xref: /petsc/src/mat/impls/aij/mpi/mpiptap.c (revision ec18b7bcaa0343907d38c35c0fddd6a26ee1f613)
1 
2 /*
3   Defines projective product routines where A is a MPIAIJ matrix
4           C = P^T * A * P
5 */
6 
7 #include <../src/mat/impls/aij/seq/aij.h>   /*I "petscmat.h" I*/
8 #include <../src/mat/utils/freespace.h>
9 #include <../src/mat/impls/aij/mpi/mpiaij.h>
10 #include <petscbt.h>
11 #include <petsctime.h>
12 
13 /* #define PTAP_PROFILE */
14 
15 PetscErrorCode MatView_MPIAIJ_PtAP(Mat A,PetscViewer viewer)
16 {
17   PetscErrorCode    ierr;
18   Mat_MPIAIJ        *a=(Mat_MPIAIJ*)A->data;
19   Mat_PtAPMPI       *ptap=a->ptap;
20   PetscBool         iascii;
21   PetscViewerFormat format;
22 
23   PetscFunctionBegin;
24   ierr = PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);CHKERRQ(ierr);
25   if (iascii) {
26     ierr = PetscViewerGetFormat(viewer,&format);CHKERRQ(ierr);
27     if (format == PETSC_VIEWER_ASCII_INFO || format == PETSC_VIEWER_ASCII_INFO_DETAIL) {
28       if (ptap->algType == 0) {
29         ierr = PetscViewerASCIIPrintf(viewer,"using scalable MatPtAP() implementation\n");CHKERRQ(ierr);
30       } else if (ptap->algType == 1) {
31         ierr = PetscViewerASCIIPrintf(viewer,"using nonscalable MatPtAP() implementation\n");CHKERRQ(ierr);
32       }
33     }
34   }
35   ierr = (ptap->view)(A,viewer);CHKERRQ(ierr);
36   PetscFunctionReturn(0);
37 }
38 
39 PetscErrorCode MatDestroy_MPIAIJ_PtAP(Mat A)
40 {
41   PetscErrorCode ierr;
42   Mat_MPIAIJ     *a=(Mat_MPIAIJ*)A->data;
43   Mat_PtAPMPI    *ptap=a->ptap;
44 
45   PetscFunctionBegin;
46   if (ptap) {
47     Mat_Merge_SeqsToMPI *merge=ptap->merge;
48     ierr = PetscFree2(ptap->startsj_s,ptap->startsj_r);CHKERRQ(ierr);
49     ierr = PetscFree(ptap->bufa);CHKERRQ(ierr);
50     ierr = MatDestroy(&ptap->P_loc);CHKERRQ(ierr);
51     ierr = MatDestroy(&ptap->P_oth);CHKERRQ(ierr);
52     ierr = MatDestroy(&ptap->A_loc);CHKERRQ(ierr); /* used by MatTransposeMatMult() */
53     ierr = MatDestroy(&ptap->Rd);CHKERRQ(ierr);
54     ierr = MatDestroy(&ptap->Ro);CHKERRQ(ierr);
55     if (ptap->AP_loc) { /* used by alg_rap */
56       Mat_SeqAIJ *ap = (Mat_SeqAIJ*)(ptap->AP_loc)->data;
57       ierr = PetscFree(ap->i);CHKERRQ(ierr);
58       ierr = PetscFree2(ap->j,ap->a);CHKERRQ(ierr);
59       ierr = MatDestroy(&ptap->AP_loc);CHKERRQ(ierr);
60     } else { /* used by alg_ptap */
61       ierr = PetscFree(ptap->api);CHKERRQ(ierr);
62       ierr = PetscFree(ptap->apj);CHKERRQ(ierr);
63     }
64     ierr = MatDestroy(&ptap->C_loc);CHKERRQ(ierr);
65     ierr = MatDestroy(&ptap->C_oth);CHKERRQ(ierr);
66     if (ptap->apa) {ierr = PetscFree(ptap->apa);CHKERRQ(ierr);}
67 
68     if (merge) { /* used by alg_ptap */
69       ierr = PetscFree(merge->id_r);CHKERRQ(ierr);
70       ierr = PetscFree(merge->len_s);CHKERRQ(ierr);
71       ierr = PetscFree(merge->len_r);CHKERRQ(ierr);
72       ierr = PetscFree(merge->bi);CHKERRQ(ierr);
73       ierr = PetscFree(merge->bj);CHKERRQ(ierr);
74       ierr = PetscFree(merge->buf_ri[0]);CHKERRQ(ierr);
75       ierr = PetscFree(merge->buf_ri);CHKERRQ(ierr);
76       ierr = PetscFree(merge->buf_rj[0]);CHKERRQ(ierr);
77       ierr = PetscFree(merge->buf_rj);CHKERRQ(ierr);
78       ierr = PetscFree(merge->coi);CHKERRQ(ierr);
79       ierr = PetscFree(merge->coj);CHKERRQ(ierr);
80       ierr = PetscFree(merge->owners_co);CHKERRQ(ierr);
81       ierr = PetscLayoutDestroy(&merge->rowmap);CHKERRQ(ierr);
82       ierr = PetscFree(ptap->merge);CHKERRQ(ierr);
83     }
84 
85     ierr = ptap->destroy(A);CHKERRQ(ierr);
86     ierr = PetscFree(ptap);CHKERRQ(ierr);
87   }
88   PetscFunctionReturn(0);
89 }
90 
91 PetscErrorCode MatDuplicate_MPIAIJ_MatPtAP(Mat A, MatDuplicateOption op, Mat *M)
92 {
93   PetscErrorCode ierr;
94   Mat_MPIAIJ     *a     = (Mat_MPIAIJ*)A->data;
95   Mat_PtAPMPI    *ptap  = a->ptap;
96 
97   PetscFunctionBegin;
98   ierr = (*ptap->duplicate)(A,op,M);CHKERRQ(ierr);
99   (*M)->ops->destroy   = ptap->destroy;
100   (*M)->ops->duplicate = ptap->duplicate;
101   (*M)->ops->view      = ptap->view;
102   PetscFunctionReturn(0);
103 }
104 
105 PETSC_INTERN PetscErrorCode MatPtAP_MPIAIJ_MPIAIJ(Mat A,Mat P,MatReuse scall,PetscReal fill,Mat *C)
106 {
107   PetscErrorCode ierr;
108   PetscBool      rap=PETSC_TRUE; /* do R=P^T locally, then C=R*A*P */
109   MPI_Comm       comm;
110 
111   PetscFunctionBegin;
112   /* check if matrix local sizes are compatible */
113   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
114   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend) SETERRQ4(comm,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, Arow (%D, %D) != Prow (%D,%D)",A->rmap->rstart,A->rmap->rend,P->rmap->rstart,P->rmap->rend);
115   if (A->cmap->rstart != P->rmap->rstart || A->cmap->rend != P->rmap->rend) SETERRQ4(comm,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, Acol (%D, %D) != Prow (%D,%D)",A->cmap->rstart,A->cmap->rend,P->rmap->rstart,P->rmap->rend);
116 
117   ierr = PetscOptionsGetBool(NULL,NULL,"-matrap",&rap,NULL);CHKERRQ(ierr);
118   if (scall == MAT_INITIAL_MATRIX) {
119     ierr = PetscLogEventBegin(MAT_PtAPSymbolic,A,P,0,0);CHKERRQ(ierr);
120     if (rap) { /* do R=P^T locally, then C=R*A*P */
121       ierr = MatPtAPSymbolic_MPIAIJ_MPIAIJ(A,P,fill,C);CHKERRQ(ierr);
122     } else {       /* do P^T*A*P */
123       ierr = MatPtAPSymbolic_MPIAIJ_MPIAIJ_ptap(A,P,fill,C);CHKERRQ(ierr);
124     }
125     ierr = PetscLogEventEnd(MAT_PtAPSymbolic,A,P,0,0);CHKERRQ(ierr);
126   }
127   ierr = PetscLogEventBegin(MAT_PtAPNumeric,A,P,0,0);CHKERRQ(ierr);
128   ierr = (*(*C)->ops->ptapnumeric)(A,P,*C);CHKERRQ(ierr);
129   ierr = PetscLogEventEnd(MAT_PtAPNumeric,A,P,0,0);CHKERRQ(ierr);
130   PetscFunctionReturn(0);
131 }
132 
133 #if defined(PETSC_HAVE_HYPRE)
134 PETSC_INTERN PetscErrorCode MatPtAPSymbolic_AIJ_AIJ_wHYPRE(Mat,Mat,PetscReal,Mat*);
135 #endif
136 
137 PetscErrorCode MatPtAPNumeric_MPIAIJ_MPIAIJ_scalable(Mat A,Mat P,Mat C)
138 {
139   PetscErrorCode    ierr;
140   Mat_MPIAIJ        *a=(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
141   Mat_SeqAIJ        *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
142   Mat_SeqAIJ        *ap,*p_loc,*p_oth,*c_seq;
143   Mat_PtAPMPI       *ptap = c->ptap;
144   Mat               AP_loc,C_loc,C_oth;
145   PetscInt          i,rstart,rend,cm,ncols,row,*api,*apj,am = A->rmap->n,apnz;
146   PetscScalar       *apa;
147   const PetscInt    *cols;
148   const PetscScalar *vals;
149 
150   PetscFunctionBegin;
151   ierr = MatZeroEntries(C);CHKERRQ(ierr);
152 
153   /* 1) get R = Pd^T,Ro = Po^T */
154   if (ptap->reuse == MAT_REUSE_MATRIX) {
155     ierr = MatTranspose_SeqAIJ(p->A,MAT_REUSE_MATRIX,&ptap->Rd);CHKERRQ(ierr);
156     ierr = MatTranspose_SeqAIJ(p->B,MAT_REUSE_MATRIX,&ptap->Ro);CHKERRQ(ierr);
157   }
158 
159   /* 2) get AP_loc */
160   AP_loc = ptap->AP_loc;
161   ap = (Mat_SeqAIJ*)AP_loc->data;
162 
163   /* 2-1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
164   /*-----------------------------------------------------*/
165   if (ptap->reuse == MAT_REUSE_MATRIX) {
166     /* P_oth and P_loc are obtained in MatPtASymbolic() when reuse == MAT_INITIAL_MATRIX */
167     ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
168     ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
169   }
170 
171   /* 2-2) compute numeric A_loc*P - dominating part */
172   /* ---------------------------------------------- */
173   /* get data from symbolic products */
174   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
175   if (ptap->P_oth) p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
176   else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"ptap->P_oth is NULL. Cannot proceed!");
177 
178   api   = ap->i;
179   apj   = ap->j;
180   for (i=0; i<am; i++) {
181     /* AP[i,:] = A[i,:]*P = Ad*P_loc Ao*P_oth */
182     apnz = api[i+1] - api[i];
183     apa = ap->a + api[i];
184     ierr = PetscMemzero(apa,sizeof(PetscScalar)*apnz);CHKERRQ(ierr);
185     AProw_scalable(i,ad,ao,p_loc,p_oth,api,apj,apa);
186     ierr = PetscLogFlops(2.0*apnz);CHKERRQ(ierr);
187   }
188 
189   /* 3) C_loc = Rd*AP_loc, C_oth = Ro*AP_loc */
190   ierr = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable(ptap->Rd,AP_loc,ptap->C_loc);CHKERRQ(ierr);
191   ierr = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable(ptap->Ro,AP_loc,ptap->C_oth);CHKERRQ(ierr);
192 
193   C_loc = ptap->C_loc;
194   C_oth = ptap->C_oth;
195 
196   /* add C_loc and Co to to C */
197   ierr = MatGetOwnershipRange(C,&rstart,&rend);CHKERRQ(ierr);
198 
199   /* C_loc -> C */
200   cm    = C_loc->rmap->N;
201   c_seq = (Mat_SeqAIJ*)C_loc->data;
202   cols = c_seq->j;
203   vals = c_seq->a;
204   for (i=0; i<cm; i++) {
205     ncols = c_seq->i[i+1] - c_seq->i[i];
206     row = rstart + i;
207     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
208     cols += ncols; vals += ncols;
209   }
210 
211   /* Co -> C, off-processor part */
212   cm = C_oth->rmap->N;
213   c_seq = (Mat_SeqAIJ*)C_oth->data;
214   cols = c_seq->j;
215   vals = c_seq->a;
216   for (i=0; i<cm; i++) {
217     ncols = c_seq->i[i+1] - c_seq->i[i];
218     row = p->garray[i];
219     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
220     cols += ncols; vals += ncols;
221   }
222   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
223   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
224 
225   ptap->reuse = MAT_REUSE_MATRIX;
226   PetscFunctionReturn(0);
227 }
228 
229 PetscErrorCode MatPtAPSymbolic_MPIAIJ_MPIAIJ_scalable(Mat A,Mat P,PetscReal fill,Mat *C)
230 {
231   PetscErrorCode      ierr;
232   Mat_PtAPMPI         *ptap;
233   Mat_MPIAIJ          *a=(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data,*c;
234   MPI_Comm            comm;
235   PetscMPIInt         size,rank;
236   Mat                 Cmpi,P_loc,P_oth;
237   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
238   PetscInt            am=A->rmap->n,pm=P->rmap->n,pN=P->cmap->N,pn=P->cmap->n;
239   PetscInt            *lnk,i,k,pnz,row,nsend;
240   PetscMPIInt         tagi,tagj,*len_si,*len_s,*len_ri,icompleted=0,nrecv;
241   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
242   PetscInt            len,proc,*dnz,*onz,*owners,nzi,nspacedouble;
243   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
244   MPI_Request         *swaits,*rwaits;
245   MPI_Status          *sstatus,rstatus;
246   PetscLayout         rowmap;
247   PetscInt            *owners_co,*coi,*coj;    /* i and j array of (p->B)^T*A*P - used in the communication */
248   PetscMPIInt         *len_r,*id_r;    /* array of length of comm->size, store send/recv matrix values */
249   PetscInt            *api,*apj,*Jptr,apnz,*prmap=p->garray,con,j,Crmax,*aj,*ai,*pi;
250   Mat_SeqAIJ          *p_loc,*p_oth=NULL,*ad=(Mat_SeqAIJ*)(a->A)->data,*ao=NULL,*c_loc,*c_oth;
251   PetscScalar         *apv;
252   PetscTable          ta;
253 #if defined(PETSC_USE_INFO)
254   PetscReal           apfill;
255 #endif
256 
257   PetscFunctionBegin;
258   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
259   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
260   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
261 
262   if (size > 1) ao = (Mat_SeqAIJ*)(a->B)->data;
263 
264   /* create symbolic parallel matrix Cmpi */
265   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
266   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
267 
268   /* create struct Mat_PtAPMPI and attached it to C later */
269   ierr        = PetscNew(&ptap);CHKERRQ(ierr);
270   ptap->reuse = MAT_INITIAL_MATRIX;
271   ptap->algType = 0;
272 
273   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
274   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&P_oth);CHKERRQ(ierr);
275   /* get P_loc by taking all local rows of P */
276   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&P_loc);CHKERRQ(ierr);
277 
278   ptap->P_loc = P_loc;
279   ptap->P_oth = P_oth;
280 
281   /* (0) compute Rd = Pd^T, Ro = Po^T  */
282   /* --------------------------------- */
283   ierr = MatTranspose_SeqAIJ(p->A,MAT_INITIAL_MATRIX,&ptap->Rd);CHKERRQ(ierr);
284   ierr = MatTranspose_SeqAIJ(p->B,MAT_INITIAL_MATRIX,&ptap->Ro);CHKERRQ(ierr);
285 
286   /* (1) compute symbolic AP = A_loc*P = Ad*P_loc + Ao*P_oth (api,apj) */
287   /* ----------------------------------------------------------------- */
288   p_loc  = (Mat_SeqAIJ*)P_loc->data;
289   if (P_oth) p_oth = (Mat_SeqAIJ*)P_oth->data;
290 
291   /* create and initialize a linked list */
292   ierr = PetscTableCreate(pn,pN,&ta);CHKERRQ(ierr); /* for compute AP_loc and Cmpi */
293   MatRowMergeMax_SeqAIJ(p_loc,P_loc->rmap->N,ta);
294   MatRowMergeMax_SeqAIJ(p_oth,P_oth->rmap->N,ta);
295   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr); /* Crmax = nnz(sum of Prows) */
296 
297   ierr = PetscLLCondensedCreate_Scalable(Crmax,&lnk);CHKERRQ(ierr);
298 
299   /* Initial FreeSpace size is fill*(nnz(A) + nnz(P)) */
300   if (ao) {
301     ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ad->i[am],PetscIntSumTruncate(ao->i[am],p_loc->i[pm]))),&free_space);CHKERRQ(ierr);
302   } else {
303     ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ad->i[am],p_loc->i[pm])),&free_space);CHKERRQ(ierr);
304   }
305   current_space = free_space;
306   nspacedouble  = 0;
307 
308   ierr   = PetscMalloc1(am+1,&api);CHKERRQ(ierr);
309   api[0] = 0;
310   for (i=0; i<am; i++) {
311     /* diagonal portion: Ad[i,:]*P */
312     ai = ad->i; pi = p_loc->i;
313     nzi = ai[i+1] - ai[i];
314     aj  = ad->j + ai[i];
315     for (j=0; j<nzi; j++) {
316       row  = aj[j];
317       pnz  = pi[row+1] - pi[row];
318       Jptr = p_loc->j + pi[row];
319       /* add non-zero cols of P into the sorted linked list lnk */
320       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
321     }
322     /* off-diagonal portion: Ao[i,:]*P */
323     if (ao) {
324       ai = ao->i; pi = p_oth->i;
325       nzi = ai[i+1] - ai[i];
326       aj  = ao->j + ai[i];
327       for (j=0; j<nzi; j++) {
328         row  = aj[j];
329         pnz  = pi[row+1] - pi[row];
330         Jptr = p_oth->j + pi[row];
331         ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
332       }
333     }
334     apnz     = lnk[0];
335     api[i+1] = api[i] + apnz;
336 
337     /* if free space is not available, double the total space in the list */
338     if (current_space->local_remaining<apnz) {
339       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
340       nspacedouble++;
341     }
342 
343     /* Copy data into free space, then initialize lnk */
344     ierr = PetscLLCondensedClean_Scalable(apnz,current_space->array,lnk);CHKERRQ(ierr);
345 
346     current_space->array           += apnz;
347     current_space->local_used      += apnz;
348     current_space->local_remaining -= apnz;
349   }
350   /* Allocate space for apj and apv, initialize apj, and */
351   /* destroy list of free space and other temporary array(s) */
352   ierr = PetscMalloc2(api[am],&apj,api[am],&apv);CHKERRQ(ierr);
353   ierr = PetscFreeSpaceContiguous(&free_space,apj);CHKERRQ(ierr);
354   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
355 
356   /* Create AP_loc for reuse */
357   ierr = MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,am,pN,api,apj,apv,&ptap->AP_loc);CHKERRQ(ierr);
358 
359 #if defined(PETSC_USE_INFO)
360   if (ao) {
361     apfill = (PetscReal)api[am]/(ad->i[am]+ao->i[am]+p_loc->i[pm]+1);
362   } else {
363     apfill = (PetscReal)api[am]/(ad->i[am]+p_loc->i[pm]+1);
364   }
365   ptap->AP_loc->info.mallocs           = nspacedouble;
366   ptap->AP_loc->info.fill_ratio_given  = fill;
367   ptap->AP_loc->info.fill_ratio_needed = apfill;
368 
369   if (api[am]) {
370     ierr = PetscInfo3(ptap->AP_loc,"Scalable algorithm, AP_loc reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)apfill);CHKERRQ(ierr);
371     ierr = PetscInfo1(ptap->AP_loc,"Use MatPtAP(A,B,MatReuse,%g,&C) for best AP_loc performance.;\n",(double)apfill);CHKERRQ(ierr);
372   } else {
373     ierr = PetscInfo(ptap->AP_loc,"Scalable algorithm, AP_loc is empty \n");CHKERRQ(ierr);
374   }
375 #endif
376 
377   /* (2-1) compute symbolic Co = Ro*AP_loc  */
378   /* ------------------------------------ */
379   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Ro,ptap->AP_loc,fill,&ptap->C_oth);CHKERRQ(ierr);
380 
381   /* (3) send coj of C_oth to other processors  */
382   /* ------------------------------------------ */
383   /* determine row ownership */
384   ierr = PetscLayoutCreate(comm,&rowmap);CHKERRQ(ierr);
385   rowmap->n  = pn;
386   rowmap->bs = 1;
387   ierr   = PetscLayoutSetUp(rowmap);CHKERRQ(ierr);
388   owners = rowmap->range;
389 
390   /* determine the number of messages to send, their lengths */
391   ierr = PetscMalloc4(size,&len_s,size,&len_si,size,&sstatus,size+2,&owners_co);CHKERRQ(ierr);
392   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
393   ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
394 
395   c_oth = (Mat_SeqAIJ*)ptap->C_oth->data;
396   coi   = c_oth->i; coj = c_oth->j;
397   con   = ptap->C_oth->rmap->n;
398   proc  = 0;
399   for (i=0; i<con; i++) {
400     while (prmap[i] >= owners[proc+1]) proc++;
401     len_si[proc]++;               /* num of rows in Co(=Pt*AP) to be sent to [proc] */
402     len_s[proc] += coi[i+1] - coi[i]; /* num of nonzeros in Co to be sent to [proc] */
403   }
404 
405   len          = 0; /* max length of buf_si[], see (4) */
406   owners_co[0] = 0;
407   nsend        = 0;
408   for (proc=0; proc<size; proc++) {
409     owners_co[proc+1] = owners_co[proc] + len_si[proc];
410     if (len_s[proc]) {
411       nsend++;
412       len_si[proc] = 2*(len_si[proc] + 1); /* length of buf_si to be sent to [proc] */
413       len         += len_si[proc];
414     }
415   }
416 
417   /* determine the number and length of messages to receive for coi and coj  */
418   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&nrecv);CHKERRQ(ierr);
419   ierr = PetscGatherMessageLengths2(comm,nsend,nrecv,len_s,len_si,&id_r,&len_r,&len_ri);CHKERRQ(ierr);
420 
421   /* post the Irecv and Isend of coj */
422   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
423   ierr = PetscPostIrecvInt(comm,tagj,nrecv,id_r,len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
424   ierr = PetscMalloc1(nsend+1,&swaits);CHKERRQ(ierr);
425   for (proc=0, k=0; proc<size; proc++) {
426     if (!len_s[proc]) continue;
427     i    = owners_co[proc];
428     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
429     k++;
430   }
431 
432   /* (2-2) compute symbolic C_loc = Rd*AP_loc */
433   /* ---------------------------------------- */
434   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Rd,ptap->AP_loc,fill,&ptap->C_loc);CHKERRQ(ierr);
435   c_loc = (Mat_SeqAIJ*)ptap->C_loc->data;
436 
437   /* receives coj are complete */
438   for (i=0; i<nrecv; i++) {
439     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
440   }
441   ierr = PetscFree(rwaits);CHKERRQ(ierr);
442   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
443 
444   /* add received column indices into ta to update Crmax */
445   for (k=0; k<nrecv; k++) {/* k-th received message */
446     Jptr = buf_rj[k];
447     for (j=0; j<len_r[k]; j++) {
448       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
449     }
450   }
451   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
452   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
453 
454   /* (4) send and recv coi */
455   /*-----------------------*/
456   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
457   ierr   = PetscPostIrecvInt(comm,tagi,nrecv,id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
458   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
459   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
460   for (proc=0,k=0; proc<size; proc++) {
461     if (!len_s[proc]) continue;
462     /* form outgoing message for i-structure:
463          buf_si[0]:                 nrows to be sent
464                [1:nrows]:           row index (global)
465                [nrows+1:2*nrows+1]: i-structure index
466     */
467     /*-------------------------------------------*/
468     nrows       = len_si[proc]/2 - 1; /* num of rows in Co to be sent to [proc] */
469     buf_si_i    = buf_si + nrows+1;
470     buf_si[0]   = nrows;
471     buf_si_i[0] = 0;
472     nrows       = 0;
473     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
474       nzi = coi[i+1] - coi[i];
475       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
476       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
477       nrows++;
478     }
479     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
480     k++;
481     buf_si += len_si[proc];
482   }
483   for (i=0; i<nrecv; i++) {
484     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
485   }
486   ierr = PetscFree(rwaits);CHKERRQ(ierr);
487   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
488 
489   ierr = PetscFree4(len_s,len_si,sstatus,owners_co);CHKERRQ(ierr);
490   ierr = PetscFree(len_ri);CHKERRQ(ierr);
491   ierr = PetscFree(swaits);CHKERRQ(ierr);
492   ierr = PetscFree(buf_s);CHKERRQ(ierr);
493 
494   /* (5) compute the local portion of Cmpi      */
495   /* ------------------------------------------ */
496   /* set initial free space to be Crmax, sufficient for holding nozeros in each row of Cmpi */
497   ierr          = PetscFreeSpaceGet(Crmax,&free_space);CHKERRQ(ierr);
498   current_space = free_space;
499 
500   ierr = PetscMalloc3(nrecv,&buf_ri_k,nrecv,&nextrow,nrecv,&nextci);CHKERRQ(ierr);
501   for (k=0; k<nrecv; k++) {
502     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
503     nrows       = *buf_ri_k[k];
504     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
505     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
506   }
507 
508   ierr = MatPreallocateInitialize(comm,pn,pn,dnz,onz);CHKERRQ(ierr);
509   ierr = PetscLLCondensedCreate_Scalable(Crmax,&lnk);CHKERRQ(ierr);
510   for (i=0; i<pn; i++) {
511     /* add C_loc into Cmpi */
512     nzi  = c_loc->i[i+1] - c_loc->i[i];
513     Jptr = c_loc->j + c_loc->i[i];
514     ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
515 
516     /* add received col data into lnk */
517     for (k=0; k<nrecv; k++) { /* k-th received message */
518       if (i == *nextrow[k]) { /* i-th row */
519         nzi  = *(nextci[k]+1) - *nextci[k];
520         Jptr = buf_rj[k] + *nextci[k];
521         ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
522         nextrow[k]++; nextci[k]++;
523       }
524     }
525     nzi = lnk[0];
526 
527     /* copy data into free space, then initialize lnk */
528     ierr = PetscLLCondensedClean_Scalable(nzi,current_space->array,lnk);CHKERRQ(ierr);
529     ierr = MatPreallocateSet(i+owners[rank],nzi,current_space->array,dnz,onz);CHKERRQ(ierr);
530   }
531   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
532   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
533   ierr = PetscFreeSpaceDestroy(free_space);CHKERRQ(ierr);
534 
535   /* local sizes and preallocation */
536   ierr = MatSetSizes(Cmpi,pn,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
537   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(P->cmap->bs));CHKERRQ(ierr);
538   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
539   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
540 
541   /* members in merge */
542   ierr = PetscFree(id_r);CHKERRQ(ierr);
543   ierr = PetscFree(len_r);CHKERRQ(ierr);
544   ierr = PetscFree(buf_ri[0]);CHKERRQ(ierr);
545   ierr = PetscFree(buf_ri);CHKERRQ(ierr);
546   ierr = PetscFree(buf_rj[0]);CHKERRQ(ierr);
547   ierr = PetscFree(buf_rj);CHKERRQ(ierr);
548   ierr = PetscLayoutDestroy(&rowmap);CHKERRQ(ierr);
549 
550   /* attach the supporting struct to Cmpi for reuse */
551   c = (Mat_MPIAIJ*)Cmpi->data;
552   c->ptap         = ptap;
553   ptap->duplicate = Cmpi->ops->duplicate;
554   ptap->destroy   = Cmpi->ops->destroy;
555   ptap->view      = Cmpi->ops->view;
556 
557   /* Cmpi is not ready for use - assembly will be done by MatPtAPNumeric() */
558   Cmpi->assembled        = PETSC_FALSE;
559   Cmpi->ops->destroy     = MatDestroy_MPIAIJ_PtAP;
560   Cmpi->ops->duplicate   = MatDuplicate_MPIAIJ_MatPtAP;
561   Cmpi->ops->view        = MatView_MPIAIJ_PtAP;
562   *C                     = Cmpi;
563   PetscFunctionReturn(0);
564 }
565 
566 PetscErrorCode MatPtAPSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat *C)
567 {
568   PetscErrorCode      ierr;
569   Mat_PtAPMPI         *ptap;
570   Mat_MPIAIJ          *a=(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data,*c;
571   MPI_Comm            comm;
572   PetscMPIInt         size,rank;
573   Mat                 Cmpi;
574   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
575   PetscInt            am=A->rmap->n,pm=P->rmap->n,pN=P->cmap->N,pn=P->cmap->n;
576   PetscInt            *lnk,i,k,pnz,row,nsend;
577   PetscBT             lnkbt;
578   PetscMPIInt         tagi,tagj,*len_si,*len_s,*len_ri,icompleted=0,nrecv;
579   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
580   PetscInt            len,proc,*dnz,*onz,*owners,nzi,nspacedouble;
581   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
582   MPI_Request         *swaits,*rwaits;
583   MPI_Status          *sstatus,rstatus;
584   PetscLayout         rowmap;
585   PetscInt            *owners_co,*coi,*coj;    /* i and j array of (p->B)^T*A*P - used in the communication */
586   PetscMPIInt         *len_r,*id_r;    /* array of length of comm->size, store send/recv matrix values */
587   PetscInt            *api,*apj,*Jptr,apnz,*prmap=p->garray,con,j,ap_rmax=0,Crmax,*aj,*ai,*pi;
588   Mat_SeqAIJ          *p_loc,*p_oth=NULL,*ad=(Mat_SeqAIJ*)(a->A)->data,*ao=NULL,*c_loc,*c_oth;
589   PetscScalar         *apv;
590   PetscTable          ta;
591 #if defined(PETSC_HAVE_HYPRE)
592   const char          *algTypes[3] = {"scalable","nonscalable","hypre"};
593   PetscInt            nalg = 3;
594 #else
595   const char          *algTypes[2] = {"scalable","nonscalable"};
596   PetscInt            nalg = 2;
597 #endif
598   PetscInt            alg = 1; /* set default algorithm */
599 #if defined(PETSC_USE_INFO)
600   PetscReal           apfill;
601 #endif
602 #if defined(PTAP_PROFILE)
603   PetscLogDouble      t0,t1,t11,t12,t2,t3,t4;
604 #endif
605   PetscBool           flg;
606 
607   PetscFunctionBegin;
608   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
609 
610   /* pick an algorithm */
611   ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
612   PetscOptionsObject->alreadyprinted = PETSC_FALSE; /* a hack to ensure the option shows in '-help' */
613   ierr = PetscOptionsEList("-matptap_via","Algorithmic approach","MatPtAP",algTypes,nalg,algTypes[1],&alg,&flg);CHKERRQ(ierr);
614   ierr = PetscOptionsEnd();CHKERRQ(ierr);
615 
616   if (!flg && pN > 100000) { /* may switch to scalable algorithm as default */
617     MatInfo     Ainfo,Pinfo;
618     PetscInt    nz_local;
619     PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
620 
621     ierr = MatGetInfo(A,MAT_LOCAL,&Ainfo);CHKERRQ(ierr);
622     ierr = MatGetInfo(P,MAT_LOCAL,&Pinfo);CHKERRQ(ierr);
623     nz_local = (PetscInt)(Ainfo.nz_allocated + Pinfo.nz_allocated);
624 
625     if (pN > fill*nz_local) alg_scalable_loc = PETSC_TRUE;
626     ierr = MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm);CHKERRQ(ierr);
627 
628     if (alg_scalable) {
629       alg  = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
630     }
631   }
632 
633   if (alg == 0) {
634     ierr = MatPtAPSymbolic_MPIAIJ_MPIAIJ_scalable(A,P,fill,C);CHKERRQ(ierr);
635     (*C)->ops->ptapnumeric = MatPtAPNumeric_MPIAIJ_MPIAIJ_scalable;
636     PetscFunctionReturn(0);
637 
638 #if defined(PETSC_HAVE_HYPRE)
639   } else if (alg == 2) {
640     /* Use boomerAMGBuildCoarseOperator */
641     ierr = MatPtAPSymbolic_AIJ_AIJ_wHYPRE(A,P,fill,C);CHKERRQ(ierr);
642     PetscFunctionReturn(0);
643 #endif
644   }
645 
646 #if defined(PTAP_PROFILE)
647   ierr = PetscTime(&t0);CHKERRQ(ierr);
648 #endif
649 
650   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
651   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
652 
653   if (size > 1) ao = (Mat_SeqAIJ*)(a->B)->data;
654 
655   /* create symbolic parallel matrix Cmpi */
656   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
657   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
658 
659   /* Do dense axpy in MatPtAPNumeric_MPIAIJ_MPIAIJ() */
660   Cmpi->ops->ptapnumeric = MatPtAPNumeric_MPIAIJ_MPIAIJ;
661 
662   /* create struct Mat_PtAPMPI and attached it to C later */
663   ierr        = PetscNew(&ptap);CHKERRQ(ierr);
664   ptap->reuse = MAT_INITIAL_MATRIX;
665   ptap->algType = alg;
666 
667   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
668   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
669   /* get P_loc by taking all local rows of P */
670   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
671 
672   /* (0) compute Rd = Pd^T, Ro = Po^T  */
673   /* --------------------------------- */
674   ierr = MatTranspose_SeqAIJ(p->A,MAT_INITIAL_MATRIX,&ptap->Rd);CHKERRQ(ierr);
675   ierr = MatTranspose_SeqAIJ(p->B,MAT_INITIAL_MATRIX,&ptap->Ro);CHKERRQ(ierr);
676 #if defined(PTAP_PROFILE)
677   ierr = PetscTime(&t11);CHKERRQ(ierr);
678 #endif
679 
680   /* (1) compute symbolic AP = A_loc*P = Ad*P_loc + Ao*P_oth (api,apj) */
681   /* ----------------------------------------------------------------- */
682   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
683   if (ptap->P_oth) p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
684 
685   /* create and initialize a linked list */
686   ierr = PetscTableCreate(pn,pN,&ta);CHKERRQ(ierr); /* for compute AP_loc and Cmpi */
687   MatRowMergeMax_SeqAIJ(p_loc,ptap->P_loc->rmap->N,ta);
688   MatRowMergeMax_SeqAIJ(p_oth,ptap->P_oth->rmap->N,ta);
689   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr); /* Crmax = nnz(sum of Prows) */
690   /* printf("[%d] est %d, Crmax %d; pN %d\n",rank,5*(p_loc->rmax+p_oth->rmax + (PetscInt)(1.e-2*pN)),Crmax,pN); */
691 
692   ierr = PetscLLCondensedCreate(Crmax,pN,&lnk,&lnkbt);CHKERRQ(ierr);
693 
694   /* Initial FreeSpace size is fill*(nnz(A) + nnz(P)) */
695   if (ao) {
696     ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ad->i[am],PetscIntSumTruncate(ao->i[am],p_loc->i[pm]))),&free_space);CHKERRQ(ierr);
697   } else {
698     ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ad->i[am],p_loc->i[pm])),&free_space);CHKERRQ(ierr);
699   }
700   current_space = free_space;
701   nspacedouble  = 0;
702 
703   ierr   = PetscMalloc1(am+1,&api);CHKERRQ(ierr);
704   api[0] = 0;
705   for (i=0; i<am; i++) {
706     /* diagonal portion: Ad[i,:]*P */
707     ai = ad->i; pi = p_loc->i;
708     nzi = ai[i+1] - ai[i];
709     aj  = ad->j + ai[i];
710     for (j=0; j<nzi; j++) {
711       row  = aj[j];
712       pnz  = pi[row+1] - pi[row];
713       Jptr = p_loc->j + pi[row];
714       /* add non-zero cols of P into the sorted linked list lnk */
715       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
716     }
717     /* off-diagonal portion: Ao[i,:]*P */
718     if (ao) {
719       ai = ao->i; pi = p_oth->i;
720       nzi = ai[i+1] - ai[i];
721       aj  = ao->j + ai[i];
722       for (j=0; j<nzi; j++) {
723         row  = aj[j];
724         pnz  = pi[row+1] - pi[row];
725         Jptr = p_oth->j + pi[row];
726         ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
727       }
728     }
729     apnz     = lnk[0];
730     api[i+1] = api[i] + apnz;
731     if (ap_rmax < apnz) ap_rmax = apnz;
732 
733     /* if free space is not available, double the total space in the list */
734     if (current_space->local_remaining<apnz) {
735       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
736       nspacedouble++;
737     }
738 
739     /* Copy data into free space, then initialize lnk */
740     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
741 
742     current_space->array           += apnz;
743     current_space->local_used      += apnz;
744     current_space->local_remaining -= apnz;
745   }
746   /* Allocate space for apj and apv, initialize apj, and */
747   /* destroy list of free space and other temporary array(s) */
748   ierr   = PetscMalloc2(api[am],&apj,api[am],&apv);CHKERRQ(ierr);
749   ierr   = PetscFreeSpaceContiguous(&free_space,apj);CHKERRQ(ierr);
750   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
751 
752   /* Create AP_loc for reuse */
753   ierr = MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,am,pN,api,apj,apv,&ptap->AP_loc);CHKERRQ(ierr);
754 
755 #if defined(PETSC_USE_INFO)
756   if (ao) {
757     apfill = (PetscReal)api[am]/(ad->i[am]+ao->i[am]+p_loc->i[pm]+1);
758   } else {
759     apfill = (PetscReal)api[am]/(ad->i[am]+p_loc->i[pm]+1);
760   }
761   ptap->AP_loc->info.mallocs           = nspacedouble;
762   ptap->AP_loc->info.fill_ratio_given  = fill;
763   ptap->AP_loc->info.fill_ratio_needed = apfill;
764 
765   if (api[am]) {
766     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);
767     ierr = PetscInfo1(ptap->AP_loc,"Use MatPtAP(A,B,MatReuse,%g,&C) for best AP_loc performance.;\n",(double)apfill);CHKERRQ(ierr);
768   } else {
769     ierr = PetscInfo(ptap->AP_loc,"Nonscalable algorithm, AP_loc is empty \n");CHKERRQ(ierr);
770   }
771 #endif
772 
773 #if defined(PTAP_PROFILE)
774   ierr = PetscTime(&t12);CHKERRQ(ierr);
775 #endif
776 
777   /* (2-1) compute symbolic Co = Ro*AP_loc  */
778   /* ------------------------------------ */
779   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Ro,ptap->AP_loc,fill,&ptap->C_oth);CHKERRQ(ierr);
780 #if defined(PTAP_PROFILE)
781   ierr = PetscTime(&t1);CHKERRQ(ierr);
782 #endif
783 
784   /* (3) send coj of C_oth to other processors  */
785   /* ------------------------------------------ */
786   /* determine row ownership */
787   ierr = PetscLayoutCreate(comm,&rowmap);CHKERRQ(ierr);
788   rowmap->n  = pn;
789   rowmap->bs = 1;
790   ierr   = PetscLayoutSetUp(rowmap);CHKERRQ(ierr);
791   owners = rowmap->range;
792 
793   /* determine the number of messages to send, their lengths */
794   ierr = PetscMalloc4(size,&len_s,size,&len_si,size,&sstatus,size+2,&owners_co);CHKERRQ(ierr);
795   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
796   ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
797 
798   c_oth = (Mat_SeqAIJ*)ptap->C_oth->data;
799   coi   = c_oth->i; coj = c_oth->j;
800   con   = ptap->C_oth->rmap->n;
801   proc  = 0;
802   for (i=0; i<con; i++) {
803     while (prmap[i] >= owners[proc+1]) proc++;
804     len_si[proc]++;               /* num of rows in Co(=Pt*AP) to be sent to [proc] */
805     len_s[proc] += coi[i+1] - coi[i]; /* num of nonzeros in Co to be sent to [proc] */
806   }
807 
808   len          = 0; /* max length of buf_si[], see (4) */
809   owners_co[0] = 0;
810   nsend        = 0;
811   for (proc=0; proc<size; proc++) {
812     owners_co[proc+1] = owners_co[proc] + len_si[proc];
813     if (len_s[proc]) {
814       nsend++;
815       len_si[proc] = 2*(len_si[proc] + 1); /* length of buf_si to be sent to [proc] */
816       len         += len_si[proc];
817     }
818   }
819 
820   /* determine the number and length of messages to receive for coi and coj  */
821   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&nrecv);CHKERRQ(ierr);
822   ierr = PetscGatherMessageLengths2(comm,nsend,nrecv,len_s,len_si,&id_r,&len_r,&len_ri);CHKERRQ(ierr);
823 
824   /* post the Irecv and Isend of coj */
825   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
826   ierr = PetscPostIrecvInt(comm,tagj,nrecv,id_r,len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
827   ierr = PetscMalloc1(nsend+1,&swaits);CHKERRQ(ierr);
828   for (proc=0, k=0; proc<size; proc++) {
829     if (!len_s[proc]) continue;
830     i    = owners_co[proc];
831     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
832     k++;
833   }
834 
835   /* (2-2) compute symbolic C_loc = Rd*AP_loc */
836   /* ---------------------------------------- */
837   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Rd,ptap->AP_loc,fill,&ptap->C_loc);CHKERRQ(ierr);
838   c_loc = (Mat_SeqAIJ*)ptap->C_loc->data;
839 
840   /* receives coj are complete */
841   for (i=0; i<nrecv; i++) {
842     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
843   }
844   ierr = PetscFree(rwaits);CHKERRQ(ierr);
845   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
846 
847   /* add received column indices into ta to update Crmax */
848   for (k=0; k<nrecv; k++) {/* k-th received message */
849     Jptr = buf_rj[k];
850     for (j=0; j<len_r[k]; j++) {
851       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
852     }
853   }
854   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
855   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
856 
857   /* (4) send and recv coi */
858   /*-----------------------*/
859   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
860   ierr   = PetscPostIrecvInt(comm,tagi,nrecv,id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
861   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
862   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
863   for (proc=0,k=0; proc<size; proc++) {
864     if (!len_s[proc]) continue;
865     /* form outgoing message for i-structure:
866          buf_si[0]:                 nrows to be sent
867                [1:nrows]:           row index (global)
868                [nrows+1:2*nrows+1]: i-structure index
869     */
870     /*-------------------------------------------*/
871     nrows       = len_si[proc]/2 - 1; /* num of rows in Co to be sent to [proc] */
872     buf_si_i    = buf_si + nrows+1;
873     buf_si[0]   = nrows;
874     buf_si_i[0] = 0;
875     nrows       = 0;
876     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
877       nzi = coi[i+1] - coi[i];
878       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
879       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
880       nrows++;
881     }
882     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
883     k++;
884     buf_si += len_si[proc];
885   }
886   for (i=0; i<nrecv; i++) {
887     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
888   }
889   ierr = PetscFree(rwaits);CHKERRQ(ierr);
890   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
891 
892   ierr = PetscFree4(len_s,len_si,sstatus,owners_co);CHKERRQ(ierr);
893   ierr = PetscFree(len_ri);CHKERRQ(ierr);
894   ierr = PetscFree(swaits);CHKERRQ(ierr);
895   ierr = PetscFree(buf_s);CHKERRQ(ierr);
896 #if defined(PTAP_PROFILE)
897   ierr = PetscTime(&t2);CHKERRQ(ierr);
898 #endif
899   /* (5) compute the local portion of Cmpi      */
900   /* ------------------------------------------ */
901   /* set initial free space to be Crmax, sufficient for holding nozeros in each row of Cmpi */
902   ierr          = PetscFreeSpaceGet(Crmax,&free_space);CHKERRQ(ierr);
903   current_space = free_space;
904 
905   ierr = PetscMalloc3(nrecv,&buf_ri_k,nrecv,&nextrow,nrecv,&nextci);CHKERRQ(ierr);
906   for (k=0; k<nrecv; k++) {
907     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
908     nrows       = *buf_ri_k[k];
909     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
910     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
911   }
912 
913   ierr = MatPreallocateInitialize(comm,pn,pn,dnz,onz);CHKERRQ(ierr);
914   ierr = PetscLLCondensedCreate(Crmax,pN,&lnk,&lnkbt);CHKERRQ(ierr);
915   for (i=0; i<pn; i++) {
916     /* add C_loc into Cmpi */
917     nzi  = c_loc->i[i+1] - c_loc->i[i];
918     Jptr = c_loc->j + c_loc->i[i];
919     ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
920 
921     /* add received col data into lnk */
922     for (k=0; k<nrecv; k++) { /* k-th received message */
923       if (i == *nextrow[k]) { /* i-th row */
924         nzi  = *(nextci[k]+1) - *nextci[k];
925         Jptr = buf_rj[k] + *nextci[k];
926         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
927         nextrow[k]++; nextci[k]++;
928       }
929     }
930     nzi = lnk[0];
931 
932     /* copy data into free space, then initialize lnk */
933     ierr = PetscLLCondensedClean(pN,nzi,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
934     ierr = MatPreallocateSet(i+owners[rank],nzi,current_space->array,dnz,onz);CHKERRQ(ierr);
935   }
936   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
937   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
938   ierr = PetscFreeSpaceDestroy(free_space);CHKERRQ(ierr);
939 #if defined(PTAP_PROFILE)
940   ierr = PetscTime(&t3);CHKERRQ(ierr);
941 #endif
942 
943   /* local sizes and preallocation */
944   ierr = MatSetSizes(Cmpi,pn,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
945   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(P->cmap->bs));CHKERRQ(ierr);
946   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
947   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
948 
949   /* members in merge */
950   ierr = PetscFree(id_r);CHKERRQ(ierr);
951   ierr = PetscFree(len_r);CHKERRQ(ierr);
952   ierr = PetscFree(buf_ri[0]);CHKERRQ(ierr);
953   ierr = PetscFree(buf_ri);CHKERRQ(ierr);
954   ierr = PetscFree(buf_rj[0]);CHKERRQ(ierr);
955   ierr = PetscFree(buf_rj);CHKERRQ(ierr);
956   ierr = PetscLayoutDestroy(&rowmap);CHKERRQ(ierr);
957 
958   /* attach the supporting struct to Cmpi for reuse */
959   c = (Mat_MPIAIJ*)Cmpi->data;
960   c->ptap         = ptap;
961   ptap->duplicate = Cmpi->ops->duplicate;
962   ptap->destroy   = Cmpi->ops->destroy;
963   ptap->view      = Cmpi->ops->view;
964 
965   if (alg == 1) {
966     ierr = PetscCalloc1(pN,&ptap->apa);CHKERRQ(ierr);
967   }
968 
969   /* Cmpi is not ready for use - assembly will be done by MatPtAPNumeric() */
970   Cmpi->assembled        = PETSC_FALSE;
971   Cmpi->ops->destroy     = MatDestroy_MPIAIJ_PtAP;
972   Cmpi->ops->duplicate   = MatDuplicate_MPIAIJ_MatPtAP;
973   Cmpi->ops->view        = MatView_MPIAIJ_PtAP;
974   *C                     = Cmpi;
975 
976 #if defined(PTAP_PROFILE)
977   ierr = PetscTime(&t4);CHKERRQ(ierr);
978   if (rank == 1) {
979     printf("PtAPSym: %g + %g + %g + %g + %g + %g = %g\n",t11-t0,t1-t11,t12-t11,t2-t2,t3-t2,t4-t3,t4-t0);
980   }
981 #endif
982   PetscFunctionReturn(0);
983 }
984 
985 PetscErrorCode MatPtAPNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
986 {
987   PetscErrorCode    ierr;
988   Mat_MPIAIJ        *a=(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
989   Mat_SeqAIJ        *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
990   Mat_SeqAIJ        *ap,*p_loc,*p_oth=NULL,*c_seq;
991   Mat_PtAPMPI       *ptap = c->ptap;
992   Mat               AP_loc,C_loc,C_oth;
993   PetscInt          i,rstart,rend,cm,ncols,row;
994   PetscInt          *api,*apj,am = A->rmap->n,j,col,apnz;
995   PetscScalar       *apa;
996   const PetscInt    *cols;
997   const PetscScalar *vals;
998 #if defined(PTAP_PROFILE)
999   PetscMPIInt       rank;
1000   MPI_Comm          comm;
1001   PetscLogDouble    t0,t1,t2,t3,t4,eR,eAP,eCseq,eCmpi;
1002 #endif
1003 
1004   PetscFunctionBegin;
1005   ierr = MatZeroEntries(C);CHKERRQ(ierr);
1006 
1007   /* 1) get R = Pd^T,Ro = Po^T */
1008 #if defined(PTAP_PROFILE)
1009   ierr = PetscTime(&t0);CHKERRQ(ierr);
1010 #endif
1011   if (ptap->reuse == MAT_REUSE_MATRIX) {
1012     ierr = MatTranspose_SeqAIJ(p->A,MAT_REUSE_MATRIX,&ptap->Rd);CHKERRQ(ierr);
1013     ierr = MatTranspose_SeqAIJ(p->B,MAT_REUSE_MATRIX,&ptap->Ro);CHKERRQ(ierr);
1014   }
1015 #if defined(PTAP_PROFILE)
1016   ierr = PetscTime(&t1);CHKERRQ(ierr);
1017   eR = t1 - t0;
1018 #endif
1019 
1020   /* 2) get AP_loc */
1021   AP_loc = ptap->AP_loc;
1022   ap = (Mat_SeqAIJ*)AP_loc->data;
1023 
1024   /* 2-1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
1025   /*-----------------------------------------------------*/
1026   if (ptap->reuse == MAT_REUSE_MATRIX) {
1027     /* P_oth and P_loc are obtained in MatPtASymbolic() when reuse == MAT_INITIAL_MATRIX */
1028     ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
1029     ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
1030   }
1031 
1032   /* 2-2) compute numeric A_loc*P - dominating part */
1033   /* ---------------------------------------------- */
1034   /* get data from symbolic products */
1035   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
1036   if (ptap->P_oth) {
1037     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
1038   }
1039   apa   = ptap->apa;
1040   api   = ap->i;
1041   apj   = ap->j;
1042   for (i=0; i<am; i++) {
1043     /* AP[i,:] = A[i,:]*P = Ad*P_loc Ao*P_oth */
1044     AProw_nonscalable(i,ad,ao,p_loc,p_oth,apa);
1045     apnz = api[i+1] - api[i];
1046     for (j=0; j<apnz; j++) {
1047       col = apj[j+api[i]];
1048       ap->a[j+ap->i[i]] = apa[col];
1049       apa[col] = 0.0;
1050     }
1051     ierr = PetscLogFlops(2.0*apnz);CHKERRQ(ierr);
1052   }
1053 #if defined(PTAP_PROFILE)
1054   ierr = PetscTime(&t2);CHKERRQ(ierr);
1055   eAP = t2 - t1;
1056 #endif
1057 
1058   /* 3) C_loc = Rd*AP_loc, C_oth = Ro*AP_loc */
1059   ierr = ((ptap->C_loc)->ops->matmultnumeric)(ptap->Rd,AP_loc,ptap->C_loc);CHKERRQ(ierr);
1060   ierr = ((ptap->C_oth)->ops->matmultnumeric)(ptap->Ro,AP_loc,ptap->C_oth);CHKERRQ(ierr);
1061   C_loc = ptap->C_loc;
1062   C_oth = ptap->C_oth;
1063 #if defined(PTAP_PROFILE)
1064   ierr = PetscTime(&t3);CHKERRQ(ierr);
1065   eCseq = t3 - t2;
1066 #endif
1067 
1068   /* add C_loc and Co to to C */
1069   ierr = MatGetOwnershipRange(C,&rstart,&rend);CHKERRQ(ierr);
1070 
1071   /* C_loc -> C */
1072   cm    = C_loc->rmap->N;
1073   c_seq = (Mat_SeqAIJ*)C_loc->data;
1074   cols = c_seq->j;
1075   vals = c_seq->a;
1076   for (i=0; i<cm; i++) {
1077     ncols = c_seq->i[i+1] - c_seq->i[i];
1078     row = rstart + i;
1079     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1080     cols += ncols; vals += ncols;
1081   }
1082 
1083   /* Co -> C, off-processor part */
1084   cm = C_oth->rmap->N;
1085   c_seq = (Mat_SeqAIJ*)C_oth->data;
1086   cols = c_seq->j;
1087   vals = c_seq->a;
1088   for (i=0; i<cm; i++) {
1089     ncols = c_seq->i[i+1] - c_seq->i[i];
1090     row = p->garray[i];
1091     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1092     cols += ncols; vals += ncols;
1093   }
1094   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1095   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1096 #if defined(PTAP_PROFILE)
1097   ierr = PetscTime(&t4);CHKERRQ(ierr);
1098   eCmpi = t4 - t3;
1099 
1100   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1101   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1102   if (rank==1) {
1103     ierr = PetscPrintf(MPI_COMM_SELF," R %g, AP %g, Cseq %g, Cmpi %g = %g\n", eR,eAP,eCseq,eCmpi,eR+eAP+eCseq+eCmpi);CHKERRQ(ierr);
1104   }
1105 #endif
1106   ptap->reuse = MAT_REUSE_MATRIX;
1107   PetscFunctionReturn(0);
1108 }
1109 
1110 PetscErrorCode MatPtAPSymbolic_MPIAIJ_MPIAIJ_ptap(Mat A,Mat P,PetscReal fill,Mat *C)
1111 {
1112   PetscErrorCode      ierr;
1113   Mat                 Cmpi;
1114   Mat_PtAPMPI         *ptap;
1115   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1116   Mat_MPIAIJ          *a        =(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data,*c;
1117   Mat_SeqAIJ          *ad       =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
1118   Mat_SeqAIJ          *p_loc,*p_oth;
1119   PetscInt            *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pdti,*pdtj,*poti,*potj,*ptJ;
1120   PetscInt            *adi=ad->i,*aj,*aoi=ao->i,nnz;
1121   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1122   PetscInt            am=A->rmap->n,pN=P->cmap->N,pm=P->rmap->n,pn=P->cmap->n;
1123   PetscBT             lnkbt;
1124   MPI_Comm            comm;
1125   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri,icompleted=0;
1126   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1127   PetscInt            len,proc,*dnz,*onz,*owners;
1128   PetscInt            nzi,*pti,*ptj;
1129   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1130   MPI_Request         *swaits,*rwaits;
1131   MPI_Status          *sstatus,rstatus;
1132   Mat_Merge_SeqsToMPI *merge;
1133   PetscInt            *api,*apj,*Jptr,apnz,*prmap=p->garray,pon,nspacedouble=0,j,ap_rmax=0;
1134   PetscReal           afill=1.0,afill_tmp;
1135 
1136   PetscFunctionBegin;
1137   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1138   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1139   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1140 
1141   /* create struct Mat_PtAPMPI and attached it to C later */
1142   ierr        = PetscNew(&ptap);CHKERRQ(ierr);
1143   ierr        = PetscNew(&merge);CHKERRQ(ierr);
1144   ptap->merge = merge;
1145   ptap->reuse = MAT_INITIAL_MATRIX;
1146 
1147   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
1148   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
1149 
1150   /* get P_loc by taking all local rows of P */
1151   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
1152 
1153   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
1154   p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
1155   pi_loc = p_loc->i; pj_loc = p_loc->j;
1156   pi_oth = p_oth->i; pj_oth = p_oth->j;
1157 
1158   /* (1) compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth (api,apj) */
1159   /*--------------------------------------------------------------------------*/
1160   ierr   = PetscMalloc1(am+1,&api);CHKERRQ(ierr);
1161   api[0] = 0;
1162 
1163   /* create and initialize a linked list */
1164   ierr = PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt);CHKERRQ(ierr);
1165 
1166   /* Initial FreeSpace size is fill*(nnz(A) + nnz(P)) -OOM for ex56, np=8k on Intrepid! */
1167   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space);CHKERRQ(ierr);
1168   current_space = free_space;
1169 
1170   for (i=0; i<am; i++) {
1171     /* diagonal portion of A */
1172     nzi = adi[i+1] - adi[i];
1173     aj  = ad->j + adi[i];
1174     for (j=0; j<nzi; j++) {
1175       row  = aj[j];
1176       pnz  = pi_loc[row+1] - pi_loc[row];
1177       Jptr = pj_loc + pi_loc[row];
1178       /* add non-zero cols of P into the sorted linked list lnk */
1179       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1180     }
1181     /* off-diagonal portion of A */
1182     nzi = aoi[i+1] - aoi[i];
1183     aj  = ao->j + aoi[i];
1184     for (j=0; j<nzi; j++) {
1185       row  = aj[j];
1186       pnz  = pi_oth[row+1] - pi_oth[row];
1187       Jptr = pj_oth + pi_oth[row];
1188       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1189     }
1190     apnz     = lnk[0];
1191     api[i+1] = api[i] + apnz;
1192     if (ap_rmax < apnz) ap_rmax = apnz;
1193 
1194     /* if free space is not available, double the total space in the list */
1195     if (current_space->local_remaining<apnz) {
1196       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1197       nspacedouble++;
1198     }
1199 
1200     /* Copy data into free space, then initialize lnk */
1201     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1202 
1203     current_space->array           += apnz;
1204     current_space->local_used      += apnz;
1205     current_space->local_remaining -= apnz;
1206   }
1207 
1208   /* Allocate space for apj, initialize apj, and */
1209   /* destroy list of free space and other temporary array(s) */
1210   ierr      = PetscMalloc1(api[am]+1,&apj);CHKERRQ(ierr);
1211   ierr      = PetscFreeSpaceContiguous(&free_space,apj);CHKERRQ(ierr);
1212   afill_tmp = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1);
1213   if (afill_tmp > afill) afill = afill_tmp;
1214 
1215   /* (2) determine symbolic Co=(p->B)^T*AP - send to others (coi,coj)*/
1216   /*-----------------------------------------------------------------*/
1217   ierr = MatGetSymbolicTranspose_SeqAIJ(p->B,&poti,&potj);CHKERRQ(ierr);
1218 
1219   /* then, compute symbolic Co = (p->B)^T*AP */
1220   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1221                          >= (num of nonzero rows of C_seq) - pn */
1222   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
1223   coi[0] = 0;
1224 
1225   /* set initial free space to be fill*(nnz(p->B) + nnz(AP)) */
1226   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(poti[pon],api[am]));
1227   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1228   current_space = free_space;
1229 
1230   for (i=0; i<pon; i++) {
1231     pnz = poti[i+1] - poti[i];
1232     ptJ = potj + poti[i];
1233     for (j=0; j<pnz; j++) {
1234       row  = ptJ[j]; /* row of AP == col of Pot */
1235       apnz = api[row+1] - api[row];
1236       Jptr = apj + api[row];
1237       /* add non-zero cols of AP into the sorted linked list lnk */
1238       ierr = PetscLLCondensedAddSorted(apnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1239     }
1240     nnz = lnk[0];
1241 
1242     /* If free space is not available, double the total space in the list */
1243     if (current_space->local_remaining<nnz) {
1244       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1245       nspacedouble++;
1246     }
1247 
1248     /* Copy data into free space, and zero out denserows */
1249     ierr = PetscLLCondensedClean(pN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1250 
1251     current_space->array           += nnz;
1252     current_space->local_used      += nnz;
1253     current_space->local_remaining -= nnz;
1254 
1255     coi[i+1] = coi[i] + nnz;
1256   }
1257 
1258   ierr      = PetscMalloc1(coi[pon],&coj);CHKERRQ(ierr);
1259   ierr      = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1260   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + api[am]+1);
1261   if (afill_tmp > afill) afill = afill_tmp;
1262   ierr = MatRestoreSymbolicTranspose_SeqAIJ(p->B,&poti,&potj);CHKERRQ(ierr);
1263 
1264   /* (3) send j-array (coj) of Co to other processors */
1265   /*--------------------------------------------------*/
1266   ierr = PetscCalloc1(size,&merge->len_s);CHKERRQ(ierr);
1267   len_s        = merge->len_s;
1268   merge->nsend = 0;
1269 
1270 
1271   /* determine row ownership */
1272   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1273   merge->rowmap->n  = pn;
1274   merge->rowmap->bs = 1;
1275 
1276   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1277   owners = merge->rowmap->range;
1278 
1279   /* determine the number of messages to send, their lengths */
1280   ierr = PetscMalloc2(size,&len_si,size,&sstatus);CHKERRQ(ierr);
1281   ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1282   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
1283 
1284   proc = 0;
1285   for (i=0; i<pon; i++) {
1286     while (prmap[i] >= owners[proc+1]) proc++;
1287     len_si[proc]++;               /* num of rows in Co(=Pt*AP) to be sent to [proc] */
1288     len_s[proc] += coi[i+1] - coi[i]; /* num of nonzeros in Co to be sent to [proc] */
1289   }
1290 
1291   len          = 0; /* max length of buf_si[], see (4) */
1292   owners_co[0] = 0;
1293   for (proc=0; proc<size; proc++) {
1294     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1295     if (len_s[proc]) {
1296       merge->nsend++;
1297       len_si[proc] = 2*(len_si[proc] + 1); /* length of buf_si to be sent to [proc] */
1298       len         += len_si[proc];
1299     }
1300   }
1301 
1302   /* determine the number and length of messages to receive for coi and coj  */
1303   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1304   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1305 
1306   /* post the Irecv and Isend of coj */
1307   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1308   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1309   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
1310   for (proc=0, k=0; proc<size; proc++) {
1311     if (!len_s[proc]) continue;
1312     i    = owners_co[proc];
1313     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1314     k++;
1315   }
1316 
1317   /* receives and sends of coj are complete */
1318   for (i=0; i<merge->nrecv; i++) {
1319     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1320   }
1321   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1322   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1323 
1324   /* (4) send and recv coi */
1325   /*-----------------------*/
1326   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1327   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1328   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1329   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1330   for (proc=0,k=0; proc<size; proc++) {
1331     if (!len_s[proc]) continue;
1332     /* form outgoing message for i-structure:
1333          buf_si[0]:                 nrows to be sent
1334                [1:nrows]:           row index (global)
1335                [nrows+1:2*nrows+1]: i-structure index
1336     */
1337     /*-------------------------------------------*/
1338     nrows       = len_si[proc]/2 - 1; /* num of rows in Co to be sent to [proc] */
1339     buf_si_i    = buf_si + nrows+1;
1340     buf_si[0]   = nrows;
1341     buf_si_i[0] = 0;
1342     nrows       = 0;
1343     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1344       nzi = coi[i+1] - coi[i];
1345       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1346       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1347       nrows++;
1348     }
1349     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1350     k++;
1351     buf_si += len_si[proc];
1352   }
1353   i = merge->nrecv;
1354   while (i--) {
1355     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1356   }
1357   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1358   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1359 
1360   ierr = PetscFree2(len_si,sstatus);CHKERRQ(ierr);
1361   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1362   ierr = PetscFree(swaits);CHKERRQ(ierr);
1363   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1364 
1365   /* (5) compute the local portion of C (mpi mat) */
1366   /*----------------------------------------------*/
1367   ierr = MatGetSymbolicTranspose_SeqAIJ(p->A,&pdti,&pdtj);CHKERRQ(ierr);
1368 
1369   /* allocate pti array and free space for accumulating nonzero column info */
1370   ierr   = PetscMalloc1(pn+1,&pti);CHKERRQ(ierr);
1371   pti[0] = 0;
1372 
1373   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
1374   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(pi_loc[pm],api[am]));
1375   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1376   current_space = free_space;
1377 
1378   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1379   for (k=0; k<merge->nrecv; k++) {
1380     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1381     nrows       = *buf_ri_k[k];
1382     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1383     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1384   }
1385   ierr = MatPreallocateInitialize(comm,pn,pn,dnz,onz);CHKERRQ(ierr);
1386 
1387   for (i=0; i<pn; i++) {
1388     /* add pdt[i,:]*AP into lnk */
1389     pnz = pdti[i+1] - pdti[i];
1390     ptJ = pdtj + pdti[i];
1391     for (j=0; j<pnz; j++) {
1392       row  = ptJ[j];  /* row of AP == col of Pt */
1393       apnz = api[row+1] - api[row];
1394       Jptr = apj + api[row];
1395       /* add non-zero cols of AP into the sorted linked list lnk */
1396       ierr = PetscLLCondensedAddSorted(apnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1397     }
1398 
1399     /* add received col data into lnk */
1400     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1401       if (i == *nextrow[k]) { /* i-th row */
1402         nzi  = *(nextci[k]+1) - *nextci[k];
1403         Jptr = buf_rj[k] + *nextci[k];
1404         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1405         nextrow[k]++; nextci[k]++;
1406       }
1407     }
1408     nnz = lnk[0];
1409 
1410     /* if free space is not available, make more free space */
1411     if (current_space->local_remaining<nnz) {
1412       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1413       nspacedouble++;
1414     }
1415     /* copy data into free space, then initialize lnk */
1416     ierr = PetscLLCondensedClean(pN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1417     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1418 
1419     current_space->array           += nnz;
1420     current_space->local_used      += nnz;
1421     current_space->local_remaining -= nnz;
1422 
1423     pti[i+1] = pti[i] + nnz;
1424   }
1425   ierr = MatRestoreSymbolicTranspose_SeqAIJ(p->A,&pdti,&pdtj);CHKERRQ(ierr);
1426   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1427 
1428   ierr      = PetscMalloc1(pti[pn]+1,&ptj);CHKERRQ(ierr);
1429   ierr      = PetscFreeSpaceContiguous(&free_space,ptj);CHKERRQ(ierr);
1430   afill_tmp = (PetscReal)pti[pn]/(pi_loc[pm] + api[am]+1);
1431   if (afill_tmp > afill) afill = afill_tmp;
1432   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1433 
1434   /* (6) create symbolic parallel matrix Cmpi */
1435   /*------------------------------------------*/
1436   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1437   ierr = MatSetSizes(Cmpi,pn,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1438   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(P->cmap->bs));CHKERRQ(ierr);
1439   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1440   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1441   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1442 
1443   merge->bi        = pti;      /* Cseq->i */
1444   merge->bj        = ptj;      /* Cseq->j */
1445   merge->coi       = coi;      /* Co->i   */
1446   merge->coj       = coj;      /* Co->j   */
1447   merge->buf_ri    = buf_ri;
1448   merge->buf_rj    = buf_rj;
1449   merge->owners_co = owners_co;
1450   merge->destroy   = Cmpi->ops->destroy;
1451 
1452   /* attach the supporting struct to Cmpi for reuse */
1453   c           = (Mat_MPIAIJ*)Cmpi->data;
1454   c->ptap     = ptap;
1455   ptap->api   = api;
1456   ptap->apj   = apj;
1457   ptap->duplicate = Cmpi->ops->duplicate;
1458   ptap->destroy   = Cmpi->ops->destroy;
1459 
1460   /* Cmpi is not ready for use - assembly will be done by MatPtAPNumeric() */
1461   Cmpi->assembled        = PETSC_FALSE;
1462   Cmpi->ops->destroy     = MatDestroy_MPIAIJ_PtAP;
1463   Cmpi->ops->duplicate   = MatDuplicate_MPIAIJ_MatPtAP;
1464   Cmpi->ops->ptapnumeric = MatPtAPNumeric_MPIAIJ_MPIAIJ_ptap;
1465   *C                     = Cmpi;
1466 
1467   /* flag 'scalable' determines which implementations to be used:
1468        0: do dense axpy in MatPtAPNumeric() - fast, but requires storage of a nonscalable dense array apa;
1469        1: do sparse axpy in MatPtAPNumeric() - might slow, uses a sparse array apa */
1470   /* set default scalable */
1471   ptap->scalable = PETSC_FALSE; /* PETSC_TRUE; */
1472 
1473   ierr = PetscOptionsGetBool(((PetscObject)Cmpi)->options,((PetscObject)Cmpi)->prefix,"-matptap_scalable",&ptap->scalable,NULL);CHKERRQ(ierr);
1474   if (!ptap->scalable) {  /* Do dense axpy */
1475     ierr = PetscCalloc1(pN,&ptap->apa);CHKERRQ(ierr);
1476   } else {
1477     ierr = PetscCalloc1(ap_rmax+1,&ptap->apa);CHKERRQ(ierr);
1478   }
1479 
1480 #if defined(PETSC_USE_INFO)
1481   if (pti[pn] != 0) {
1482     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1483     ierr = PetscInfo1(Cmpi,"Use MatPtAP(A,P,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
1484   } else {
1485     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1486   }
1487 #endif
1488   PetscFunctionReturn(0);
1489 }
1490 
1491 PetscErrorCode MatPtAPNumeric_MPIAIJ_MPIAIJ_ptap(Mat A,Mat P,Mat C)
1492 {
1493   PetscErrorCode      ierr;
1494   Mat_MPIAIJ          *a =(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1495   Mat_SeqAIJ          *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
1496   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
1497   Mat_SeqAIJ          *p_loc,*p_oth;
1498   Mat_PtAPMPI         *ptap;
1499   Mat_Merge_SeqsToMPI *merge;
1500   PetscInt            *adi=ad->i,*aoi=ao->i,*adj,*aoj,*apJ,nextp;
1501   PetscInt            *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pJ,*pj;
1502   PetscInt            i,j,k,anz,pnz,apnz,nextap,row,*cj;
1503   MatScalar           *ada,*aoa,*apa,*pa,*ca,*pa_loc,*pa_oth,valtmp;
1504   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
1505   MPI_Comm            comm;
1506   PetscMPIInt         size,rank,taga,*len_s;
1507   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
1508   PetscInt            **buf_ri,**buf_rj;
1509   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
1510   MPI_Request         *s_waits,*r_waits;
1511   MPI_Status          *status;
1512   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
1513   PetscInt            *api,*apj,*coi,*coj;
1514   PetscInt            *poJ=po->j,*pdJ=pd->j,pcstart=P->cmap->rstart,pcend=P->cmap->rend;
1515   PetscBool           scalable;
1516 #if defined(PTAP_PROFILE)
1517   PetscLogDouble t0,t1,t2,eP,t3,t4,et2_AP=0.0,ePtAP=0.0,t2_0,t2_1,t2_2;
1518 #endif
1519 
1520   PetscFunctionBegin;
1521   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1522 #if defined(PTAP_PROFILE)
1523   ierr = PetscTime(&t0);CHKERRQ(ierr);
1524 #endif
1525   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1526   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1527 
1528   ptap = c->ptap;
1529   if (!ptap) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_INCOMP,"MatPtAP() has not been called to create matrix C yet, cannot use MAT_REUSE_MATRIX");
1530   merge    = ptap->merge;
1531   apa      = ptap->apa;
1532   scalable = ptap->scalable;
1533 
1534   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
1535   /*-----------------------------------------------------*/
1536   if (ptap->reuse == MAT_INITIAL_MATRIX) {
1537     /* P_oth and P_loc are obtained in MatPtASymbolic(), skip calling MatGetBrowsOfAoCols() and MatMPIAIJGetLocalMat() */
1538     ptap->reuse = MAT_REUSE_MATRIX;
1539   } else { /* update numerical values of P_oth and P_loc */
1540     ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
1541     ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
1542   }
1543 #if defined(PTAP_PROFILE)
1544   ierr = PetscTime(&t1);CHKERRQ(ierr);
1545   eP = t1-t0;
1546 #endif
1547   /*
1548   printf("[%d] Ad: %d, %d; Ao: %d, %d; P_loc: %d, %d; P_oth %d, %d;\n",rank,
1549          a->A->rmap->N,a->A->cmap->N,a->B->rmap->N,a->B->cmap->N,
1550          ptap->P_loc->rmap->N,ptap->P_loc->cmap->N,
1551          ptap->P_oth->rmap->N,ptap->P_oth->cmap->N);
1552    */
1553 
1554   /* 2) compute numeric C_seq = P_loc^T*A_loc*P - dominating part */
1555   /*--------------------------------------------------------------*/
1556   /* get data from symbolic products */
1557   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
1558   p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
1559   pi_loc=p_loc->i; pj_loc=p_loc->j; pa_loc=p_loc->a;
1560   pi_oth=p_oth->i; pj_oth=p_oth->j; pa_oth=p_oth->a;
1561 
1562   coi  = merge->coi; coj = merge->coj;
1563   ierr = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
1564 
1565   bi     = merge->bi; bj = merge->bj;
1566   owners = merge->rowmap->range;
1567   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);  /* ba: Cseq->a */
1568 
1569   api = ptap->api; apj = ptap->apj;
1570 
1571   if (!scalable) { /* Do dense axpy on apa (length of pN, stores A[i,:]*P) - nonscalable, but faster (could take 1/3 scalable time) */
1572     ierr = PetscInfo(C,"Using non-scalable dense axpy\n");CHKERRQ(ierr);
1573 #if defined(PTAP_PROFILE)
1574     ierr = PetscTime(&t1);CHKERRQ(ierr);
1575 #endif
1576     for (i=0; i<am; i++) {
1577 #if defined(PTAP_PROFILE)
1578       ierr = PetscTime(&t2_0);CHKERRQ(ierr);
1579 #endif
1580       /* 2-a) form i-th sparse row of A_loc*P = Ad*P_loc + Ao*P_oth */
1581       /*------------------------------------------------------------*/
1582       apJ = apj + api[i];
1583 
1584       /* diagonal portion of A */
1585       anz = adi[i+1] - adi[i];
1586       adj = ad->j + adi[i];
1587       ada = ad->a + adi[i];
1588       for (j=0; j<anz; j++) {
1589         row = adj[j];
1590         pnz = pi_loc[row+1] - pi_loc[row];
1591         pj  = pj_loc + pi_loc[row];
1592         pa  = pa_loc + pi_loc[row];
1593 
1594         /* perform dense axpy */
1595         valtmp = ada[j];
1596         for (k=0; k<pnz; k++) {
1597           apa[pj[k]] += valtmp*pa[k];
1598         }
1599         ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
1600       }
1601 
1602       /* off-diagonal portion of A */
1603       anz = aoi[i+1] - aoi[i];
1604       aoj = ao->j + aoi[i];
1605       aoa = ao->a + aoi[i];
1606       for (j=0; j<anz; j++) {
1607         row = aoj[j];
1608         pnz = pi_oth[row+1] - pi_oth[row];
1609         pj  = pj_oth + pi_oth[row];
1610         pa  = pa_oth + pi_oth[row];
1611 
1612         /* perform dense axpy */
1613         valtmp = aoa[j];
1614         for (k=0; k<pnz; k++) {
1615           apa[pj[k]] += valtmp*pa[k];
1616         }
1617         ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
1618       }
1619 #if defined(PTAP_PROFILE)
1620       ierr    = PetscTime(&t2_1);CHKERRQ(ierr);
1621       et2_AP += t2_1 - t2_0;
1622 #endif
1623 
1624       /* 2-b) Compute Cseq = P_loc[i,:]^T*AP[i,:] using outer product */
1625       /*--------------------------------------------------------------*/
1626       apnz = api[i+1] - api[i];
1627       /* put the value into Co=(p->B)^T*AP (off-diagonal part, send to others) */
1628       pnz = po->i[i+1] - po->i[i];
1629       poJ = po->j + po->i[i];
1630       pA  = po->a + po->i[i];
1631       for (j=0; j<pnz; j++) {
1632         row = poJ[j];
1633         cnz = coi[row+1] - coi[row];
1634         cj  = coj + coi[row];
1635         ca  = coa + coi[row];
1636         /* perform dense axpy */
1637         valtmp = pA[j];
1638         for (k=0; k<cnz; k++) {
1639           ca[k] += valtmp*apa[cj[k]];
1640         }
1641         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1642       }
1643       /* put the value into Cd (diagonal part) */
1644       pnz = pd->i[i+1] - pd->i[i];
1645       pdJ = pd->j + pd->i[i];
1646       pA  = pd->a + pd->i[i];
1647       for (j=0; j<pnz; j++) {
1648         row = pdJ[j];
1649         cnz = bi[row+1] - bi[row];
1650         cj  = bj + bi[row];
1651         ca  = ba + bi[row];
1652         /* perform dense axpy */
1653         valtmp = pA[j];
1654         for (k=0; k<cnz; k++) {
1655           ca[k] += valtmp*apa[cj[k]];
1656         }
1657         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1658       }
1659 
1660       /* zero the current row of A*P */
1661       for (k=0; k<apnz; k++) apa[apJ[k]] = 0.0;
1662 #if defined(PTAP_PROFILE)
1663       ierr      = PetscTime(&t2_2);CHKERRQ(ierr);
1664       ePtAP += t2_2 - t2_1;
1665 #endif
1666     }
1667   } else { /* Do sparse axpy on apa (length of ap_rmax, stores A[i,:]*P) - scalable, but slower */
1668     ierr = PetscInfo(C,"Using scalable sparse axpy\n");CHKERRQ(ierr);
1669     /*-----------------------------------------------------------------------------------------*/
1670     pA=pa_loc;
1671     for (i=0; i<am; i++) {
1672 #if defined(PTAP_PROFILE)
1673       ierr = PetscTime(&t2_0);CHKERRQ(ierr);
1674 #endif
1675       /* form i-th sparse row of A*P */
1676       apnz = api[i+1] - api[i];
1677       apJ  = apj + api[i];
1678       /* diagonal portion of A */
1679       anz = adi[i+1] - adi[i];
1680       adj = ad->j + adi[i];
1681       ada = ad->a + adi[i];
1682       for (j=0; j<anz; j++) {
1683         row    = adj[j];
1684         pnz    = pi_loc[row+1] - pi_loc[row];
1685         pj     = pj_loc + pi_loc[row];
1686         pa     = pa_loc + pi_loc[row];
1687         valtmp = ada[j];
1688         nextp  = 0;
1689         for (k=0; nextp<pnz; k++) {
1690           if (apJ[k] == pj[nextp]) { /* col of AP == col of P */
1691             apa[k] += valtmp*pa[nextp++];
1692           }
1693         }
1694         ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
1695       }
1696       /* off-diagonal portion of A */
1697       anz = aoi[i+1] - aoi[i];
1698       aoj = ao->j + aoi[i];
1699       aoa = ao->a + aoi[i];
1700       for (j=0; j<anz; j++) {
1701         row    = aoj[j];
1702         pnz    = pi_oth[row+1] - pi_oth[row];
1703         pj     = pj_oth + pi_oth[row];
1704         pa     = pa_oth + pi_oth[row];
1705         valtmp = aoa[j];
1706         nextp  = 0;
1707         for (k=0; nextp<pnz; k++) {
1708           if (apJ[k] == pj[nextp]) { /* col of AP == col of P */
1709             apa[k] += valtmp*pa[nextp++];
1710           }
1711         }
1712         ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
1713       }
1714 #if defined(PTAP_PROFILE)
1715       ierr    = PetscTime(&t2_1);CHKERRQ(ierr);
1716       et2_AP += t2_1 - t2_0;
1717 #endif
1718 
1719       /* 2-b) Compute Cseq = P_loc[i,:]^T*AP[i,:] using outer product */
1720       /*--------------------------------------------------------------*/
1721       pnz = pi_loc[i+1] - pi_loc[i];
1722       pJ  = pj_loc + pi_loc[i];
1723       for (j=0; j<pnz; j++) {
1724         nextap = 0;
1725         row    = pJ[j]; /* global index */
1726         if (row < pcstart || row >=pcend) { /* put the value into Co */
1727           row = *poJ;
1728           cj  = coj + coi[row];
1729           ca  = coa + coi[row]; poJ++;
1730         } else {                            /* put the value into Cd */
1731           row = *pdJ;
1732           cj  = bj + bi[row];
1733           ca  = ba + bi[row]; pdJ++;
1734         }
1735         valtmp = pA[j];
1736         for (k=0; nextap<apnz; k++) {
1737           if (cj[k]==apJ[nextap]) ca[k] += valtmp*apa[nextap++];
1738         }
1739         ierr = PetscLogFlops(2.0*apnz);CHKERRQ(ierr);
1740       }
1741       pA += pnz;
1742       /* zero the current row info for A*P */
1743       ierr = PetscMemzero(apa,apnz*sizeof(MatScalar));CHKERRQ(ierr);
1744 #if defined(PTAP_PROFILE)
1745       ierr      = PetscTime(&t2_2);CHKERRQ(ierr);
1746       ePtAP += t2_2 - t2_1;
1747 #endif
1748     }
1749   }
1750 #if defined(PTAP_PROFILE)
1751   ierr = PetscTime(&t2);CHKERRQ(ierr);
1752 #endif
1753 
1754   /* 3) send and recv matrix values coa */
1755   /*------------------------------------*/
1756   buf_ri = merge->buf_ri;
1757   buf_rj = merge->buf_rj;
1758   len_s  = merge->len_s;
1759   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1760   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1761 
1762   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1763   for (proc=0,k=0; proc<size; proc++) {
1764     if (!len_s[proc]) continue;
1765     i    = merge->owners_co[proc];
1766     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1767     k++;
1768   }
1769   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1770   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1771 
1772   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1773   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1774   ierr = PetscFree(coa);CHKERRQ(ierr);
1775 #if defined(PTAP_PROFILE)
1776   ierr = PetscTime(&t3);CHKERRQ(ierr);
1777 #endif
1778 
1779   /* 4) insert local Cseq and received values into Cmpi */
1780   /*------------------------------------------------------*/
1781   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1782   for (k=0; k<merge->nrecv; k++) {
1783     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1784     nrows       = *(buf_ri_k[k]);
1785     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1786     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1787   }
1788 
1789   for (i=0; i<cm; i++) {
1790     row  = owners[rank] + i; /* global row index of C_seq */
1791     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1792     ba_i = ba + bi[i];
1793     bnz  = bi[i+1] - bi[i];
1794     /* add received vals into ba */
1795     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1796       /* i-th row */
1797       if (i == *nextrow[k]) {
1798         cnz    = *(nextci[k]+1) - *nextci[k];
1799         cj     = buf_rj[k] + *(nextci[k]);
1800         ca     = abuf_r[k] + *(nextci[k]);
1801         nextcj = 0;
1802         for (j=0; nextcj<cnz; j++) {
1803           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1804             ba_i[j] += ca[nextcj++];
1805           }
1806         }
1807         nextrow[k]++; nextci[k]++;
1808         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1809       }
1810     }
1811     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1812   }
1813   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1814   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1815 
1816   ierr = PetscFree(ba);CHKERRQ(ierr);
1817   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1818   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1819   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1820 #if defined(PTAP_PROFILE)
1821   ierr = PetscTime(&t4);CHKERRQ(ierr);
1822   if (rank==1) {
1823     ierr = PetscPrintf(MPI_COMM_SELF,"  [%d] PtAPNum %g/P + %g/PtAP( %g/A*P + %g/Pt*AP ) + %g/comm + %g/Cloc = %g\n\n",rank,eP,t2-t1,et2_AP,ePtAP,t3-t2,t4-t3,t4-t0);CHKERRQ(ierr);
1824   }
1825 #endif
1826   PetscFunctionReturn(0);
1827 }
1828