xref: /petsc/src/mat/impls/aij/mpi/mpiptap.c (revision 445158ff53a04af74a848b0e5b9ce8938ae41542)
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 extern PetscErrorCode MatDestroy_MPIAIJ(Mat);
16 #undef __FUNCT__
17 #define __FUNCT__ "MatDestroy_MPIAIJ_PtAP"
18 PetscErrorCode MatDestroy_MPIAIJ_PtAP(Mat A)
19 {
20   PetscErrorCode ierr;
21   Mat_MPIAIJ     *a   =(Mat_MPIAIJ*)A->data;
22   Mat_PtAPMPI    *ptap=a->ptap;
23 
24   PetscFunctionBegin;
25   if (ptap) {
26     Mat_Merge_SeqsToMPI *merge=ptap->merge;
27     ierr = PetscFree2(ptap->startsj_s,ptap->startsj_r);CHKERRQ(ierr);
28     ierr = PetscFree(ptap->bufa);CHKERRQ(ierr);
29     ierr = MatDestroy(&ptap->P_loc);CHKERRQ(ierr);
30     ierr = MatDestroy(&ptap->P_oth);CHKERRQ(ierr);
31     ierr = MatDestroy(&ptap->A_loc);CHKERRQ(ierr); /* used by MatTransposeMatMult() */
32 
33     ierr = MatDestroy(&ptap->Rd);CHKERRQ(ierr);
34     ierr = MatDestroy(&ptap->Ro);CHKERRQ(ierr);
35 
36     ierr = PetscFree(ptap->api);CHKERRQ(ierr);
37     ierr = PetscFree2(ptap->apj,ptap->apv);CHKERRQ(ierr);
38     ierr = MatDestroy(&ptap->AP_loc);CHKERRQ(ierr);
39     ierr = MatDestroy(&ptap->C_loc);CHKERRQ(ierr);
40     ierr = MatDestroy(&ptap->C_oth);CHKERRQ(ierr);
41 
42     if (ptap->api) {ierr = PetscFree(ptap->api);CHKERRQ(ierr);}
43     if (ptap->apj) {ierr = PetscFree(ptap->apj);CHKERRQ(ierr);}
44     if (ptap->apa) {ierr = PetscFree(ptap->apa);CHKERRQ(ierr);}
45     if (merge) {
46       ierr = PetscFree(merge->id_r);CHKERRQ(ierr);
47       ierr = PetscFree(merge->len_s);CHKERRQ(ierr);
48       ierr = PetscFree(merge->len_r);CHKERRQ(ierr);
49       ierr = PetscFree(merge->bi);CHKERRQ(ierr);
50       ierr = PetscFree(merge->bj);CHKERRQ(ierr);
51       ierr = PetscFree(merge->buf_ri[0]);CHKERRQ(ierr);
52       ierr = PetscFree(merge->buf_ri);CHKERRQ(ierr);
53       ierr = PetscFree(merge->buf_rj[0]);CHKERRQ(ierr);
54       ierr = PetscFree(merge->buf_rj);CHKERRQ(ierr);
55       ierr = PetscFree(merge->coi);CHKERRQ(ierr);
56       ierr = PetscFree(merge->coj);CHKERRQ(ierr);
57       ierr = PetscFree(merge->owners_co);CHKERRQ(ierr);
58       ierr = PetscLayoutDestroy(&merge->rowmap);CHKERRQ(ierr);
59       ierr = merge->destroy(A);CHKERRQ(ierr);
60       ierr = PetscFree(ptap->merge);CHKERRQ(ierr);
61     } else {
62       ierr = MatDestroy_MPIAIJ(A);CHKERRQ(ierr);
63     }
64     ierr = PetscFree(ptap);CHKERRQ(ierr);
65   }
66   PetscFunctionReturn(0);
67 }
68 
69 #undef __FUNCT__
70 #define __FUNCT__ "MatDuplicate_MPIAIJ_MatPtAP"
71 PetscErrorCode MatDuplicate_MPIAIJ_MatPtAP(Mat A, MatDuplicateOption op, Mat *M)
72 {
73   PetscErrorCode      ierr;
74   Mat_MPIAIJ          *a     = (Mat_MPIAIJ*)A->data;
75   Mat_PtAPMPI         *ptap  = a->ptap;
76   Mat_Merge_SeqsToMPI *merge = ptap->merge;
77 
78   PetscFunctionBegin;
79   ierr = (*merge->duplicate)(A,op,M);CHKERRQ(ierr);
80 
81   (*M)->ops->destroy   = merge->destroy;
82   (*M)->ops->duplicate = merge->duplicate;
83   PetscFunctionReturn(0);
84 }
85 
86 #undef __FUNCT__
87 #define __FUNCT__ "MatPtAP_MPIAIJ_MPIAIJ"
88 PetscErrorCode MatPtAP_MPIAIJ_MPIAIJ(Mat A,Mat P,MatReuse scall,PetscReal fill,Mat *C)
89 {
90   PetscErrorCode ierr;
91   PetscBool      newalg=PETSC_FALSE;
92 
93   PetscFunctionBegin;
94   ierr = PetscOptionsGetBool(NULL,"-matptap_new",&newalg,NULL);CHKERRQ(ierr);
95   if (scall == MAT_INITIAL_MATRIX) {
96     ierr = PetscLogEventBegin(MAT_PtAPSymbolic,A,P,0,0);CHKERRQ(ierr);
97     if (newalg) {
98       ierr = MatPtAPSymbolic_MPIAIJ_MPIAIJ_new(A,P,fill,C);CHKERRQ(ierr);
99     } else {
100       ierr = MatPtAPSymbolic_MPIAIJ_MPIAIJ(A,P,fill,C);CHKERRQ(ierr);
101     }
102     ierr = PetscLogEventEnd(MAT_PtAPSymbolic,A,P,0,0);CHKERRQ(ierr);
103   }
104   ierr = PetscLogEventBegin(MAT_PtAPNumeric,A,P,0,0);CHKERRQ(ierr);
105   if (newalg) {
106     ierr = MatPtAPNumeric_MPIAIJ_MPIAIJ_new(A,P,*C);CHKERRQ(ierr);
107   } else {
108     ierr = MatPtAPNumeric_MPIAIJ_MPIAIJ(A,P,*C);CHKERRQ(ierr);
109   }
110   ierr = PetscLogEventEnd(MAT_PtAPNumeric,A,P,0,0);CHKERRQ(ierr);
111   PetscFunctionReturn(0);
112 }
113 
114 #undef __FUNCT__
115 #define __FUNCT__ "MatPtAPSymbolic_MPIAIJ_MPIAIJ_new"
116 PetscErrorCode MatPtAPSymbolic_MPIAIJ_MPIAIJ_new(Mat A,Mat P,PetscReal fill,Mat *C)
117 {
118   PetscErrorCode ierr;
119   Mat_PtAPMPI    *ptap;
120   Mat_MPIAIJ     *p=(Mat_MPIAIJ*)P->data;
121   Mat_MPIAIJ     *c;
122   MPI_Comm       comm;
123   PetscMPIInt    size,rank;
124   PetscLogDouble      t0,t1,t11,t12,t2,t3,t4;
125 
126   PetscFunctionBegin;
127   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
128   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
129   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
130 
131   /* check if matrix local sizes are compatible -- MV! */
132   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend) {
133     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);
134   }
135   if (A->cmap->rstart != P->rmap->rstart || A->cmap->rend != P->rmap->rend) {
136     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);
137   }
138 
139   //=============================================================================
140   Mat                 Cmpi;
141   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
142   PetscInt            *pdti,*pdtj,*ptJ,nnz;
143   PetscInt            *lnk,i,k,pnz,row;
144   PetscInt            pN=P->cmap->N,pn=P->cmap->n;
145   PetscBT             lnkbt;
146   PetscMPIInt         tagi,tagj,*len_si,*len_s,*len_ri,icompleted=0;
147   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
148   PetscInt            len,proc,*dnz,*onz,*owners;
149   PetscInt            nzi;
150   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
151   MPI_Request         *swaits,*rwaits;
152   MPI_Status          *sstatus,rstatus;
153   PetscLayout rowmap;
154   PetscInt    *owners_co,*coi,*coj;    /* i and j array of (p->B)^T*A*P - used in the communication */
155   PetscInt    nsend,nrecv;
156   PetscMPIInt *len_r,*id_r;    /* array of length of comm->size, store send/recv matrix values */
157   PetscInt            *api,*apj,*Jptr,apnz,*prmap=p->garray,pon,j,ap_rmax=0;
158   PetscInt            rmax;
159 
160   ierr = PetscTime(&t0);CHKERRQ(ierr);
161   /* create struct Mat_PtAPMPI and attached it to C later */
162   ierr        = PetscNew(&ptap);CHKERRQ(ierr);
163   ptap->reuse = MAT_INITIAL_MATRIX;
164 
165   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
166   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
167 
168   /* get P_loc by taking all local rows of P */
169   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
170 
171   ptap->reuse = MAT_INITIAL_MATRIX;
172 
173   /* (1) compute symbolic AP = A*P, then get AP_loc */
174   /*--------------------------------------------------------------------------*/
175   ierr = MatTranspose_SeqAIJ(p->A,MAT_INITIAL_MATRIX,&ptap->Rd);CHKERRQ(ierr);
176   ierr = MatTranspose_SeqAIJ(p->B,MAT_INITIAL_MATRIX,&ptap->Ro);CHKERRQ(ierr);
177   ierr = PetscTime(&t11);CHKERRQ(ierr);
178 
179   // optiomize!!!
180   //===========================
181 #if 0
182   ierr = MatMatMult(A,P,MAT_INITIAL_MATRIX,2.0,&AP);CHKERRQ(ierr);
183   ierr = MatMPIAIJGetLocalMat(AP,MAT_INITIAL_MATRIX,&ptap->AP_loc);CHKERRQ(ierr);
184   ierr = MatDestroy(&AP);CHKERRQ(ierr);
185   Mat_SeqAIJ *ap = (Mat_SeqAIJ*)ptap->AP_loc->data;
186 #endif
187   /* (1) compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth (api,apj) */
188   /*--------------------------------------------------------------------------*/
189   Mat_SeqAIJ *p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
190   Mat_SeqAIJ *p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
191   PetscInt   am=A->rmap->n,pm=P->rmap->n;
192 
193   Mat_MPIAIJ          *a=(Mat_MPIAIJ*)A->data;
194   Mat_SeqAIJ          *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
195   PetscInt   *adi=ad->i,*aoi=ao->i,*pi_loc=p_loc->i,*aj;
196 
197   ierr   = PetscMalloc1(am+1,&api);CHKERRQ(ierr);
198   api[0] = 0;
199 
200   /* create and initialize a linked list */
201   ierr = PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt);CHKERRQ(ierr);
202 
203   /* Initial FreeSpace size is fill*(nnz(A) + nnz(P)) -OOM for ex56, np=8k on Intrepid! */
204   ierr = PetscFreeSpaceGet((PetscInt)(fill*(adi[am]+aoi[am]+pi_loc[pm])),&free_space);CHKERRQ(ierr);
205 
206   current_space = free_space;
207   PetscInt *pj_loc = p_loc->j;
208   PetscInt *pi_oth = p_oth->i;
209   PetscInt *pj_oth = p_oth->j,nspacedouble=0;
210   for (i=0; i<am; i++) {
211     /* diagonal portion of A */
212     nzi = adi[i+1] - adi[i];
213     aj  = ad->j + adi[i];
214     for (j=0; j<nzi; j++) {
215       row  = aj[j];
216       pnz  = pi_loc[row+1] - pi_loc[row];
217       Jptr = pj_loc + pi_loc[row];
218       /* add non-zero cols of P into the sorted linked list lnk */
219       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
220     }
221 
222     /* off-diagonal portion of A */
223     nzi = aoi[i+1] - aoi[i];
224     aj  = ao->j + aoi[i];
225     for (j=0; j<nzi; j++) {
226       row  = aj[j];
227       pnz  = pi_oth[row+1] - pi_oth[row];
228       Jptr = pj_oth + pi_oth[row];
229       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
230     }
231 
232     apnz     = lnk[0];
233     api[i+1] = api[i] + apnz;
234     //    if (api[i+1] != ap->i[i+1]) SETERRQ2(comm,PETSC_ERR_ARG_SIZ,"api %d != ap->i %d",api[i+1],ap->i[i+1]);
235     if (ap_rmax < apnz) ap_rmax = apnz;
236 
237     /* if free space is not available, double the total space in the list */
238     if (current_space->local_remaining<apnz) {
239       ierr = PetscFreeSpaceGet(apnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
240       nspacedouble++;
241     }
242 
243     /* Copy data into free space, then initialize lnk */
244     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
245 
246     current_space->array           += apnz;
247     current_space->local_used      += apnz;
248     current_space->local_remaining -= apnz;
249   }
250   /* Allocate space for apj, initialize apj, and */
251   /* destroy list of free space and other temporary array(s) */
252   PetscScalar *apv;
253   ierr      = PetscMalloc2(api[am],&apj,api[am],&apv);CHKERRQ(ierr);
254   ierr      = PetscFreeSpaceContiguous(&free_space,apj);CHKERRQ(ierr);
255   //afill_tmp = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1);
256   //if (afill_tmp > afill) afill = afill_tmp;
257 
258   //for (j=0; j<api[am]; j++) {
259   //  if (apj[j] != ap->j[j]) SETERRQ3(comm,PETSC_ERR_ARG_SIZ,"%d: apj %d != ap->j %d",j,apj[j],ap->j[j]);
260   //}
261 
262   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
263 
264   ierr = MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,am,pN,api,apj,apv,&ptap->AP_loc);CHKERRQ(ierr);
265   ierr = PetscTime(&t12);CHKERRQ(ierr);
266   //==========================================
267 
268   /* (2) compute C_loc=Rd*AP_loc, Co=Ro*AP_loc */
269   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Rd,ptap->AP_loc,2.0,&ptap->C_loc);CHKERRQ(ierr);
270   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Ro,ptap->AP_loc,2.0,&ptap->C_oth);CHKERRQ(ierr);
271   ierr = PetscTime(&t1);CHKERRQ(ierr);
272 
273   //Mat_SeqAIJ *ap = (Mat_SeqAIJ*)ptap->AP_loc->data;
274   Mat_SeqAIJ *c_oth  = (Mat_SeqAIJ*)ptap->C_oth->data;
275 
276   /* (2) determine symbolic Co=(p->B)^T*AP - send to others (coi,coj)*/
277   /*-----------------------------------------------------------------*/
278   coi = c_oth->i; coj =  c_oth->j;
279 
280   /* then, compute symbolic Co = (p->B)^T*AP */
281   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors
282                          >= (num of nonzero rows of C_seq) - pn */
283 
284   /* (3) send j-array (coj) of Co to other processors */
285   /*--------------------------------------------------*/
286   ierr = PetscCalloc1(size,&len_s);CHKERRQ(ierr);
287   len_s        = len_s;
288   nsend = 0;
289 
290   /* determine row ownership */
291   ierr = PetscLayoutCreate(comm,&rowmap);CHKERRQ(ierr);
292   rowmap->n  = pn;
293   rowmap->bs = 1;
294 
295   ierr   = PetscLayoutSetUp(rowmap);CHKERRQ(ierr);
296   owners = rowmap->range;
297 
298   /* determine the number of messages to send, their lengths */
299   ierr = PetscMalloc2(size,&len_si,size,&sstatus);CHKERRQ(ierr);
300   ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
301   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
302 
303   proc = 0;
304   for (i=0; i<pon; i++) {
305     while (prmap[i] >= owners[proc+1]) proc++;
306     len_si[proc]++;               /* num of rows in Co(=Pt*AP) to be sent to [proc] */
307     len_s[proc] += coi[i+1] - coi[i]; /* num of nonzeros in Co to be sent to [proc] */
308   }
309 
310   len          = 0; /* max length of buf_si[], see (4) */
311   owners_co[0] = 0;
312   for (proc=0; proc<size; proc++) {
313     owners_co[proc+1] = owners_co[proc] + len_si[proc];
314     if (len_s[proc]) {
315       nsend++;
316       len_si[proc] = 2*(len_si[proc] + 1); /* length of buf_si to be sent to [proc] */
317       len         += len_si[proc];
318     }
319   }
320 
321   /* determine the number and length of messages to receive for coi and coj  */
322   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&nrecv);CHKERRQ(ierr);
323   ierr = PetscGatherMessageLengths2(comm,nsend,nrecv,len_s,len_si,&id_r,&len_r,&len_ri);CHKERRQ(ierr);
324 
325   /* post the Irecv and Isend of coj */
326   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
327   ierr = PetscPostIrecvInt(comm,tagj,nrecv,id_r,len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
328   ierr = PetscMalloc1(nsend+1,&swaits);CHKERRQ(ierr);
329   for (proc=0, k=0; proc<size; proc++) {
330     if (!len_s[proc]) continue;
331     i    = owners_co[proc];
332     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
333     k++;
334   }
335 
336   /* receives and sends of coj are complete */
337   for (i=0; i<nrecv; i++) {
338     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
339   }
340   ierr = PetscFree(rwaits);CHKERRQ(ierr);
341   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
342 
343   /* (4) send and recv coi */
344   /*-----------------------*/
345   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
346   ierr   = PetscPostIrecvInt(comm,tagi,nrecv,id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
347   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
348   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
349   for (proc=0,k=0; proc<size; proc++) {
350     if (!len_s[proc]) continue;
351     /* form outgoing message for i-structure:
352          buf_si[0]:                 nrows to be sent
353                [1:nrows]:           row index (global)
354                [nrows+1:2*nrows+1]: i-structure index
355     */
356     /*-------------------------------------------*/
357     nrows       = len_si[proc]/2 - 1; /* num of rows in Co to be sent to [proc] */
358     buf_si_i    = buf_si + nrows+1;
359     buf_si[0]   = nrows;
360     buf_si_i[0] = 0;
361     nrows       = 0;
362     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
363       nzi = coi[i+1] - coi[i];
364       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
365       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
366       nrows++;
367     }
368     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
369     k++;
370     buf_si += len_si[proc];
371   }
372   i = nrecv;
373   while (i--) {
374     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
375   }
376   ierr = PetscFree(rwaits);CHKERRQ(ierr);
377   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
378 
379   ierr = PetscFree2(len_si,sstatus);CHKERRQ(ierr);
380   ierr = PetscFree(len_ri);CHKERRQ(ierr);
381   ierr = PetscFree(swaits);CHKERRQ(ierr);
382   ierr = PetscFree(buf_s);CHKERRQ(ierr);
383 
384   ierr = PetscTime(&t2);CHKERRQ(ierr);
385 
386   /* (5) compute the local portion of C (mpi mat) */
387   /*----------------------------------------------*/
388 
389   Mat_SeqAIJ *pd = (Mat_SeqAIJ *)ptap->Rd->data;
390   pdti = pd->i; pdtj = pd->j;
391 
392   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
393   //nnz           = fill*(pi_loc[pm] + api[am]);
394   nnz = pN;  //new - nonscalable!!!
395   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
396   current_space = free_space;
397 
398   ierr = PetscMalloc3(nrecv,&buf_ri_k,nrecv,&nextrow,nrecv,&nextci);CHKERRQ(ierr);
399   for (k=0; k<nrecv; k++) {
400     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
401     nrows       = *buf_ri_k[k];
402     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
403     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
404   }
405   ierr = MatPreallocateInitialize(comm,pn,pn,dnz,onz);CHKERRQ(ierr);
406 
407   /* create and initialize a linked list */
408   ierr = PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt);CHKERRQ(ierr);
409 
410   rmax = 0;
411   for (i=0; i<pn; i++) {
412     /* add pdt[i,:]*AP into lnk */
413     pnz = pdti[i+1] - pdti[i];
414     ptJ = pdtj + pdti[i];
415     for (j=0; j<pnz; j++) {
416       row  = ptJ[j];  /* row of AP == col of Pt */
417       apnz = api[row+1] - api[row];
418       Jptr = apj + api[row];
419       /* add non-zero cols of AP into the sorted linked list lnk */
420       ierr = PetscLLCondensedAddSorted(apnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
421     }
422 
423     /* add received col data into lnk */
424     for (k=0; k<nrecv; k++) { /* k-th received message */
425       if (i == *nextrow[k]) { /* i-th row */
426         nzi  = *(nextci[k]+1) - *nextci[k];
427         Jptr = buf_rj[k] + *nextci[k];
428         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
429         nextrow[k]++; nextci[k]++;
430       }
431     }
432     nnz = lnk[0];
433 #if 0
434     /* if free space is not available, make more free space */
435     if (current_space->local_remaining<nnz) {
436       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
437       nspacedouble++;
438     }
439 #endif
440     /* copy data into free space, then initialize lnk */
441     ierr = PetscLLCondensedClean(pN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
442     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
443 
444     if (nnz > rmax) rmax = nnz;
445   }
446   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
447   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
448   ierr = PetscFreeSpaceDestroy(free_space);CHKERRQ(ierr);
449 
450   ierr = PetscTime(&t3);CHKERRQ(ierr);
451 
452   /* (6) create symbolic parallel matrix Cmpi */
453   /*------------------------------------------*/
454   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
455   ierr = MatSetSizes(Cmpi,pn,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
456   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(P->cmap->bs));CHKERRQ(ierr);
457   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
458   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
459   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
460 
461   /* members in merge */
462   ierr = PetscFree(id_r);CHKERRQ(ierr);
463   ierr = PetscFree(len_s);CHKERRQ(ierr);
464   ierr = PetscFree(len_r);CHKERRQ(ierr);
465   ierr = PetscFree(buf_ri[0]);CHKERRQ(ierr);
466   ierr = PetscFree(buf_ri);CHKERRQ(ierr);
467   ierr = PetscFree(buf_rj[0]);CHKERRQ(ierr);
468   ierr = PetscFree(buf_rj);CHKERRQ(ierr);
469   ierr = PetscFree(owners_co);CHKERRQ(ierr);
470   ierr = PetscLayoutDestroy(&rowmap);CHKERRQ(ierr);
471 
472   /* Cmpi is not ready for use - assembly will be done by MatPtAPNumeric() */
473   Cmpi->assembled        = PETSC_FALSE;
474   Cmpi->ops->destroy     = MatDestroy_MPIAIJ_PtAP;
475   Cmpi->ops->duplicate   = 0; //MatDuplicate_MPIAIJ_MatPtAP; //not done yet
476   Cmpi->ops->ptapnumeric = MatPtAPNumeric_MPIAIJ_MPIAIJ_new;
477 
478   /* attach the supporting struct to Cmpi for reuse */
479   c           = (Mat_MPIAIJ*)Cmpi->data;
480   c->ptap     = ptap;
481   ptap->api   = api;
482   ptap->apj   = apj;
483   ptap->apv   = apv;
484   ptap->rmax  = ap_rmax;
485   *C          = Cmpi;
486 
487   /* flag 'scalable' determines which implementations to be used:
488        0: do dense axpy in MatPtAPNumeric() - fast, but requires storage of a nonscalable dense array apa;
489        1: do sparse axpy in MatPtAPNumeric() - might slow, uses a sparse array apa */
490   /* set default scalable */
491   ptap->scalable = PETSC_FALSE; //PETSC_TRUE;
492 
493   ierr = PetscOptionsGetBool(((PetscObject)(*C))->prefix,"-matptap_scalable",&ptap->scalable,NULL);CHKERRQ(ierr);
494   if (!ptap->scalable) {  /* Do dense axpy */
495     ierr = PetscCalloc1(P->cmap->N,&ptap->apa);CHKERRQ(ierr);
496   } else {
497     //ierr = PetscCalloc1(ap_rmax+1,&ptap->apa);CHKERRQ(ierr);
498   }
499 
500   ierr = PetscTime(&t4);CHKERRQ(ierr);
501   if (rank == 1) {
502     printf("PtAPSym: %g + %g + %g + %g + %g + %g = %g\n",t11-t0,t1-t11,t12-t11,t2-t2,t3-t2,t4-t3,t4-t0);
503   }
504   PetscFunctionReturn(0);
505 }
506 
507 #undef __FUNCT__
508 #define __FUNCT__ "MatPtAPNumeric_MPIAIJ_MPIAIJ_new"
509 PetscErrorCode MatPtAPNumeric_MPIAIJ_MPIAIJ_new(Mat A,Mat P,Mat C)
510 {
511   PetscErrorCode    ierr;
512   Mat_MPIAIJ        *a=(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
513   Mat_SeqAIJ        *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
514   Mat_PtAPMPI       *ptap = c->ptap;
515   Mat               AP_loc,C_loc,C_oth;
516   PetscInt          i,rstart,rend,cm,ncols,row;
517   PetscMPIInt       rank;
518   MPI_Comm          comm;
519   const PetscInt    *cols;
520   const PetscScalar *vals;
521   PetscLogDouble    t0,t1,t2,t3,t4,eR,eAP,eCseq,eCmpi;
522 
523   PetscFunctionBegin;
524   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
525   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
526 
527   ierr = MatZeroEntries(C);CHKERRQ(ierr);
528 
529   /* 1) get R = Pd^T,Ro = Po^T */
530   ierr = PetscTime(&t0);CHKERRQ(ierr);
531   if (ptap->reuse == MAT_REUSE_MATRIX) {
532     ierr = MatTranspose_SeqAIJ(p->A,MAT_REUSE_MATRIX,&ptap->Rd);CHKERRQ(ierr);
533     ierr = MatTranspose_SeqAIJ(p->B,MAT_REUSE_MATRIX,&ptap->Ro);CHKERRQ(ierr);
534   }
535   ierr = PetscTime(&t1);CHKERRQ(ierr);
536   eR = t1 - t0;
537 
538   /* 2) get AP_loc */
539   AP_loc = ptap->AP_loc;
540   Mat_SeqAIJ    *ap=(Mat_SeqAIJ*)AP_loc->data;
541 
542   /* 2-1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
543   /*-----------------------------------------------------*/
544   if (ptap->reuse == MAT_REUSE_MATRIX) {
545     /* P_oth and P_loc are obtained in MatPtASymbolic() when reuse == MAT_INITIAL_MATRIX */
546     ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
547     ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
548   }
549 
550   /* 2-2) compute numeric C_seq = P_loc^T*A_loc*P - dominating part */
551   /*--------------------------------------------------------------*/
552   /* get data from symbolic products */
553   Mat_SeqAIJ  *p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
554   Mat_SeqAIJ  *p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
555   PetscInt    *api,*apj,am = A->rmap->n,j,col,apnz;
556   PetscScalar *apa = ptap->apa;
557   //if (ptap->reuse == MAT_REUSE_MATRIX) {
558     api = ptap->api; //ap->i;
559     apj = ptap->apj; //ap->j;
560     for (i=0; i<am; i++) {
561       /* AP[i,:] = A[i,:]*P = Ad*P_loc Ao*P_oth */
562       AProw_nonscalable(i,ad,ao,p_loc,p_oth,apa);
563       apnz = api[i+1] - api[i];
564       for (j=0; j<apnz; j++) {
565         col = apj[j+api[i]];
566         ap->a[j+ap->i[i]] = apa[col];
567         apa[col] = 0.0;
568       }
569     }
570     //}
571 
572   ierr = PetscTime(&t2);CHKERRQ(ierr);
573   eAP = t2 - t1;
574 
575   /* 3) C_loc = R*AP_loc, Co = Ro*AP_loc */
576   ierr = ((ptap->C_loc)->ops->matmultnumeric)(ptap->Rd,AP_loc,ptap->C_loc);CHKERRQ(ierr);
577   ierr = ((ptap->C_oth)->ops->matmultnumeric)(ptap->Ro,AP_loc,ptap->C_oth);CHKERRQ(ierr);
578 
579   C_loc = ptap->C_loc;
580   C_oth = ptap->C_oth;
581   //printf("[%d] Co %d, %d\n", rank,Co->rmap->N,Co->cmap->N);
582   ierr = PetscTime(&t3);CHKERRQ(ierr);
583   eCseq = t3 - t2;
584 
585   /* add C_loc and Co to to C */
586   ierr = MatGetOwnershipRange(C,&rstart,&rend);CHKERRQ(ierr);
587 
588   /* C_loc -> C */
589   cm = C_loc->rmap->N;
590   Mat_SeqAIJ *c_seq;
591   c_seq = (Mat_SeqAIJ*)C_loc->data;
592   for (i=0; i<cm; i++) {
593     ncols = c_seq->i[i+1] - c_seq->i[i];
594     row = rstart + i;
595     cols = c_seq->j + c_seq->i[i];
596     vals = c_seq->a + c_seq->i[i];
597     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
598   }
599 
600   /* Co -> C, off-processor part */
601   //printf("[%d] p->B %d, %d\n",rank,p->B->rmap->N,p->B->cmap->N);
602   cm = C_oth->rmap->N;
603   c_seq = (Mat_SeqAIJ*)C_oth->data;
604   for (i=0; i<cm; i++) {
605     ncols = c_seq->i[i+1] - c_seq->i[i];
606     row = p->garray[i];
607     cols = c_seq->j + c_seq->i[i];
608     vals = c_seq->a + c_seq->i[i];
609     //printf("[%d] row[%d] = %d\n",rank,i,row);
610     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
611   }
612   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
613   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
614   ierr = PetscTime(&t4);CHKERRQ(ierr);
615   eCmpi = t4 - t3;
616 
617   if (rank==1) {
618     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);
619   }
620   ptap->reuse = MAT_REUSE_MATRIX;
621   PetscFunctionReturn(0);
622 }
623 
624 #undef __FUNCT__
625 #define __FUNCT__ "MatPtAPSymbolic_MPIAIJ_MPIAIJ"
626 PetscErrorCode MatPtAPSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat *C)
627 {
628   PetscErrorCode      ierr;
629   Mat                 Cmpi;
630   Mat_PtAPMPI         *ptap;
631   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
632   Mat_MPIAIJ          *a        =(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data,*c;
633   Mat_SeqAIJ          *ad       =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
634   Mat_SeqAIJ          *p_loc,*p_oth;
635   PetscInt            *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pdti,*pdtj,*poti,*potj,*ptJ;
636   PetscInt            *adi=ad->i,*aj,*aoi=ao->i,nnz;
637   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
638   PetscInt            am=A->rmap->n,pN=P->cmap->N,pm=P->rmap->n,pn=P->cmap->n;
639   PetscBT             lnkbt;
640   MPI_Comm            comm;
641   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri,icompleted=0;
642   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
643   PetscInt            len,proc,*dnz,*onz,*owners;
644   PetscInt            nzi,*pti,*ptj;
645   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
646   MPI_Request         *swaits,*rwaits;
647   MPI_Status          *sstatus,rstatus;
648   Mat_Merge_SeqsToMPI *merge;
649   PetscInt            *api,*apj,*Jptr,apnz,*prmap=p->garray,pon,nspacedouble=0,j,ap_rmax=0;
650   PetscReal           afill=1.0,afill_tmp;
651   PetscInt            rmax;
652 
653   PetscFunctionBegin;
654   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
655 
656   /* check if matrix local sizes are compatible */
657   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend) {
658     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);
659   }
660   if (A->cmap->rstart != P->rmap->rstart || A->cmap->rend != P->rmap->rend) {
661     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);
662   }
663 
664   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
665   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
666 
667   /* create struct Mat_PtAPMPI and attached it to C later */
668   ierr        = PetscNew(&ptap);CHKERRQ(ierr);
669   ierr        = PetscNew(&merge);CHKERRQ(ierr);
670   ptap->merge = merge;
671   ptap->reuse = MAT_INITIAL_MATRIX;
672 
673   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
674   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
675 
676   /* get P_loc by taking all local rows of P */
677   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
678 
679   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
680   p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
681   pi_loc = p_loc->i; pj_loc = p_loc->j;
682   pi_oth = p_oth->i; pj_oth = p_oth->j;
683 
684   /* (1) compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth (api,apj) */
685   /*--------------------------------------------------------------------------*/
686   ierr   = PetscMalloc1(am+1,&api);CHKERRQ(ierr);
687   api[0] = 0;
688 
689   /* create and initialize a linked list */
690   ierr = PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt);CHKERRQ(ierr);
691 
692   /* Initial FreeSpace size is fill*(nnz(A) + nnz(P)) -OOM for ex56, np=8k on Intrepid! */
693   ierr = PetscFreeSpaceGet((PetscInt)(fill*(adi[am]+aoi[am]+pi_loc[pm])),&free_space);CHKERRQ(ierr);
694 
695   current_space = free_space;
696 
697   for (i=0; i<am; i++) {
698     /* diagonal portion of A */
699     nzi = adi[i+1] - adi[i];
700     aj  = ad->j + adi[i];
701     for (j=0; j<nzi; j++) {
702       row  = aj[j];
703       pnz  = pi_loc[row+1] - pi_loc[row];
704       Jptr = pj_loc + pi_loc[row];
705       /* add non-zero cols of P into the sorted linked list lnk */
706       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
707     }
708     /* off-diagonal portion of A */
709     nzi = aoi[i+1] - aoi[i];
710     aj  = ao->j + aoi[i];
711     for (j=0; j<nzi; j++) {
712       row  = aj[j];
713       pnz  = pi_oth[row+1] - pi_oth[row];
714       Jptr = pj_oth + pi_oth[row];
715       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
716     }
717     apnz     = lnk[0];
718     api[i+1] = api[i] + apnz;
719     if (ap_rmax < apnz) ap_rmax = apnz;
720 
721     /* if free space is not available, double the total space in the list */
722     if (current_space->local_remaining<apnz) {
723       ierr = PetscFreeSpaceGet(apnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
724       nspacedouble++;
725     }
726 
727     /* Copy data into free space, then initialize lnk */
728     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
729 
730     current_space->array           += apnz;
731     current_space->local_used      += apnz;
732     current_space->local_remaining -= apnz;
733   }
734 
735   /* Allocate space for apj, initialize apj, and */
736   /* destroy list of free space and other temporary array(s) */
737   ierr      = PetscMalloc1(api[am]+1,&apj);CHKERRQ(ierr);
738   ierr      = PetscFreeSpaceContiguous(&free_space,apj);CHKERRQ(ierr);
739   afill_tmp = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1);
740   if (afill_tmp > afill) afill = afill_tmp;
741 
742   /* (2) determine symbolic Co=(p->B)^T*AP - send to others (coi,coj)*/
743   /*-----------------------------------------------------------------*/
744   ierr = MatGetSymbolicTranspose_SeqAIJ(p->B,&poti,&potj);CHKERRQ(ierr);
745 
746   /* then, compute symbolic Co = (p->B)^T*AP */
747   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors
748                          >= (num of nonzero rows of C_seq) - pn */
749   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
750   coi[0] = 0;
751 
752   /* set initial free space to be fill*(nnz(p->B) + nnz(AP)) */
753   nnz           = fill*(poti[pon] + api[am]);
754   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
755   current_space = free_space;
756 
757   for (i=0; i<pon; i++) {
758     pnz = poti[i+1] - poti[i];
759     ptJ = potj + poti[i];
760     for (j=0; j<pnz; j++) {
761       row  = ptJ[j]; /* row of AP == col of Pot */
762       apnz = api[row+1] - api[row];
763       Jptr = apj + api[row];
764       /* add non-zero cols of AP into the sorted linked list lnk */
765       ierr = PetscLLCondensedAddSorted(apnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
766     }
767     nnz = lnk[0];
768 
769     /* If free space is not available, double the total space in the list */
770     if (current_space->local_remaining<nnz) {
771       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
772       nspacedouble++;
773     }
774 
775     /* Copy data into free space, and zero out denserows */
776     ierr = PetscLLCondensedClean(pN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
777 
778     current_space->array           += nnz;
779     current_space->local_used      += nnz;
780     current_space->local_remaining -= nnz;
781 
782     coi[i+1] = coi[i] + nnz;
783   }
784 
785   ierr      = PetscMalloc1(coi[pon],&coj);CHKERRQ(ierr);
786   ierr      = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
787   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + api[am]+1);
788   if (afill_tmp > afill) afill = afill_tmp;
789   ierr = MatRestoreSymbolicTranspose_SeqAIJ(p->B,&poti,&potj);CHKERRQ(ierr);
790 
791   /* (3) send j-array (coj) of Co to other processors */
792   /*--------------------------------------------------*/
793   ierr = PetscCalloc1(size,&merge->len_s);CHKERRQ(ierr);
794   len_s        = merge->len_s;
795   merge->nsend = 0;
796 
797 
798   /* determine row ownership */
799   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
800   merge->rowmap->n  = pn;
801   merge->rowmap->bs = 1;
802 
803   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
804   owners = merge->rowmap->range;
805 
806   /* determine the number of messages to send, their lengths */
807   ierr = PetscMalloc2(size,&len_si,size,&sstatus);CHKERRQ(ierr);
808   ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
809   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
810 
811   proc = 0;
812   for (i=0; i<pon; i++) {
813     while (prmap[i] >= owners[proc+1]) proc++;
814     len_si[proc]++;               /* num of rows in Co(=Pt*AP) to be sent to [proc] */
815     len_s[proc] += coi[i+1] - coi[i]; /* num of nonzeros in Co to be sent to [proc] */
816   }
817 
818   len          = 0; /* max length of buf_si[], see (4) */
819   owners_co[0] = 0;
820   for (proc=0; proc<size; proc++) {
821     owners_co[proc+1] = owners_co[proc] + len_si[proc];
822     if (len_s[proc]) {
823       merge->nsend++;
824       len_si[proc] = 2*(len_si[proc] + 1); /* length of buf_si to be sent to [proc] */
825       len         += len_si[proc];
826     }
827   }
828 
829   /* determine the number and length of messages to receive for coi and coj  */
830   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
831   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
832 
833   /* post the Irecv and Isend of coj */
834   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
835   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
836   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
837   for (proc=0, k=0; proc<size; proc++) {
838     if (!len_s[proc]) continue;
839     i    = owners_co[proc];
840     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
841     k++;
842   }
843 
844   /* receives and sends of coj are complete */
845   for (i=0; i<merge->nrecv; i++) {
846     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
847   }
848   ierr = PetscFree(rwaits);CHKERRQ(ierr);
849   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
850 
851   /* (4) send and recv coi */
852   /*-----------------------*/
853   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
854   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
855   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
856   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
857   for (proc=0,k=0; proc<size; proc++) {
858     if (!len_s[proc]) continue;
859     /* form outgoing message for i-structure:
860          buf_si[0]:                 nrows to be sent
861                [1:nrows]:           row index (global)
862                [nrows+1:2*nrows+1]: i-structure index
863     */
864     /*-------------------------------------------*/
865     nrows       = len_si[proc]/2 - 1; /* num of rows in Co to be sent to [proc] */
866     buf_si_i    = buf_si + nrows+1;
867     buf_si[0]   = nrows;
868     buf_si_i[0] = 0;
869     nrows       = 0;
870     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
871       nzi = coi[i+1] - coi[i];
872       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
873       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
874       nrows++;
875     }
876     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
877     k++;
878     buf_si += len_si[proc];
879   }
880   i = merge->nrecv;
881   while (i--) {
882     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
883   }
884   ierr = PetscFree(rwaits);CHKERRQ(ierr);
885   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
886 
887   ierr = PetscFree2(len_si,sstatus);CHKERRQ(ierr);
888   ierr = PetscFree(len_ri);CHKERRQ(ierr);
889   ierr = PetscFree(swaits);CHKERRQ(ierr);
890   ierr = PetscFree(buf_s);CHKERRQ(ierr);
891 
892   /* (5) compute the local portion of C (mpi mat) */
893   /*----------------------------------------------*/
894   ierr = MatGetSymbolicTranspose_SeqAIJ(p->A,&pdti,&pdtj);CHKERRQ(ierr);
895 
896   /* allocate pti array and free space for accumulating nonzero column info */
897   ierr   = PetscMalloc1(pn+1,&pti);CHKERRQ(ierr);
898   pti[0] = 0;
899 
900   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
901   nnz           = fill*(pi_loc[pm] + api[am]);
902   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
903   current_space = free_space;
904 
905   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
906   for (k=0; k<merge->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   ierr = MatPreallocateInitialize(comm,pn,pn,dnz,onz);CHKERRQ(ierr);
913   rmax = 0;
914   for (i=0; i<pn; i++) {
915     /* add pdt[i,:]*AP into lnk */
916     pnz = pdti[i+1] - pdti[i];
917     ptJ = pdtj + pdti[i];
918     for (j=0; j<pnz; j++) {
919       row  = ptJ[j];  /* row of AP == col of Pt */
920       apnz = api[row+1] - api[row];
921       Jptr = apj + api[row];
922       /* add non-zero cols of AP into the sorted linked list lnk */
923       ierr = PetscLLCondensedAddSorted(apnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
924     }
925 
926     /* add received col data into lnk */
927     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
928       if (i == *nextrow[k]) { /* i-th row */
929         nzi  = *(nextci[k]+1) - *nextci[k];
930         Jptr = buf_rj[k] + *nextci[k];
931         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
932         nextrow[k]++; nextci[k]++;
933       }
934     }
935     nnz = lnk[0];
936 
937     /* if free space is not available, make more free space */
938     if (current_space->local_remaining<nnz) {
939       ierr = PetscFreeSpaceGet(nnz+current_space->total_array_size,&current_space);CHKERRQ(ierr);
940       nspacedouble++;
941     }
942     /* copy data into free space, then initialize lnk */
943     ierr = PetscLLCondensedClean(pN,nnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
944     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
945 
946     current_space->array           += nnz;
947     current_space->local_used      += nnz;
948     current_space->local_remaining -= nnz;
949 
950     pti[i+1] = pti[i] + nnz;
951     if (nnz > rmax) rmax = nnz;
952   }
953   ierr = MatRestoreSymbolicTranspose_SeqAIJ(p->A,&pdti,&pdtj);CHKERRQ(ierr);
954   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
955 
956   ierr      = PetscMalloc1(pti[pn]+1,&ptj);CHKERRQ(ierr);
957   ierr      = PetscFreeSpaceContiguous(&free_space,ptj);CHKERRQ(ierr);
958   afill_tmp = (PetscReal)pti[pn]/(pi_loc[pm] + api[am]+1);
959   if (afill_tmp > afill) afill = afill_tmp;
960   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
961 
962   /* (6) create symbolic parallel matrix Cmpi */
963   /*------------------------------------------*/
964   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
965   ierr = MatSetSizes(Cmpi,pn,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
966   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(P->cmap->bs));CHKERRQ(ierr);
967   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
968   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
969   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
970 
971   merge->bi        = pti;      /* Cseq->i */
972   merge->bj        = ptj;      /* Cseq->j */
973   merge->coi       = coi;      /* Co->i   */
974   merge->coj       = coj;      /* Co->j   */
975   merge->buf_ri    = buf_ri;
976   merge->buf_rj    = buf_rj;
977   merge->owners_co = owners_co;
978   merge->destroy   = Cmpi->ops->destroy;
979   merge->duplicate = Cmpi->ops->duplicate;
980 
981   /* Cmpi is not ready for use - assembly will be done by MatPtAPNumeric() */
982   Cmpi->assembled      = PETSC_FALSE;
983   Cmpi->ops->destroy   = MatDestroy_MPIAIJ_PtAP;
984   Cmpi->ops->duplicate = MatDuplicate_MPIAIJ_MatPtAP;
985 
986   /* attach the supporting struct to Cmpi for reuse */
987   c           = (Mat_MPIAIJ*)Cmpi->data;
988   c->ptap     = ptap;
989   ptap->api   = api;
990   ptap->apj   = apj;
991   ptap->rmax  = ap_rmax;
992   *C          = Cmpi;
993 
994   /* flag 'scalable' determines which implementations to be used:
995        0: do dense axpy in MatPtAPNumeric() - fast, but requires storage of a nonscalable dense array apa;
996        1: do sparse axpy in MatPtAPNumeric() - might slow, uses a sparse array apa */
997   /* set default scalable */
998   ptap->scalable = PETSC_FALSE; //PETSC_TRUE;
999 
1000   ierr = PetscOptionsGetBool(((PetscObject)Cmpi)->prefix,"-matptap_scalable",&ptap->scalable,NULL);CHKERRQ(ierr);
1001   if (!ptap->scalable) {  /* Do dense axpy */
1002     ierr = PetscCalloc1(pN,&ptap->apa);CHKERRQ(ierr);
1003   } else {
1004     ierr = PetscCalloc1(ap_rmax+1,&ptap->apa);CHKERRQ(ierr);
1005   }
1006 
1007 #if defined(PETSC_USE_INFO)
1008   if (pti[pn] != 0) {
1009     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1010     ierr = PetscInfo1(Cmpi,"Use MatPtAP(A,P,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
1011   } else {
1012     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1013   }
1014 #endif
1015   PetscFunctionReturn(0);
1016 }
1017 
1018 #undef __FUNCT__
1019 #define __FUNCT__ "MatPtAPNumeric_MPIAIJ_MPIAIJ"
1020 PetscErrorCode MatPtAPNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
1021 {
1022   PetscErrorCode      ierr;
1023   Mat_MPIAIJ          *a =(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1024   Mat_SeqAIJ          *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
1025   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
1026   Mat_SeqAIJ          *p_loc,*p_oth;
1027   Mat_PtAPMPI         *ptap;
1028   Mat_Merge_SeqsToMPI *merge;
1029   PetscInt            *adi=ad->i,*aoi=ao->i,*adj,*aoj,*apJ,nextp;
1030   PetscInt            *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pJ,*pj;
1031   PetscInt            i,j,k,anz,pnz,apnz,nextap,row,*cj;
1032   MatScalar           *ada,*aoa,*apa,*pa,*ca,*pa_loc,*pa_oth,valtmp;
1033   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
1034   MPI_Comm            comm;
1035   PetscMPIInt         size,rank,taga,*len_s;
1036   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
1037   PetscInt            **buf_ri,**buf_rj;
1038   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
1039   MPI_Request         *s_waits,*r_waits;
1040   MPI_Status          *status;
1041   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
1042   PetscInt            *api,*apj,*coi,*coj;
1043   PetscInt            *poJ=po->j,*pdJ=pd->j,pcstart=P->cmap->rstart,pcend=P->cmap->rend;
1044   PetscBool           scalable;
1045 #if defined(PTAP_PROFILE)
1046   PetscLogDouble t0,t1,t2,eP,t3,t4,et2_AP=0.0,ePtAP=0.0,t2_0,t2_1,t2_2;
1047 #endif
1048 
1049   PetscFunctionBegin;
1050   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1051 #if defined(PTAP_PROFILE)
1052   ierr = PetscTime(&t0);CHKERRQ(ierr);
1053 #endif
1054   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1055   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1056 
1057   ptap = c->ptap;
1058   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");
1059   merge    = ptap->merge;
1060   apa      = ptap->apa;
1061   scalable = ptap->scalable;
1062 
1063   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
1064   /*-----------------------------------------------------*/
1065   if (ptap->reuse == MAT_INITIAL_MATRIX) {
1066     /* P_oth and P_loc are obtained in MatPtASymbolic(), skip calling MatGetBrowsOfAoCols() and MatMPIAIJGetLocalMat() */
1067     ptap->reuse = MAT_REUSE_MATRIX;
1068   } else { /* update numerical values of P_oth and P_loc */
1069     ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
1070     ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
1071   }
1072 #if defined(PTAP_PROFILE)
1073   ierr = PetscTime(&t1);CHKERRQ(ierr);
1074   eP = t1-t0;
1075 #endif
1076   /*
1077   printf("[%d] Ad: %d, %d; Ao: %d, %d; P_loc: %d, %d; P_oth %d, %d;\n",rank,
1078          a->A->rmap->N,a->A->cmap->N,a->B->rmap->N,a->B->cmap->N,
1079          ptap->P_loc->rmap->N,ptap->P_loc->cmap->N,
1080          ptap->P_oth->rmap->N,ptap->P_oth->cmap->N);
1081    */
1082 
1083   /* 2) compute numeric C_seq = P_loc^T*A_loc*P - dominating part */
1084   /*--------------------------------------------------------------*/
1085   /* get data from symbolic products */
1086   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
1087   p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
1088   pi_loc=p_loc->i; pj_loc=p_loc->j; pJ=pj_loc; pa_loc=p_loc->a;
1089   pi_oth=p_oth->i; pj_oth=p_oth->j; pa_oth=p_oth->a;
1090 
1091   coi  = merge->coi; coj = merge->coj;
1092   ierr = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
1093 
1094   bi     = merge->bi; bj = merge->bj;
1095   owners = merge->rowmap->range;
1096   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);  /* ba: Cseq->a */
1097 
1098   api = ptap->api; apj = ptap->apj;
1099 
1100   if (!scalable) { /* Do dense axpy on apa (length of pN, stores A[i,:]*P) - nonscalable, but faster (could take 1/3 scalable time) */
1101     ierr = PetscInfo(C,"Using non-scalable dense axpy\n");CHKERRQ(ierr);
1102 #if 0
1103     /* ------ 10x slower -------------- */
1104     /*==================================*/
1105     Mat         R = ptap->R;
1106     Mat_SeqAIJ  *r = (Mat_SeqAIJ*)R->data;
1107     PetscInt    *ri=r->i,*rj=r->j,rnz,arow,l,prow,pcol,pN=P->cmap->N;
1108     PetscScalar *ra=r->a,tmp,cdense[pN];
1109 
1110     ierr = PetscMemzero(cdense,pN*sizeof(PetscScalar));CHKERRQ(ierr);
1111     for (i=0; i<cm; i++) { /* each row of C or R */
1112       rnz = ri[i+1] - ri[i];
1113 
1114       for (j=0; j<rnz; j++) { /* each nz of R */
1115         arow = rj[ri[i] + j];
1116 
1117         /* diagonal portion of A */
1118         anz  = ad->i[arow+1] - ad->i[arow];
1119         for (k=0; k<anz; k++) { /* each nz of Ad */
1120           tmp  = ra[ri[i] + j]*ad->a[ad->i[arow] + k];
1121           prow = ad->j[ad->i[arow] + k];
1122           pnz  = pi_loc[prow+1] - pi_loc[prow];
1123 
1124           for (l=0; l<pnz; l++) { /* each nz of P_loc */
1125             pcol = pj_loc[pi_loc[prow] + l];
1126             cdense[pcol] += tmp*pa_loc[pi_loc[prow] + l];
1127           }
1128         }
1129 
1130         /* off-diagonal portion of A */
1131         anz  = ao->i[arow+1] - ao->i[arow];
1132         for (k=0; k<anz; k++) { /* each nz of Ao */
1133           tmp  = ra[ri[i] + j]*ao->a[ao->i[arow] + k];
1134           prow = ao->j[ao->i[arow] + k];
1135           pnz  = pi_oth[prow+1] - pi_oth[prow];
1136 
1137           for (l=0; l<pnz; l++) { /* each nz of P_oth */
1138             pcol = pj_oth[pi_oth[prow] + l];
1139             cdense[pcol] += tmp*pa_oth[pi_oth[prow] + l];
1140           }
1141         }
1142 
1143       } //for (j=0; j<rnz; j++)
1144 
1145       /* copy cdense[] into ca; zero cdense[] */
1146       cnz = bi[i+1] - bi[i];
1147       cj  = bj + bi[i];
1148       ca  = ba + bi[i];
1149       for (j=0; j<cnz; j++) {
1150         ca[j] += cdense[cj[j]];
1151         cdense[cj[j]] = 0.0;
1152       }
1153 #if 0
1154       if (rank == 0) {
1155         printf("[%d] row %d: ",rank,i);
1156         for (j=0; j<pN; j++) printf(" %g,",cdense[j]);
1157         printf("\n");
1158       }
1159       for (j=0; j<pN; j++) cdense[j]=0.0; // zero cdnese[]
1160 #endif
1161     } //for (i=0; i<cm; i++) {
1162 #endif
1163 
1164     //==========================================
1165 
1166     ierr = PetscTime(&t1);CHKERRQ(ierr);
1167     for (i=0; i<am; i++) {
1168 #if defined(PTAP_PROFILE)
1169       ierr = PetscTime(&t2_0);CHKERRQ(ierr);
1170 #endif
1171       /* 2-a) form i-th sparse row of A_loc*P = Ad*P_loc + Ao*P_oth */
1172       /*------------------------------------------------------------*/
1173       apJ = apj + api[i];
1174 
1175       /* diagonal portion of A */
1176       anz = adi[i+1] - adi[i];
1177       adj = ad->j + adi[i];
1178       ada = ad->a + adi[i];
1179       for (j=0; j<anz; j++) {
1180         row = adj[j];
1181         pnz = pi_loc[row+1] - pi_loc[row];
1182         pj  = pj_loc + pi_loc[row];
1183         pa  = pa_loc + pi_loc[row];
1184 
1185         /* perform dense axpy */
1186         valtmp = ada[j];
1187         for (k=0; k<pnz; k++) {
1188           apa[pj[k]] += valtmp*pa[k];
1189         }
1190         ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
1191       }
1192 
1193       /* off-diagonal portion of A */
1194       anz = aoi[i+1] - aoi[i];
1195       aoj = ao->j + aoi[i];
1196       aoa = ao->a + aoi[i];
1197       for (j=0; j<anz; j++) {
1198         row = aoj[j];
1199         pnz = pi_oth[row+1] - pi_oth[row];
1200         pj  = pj_oth + pi_oth[row];
1201         pa  = pa_oth + pi_oth[row];
1202 
1203         /* perform dense axpy */
1204         valtmp = aoa[j];
1205         for (k=0; k<pnz; k++) {
1206           apa[pj[k]] += valtmp*pa[k];
1207         }
1208         ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
1209       }
1210 #if defined(PTAP_PROFILE)
1211       ierr    = PetscTime(&t2_1);CHKERRQ(ierr);
1212       et2_AP += t2_1 - t2_0;
1213 #endif
1214 
1215       /* 2-b) Compute Cseq = P_loc[i,:]^T*AP[i,:] using outer product */
1216       /*--------------------------------------------------------------*/
1217       apnz = api[i+1] - api[i];
1218       /* put the value into Co=(p->B)^T*AP (off-diagonal part, send to others) */
1219       pnz = po->i[i+1] - po->i[i];
1220       poJ = po->j + po->i[i];
1221       pA  = po->a + po->i[i];
1222       for (j=0; j<pnz; j++) {
1223         row = poJ[j];
1224         cnz = coi[row+1] - coi[row];
1225         cj  = coj + coi[row];
1226         ca  = coa + coi[row];
1227         /* perform dense axpy */
1228         valtmp = pA[j];
1229         for (k=0; k<cnz; k++) {
1230           ca[k] += valtmp*apa[cj[k]];
1231         }
1232         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1233       }
1234 #if 1
1235       /* put the value into Cd (diagonal part) */
1236       pnz = pd->i[i+1] - pd->i[i];
1237       pdJ = pd->j + pd->i[i];
1238       pA  = pd->a + pd->i[i];
1239       for (j=0; j<pnz; j++) {
1240         row = pdJ[j];
1241         cnz = bi[row+1] - bi[row];
1242         cj  = bj + bi[row];
1243         ca  = ba + bi[row];
1244         /* perform dense axpy */
1245         valtmp = pA[j];
1246         for (k=0; k<cnz; k++) {
1247           ca[k] += valtmp*apa[cj[k]];
1248         }
1249         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1250       }
1251 #endif
1252       /* zero the current row of A*P */
1253       for (k=0; k<apnz; k++) apa[apJ[k]] = 0.0;
1254 #if defined(PTAP_PROFILE)
1255       ierr      = PetscTime(&t2_2);CHKERRQ(ierr);
1256       ePtAP += t2_2 - t2_1;
1257 #endif
1258     }
1259 
1260     if (rank == 100) {
1261     for (row=0; row<cm; row++) {
1262       printf("[%d] row %d: ",rank,row);
1263       cnz = bi[row+1] - bi[row];
1264       for (j=0; j<cnz; j++) printf(" %g,",ba[bi[row]+j]);
1265       printf("\n");
1266     }
1267     }
1268 
1269   } else { /* Do sparse axpy on apa (length of ap_rmax, stores A[i,:]*P) - scalable, but slower */
1270     ierr = PetscInfo(C,"Using scalable sparse axpy\n");CHKERRQ(ierr);
1271     /*-----------------------------------------------------------------------------------------*/
1272     pA=pa_loc;
1273     for (i=0; i<am; i++) {
1274 #if defined(PTAP_PROFILE)
1275       ierr = PetscTime(&t2_0);CHKERRQ(ierr);
1276 #endif
1277       /* form i-th sparse row of A*P */
1278       apnz = api[i+1] - api[i];
1279       apJ  = apj + api[i];
1280       /* diagonal portion of A */
1281       anz = adi[i+1] - adi[i];
1282       adj = ad->j + adi[i];
1283       ada = ad->a + adi[i];
1284       for (j=0; j<anz; j++) {
1285         row    = adj[j];
1286         pnz    = pi_loc[row+1] - pi_loc[row];
1287         pj     = pj_loc + pi_loc[row];
1288         pa     = pa_loc + pi_loc[row];
1289         valtmp = ada[j];
1290         nextp  = 0;
1291         for (k=0; nextp<pnz; k++) {
1292           if (apJ[k] == pj[nextp]) { /* col of AP == col of P */
1293             apa[k] += valtmp*pa[nextp++];
1294           }
1295         }
1296         ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
1297       }
1298       /* off-diagonal portion of A */
1299       anz = aoi[i+1] - aoi[i];
1300       aoj = ao->j + aoi[i];
1301       aoa = ao->a + aoi[i];
1302       for (j=0; j<anz; j++) {
1303         row    = aoj[j];
1304         pnz    = pi_oth[row+1] - pi_oth[row];
1305         pj     = pj_oth + pi_oth[row];
1306         pa     = pa_oth + pi_oth[row];
1307         valtmp = aoa[j];
1308         nextp  = 0;
1309         for (k=0; nextp<pnz; k++) {
1310           if (apJ[k] == pj[nextp]) { /* col of AP == col of P */
1311             apa[k] += valtmp*pa[nextp++];
1312           }
1313         }
1314         ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
1315       }
1316 #if defined(PTAP_PROFILE)
1317       ierr    = PetscTime(&t2_1);CHKERRQ(ierr);
1318       et2_AP += t2_1 - t2_0;
1319 #endif
1320 
1321       /* 2-b) Compute Cseq = P_loc[i,:]^T*AP[i,:] using outer product */
1322       /*--------------------------------------------------------------*/
1323       pnz = pi_loc[i+1] - pi_loc[i];
1324       pJ  = pj_loc + pi_loc[i];
1325       for (j=0; j<pnz; j++) {
1326         nextap = 0;
1327         row    = pJ[j]; /* global index */
1328         if (row < pcstart || row >=pcend) { /* put the value into Co */
1329           row = *poJ;
1330           cj  = coj + coi[row];
1331           ca  = coa + coi[row]; poJ++;
1332         } else {                            /* put the value into Cd */
1333           row = *pdJ;
1334           cj  = bj + bi[row];
1335           ca  = ba + bi[row]; pdJ++;
1336         }
1337         valtmp = pA[j];
1338         for (k=0; nextap<apnz; k++) {
1339           if (cj[k]==apJ[nextap]) ca[k] += valtmp*apa[nextap++];
1340         }
1341         ierr = PetscLogFlops(2.0*apnz);CHKERRQ(ierr);
1342       }
1343       pA += pnz;
1344       /* zero the current row info for A*P */
1345       ierr = PetscMemzero(apa,apnz*sizeof(MatScalar));CHKERRQ(ierr);
1346 #if defined(PTAP_PROFILE)
1347       ierr      = PetscTime(&t2_2);CHKERRQ(ierr);
1348       ePtAP += t2_2 - t2_1;
1349 #endif
1350     }
1351   }
1352 #if defined(PTAP_PROFILE)
1353   ierr = PetscTime(&t2);CHKERRQ(ierr);
1354 #endif
1355 
1356   /* 3) send and recv matrix values coa */
1357   /*------------------------------------*/
1358   buf_ri = merge->buf_ri;
1359   buf_rj = merge->buf_rj;
1360   len_s  = merge->len_s;
1361   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1362   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1363 
1364   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1365   for (proc=0,k=0; proc<size; proc++) {
1366     if (!len_s[proc]) continue;
1367     i    = merge->owners_co[proc];
1368     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1369     k++;
1370   }
1371   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1372   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1373 
1374   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1375   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1376   ierr = PetscFree(coa);CHKERRQ(ierr);
1377 #if defined(PTAP_PROFILE)
1378   ierr = PetscTime(&t3);CHKERRQ(ierr);
1379 #endif
1380 
1381   /* 4) insert local Cseq and received values into Cmpi */
1382   /*------------------------------------------------------*/
1383   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1384   for (k=0; k<merge->nrecv; k++) {
1385     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1386     nrows       = *(buf_ri_k[k]);
1387     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1388     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1389   }
1390 
1391   for (i=0; i<cm; i++) {
1392     row  = owners[rank] + i; /* global row index of C_seq */
1393     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1394     ba_i = ba + bi[i];
1395     bnz  = bi[i+1] - bi[i];
1396     /* add received vals into ba */
1397     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1398       /* i-th row */
1399       if (i == *nextrow[k]) {
1400         cnz    = *(nextci[k]+1) - *nextci[k];
1401         cj     = buf_rj[k] + *(nextci[k]);
1402         ca     = abuf_r[k] + *(nextci[k]);
1403         nextcj = 0;
1404         for (j=0; nextcj<cnz; j++) {
1405           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1406             ba_i[j] += ca[nextcj++];
1407           }
1408         }
1409         nextrow[k]++; nextci[k]++;
1410         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1411       }
1412     }
1413     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1414   }
1415   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1416   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1417 
1418   ierr = PetscFree(ba);CHKERRQ(ierr);
1419   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1420   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1421   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1422 #if defined(PTAP_PROFILE)
1423   ierr = PetscTime(&t4);CHKERRQ(ierr);
1424   if (rank==1) {
1425     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);
1426   }
1427 #endif
1428   PetscFunctionReturn(0);
1429 }
1430