xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision 912837bb8b09b4e0c9633a5866494e6b9a8a97a4)
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          *a=(Mat_MPIAIJ*)A->data,*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            am=A->rmap->n,pm=P->rmap->n,pN=P->cmap->N,pn=P->cmap->n;
1440   PetscInt            *lnk,i,k,pnz,row,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,nspacedouble;
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            *api,*apj,*Jptr,apnz,*prmap=p->garray,con,j,ap_rmax=0,Crmax,*aj,*ai,*pi;
1452   Mat_SeqAIJ          *p_loc,*p_oth=NULL,*ad=(Mat_SeqAIJ*)(a->A)->data,*ao=NULL,*c_loc,*c_oth;
1453   PetscScalar         *apv;
1454   PetscTable          ta;
1455 #if defined(PETSC_HAVE_HYPRE)
1456   const char          *algTypes[3] = {"scalable","nonscalable","hypre"};
1457   PetscInt            nalg = 3;
1458 #else
1459   const char          *algTypes[2] = {"scalable","nonscalable"};
1460   PetscInt            nalg = 2;
1461 #endif
1462   PetscInt            alg = 1; /* set default algorithm */
1463 #if defined(PETSC_USE_INFO)
1464   PetscReal           apfill;
1465 #endif
1466   PetscBool           flg;
1467 
1468   PetscFunctionBegin;
1469   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1470   printf("MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable..\n");
1471 #if 0
1472   /* pick an algorithm */
1473   ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
1474   PetscOptionsObject->alreadyprinted = PETSC_FALSE; /* a hack to ensure the option shows in '-help' */
1475   ierr = PetscOptionsEList("-matptap_via","Algorithmic approach","MatPtAP",algTypes,nalg,algTypes[1],&alg,&flg);CHKERRQ(ierr);
1476   ierr = PetscOptionsEnd();CHKERRQ(ierr);
1477 
1478   if (!flg && pN > 100000) { /* may switch to scalable algorithm as default */
1479     MatInfo     Ainfo,Pinfo;
1480     PetscInt    nz_local;
1481     PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
1482 
1483     ierr = MatGetInfo(A,MAT_LOCAL,&Ainfo);CHKERRQ(ierr);
1484     ierr = MatGetInfo(P,MAT_LOCAL,&Pinfo);CHKERRQ(ierr);
1485     nz_local = (PetscInt)(Ainfo.nz_allocated + Pinfo.nz_allocated);
1486 
1487     if (pN > fill*nz_local) alg_scalable_loc = PETSC_TRUE;
1488     ierr = MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm);CHKERRQ(ierr);
1489 
1490     if (alg_scalable) {
1491       alg  = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
1492     }
1493   }
1494 
1495   if (alg == 0) {
1496     ierr = MatPtAPSymbolic_MPIAIJ_MPIAIJ_scalable(A,P,fill,C);CHKERRQ(ierr);
1497     (*C)->ops->ptapnumeric = MatPtAPNumeric_MPIAIJ_MPIAIJ_scalable;
1498     PetscFunctionReturn(0);
1499 
1500 #if defined(PETSC_HAVE_HYPRE)
1501   } else if (alg == 2) {
1502     /* Use boomerAMGBuildCoarseOperator */
1503     ierr = MatPtAPSymbolic_AIJ_AIJ_wHYPRE(A,P,fill,C);CHKERRQ(ierr);
1504     PetscFunctionReturn(0);
1505 #endif
1506   }
1507 
1508 #endif
1509 
1510   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1511   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1512 
1513   if (size > 1) ao = (Mat_SeqAIJ*)(a->B)->data;
1514 
1515   /* create symbolic parallel matrix Cmpi */
1516   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1517   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1518 
1519   /* Do dense axpy in MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ() */
1520   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1521 
1522   /* create struct Mat_PtAPMPI and attached it to C later */
1523   ierr        = PetscNew(&ptap);CHKERRQ(ierr);
1524   ptap->reuse = MAT_INITIAL_MATRIX;
1525   ptap->algType = alg;
1526 
1527   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
1528   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
1529   /* get P_loc by taking all local rows of P */
1530   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
1531 
1532   /* (0) compute Rd = Pd^T, Ro = Po^T  */
1533   /* --------------------------------- */
1534   ierr = MatTranspose_SeqAIJ(p->A,MAT_INITIAL_MATRIX,&ptap->Rd);CHKERRQ(ierr);
1535   ierr = MatTranspose_SeqAIJ(p->B,MAT_INITIAL_MATRIX,&ptap->Ro);CHKERRQ(ierr);
1536 
1537   /* (1) compute symbolic A_loc [AP = A_loc*P = Ad*P_loc + Ao*P_oth (api,apj)] */
1538   /* ----------------------------------------------------------------- */
1539   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
1540   if (ptap->P_oth) p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
1541 
1542   /* create and initialize a linked list */
1543   ierr = PetscTableCreate(pn,pN,&ta);CHKERRQ(ierr); /* for compute AP_loc and Cmpi */
1544   MatRowMergeMax_SeqAIJ(p_loc,ptap->P_loc->rmap->N,ta);
1545   MatRowMergeMax_SeqAIJ(p_oth,ptap->P_oth->rmap->N,ta);
1546   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr); /* Crmax = nnz(sum of Prows) */
1547   /* printf("[%d] est %d, Crmax %d; pN %d\n",rank,5*(p_loc->rmax+p_oth->rmax + (PetscInt)(1.e-2*pN)),Crmax,pN); */
1548 
1549   ierr = PetscLLCondensedCreate(Crmax,pN,&lnk,&lnkbt);CHKERRQ(ierr);
1550 
1551   /* Initial FreeSpace size is fill*(nnz(A) + nnz(P)) */
1552   if (ao) {
1553     ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ad->i[am],PetscIntSumTruncate(ao->i[am],p_loc->i[pm]))),&free_space);CHKERRQ(ierr);
1554   } else {
1555     ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ad->i[am],p_loc->i[pm])),&free_space);CHKERRQ(ierr);
1556   }
1557   current_space = free_space;
1558   nspacedouble  = 0;
1559 
1560   ierr   = PetscMalloc1(am+1,&api);CHKERRQ(ierr);
1561   api[0] = 0;
1562   for (i=0; i<am; i++) {
1563     /* diagonal portion: Ad[i,:]*P */
1564     ai = ad->i; pi = p_loc->i;
1565     nzi = ai[i+1] - ai[i];
1566     aj  = ad->j + ai[i];
1567     for (j=0; j<nzi; j++) {
1568       row  = aj[j];
1569       pnz  = pi[row+1] - pi[row];
1570       Jptr = p_loc->j + pi[row];
1571       /* add non-zero cols of P into the sorted linked list lnk */
1572       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1573     }
1574     /* off-diagonal portion: Ao[i,:]*P */
1575     if (ao) {
1576       ai = ao->i; pi = p_oth->i;
1577       nzi = ai[i+1] - ai[i];
1578       aj  = ao->j + ai[i];
1579       for (j=0; j<nzi; j++) {
1580         row  = aj[j];
1581         pnz  = pi[row+1] - pi[row];
1582         Jptr = p_oth->j + pi[row];
1583         ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1584       }
1585     }
1586     apnz     = lnk[0];
1587     api[i+1] = api[i] + apnz;
1588     if (ap_rmax < apnz) ap_rmax = apnz;
1589 
1590     /* if free space is not available, double the total space in the list */
1591     if (current_space->local_remaining<apnz) {
1592       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1593       nspacedouble++;
1594     }
1595 
1596     /* Copy data into free space, then initialize lnk */
1597     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1598 
1599     current_space->array           += apnz;
1600     current_space->local_used      += apnz;
1601     current_space->local_remaining -= apnz;
1602   }
1603   /* Allocate space for apj and apv, initialize apj, and */
1604   /* destroy list of free space and other temporary array(s) */
1605   ierr   = PetscMalloc2(api[am],&apj,api[am],&apv);CHKERRQ(ierr);
1606   ierr   = PetscFreeSpaceContiguous(&free_space,apj);CHKERRQ(ierr);
1607   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1608 
1609   /* Create AP_loc for reuse */
1610   ierr = MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,am,pN,api,apj,apv,&ptap->AP_loc);CHKERRQ(ierr);
1611 
1612 #if defined(PETSC_USE_INFO)
1613   if (ao) {
1614     apfill = (PetscReal)api[am]/(ad->i[am]+ao->i[am]+p_loc->i[pm]+1);
1615   } else {
1616     apfill = (PetscReal)api[am]/(ad->i[am]+p_loc->i[pm]+1);
1617   }
1618   ptap->AP_loc->info.mallocs           = nspacedouble;
1619   ptap->AP_loc->info.fill_ratio_given  = fill;
1620   ptap->AP_loc->info.fill_ratio_needed = apfill;
1621 
1622   if (api[am]) {
1623     ierr = PetscInfo3(ptap->AP_loc,"Nonscalable algorithm, AP_loc reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)apfill);CHKERRQ(ierr);
1624     ierr = PetscInfo1(ptap->AP_loc,"Use MatPtAP(A,B,MatReuse,%g,&C) for best AP_loc performance.;\n",(double)apfill);CHKERRQ(ierr);
1625   } else {
1626     ierr = PetscInfo(ptap->AP_loc,"Nonscalable algorithm, AP_loc is empty \n");CHKERRQ(ierr);
1627   }
1628 #endif
1629 
1630   /* (2-1) compute symbolic Co = Ro*AP_loc  */
1631   /* ------------------------------------ */
1632   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Ro,ptap->AP_loc,fill,&ptap->C_oth);CHKERRQ(ierr);
1633 #if defined(PTAP_PROFILE)
1634   ierr = PetscTime(&t1);CHKERRQ(ierr);
1635 #endif
1636 
1637   /* (3) send coj of C_oth to other processors  */
1638   /* ------------------------------------------ */
1639   /* determine row ownership */
1640   ierr = PetscLayoutCreate(comm,&rowmap);CHKERRQ(ierr);
1641   rowmap->n  = pn;
1642   rowmap->bs = 1;
1643   ierr   = PetscLayoutSetUp(rowmap);CHKERRQ(ierr);
1644   owners = rowmap->range;
1645 
1646   /* determine the number of messages to send, their lengths */
1647   ierr = PetscMalloc4(size,&len_s,size,&len_si,size,&sstatus,size+2,&owners_co);CHKERRQ(ierr);
1648   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1649   ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1650 
1651   c_oth = (Mat_SeqAIJ*)ptap->C_oth->data;
1652   coi   = c_oth->i; coj = c_oth->j;
1653   con   = ptap->C_oth->rmap->n;
1654   proc  = 0;
1655   for (i=0; i<con; i++) {
1656     while (prmap[i] >= owners[proc+1]) proc++;
1657     len_si[proc]++;               /* num of rows in Co(=Pt*AP) to be sent to [proc] */
1658     len_s[proc] += coi[i+1] - coi[i]; /* num of nonzeros in Co to be sent to [proc] */
1659   }
1660 
1661   len          = 0; /* max length of buf_si[], see (4) */
1662   owners_co[0] = 0;
1663   nsend        = 0;
1664   for (proc=0; proc<size; proc++) {
1665     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1666     if (len_s[proc]) {
1667       nsend++;
1668       len_si[proc] = 2*(len_si[proc] + 1); /* length of buf_si to be sent to [proc] */
1669       len         += len_si[proc];
1670     }
1671   }
1672 
1673   /* determine the number and length of messages to receive for coi and coj  */
1674   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&nrecv);CHKERRQ(ierr);
1675   ierr = PetscGatherMessageLengths2(comm,nsend,nrecv,len_s,len_si,&id_r,&len_r,&len_ri);CHKERRQ(ierr);
1676 
1677   /* post the Irecv and Isend of coj */
1678   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1679   ierr = PetscPostIrecvInt(comm,tagj,nrecv,id_r,len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1680   ierr = PetscMalloc1(nsend+1,&swaits);CHKERRQ(ierr);
1681   for (proc=0, k=0; proc<size; proc++) {
1682     if (!len_s[proc]) continue;
1683     i    = owners_co[proc];
1684     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1685     k++;
1686   }
1687 
1688   /* (2-2) compute symbolic C_loc = Rd*AP_loc */
1689   /* ---------------------------------------- */
1690   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Rd,ptap->AP_loc,fill,&ptap->C_loc);CHKERRQ(ierr);
1691   c_loc = (Mat_SeqAIJ*)ptap->C_loc->data;
1692 
1693   /* receives coj are complete */
1694   for (i=0; i<nrecv; i++) {
1695     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1696   }
1697   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1698   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
1699 
1700   /* add received column indices into ta to update Crmax */
1701   for (k=0; k<nrecv; k++) {/* k-th received message */
1702     Jptr = buf_rj[k];
1703     for (j=0; j<len_r[k]; j++) {
1704       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
1705     }
1706   }
1707   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
1708   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
1709 
1710   /* (4) send and recv coi */
1711   /*-----------------------*/
1712   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1713   ierr   = PetscPostIrecvInt(comm,tagi,nrecv,id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1714   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1715   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1716   for (proc=0,k=0; proc<size; proc++) {
1717     if (!len_s[proc]) continue;
1718     /* form outgoing message for i-structure:
1719          buf_si[0]:                 nrows to be sent
1720                [1:nrows]:           row index (global)
1721                [nrows+1:2*nrows+1]: i-structure index
1722     */
1723     /*-------------------------------------------*/
1724     nrows       = len_si[proc]/2 - 1; /* num of rows in Co to be sent to [proc] */
1725     buf_si_i    = buf_si + nrows+1;
1726     buf_si[0]   = nrows;
1727     buf_si_i[0] = 0;
1728     nrows       = 0;
1729     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1730       nzi = coi[i+1] - coi[i];
1731       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1732       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1733       nrows++;
1734     }
1735     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1736     k++;
1737     buf_si += len_si[proc];
1738   }
1739   for (i=0; i<nrecv; i++) {
1740     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1741   }
1742   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1743   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
1744 
1745   ierr = PetscFree4(len_s,len_si,sstatus,owners_co);CHKERRQ(ierr);
1746   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1747   ierr = PetscFree(swaits);CHKERRQ(ierr);
1748   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1749 
1750   /* (5) compute the local portion of Cmpi      */
1751   /* ------------------------------------------ */
1752   /* set initial free space to be Crmax, sufficient for holding nozeros in each row of Cmpi */
1753   ierr          = PetscFreeSpaceGet(Crmax,&free_space);CHKERRQ(ierr);
1754   current_space = free_space;
1755 
1756   ierr = PetscMalloc3(nrecv,&buf_ri_k,nrecv,&nextrow,nrecv,&nextci);CHKERRQ(ierr);
1757   for (k=0; k<nrecv; k++) {
1758     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1759     nrows       = *buf_ri_k[k];
1760     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1761     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1762   }
1763 
1764   ierr = MatPreallocateInitialize(comm,pn,pn,dnz,onz);CHKERRQ(ierr);
1765   ierr = PetscLLCondensedCreate(Crmax,pN,&lnk,&lnkbt);CHKERRQ(ierr);
1766   for (i=0; i<pn; i++) {
1767     /* add C_loc into Cmpi */
1768     nzi  = c_loc->i[i+1] - c_loc->i[i];
1769     Jptr = c_loc->j + c_loc->i[i];
1770     ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1771 
1772     /* add received col data into lnk */
1773     for (k=0; k<nrecv; k++) { /* k-th received message */
1774       if (i == *nextrow[k]) { /* i-th row */
1775         nzi  = *(nextci[k]+1) - *nextci[k];
1776         Jptr = buf_rj[k] + *nextci[k];
1777         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1778         nextrow[k]++; nextci[k]++;
1779       }
1780     }
1781     nzi = lnk[0];
1782 
1783     /* copy data into free space, then initialize lnk */
1784     ierr = PetscLLCondensedClean(pN,nzi,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1785     ierr = MatPreallocateSet(i+owners[rank],nzi,current_space->array,dnz,onz);CHKERRQ(ierr);
1786   }
1787   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1788   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1789   ierr = PetscFreeSpaceDestroy(free_space);CHKERRQ(ierr);
1790 
1791   /* local sizes and preallocation */
1792   ierr = MatSetSizes(Cmpi,pn,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1793   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(P->cmap->bs));CHKERRQ(ierr);
1794   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1795   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1796 
1797   /* members in merge */
1798   ierr = PetscFree(id_r);CHKERRQ(ierr);
1799   ierr = PetscFree(len_r);CHKERRQ(ierr);
1800   ierr = PetscFree(buf_ri[0]);CHKERRQ(ierr);
1801   ierr = PetscFree(buf_ri);CHKERRQ(ierr);
1802   ierr = PetscFree(buf_rj[0]);CHKERRQ(ierr);
1803   ierr = PetscFree(buf_rj);CHKERRQ(ierr);
1804   ierr = PetscLayoutDestroy(&rowmap);CHKERRQ(ierr);
1805 
1806   /* attach the supporting struct to Cmpi for reuse */
1807   c = (Mat_MPIAIJ*)Cmpi->data;
1808   c->ptap         = ptap;
1809   ptap->duplicate = Cmpi->ops->duplicate;
1810   ptap->destroy   = Cmpi->ops->destroy;
1811   ptap->view      = Cmpi->ops->view;
1812 
1813   if (alg == 1) {
1814     ierr = PetscCalloc1(pN,&ptap->apa);CHKERRQ(ierr);
1815   }
1816 
1817   /* Cmpi is not ready for use - assembly will be done by MatPtAPNumeric() */
1818   Cmpi->assembled        = PETSC_FALSE;
1819   Cmpi->ops->destroy     = MatDestroy_MPIAIJ_PtAP;
1820   //Cmpi->ops->duplicate   = MatDuplicate_MPIAIJ_MatPtAP;
1821   //Cmpi->ops->view        = MatView_MPIAIJ_PtAP;
1822   *C                     = Cmpi;
1823   PetscFunctionReturn(0);
1824 }
1825 
1826 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,Mat C)
1827 {
1828   PetscErrorCode    ierr;
1829   Mat_MPIAIJ        *a=(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1830   Mat_SeqAIJ        *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
1831   Mat_SeqAIJ        *ap,*p_loc,*p_oth=NULL,*c_seq;
1832   Mat_PtAPMPI       *ptap = c->ptap;
1833   Mat               AP_loc,C_loc,C_oth;
1834   PetscInt          i,rstart,rend,cm,ncols,row;
1835   PetscInt          *api,*apj,am = A->rmap->n,j,col,apnz;
1836   PetscScalar       *apa;
1837   const PetscInt    *cols;
1838   const PetscScalar *vals;
1839 
1840   PetscFunctionBegin;
1841   ierr = MatZeroEntries(C);CHKERRQ(ierr);
1842   printf("MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable...\n");
1843 
1844   /* 1) get R = Pd^T,Ro = Po^T */
1845   if (ptap->reuse == MAT_REUSE_MATRIX) {
1846     ierr = MatTranspose_SeqAIJ(p->A,MAT_REUSE_MATRIX,&ptap->Rd);CHKERRQ(ierr);
1847     ierr = MatTranspose_SeqAIJ(p->B,MAT_REUSE_MATRIX,&ptap->Ro);CHKERRQ(ierr);
1848   }
1849 
1850   /* 2) get AP_loc */
1851   AP_loc = ptap->AP_loc;
1852   ap = (Mat_SeqAIJ*)AP_loc->data;
1853 
1854   /* 2-1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
1855   /*-----------------------------------------------------*/
1856   if (ptap->reuse == MAT_REUSE_MATRIX) {
1857     /* P_oth and P_loc are obtained in MatPtASymbolic() when reuse == MAT_INITIAL_MATRIX */
1858     ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
1859     ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
1860   }
1861 
1862   /* 2-2) compute numeric A_loc*P - dominating part */
1863   /* ---------------------------------------------- */
1864   /* get data from symbolic products */
1865   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
1866   if (ptap->P_oth) {
1867     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
1868   }
1869   apa   = ptap->apa;
1870   api   = ap->i;
1871   apj   = ap->j;
1872 
1873   for (i=0; i<am; i++) {
1874     /* AP[i,:] = A[i,:]*P = Ad*P_loc Ao*P_oth */
1875     AProw_nonscalable(i,ad,ao,p_loc,p_oth,apa);
1876     apnz = api[i+1] - api[i];
1877     for (j=0; j<apnz; j++) {
1878       col = apj[j+api[i]];
1879       ap->a[j+ap->i[i]] = apa[col];
1880       apa[col] = 0.0;
1881     }
1882     ierr = PetscLogFlops(2.0*apnz);CHKERRQ(ierr);
1883   }
1884 
1885   /* 3) C_loc = Rd*AP_loc, C_oth = Ro*AP_loc */
1886   ierr = ((ptap->C_loc)->ops->matmultnumeric)(ptap->Rd,AP_loc,ptap->C_loc);CHKERRQ(ierr);
1887   ierr = ((ptap->C_oth)->ops->matmultnumeric)(ptap->Ro,AP_loc,ptap->C_oth);CHKERRQ(ierr);
1888   C_loc = ptap->C_loc;
1889   C_oth = ptap->C_oth;
1890 
1891   /* add C_loc and Co to to C */
1892   ierr = MatGetOwnershipRange(C,&rstart,&rend);CHKERRQ(ierr);
1893 
1894   /* C_loc -> C */
1895   cm    = C_loc->rmap->N;
1896   c_seq = (Mat_SeqAIJ*)C_loc->data;
1897   cols = c_seq->j;
1898   vals = c_seq->a;
1899   for (i=0; i<cm; i++) {
1900     ncols = c_seq->i[i+1] - c_seq->i[i];
1901     row = rstart + i;
1902     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1903     cols += ncols; vals += ncols;
1904   }
1905 
1906   /* Co -> C, off-processor part */
1907   cm = C_oth->rmap->N;
1908   c_seq = (Mat_SeqAIJ*)C_oth->data;
1909   cols = c_seq->j;
1910   vals = c_seq->a;
1911   for (i=0; i<cm; i++) {
1912     ncols = c_seq->i[i+1] - c_seq->i[i];
1913     row = p->garray[i];
1914     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1915     cols += ncols; vals += ncols;
1916   }
1917   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1918   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1919 
1920   ptap->reuse = MAT_REUSE_MATRIX;
1921   PetscFunctionReturn(0);
1922 }
1923 //=============================
1924 
1925 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ(Mat P,Mat A,Mat C)
1926 {
1927   PetscErrorCode      ierr;
1928   Mat_Merge_SeqsToMPI *merge;
1929   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1930   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
1931   Mat_PtAPMPI         *ptap;
1932   PetscInt            *adj;
1933   PetscInt            i,j,k,anz,pnz,row,*cj,nexta;
1934   MatScalar           *ada,*ca,valtmp;
1935   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
1936   MPI_Comm            comm;
1937   PetscMPIInt         size,rank,taga,*len_s;
1938   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
1939   PetscInt            **buf_ri,**buf_rj;
1940   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
1941   MPI_Request         *s_waits,*r_waits;
1942   MPI_Status          *status;
1943   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
1944   PetscInt            *ai,*aj,*coi,*coj;
1945   PetscInt            *poJ,*pdJ;
1946   Mat                 A_loc;
1947   Mat_SeqAIJ          *a_loc;
1948 
1949   PetscFunctionBegin;
1950   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1951   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1952   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1953 
1954   ptap  = c->ptap;
1955   merge = ptap->merge;
1956 
1957   /* 2) compute numeric C_seq = P_loc^T*A_loc */
1958   /*------------------------------------------*/
1959   /* get data from symbolic products */
1960   coi    = merge->coi; coj = merge->coj;
1961   ierr   = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
1962   bi     = merge->bi; bj = merge->bj;
1963   owners = merge->rowmap->range;
1964   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
1965 
1966   /* get A_loc by taking all local rows of A */
1967   A_loc = ptap->A_loc;
1968   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
1969   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1970   ai    = a_loc->i;
1971   aj    = a_loc->j;
1972 
1973   for (i=0; i<am; i++) {
1974     anz = ai[i+1] - ai[i];
1975     adj = aj + ai[i];
1976     ada = a_loc->a + ai[i];
1977 
1978     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1979     /*-------------------------------------------------------------*/
1980     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1981     pnz = po->i[i+1] - po->i[i];
1982     poJ = po->j + po->i[i];
1983     pA  = po->a + po->i[i];
1984     for (j=0; j<pnz; j++) {
1985       row = poJ[j];
1986       cj  = coj + coi[row];
1987       ca  = coa + coi[row];
1988       /* perform sparse axpy */
1989       nexta  = 0;
1990       valtmp = pA[j];
1991       for (k=0; nexta<anz; k++) {
1992         if (cj[k] == adj[nexta]) {
1993           ca[k] += valtmp*ada[nexta];
1994           nexta++;
1995         }
1996       }
1997       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1998     }
1999 
2000     /* put the value into Cd (diagonal part) */
2001     pnz = pd->i[i+1] - pd->i[i];
2002     pdJ = pd->j + pd->i[i];
2003     pA  = pd->a + pd->i[i];
2004     for (j=0; j<pnz; j++) {
2005       row = pdJ[j];
2006       cj  = bj + bi[row];
2007       ca  = ba + bi[row];
2008       /* perform sparse axpy */
2009       nexta  = 0;
2010       valtmp = pA[j];
2011       for (k=0; nexta<anz; k++) {
2012         if (cj[k] == adj[nexta]) {
2013           ca[k] += valtmp*ada[nexta];
2014           nexta++;
2015         }
2016       }
2017       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
2018     }
2019   }
2020 
2021   /* 3) send and recv matrix values coa */
2022   /*------------------------------------*/
2023   buf_ri = merge->buf_ri;
2024   buf_rj = merge->buf_rj;
2025   len_s  = merge->len_s;
2026   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
2027   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
2028 
2029   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
2030   for (proc=0,k=0; proc<size; proc++) {
2031     if (!len_s[proc]) continue;
2032     i    = merge->owners_co[proc];
2033     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
2034     k++;
2035   }
2036   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
2037   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
2038 
2039   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
2040   ierr = PetscFree(r_waits);CHKERRQ(ierr);
2041   ierr = PetscFree(coa);CHKERRQ(ierr);
2042 
2043   /* 4) insert local Cseq and received values into Cmpi */
2044   /*----------------------------------------------------*/
2045   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
2046   for (k=0; k<merge->nrecv; k++) {
2047     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
2048     nrows       = *(buf_ri_k[k]);
2049     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
2050     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
2051   }
2052 
2053   for (i=0; i<cm; i++) {
2054     row  = owners[rank] + i; /* global row index of C_seq */
2055     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
2056     ba_i = ba + bi[i];
2057     bnz  = bi[i+1] - bi[i];
2058     /* add received vals into ba */
2059     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
2060       /* i-th row */
2061       if (i == *nextrow[k]) {
2062         cnz    = *(nextci[k]+1) - *nextci[k];
2063         cj     = buf_rj[k] + *(nextci[k]);
2064         ca     = abuf_r[k] + *(nextci[k]);
2065         nextcj = 0;
2066         for (j=0; nextcj<cnz; j++) {
2067           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
2068             ba_i[j] += ca[nextcj++];
2069           }
2070         }
2071         nextrow[k]++; nextci[k]++;
2072         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
2073       }
2074     }
2075     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
2076   }
2077   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2078   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2079 
2080   ierr = PetscFree(ba);CHKERRQ(ierr);
2081   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
2082   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
2083   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
2084   PetscFunctionReturn(0);
2085 }
2086 
2087 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ();
2088    differ from MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable in using LLCondensedCreate_Scalable() */
2089 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P,Mat A,PetscReal fill,Mat *C)
2090 {
2091   PetscErrorCode      ierr;
2092   Mat                 Cmpi,A_loc,POt,PDt;
2093   Mat_PtAPMPI         *ptap;
2094   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
2095   Mat_MPIAIJ          *p=(Mat_MPIAIJ*)P->data,*a=(Mat_MPIAIJ*)A->data,*c;
2096   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
2097   PetscInt            nnz;
2098   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
2099   PetscInt            am  =A->rmap->n,pn=P->cmap->n;
2100   MPI_Comm            comm;
2101   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
2102   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
2103   PetscInt            len,proc,*dnz,*onz,*owners;
2104   PetscInt            nzi,*bi,*bj;
2105   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
2106   MPI_Request         *swaits,*rwaits;
2107   MPI_Status          *sstatus,rstatus;
2108   Mat_Merge_SeqsToMPI *merge;
2109   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
2110   PetscReal           afill  =1.0,afill_tmp;
2111   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Armax;
2112   PetscScalar         *vals;
2113   Mat_SeqAIJ          *a_loc,*pdt,*pot;
2114   PetscTable          ta;
2115 
2116   PetscFunctionBegin;
2117   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
2118   /* check if matrix local sizes are compatible */
2119   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);
2120 
2121   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
2122   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
2123 
2124   /* create struct Mat_PtAPMPI and attached it to C later */
2125   ierr = PetscNew(&ptap);CHKERRQ(ierr);
2126 
2127   /* get A_loc by taking all local rows of A */
2128   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
2129 
2130   ptap->A_loc = A_loc;
2131   a_loc       = (Mat_SeqAIJ*)(A_loc)->data;
2132   ai          = a_loc->i;
2133   aj          = a_loc->j;
2134 
2135   /* determine symbolic Co=(p->B)^T*A - send to others */
2136   /*----------------------------------------------------*/
2137   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
2138   pdt  = (Mat_SeqAIJ*)PDt->data;
2139   pdti = pdt->i; pdtj = pdt->j;
2140 
2141   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
2142   pot  = (Mat_SeqAIJ*)POt->data;
2143   poti = pot->i; potj = pot->j;
2144 
2145   /* then, compute symbolic Co = (p->B)^T*A */
2146   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors
2147                          >= (num of nonzero rows of C_seq) - pn */
2148   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
2149   coi[0] = 0;
2150 
2151   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
2152   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(poti[pon],ai[am]));
2153   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
2154   current_space = free_space;
2155 
2156   /* create and initialize a linked list */
2157   ierr = PetscTableCreate(A->cmap->n + a->B->cmap->N,aN,&ta);CHKERRQ(ierr);
2158   MatRowMergeMax_SeqAIJ(a_loc,am,ta);
2159   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
2160 
2161   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
2162 
2163   for (i=0; i<pon; i++) {
2164     pnz = poti[i+1] - poti[i];
2165     ptJ = potj + poti[i];
2166     for (j=0; j<pnz; j++) {
2167       row  = ptJ[j]; /* row of A_loc == col of Pot */
2168       anz  = ai[row+1] - ai[row];
2169       Jptr = aj + ai[row];
2170       /* add non-zero cols of AP into the sorted linked list lnk */
2171       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
2172     }
2173     nnz = lnk[0];
2174 
2175     /* If free space is not available, double the total space in the list */
2176     if (current_space->local_remaining<nnz) {
2177       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
2178       nspacedouble++;
2179     }
2180 
2181     /* Copy data into free space, and zero out denserows */
2182     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
2183 
2184     current_space->array           += nnz;
2185     current_space->local_used      += nnz;
2186     current_space->local_remaining -= nnz;
2187 
2188     coi[i+1] = coi[i] + nnz;
2189   }
2190 
2191   ierr = PetscMalloc1(coi[pon]+1,&coj);CHKERRQ(ierr);
2192   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
2193   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr); /* must destroy to get a new one for C */
2194 
2195   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
2196   if (afill_tmp > afill) afill = afill_tmp;
2197 
2198   /* send j-array (coj) of Co to other processors */
2199   /*----------------------------------------------*/
2200   /* determine row ownership */
2201   ierr = PetscNew(&merge);CHKERRQ(ierr);
2202   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
2203 
2204   merge->rowmap->n  = pn;
2205   merge->rowmap->bs = 1;
2206 
2207   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
2208   owners = merge->rowmap->range;
2209 
2210   /* determine the number of messages to send, their lengths */
2211   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
2212   ierr = PetscMalloc1(size,&merge->len_s);CHKERRQ(ierr);
2213 
2214   len_s        = merge->len_s;
2215   merge->nsend = 0;
2216 
2217   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
2218   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
2219 
2220   proc = 0;
2221   for (i=0; i<pon; i++) {
2222     while (prmap[i] >= owners[proc+1]) proc++;
2223     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
2224     len_s[proc] += coi[i+1] - coi[i];
2225   }
2226 
2227   len          = 0; /* max length of buf_si[] */
2228   owners_co[0] = 0;
2229   for (proc=0; proc<size; proc++) {
2230     owners_co[proc+1] = owners_co[proc] + len_si[proc];
2231     if (len_si[proc]) {
2232       merge->nsend++;
2233       len_si[proc] = 2*(len_si[proc] + 1);
2234       len         += len_si[proc];
2235     }
2236   }
2237 
2238   /* determine the number and length of messages to receive for coi and coj  */
2239   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
2240   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
2241 
2242   /* post the Irecv and Isend of coj */
2243   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
2244   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
2245   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
2246   for (proc=0, k=0; proc<size; proc++) {
2247     if (!len_s[proc]) continue;
2248     i    = owners_co[proc];
2249     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
2250     k++;
2251   }
2252 
2253   /* receives and sends of coj are complete */
2254   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
2255   for (i=0; i<merge->nrecv; i++) {
2256     PetscMPIInt icompleted;
2257     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
2258   }
2259   ierr = PetscFree(rwaits);CHKERRQ(ierr);
2260   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
2261 
2262   /* add received column indices into table to update Armax */
2263   /* Armax can be as large as aN if a P[row,:] is dense, see src/ksp/ksp/examples/tutorials/ex56.c! */
2264   for (k=0; k<merge->nrecv; k++) {/* k-th received message */
2265     Jptr = buf_rj[k];
2266     for (j=0; j<merge->len_r[k]; j++) {
2267       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
2268     }
2269   }
2270   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
2271   /* 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); */
2272 
2273   /* send and recv coi */
2274   /*-------------------*/
2275   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
2276   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
2277   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
2278   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
2279   for (proc=0,k=0; proc<size; proc++) {
2280     if (!len_s[proc]) continue;
2281     /* form outgoing message for i-structure:
2282          buf_si[0]:                 nrows to be sent
2283                [1:nrows]:           row index (global)
2284                [nrows+1:2*nrows+1]: i-structure index
2285     */
2286     /*-------------------------------------------*/
2287     nrows       = len_si[proc]/2 - 1;
2288     buf_si_i    = buf_si + nrows+1;
2289     buf_si[0]   = nrows;
2290     buf_si_i[0] = 0;
2291     nrows       = 0;
2292     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
2293       nzi               = coi[i+1] - coi[i];
2294       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
2295       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
2296       nrows++;
2297     }
2298     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
2299     k++;
2300     buf_si += len_si[proc];
2301   }
2302   i = merge->nrecv;
2303   while (i--) {
2304     PetscMPIInt icompleted;
2305     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
2306   }
2307   ierr = PetscFree(rwaits);CHKERRQ(ierr);
2308   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
2309   ierr = PetscFree(len_si);CHKERRQ(ierr);
2310   ierr = PetscFree(len_ri);CHKERRQ(ierr);
2311   ierr = PetscFree(swaits);CHKERRQ(ierr);
2312   ierr = PetscFree(sstatus);CHKERRQ(ierr);
2313   ierr = PetscFree(buf_s);CHKERRQ(ierr);
2314 
2315   /* compute the local portion of C (mpi mat) */
2316   /*------------------------------------------*/
2317   /* allocate bi array and free space for accumulating nonzero column info */
2318   ierr  = PetscMalloc1(pn+1,&bi);CHKERRQ(ierr);
2319   bi[0] = 0;
2320 
2321   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
2322   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(pdti[pn],PetscIntSumTruncate(poti[pon],ai[am])));
2323   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
2324   current_space = free_space;
2325 
2326   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
2327   for (k=0; k<merge->nrecv; k++) {
2328     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
2329     nrows       = *buf_ri_k[k];
2330     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
2331     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th recieved i-structure  */
2332   }
2333 
2334   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
2335   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
2336   rmax = 0;
2337   for (i=0; i<pn; i++) {
2338     /* add pdt[i,:]*AP into lnk */
2339     pnz = pdti[i+1] - pdti[i];
2340     ptJ = pdtj + pdti[i];
2341     for (j=0; j<pnz; j++) {
2342       row  = ptJ[j];  /* row of AP == col of Pt */
2343       anz  = ai[row+1] - ai[row];
2344       Jptr = aj + ai[row];
2345       /* add non-zero cols of AP into the sorted linked list lnk */
2346       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
2347     }
2348 
2349     /* add received col data into lnk */
2350     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
2351       if (i == *nextrow[k]) { /* i-th row */
2352         nzi  = *(nextci[k]+1) - *nextci[k];
2353         Jptr = buf_rj[k] + *nextci[k];
2354         ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
2355         nextrow[k]++; nextci[k]++;
2356       }
2357     }
2358     nnz = lnk[0];
2359 
2360     /* if free space is not available, make more free space */
2361     if (current_space->local_remaining<nnz) {
2362       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
2363       nspacedouble++;
2364     }
2365     /* copy data into free space, then initialize lnk */
2366     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
2367     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
2368 
2369     current_space->array           += nnz;
2370     current_space->local_used      += nnz;
2371     current_space->local_remaining -= nnz;
2372 
2373     bi[i+1] = bi[i] + nnz;
2374     if (nnz > rmax) rmax = nnz;
2375   }
2376   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
2377 
2378   ierr      = PetscMalloc1(bi[pn]+1,&bj);CHKERRQ(ierr);
2379   ierr      = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
2380   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
2381   if (afill_tmp > afill) afill = afill_tmp;
2382   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
2383   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
2384 
2385   ierr = MatDestroy(&POt);CHKERRQ(ierr);
2386   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
2387 
2388   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
2389   /*----------------------------------------------------------------------------------*/
2390   ierr = PetscCalloc1(rmax+1,&vals);CHKERRQ(ierr);
2391 
2392   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
2393   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
2394   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
2395   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
2396   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
2397   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
2398   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
2399   for (i=0; i<pn; i++) {
2400     row  = i + rstart;
2401     nnz  = bi[i+1] - bi[i];
2402     Jptr = bj + bi[i];
2403     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
2404   }
2405   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2406   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2407   ierr = PetscFree(vals);CHKERRQ(ierr);
2408 
2409   merge->bi        = bi;
2410   merge->bj        = bj;
2411   merge->coi       = coi;
2412   merge->coj       = coj;
2413   merge->buf_ri    = buf_ri;
2414   merge->buf_rj    = buf_rj;
2415   merge->owners_co = owners_co;
2416 
2417   /* attach the supporting struct to Cmpi for reuse */
2418   c = (Mat_MPIAIJ*)Cmpi->data;
2419 
2420   c->ptap     = ptap;
2421   ptap->api   = NULL;
2422   ptap->apj   = NULL;
2423   ptap->merge = merge;
2424   ptap->apa   = NULL;
2425   ptap->destroy   = Cmpi->ops->destroy;
2426   ptap->duplicate = Cmpi->ops->duplicate;
2427 
2428   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ;
2429   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
2430   Cmpi->ops->duplicate               = MatDuplicate_MPIAIJ_MatPtAP;
2431 
2432   *C = Cmpi;
2433 #if defined(PETSC_USE_INFO)
2434   if (bi[pn] != 0) {
2435     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
2436     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
2437   } else {
2438     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
2439   }
2440 #endif
2441   PetscFunctionReturn(0);
2442 }
2443