xref: /petsc/src/mat/impls/aij/mpi/mpiptap.c (revision e9ede7d0480700edd5b61d6f5d5ea240462b08e4)
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      flg;
109   MPI_Comm       comm;
110 #if !defined(PETSC_HAVE_HYPRE)
111   const char          *algTypes[2] = {"scalable","nonscalable"};
112   PetscInt            nalg=2;
113 #else
114   const char          *algTypes[3] = {"scalable","nonscalable","hypre"};
115   PetscInt            nalg=3;
116 #endif
117   PetscInt            pN=P->cmap->N,alg=1; /* set default algorithm */
118 
119   PetscFunctionBegin;
120   /* check if matrix local sizes are compatible */
121   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
122   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);
123   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);
124 
125   if (scall == MAT_INITIAL_MATRIX) {
126     /* pick an algorithm */
127     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
128     PetscOptionsObject->alreadyprinted = PETSC_FALSE; /* a hack to ensure the option shows in '-help' */
129     ierr = PetscOptionsEList("-matptap_via","Algorithmic approach","MatPtAP",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
130     ierr = PetscOptionsEnd();CHKERRQ(ierr);
131 
132     if (!flg && pN > 100000) { /* may switch to scalable algorithm as default */
133       MatInfo     Ainfo,Pinfo;
134       PetscInt    nz_local;
135       PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
136 
137       ierr = MatGetInfo(A,MAT_LOCAL,&Ainfo);CHKERRQ(ierr);
138       ierr = MatGetInfo(P,MAT_LOCAL,&Pinfo);CHKERRQ(ierr);
139       nz_local = (PetscInt)(Ainfo.nz_allocated + Pinfo.nz_allocated);
140 
141       if (pN > fill*nz_local) alg_scalable_loc = PETSC_TRUE;
142       ierr = MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm);CHKERRQ(ierr);
143 
144       if (alg_scalable) {
145         alg = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
146       }
147     }
148 
149     switch (alg) {
150     case 1:
151       /* do R=P^T locally, then C=R*A*P */
152       ierr = PetscLogEventBegin(MAT_PtAPSymbolic,A,P,0,0);CHKERRQ(ierr);
153       ierr = MatPtAPSymbolic_MPIAIJ_MPIAIJ(A,P,fill,C);CHKERRQ(ierr);
154       ierr = PetscLogEventEnd(MAT_PtAPSymbolic,A,P,0,0);CHKERRQ(ierr);
155       break;
156 #if defined(PETSC_HAVE_HYPRE)
157     case 2:
158       /* Use boomerAMGBuildCoarseOperator */
159       ierr = MatPtAPSymbolic_AIJ_AIJ_wHYPRE(A,P,fill,C);CHKERRQ(ierr);
160       PetscFunctionReturn(0);
161       break;
162 #endif
163     default:
164       /* do R=P^T locally, then C=R*A*P */
165       ierr = PetscLogEventBegin(MAT_PtAPSymbolic,A,P,0,0);CHKERRQ(ierr);
166       ierr = MatPtAPSymbolic_MPIAIJ_MPIAIJ_scalable(A,P,fill,C);CHKERRQ(ierr);
167       ierr = PetscLogEventEnd(MAT_PtAPSymbolic,A,P,0,0);CHKERRQ(ierr);
168       break;
169     }
170   }
171   ierr = PetscLogEventBegin(MAT_PtAPNumeric,A,P,0,0);CHKERRQ(ierr);
172   ierr = (*(*C)->ops->ptapnumeric)(A,P,*C);CHKERRQ(ierr);
173   ierr = PetscLogEventEnd(MAT_PtAPNumeric,A,P,0,0);CHKERRQ(ierr);
174   PetscFunctionReturn(0);
175 }
176 
177 PetscErrorCode MatPtAPNumeric_MPIAIJ_MPIAIJ_scalable(Mat A,Mat P,Mat C)
178 {
179   PetscErrorCode    ierr;
180   Mat_MPIAIJ        *a=(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
181   Mat_SeqAIJ        *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
182   Mat_SeqAIJ        *ap,*p_loc,*p_oth,*c_seq;
183   Mat_PtAPMPI       *ptap = c->ptap;
184   Mat               AP_loc,C_loc,C_oth;
185   PetscInt          i,rstart,rend,cm,ncols,row,*api,*apj,am = A->rmap->n,apnz;
186   PetscScalar       *apa;
187   const PetscInt    *cols;
188   const PetscScalar *vals;
189 
190   PetscFunctionBegin;
191   ierr = MatZeroEntries(C);CHKERRQ(ierr);
192 
193   /* 1) get R = Pd^T,Ro = Po^T */
194   if (ptap->reuse == MAT_REUSE_MATRIX) {
195     ierr = MatTranspose_SeqAIJ(p->A,MAT_REUSE_MATRIX,&ptap->Rd);CHKERRQ(ierr);
196     ierr = MatTranspose_SeqAIJ(p->B,MAT_REUSE_MATRIX,&ptap->Ro);CHKERRQ(ierr);
197   }
198 
199   /* 2) get AP_loc */
200   AP_loc = ptap->AP_loc;
201   ap = (Mat_SeqAIJ*)AP_loc->data;
202 
203   /* 2-1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
204   /*-----------------------------------------------------*/
205   if (ptap->reuse == MAT_REUSE_MATRIX) {
206     /* P_oth and P_loc are obtained in MatPtASymbolic() when reuse == MAT_INITIAL_MATRIX */
207     ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
208     ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
209   }
210 
211   /* 2-2) compute numeric A_loc*P - dominating part */
212   /* ---------------------------------------------- */
213   /* get data from symbolic products */
214   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
215   if (ptap->P_oth) p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
216   else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"ptap->P_oth is NULL. Cannot proceed!");
217 
218   api   = ap->i;
219   apj   = ap->j;
220   for (i=0; i<am; i++) {
221     /* AP[i,:] = A[i,:]*P = Ad*P_loc Ao*P_oth */
222     apnz = api[i+1] - api[i];
223     apa = ap->a + api[i];
224     ierr = PetscMemzero(apa,sizeof(PetscScalar)*apnz);CHKERRQ(ierr);
225     AProw_scalable(i,ad,ao,p_loc,p_oth,api,apj,apa);
226     ierr = PetscLogFlops(2.0*apnz);CHKERRQ(ierr);
227   }
228 
229   /* 3) C_loc = Rd*AP_loc, C_oth = Ro*AP_loc */
230   ierr = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable(ptap->Rd,AP_loc,ptap->C_loc);CHKERRQ(ierr);
231   ierr = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable(ptap->Ro,AP_loc,ptap->C_oth);CHKERRQ(ierr);
232 
233   C_loc = ptap->C_loc;
234   C_oth = ptap->C_oth;
235 
236   /* add C_loc and Co to to C */
237   ierr = MatGetOwnershipRange(C,&rstart,&rend);CHKERRQ(ierr);
238 
239   /* C_loc -> C */
240   cm    = C_loc->rmap->N;
241   c_seq = (Mat_SeqAIJ*)C_loc->data;
242   cols = c_seq->j;
243   vals = c_seq->a;
244 
245   /* The (fast) MatSetValues_MPIAIJ_CopyFromCSRFormat function can only be used when C->was_assembled is PETSC_FALSE and */
246   /* when there are no off-processor parts.  */
247   if (C->assembled) {
248     C->was_assembled = PETSC_TRUE;
249     C->assembled     = PETSC_FALSE;
250   }
251   if (C->was_assembled) {
252     for (i=0; i<cm; i++) {
253       ncols = c_seq->i[i+1] - c_seq->i[i];
254       row = rstart + i;
255       ierr = MatSetValues_MPIAIJ(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
256       cols += ncols; vals += ncols;
257     }
258   } else {
259     ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat(C,c_seq->j,c_seq->i,c_seq->a);CHKERRQ(ierr);
260   }
261 
262   /* Co -> C, off-processor part */
263   cm = C_oth->rmap->N;
264   c_seq = (Mat_SeqAIJ*)C_oth->data;
265   cols = c_seq->j;
266   vals = c_seq->a;
267   for (i=0; i<cm; i++) {
268     ncols = c_seq->i[i+1] - c_seq->i[i];
269     row = p->garray[i];
270     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
271     cols += ncols; vals += ncols;
272   }
273   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
274   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
275 
276   ptap->reuse = MAT_REUSE_MATRIX;
277   PetscFunctionReturn(0);
278 }
279 
280 PetscErrorCode MatPtAPSymbolic_MPIAIJ_MPIAIJ_scalable(Mat A,Mat P,PetscReal fill,Mat *C)
281 {
282   PetscErrorCode      ierr;
283   Mat_PtAPMPI         *ptap;
284   Mat_MPIAIJ          *a=(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data,*c;
285   MPI_Comm            comm;
286   PetscMPIInt         size,rank;
287   Mat                 Cmpi,P_loc,P_oth;
288   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
289   PetscInt            am=A->rmap->n,pm=P->rmap->n,pN=P->cmap->N,pn=P->cmap->n;
290   PetscInt            *lnk,i,k,pnz,row,nsend;
291   PetscMPIInt         tagi,tagj,*len_si,*len_s,*len_ri,icompleted=0,nrecv;
292   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
293   PetscInt            len,proc,*dnz,*onz,*owners,nzi,nspacedouble;
294   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
295   MPI_Request         *swaits,*rwaits;
296   MPI_Status          *sstatus,rstatus;
297   PetscLayout         rowmap;
298   PetscInt            *owners_co,*coi,*coj;    /* i and j array of (p->B)^T*A*P - used in the communication */
299   PetscMPIInt         *len_r,*id_r;    /* array of length of comm->size, store send/recv matrix values */
300   PetscInt            *api,*apj,*Jptr,apnz,*prmap=p->garray,con,j,Crmax,*aj,*ai,*pi;
301   Mat_SeqAIJ          *p_loc,*p_oth=NULL,*ad=(Mat_SeqAIJ*)(a->A)->data,*ao=NULL,*c_loc,*c_oth;
302   PetscScalar         *apv;
303   PetscTable          ta;
304 #if defined(PETSC_USE_INFO)
305   PetscReal           apfill;
306 #endif
307 
308   PetscFunctionBegin;
309   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
310   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
311   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
312 
313   if (size > 1) ao = (Mat_SeqAIJ*)(a->B)->data;
314 
315   /* create symbolic parallel matrix Cmpi */
316   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
317   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
318 
319   /* create struct Mat_PtAPMPI and attached it to C later */
320   ierr        = PetscNew(&ptap);CHKERRQ(ierr);
321   ptap->reuse = MAT_INITIAL_MATRIX;
322   ptap->algType = 0;
323 
324   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
325   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&P_oth);CHKERRQ(ierr);
326   /* get P_loc by taking all local rows of P */
327   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&P_loc);CHKERRQ(ierr);
328 
329   ptap->P_loc = P_loc;
330   ptap->P_oth = P_oth;
331 
332   /* (0) compute Rd = Pd^T, Ro = Po^T  */
333   /* --------------------------------- */
334   ierr = MatTranspose_SeqAIJ(p->A,MAT_INITIAL_MATRIX,&ptap->Rd);CHKERRQ(ierr);
335   ierr = MatTranspose_SeqAIJ(p->B,MAT_INITIAL_MATRIX,&ptap->Ro);CHKERRQ(ierr);
336 
337   /* (1) compute symbolic AP = A_loc*P = Ad*P_loc + Ao*P_oth (api,apj) */
338   /* ----------------------------------------------------------------- */
339   p_loc  = (Mat_SeqAIJ*)P_loc->data;
340   if (P_oth) p_oth = (Mat_SeqAIJ*)P_oth->data;
341 
342   /* create and initialize a linked list */
343   ierr = PetscTableCreate(pn,pN,&ta);CHKERRQ(ierr); /* for compute AP_loc and Cmpi */
344   MatRowMergeMax_SeqAIJ(p_loc,P_loc->rmap->N,ta);
345   MatRowMergeMax_SeqAIJ(p_oth,P_oth->rmap->N,ta);
346   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr); /* Crmax = nnz(sum of Prows) */
347 
348   ierr = PetscLLCondensedCreate_Scalable(Crmax,&lnk);CHKERRQ(ierr);
349 
350   /* Initial FreeSpace size is fill*(nnz(A) + nnz(P)) */
351   if (ao) {
352     ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ad->i[am],PetscIntSumTruncate(ao->i[am],p_loc->i[pm]))),&free_space);CHKERRQ(ierr);
353   } else {
354     ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ad->i[am],p_loc->i[pm])),&free_space);CHKERRQ(ierr);
355   }
356   current_space = free_space;
357   nspacedouble  = 0;
358 
359   ierr   = PetscMalloc1(am+1,&api);CHKERRQ(ierr);
360   api[0] = 0;
361   for (i=0; i<am; i++) {
362     /* diagonal portion: Ad[i,:]*P */
363     ai = ad->i; pi = p_loc->i;
364     nzi = ai[i+1] - ai[i];
365     aj  = ad->j + ai[i];
366     for (j=0; j<nzi; j++) {
367       row  = aj[j];
368       pnz  = pi[row+1] - pi[row];
369       Jptr = p_loc->j + pi[row];
370       /* add non-zero cols of P into the sorted linked list lnk */
371       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
372     }
373     /* off-diagonal portion: Ao[i,:]*P */
374     if (ao) {
375       ai = ao->i; pi = p_oth->i;
376       nzi = ai[i+1] - ai[i];
377       aj  = ao->j + ai[i];
378       for (j=0; j<nzi; j++) {
379         row  = aj[j];
380         pnz  = pi[row+1] - pi[row];
381         Jptr = p_oth->j + pi[row];
382         ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
383       }
384     }
385     apnz     = lnk[0];
386     api[i+1] = api[i] + apnz;
387 
388     /* if free space is not available, double the total space in the list */
389     if (current_space->local_remaining<apnz) {
390       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
391       nspacedouble++;
392     }
393 
394     /* Copy data into free space, then initialize lnk */
395     ierr = PetscLLCondensedClean_Scalable(apnz,current_space->array,lnk);CHKERRQ(ierr);
396 
397     current_space->array           += apnz;
398     current_space->local_used      += apnz;
399     current_space->local_remaining -= apnz;
400   }
401   /* Allocate space for apj and apv, initialize apj, and */
402   /* destroy list of free space and other temporary array(s) */
403   ierr = PetscMalloc2(api[am],&apj,api[am],&apv);CHKERRQ(ierr);
404   ierr = PetscFreeSpaceContiguous(&free_space,apj);CHKERRQ(ierr);
405   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
406 
407   /* Create AP_loc for reuse */
408   ierr = MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,am,pN,api,apj,apv,&ptap->AP_loc);CHKERRQ(ierr);
409 
410 #if defined(PETSC_USE_INFO)
411   if (ao) {
412     apfill = (PetscReal)api[am]/(ad->i[am]+ao->i[am]+p_loc->i[pm]+1);
413   } else {
414     apfill = (PetscReal)api[am]/(ad->i[am]+p_loc->i[pm]+1);
415   }
416   ptap->AP_loc->info.mallocs           = nspacedouble;
417   ptap->AP_loc->info.fill_ratio_given  = fill;
418   ptap->AP_loc->info.fill_ratio_needed = apfill;
419 
420   if (api[am]) {
421     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);
422     ierr = PetscInfo1(ptap->AP_loc,"Use MatPtAP(A,B,MatReuse,%g,&C) for best AP_loc performance.;\n",(double)apfill);CHKERRQ(ierr);
423   } else {
424     ierr = PetscInfo(ptap->AP_loc,"Scalable algorithm, AP_loc is empty \n");CHKERRQ(ierr);
425   }
426 #endif
427 
428   /* (2-1) compute symbolic Co = Ro*AP_loc  */
429   /* ------------------------------------ */
430   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Ro,ptap->AP_loc,fill,&ptap->C_oth);CHKERRQ(ierr);
431 
432   /* (3) send coj of C_oth to other processors  */
433   /* ------------------------------------------ */
434   /* determine row ownership */
435   ierr = PetscLayoutCreate(comm,&rowmap);CHKERRQ(ierr);
436   rowmap->n  = pn;
437   rowmap->bs = 1;
438   ierr   = PetscLayoutSetUp(rowmap);CHKERRQ(ierr);
439   owners = rowmap->range;
440 
441   /* determine the number of messages to send, their lengths */
442   ierr = PetscMalloc4(size,&len_s,size,&len_si,size,&sstatus,size+2,&owners_co);CHKERRQ(ierr);
443   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
444   ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
445 
446   c_oth = (Mat_SeqAIJ*)ptap->C_oth->data;
447   coi   = c_oth->i; coj = c_oth->j;
448   con   = ptap->C_oth->rmap->n;
449   proc  = 0;
450   for (i=0; i<con; i++) {
451     while (prmap[i] >= owners[proc+1]) proc++;
452     len_si[proc]++;               /* num of rows in Co(=Pt*AP) to be sent to [proc] */
453     len_s[proc] += coi[i+1] - coi[i]; /* num of nonzeros in Co to be sent to [proc] */
454   }
455 
456   len          = 0; /* max length of buf_si[], see (4) */
457   owners_co[0] = 0;
458   nsend        = 0;
459   for (proc=0; proc<size; proc++) {
460     owners_co[proc+1] = owners_co[proc] + len_si[proc];
461     if (len_s[proc]) {
462       nsend++;
463       len_si[proc] = 2*(len_si[proc] + 1); /* length of buf_si to be sent to [proc] */
464       len         += len_si[proc];
465     }
466   }
467 
468   /* determine the number and length of messages to receive for coi and coj  */
469   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&nrecv);CHKERRQ(ierr);
470   ierr = PetscGatherMessageLengths2(comm,nsend,nrecv,len_s,len_si,&id_r,&len_r,&len_ri);CHKERRQ(ierr);
471 
472   /* post the Irecv and Isend of coj */
473   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
474   ierr = PetscPostIrecvInt(comm,tagj,nrecv,id_r,len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
475   ierr = PetscMalloc1(nsend+1,&swaits);CHKERRQ(ierr);
476   for (proc=0, k=0; proc<size; proc++) {
477     if (!len_s[proc]) continue;
478     i    = owners_co[proc];
479     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
480     k++;
481   }
482 
483   /* (2-2) compute symbolic C_loc = Rd*AP_loc */
484   /* ---------------------------------------- */
485   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Rd,ptap->AP_loc,fill,&ptap->C_loc);CHKERRQ(ierr);
486   c_loc = (Mat_SeqAIJ*)ptap->C_loc->data;
487 
488   /* receives coj are complete */
489   for (i=0; i<nrecv; i++) {
490     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
491   }
492   ierr = PetscFree(rwaits);CHKERRQ(ierr);
493   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
494 
495   /* add received column indices into ta to update Crmax */
496   for (k=0; k<nrecv; k++) {/* k-th received message */
497     Jptr = buf_rj[k];
498     for (j=0; j<len_r[k]; j++) {
499       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
500     }
501   }
502   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
503   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
504 
505   /* (4) send and recv coi */
506   /*-----------------------*/
507   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
508   ierr   = PetscPostIrecvInt(comm,tagi,nrecv,id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
509   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
510   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
511   for (proc=0,k=0; proc<size; proc++) {
512     if (!len_s[proc]) continue;
513     /* form outgoing message for i-structure:
514          buf_si[0]:                 nrows to be sent
515                [1:nrows]:           row index (global)
516                [nrows+1:2*nrows+1]: i-structure index
517     */
518     /*-------------------------------------------*/
519     nrows       = len_si[proc]/2 - 1; /* num of rows in Co to be sent to [proc] */
520     buf_si_i    = buf_si + nrows+1;
521     buf_si[0]   = nrows;
522     buf_si_i[0] = 0;
523     nrows       = 0;
524     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
525       nzi = coi[i+1] - coi[i];
526       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
527       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
528       nrows++;
529     }
530     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
531     k++;
532     buf_si += len_si[proc];
533   }
534   for (i=0; i<nrecv; i++) {
535     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
536   }
537   ierr = PetscFree(rwaits);CHKERRQ(ierr);
538   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
539 
540   ierr = PetscFree4(len_s,len_si,sstatus,owners_co);CHKERRQ(ierr);
541   ierr = PetscFree(len_ri);CHKERRQ(ierr);
542   ierr = PetscFree(swaits);CHKERRQ(ierr);
543   ierr = PetscFree(buf_s);CHKERRQ(ierr);
544 
545   /* (5) compute the local portion of Cmpi      */
546   /* ------------------------------------------ */
547   /* set initial free space to be Crmax, sufficient for holding nozeros in each row of Cmpi */
548   ierr          = PetscFreeSpaceGet(Crmax,&free_space);CHKERRQ(ierr);
549   current_space = free_space;
550 
551   ierr = PetscMalloc3(nrecv,&buf_ri_k,nrecv,&nextrow,nrecv,&nextci);CHKERRQ(ierr);
552   for (k=0; k<nrecv; k++) {
553     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
554     nrows       = *buf_ri_k[k];
555     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
556     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
557   }
558 
559   ierr = MatPreallocateInitialize(comm,pn,pn,dnz,onz);CHKERRQ(ierr);
560   ierr = PetscLLCondensedCreate_Scalable(Crmax,&lnk);CHKERRQ(ierr);
561   for (i=0; i<pn; i++) {
562     /* add C_loc into Cmpi */
563     nzi  = c_loc->i[i+1] - c_loc->i[i];
564     Jptr = c_loc->j + c_loc->i[i];
565     ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
566 
567     /* add received col data into lnk */
568     for (k=0; k<nrecv; k++) { /* k-th received message */
569       if (i == *nextrow[k]) { /* i-th row */
570         nzi  = *(nextci[k]+1) - *nextci[k];
571         Jptr = buf_rj[k] + *nextci[k];
572         ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
573         nextrow[k]++; nextci[k]++;
574       }
575     }
576     nzi = lnk[0];
577 
578     /* copy data into free space, then initialize lnk */
579     ierr = PetscLLCondensedClean_Scalable(nzi,current_space->array,lnk);CHKERRQ(ierr);
580     ierr = MatPreallocateSet(i+owners[rank],nzi,current_space->array,dnz,onz);CHKERRQ(ierr);
581   }
582   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
583   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
584   ierr = PetscFreeSpaceDestroy(free_space);CHKERRQ(ierr);
585 
586   /* local sizes and preallocation */
587   ierr = MatSetSizes(Cmpi,pn,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
588   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(P->cmap->bs));CHKERRQ(ierr);
589   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
590   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
591 
592   /* members in merge */
593   ierr = PetscFree(id_r);CHKERRQ(ierr);
594   ierr = PetscFree(len_r);CHKERRQ(ierr);
595   ierr = PetscFree(buf_ri[0]);CHKERRQ(ierr);
596   ierr = PetscFree(buf_ri);CHKERRQ(ierr);
597   ierr = PetscFree(buf_rj[0]);CHKERRQ(ierr);
598   ierr = PetscFree(buf_rj);CHKERRQ(ierr);
599   ierr = PetscLayoutDestroy(&rowmap);CHKERRQ(ierr);
600 
601   /* attach the supporting struct to Cmpi for reuse */
602   c = (Mat_MPIAIJ*)Cmpi->data;
603   c->ptap         = ptap;
604   ptap->duplicate = Cmpi->ops->duplicate;
605   ptap->destroy   = Cmpi->ops->destroy;
606   ptap->view      = Cmpi->ops->view;
607 
608   /* Cmpi is not ready for use - assembly will be done by MatPtAPNumeric() */
609   Cmpi->assembled        = PETSC_FALSE;
610   Cmpi->ops->ptapnumeric = MatPtAPNumeric_MPIAIJ_MPIAIJ_scalable;
611   Cmpi->ops->destroy     = MatDestroy_MPIAIJ_PtAP;
612   Cmpi->ops->duplicate   = MatDuplicate_MPIAIJ_MatPtAP;
613   Cmpi->ops->view        = MatView_MPIAIJ_PtAP;
614   *C                     = Cmpi;
615   PetscFunctionReturn(0);
616 }
617 
618 PetscErrorCode MatPtAPSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat *C)
619 {
620   PetscErrorCode      ierr;
621   Mat_PtAPMPI         *ptap;
622   Mat_MPIAIJ          *a=(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data,*c;
623   MPI_Comm            comm;
624   PetscMPIInt         size,rank;
625   Mat                 Cmpi;
626   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
627   PetscInt            am=A->rmap->n,pm=P->rmap->n,pN=P->cmap->N,pn=P->cmap->n;
628   PetscInt            *lnk,i,k,pnz,row,nsend;
629   PetscBT             lnkbt;
630   PetscMPIInt         tagi,tagj,*len_si,*len_s,*len_ri,icompleted=0,nrecv;
631   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
632   PetscInt            len,proc,*dnz,*onz,*owners,nzi,nspacedouble;
633   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
634   MPI_Request         *swaits,*rwaits;
635   MPI_Status          *sstatus,rstatus;
636   PetscLayout         rowmap;
637   PetscInt            *owners_co,*coi,*coj;    /* i and j array of (p->B)^T*A*P - used in the communication */
638   PetscMPIInt         *len_r,*id_r;    /* array of length of comm->size, store send/recv matrix values */
639   PetscInt            *api,*apj,*Jptr,apnz,*prmap=p->garray,con,j,ap_rmax=0,Crmax,*aj,*ai,*pi;
640   Mat_SeqAIJ          *p_loc,*p_oth=NULL,*ad=(Mat_SeqAIJ*)(a->A)->data,*ao=NULL,*c_loc,*c_oth;
641   PetscScalar         *apv;
642   PetscTable          ta;
643 #if defined(PETSC_USE_INFO)
644   PetscReal           apfill;
645 #endif
646 
647   PetscFunctionBegin;
648   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
649   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
650   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
651 
652   if (size > 1) ao = (Mat_SeqAIJ*)(a->B)->data;
653 
654   /* create symbolic parallel matrix Cmpi */
655   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
656   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
657 
658   /* Do dense axpy in MatPtAPNumeric_MPIAIJ_MPIAIJ() */
659   Cmpi->ops->ptapnumeric = MatPtAPNumeric_MPIAIJ_MPIAIJ;
660 
661   /* create struct Mat_PtAPMPI and attached it to C later */
662   ierr        = PetscNew(&ptap);CHKERRQ(ierr);
663   ptap->reuse = MAT_INITIAL_MATRIX;
664   ptap->algType = 1;
665 
666   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
667   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
668   /* get P_loc by taking all local rows of P */
669   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
670 
671   /* (0) compute Rd = Pd^T, Ro = Po^T  */
672   /* --------------------------------- */
673   ierr = MatTranspose_SeqAIJ(p->A,MAT_INITIAL_MATRIX,&ptap->Rd);CHKERRQ(ierr);
674   ierr = MatTranspose_SeqAIJ(p->B,MAT_INITIAL_MATRIX,&ptap->Ro);CHKERRQ(ierr);
675 
676   /* (1) compute symbolic AP = A_loc*P = Ad*P_loc + Ao*P_oth (api,apj) */
677   /* ----------------------------------------------------------------- */
678   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
679   if (ptap->P_oth) p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
680 
681   /* create and initialize a linked list */
682   ierr = PetscTableCreate(pn,pN,&ta);CHKERRQ(ierr); /* for compute AP_loc and Cmpi */
683   MatRowMergeMax_SeqAIJ(p_loc,ptap->P_loc->rmap->N,ta);
684   MatRowMergeMax_SeqAIJ(p_oth,ptap->P_oth->rmap->N,ta);
685   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr); /* Crmax = nnz(sum of Prows) */
686   /* printf("[%d] est %d, Crmax %d; pN %d\n",rank,5*(p_loc->rmax+p_oth->rmax + (PetscInt)(1.e-2*pN)),Crmax,pN); */
687 
688   ierr = PetscLLCondensedCreate(Crmax,pN,&lnk,&lnkbt);CHKERRQ(ierr);
689 
690   /* Initial FreeSpace size is fill*(nnz(A) + nnz(P)) */
691   if (ao) {
692     ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ad->i[am],PetscIntSumTruncate(ao->i[am],p_loc->i[pm]))),&free_space);CHKERRQ(ierr);
693   } else {
694     ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ad->i[am],p_loc->i[pm])),&free_space);CHKERRQ(ierr);
695   }
696   current_space = free_space;
697   nspacedouble  = 0;
698 
699   ierr   = PetscMalloc1(am+1,&api);CHKERRQ(ierr);
700   api[0] = 0;
701   for (i=0; i<am; i++) {
702     /* diagonal portion: Ad[i,:]*P */
703     ai = ad->i; pi = p_loc->i;
704     nzi = ai[i+1] - ai[i];
705     aj  = ad->j + ai[i];
706     for (j=0; j<nzi; j++) {
707       row  = aj[j];
708       pnz  = pi[row+1] - pi[row];
709       Jptr = p_loc->j + pi[row];
710       /* add non-zero cols of P into the sorted linked list lnk */
711       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
712     }
713     /* off-diagonal portion: Ao[i,:]*P */
714     if (ao) {
715       ai = ao->i; pi = p_oth->i;
716       nzi = ai[i+1] - ai[i];
717       aj  = ao->j + ai[i];
718       for (j=0; j<nzi; j++) {
719         row  = aj[j];
720         pnz  = pi[row+1] - pi[row];
721         Jptr = p_oth->j + pi[row];
722         ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
723       }
724     }
725     apnz     = lnk[0];
726     api[i+1] = api[i] + apnz;
727     if (ap_rmax < apnz) ap_rmax = apnz;
728 
729     /* if free space is not available, double the total space in the list */
730     if (current_space->local_remaining<apnz) {
731       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
732       nspacedouble++;
733     }
734 
735     /* Copy data into free space, then initialize lnk */
736     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
737 
738     current_space->array           += apnz;
739     current_space->local_used      += apnz;
740     current_space->local_remaining -= apnz;
741   }
742   /* Allocate space for apj and apv, initialize apj, and */
743   /* destroy list of free space and other temporary array(s) */
744   ierr   = PetscMalloc2(api[am],&apj,api[am],&apv);CHKERRQ(ierr);
745   ierr   = PetscFreeSpaceContiguous(&free_space,apj);CHKERRQ(ierr);
746   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
747 
748   /* Create AP_loc for reuse */
749   ierr = MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,am,pN,api,apj,apv,&ptap->AP_loc);CHKERRQ(ierr);
750 
751 #if defined(PETSC_USE_INFO)
752   if (ao) {
753     apfill = (PetscReal)api[am]/(ad->i[am]+ao->i[am]+p_loc->i[pm]+1);
754   } else {
755     apfill = (PetscReal)api[am]/(ad->i[am]+p_loc->i[pm]+1);
756   }
757   ptap->AP_loc->info.mallocs           = nspacedouble;
758   ptap->AP_loc->info.fill_ratio_given  = fill;
759   ptap->AP_loc->info.fill_ratio_needed = apfill;
760 
761   if (api[am]) {
762     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);
763     ierr = PetscInfo1(ptap->AP_loc,"Use MatPtAP(A,B,MatReuse,%g,&C) for best AP_loc performance.;\n",(double)apfill);CHKERRQ(ierr);
764   } else {
765     ierr = PetscInfo(ptap->AP_loc,"Nonscalable algorithm, AP_loc is empty \n");CHKERRQ(ierr);
766   }
767 #endif
768 
769   /* (2-1) compute symbolic Co = Ro*AP_loc  */
770   /* ------------------------------------ */
771   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Ro,ptap->AP_loc,fill,&ptap->C_oth);CHKERRQ(ierr);
772 
773   /* (3) send coj of C_oth to other processors  */
774   /* ------------------------------------------ */
775   /* determine row ownership */
776   ierr = PetscLayoutCreate(comm,&rowmap);CHKERRQ(ierr);
777   rowmap->n  = pn;
778   rowmap->bs = 1;
779   ierr   = PetscLayoutSetUp(rowmap);CHKERRQ(ierr);
780   owners = rowmap->range;
781 
782   /* determine the number of messages to send, their lengths */
783   ierr = PetscMalloc4(size,&len_s,size,&len_si,size,&sstatus,size+2,&owners_co);CHKERRQ(ierr);
784   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
785   ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
786 
787   c_oth = (Mat_SeqAIJ*)ptap->C_oth->data;
788   coi   = c_oth->i; coj = c_oth->j;
789   con   = ptap->C_oth->rmap->n;
790   proc  = 0;
791   for (i=0; i<con; i++) {
792     while (prmap[i] >= owners[proc+1]) proc++;
793     len_si[proc]++;               /* num of rows in Co(=Pt*AP) to be sent to [proc] */
794     len_s[proc] += coi[i+1] - coi[i]; /* num of nonzeros in Co to be sent to [proc] */
795   }
796 
797   len          = 0; /* max length of buf_si[], see (4) */
798   owners_co[0] = 0;
799   nsend        = 0;
800   for (proc=0; proc<size; proc++) {
801     owners_co[proc+1] = owners_co[proc] + len_si[proc];
802     if (len_s[proc]) {
803       nsend++;
804       len_si[proc] = 2*(len_si[proc] + 1); /* length of buf_si to be sent to [proc] */
805       len         += len_si[proc];
806     }
807   }
808 
809   /* determine the number and length of messages to receive for coi and coj  */
810   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&nrecv);CHKERRQ(ierr);
811   ierr = PetscGatherMessageLengths2(comm,nsend,nrecv,len_s,len_si,&id_r,&len_r,&len_ri);CHKERRQ(ierr);
812 
813   /* post the Irecv and Isend of coj */
814   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
815   ierr = PetscPostIrecvInt(comm,tagj,nrecv,id_r,len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
816   ierr = PetscMalloc1(nsend+1,&swaits);CHKERRQ(ierr);
817   for (proc=0, k=0; proc<size; proc++) {
818     if (!len_s[proc]) continue;
819     i    = owners_co[proc];
820     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
821     k++;
822   }
823 
824   /* (2-2) compute symbolic C_loc = Rd*AP_loc */
825   /* ---------------------------------------- */
826   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Rd,ptap->AP_loc,fill,&ptap->C_loc);CHKERRQ(ierr);
827   c_loc = (Mat_SeqAIJ*)ptap->C_loc->data;
828 
829   /* receives coj are complete */
830   for (i=0; i<nrecv; i++) {
831     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
832   }
833   ierr = PetscFree(rwaits);CHKERRQ(ierr);
834   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
835 
836   /* add received column indices into ta to update Crmax */
837   for (k=0; k<nrecv; k++) {/* k-th received message */
838     Jptr = buf_rj[k];
839     for (j=0; j<len_r[k]; j++) {
840       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
841     }
842   }
843   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
844   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
845 
846   /* (4) send and recv coi */
847   /*-----------------------*/
848   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
849   ierr   = PetscPostIrecvInt(comm,tagi,nrecv,id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
850   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
851   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
852   for (proc=0,k=0; proc<size; proc++) {
853     if (!len_s[proc]) continue;
854     /* form outgoing message for i-structure:
855          buf_si[0]:                 nrows to be sent
856                [1:nrows]:           row index (global)
857                [nrows+1:2*nrows+1]: i-structure index
858     */
859     /*-------------------------------------------*/
860     nrows       = len_si[proc]/2 - 1; /* num of rows in Co to be sent to [proc] */
861     buf_si_i    = buf_si + nrows+1;
862     buf_si[0]   = nrows;
863     buf_si_i[0] = 0;
864     nrows       = 0;
865     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
866       nzi = coi[i+1] - coi[i];
867       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
868       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
869       nrows++;
870     }
871     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
872     k++;
873     buf_si += len_si[proc];
874   }
875   for (i=0; i<nrecv; i++) {
876     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
877   }
878   ierr = PetscFree(rwaits);CHKERRQ(ierr);
879   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
880 
881   ierr = PetscFree4(len_s,len_si,sstatus,owners_co);CHKERRQ(ierr);
882   ierr = PetscFree(len_ri);CHKERRQ(ierr);
883   ierr = PetscFree(swaits);CHKERRQ(ierr);
884   ierr = PetscFree(buf_s);CHKERRQ(ierr);
885 
886   /* (5) compute the local portion of Cmpi      */
887   /* ------------------------------------------ */
888   /* set initial free space to be Crmax, sufficient for holding nozeros in each row of Cmpi */
889   ierr          = PetscFreeSpaceGet(Crmax,&free_space);CHKERRQ(ierr);
890   current_space = free_space;
891 
892   ierr = PetscMalloc3(nrecv,&buf_ri_k,nrecv,&nextrow,nrecv,&nextci);CHKERRQ(ierr);
893   for (k=0; k<nrecv; k++) {
894     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
895     nrows       = *buf_ri_k[k];
896     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
897     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
898   }
899 
900   ierr = MatPreallocateInitialize(comm,pn,pn,dnz,onz);CHKERRQ(ierr);
901   ierr = PetscLLCondensedCreate(Crmax,pN,&lnk,&lnkbt);CHKERRQ(ierr);
902   for (i=0; i<pn; i++) {
903     /* add C_loc into Cmpi */
904     nzi  = c_loc->i[i+1] - c_loc->i[i];
905     Jptr = c_loc->j + c_loc->i[i];
906     ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
907 
908     /* add received col data into lnk */
909     for (k=0; k<nrecv; k++) { /* k-th received message */
910       if (i == *nextrow[k]) { /* i-th row */
911         nzi  = *(nextci[k]+1) - *nextci[k];
912         Jptr = buf_rj[k] + *nextci[k];
913         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
914         nextrow[k]++; nextci[k]++;
915       }
916     }
917     nzi = lnk[0];
918 
919     /* copy data into free space, then initialize lnk */
920     ierr = PetscLLCondensedClean(pN,nzi,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
921     ierr = MatPreallocateSet(i+owners[rank],nzi,current_space->array,dnz,onz);CHKERRQ(ierr);
922   }
923   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
924   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
925   ierr = PetscFreeSpaceDestroy(free_space);CHKERRQ(ierr);
926 
927   /* local sizes and preallocation */
928   ierr = MatSetSizes(Cmpi,pn,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
929   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(P->cmap->bs));CHKERRQ(ierr);
930   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
931   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
932 
933   /* members in merge */
934   ierr = PetscFree(id_r);CHKERRQ(ierr);
935   ierr = PetscFree(len_r);CHKERRQ(ierr);
936   ierr = PetscFree(buf_ri[0]);CHKERRQ(ierr);
937   ierr = PetscFree(buf_ri);CHKERRQ(ierr);
938   ierr = PetscFree(buf_rj[0]);CHKERRQ(ierr);
939   ierr = PetscFree(buf_rj);CHKERRQ(ierr);
940   ierr = PetscLayoutDestroy(&rowmap);CHKERRQ(ierr);
941 
942   /* attach the supporting struct to Cmpi for reuse */
943   c = (Mat_MPIAIJ*)Cmpi->data;
944   c->ptap         = ptap;
945   ptap->duplicate = Cmpi->ops->duplicate;
946   ptap->destroy   = Cmpi->ops->destroy;
947   ptap->view      = Cmpi->ops->view;
948   ierr = PetscCalloc1(pN,&ptap->apa);CHKERRQ(ierr);
949 
950   /* Cmpi is not ready for use - assembly will be done by MatPtAPNumeric() */
951   Cmpi->assembled        = PETSC_FALSE;
952   Cmpi->ops->destroy     = MatDestroy_MPIAIJ_PtAP;
953   Cmpi->ops->duplicate   = MatDuplicate_MPIAIJ_MatPtAP;
954   Cmpi->ops->view        = MatView_MPIAIJ_PtAP;
955   *C                     = Cmpi;
956   PetscFunctionReturn(0);
957 }
958 
959 
960 PetscErrorCode MatPtAPNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
961 {
962   PetscErrorCode    ierr;
963   Mat_MPIAIJ        *a=(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
964   Mat_SeqAIJ        *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
965   Mat_SeqAIJ        *ap,*p_loc,*p_oth=NULL,*c_seq;
966   Mat_PtAPMPI       *ptap = c->ptap;
967   Mat               AP_loc,C_loc,C_oth;
968   PetscInt          i,rstart,rend,cm,ncols,row;
969   PetscInt          *api,*apj,am = A->rmap->n,j,col,apnz;
970   PetscScalar       *apa;
971   const PetscInt    *cols;
972   const PetscScalar *vals;
973 
974   PetscFunctionBegin;
975   ierr = MatZeroEntries(C);CHKERRQ(ierr);
976   /* 1) get R = Pd^T,Ro = Po^T */
977   if (ptap->reuse == MAT_REUSE_MATRIX) {
978     ierr = MatTranspose_SeqAIJ(p->A,MAT_REUSE_MATRIX,&ptap->Rd);CHKERRQ(ierr);
979     ierr = MatTranspose_SeqAIJ(p->B,MAT_REUSE_MATRIX,&ptap->Ro);CHKERRQ(ierr);
980   }
981 
982   /* 2) get AP_loc */
983   AP_loc = ptap->AP_loc;
984   ap = (Mat_SeqAIJ*)AP_loc->data;
985 
986   /* 2-1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
987   /*-----------------------------------------------------*/
988   if (ptap->reuse == MAT_REUSE_MATRIX) {
989     /* P_oth and P_loc are obtained in MatPtASymbolic() when reuse == MAT_INITIAL_MATRIX */
990     ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
991     ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
992   }
993 
994   /* 2-2) compute numeric A_loc*P - dominating part */
995   /* ---------------------------------------------- */
996   /* get data from symbolic products */
997   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
998   if (ptap->P_oth) {
999     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
1000   }
1001   apa   = ptap->apa;
1002   api   = ap->i;
1003   apj   = ap->j;
1004   for (i=0; i<am; i++) {
1005     /* AP[i,:] = A[i,:]*P = Ad*P_loc Ao*P_oth */
1006     AProw_nonscalable(i,ad,ao,p_loc,p_oth,apa);
1007     apnz = api[i+1] - api[i];
1008     for (j=0; j<apnz; j++) {
1009       col = apj[j+api[i]];
1010       ap->a[j+ap->i[i]] = apa[col];
1011       apa[col] = 0.0;
1012     }
1013     ierr = PetscLogFlops(2.0*apnz);CHKERRQ(ierr);
1014   }
1015 
1016   /* 3) C_loc = Rd*AP_loc, C_oth = Ro*AP_loc */
1017   ierr = ((ptap->C_loc)->ops->matmultnumeric)(ptap->Rd,AP_loc,ptap->C_loc);CHKERRQ(ierr);
1018   ierr = ((ptap->C_oth)->ops->matmultnumeric)(ptap->Ro,AP_loc,ptap->C_oth);CHKERRQ(ierr);
1019   C_loc = ptap->C_loc;
1020   C_oth = ptap->C_oth;
1021 
1022   /* add C_loc and Co to to C */
1023   ierr = MatGetOwnershipRange(C,&rstart,&rend);CHKERRQ(ierr);
1024 
1025   /* C_loc -> C */
1026   cm    = C_loc->rmap->N;
1027   c_seq = (Mat_SeqAIJ*)C_loc->data;
1028   cols = c_seq->j;
1029   vals = c_seq->a;
1030 
1031 
1032   /* The (fast) MatSetValues_MPIAIJ_CopyFromCSRFormat function can only be used when C->was_assembled is PETSC_FALSE and */
1033   /* when there are no off-processor parts.  */
1034   if (C->assembled) {
1035     C->was_assembled = PETSC_TRUE;
1036     C->assembled     = PETSC_FALSE;
1037   }
1038   if (C->was_assembled) {
1039     for (i=0; i<cm; i++) {
1040       ncols = c_seq->i[i+1] - c_seq->i[i];
1041       row = rstart + i;
1042       ierr = MatSetValues_MPIAIJ(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1043       cols += ncols; vals += ncols;
1044     }
1045   } else {
1046     ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat(C,c_seq->j,c_seq->i,c_seq->a);CHKERRQ(ierr);
1047   }
1048 
1049   /* Co -> C, off-processor part */
1050   cm = C_oth->rmap->N;
1051   c_seq = (Mat_SeqAIJ*)C_oth->data;
1052   cols = c_seq->j;
1053   vals = c_seq->a;
1054   for (i=0; i<cm; i++) {
1055     ncols = c_seq->i[i+1] - c_seq->i[i];
1056     row = p->garray[i];
1057     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1058     cols += ncols; vals += ncols;
1059   }
1060 
1061   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1062   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1063 
1064   ptap->reuse = MAT_REUSE_MATRIX;
1065   PetscFunctionReturn(0);
1066 }
1067