
2079     k++;
2080   }
2081 
2082   /* receives and sends of coj are complete */
2083   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
2084   for (i=0; i<merge->nrecv; i++) {
2085     PetscMPIInt icompleted;
2086     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
2087   }
2088   ierr = PetscFree(rwaits);CHKERRQ(ierr);
2089   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
2090 
2091   /* add received column indices into table to update Armax */
2092   /* Armax can be as large as aN if a P[row,:] is dense, see src/ksp/ksp/examples/tutorials/ex56.c! */
2093   for (k=0; k<merge->nrecv; k++) {/* k-th received message */
2094     Jptr = buf_rj[k];
2095     for (j=0; j<merge->len_r[k]; j++) {
2096       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
2097     }
2098   }
2099   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
2100   /* 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); */
2101 
2102   /* send and recv coi */
2103   /*-------------------*/
2104   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
2105   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
2106   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
2107   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
2108   for (proc=0,k=0; proc<size; proc++) {
2109     if (!len_s[proc]) continue;
2110     /* form outgoing message for i-structure:
2111          buf_si[0]:                 nrows to be sent
2112                [1:nrows]:           row index (global)
2113                [nrows+1:2*nrows+1]: i-structure index
2114     */
2115     /*-------------------------------------------*/
2116     nrows       = len_si[proc]/2 - 1;
2117     buf_si_i    = buf_si + nrows+1;
2118     buf_si[0]   = nrows;
2119     buf_si_i[0] = 0;
2120     nrows       = 0;
2121     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
2122       nzi               = coi[i+1] - coi[i];
2123       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
2124       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
2125       nrows++;
2126     }
2127     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
2128     k++;
2129     buf_si += len_si[proc];
2130   }
2131   i = merge->nrecv;
2132   while (i--) {
2133     PetscMPIInt icompleted;
2134     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
2135   }
2136   ierr = PetscFree(rwaits);CHKERRQ(ierr);
2137   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
2138   ierr = PetscFree(len_si);CHKERRQ(ierr);
2139   ierr = PetscFree(len_ri);CHKERRQ(ierr);
2140   ierr = PetscFree(swaits);CHKERRQ(ierr);
2141   ierr = PetscFree(sstatus);CHKERRQ(ierr);
2142   ierr = PetscFree(buf_s);CHKERRQ(ierr);
2143 
2144   /* compute the local portion of C (mpi mat) */
2145   /*------------------------------------------*/
2146   /* allocate bi array and free space for accumulating nonzero column info */
2147   ierr  = PetscMalloc1(pn+1,&bi);CHKERRQ(ierr);
2148   bi[0] = 0;
2149 
2150   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
2151   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(pdti[pn],PetscIntSumTruncate(poti[pon],ai[am])));
2152   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
2153   current_space = free_space;
2154 
2155   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
2156   for (k=0; k<merge->nrecv; k++) {
2157     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
2158     nrows       = *buf_ri_k[k];
2159     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
2160     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th recieved i-structure  */
2161   }
2162 
2163   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
2164   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
2165   rmax = 0;
2166   for (i=0; i<pn; i++) {
2167     /* add pdt[i,:]*AP into lnk */
2168     pnz = pdti[i+1] - pdti[i];
2169     ptJ = pdtj + pdti[i];
2170     for (j=0; j<pnz; j++) {
2171       row  = ptJ[j];  /* row of AP == col of Pt */
2172       anz  = ai[row+1] - ai[row];
2173       Jptr = aj + ai[row];
2174       /* add non-zero cols of AP into the sorted linked list lnk */
2175       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
2176     }
2177 
2178     /* add received col data into lnk */
2179     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
2180       if (i == *nextrow[k]) { /* i-th row */
2181         nzi  = *(nextci[k]+1) - *nextci[k];
2182         Jptr = buf_rj[k] + *nextci[k];
2183         ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
2184         nextrow[k]++; nextci[k]++;
2185       }
2186     }
2187     nnz = lnk[0];
2188 
2189     /* if free space is not available, make more free space */
2190     if (current_space->local_remaining<nnz) {
2191       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
2192       nspacedouble++;
2193     }
2194     /* copy data into free space, then initialize lnk */
2195     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
2196     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
2197 
2198     current_space->array           += nnz;
2199     current_space->local_used      += nnz;
2200     current_space->local_remaining -= nnz;
2201 
2202     bi[i+1] = bi[i] + nnz;
2203     if (nnz > rmax) rmax = nnz;
2204   }
2205   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
2206 
2207   ierr      = PetscMalloc1(bi[pn]+1,&bj);CHKERRQ(ierr);
2208   ierr      = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
2209   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
2210   if (afill_tmp > afill) afill = afill_tmp;
2211   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
2212   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
2213 
2214   ierr = MatDestroy(&POt);CHKERRQ(ierr);
2215   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
2216 
2217   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
2218   /*----------------------------------------------------------------------------------*/
2219   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
2220   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
2221   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
2222   ierr = MatGetType(A,&mtype);CHKERRQ(ierr);
2223   ierr = MatSetType(Cmpi,mtype);CHKERRQ(ierr);
2224   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
2225   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
2226   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
2227   for (i=0; i<pn; i++) {
2228     row  = i + rstart;
2229     nnz  = bi[i+1] - bi[i];
2230     Jptr = bj + bi[i];
2231     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,NULL,INSERT_VALUES);CHKERRQ(ierr);
2232   }
2233   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2234   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2235   merge->bi        = bi;
2236   merge->bj        = bj;
2237   merge->coi       = coi;
2238   merge->coj       = coj;
2239   merge->buf_ri    = buf_ri;
2240   merge->buf_rj    = buf_rj;
2241   merge->owners_co = owners_co;
2242 
2243   /* attach the supporting struct to Cmpi for reuse */
2244   c = (Mat_MPIAIJ*)Cmpi->data;
2245 
2246   c->ap       = ptap;
2247   ptap->api   = NULL;
2248   ptap->apj   = NULL;
2249   ptap->merge = merge;
2250   ptap->apa   = NULL;
2251   ptap->destroy   = Cmpi->ops->destroy;
2252   ptap->duplicate = Cmpi->ops->duplicate;
2253 
2254   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ;
2255   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
2256   Cmpi->ops->freeintermediatedatastructures = MatFreeIntermediateDataStructures_MPIAIJ_AP;
2257 
2258   *C = Cmpi;
2259 #if defined(PETSC_USE_INFO)
2260   if (bi[pn] != 0) {
2261     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
2262     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
2263   } else {
2264     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
2265   }
2266 #endif
2267   PetscFunctionReturn(0);
2268 }
2269