xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision 7f3e1f30fe01cfc2e350e00f4071ec3caee970a1)
1 
2 /*
3   Defines matrix-matrix product routines for pairs of MPIAIJ matrices
4           C = A * B
5 */
6 #include <../src/mat/impls/aij/seq/aij.h> /*I "petscmat.h" I*/
7 #include <../src/mat/utils/freespace.h>
8 #include <../src/mat/impls/aij/mpi/mpiaij.h>
9 #include <petscbt.h>
10 #include <../src/mat/impls/dense/mpi/mpidense.h>
11 #include <petsc/private/vecimpl.h>
12 
13 #if defined(PETSC_HAVE_HYPRE)
14 PETSC_INTERN PetscErrorCode MatMatMultSymbolic_AIJ_AIJ_wHYPRE(Mat,Mat,PetscReal,Mat*);
15 #endif
16 
17 PETSC_INTERN PetscErrorCode MatMatMult_MPIAIJ_MPIAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill, Mat *C)
18 {
19   PetscErrorCode ierr;
20 #if defined(PETSC_HAVE_HYPRE)
21   const char     *algTypes[3] = {"scalable","nonscalable","hypre"};
22   PetscInt       nalg = 3;
23 #else
24   const char     *algTypes[2] = {"scalable","nonscalable"};
25   PetscInt       nalg = 2;
26 #endif
27   PetscInt       alg = 1; /* set nonscalable algorithm as default */
28   MPI_Comm       comm;
29   PetscBool      flg;
30 
31   PetscFunctionBegin;
32   if (scall == MAT_INITIAL_MATRIX) {
33     ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
34     if (A->cmap->rstart != B->rmap->rstart || A->cmap->rend != B->rmap->rend) SETERRQ4(comm,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, (%D, %D) != (%D,%D)",A->cmap->rstart,A->cmap->rend,B->rmap->rstart,B->rmap->rend);
35 
36     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
37     PetscOptionsObject->alreadyprinted = PETSC_FALSE; /* a hack to ensure the option shows in '-help' */
38     ierr = PetscOptionsEList("-matmatmult_via","Algorithmic approach","MatMatMult",algTypes,nalg,algTypes[1],&alg,&flg);CHKERRQ(ierr);
39     ierr = PetscOptionsEnd();CHKERRQ(ierr);
40 
41     if (!flg && B->cmap->N > 100000) { /* may switch to scalable algorithm as default */
42       MatInfo     Ainfo,Binfo;
43       PetscInt    nz_local;
44       PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
45 
46       ierr = MatGetInfo(A,MAT_LOCAL,&Ainfo);CHKERRQ(ierr);
47       ierr = MatGetInfo(B,MAT_LOCAL,&Binfo);CHKERRQ(ierr);
48       nz_local = (PetscInt)(Ainfo.nz_allocated + Binfo.nz_allocated);
49 
50       if (B->cmap->N > fill*nz_local) alg_scalable_loc = PETSC_TRUE;
51       ierr = MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm);CHKERRQ(ierr);
52 
53       if (alg_scalable) {
54         alg  = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
55         ierr = PetscInfo2(B,"Use scalable algorithm, BN %D, fill*nz_allocated %g\n",B->cmap->N,fill*nz_local);CHKERRQ(ierr);
56       }
57     }
58 
59     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
60     switch (alg) {
61     case 1:
62       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(A,B,fill,C);CHKERRQ(ierr);
63       break;
64 #if defined(PETSC_HAVE_HYPRE)
65     case 2:
66       ierr = MatMatMultSymbolic_AIJ_AIJ_wHYPRE(A,B,fill,C);CHKERRQ(ierr);
67       break;
68 #endif
69     default:
70       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ(A,B,fill,C);CHKERRQ(ierr);
71       break;
72     }
73     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
74   }
75   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
76   ierr = (*(*C)->ops->matmultnumeric)(A,B,*C);CHKERRQ(ierr);
77   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
78   PetscFunctionReturn(0);
79 }
80 
81 PetscErrorCode MatDestroy_MPIAIJ_MatMatMult(Mat A)
82 {
83   PetscErrorCode ierr;
84   Mat_MPIAIJ     *a    = (Mat_MPIAIJ*)A->data;
85   Mat_PtAPMPI    *ptap = a->ptap;
86 
87   PetscFunctionBegin;
88   ierr = PetscFree2(ptap->startsj_s,ptap->startsj_r);CHKERRQ(ierr);
89   ierr = PetscFree(ptap->bufa);CHKERRQ(ierr);
90   ierr = MatDestroy(&ptap->P_loc);CHKERRQ(ierr);
91   ierr = MatDestroy(&ptap->P_oth);CHKERRQ(ierr);
92   ierr = MatDestroy(&ptap->Pt);CHKERRQ(ierr);
93   ierr = PetscFree(ptap->api);CHKERRQ(ierr);
94   ierr = PetscFree(ptap->apj);CHKERRQ(ierr);
95   ierr = PetscFree(ptap->apa);CHKERRQ(ierr);
96   ierr = ptap->destroy(A);CHKERRQ(ierr);
97   ierr = PetscFree(ptap);CHKERRQ(ierr);
98   PetscFunctionReturn(0);
99 }
100 
101 PetscErrorCode MatDuplicate_MPIAIJ_MatMatMult(Mat A, MatDuplicateOption op, Mat *M)
102 {
103   PetscErrorCode ierr;
104   Mat_MPIAIJ     *a    = (Mat_MPIAIJ*)A->data;
105   Mat_PtAPMPI    *ptap = a->ptap;
106 
107   PetscFunctionBegin;
108   ierr = (*ptap->duplicate)(A,op,M);CHKERRQ(ierr);
109 
110   (*M)->ops->destroy   = ptap->destroy;   /* = MatDestroy_MPIAIJ, *M doesn't duplicate A's special structure! */
111   (*M)->ops->duplicate = ptap->duplicate; /* = MatDuplicate_MPIAIJ */
112   PetscFunctionReturn(0);
113 }
114 
115 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,Mat C)
116 {
117   PetscErrorCode ierr;
118   Mat_MPIAIJ     *a  =(Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
119   Mat_SeqAIJ     *ad =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
120   Mat_SeqAIJ     *cd =(Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
121   PetscScalar    *cda=cd->a,*coa=co->a;
122   Mat_SeqAIJ     *p_loc,*p_oth;
123   PetscScalar    *apa,*ca;
124   PetscInt       cm   =C->rmap->n;
125   Mat_PtAPMPI    *ptap=c->ptap;
126   PetscInt       *api,*apj,*apJ,i,k;
127   PetscInt       cstart=C->cmap->rstart;
128   PetscInt       cdnz,conz,k0,k1;
129   MPI_Comm       comm;
130   PetscMPIInt    size;
131 
132   PetscFunctionBegin;
133   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
134   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
135 
136   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
137   /*-----------------------------------------------------*/
138   /* update numerical values of P_oth and P_loc */
139   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
140   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
141 
142   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
143   /*----------------------------------------------------------*/
144   /* get data from symbolic products */
145   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
146   p_oth = NULL;
147   if (size >1) {
148     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
149   }
150 
151   /* get apa for storing dense row A[i,:]*P */
152   apa = ptap->apa;
153 
154   api = ptap->api;
155   apj = ptap->apj;
156   for (i=0; i<cm; i++) {
157     /* compute apa = A[i,:]*P */
158     AProw_nonscalable(i,ad,ao,p_loc,p_oth,apa);
159 
160     /* set values in C */
161     apJ  = apj + api[i];
162     cdnz = cd->i[i+1] - cd->i[i];
163     conz = co->i[i+1] - co->i[i];
164 
165     /* 1st off-diagoanl part of C */
166     ca = coa + co->i[i];
167     k  = 0;
168     for (k0=0; k0<conz; k0++) {
169       if (apJ[k] >= cstart) break;
170       ca[k0]      = apa[apJ[k]];
171       apa[apJ[k++]] = 0.0;
172     }
173 
174     /* diagonal part of C */
175     ca = cda + cd->i[i];
176     for (k1=0; k1<cdnz; k1++) {
177       ca[k1]      = apa[apJ[k]];
178       apa[apJ[k++]] = 0.0;
179     }
180 
181     /* 2nd off-diagoanl part of C */
182     ca = coa + co->i[i];
183     for (; k0<conz; k0++) {
184       ca[k0]      = apa[apJ[k]];
185       apa[apJ[k++]] = 0.0;
186     }
187   }
188   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
189   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
190   PetscFunctionReturn(0);
191 }
192 
193 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,PetscReal fill,Mat *C)
194 {
195   PetscErrorCode     ierr;
196   MPI_Comm           comm;
197   PetscMPIInt        size;
198   Mat                Cmpi;
199   Mat_PtAPMPI        *ptap;
200   PetscFreeSpaceList free_space=NULL,current_space=NULL;
201   Mat_MPIAIJ         *a        =(Mat_MPIAIJ*)A->data,*c;
202   Mat_SeqAIJ         *ad       =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
203   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
204   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
205   PetscInt           *lnk,i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi;
206   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
207   PetscBT            lnkbt;
208   PetscScalar        *apa;
209   PetscReal          afill;
210 
211   PetscFunctionBegin;
212   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
213   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
214 
215   /* create struct Mat_PtAPMPI and attached it to C later */
216   ierr = PetscNew(&ptap);CHKERRQ(ierr);
217 
218   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
219   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
220 
221   /* get P_loc by taking all local rows of P */
222   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
223 
224   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
225   pi_loc = p_loc->i; pj_loc = p_loc->j;
226   if (size > 1) {
227     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
228     pi_oth = p_oth->i; pj_oth = p_oth->j;
229   } else {
230     p_oth = NULL;
231     pi_oth = NULL; pj_oth = NULL;
232   }
233 
234   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
235   /*-------------------------------------------------------------------*/
236   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
237   ptap->api = api;
238   api[0]    = 0;
239 
240   /* create and initialize a linked list */
241   ierr = PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt);CHKERRQ(ierr);
242 
243   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
244   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space);CHKERRQ(ierr);
245   current_space = free_space;
246 
247   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
248   for (i=0; i<am; i++) {
249     /* diagonal portion of A */
250     nzi = adi[i+1] - adi[i];
251     for (j=0; j<nzi; j++) {
252       row  = *adj++;
253       pnz  = pi_loc[row+1] - pi_loc[row];
254       Jptr = pj_loc + pi_loc[row];
255       /* add non-zero cols of P into the sorted linked list lnk */
256       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
257     }
258     /* off-diagonal portion of A */
259     nzi = aoi[i+1] - aoi[i];
260     for (j=0; j<nzi; j++) {
261       row  = *aoj++;
262       pnz  = pi_oth[row+1] - pi_oth[row];
263       Jptr = pj_oth + pi_oth[row];
264       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
265     }
266 
267     apnz     = lnk[0];
268     api[i+1] = api[i] + apnz;
269 
270     /* if free space is not available, double the total space in the list */
271     if (current_space->local_remaining<apnz) {
272       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
273       nspacedouble++;
274     }
275 
276     /* Copy data into free space, then initialize lnk */
277     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
278     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
279 
280     current_space->array           += apnz;
281     current_space->local_used      += apnz;
282     current_space->local_remaining -= apnz;
283   }
284 
285   /* Allocate space for apj, initialize apj, and */
286   /* destroy list of free space and other temporary array(s) */
287   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
288   apj  = ptap->apj;
289   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
290   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
291 
292   /* malloc apa to store dense row A[i,:]*P */
293   ierr = PetscCalloc1(pN,&apa);CHKERRQ(ierr);
294 
295   ptap->apa = apa;
296 
297   /* create and assemble symbolic parallel matrix Cmpi */
298   /*----------------------------------------------------*/
299   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
300   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
301   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
302 
303   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
304   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
305   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
306   for (i=0; i<am; i++) {
307     row  = i + rstart;
308     apnz = api[i+1] - api[i];
309     ierr = MatSetValues(Cmpi,1,&row,apnz,apj,apa,INSERT_VALUES);CHKERRQ(ierr);
310     apj += apnz;
311   }
312   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
313   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
314 
315   ptap->destroy        = Cmpi->ops->destroy;
316   ptap->duplicate      = Cmpi->ops->duplicate;
317   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
318   Cmpi->ops->destroy   = MatDestroy_MPIAIJ_MatMatMult;
319   Cmpi->ops->duplicate = MatDuplicate_MPIAIJ_MatMatMult;
320 
321   /* attach the supporting struct to Cmpi for reuse */
322   c       = (Mat_MPIAIJ*)Cmpi->data;
323   c->ptap = ptap;
324 
325   *C = Cmpi;
326 
327   /* set MatInfo */
328   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
329   if (afill < 1.0) afill = 1.0;
330   Cmpi->info.mallocs           = nspacedouble;
331   Cmpi->info.fill_ratio_given  = fill;
332   Cmpi->info.fill_ratio_needed = afill;
333 
334 #if defined(PETSC_USE_INFO)
335   if (api[am]) {
336     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
337     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
338   } else {
339     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
340   }
341 #endif
342   PetscFunctionReturn(0);
343 }
344 
345 PETSC_INTERN PetscErrorCode MatMatMult_MPIAIJ_MPIDense(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
346 {
347   PetscErrorCode ierr;
348 
349   PetscFunctionBegin;
350   if (scall == MAT_INITIAL_MATRIX) {
351     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
352     ierr = MatMatMultSymbolic_MPIAIJ_MPIDense(A,B,fill,C);CHKERRQ(ierr);
353     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
354   }
355   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
356   ierr = MatMatMultNumeric_MPIAIJ_MPIDense(A,B,*C);CHKERRQ(ierr);
357   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
358   PetscFunctionReturn(0);
359 }
360 
361 typedef struct {
362   Mat         workB;
363   PetscScalar *rvalues,*svalues;
364   MPI_Request *rwaits,*swaits;
365 } MPIAIJ_MPIDense;
366 
367 PetscErrorCode MatMPIAIJ_MPIDenseDestroy(void *ctx)
368 {
369   MPIAIJ_MPIDense *contents = (MPIAIJ_MPIDense*) ctx;
370   PetscErrorCode  ierr;
371 
372   PetscFunctionBegin;
373   ierr = MatDestroy(&contents->workB);CHKERRQ(ierr);
374   ierr = PetscFree4(contents->rvalues,contents->svalues,contents->rwaits,contents->swaits);CHKERRQ(ierr);
375   ierr = PetscFree(contents);CHKERRQ(ierr);
376   PetscFunctionReturn(0);
377 }
378 
379 /*
380     This is a "dummy function" that handles the case where matrix C was created as a dense matrix
381   directly by the user and passed to MatMatMult() with the MAT_REUSE_MATRIX option
382 
383   It is the same as MatMatMultSymbolic_MPIAIJ_MPIDense() except does not create C
384 */
385 PetscErrorCode MatMatMultNumeric_MPIDense(Mat A,Mat B,Mat C)
386 {
387   PetscErrorCode         ierr;
388   PetscBool              flg;
389   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*) A->data;
390   PetscInt               nz   = aij->B->cmap->n;
391   PetscContainer         container;
392   MPIAIJ_MPIDense        *contents;
393   VecScatter             ctx   = aij->Mvctx;
394   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
395   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
396 
397   PetscFunctionBegin;
398   ierr = PetscObjectTypeCompare((PetscObject)B,MATMPIDENSE,&flg);CHKERRQ(ierr);
399   if (!flg) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Second matrix must be mpidense");
400 
401   /* Handle case where where user provided the final C matrix rather than calling MatMatMult() with MAT_INITIAL_MATRIX*/
402   ierr = PetscObjectTypeCompare((PetscObject)A,MATMPIAIJ,&flg);CHKERRQ(ierr);
403   if (!flg) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"First matrix must be MPIAIJ");
404 
405   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
406 
407   ierr = PetscNew(&contents);CHKERRQ(ierr);
408   /* Create work matrix used to store off processor rows of B needed for local product */
409   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,B->cmap->N,NULL,&contents->workB);CHKERRQ(ierr);
410   /* Create work arrays needed */
411   ierr = PetscMalloc4(B->cmap->N*from->starts[from->n],&contents->rvalues,
412                       B->cmap->N*to->starts[to->n],&contents->svalues,
413                       from->n,&contents->rwaits,
414                       to->n,&contents->swaits);CHKERRQ(ierr);
415 
416   ierr = PetscContainerCreate(PetscObjectComm((PetscObject)A),&container);CHKERRQ(ierr);
417   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
418   ierr = PetscContainerSetUserDestroy(container,MatMPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
419   ierr = PetscObjectCompose((PetscObject)C,"workB",(PetscObject)container);CHKERRQ(ierr);
420   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
421 
422   ierr = (*C->ops->matmultnumeric)(A,B,C);CHKERRQ(ierr);
423   PetscFunctionReturn(0);
424 }
425 
426 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat A,Mat B,PetscReal fill,Mat *C)
427 {
428   PetscErrorCode         ierr;
429   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*) A->data;
430   PetscInt               nz   = aij->B->cmap->n;
431   PetscContainer         container;
432   MPIAIJ_MPIDense        *contents;
433   VecScatter             ctx   = aij->Mvctx;
434   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
435   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
436   PetscInt               m     = A->rmap->n,n=B->cmap->n;
437 
438   PetscFunctionBegin;
439   ierr = MatCreate(PetscObjectComm((PetscObject)B),C);CHKERRQ(ierr);
440   ierr = MatSetSizes(*C,m,n,A->rmap->N,B->cmap->N);CHKERRQ(ierr);
441   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
442   ierr = MatSetType(*C,MATMPIDENSE);CHKERRQ(ierr);
443   ierr = MatMPIDenseSetPreallocation(*C,NULL);CHKERRQ(ierr);
444   ierr = MatAssemblyBegin(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
445   ierr = MatAssemblyEnd(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
446 
447   (*C)->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
448 
449   ierr = PetscNew(&contents);CHKERRQ(ierr);
450   /* Create work matrix used to store off processor rows of B needed for local product */
451   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,B->cmap->N,NULL,&contents->workB);CHKERRQ(ierr);
452   /* Create work arrays needed */
453   ierr = PetscMalloc4(B->cmap->N*from->starts[from->n],&contents->rvalues,
454                       B->cmap->N*to->starts[to->n],&contents->svalues,
455                       from->n,&contents->rwaits,
456                       to->n,&contents->swaits);CHKERRQ(ierr);
457 
458   ierr = PetscContainerCreate(PetscObjectComm((PetscObject)A),&container);CHKERRQ(ierr);
459   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
460   ierr = PetscContainerSetUserDestroy(container,MatMPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
461   ierr = PetscObjectCompose((PetscObject)(*C),"workB",(PetscObject)container);CHKERRQ(ierr);
462   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
463   PetscFunctionReturn(0);
464 }
465 
466 /*
467     Performs an efficient scatter on the rows of B needed by this process; this is
468     a modification of the VecScatterBegin_() routines.
469 */
470 PetscErrorCode MatMPIDenseScatter(Mat A,Mat B,Mat C,Mat *outworkB)
471 {
472   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*)A->data;
473   PetscErrorCode         ierr;
474   PetscScalar            *b,*w,*svalues,*rvalues;
475   VecScatter             ctx   = aij->Mvctx;
476   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
477   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
478   PetscInt               i,j,k;
479   PetscInt               *sindices,*sstarts,*rindices,*rstarts;
480   PetscMPIInt            *sprocs,*rprocs,nrecvs;
481   MPI_Request            *swaits,*rwaits;
482   MPI_Comm               comm;
483   PetscMPIInt            tag  = ((PetscObject)ctx)->tag,ncols = B->cmap->N, nrows = aij->B->cmap->n,imdex,nrowsB = B->rmap->n;
484   MPI_Status             status;
485   MPIAIJ_MPIDense        *contents;
486   PetscContainer         container;
487   Mat                    workB;
488 
489   PetscFunctionBegin;
490   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
491   ierr = PetscObjectQuery((PetscObject)C,"workB",(PetscObject*)&container);CHKERRQ(ierr);
492   if (!container) SETERRQ(comm,PETSC_ERR_PLIB,"Container does not exist");
493   ierr = PetscContainerGetPointer(container,(void**)&contents);CHKERRQ(ierr);
494 
495   workB = *outworkB = contents->workB;
496   if (nrows != workB->rmap->n) SETERRQ2(comm,PETSC_ERR_PLIB,"Number of rows of workB %D not equal to columns of aij->B %D",nrows,workB->cmap->n);
497   sindices = to->indices;
498   sstarts  = to->starts;
499   sprocs   = to->procs;
500   swaits   = contents->swaits;
501   svalues  = contents->svalues;
502 
503   rindices = from->indices;
504   rstarts  = from->starts;
505   rprocs   = from->procs;
506   rwaits   = contents->rwaits;
507   rvalues  = contents->rvalues;
508 
509   ierr = MatDenseGetArray(B,&b);CHKERRQ(ierr);
510   ierr = MatDenseGetArray(workB,&w);CHKERRQ(ierr);
511 
512   for (i=0; i<from->n; i++) {
513     ierr = MPI_Irecv(rvalues+ncols*rstarts[i],ncols*(rstarts[i+1]-rstarts[i]),MPIU_SCALAR,rprocs[i],tag,comm,rwaits+i);CHKERRQ(ierr);
514   }
515 
516   for (i=0; i<to->n; i++) {
517     /* pack a message at a time */
518     for (j=0; j<sstarts[i+1]-sstarts[i]; j++) {
519       for (k=0; k<ncols; k++) {
520         svalues[ncols*(sstarts[i] + j) + k] = b[sindices[sstarts[i]+j] + nrowsB*k];
521       }
522     }
523     ierr = MPI_Isend(svalues+ncols*sstarts[i],ncols*(sstarts[i+1]-sstarts[i]),MPIU_SCALAR,sprocs[i],tag,comm,swaits+i);CHKERRQ(ierr);
524   }
525 
526   nrecvs = from->n;
527   while (nrecvs) {
528     ierr = MPI_Waitany(from->n,rwaits,&imdex,&status);CHKERRQ(ierr);
529     nrecvs--;
530     /* unpack a message at a time */
531     for (j=0; j<rstarts[imdex+1]-rstarts[imdex]; j++) {
532       for (k=0; k<ncols; k++) {
533         w[rindices[rstarts[imdex]+j] + nrows*k] = rvalues[ncols*(rstarts[imdex] + j) + k];
534       }
535     }
536   }
537   if (to->n) {ierr = MPI_Waitall(to->n,swaits,to->sstatus);CHKERRQ(ierr);}
538 
539   ierr = MatDenseRestoreArray(B,&b);CHKERRQ(ierr);
540   ierr = MatDenseRestoreArray(workB,&w);CHKERRQ(ierr);
541   ierr = MatAssemblyBegin(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
542   ierr = MatAssemblyEnd(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
543   PetscFunctionReturn(0);
544 }
545 extern PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat,Mat,Mat);
546 
547 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat A,Mat B,Mat C)
548 {
549   PetscErrorCode ierr;
550   Mat_MPIAIJ     *aij    = (Mat_MPIAIJ*)A->data;
551   Mat_MPIDense   *bdense = (Mat_MPIDense*)B->data;
552   Mat_MPIDense   *cdense = (Mat_MPIDense*)C->data;
553   Mat            workB;
554 
555   PetscFunctionBegin;
556   /* diagonal block of A times all local rows of B*/
557   ierr = MatMatMultNumeric_SeqAIJ_SeqDense(aij->A,bdense->A,cdense->A);CHKERRQ(ierr);
558 
559   /* get off processor parts of B needed to complete the product */
560   ierr = MatMPIDenseScatter(A,B,C,&workB);CHKERRQ(ierr);
561 
562   /* off-diagonal block of A times nonlocal rows of B */
563   ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A);CHKERRQ(ierr);
564   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
565   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
566   PetscFunctionReturn(0);
567 }
568 
569 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
570 {
571   PetscErrorCode ierr;
572   Mat_MPIAIJ     *a   = (Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
573   Mat_SeqAIJ     *ad  = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
574   Mat_SeqAIJ     *cd  = (Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
575   PetscInt       *adi = ad->i,*adj,*aoi=ao->i,*aoj;
576   PetscScalar    *ada,*aoa,*cda=cd->a,*coa=co->a;
577   Mat_SeqAIJ     *p_loc,*p_oth;
578   PetscInt       *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
579   PetscScalar    *pa_loc,*pa_oth,*pa,valtmp,*ca;
580   PetscInt       cm          = C->rmap->n,anz,pnz;
581   Mat_PtAPMPI    *ptap       = c->ptap;
582   PetscScalar    *apa_sparse = ptap->apa;
583   PetscInt       *api,*apj,*apJ,i,j,k,row;
584   PetscInt       cstart = C->cmap->rstart;
585   PetscInt       cdnz,conz,k0,k1,nextp;
586   MPI_Comm       comm;
587   PetscMPIInt    size;
588 
589   PetscFunctionBegin;
590   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
591   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
592 
593   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
594   /*-----------------------------------------------------*/
595   /* update numerical values of P_oth and P_loc */
596   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
597   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
598 
599   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
600   /*----------------------------------------------------------*/
601   /* get data from symbolic products */
602   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
603   pi_loc = p_loc->i; pj_loc = p_loc->j; pa_loc = p_loc->a;
604   if (size >1) {
605     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
606     pi_oth = p_oth->i; pj_oth = p_oth->j; pa_oth = p_oth->a;
607   } else {
608     p_oth = NULL; pi_oth = NULL; pj_oth = NULL; pa_oth = NULL;
609   }
610 
611   api = ptap->api;
612   apj = ptap->apj;
613   for (i=0; i<cm; i++) {
614     apJ = apj + api[i];
615 
616     /* diagonal portion of A */
617     anz = adi[i+1] - adi[i];
618     adj = ad->j + adi[i];
619     ada = ad->a + adi[i];
620     for (j=0; j<anz; j++) {
621       row = adj[j];
622       pnz = pi_loc[row+1] - pi_loc[row];
623       pj  = pj_loc + pi_loc[row];
624       pa  = pa_loc + pi_loc[row];
625       /* perform sparse axpy */
626       valtmp = ada[j];
627       nextp  = 0;
628       for (k=0; nextp<pnz; k++) {
629         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
630           apa_sparse[k] += valtmp*pa[nextp++];
631         }
632       }
633       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
634     }
635 
636     /* off-diagonal portion of A */
637     anz = aoi[i+1] - aoi[i];
638     aoj = ao->j + aoi[i];
639     aoa = ao->a + aoi[i];
640     for (j=0; j<anz; j++) {
641       row = aoj[j];
642       pnz = pi_oth[row+1] - pi_oth[row];
643       pj  = pj_oth + pi_oth[row];
644       pa  = pa_oth + pi_oth[row];
645       /* perform sparse axpy */
646       valtmp = aoa[j];
647       nextp  = 0;
648       for (k=0; nextp<pnz; k++) {
649         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
650           apa_sparse[k] += valtmp*pa[nextp++];
651         }
652       }
653       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
654     }
655 
656     /* set values in C */
657     cdnz = cd->i[i+1] - cd->i[i];
658     conz = co->i[i+1] - co->i[i];
659 
660     /* 1st off-diagoanl part of C */
661     ca = coa + co->i[i];
662     k  = 0;
663     for (k0=0; k0<conz; k0++) {
664       if (apJ[k] >= cstart) break;
665       ca[k0]        = apa_sparse[k];
666       apa_sparse[k] = 0.0;
667       k++;
668     }
669 
670     /* diagonal part of C */
671     ca = cda + cd->i[i];
672     for (k1=0; k1<cdnz; k1++) {
673       ca[k1]        = apa_sparse[k];
674       apa_sparse[k] = 0.0;
675       k++;
676     }
677 
678     /* 2nd off-diagoanl part of C */
679     ca = coa + co->i[i];
680     for (; k0<conz; k0++) {
681       ca[k0]        = apa_sparse[k];
682       apa_sparse[k] = 0.0;
683       k++;
684     }
685   }
686   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
687   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
688   PetscFunctionReturn(0);
689 }
690 
691 /* same as MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(), except using LLCondensed to avoid O(BN) memory requirement */
692 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat *C)
693 {
694   PetscErrorCode     ierr;
695   MPI_Comm           comm;
696   PetscMPIInt        size;
697   Mat                Cmpi;
698   Mat_PtAPMPI        *ptap;
699   PetscFreeSpaceList free_space = NULL,current_space=NULL;
700   Mat_MPIAIJ         *a         = (Mat_MPIAIJ*)A->data,*c;
701   Mat_SeqAIJ         *ad        = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
702   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
703   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
704   PetscInt           i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi,*lnk,apnz_max;
705   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
706   PetscReal          afill;
707   PetscScalar        *apa;
708   PetscTable         ta;
709 
710   PetscFunctionBegin;
711   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
712   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
713 
714   /* create struct Mat_PtAPMPI and attached it to C later */
715   ierr = PetscNew(&ptap);CHKERRQ(ierr);
716 
717   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
718   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
719 
720   /* get P_loc by taking all local rows of P */
721   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
722 
723   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
724   pi_loc = p_loc->i; pj_loc = p_loc->j;
725   if (size > 1) {
726     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
727     pi_oth = p_oth->i; pj_oth = p_oth->j;
728   } else {
729     p_oth  = NULL;
730     pi_oth = NULL; pj_oth = NULL;
731   }
732 
733   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
734   /*-------------------------------------------------------------------*/
735   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
736   ptap->api = api;
737   api[0]    = 0;
738 
739   /* create and initialize a linked list */
740   ierr = PetscTableCreate(pn,pN,&ta);CHKERRQ(ierr);
741 
742   /* Calculate apnz_max */
743   apnz_max = 0;
744   for (i=0; i<am; i++) {
745     ierr = PetscTableRemoveAll(ta);CHKERRQ(ierr);
746     /* diagonal portion of A */
747     nzi  = adi[i+1] - adi[i];
748     Jptr = adj+adi[i];  /* cols of A_diag */
749     MatMergeRows_SeqAIJ(p_loc,nzi,Jptr,ta);
750     ierr = PetscTableGetCount(ta,&apnz);CHKERRQ(ierr);
751     if (apnz_max < apnz) apnz_max = apnz;
752 
753     /*  off-diagonal portion of A */
754     nzi = aoi[i+1] - aoi[i];
755     Jptr = aoj+aoi[i];  /* cols of A_off */
756     MatMergeRows_SeqAIJ(p_oth,nzi,Jptr,ta);
757     ierr = PetscTableGetCount(ta,&apnz);CHKERRQ(ierr);
758     if (apnz_max < apnz) apnz_max = apnz;
759   }
760   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
761 
762   ierr = PetscLLCondensedCreate_Scalable(apnz_max,&lnk);CHKERRQ(ierr);
763 
764   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
765   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space);CHKERRQ(ierr);
766   current_space = free_space;
767   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
768   for (i=0; i<am; i++) {
769     /* diagonal portion of A */
770     nzi = adi[i+1] - adi[i];
771     for (j=0; j<nzi; j++) {
772       row  = *adj++;
773       pnz  = pi_loc[row+1] - pi_loc[row];
774       Jptr = pj_loc + pi_loc[row];
775       /* add non-zero cols of P into the sorted linked list lnk */
776       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
777     }
778     /* off-diagonal portion of A */
779     nzi = aoi[i+1] - aoi[i];
780     for (j=0; j<nzi; j++) {
781       row  = *aoj++;
782       pnz  = pi_oth[row+1] - pi_oth[row];
783       Jptr = pj_oth + pi_oth[row];
784       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
785     }
786 
787     apnz     = *lnk;
788     api[i+1] = api[i] + apnz;
789 
790     /* if free space is not available, double the total space in the list */
791     if (current_space->local_remaining<apnz) {
792       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
793       nspacedouble++;
794     }
795 
796     /* Copy data into free space, then initialize lnk */
797     ierr = PetscLLCondensedClean_Scalable(apnz,current_space->array,lnk);CHKERRQ(ierr);
798     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
799 
800     current_space->array           += apnz;
801     current_space->local_used      += apnz;
802     current_space->local_remaining -= apnz;
803   }
804 
805   /* Allocate space for apj, initialize apj, and */
806   /* destroy list of free space and other temporary array(s) */
807   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
808   apj  = ptap->apj;
809   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
810   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
811 
812   /* create and assemble symbolic parallel matrix Cmpi */
813   /*----------------------------------------------------*/
814   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
815   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
816   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
817   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
818   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
819   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
820 
821   /* malloc apa for assembly Cmpi */
822   ierr = PetscCalloc1(apnz_max,&apa);CHKERRQ(ierr);
823 
824   ptap->apa = apa;
825   for (i=0; i<am; i++) {
826     row  = i + rstart;
827     apnz = api[i+1] - api[i];
828     ierr = MatSetValues(Cmpi,1,&row,apnz,apj,apa,INSERT_VALUES);CHKERRQ(ierr);
829     apj += apnz;
830   }
831   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
832   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
833 
834   ptap->destroy             = Cmpi->ops->destroy;
835   ptap->duplicate           = Cmpi->ops->duplicate;
836   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ;
837   Cmpi->ops->destroy        = MatDestroy_MPIAIJ_MatMatMult;
838   Cmpi->ops->duplicate      = MatDuplicate_MPIAIJ_MatMatMult;
839 
840   /* attach the supporting struct to Cmpi for reuse */
841   c       = (Mat_MPIAIJ*)Cmpi->data;
842   c->ptap = ptap;
843 
844   *C = Cmpi;
845 
846   /* set MatInfo */
847   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
848   if (afill < 1.0) afill = 1.0;
849   Cmpi->info.mallocs           = nspacedouble;
850   Cmpi->info.fill_ratio_given  = fill;
851   Cmpi->info.fill_ratio_needed = afill;
852 
853 #if defined(PETSC_USE_INFO)
854   if (api[am]) {
855     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
856     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
857   } else {
858     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
859   }
860 #endif
861   PetscFunctionReturn(0);
862 }
863 
864 /*-------------------------------------------------------------------------*/
865 PetscErrorCode MatTransposeMatMult_MPIAIJ_MPIAIJ(Mat P,Mat A,MatReuse scall,PetscReal fill,Mat *C)
866 {
867   PetscErrorCode ierr;
868   const char     *algTypes[3] = {"scalable","nonscalable","matmatmult"};
869   PetscInt       alg=0; /* set default algorithm */
870 
871   PetscFunctionBegin;
872   if (scall == MAT_INITIAL_MATRIX) {
873     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
874     PetscOptionsObject->alreadyprinted = PETSC_FALSE; /* a hack to ensure the option shows in '-help' */
875     ierr = PetscOptionsEList("-mattransposematmult_via","Algorithmic approach","MatTransposeMatMult",algTypes,3,algTypes[0],&alg,NULL);CHKERRQ(ierr);
876     ierr = PetscOptionsEnd();CHKERRQ(ierr);
877 
878     //ierr = PetscLogEventBegin(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
879     switch (alg) {
880     case 1:
881       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(P,A,fill,C);CHKERRQ(ierr);
882       break;
883     case 2:
884     {
885       Mat         Pt;
886       Mat_PtAPMPI *ptap;
887       Mat_MPIAIJ  *c;
888       ierr = MatTranspose(P,MAT_INITIAL_MATRIX,&Pt);CHKERRQ(ierr);
889       ierr = MatMatMult(Pt,A,MAT_INITIAL_MATRIX,fill,C);CHKERRQ(ierr);
890       c        = (Mat_MPIAIJ*)(*C)->data;
891       ptap     = c->ptap;
892       ptap->Pt = Pt;
893       (*C)->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult;
894       PetscFunctionReturn(0);
895     }
896       break;
897     default:
898       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(P,A,fill,C);CHKERRQ(ierr);
899       break;
900     }
901     //ierr = PetscLogEventEnd(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
902   }
903   //ierr = PetscLogEventBegin(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
904   ierr = (*(*C)->ops->mattransposemultnumeric)(P,A,*C);CHKERRQ(ierr);
905   //ierr = PetscLogEventEnd(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
906   PetscFunctionReturn(0);
907 }
908 
909 /* This routine only works when scall=MAT_REUSE_MATRIX! */
910 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult(Mat P,Mat A,Mat C)
911 {
912   PetscErrorCode ierr;
913   Mat_MPIAIJ     *c=(Mat_MPIAIJ*)C->data;
914   Mat_PtAPMPI    *ptap= c->ptap;
915   Mat            Pt=ptap->Pt;
916 
917   PetscFunctionBegin;
918   ierr = MatTranspose(P,MAT_REUSE_MATRIX,&Pt);CHKERRQ(ierr);
919   ierr = MatMatMultNumeric(Pt,A,C);CHKERRQ(ierr);
920   PetscFunctionReturn(0);
921 }
922 
923 /* Non-scalable version, use dense axpy */
924 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable_old(Mat P,Mat A,Mat C)
925 {
926   PetscErrorCode      ierr;
927   Mat_Merge_SeqsToMPI *merge;
928   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
929   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
930   Mat_PtAPMPI         *ptap;
931   PetscInt            *adj,*aJ;
932   PetscInt            i,j,k,anz,pnz,row,*cj;
933   MatScalar           *ada,*aval,*ca,valtmp;
934   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
935   MPI_Comm            comm;
936   PetscMPIInt         size,rank,taga,*len_s;
937   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
938   PetscInt            **buf_ri,**buf_rj;
939   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
940   MPI_Request         *s_waits,*r_waits;
941   MPI_Status          *status;
942   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
943   PetscInt            *ai,*aj,*coi,*coj;
944   PetscInt            *poJ,*pdJ;
945   Mat                 A_loc;
946   Mat_SeqAIJ          *a_loc;
947 
948   PetscFunctionBegin;
949   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
950   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
951   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
952 
953   ptap  = c->ptap;
954   merge = ptap->merge;
955 
956   /* 2) compute numeric C_seq = P_loc^T*A_loc - dominating part */
957   /*------------------------------------------------------------*/
958   /* get data from symbolic products */
959   coi  = merge->coi; coj = merge->coj;
960   ierr = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
961 
962   bi     = merge->bi; bj = merge->bj;
963   owners = merge->rowmap->range;
964   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
965 
966   /* get A_loc by taking all local rows of A */
967   A_loc = ptap->A_loc;
968   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
969   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
970   ai    = a_loc->i;
971   aj    = a_loc->j;
972 
973   ierr = PetscCalloc1(A->cmap->N,&aval);CHKERRQ(ierr); /* non-scalable!!! */
974 
975   for (i=0; i<am; i++) {
976     /* 2-a) put A[i,:] to dense array aval */
977     anz = ai[i+1] - ai[i];
978     adj = aj + ai[i];
979     ada = a_loc->a + ai[i];
980     for (j=0; j<anz; j++) {
981       aval[adj[j]] = ada[j];
982     }
983 
984     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
985     /*--------------------------------------------------------------*/
986     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
987     pnz = po->i[i+1] - po->i[i];
988     poJ = po->j + po->i[i];
989     pA  = po->a + po->i[i];
990     for (j=0; j<pnz; j++) {
991       row = poJ[j];
992       cnz = coi[row+1] - coi[row];
993       cj  = coj + coi[row];
994       ca  = coa + coi[row];
995       /* perform dense axpy */
996       valtmp = pA[j];
997       for (k=0; k<cnz; k++) {
998         ca[k] += valtmp*aval[cj[k]];
999       }
1000       ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1001     }
1002 
1003     /* put the value into Cd (diagonal part) */
1004     pnz = pd->i[i+1] - pd->i[i];
1005     pdJ = pd->j + pd->i[i];
1006     pA  = pd->a + pd->i[i];
1007     for (j=0; j<pnz; j++) {
1008       row = pdJ[j];
1009       cnz = bi[row+1] - bi[row];
1010       cj  = bj + bi[row];
1011       ca  = ba + bi[row];
1012       /* perform dense axpy */
1013       valtmp = pA[j];
1014       for (k=0; k<cnz; k++) {
1015         ca[k] += valtmp*aval[cj[k]];
1016       }
1017       ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1018     }
1019 
1020     /* zero the current row of Pt*A */
1021     aJ = aj + ai[i];
1022     for (k=0; k<anz; k++) aval[aJ[k]] = 0.0;
1023   }
1024 
1025   /* 3) send and recv matrix values coa */
1026   /*------------------------------------*/
1027   buf_ri = merge->buf_ri;
1028   buf_rj = merge->buf_rj;
1029   len_s  = merge->len_s;
1030   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1031   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1032 
1033   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1034   for (proc=0,k=0; proc<size; proc++) {
1035     if (!len_s[proc]) continue;
1036     i    = merge->owners_co[proc];
1037     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1038     k++;
1039   }
1040   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1041   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1042 
1043   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1044   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1045   ierr = PetscFree(coa);CHKERRQ(ierr);
1046 
1047   /* 4) insert local Cseq and received values into Cmpi */
1048   /*----------------------------------------------------*/
1049   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1050   for (k=0; k<merge->nrecv; k++) {
1051     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1052     nrows       = *(buf_ri_k[k]);
1053     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1054     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1055   }
1056 
1057   for (i=0; i<cm; i++) {
1058     row  = owners[rank] + i; /* global row index of C_seq */
1059     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1060     ba_i = ba + bi[i];
1061     bnz  = bi[i+1] - bi[i];
1062     /* add received vals into ba */
1063     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1064       /* i-th row */
1065       if (i == *nextrow[k]) {
1066         cnz    = *(nextci[k]+1) - *nextci[k];
1067         cj     = buf_rj[k] + *(nextci[k]);
1068         ca     = abuf_r[k] + *(nextci[k]);
1069         nextcj = 0;
1070         for (j=0; nextcj<cnz; j++) {
1071           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1072             ba_i[j] += ca[nextcj++];
1073           }
1074         }
1075         nextrow[k]++; nextci[k]++;
1076         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1077       }
1078     }
1079     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1080   }
1081   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1082   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1083 
1084   ierr = PetscFree(ba);CHKERRQ(ierr);
1085   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1086   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1087   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1088   ierr = PetscFree(aval);CHKERRQ(ierr);
1089   PetscFunctionReturn(0);
1090 }
1091 
1092 PetscErrorCode MatDuplicate_MPIAIJ_MatPtAP(Mat, MatDuplicateOption,Mat*);
1093 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1094 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable_old(Mat P,Mat A,PetscReal fill,Mat *C)
1095 {
1096   PetscErrorCode      ierr;
1097   Mat                 Cmpi,A_loc,POt,PDt;
1098   Mat_PtAPMPI         *ptap;
1099   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1100   Mat_MPIAIJ          *p        =(Mat_MPIAIJ*)P->data,*c;
1101   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1102   PetscInt            nnz;
1103   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1104   PetscInt            am=A->rmap->n,pn=P->cmap->n;
1105   PetscBT             lnkbt;
1106   MPI_Comm            comm;
1107   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1108   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1109   PetscInt            len,proc,*dnz,*onz,*owners;
1110   PetscInt            nzi,*bi,*bj;
1111   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1112   MPI_Request         *swaits,*rwaits;
1113   MPI_Status          *sstatus,rstatus;
1114   Mat_Merge_SeqsToMPI *merge;
1115   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1116   PetscReal           afill  =1.0,afill_tmp;
1117   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N;
1118   PetscScalar         *vals;
1119   Mat_SeqAIJ          *a_loc, *pdt,*pot;
1120 
1121   PetscFunctionBegin;
1122   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1123   /* check if matrix local sizes are compatible */
1124   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);
1125 
1126   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1127   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1128 
1129   /* create struct Mat_PtAPMPI and attached it to C later */
1130   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1131 
1132   /* get A_loc by taking all local rows of A */
1133   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1134 
1135   ptap->A_loc = A_loc;
1136 
1137   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1138   ai    = a_loc->i;
1139   aj    = a_loc->j;
1140 
1141   /* determine symbolic Co=(p->B)^T*A - send to others */
1142   /*----------------------------------------------------*/
1143   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1144   pdt  = (Mat_SeqAIJ*)PDt->data;
1145   pdti = pdt->i; pdtj = pdt->j;
1146 
1147   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1148   pot  = (Mat_SeqAIJ*)POt->data;
1149   poti = pot->i; potj = pot->j;
1150 
1151   /* then, compute symbolic Co = (p->B)^T*A */
1152   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors >= (num of nonzero rows of C_seq) - pn */
1153   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
1154   coi[0] = 0;
1155 
1156   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1157   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(poti[pon],ai[am]));
1158   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1159   current_space = free_space;
1160 
1161   /* create and initialize a linked list */
1162   ierr = PetscLLCondensedCreate(aN,aN,&lnk,&lnkbt);CHKERRQ(ierr);
1163 
1164   for (i=0; i<pon; i++) {
1165     pnz = poti[i+1] - poti[i];
1166     ptJ = potj + poti[i];
1167     for (j=0; j<pnz; j++) {
1168       row  = ptJ[j]; /* row of A_loc == col of Pot */
1169       anz  = ai[row+1] - ai[row];
1170       Jptr = aj + ai[row];
1171       /* add non-zero cols of AP into the sorted linked list lnk */
1172       ierr = PetscLLCondensedAddSorted(anz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1173     }
1174     nnz = lnk[0];
1175 
1176     /* If free space is not available, double the total space in the list */
1177     if (current_space->local_remaining<nnz) {
1178       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1179       nspacedouble++;
1180     }
1181 
1182     /* Copy data into free space, and zero out denserows */
1183     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1184 
1185     current_space->array           += nnz;
1186     current_space->local_used      += nnz;
1187     current_space->local_remaining -= nnz;
1188 
1189     coi[i+1] = coi[i] + nnz;
1190   }
1191 
1192   ierr = PetscMalloc1(coi[pon]+1,&coj);CHKERRQ(ierr);
1193   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1194 
1195   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1196   if (afill_tmp > afill) afill = afill_tmp;
1197 
1198   /* send j-array (coj) of Co to other processors */
1199   /*----------------------------------------------*/
1200   /* determine row ownership */
1201   ierr = PetscNew(&merge);CHKERRQ(ierr);
1202   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1203 
1204   merge->rowmap->n  = pn;
1205   merge->rowmap->bs = 1;
1206 
1207   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1208   owners = merge->rowmap->range;
1209 
1210   /* determine the number of messages to send, their lengths */
1211   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
1212   ierr = PetscMalloc1(size,&merge->len_s);CHKERRQ(ierr);
1213 
1214   len_s        = merge->len_s;
1215   merge->nsend = 0;
1216 
1217   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
1218   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1219 
1220   proc = 0;
1221   for (i=0; i<pon; i++) {
1222     while (prmap[i] >= owners[proc+1]) proc++;
1223     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1224     len_s[proc] += coi[i+1] - coi[i];
1225   }
1226 
1227   len          = 0; /* max length of buf_si[] */
1228   owners_co[0] = 0;
1229   for (proc=0; proc<size; proc++) {
1230     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1231     if (len_si[proc]) {
1232       merge->nsend++;
1233       len_si[proc] = 2*(len_si[proc] + 1);
1234       len         += len_si[proc];
1235     }
1236   }
1237 
1238   /* determine the number and length of messages to receive for coi and coj  */
1239   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1240   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1241 
1242   /* post the Irecv and Isend of coj */
1243   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1244   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1245   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
1246   for (proc=0, k=0; proc<size; proc++) {
1247     if (!len_s[proc]) continue;
1248     i    = owners_co[proc];
1249     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1250     k++;
1251   }
1252 
1253   /* receives and sends of coj are complete */
1254   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
1255   for (i=0; i<merge->nrecv; i++) {
1256     PetscMPIInt icompleted;
1257     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1258   }
1259   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1260   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1261 
1262   /* send and recv coi */
1263   /*-------------------*/
1264   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1265   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1266   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1267   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1268   for (proc=0,k=0; proc<size; proc++) {
1269     if (!len_s[proc]) continue;
1270     /* form outgoing message for i-structure:
1271          buf_si[0]:                 nrows to be sent
1272                [1:nrows]:           row index (global)
1273                [nrows+1:2*nrows+1]: i-structure index
1274     */
1275     /*-------------------------------------------*/
1276     nrows       = len_si[proc]/2 - 1;
1277     buf_si_i    = buf_si + nrows+1;
1278     buf_si[0]   = nrows;
1279     buf_si_i[0] = 0;
1280     nrows       = 0;
1281     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1282       nzi               = coi[i+1] - coi[i];
1283       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi; /* i-structure */
1284       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1285       nrows++;
1286     }
1287     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1288     k++;
1289     buf_si += len_si[proc];
1290   }
1291   i = merge->nrecv;
1292   while (i--) {
1293     PetscMPIInt icompleted;
1294     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1295   }
1296   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1297   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1298   ierr = PetscFree(len_si);CHKERRQ(ierr);
1299   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1300   ierr = PetscFree(swaits);CHKERRQ(ierr);
1301   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1302   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1303 
1304   /* compute the local portion of C (mpi mat) */
1305   /*------------------------------------------*/
1306   /* allocate bi array and free space for accumulating nonzero column info */
1307   ierr  = PetscMalloc1(pn+1,&bi);CHKERRQ(ierr);
1308   bi[0] = 0;
1309 
1310   /* set initial free space to be fill*(nnz(P) + nnz(A)) */
1311   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(pdti[pn],PetscIntSumTruncate(poti[pon],ai[am])));
1312   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1313   current_space = free_space;
1314 
1315   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1316   for (k=0; k<merge->nrecv; k++) {
1317     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1318     nrows       = *buf_ri_k[k];
1319     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1320     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1321   }
1322 
1323   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1324   rmax = 0;
1325   for (i=0; i<pn; i++) {
1326     /* add pdt[i,:]*AP into lnk */
1327     pnz = pdti[i+1] - pdti[i];
1328     ptJ = pdtj + pdti[i];
1329     for (j=0; j<pnz; j++) {
1330       row  = ptJ[j];  /* row of AP == col of Pt */
1331       anz  = ai[row+1] - ai[row];
1332       Jptr = aj + ai[row];
1333       /* add non-zero cols of AP into the sorted linked list lnk */
1334       ierr = PetscLLCondensedAddSorted(anz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1335     }
1336 
1337     /* add received col data into lnk */
1338     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1339       if (i == *nextrow[k]) { /* i-th row */
1340         nzi  = *(nextci[k]+1) - *nextci[k];
1341         Jptr = buf_rj[k] + *nextci[k];
1342         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1343         nextrow[k]++; nextci[k]++;
1344       }
1345     }
1346     nnz = lnk[0];
1347 
1348     /* if free space is not available, make more free space */
1349     if (current_space->local_remaining<nnz) {
1350       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1351       nspacedouble++;
1352     }
1353     /* copy data into free space, then initialize lnk */
1354     ierr = PetscLLCondensedClean(aN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1355     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1356 
1357     current_space->array           += nnz;
1358     current_space->local_used      += nnz;
1359     current_space->local_remaining -= nnz;
1360 
1361     bi[i+1] = bi[i] + nnz;
1362     if (nnz > rmax) rmax = nnz;
1363   }
1364   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1365 
1366   ierr = PetscMalloc1(bi[pn]+1,&bj);CHKERRQ(ierr);
1367   ierr = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1368 
1369   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
1370   if (afill_tmp > afill) afill = afill_tmp;
1371   ierr = PetscLLCondensedDestroy(lnk,lnkbt);CHKERRQ(ierr);
1372   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1373   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1374 
1375   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
1376   /*----------------------------------------------------------------------------------*/
1377   ierr = PetscCalloc1(rmax+1,&vals);CHKERRQ(ierr);
1378 
1379   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1380   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1381   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
1382   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1383   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1384   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1385   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
1386   for (i=0; i<pn; i++) {
1387     row  = i + rstart;
1388     nnz  = bi[i+1] - bi[i];
1389     Jptr = bj + bi[i];
1390     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
1391   }
1392   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1393   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1394   ierr = PetscFree(vals);CHKERRQ(ierr);
1395 
1396   merge->bi        = bi;
1397   merge->bj        = bj;
1398   merge->coi       = coi;
1399   merge->coj       = coj;
1400   merge->buf_ri    = buf_ri;
1401   merge->buf_rj    = buf_rj;
1402   merge->owners_co = owners_co;
1403 
1404   /* attach the supporting struct to Cmpi for reuse */
1405   c           = (Mat_MPIAIJ*)Cmpi->data;
1406   c->ptap     = ptap;
1407   ptap->api   = NULL;
1408   ptap->apj   = NULL;
1409   ptap->merge = merge;
1410   ptap->destroy   = Cmpi->ops->destroy;
1411   ptap->duplicate = Cmpi->ops->duplicate;
1412 
1413   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable_old;
1414   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
1415   Cmpi->ops->duplicate               = MatDuplicate_MPIAIJ_MatPtAP;
1416 
1417   *C = Cmpi;
1418 #if defined(PETSC_USE_INFO)
1419   if (bi[pn] != 0) {
1420     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1421     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
1422   } else {
1423     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1424   }
1425 #endif
1426   PetscFunctionReturn(0);
1427 }
1428 
1429 //================================
1430 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,PetscReal fill,Mat *C)
1431 {
1432   PetscErrorCode      ierr;
1433   Mat_PtAPMPI         *ptap;
1434   Mat_MPIAIJ          *p=(Mat_MPIAIJ*)P->data,*c;
1435   MPI_Comm            comm;
1436   PetscMPIInt         size,rank;
1437   Mat                 Cmpi;
1438   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1439   PetscInt            pN=P->cmap->N,pn=P->cmap->n;
1440   PetscInt            *lnk,i,k,nsend;
1441   PetscBT             lnkbt;
1442   PetscMPIInt         tagi,tagj,*len_si,*len_s,*len_ri,icompleted=0,nrecv;
1443   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1444   PetscInt            len,proc,*dnz,*onz,*owners,nzi;
1445   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1446   MPI_Request         *swaits,*rwaits;
1447   MPI_Status          *sstatus,rstatus;
1448   PetscLayout         rowmap;
1449   PetscInt            *owners_co,*coi,*coj;    /* i and j array of (p->B)^T*A*P - used in the communication */
1450   PetscMPIInt         *len_r,*id_r;    /* array of length of comm->size, store send/recv matrix values */
1451   PetscInt            *Jptr,*prmap=p->garray,con,j,Crmax;
1452   Mat_SeqAIJ          *p_loc,*p_oth=NULL,*c_loc,*c_oth;
1453   PetscTable          ta;
1454 #if defined(PETSC_HAVE_HYPRE)
1455   const char          *algTypes[3] = {"scalable","nonscalable","hypre"};
1456   PetscInt            nalg = 3;
1457 #else
1458   const char          *algTypes[2] = {"scalable","nonscalable"};
1459   PetscInt            nalg = 2;
1460 #endif
1461   PetscInt            alg = 1; /* set default algorithm */
1462 #if defined(PETSC_USE_INFO)
1463   PetscReal           apfill;
1464 #endif
1465   PetscBool           flg;
1466 
1467   PetscFunctionBegin;
1468   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1469   printf("MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable..\n");
1470 #if 0
1471   /* pick an algorithm */
1472   ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
1473   PetscOptionsObject->alreadyprinted = PETSC_FALSE; /* a hack to ensure the option shows in '-help' */
1474   ierr = PetscOptionsEList("-matptap_via","Algorithmic approach","MatPtAP",algTypes,nalg,algTypes[1],&alg,&flg);CHKERRQ(ierr);
1475   ierr = PetscOptionsEnd();CHKERRQ(ierr);
1476 
1477   if (!flg && pN > 100000) { /* may switch to scalable algorithm as default */
1478     MatInfo     Ainfo,Pinfo;
1479     PetscInt    nz_local;
1480     PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
1481 
1482     ierr = MatGetInfo(A,MAT_LOCAL,&Ainfo);CHKERRQ(ierr);
1483     ierr = MatGetInfo(P,MAT_LOCAL,&Pinfo);CHKERRQ(ierr);
1484     nz_local = (PetscInt)(Ainfo.nz_allocated + Pinfo.nz_allocated);
1485 
1486     if (pN > fill*nz_local) alg_scalable_loc = PETSC_TRUE;
1487     ierr = MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm);CHKERRQ(ierr);
1488 
1489     if (alg_scalable) {
1490       alg  = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
1491     }
1492   }
1493 
1494   if (alg == 0) {
1495     ierr = MatPtAPSymbolic_MPIAIJ_MPIAIJ_scalable(A,P,fill,C);CHKERRQ(ierr);
1496     (*C)->ops->ptapnumeric = MatPtAPNumeric_MPIAIJ_MPIAIJ_scalable;
1497     PetscFunctionReturn(0);
1498 
1499 #if defined(PETSC_HAVE_HYPRE)
1500   } else if (alg == 2) {
1501     /* Use boomerAMGBuildCoarseOperator */
1502     ierr = MatPtAPSymbolic_AIJ_AIJ_wHYPRE(A,P,fill,C);CHKERRQ(ierr);
1503     PetscFunctionReturn(0);
1504 #endif
1505   }
1506 
1507 #endif
1508 
1509   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1510   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1511 
1512   /* create symbolic parallel matrix Cmpi */
1513   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1514   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1515 
1516   /* Do dense axpy in MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ() */
1517   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1518 
1519   /* create struct Mat_PtAPMPI and attached it to C later */
1520   ierr        = PetscNew(&ptap);CHKERRQ(ierr);
1521   ptap->reuse = MAT_INITIAL_MATRIX;
1522   ptap->algType = alg;
1523 
1524   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
1525   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
1526   /* get P_loc by taking all local rows of P */
1527   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
1528 
1529   /* (0) compute Rd = Pd^T, Ro = Po^T  */
1530   /* --------------------------------- */
1531   ierr = MatTranspose_SeqAIJ(p->A,MAT_INITIAL_MATRIX,&ptap->Rd);CHKERRQ(ierr);
1532   ierr = MatTranspose_SeqAIJ(p->B,MAT_INITIAL_MATRIX,&ptap->Ro);CHKERRQ(ierr);
1533 
1534   /* (1) compute symbolic A_loc (A_loc = P_loc in ex209!!!) */
1535   /* -------------------------------------------------------*/
1536   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
1537   if (ptap->P_oth) p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
1538 
1539   /* create and initialize a linked list ??? */
1540   ierr = PetscTableCreate(pn,pN,&ta);CHKERRQ(ierr); /* for compute AP_loc and Cmpi */
1541   MatRowMergeMax_SeqAIJ(p_loc,ptap->P_loc->rmap->N,ta);
1542   MatRowMergeMax_SeqAIJ(p_oth,ptap->P_oth->rmap->N,ta);
1543   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr); /* Crmax = nnz(sum of Prows) */
1544   /* printf("[%d] est %d, Crmax %d; pN %d\n",rank,5*(p_loc->rmax+p_oth->rmax + (PetscInt)(1.e-2*pN)),Crmax,pN); */
1545 
1546   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->AP_loc);CHKERRQ(ierr); //ptap->AP_loc = A_loc
1547 
1548   /* (2-1) compute symbolic Co = Ro*A_loc  */
1549   /* ------------------------------------ */
1550   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Ro,ptap->AP_loc,fill,&ptap->C_oth);CHKERRQ(ierr);
1551 
1552   /* (3) send coj of C_oth to other processors  */
1553   /* ------------------------------------------ */
1554   /* determine row ownership */
1555   ierr = PetscLayoutCreate(comm,&rowmap);CHKERRQ(ierr);
1556   rowmap->n  = pn;
1557   rowmap->bs = 1;
1558   ierr   = PetscLayoutSetUp(rowmap);CHKERRQ(ierr);
1559   owners = rowmap->range;
1560 
1561   /* determine the number of messages to send, their lengths */
1562   ierr = PetscMalloc4(size,&len_s,size,&len_si,size,&sstatus,size+2,&owners_co);CHKERRQ(ierr);
1563   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1564   ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1565 
1566   c_oth = (Mat_SeqAIJ*)ptap->C_oth->data;
1567   coi   = c_oth->i; coj = c_oth->j;
1568   con   = ptap->C_oth->rmap->n;
1569   proc  = 0;
1570   for (i=0; i<con; i++) {
1571     while (prmap[i] >= owners[proc+1]) proc++;
1572     len_si[proc]++;               /* num of rows in Co(=Pt*A) to be sent to [proc] */
1573     len_s[proc] += coi[i+1] - coi[i]; /* num of nonzeros in Co to be sent to [proc] */
1574   }
1575 
1576   len          = 0; /* max length of buf_si[], see (4) */
1577   owners_co[0] = 0;
1578   nsend        = 0;
1579   for (proc=0; proc<size; proc++) {
1580     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1581     if (len_s[proc]) {
1582       nsend++;
1583       len_si[proc] = 2*(len_si[proc] + 1); /* length of buf_si to be sent to [proc] */
1584       len         += len_si[proc];
1585     }
1586   }
1587 
1588   /* determine the number and length of messages to receive for coi and coj  */
1589   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&nrecv);CHKERRQ(ierr);
1590   ierr = PetscGatherMessageLengths2(comm,nsend,nrecv,len_s,len_si,&id_r,&len_r,&len_ri);CHKERRQ(ierr);
1591 
1592   /* post the Irecv and Isend of coj */
1593   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1594   ierr = PetscPostIrecvInt(comm,tagj,nrecv,id_r,len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1595   ierr = PetscMalloc1(nsend+1,&swaits);CHKERRQ(ierr);
1596   for (proc=0, k=0; proc<size; proc++) {
1597     if (!len_s[proc]) continue;
1598     i    = owners_co[proc];
1599     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1600     k++;
1601   }
1602 
1603   /* (2-2) compute symbolic C_loc = Rd*A_loc */
1604   /* ---------------------------------------- */
1605   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Rd,ptap->AP_loc,fill,&ptap->C_loc);CHKERRQ(ierr);
1606   c_loc = (Mat_SeqAIJ*)ptap->C_loc->data;
1607 
1608   /* receives coj are complete */
1609   for (i=0; i<nrecv; i++) {
1610     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1611   }
1612   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1613   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
1614 
1615   /* add received column indices into ta to update Crmax */
1616   for (k=0; k<nrecv; k++) {/* k-th received message */
1617     Jptr = buf_rj[k];
1618     for (j=0; j<len_r[k]; j++) {
1619       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
1620     }
1621   }
1622   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
1623   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
1624 
1625   /* (4) send and recv coi */
1626   /*-----------------------*/
1627   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1628   ierr   = PetscPostIrecvInt(comm,tagi,nrecv,id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1629   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1630   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1631   for (proc=0,k=0; proc<size; proc++) {
1632     if (!len_s[proc]) continue;
1633     /* form outgoing message for i-structure:
1634          buf_si[0]:                 nrows to be sent
1635                [1:nrows]:           row index (global)
1636                [nrows+1:2*nrows+1]: i-structure index
1637     */
1638     /*-------------------------------------------*/
1639     nrows       = len_si[proc]/2 - 1; /* num of rows in Co to be sent to [proc] */
1640     buf_si_i    = buf_si + nrows+1;
1641     buf_si[0]   = nrows;
1642     buf_si_i[0] = 0;
1643     nrows       = 0;
1644     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1645       nzi = coi[i+1] - coi[i];
1646       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1647       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1648       nrows++;
1649     }
1650     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1651     k++;
1652     buf_si += len_si[proc];
1653   }
1654   for (i=0; i<nrecv; i++) {
1655     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1656   }
1657   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1658   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
1659 
1660   ierr = PetscFree4(len_s,len_si,sstatus,owners_co);CHKERRQ(ierr);
1661   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1662   ierr = PetscFree(swaits);CHKERRQ(ierr);
1663   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1664 
1665   /* (5) compute the local portion of Cmpi      */
1666   /* ------------------------------------------ */
1667   /* set initial free space to be Crmax, sufficient for holding nozeros in each row of Cmpi */
1668   ierr          = PetscFreeSpaceGet(Crmax,&free_space);CHKERRQ(ierr);
1669   current_space = free_space;
1670 
1671   ierr = PetscMalloc3(nrecv,&buf_ri_k,nrecv,&nextrow,nrecv,&nextci);CHKERRQ(ierr);
1672   for (k=0; k<nrecv; k++) {
1673     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1674     nrows       = *buf_ri_k[k];
1675     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1676     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1677   }
1678 
1679   ierr = MatPreallocateInitialize(comm,pn,pn,dnz,onz);CHKERRQ(ierr);
1680   ierr = PetscLLCondensedCreate(Crmax,pN,&lnk,&lnkbt);CHKERRQ(ierr);
1681   for (i=0; i<pn; i++) {
1682     /* add C_loc into Cmpi */
1683     nzi  = c_loc->i[i+1] - c_loc->i[i];
1684     Jptr = c_loc->j + c_loc->i[i];
1685     ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1686 
1687     /* add received col data into lnk */
1688     for (k=0; k<nrecv; k++) { /* k-th received message */
1689       if (i == *nextrow[k]) { /* i-th row */
1690         nzi  = *(nextci[k]+1) - *nextci[k];
1691         Jptr = buf_rj[k] + *nextci[k];
1692         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1693         nextrow[k]++; nextci[k]++;
1694       }
1695     }
1696     nzi = lnk[0];
1697 
1698     /* copy data into free space, then initialize lnk */
1699     ierr = PetscLLCondensedClean(pN,nzi,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1700     ierr = MatPreallocateSet(i+owners[rank],nzi,current_space->array,dnz,onz);CHKERRQ(ierr);
1701   }
1702   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1703   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1704   ierr = PetscFreeSpaceDestroy(free_space);CHKERRQ(ierr);
1705 
1706   /* local sizes and preallocation */
1707   ierr = MatSetSizes(Cmpi,pn,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1708   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(P->cmap->bs));CHKERRQ(ierr);
1709   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1710   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1711 
1712   /* members in merge */
1713   ierr = PetscFree(id_r);CHKERRQ(ierr);
1714   ierr = PetscFree(len_r);CHKERRQ(ierr);
1715   ierr = PetscFree(buf_ri[0]);CHKERRQ(ierr);
1716   ierr = PetscFree(buf_ri);CHKERRQ(ierr);
1717   ierr = PetscFree(buf_rj[0]);CHKERRQ(ierr);
1718   ierr = PetscFree(buf_rj);CHKERRQ(ierr);
1719   ierr = PetscLayoutDestroy(&rowmap);CHKERRQ(ierr);
1720 
1721   /* attach the supporting struct to Cmpi for reuse */
1722   c = (Mat_MPIAIJ*)Cmpi->data;
1723   c->ptap         = ptap;
1724   ptap->duplicate = Cmpi->ops->duplicate;
1725   ptap->destroy   = Cmpi->ops->destroy;
1726   ptap->view      = Cmpi->ops->view;
1727 
1728   if (alg == 1) {
1729     ierr = PetscCalloc1(pN,&ptap->apa);CHKERRQ(ierr);
1730   }
1731 
1732   /* Cmpi is not ready for use - assembly will be done by MatPtAPNumeric() */
1733   Cmpi->assembled        = PETSC_FALSE;
1734   Cmpi->ops->destroy     = MatDestroy_MPIAIJ_PtAP;
1735   //Cmpi->ops->duplicate   = MatDuplicate_MPIAIJ_MatPtAP;
1736   //Cmpi->ops->view        = MatView_MPIAIJ_PtAP;
1737   *C                     = Cmpi;
1738   PetscFunctionReturn(0);
1739 }
1740 
1741 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,Mat C)
1742 {
1743   PetscErrorCode    ierr;
1744   Mat_MPIAIJ        *a=(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1745   Mat_SeqAIJ        *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
1746   Mat_SeqAIJ        *ap,*p_loc,*p_oth=NULL,*c_seq;
1747   Mat_PtAPMPI       *ptap = c->ptap;
1748   Mat               AP_loc,C_loc,C_oth;
1749   PetscInt          i,rstart,rend,cm,ncols,row;
1750   PetscInt          *api,*apj,am = A->rmap->n,j,col,apnz;
1751   PetscScalar       *apa;
1752   const PetscInt    *cols;
1753   const PetscScalar *vals;
1754 
1755   PetscFunctionBegin;
1756   ierr = MatZeroEntries(C);CHKERRQ(ierr);
1757   printf("MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable...\n");
1758 
1759   /* 1) get R = Pd^T,Ro = Po^T */
1760   if (ptap->reuse == MAT_REUSE_MATRIX) {
1761     ierr = MatTranspose_SeqAIJ(p->A,MAT_REUSE_MATRIX,&ptap->Rd);CHKERRQ(ierr);
1762     ierr = MatTranspose_SeqAIJ(p->B,MAT_REUSE_MATRIX,&ptap->Ro);CHKERRQ(ierr);
1763   }
1764 
1765   /* 2) get AP_loc */
1766   AP_loc = ptap->AP_loc;
1767   ap = (Mat_SeqAIJ*)AP_loc->data;
1768 
1769   /* 2-1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
1770   /*-----------------------------------------------------*/
1771   if (ptap->reuse == MAT_REUSE_MATRIX) {
1772     /* P_oth and P_loc are obtained in MatPtASymbolic() when reuse == MAT_INITIAL_MATRIX */
1773     ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
1774     ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
1775   }
1776 
1777   /* 2-2) compute numeric A_loc*P - dominating part */
1778   /* ---------------------------------------------- */
1779   /* get data from symbolic products */
1780   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
1781   if (ptap->P_oth) {
1782     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
1783   }
1784   apa   = ptap->apa;
1785   api   = ap->i;
1786   apj   = ap->j;
1787 
1788   for (i=0; i<am; i++) {
1789     /* AP[i,:] = A[i,:]*P = Ad*P_loc Ao*P_oth */
1790     AProw_nonscalable(i,ad,ao,p_loc,p_oth,apa);
1791     apnz = api[i+1] - api[i];
1792     for (j=0; j<apnz; j++) {
1793       col = apj[j+api[i]];
1794       ap->a[j+ap->i[i]] = apa[col];
1795       apa[col] = 0.0;
1796     }
1797     ierr = PetscLogFlops(2.0*apnz);CHKERRQ(ierr);
1798   }
1799 
1800   /* 3) C_loc = Rd*AP_loc, C_oth = Ro*AP_loc */
1801   ierr = ((ptap->C_loc)->ops->matmultnumeric)(ptap->Rd,AP_loc,ptap->C_loc);CHKERRQ(ierr);
1802   ierr = ((ptap->C_oth)->ops->matmultnumeric)(ptap->Ro,AP_loc,ptap->C_oth);CHKERRQ(ierr);
1803   C_loc = ptap->C_loc;
1804   C_oth = ptap->C_oth;
1805 
1806   /* add C_loc and Co to to C */
1807   ierr = MatGetOwnershipRange(C,&rstart,&rend);CHKERRQ(ierr);
1808 
1809   /* C_loc -> C */
1810   cm    = C_loc->rmap->N;
1811   c_seq = (Mat_SeqAIJ*)C_loc->data;
1812   cols = c_seq->j;
1813   vals = c_seq->a;
1814   for (i=0; i<cm; i++) {
1815     ncols = c_seq->i[i+1] - c_seq->i[i];
1816     row = rstart + i;
1817     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1818     cols += ncols; vals += ncols;
1819   }
1820 
1821   /* Co -> C, off-processor part */
1822   cm = C_oth->rmap->N;
1823   c_seq = (Mat_SeqAIJ*)C_oth->data;
1824   cols = c_seq->j;
1825   vals = c_seq->a;
1826   for (i=0; i<cm; i++) {
1827     ncols = c_seq->i[i+1] - c_seq->i[i];
1828     row = p->garray[i];
1829     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1830     cols += ncols; vals += ncols;
1831   }
1832   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1833   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1834 
1835   ptap->reuse = MAT_REUSE_MATRIX;
1836   PetscFunctionReturn(0);
1837 }
1838 //=============================
1839 
1840 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ(Mat P,Mat A,Mat C)
1841 {
1842   PetscErrorCode      ierr;
1843   Mat_Merge_SeqsToMPI *merge;
1844   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1845   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
1846   Mat_PtAPMPI         *ptap;
1847   PetscInt            *adj;
1848   PetscInt            i,j,k,anz,pnz,row,*cj,nexta;
1849   MatScalar           *ada,*ca,valtmp;
1850   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
1851   MPI_Comm            comm;
1852   PetscMPIInt         size,rank,taga,*len_s;
1853   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
1854   PetscInt            **buf_ri,**buf_rj;
1855   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
1856   MPI_Request         *s_waits,*r_waits;
1857   MPI_Status          *status;
1858   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
1859   PetscInt            *ai,*aj,*coi,*coj;
1860   PetscInt            *poJ,*pdJ;
1861   Mat                 A_loc;
1862   Mat_SeqAIJ          *a_loc;
1863 
1864   PetscFunctionBegin;
1865   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1866   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1867   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1868 
1869   ptap  = c->ptap;
1870   merge = ptap->merge;
1871 
1872   /* 2) compute numeric C_seq = P_loc^T*A_loc */
1873   /*------------------------------------------*/
1874   /* get data from symbolic products */
1875   coi    = merge->coi; coj = merge->coj;
1876   ierr   = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
1877   bi     = merge->bi; bj = merge->bj;
1878   owners = merge->rowmap->range;
1879   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
1880 
1881   /* get A_loc by taking all local rows of A */
1882   A_loc = ptap->A_loc;
1883   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
1884   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1885   ai    = a_loc->i;
1886   aj    = a_loc->j;
1887 
1888   for (i=0; i<am; i++) {
1889     anz = ai[i+1] - ai[i];
1890     adj = aj + ai[i];
1891     ada = a_loc->a + ai[i];
1892 
1893     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1894     /*-------------------------------------------------------------*/
1895     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1896     pnz = po->i[i+1] - po->i[i];
1897     poJ = po->j + po->i[i];
1898     pA  = po->a + po->i[i];
1899     for (j=0; j<pnz; j++) {
1900       row = poJ[j];
1901       cj  = coj + coi[row];
1902       ca  = coa + coi[row];
1903       /* perform sparse axpy */
1904       nexta  = 0;
1905       valtmp = pA[j];
1906       for (k=0; nexta<anz; k++) {
1907         if (cj[k] == adj[nexta]) {
1908           ca[k] += valtmp*ada[nexta];
1909           nexta++;
1910         }
1911       }
1912       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1913     }
1914 
1915     /* put the value into Cd (diagonal part) */
1916     pnz = pd->i[i+1] - pd->i[i];
1917     pdJ = pd->j + pd->i[i];
1918     pA  = pd->a + pd->i[i];
1919     for (j=0; j<pnz; j++) {
1920       row = pdJ[j];
1921       cj  = bj + bi[row];
1922       ca  = ba + bi[row];
1923       /* perform sparse axpy */
1924       nexta  = 0;
1925       valtmp = pA[j];
1926       for (k=0; nexta<anz; k++) {
1927         if (cj[k] == adj[nexta]) {
1928           ca[k] += valtmp*ada[nexta];
1929           nexta++;
1930         }
1931       }
1932       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1933     }
1934   }
1935 
1936   /* 3) send and recv matrix values coa */
1937   /*------------------------------------*/
1938   buf_ri = merge->buf_ri;
1939   buf_rj = merge->buf_rj;
1940   len_s  = merge->len_s;
1941   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1942   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1943 
1944   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1945   for (proc=0,k=0; proc<size; proc++) {
1946     if (!len_s[proc]) continue;
1947     i    = merge->owners_co[proc];
1948     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1949     k++;
1950   }
1951   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1952   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1953 
1954   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1955   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1956   ierr = PetscFree(coa);CHKERRQ(ierr);
1957 
1958   /* 4) insert local Cseq and received values into Cmpi */
1959   /*----------------------------------------------------*/
1960   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1961   for (k=0; k<merge->nrecv; k++) {
1962     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1963     nrows       = *(buf_ri_k[k]);
1964     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1965     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1966   }
1967 
1968   for (i=0; i<cm; i++) {
1969     row  = owners[rank] + i; /* global row index of C_seq */
1970     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1971     ba_i = ba + bi[i];
1972     bnz  = bi[i+1] - bi[i];
1973     /* add received vals into ba */
1974     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1975       /* i-th row */
1976       if (i == *nextrow[k]) {
1977         cnz    = *(nextci[k]+1) - *nextci[k];
1978         cj     = buf_rj[k] + *(nextci[k]);
1979         ca     = abuf_r[k] + *(nextci[k]);
1980         nextcj = 0;
1981         for (j=0; nextcj<cnz; j++) {
1982           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1983             ba_i[j] += ca[nextcj++];
1984           }
1985         }
1986         nextrow[k]++; nextci[k]++;
1987         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1988       }
1989     }
1990     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1991   }
1992   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1993   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1994 
1995   ierr = PetscFree(ba);CHKERRQ(ierr);
1996   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1997   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1998   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1999   PetscFunctionReturn(0);
2000 }
2001 
2002 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ();
2003    differ from MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable in using LLCondensedCreate_Scalable() */
2004 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P,Mat A,PetscReal fill,Mat *C)
2005 {
2006   PetscErrorCode      ierr;
2007   Mat                 Cmpi,A_loc,POt,PDt;
2008   Mat_PtAPMPI         *ptap;
2009   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
2010   Mat_MPIAIJ          *p=(Mat_MPIAIJ*)P->data,*a=(Mat_MPIAIJ*)A->data,*c;
2011   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
2012   PetscInt            nnz;
2013   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
2014   PetscInt            am  =A->rmap->n,pn=P->cmap->n;
2015   MPI_Comm            comm;
2016   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
2017   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
2018   PetscInt            len,proc,*dnz,*onz,*owners;
2019   PetscInt            nzi,*bi,*bj;
2020   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
2021   MPI_Request         *swaits,*rwaits;
2022   MPI_Status          *sstatus,rstatus;
2023   Mat_Merge_SeqsToMPI *merge;
2024   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
2025   PetscReal           afill  =1.0,afill_tmp;
2026   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Armax;
2027   PetscScalar         *vals;
2028   Mat_SeqAIJ          *a_loc,*pdt,*pot;
2029   PetscTable          ta;
2030 
2031   PetscFunctionBegin;
2032   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
2033   /* check if matrix local sizes are compatible */
2034   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);
2035 
2036   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
2037   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
2038 
2039   /* create struct Mat_PtAPMPI and attached it to C later */
2040   ierr = PetscNew(&ptap);CHKERRQ(ierr);
2041 
2042   /* get A_loc by taking all local rows of A */
2043   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
2044 
2045   ptap->A_loc = A_loc;
2046   a_loc       = (Mat_SeqAIJ*)(A_loc)->data;
2047   ai          = a_loc->i;
2048   aj          = a_loc->j;
2049 
2050   /* determine symbolic Co=(p->B)^T*A - send to others */
2051   /*----------------------------------------------------*/
2052   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
2053   pdt  = (Mat_SeqAIJ*)PDt->data;
2054   pdti = pdt->i; pdtj = pdt->j;
2055 
2056   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
2057   pot  = (Mat_SeqAIJ*)POt->data;
2058   poti = pot->i; potj = pot->j;
2059 
2060   /* then, compute symbolic Co = (p->B)^T*A */
2061   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors
2062                          >= (num of nonzero rows of C_seq) - pn */
2063   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
2064   coi[0] = 0;
2065 
2066   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
2067   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(poti[pon],ai[am]));
2068   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
2069   current_space = free_space;
2070 
2071   /* create and initialize a linked list */
2072   ierr = PetscTableCreate(A->cmap->n + a->B->cmap->N,aN,&ta);CHKERRQ(ierr);
2073   MatRowMergeMax_SeqAIJ(a_loc,am,ta);
2074   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
2075 
2076   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
2077 
2078   for (i=0; i<pon; i++) {
2079     pnz = poti[i+1] - poti[i];
2080     ptJ = potj + poti[i];
2081     for (j=0; j<pnz; j++) {
2082       row  = ptJ[j]; /* row of A_loc == col of Pot */
2083       anz  = ai[row+1] - ai[row];
2084       Jptr = aj + ai[row];
2085       /* add non-zero cols of AP into the sorted linked list lnk */
2086       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
2087     }
2088     nnz = lnk[0];
2089 
2090     /* If free space is not available, double the total space in the list */
2091     if (current_space->local_remaining<nnz) {
2092       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
2093       nspacedouble++;
2094     }
2095 
2096     /* Copy data into free space, and zero out denserows */
2097     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
2098 
2099     current_space->array           += nnz;
2100     current_space->local_used      += nnz;
2101     current_space->local_remaining -= nnz;
2102 
2103     coi[i+1] = coi[i] + nnz;
2104   }
2105 
2106   ierr = PetscMalloc1(coi[pon]+1,&coj);CHKERRQ(ierr);
2107   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
2108   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr); /* must destroy to get a new one for C */
2109 
2110   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
2111   if (afill_tmp > afill) afill = afill_tmp;
2112 
2113   /* send j-array (coj) of Co to other processors */
2114   /*----------------------------------------------*/
2115   /* determine row ownership */
2116   ierr = PetscNew(&merge);CHKERRQ(ierr);
2117   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
2118 
2119   merge->rowmap->n  = pn;
2120   merge->rowmap->bs = 1;
2121 
2122   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
2123   owners = merge->rowmap->range;
2124 
2125   /* determine the number of messages to send, their lengths */
2126   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
2127   ierr = PetscMalloc1(size,&merge->len_s);CHKERRQ(ierr);
2128 
2129   len_s        = merge->len_s;
2130   merge->nsend = 0;
2131 
2132   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
2133   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
2134 
2135   proc = 0;
2136   for (i=0; i<pon; i++) {
2137     while (prmap[i] >= owners[proc+1]) proc++;
2138     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
2139     len_s[proc] += coi[i+1] - coi[i];
2140   }
2141 
2142   len          = 0; /* max length of buf_si[] */
2143   owners_co[0] = 0;
2144   for (proc=0; proc<size; proc++) {
2145     owners_co[proc+1] = owners_co[proc] + len_si[proc];
2146     if (len_si[proc]) {
2147       merge->nsend++;
2148       len_si[proc] = 2*(len_si[proc] + 1);
2149       len         += len_si[proc];
2150     }
2151   }
2152 
2153   /* determine the number and length of messages to receive for coi and coj  */
2154   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
2155   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
2156 
2157   /* post the Irecv and Isend of coj */
2158   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
2159   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
2160   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
2161   for (proc=0, k=0; proc<size; proc++) {
2162     if (!len_s[proc]) continue;
2163     i    = owners_co[proc];
2164     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
2165     k++;
2166   }
2167 
2168   /* receives and sends of coj are complete */
2169   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
2170   for (i=0; i<merge->nrecv; i++) {
2171     PetscMPIInt icompleted;
2172     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
2173   }
2174   ierr = PetscFree(rwaits);CHKERRQ(ierr);
2175   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
2176 
2177   /* add received column indices into table to update Armax */
2178   /* Armax can be as large as aN if a P[row,:] is dense, see src/ksp/ksp/examples/tutorials/ex56.c! */
2179   for (k=0; k<merge->nrecv; k++) {/* k-th received message */
2180     Jptr = buf_rj[k];
2181     for (j=0; j<merge->len_r[k]; j++) {
2182       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
2183     }
2184   }
2185   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
2186   /* 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); */
2187 
2188   /* send and recv coi */
2189   /*-------------------*/
2190   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
2191   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
2192   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
2193   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
2194   for (proc=0,k=0; proc<size; proc++) {
2195     if (!len_s[proc]) continue;
2196     /* form outgoing message for i-structure:
2197          buf_si[0]:                 nrows to be sent
2198                [1:nrows]:           row index (global)
2199                [nrows+1:2*nrows+1]: i-structure index
2200     */
2201     /*-------------------------------------------*/
2202     nrows       = len_si[proc]/2 - 1;
2203     buf_si_i    = buf_si + nrows+1;
2204     buf_si[0]   = nrows;
2205     buf_si_i[0] = 0;
2206     nrows       = 0;
2207     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
2208       nzi               = coi[i+1] - coi[i];
2209       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
2210       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
2211       nrows++;
2212     }
2213     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
2214     k++;
2215     buf_si += len_si[proc];
2216   }
2217   i = merge->nrecv;
2218   while (i--) {
2219     PetscMPIInt icompleted;
2220     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
2221   }
2222   ierr = PetscFree(rwaits);CHKERRQ(ierr);
2223   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
2224   ierr = PetscFree(len_si);CHKERRQ(ierr);
2225   ierr = PetscFree(len_ri);CHKERRQ(ierr);
2226   ierr = PetscFree(swaits);CHKERRQ(ierr);
2227   ierr = PetscFree(sstatus);CHKERRQ(ierr);
2228   ierr = PetscFree(buf_s);CHKERRQ(ierr);
2229 
2230   /* compute the local portion of C (mpi mat) */
2231   /*------------------------------------------*/
2232   /* allocate bi array and free space for accumulating nonzero column info */
2233   ierr  = PetscMalloc1(pn+1,&bi);CHKERRQ(ierr);
2234   bi[0] = 0;
2235 
2236   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
2237   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(pdti[pn],PetscIntSumTruncate(poti[pon],ai[am])));
2238   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
2239   current_space = free_space;
2240 
2241   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
2242   for (k=0; k<merge->nrecv; k++) {
2243     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
2244     nrows       = *buf_ri_k[k];
2245     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
2246     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th recieved i-structure  */
2247   }
2248 
2249   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
2250   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
2251   rmax = 0;
2252   for (i=0; i<pn; i++) {
2253     /* add pdt[i,:]*AP into lnk */
2254     pnz = pdti[i+1] - pdti[i];
2255     ptJ = pdtj + pdti[i];
2256     for (j=0; j<pnz; j++) {
2257       row  = ptJ[j];  /* row of AP == col of Pt */
2258       anz  = ai[row+1] - ai[row];
2259       Jptr = aj + ai[row];
2260       /* add non-zero cols of AP into the sorted linked list lnk */
2261       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
2262     }
2263 
2264     /* add received col data into lnk */
2265     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
2266       if (i == *nextrow[k]) { /* i-th row */
2267         nzi  = *(nextci[k]+1) - *nextci[k];
2268         Jptr = buf_rj[k] + *nextci[k];
2269         ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
2270         nextrow[k]++; nextci[k]++;
2271       }
2272     }
2273     nnz = lnk[0];
2274 
2275     /* if free space is not available, make more free space */
2276     if (current_space->local_remaining<nnz) {
2277       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
2278       nspacedouble++;
2279     }
2280     /* copy data into free space, then initialize lnk */
2281     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
2282     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
2283 
2284     current_space->array           += nnz;
2285     current_space->local_used      += nnz;
2286     current_space->local_remaining -= nnz;
2287 
2288     bi[i+1] = bi[i] + nnz;
2289     if (nnz > rmax) rmax = nnz;
2290   }
2291   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
2292 
2293   ierr      = PetscMalloc1(bi[pn]+1,&bj);CHKERRQ(ierr);
2294   ierr      = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
2295   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
2296   if (afill_tmp > afill) afill = afill_tmp;
2297   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
2298   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
2299 
2300   ierr = MatDestroy(&POt);CHKERRQ(ierr);
2301   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
2302 
2303   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
2304   /*----------------------------------------------------------------------------------*/
2305   ierr = PetscCalloc1(rmax+1,&vals);CHKERRQ(ierr);
2306 
2307   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
2308   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
2309   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
2310   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
2311   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
2312   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
2313   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
2314   for (i=0; i<pn; i++) {
2315     row  = i + rstart;
2316     nnz  = bi[i+1] - bi[i];
2317     Jptr = bj + bi[i];
2318     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
2319   }
2320   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2321   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2322   ierr = PetscFree(vals);CHKERRQ(ierr);
2323 
2324   merge->bi        = bi;
2325   merge->bj        = bj;
2326   merge->coi       = coi;
2327   merge->coj       = coj;
2328   merge->buf_ri    = buf_ri;
2329   merge->buf_rj    = buf_rj;
2330   merge->owners_co = owners_co;
2331 
2332   /* attach the supporting struct to Cmpi for reuse */
2333   c = (Mat_MPIAIJ*)Cmpi->data;
2334 
2335   c->ptap     = ptap;
2336   ptap->api   = NULL;
2337   ptap->apj   = NULL;
2338   ptap->merge = merge;
2339   ptap->apa   = NULL;
2340   ptap->destroy   = Cmpi->ops->destroy;
2341   ptap->duplicate = Cmpi->ops->duplicate;
2342 
2343   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ;
2344   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
2345   Cmpi->ops->duplicate               = MatDuplicate_MPIAIJ_MatPtAP;
2346 
2347   *C = Cmpi;
2348 #if defined(PETSC_USE_INFO)
2349   if (bi[pn] != 0) {
2350     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
2351     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
2352   } else {
2353     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
2354   }
2355 #endif
2356   PetscFunctionReturn(0);
2357 }
2358