xref: /petsc/src/mat/impls/aij/mpi/mpiptap.c (revision 154d0d783dcbe04e5267aa8d1a7b3398a5442799) 
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_APMPI         *ptap=a->ap;
20   PetscBool         iascii;
21   PetscViewerFormat format;
22 
23   PetscFunctionBegin;
24   if (!ptap) {
25     /* hack: MatDuplicate() sets oldmat->ops->view to newmat which is a base mat class with null ptpa! */
26     A->ops->view = MatView_MPIAIJ;
27     ierr = (A->ops->view)(A,viewer);CHKERRQ(ierr);
28     PetscFunctionReturn(0);
29   }
30 
31   ierr = PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);CHKERRQ(ierr);
32   if (iascii) {
33     ierr = PetscViewerGetFormat(viewer,&format);CHKERRQ(ierr);
34     if (format == PETSC_VIEWER_ASCII_INFO || format == PETSC_VIEWER_ASCII_INFO_DETAIL) {
35       if (ptap->algType == 0) {
36         ierr = PetscViewerASCIIPrintf(viewer,"using scalable MatPtAP() implementation\n");CHKERRQ(ierr);
37       } else if (ptap->algType == 1) {
38         ierr = PetscViewerASCIIPrintf(viewer,"using nonscalable MatPtAP() implementation\n");CHKERRQ(ierr);
39       }
40     }
41   }
42   ierr = (ptap->view)(A,viewer);CHKERRQ(ierr);
43   PetscFunctionReturn(0);
44 }
45 
46 PetscErrorCode MatFreeIntermediateDataStructures_MPIAIJ_AP(Mat A)
47 {
48   PetscErrorCode      ierr;
49   Mat_MPIAIJ          *a=(Mat_MPIAIJ*)A->data;
50   Mat_APMPI           *ptap=a->ap;
51   Mat_Merge_SeqsToMPI *merge;
52 
53   PetscFunctionBegin;
54   if (!ptap) PetscFunctionReturn(0);
55 
56   ierr = PetscFree2(ptap->startsj_s,ptap->startsj_r);CHKERRQ(ierr);
57   ierr = PetscFree(ptap->bufa);CHKERRQ(ierr);
58   ierr = MatDestroy(&ptap->P_loc);CHKERRQ(ierr);
59   ierr = MatDestroy(&ptap->P_oth);CHKERRQ(ierr);
60   ierr = MatDestroy(&ptap->A_loc);CHKERRQ(ierr); /* used by MatTransposeMatMult() */
61   ierr = MatDestroy(&ptap->Rd);CHKERRQ(ierr);
62   ierr = MatDestroy(&ptap->Ro);CHKERRQ(ierr);
63   if (ptap->AP_loc) { /* used by alg_rap */
64     Mat_SeqAIJ *ap = (Mat_SeqAIJ*)(ptap->AP_loc)->data;
65     ierr = PetscFree(ap->i);CHKERRQ(ierr);
66     ierr = PetscFree2(ap->j,ap->a);CHKERRQ(ierr);
67     ierr = MatDestroy(&ptap->AP_loc);CHKERRQ(ierr);
68   } else { /* used by alg_ptap */
69     ierr = PetscFree(ptap->api);CHKERRQ(ierr);
70     ierr = PetscFree(ptap->apj);CHKERRQ(ierr);
71   }
72   ierr = MatDestroy(&ptap->C_loc);CHKERRQ(ierr);
73   ierr = MatDestroy(&ptap->C_oth);CHKERRQ(ierr);
74   if (ptap->apa) {ierr = PetscFree(ptap->apa);CHKERRQ(ierr);}
75 
76   ierr = MatDestroy(&ptap->Pt);CHKERRQ(ierr);
77 
78   merge=ptap->merge;
79   if (merge) { /* used by alg_ptap */
80     ierr = PetscFree(merge->id_r);CHKERRQ(ierr);
81     ierr = PetscFree(merge->len_s);CHKERRQ(ierr);
82     ierr = PetscFree(merge->len_r);CHKERRQ(ierr);
83     ierr = PetscFree(merge->bi);CHKERRQ(ierr);
84     ierr = PetscFree(merge->bj);CHKERRQ(ierr);
85     ierr = PetscFree(merge->buf_ri[0]);CHKERRQ(ierr);
86     ierr = PetscFree(merge->buf_ri);CHKERRQ(ierr);
87     ierr = PetscFree(merge->buf_rj[0]);CHKERRQ(ierr);
88     ierr = PetscFree(merge->buf_rj);CHKERRQ(ierr);
89     ierr = PetscFree(merge->coi);CHKERRQ(ierr);
90     ierr = PetscFree(merge->coj);CHKERRQ(ierr);
91     ierr = PetscFree(merge->owners_co);CHKERRQ(ierr);
92     ierr = PetscLayoutDestroy(&merge->rowmap);CHKERRQ(ierr);
93     ierr = PetscFree(ptap->merge);CHKERRQ(ierr);
94   }
95   ierr = ISLocalToGlobalMappingDestroy(&ptap->ltog);CHKERRQ(ierr);
96 
97   A->ops->destroy = ptap->destroy;
98   ierr = PetscFree(a->ap);CHKERRQ(ierr);
99   PetscFunctionReturn(0);
100 }
101 
102 PetscErrorCode MatDestroy_MPIAIJ_PtAP(Mat A)
103 {
104   PetscErrorCode ierr;
105 
106   PetscFunctionBegin;
107   ierr = (*A->ops->freeintermediatedatastructures)(A);CHKERRQ(ierr);
108   ierr = (*A->ops->destroy)(A);CHKERRQ(ierr);
109   PetscFunctionReturn(0);
110 }
111 
112 PETSC_INTERN PetscErrorCode MatPtAP_MPIAIJ_MPIAIJ(Mat A,Mat P,MatReuse scall,PetscReal fill,Mat *C)
113 {
114   PetscErrorCode ierr;
115   PetscBool      flg;
116   MPI_Comm       comm;
117 #if !defined(PETSC_HAVE_HYPRE)
118   const char          *algTypes[2] = {"scalable","nonscalable"};
119   PetscInt            nalg=2;
120 #else
121   const char          *algTypes[3] = {"scalable","nonscalable","hypre"};
122   PetscInt            nalg=3;
123 #endif
124   PetscInt            pN=P->cmap->N,alg=1; /* set default algorithm */
125 
126   PetscFunctionBegin;
127   /* check if matrix local sizes are compatible */
128   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
129   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);
130   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);
131 
132   if (scall == MAT_INITIAL_MATRIX) {
133     /* pick an algorithm */
134     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)A),((PetscObject)A)->prefix,"MatPtAP","Mat");CHKERRQ(ierr);
135     ierr = PetscOptionsEList("-matptap_via","Algorithmic approach","MatPtAP",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
136     ierr = PetscOptionsEnd();CHKERRQ(ierr);
137 
138     if (!flg && pN > 100000) { /* may switch to scalable algorithm as default */
139       MatInfo     Ainfo,Pinfo;
140       PetscInt    nz_local;
141       PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
142 
143       ierr = MatGetInfo(A,MAT_LOCAL,&Ainfo);CHKERRQ(ierr);
144       ierr = MatGetInfo(P,MAT_LOCAL,&Pinfo);CHKERRQ(ierr);
145       nz_local = (PetscInt)(Ainfo.nz_allocated + Pinfo.nz_allocated);
146 
147       if (pN > fill*nz_local) alg_scalable_loc = PETSC_TRUE;
148       ierr = MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm);CHKERRQ(ierr);
149 
150       if (alg_scalable) {
151         alg = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
152       }
153     }
154 
155     switch (alg) {
156     case 1:
157       /* do R=P^T locally, then C=R*A*P -- nonscalable */
158       ierr = PetscLogEventBegin(MAT_PtAPSymbolic,A,P,0,0);CHKERRQ(ierr);
159       ierr = MatPtAPSymbolic_MPIAIJ_MPIAIJ(A,P,fill,C);CHKERRQ(ierr);
160       ierr = PetscLogEventEnd(MAT_PtAPSymbolic,A,P,0,0);CHKERRQ(ierr);
161       break;
162 #if defined(PETSC_HAVE_HYPRE)
163     case 2:
164       /* Use boomerAMGBuildCoarseOperator */
165       ierr = PetscLogEventBegin(MAT_PtAPSymbolic,A,P,0,0);CHKERRQ(ierr);
166       ierr = MatPtAPSymbolic_AIJ_AIJ_wHYPRE(A,P,fill,C);CHKERRQ(ierr);
167       ierr = PetscLogEventEnd(MAT_PtAPSymbolic,A,P,0,0);CHKERRQ(ierr);
168       break;
169 #endif
170     default:
171       /* do R=P^T locally, then C=R*A*P */
172       ierr = PetscLogEventBegin(MAT_PtAPSymbolic,A,P,0,0);CHKERRQ(ierr);
173       ierr = MatPtAPSymbolic_MPIAIJ_MPIAIJ_scalable(A,P,fill,C);CHKERRQ(ierr);
174       ierr = PetscLogEventEnd(MAT_PtAPSymbolic,A,P,0,0);CHKERRQ(ierr);
175       break;
176     }
177 
178     if (alg == 0 || alg == 1) {
179       Mat_MPIAIJ *c  = (Mat_MPIAIJ*)(*C)->data;
180       Mat_APMPI  *ap = c->ap;
181       ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)(*C)),((PetscObject)(*C))->prefix,"MatFreeIntermediateDataStructures","Mat");CHKERRQ(ierr);
182       ap->freestruct = PETSC_FALSE;
183       ierr = PetscOptionsBool("-mat_freeintermediatedatastructures","Free intermediate data structures", "MatFreeIntermediateDataStructures",ap->freestruct,&ap->freestruct, NULL);CHKERRQ(ierr);
184       ierr = PetscOptionsEnd();CHKERRQ(ierr);
185     }
186   }
187 
188   ierr = PetscLogEventBegin(MAT_PtAPNumeric,A,P,0,0);CHKERRQ(ierr);
189   ierr = (*(*C)->ops->ptapnumeric)(A,P,*C);CHKERRQ(ierr);
190   ierr = PetscLogEventEnd(MAT_PtAPNumeric,A,P,0,0);CHKERRQ(ierr);
191   PetscFunctionReturn(0);
192 }
193 
194 PetscErrorCode MatPtAPNumeric_MPIAIJ_MPIAIJ_scalable(Mat A,Mat P,Mat C)
195 {
196   PetscErrorCode    ierr;
197   Mat_MPIAIJ        *a=(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
198   Mat_SeqAIJ        *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
199   Mat_SeqAIJ        *ap,*p_loc,*p_oth,*c_seq;
200   Mat_APMPI         *ptap = c->ap;
201   Mat               AP_loc,C_loc,C_oth;
202   PetscInt          i,rstart,rend,cm,ncols,row,*api,*apj,am = A->rmap->n,apnz,nout;
203   PetscScalar       *apa;
204   const PetscInt    *cols;
205   const PetscScalar *vals;
206 
207   PetscFunctionBegin;
208   if (!ptap) {
209     MPI_Comm comm;
210     ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
211     SETERRQ(comm,PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be reused. Do not call MatFreeIntermediateDataStructures() or use '-mat_freeintermediatedatastructures'");
212   }
213 
214   ierr = MatZeroEntries(C);CHKERRQ(ierr);
215 
216   /* 1) get R = Pd^T,Ro = Po^T */
217   if (ptap->reuse == MAT_REUSE_MATRIX) {
218     ierr = MatTranspose(p->A,MAT_REUSE_MATRIX,&ptap->Rd);CHKERRQ(ierr);
219     ierr = MatTranspose(p->B,MAT_REUSE_MATRIX,&ptap->Ro);CHKERRQ(ierr);
220   }
221 
222   /* 2) get AP_loc */
223   AP_loc = ptap->AP_loc;
224   ap = (Mat_SeqAIJ*)AP_loc->data;
225 
226   /* 2-1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
227   /*-----------------------------------------------------*/
228   if (ptap->reuse == MAT_REUSE_MATRIX) {
229     /* P_oth and P_loc are obtained in MatPtASymbolic() when reuse == MAT_INITIAL_MATRIX */
230     ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
231     ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
232   }
233 
234   /* 2-2) compute numeric A_loc*P - dominating part */
235   /* ---------------------------------------------- */
236   /* get data from symbolic products */
237   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
238   if (ptap->P_oth) p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
239   else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"ptap->P_oth is NULL. Cannot proceed!");
240 
241   api   = ap->i;
242   apj   = ap->j;
243   ierr = ISLocalToGlobalMappingApply(ptap->ltog,api[AP_loc->rmap->n],apj,apj);CHKERRQ(ierr);
244   for (i=0; i<am; i++) {
245     /* AP[i,:] = A[i,:]*P = Ad*P_loc Ao*P_oth */
246     apnz = api[i+1] - api[i];
247     apa = ap->a + api[i];
248     ierr = PetscMemzero(apa,sizeof(PetscScalar)*apnz);CHKERRQ(ierr);
249     AProw_scalable(i,ad,ao,p_loc,p_oth,api,apj,apa);
250     ierr = PetscLogFlops(2.0*apnz);CHKERRQ(ierr);
251   }
252   ierr = ISGlobalToLocalMappingApply(ptap->ltog,IS_GTOLM_DROP,api[AP_loc->rmap->n],apj,&nout,apj);CHKERRQ(ierr);
253   if (api[AP_loc->rmap->n] != nout) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_INCOMP,"Incorrect mapping %D != %D\n",api[AP_loc->rmap->n],nout);
254 
255   /* 3) C_loc = Rd*AP_loc, C_oth = Ro*AP_loc */
256   /* Always use scalable version since we are in the MPI scalable version */
257   ierr = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable(ptap->Rd,AP_loc,ptap->C_loc);CHKERRQ(ierr);
258   ierr = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable(ptap->Ro,AP_loc,ptap->C_oth);CHKERRQ(ierr);
259 
260   C_loc = ptap->C_loc;
261   C_oth = ptap->C_oth;
262 
263   /* add C_loc and Co to to C */
264   ierr = MatGetOwnershipRange(C,&rstart,&rend);CHKERRQ(ierr);
265 
266   /* C_loc -> C */
267   cm    = C_loc->rmap->N;
268   c_seq = (Mat_SeqAIJ*)C_loc->data;
269   cols = c_seq->j;
270   vals = c_seq->a;
271   ierr = ISLocalToGlobalMappingApply(ptap->ltog,c_seq->i[C_loc->rmap->n],c_seq->j,c_seq->j);CHKERRQ(ierr);
272 
273   /* The (fast) MatSetValues_MPIAIJ_CopyFromCSRFormat function can only be used when C->was_assembled is PETSC_FALSE and */
274   /* when there are no off-processor parts.  */
275   /* If was_assembled is true, then the statement aj[rowstart_diag+dnz_row] = mat_j[col] - cstart; in MatSetValues_MPIAIJ_CopyFromCSRFormat */
276   /* is no longer true. Then the more complex function MatSetValues_MPIAIJ() has to be used, where the column index is looked up from */
277   /* a table, and other, more complex stuff has to be done. */
278   if (C->assembled) {
279     C->was_assembled = PETSC_TRUE;
280     C->assembled     = PETSC_FALSE;
281   }
282   if (C->was_assembled) {
283     for (i=0; i<cm; i++) {
284       ncols = c_seq->i[i+1] - c_seq->i[i];
285       row = rstart + i;
286       ierr = MatSetValues_MPIAIJ(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
287       cols += ncols; vals += ncols;
288     }
289   } else {
290     ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat(C,c_seq->j,c_seq->i,c_seq->a);CHKERRQ(ierr);
291   }
292   ierr = ISGlobalToLocalMappingApply(ptap->ltog,IS_GTOLM_DROP,c_seq->i[C_loc->rmap->n],c_seq->j,&nout,c_seq->j);CHKERRQ(ierr);
293   if (c_seq->i[C_loc->rmap->n] != nout) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_INCOMP,"Incorrect mapping %D != %D\n",c_seq->i[C_loc->rmap->n],nout);
294 
295   /* Co -> C, off-processor part */
296   cm = C_oth->rmap->N;
297   c_seq = (Mat_SeqAIJ*)C_oth->data;
298   cols = c_seq->j;
299   vals = c_seq->a;
300   ierr = ISLocalToGlobalMappingApply(ptap->ltog,c_seq->i[C_oth->rmap->n],c_seq->j,c_seq->j);CHKERRQ(ierr);
301   for (i=0; i<cm; i++) {
302     ncols = c_seq->i[i+1] - c_seq->i[i];
303     row = p->garray[i];
304     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
305     cols += ncols; vals += ncols;
306   }
307   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
308   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
309 
310   ptap->reuse = MAT_REUSE_MATRIX;
311 
312   ierr = ISGlobalToLocalMappingApply(ptap->ltog,IS_GTOLM_DROP,c_seq->i[C_oth->rmap->n],c_seq->j,&nout,c_seq->j);CHKERRQ(ierr);
313   if (c_seq->i[C_oth->rmap->n] != nout) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_INCOMP,"Incorrect mapping %D != %D\n",c_seq->i[C_loc->rmap->n],nout);
314 
315   /* supporting struct ptap consumes almost same amount of memory as C=PtAP, release it if C will not be updated by A and P */
316   if (ptap->freestruct) {
317     ierr = MatFreeIntermediateDataStructures(C);CHKERRQ(ierr);
318   }
319   PetscFunctionReturn(0);
320 }
321 
322 PetscErrorCode MatPtAPSymbolic_MPIAIJ_MPIAIJ_scalable(Mat A,Mat P,PetscReal fill,Mat *C)
323 {
324   PetscErrorCode      ierr;
325   Mat_APMPI           *ptap;
326   Mat_MPIAIJ          *a=(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data,*c;
327   MPI_Comm            comm;
328   PetscMPIInt         size,rank;
329   Mat                 Cmpi,P_loc,P_oth;
330   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
331   PetscInt            am=A->rmap->n,pm=P->rmap->n,pN=P->cmap->N,pn=P->cmap->n;
332   PetscInt            *lnk,i,k,pnz,row,nsend;
333   PetscMPIInt         tagi,tagj,*len_si,*len_s,*len_ri,icompleted=0,nrecv;
334   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
335   PetscInt            len,proc,*dnz,*onz,*owners,nzi,nspacedouble;
336   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
337   MPI_Request         *swaits,*rwaits;
338   MPI_Status          *sstatus,rstatus;
339   PetscLayout         rowmap;
340   PetscInt            *owners_co,*coi,*coj;    /* i and j array of (p->B)^T*A*P - used in the communication */
341   PetscMPIInt         *len_r,*id_r;    /* array of length of comm->size, store send/recv matrix values */
342   PetscInt            *api,*apj,*Jptr,apnz,*prmap=p->garray,con,j,Crmax,*aj,*ai,*pi,nout;
343   Mat_SeqAIJ          *p_loc,*p_oth=NULL,*ad=(Mat_SeqAIJ*)(a->A)->data,*ao=NULL,*c_loc,*c_oth;
344   PetscScalar         *apv;
345   PetscTable          ta;
346   MatType             mtype;
347 #if defined(PETSC_USE_INFO)
348   PetscReal           apfill;
349 #endif
350 
351   PetscFunctionBegin;
352   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
353   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
354   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
355 
356   if (size > 1) ao = (Mat_SeqAIJ*)(a->B)->data;
357 
358   /* create symbolic parallel matrix Cmpi */
359   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
360   ierr = MatGetType(A,&mtype);CHKERRQ(ierr);
361   ierr = MatSetType(Cmpi,mtype);CHKERRQ(ierr);
362 
363   /* create struct Mat_APMPI and attached it to C later */
364   ierr        = PetscNew(&ptap);CHKERRQ(ierr);
365   ptap->reuse = MAT_INITIAL_MATRIX;
366   ptap->algType = 0;
367 
368   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
369   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&P_oth);CHKERRQ(ierr);
370   /* get P_loc by taking all local rows of P */
371   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&P_loc);CHKERRQ(ierr);
372 
373   ptap->P_loc = P_loc;
374   ptap->P_oth = P_oth;
375 
376   /* (0) compute Rd = Pd^T, Ro = Po^T  */
377   /* --------------------------------- */
378   ierr = MatTranspose(p->A,MAT_INITIAL_MATRIX,&ptap->Rd);CHKERRQ(ierr);
379   ierr = MatTranspose(p->B,MAT_INITIAL_MATRIX,&ptap->Ro);CHKERRQ(ierr);
380 
381   /* (1) compute symbolic AP = A_loc*P = Ad*P_loc + Ao*P_oth (api,apj) */
382   /* ----------------------------------------------------------------- */
383   p_loc  = (Mat_SeqAIJ*)P_loc->data;
384   if (P_oth) p_oth = (Mat_SeqAIJ*)P_oth->data;
385 
386   /* create and initialize a linked list */
387   ierr = PetscTableCreate(pn,pN,&ta);CHKERRQ(ierr); /* for compute AP_loc and Cmpi */
388   MatRowMergeMax_SeqAIJ(p_loc,P_loc->rmap->N,ta);
389   MatRowMergeMax_SeqAIJ(p_oth,P_oth->rmap->N,ta);
390   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr); /* Crmax = nnz(sum of Prows) */
391 
392   ierr = PetscLLCondensedCreate_Scalable(Crmax,&lnk);CHKERRQ(ierr);
393 
394   /* Initial FreeSpace size is fill*(nnz(A) + nnz(P)) */
395   if (ao) {
396     ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ad->i[am],PetscIntSumTruncate(ao->i[am],p_loc->i[pm]))),&free_space);CHKERRQ(ierr);
397   } else {
398     ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ad->i[am],p_loc->i[pm])),&free_space);CHKERRQ(ierr);
399   }
400   current_space = free_space;
401   nspacedouble  = 0;
402 
403   ierr   = PetscMalloc1(am+1,&api);CHKERRQ(ierr);
404   api[0] = 0;
405   for (i=0; i<am; i++) {
406     /* diagonal portion: Ad[i,:]*P */
407     ai = ad->i; pi = p_loc->i;
408     nzi = ai[i+1] - ai[i];
409     aj  = ad->j + ai[i];
410     for (j=0; j<nzi; j++) {
411       row  = aj[j];
412       pnz  = pi[row+1] - pi[row];
413       Jptr = p_loc->j + pi[row];
414       /* add non-zero cols of P into the sorted linked list lnk */
415       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
416     }
417     /* off-diagonal portion: Ao[i,:]*P */
418     if (ao) {
419       ai = ao->i; pi = p_oth->i;
420       nzi = ai[i+1] - ai[i];
421       aj  = ao->j + ai[i];
422       for (j=0; j<nzi; j++) {
423         row  = aj[j];
424         pnz  = pi[row+1] - pi[row];
425         Jptr = p_oth->j + pi[row];
426         ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
427       }
428     }
429     apnz     = lnk[0];
430     api[i+1] = api[i] + apnz;
431 
432     /* if free space is not available, double the total space in the list */
433     if (current_space->local_remaining<apnz) {
434       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
435       nspacedouble++;
436     }
437 
438     /* Copy data into free space, then initialize lnk */
439     ierr = PetscLLCondensedClean_Scalable(apnz,current_space->array,lnk);CHKERRQ(ierr);
440 
441     current_space->array           += apnz;
442     current_space->local_used      += apnz;
443     current_space->local_remaining -= apnz;
444   }
445   /* Allocate space for apj and apv, initialize apj, and */
446   /* destroy list of free space and other temporary array(s) */
447   ierr = PetscCalloc2(api[am],&apj,api[am],&apv);CHKERRQ(ierr);
448   ierr = PetscFreeSpaceContiguous(&free_space,apj);CHKERRQ(ierr);
449   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
450 
451   /* Create AP_loc for reuse */
452   ierr = MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,am,pN,api,apj,apv,&ptap->AP_loc);CHKERRQ(ierr);
453   ierr = MatSeqAIJCompactOutExtraColumns_SeqAIJ(ptap->AP_loc, &ptap->ltog);CHKERRQ(ierr);
454 
455 #if defined(PETSC_USE_INFO)
456   if (ao) {
457     apfill = (PetscReal)api[am]/(ad->i[am]+ao->i[am]+p_loc->i[pm]+1);
458   } else {
459     apfill = (PetscReal)api[am]/(ad->i[am]+p_loc->i[pm]+1);
460   }
461   ptap->AP_loc->info.mallocs           = nspacedouble;
462   ptap->AP_loc->info.fill_ratio_given  = fill;
463   ptap->AP_loc->info.fill_ratio_needed = apfill;
464 
465   if (api[am]) {
466     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);
467     ierr = PetscInfo1(ptap->AP_loc,"Use MatPtAP(A,B,MatReuse,%g,&C) for best AP_loc performance.;\n",(double)apfill);CHKERRQ(ierr);
468   } else {
469     ierr = PetscInfo(ptap->AP_loc,"Scalable algorithm, AP_loc is empty \n");CHKERRQ(ierr);
470   }
471 #endif
472 
473   /* (2-1) compute symbolic Co = Ro*AP_loc  */
474   /* ------------------------------------ */
475   ierr = MatSetOptionsPrefix(ptap->Ro,"inner_offdiag_");CHKERRQ(ierr);
476   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Ro,ptap->AP_loc,fill,&ptap->C_oth);CHKERRQ(ierr);
477 
478   /* (3) send coj of C_oth to other processors  */
479   /* ------------------------------------------ */
480   /* determine row ownership */
481   ierr = PetscLayoutCreate(comm,&rowmap);CHKERRQ(ierr);
482   rowmap->n  = pn;
483   rowmap->bs = 1;
484   ierr   = PetscLayoutSetUp(rowmap);CHKERRQ(ierr);
485   owners = rowmap->range;
486 
487   /* determine the number of messages to send, their lengths */
488   ierr = PetscMalloc4(size,&len_s,size,&len_si,size,&sstatus,size+2,&owners_co);CHKERRQ(ierr);
489   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
490   ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
491 
492   c_oth = (Mat_SeqAIJ*)ptap->C_oth->data;
493   coi   = c_oth->i; coj = c_oth->j;
494   con   = ptap->C_oth->rmap->n;
495   proc  = 0;
496   ierr = ISLocalToGlobalMappingApply(ptap->ltog,coi[con],coj,coj);CHKERRQ(ierr);
497   for (i=0; i<con; i++) {
498     while (prmap[i] >= owners[proc+1]) proc++;
499     len_si[proc]++;               /* num of rows in Co(=Pt*AP) to be sent to [proc] */
500     len_s[proc] += coi[i+1] - coi[i]; /* num of nonzeros in Co to be sent to [proc] */
501   }
502 
503   len          = 0; /* max length of buf_si[], see (4) */
504   owners_co[0] = 0;
505   nsend        = 0;
506   for (proc=0; proc<size; proc++) {
507     owners_co[proc+1] = owners_co[proc] + len_si[proc];
508     if (len_s[proc]) {
509       nsend++;
510       len_si[proc] = 2*(len_si[proc] + 1); /* length of buf_si to be sent to [proc] */
511       len         += len_si[proc];
512     }
513   }
514 
515   /* determine the number and length of messages to receive for coi and coj  */
516   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&nrecv);CHKERRQ(ierr);
517   ierr = PetscGatherMessageLengths2(comm,nsend,nrecv,len_s,len_si,&id_r,&len_r,&len_ri);CHKERRQ(ierr);
518 
519   /* post the Irecv and Isend of coj */
520   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
521   ierr = PetscPostIrecvInt(comm,tagj,nrecv,id_r,len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
522   ierr = PetscMalloc1(nsend+1,&swaits);CHKERRQ(ierr);
523   for (proc=0, k=0; proc<size; proc++) {
524     if (!len_s[proc]) continue;
525     i    = owners_co[proc];
526     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
527     k++;
528   }
529 
530   /* (2-2) compute symbolic C_loc = Rd*AP_loc */
531   /* ---------------------------------------- */
532   ierr = MatSetOptionsPrefix(ptap->Rd,"inner_diag_");CHKERRQ(ierr);
533   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Rd,ptap->AP_loc,fill,&ptap->C_loc);CHKERRQ(ierr);
534   c_loc = (Mat_SeqAIJ*)ptap->C_loc->data;
535   ierr = ISLocalToGlobalMappingApply(ptap->ltog,c_loc->i[ptap->C_loc->rmap->n],c_loc->j,c_loc->j);CHKERRQ(ierr);
536 
537   /* receives coj are complete */
538   for (i=0; i<nrecv; i++) {
539     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
540   }
541   ierr = PetscFree(rwaits);CHKERRQ(ierr);
542   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
543 
544   /* add received column indices into ta to update Crmax */
545   for (k=0; k<nrecv; k++) {/* k-th received message */
546     Jptr = buf_rj[k];
547     for (j=0; j<len_r[k]; j++) {
548       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
549     }
550   }
551   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
552   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
553 
554   /* (4) send and recv coi */
555   /*-----------------------*/
556   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
557   ierr   = PetscPostIrecvInt(comm,tagi,nrecv,id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
558   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
559   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
560   for (proc=0,k=0; proc<size; proc++) {
561     if (!len_s[proc]) continue;
562     /* form outgoing message for i-structure:
563          buf_si[0]:                 nrows to be sent
564                [1:nrows]:           row index (global)
565                [nrows+1:2*nrows+1]: i-structure index
566     */
567     /*-------------------------------------------*/
568     nrows       = len_si[proc]/2 - 1; /* num of rows in Co to be sent to [proc] */
569     buf_si_i    = buf_si + nrows+1;
570     buf_si[0]   = nrows;
571     buf_si_i[0] = 0;
572     nrows       = 0;
573     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
574       nzi = coi[i+1] - coi[i];
575       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
576       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
577       nrows++;
578     }
579     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
580     k++;
581     buf_si += len_si[proc];
582   }
583   for (i=0; i<nrecv; i++) {
584     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
585   }
586   ierr = PetscFree(rwaits);CHKERRQ(ierr);
587   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
588 
589   ierr = PetscFree4(len_s,len_si,sstatus,owners_co);CHKERRQ(ierr);
590   ierr = PetscFree(len_ri);CHKERRQ(ierr);
591   ierr = PetscFree(swaits);CHKERRQ(ierr);
592   ierr = PetscFree(buf_s);CHKERRQ(ierr);
593 
594   /* (5) compute the local portion of Cmpi      */
595   /* ------------------------------------------ */
596   /* set initial free space to be Crmax, sufficient for holding nozeros in each row of Cmpi */
597   ierr          = PetscFreeSpaceGet(Crmax,&free_space);CHKERRQ(ierr);
598   current_space = free_space;
599 
600   ierr = PetscMalloc3(nrecv,&buf_ri_k,nrecv,&nextrow,nrecv,&nextci);CHKERRQ(ierr);
601   for (k=0; k<nrecv; k++) {
602     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
603     nrows       = *buf_ri_k[k];
604     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
605     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
606   }
607 
608   ierr = MatPreallocateInitialize(comm,pn,pn,dnz,onz);CHKERRQ(ierr);
609   ierr = PetscLLCondensedCreate_Scalable(Crmax,&lnk);CHKERRQ(ierr);
610   for (i=0; i<pn; i++) {
611     /* add C_loc into Cmpi */
612     nzi  = c_loc->i[i+1] - c_loc->i[i];
613     Jptr = c_loc->j + c_loc->i[i];
614     ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
615 
616     /* add received col data into lnk */
617     for (k=0; k<nrecv; k++) { /* k-th received message */
618       if (i == *nextrow[k]) { /* i-th row */
619         nzi  = *(nextci[k]+1) - *nextci[k];
620         Jptr = buf_rj[k] + *nextci[k];
621         ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
622         nextrow[k]++; nextci[k]++;
623       }
624     }
625     nzi = lnk[0];
626 
627     /* copy data into free space, then initialize lnk */
628     ierr = PetscLLCondensedClean_Scalable(nzi,current_space->array,lnk);CHKERRQ(ierr);
629     ierr = MatPreallocateSet(i+owners[rank],nzi,current_space->array,dnz,onz);CHKERRQ(ierr);
630   }
631   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
632   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
633   ierr = PetscFreeSpaceDestroy(free_space);CHKERRQ(ierr);
634 
635   /* local sizes and preallocation */
636   ierr = MatSetSizes(Cmpi,pn,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
637   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(P->cmap->bs));CHKERRQ(ierr);
638   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
639   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
640 
641   /* members in merge */
642   ierr = PetscFree(id_r);CHKERRQ(ierr);
643   ierr = PetscFree(len_r);CHKERRQ(ierr);
644   ierr = PetscFree(buf_ri[0]);CHKERRQ(ierr);
645   ierr = PetscFree(buf_ri);CHKERRQ(ierr);
646   ierr = PetscFree(buf_rj[0]);CHKERRQ(ierr);
647   ierr = PetscFree(buf_rj);CHKERRQ(ierr);
648   ierr = PetscLayoutDestroy(&rowmap);CHKERRQ(ierr);
649 
650   /* attach the supporting struct to Cmpi for reuse */
651   c = (Mat_MPIAIJ*)Cmpi->data;
652   c->ap           = ptap;
653   ptap->duplicate = Cmpi->ops->duplicate;
654   ptap->destroy   = Cmpi->ops->destroy;
655   ptap->view      = Cmpi->ops->view;
656 
657   /* Cmpi is not ready for use - assembly will be done by MatPtAPNumeric() */
658   Cmpi->assembled        = PETSC_FALSE;
659   Cmpi->ops->ptapnumeric = MatPtAPNumeric_MPIAIJ_MPIAIJ_scalable;
660   Cmpi->ops->destroy     = MatDestroy_MPIAIJ_PtAP;
661   Cmpi->ops->view        = MatView_MPIAIJ_PtAP;
662   Cmpi->ops->freeintermediatedatastructures = MatFreeIntermediateDataStructures_MPIAIJ_AP;
663   *C                     = Cmpi;
664 
665    nout = 0;
666    ierr = ISGlobalToLocalMappingApply(ptap->ltog,IS_GTOLM_DROP,c_oth->i[ptap->C_oth->rmap->n],c_oth->j,&nout,c_oth->j);CHKERRQ(ierr);
667    if (c_oth->i[ptap->C_oth->rmap->n] != nout) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_INCOMP,"Incorrect mapping %D != %D\n",c_oth->i[ptap->C_oth->rmap->n],nout);
668    ierr = ISGlobalToLocalMappingApply(ptap->ltog,IS_GTOLM_DROP,c_loc->i[ptap->C_loc->rmap->n],c_loc->j,&nout,c_loc->j);CHKERRQ(ierr);
669    if (c_loc->i[ptap->C_loc->rmap->n] != nout) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_INCOMP,"Incorrect mapping %D != %D\n",c_loc->i[ptap->C_loc->rmap->n],nout);
670 
671   PetscFunctionReturn(0);
672 }
673 
674 PetscErrorCode MatPtAPSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat *C)
675 {
676   PetscErrorCode      ierr;
677   Mat_APMPI           *ptap;
678   Mat_MPIAIJ          *a=(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data,*c;
679   MPI_Comm            comm;
680   PetscMPIInt         size,rank;
681   Mat                 Cmpi;
682   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
683   PetscInt            am=A->rmap->n,pm=P->rmap->n,pN=P->cmap->N,pn=P->cmap->n;
684   PetscInt            *lnk,i,k,pnz,row,nsend;
685   PetscBT             lnkbt;
686   PetscMPIInt         tagi,tagj,*len_si,*len_s,*len_ri,icompleted=0,nrecv;
687   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
688   PetscInt            len,proc,*dnz,*onz,*owners,nzi,nspacedouble;
689   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
690   MPI_Request         *swaits,*rwaits;
691   MPI_Status          *sstatus,rstatus;
692   PetscLayout         rowmap;
693   PetscInt            *owners_co,*coi,*coj;    /* i and j array of (p->B)^T*A*P - used in the communication */
694   PetscMPIInt         *len_r,*id_r;    /* array of length of comm->size, store send/recv matrix values */
695   PetscInt            *api,*apj,*Jptr,apnz,*prmap=p->garray,con,j,ap_rmax=0,Crmax,*aj,*ai,*pi;
696   Mat_SeqAIJ          *p_loc,*p_oth=NULL,*ad=(Mat_SeqAIJ*)(a->A)->data,*ao=NULL,*c_loc,*c_oth;
697   PetscScalar         *apv;
698   PetscTable          ta;
699   MatType             mtype;
700 #if defined(PETSC_USE_INFO)
701   PetscReal           apfill;
702 #endif
703 
704   PetscFunctionBegin;
705   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
706   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
707   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
708 
709   if (size > 1) ao = (Mat_SeqAIJ*)(a->B)->data;
710 
711   /* create symbolic parallel matrix Cmpi */
712   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
713   ierr = MatGetType(A,&mtype);CHKERRQ(ierr);
714   ierr = MatSetType(Cmpi,mtype);CHKERRQ(ierr);
715 
716   /* Do dense axpy in MatPtAPNumeric_MPIAIJ_MPIAIJ() */
717   Cmpi->ops->ptapnumeric = MatPtAPNumeric_MPIAIJ_MPIAIJ;
718 
719   /* create struct Mat_APMPI and attached it to C later */
720   ierr        = PetscNew(&ptap);CHKERRQ(ierr);
721   ptap->reuse = MAT_INITIAL_MATRIX;
722   ptap->algType = 1;
723 
724   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
725   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
726   /* get P_loc by taking all local rows of P */
727   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
728 
729   /* (0) compute Rd = Pd^T, Ro = Po^T  */
730   /* --------------------------------- */
731   ierr = MatTranspose(p->A,MAT_INITIAL_MATRIX,&ptap->Rd);CHKERRQ(ierr);
732   ierr = MatTranspose(p->B,MAT_INITIAL_MATRIX,&ptap->Ro);CHKERRQ(ierr);
733 
734   /* (1) compute symbolic AP = A_loc*P = Ad*P_loc + Ao*P_oth (api,apj) */
735   /* ----------------------------------------------------------------- */
736   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
737   if (ptap->P_oth) p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
738 
739   /* create and initialize a linked list */
740   ierr = PetscTableCreate(pn,pN,&ta);CHKERRQ(ierr); /* for compute AP_loc and Cmpi */
741   MatRowMergeMax_SeqAIJ(p_loc,ptap->P_loc->rmap->N,ta);
742   MatRowMergeMax_SeqAIJ(p_oth,ptap->P_oth->rmap->N,ta);
743   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr); /* Crmax = nnz(sum of Prows) */
744   /* printf("[%d] est %d, Crmax %d; pN %d\n",rank,5*(p_loc->rmax+p_oth->rmax + (PetscInt)(1.e-2*pN)),Crmax,pN); */
745 
746   ierr = PetscLLCondensedCreate(Crmax,pN,&lnk,&lnkbt);CHKERRQ(ierr);
747 
748   /* Initial FreeSpace size is fill*(nnz(A) + nnz(P)) */
749   if (ao) {
750     ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ad->i[am],PetscIntSumTruncate(ao->i[am],p_loc->i[pm]))),&free_space);CHKERRQ(ierr);
751   } else {
752     ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ad->i[am],p_loc->i[pm])),&free_space);CHKERRQ(ierr);
753   }
754   current_space = free_space;
755   nspacedouble  = 0;
756 
757   ierr   = PetscMalloc1(am+1,&api);CHKERRQ(ierr);
758   api[0] = 0;
759   for (i=0; i<am; i++) {
760     /* diagonal portion: Ad[i,:]*P */
761     ai = ad->i; pi = p_loc->i;
762     nzi = ai[i+1] - ai[i];
763     aj  = ad->j + ai[i];
764     for (j=0; j<nzi; j++) {
765       row  = aj[j];
766       pnz  = pi[row+1] - pi[row];
767       Jptr = p_loc->j + pi[row];
768       /* add non-zero cols of P into the sorted linked list lnk */
769       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
770     }
771     /* off-diagonal portion: Ao[i,:]*P */
772     if (ao) {
773       ai = ao->i; pi = p_oth->i;
774       nzi = ai[i+1] - ai[i];
775       aj  = ao->j + ai[i];
776       for (j=0; j<nzi; j++) {
777         row  = aj[j];
778         pnz  = pi[row+1] - pi[row];
779         Jptr = p_oth->j + pi[row];
780         ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
781       }
782     }
783     apnz     = lnk[0];
784     api[i+1] = api[i] + apnz;
785     if (ap_rmax < apnz) ap_rmax = apnz;
786 
787     /* if free space is not available, double the total space in the list */
788     if (current_space->local_remaining<apnz) {
789       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
790       nspacedouble++;
791     }
792 
793     /* Copy data into free space, then initialize lnk */
794     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
795 
796     current_space->array           += apnz;
797     current_space->local_used      += apnz;
798     current_space->local_remaining -= apnz;
799   }
800   /* Allocate space for apj and apv, initialize apj, and */
801   /* destroy list of free space and other temporary array(s) */
802   ierr   = PetscMalloc2(api[am],&apj,api[am],&apv);CHKERRQ(ierr);
803   ierr   = PetscFreeSpaceContiguous(&free_space,apj);CHKERRQ(ierr);
804   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
805 
806   /* Create AP_loc for reuse */
807   ierr = MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,am,pN,api,apj,apv,&ptap->AP_loc);CHKERRQ(ierr);
808 
809 #if defined(PETSC_USE_INFO)
810   if (ao) {
811     apfill = (PetscReal)api[am]/(ad->i[am]+ao->i[am]+p_loc->i[pm]+1);
812   } else {
813     apfill = (PetscReal)api[am]/(ad->i[am]+p_loc->i[pm]+1);
814   }
815   ptap->AP_loc->info.mallocs           = nspacedouble;
816   ptap->AP_loc->info.fill_ratio_given  = fill;
817   ptap->AP_loc->info.fill_ratio_needed = apfill;
818 
819   if (api[am]) {
820     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);
821     ierr = PetscInfo1(ptap->AP_loc,"Use MatPtAP(A,B,MatReuse,%g,&C) for best AP_loc performance.;\n",(double)apfill);CHKERRQ(ierr);
822   } else {
823     ierr = PetscInfo(ptap->AP_loc,"Nonscalable algorithm, AP_loc is empty \n");CHKERRQ(ierr);
824   }
825 #endif
826 
827   /* (2-1) compute symbolic Co = Ro*AP_loc  */
828   /* ------------------------------------ */
829   ierr = MatSetOptionsPrefix(ptap->Ro,"inner_offdiag_");CHKERRQ(ierr);
830   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Ro,ptap->AP_loc,fill,&ptap->C_oth);CHKERRQ(ierr);
831 
832   /* (3) send coj of C_oth to other processors  */
833   /* ------------------------------------------ */
834   /* determine row ownership */
835   ierr = PetscLayoutCreate(comm,&rowmap);CHKERRQ(ierr);
836   rowmap->n  = pn;
837   rowmap->bs = 1;
838   ierr   = PetscLayoutSetUp(rowmap);CHKERRQ(ierr);
839   owners = rowmap->range;
840 
841   /* determine the number of messages to send, their lengths */
842   ierr = PetscMalloc4(size,&len_s,size,&len_si,size,&sstatus,size+2,&owners_co);CHKERRQ(ierr);
843   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
844   ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
845 
846   c_oth = (Mat_SeqAIJ*)ptap->C_oth->data;
847   coi   = c_oth->i; coj = c_oth->j;
848   con   = ptap->C_oth->rmap->n;
849   proc  = 0;
850   for (i=0; i<con; i++) {
851     while (prmap[i] >= owners[proc+1]) proc++;
852     len_si[proc]++;               /* num of rows in Co(=Pt*AP) to be sent to [proc] */
853     len_s[proc] += coi[i+1] - coi[i]; /* num of nonzeros in Co to be sent to [proc] */
854   }
855 
856   len          = 0; /* max length of buf_si[], see (4) */
857   owners_co[0] = 0;
858   nsend        = 0;
859   for (proc=0; proc<size; proc++) {
860     owners_co[proc+1] = owners_co[proc] + len_si[proc];
861     if (len_s[proc]) {
862       nsend++;
863       len_si[proc] = 2*(len_si[proc] + 1); /* length of buf_si to be sent to [proc] */
864       len         += len_si[proc];
865     }
866   }
867 
868   /* determine the number and length of messages to receive for coi and coj  */
869   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&nrecv);CHKERRQ(ierr);
870   ierr = PetscGatherMessageLengths2(comm,nsend,nrecv,len_s,len_si,&id_r,&len_r,&len_ri);CHKERRQ(ierr);
871 
872   /* post the Irecv and Isend of coj */
873   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
874   ierr = PetscPostIrecvInt(comm,tagj,nrecv,id_r,len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
875   ierr = PetscMalloc1(nsend+1,&swaits);CHKERRQ(ierr);
876   for (proc=0, k=0; proc<size; proc++) {
877     if (!len_s[proc]) continue;
878     i    = owners_co[proc];
879     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
880     k++;
881   }
882 
883   /* (2-2) compute symbolic C_loc = Rd*AP_loc */
884   /* ---------------------------------------- */
885   ierr = MatSetOptionsPrefix(ptap->Rd,"inner_diag_");CHKERRQ(ierr);
886   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Rd,ptap->AP_loc,fill,&ptap->C_loc);CHKERRQ(ierr);
887   c_loc = (Mat_SeqAIJ*)ptap->C_loc->data;
888 
889   /* receives coj are complete */
890   for (i=0; i<nrecv; i++) {
891     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
892   }
893   ierr = PetscFree(rwaits);CHKERRQ(ierr);
894   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
895 
896   /* add received column indices into ta to update Crmax */
897   for (k=0; k<nrecv; k++) {/* k-th received message */
898     Jptr = buf_rj[k];
899     for (j=0; j<len_r[k]; j++) {
900       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
901     }
902   }
903   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
904   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
905 
906   /* (4) send and recv coi */
907   /*-----------------------*/
908   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
909   ierr   = PetscPostIrecvInt(comm,tagi,nrecv,id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
910   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
911   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
912   for (proc=0,k=0; proc<size; proc++) {
913     if (!len_s[proc]) continue;
914     /* form outgoing message for i-structure:
915          buf_si[0]:                 nrows to be sent
916                [1:nrows]:           row index (global)
917                [nrows+1:2*nrows+1]: i-structure index
918     */
919     /*-------------------------------------------*/
920     nrows       = len_si[proc]/2 - 1; /* num of rows in Co to be sent to [proc] */
921     buf_si_i    = buf_si + nrows+1;
922     buf_si[0]   = nrows;
923     buf_si_i[0] = 0;
924     nrows       = 0;
925     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
926       nzi = coi[i+1] - coi[i];
927       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
928       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
929       nrows++;
930     }
931     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
932     k++;
933     buf_si += len_si[proc];
934   }
935   for (i=0; i<nrecv; i++) {
936     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
937   }
938   ierr = PetscFree(rwaits);CHKERRQ(ierr);
939   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
940 
941   ierr = PetscFree4(len_s,len_si,sstatus,owners_co);CHKERRQ(ierr);
942   ierr = PetscFree(len_ri);CHKERRQ(ierr);
943   ierr = PetscFree(swaits);CHKERRQ(ierr);
944   ierr = PetscFree(buf_s);CHKERRQ(ierr);
945 
946   /* (5) compute the local portion of Cmpi      */
947   /* ------------------------------------------ */
948   /* set initial free space to be Crmax, sufficient for holding nozeros in each row of Cmpi */
949   ierr          = PetscFreeSpaceGet(Crmax,&free_space);CHKERRQ(ierr);
950   current_space = free_space;
951 
952   ierr = PetscMalloc3(nrecv,&buf_ri_k,nrecv,&nextrow,nrecv,&nextci);CHKERRQ(ierr);
953   for (k=0; k<nrecv; k++) {
954     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
955     nrows       = *buf_ri_k[k];
956     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
957     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
958   }
959 
960   ierr = MatPreallocateInitialize(comm,pn,pn,dnz,onz);CHKERRQ(ierr);
961   ierr = PetscLLCondensedCreate(Crmax,pN,&lnk,&lnkbt);CHKERRQ(ierr);
962   for (i=0; i<pn; i++) {
963     /* add C_loc into Cmpi */
964     nzi  = c_loc->i[i+1] - c_loc->i[i];
965     Jptr = c_loc->j + c_loc->i[i];
966     ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
967 
968     /* add received col data into lnk */
969     for (k=0; k<nrecv; k++) { /* k-th received message */
970       if (i == *nextrow[k]) { /* i-th row */
971         nzi  = *(nextci[k]+1) - *nextci[k];
972         Jptr = buf_rj[k] + *nextci[k];
973         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
974         nextrow[k]++; nextci[k]++;
975       }
976     }
977     nzi = lnk[0];
978 
979     /* copy data into free space, then initialize lnk */
980     ierr = PetscLLCondensedClean(pN,nzi,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
981     ierr = MatPreallocateSet(i+owners[rank],nzi,current_space->array,dnz,onz);CHKERRQ(ierr);
982   }
983   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
984   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
985   ierr = PetscFreeSpaceDestroy(free_space);CHKERRQ(ierr);
986 
987   /* local sizes and preallocation */
988   ierr = MatSetSizes(Cmpi,pn,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
989   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(P->cmap->bs));CHKERRQ(ierr);
990   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
991   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
992 
993   /* members in merge */
994   ierr = PetscFree(id_r);CHKERRQ(ierr);
995   ierr = PetscFree(len_r);CHKERRQ(ierr);
996   ierr = PetscFree(buf_ri[0]);CHKERRQ(ierr);
997   ierr = PetscFree(buf_ri);CHKERRQ(ierr);
998   ierr = PetscFree(buf_rj[0]);CHKERRQ(ierr);
999   ierr = PetscFree(buf_rj);CHKERRQ(ierr);
1000   ierr = PetscLayoutDestroy(&rowmap);CHKERRQ(ierr);
1001 
1002   /* attach the supporting struct to Cmpi for reuse */
1003   c = (Mat_MPIAIJ*)Cmpi->data;
1004   c->ap           = ptap;
1005   ptap->duplicate = Cmpi->ops->duplicate;
1006   ptap->destroy   = Cmpi->ops->destroy;
1007   ptap->view      = Cmpi->ops->view;
1008   ierr = PetscCalloc1(pN,&ptap->apa);CHKERRQ(ierr);
1009 
1010   /* Cmpi is not ready for use - assembly will be done by MatPtAPNumeric() */
1011   Cmpi->assembled        = PETSC_FALSE;
1012   Cmpi->ops->destroy     = MatDestroy_MPIAIJ_PtAP;
1013   Cmpi->ops->view        = MatView_MPIAIJ_PtAP;
1014   Cmpi->ops->freeintermediatedatastructures = MatFreeIntermediateDataStructures_MPIAIJ_AP;
1015   *C                     = Cmpi;
1016   PetscFunctionReturn(0);
1017 }
1018 
1019 PetscErrorCode MatPtAPNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
1020 {
1021   PetscErrorCode    ierr;
1022   Mat_MPIAIJ        *a=(Mat_MPIAIJ*)A->data,*p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1023   Mat_SeqAIJ        *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
1024   Mat_SeqAIJ        *ap,*p_loc,*p_oth=NULL,*c_seq;
1025   Mat_APMPI         *ptap = c->ap;
1026   Mat               AP_loc,C_loc,C_oth;
1027   PetscInt          i,rstart,rend,cm,ncols,row;
1028   PetscInt          *api,*apj,am = A->rmap->n,j,col,apnz;
1029   PetscScalar       *apa;
1030   const PetscInt    *cols;
1031   const PetscScalar *vals;
1032 
1033   PetscFunctionBegin;
1034   if (!ptap) {
1035     MPI_Comm comm;
1036     ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1037     SETERRQ(comm,PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be reused. Do not call MatFreeIntermediateDataStructures() or use '-mat_freeintermediatedatastructures'");
1038   }
1039 
1040   ierr = MatZeroEntries(C);CHKERRQ(ierr);
1041   /* 1) get R = Pd^T,Ro = Po^T */
1042   if (ptap->reuse == MAT_REUSE_MATRIX) {
1043     ierr = MatTranspose(p->A,MAT_REUSE_MATRIX,&ptap->Rd);CHKERRQ(ierr);
1044     ierr = MatTranspose(p->B,MAT_REUSE_MATRIX,&ptap->Ro);CHKERRQ(ierr);
1045   }
1046 
1047   /* 2) get AP_loc */
1048   AP_loc = ptap->AP_loc;
1049   ap = (Mat_SeqAIJ*)AP_loc->data;
1050 
1051   /* 2-1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
1052   /*-----------------------------------------------------*/
1053   if (ptap->reuse == MAT_REUSE_MATRIX) {
1054     /* P_oth and P_loc are obtained in MatPtASymbolic() when reuse == MAT_INITIAL_MATRIX */
1055     ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
1056     ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
1057   }
1058 
1059   /* 2-2) compute numeric A_loc*P - dominating part */
1060   /* ---------------------------------------------- */
1061   /* get data from symbolic products */
1062   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
1063   if (ptap->P_oth) {
1064     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
1065   }
1066   apa   = ptap->apa;
1067   api   = ap->i;
1068   apj   = ap->j;
1069   for (i=0; i<am; i++) {
1070     /* AP[i,:] = A[i,:]*P = Ad*P_loc Ao*P_oth */
1071     AProw_nonscalable(i,ad,ao,p_loc,p_oth,apa);
1072     apnz = api[i+1] - api[i];
1073     for (j=0; j<apnz; j++) {
1074       col = apj[j+api[i]];
1075       ap->a[j+ap->i[i]] = apa[col];
1076       apa[col] = 0.0;
1077     }
1078     ierr = PetscLogFlops(2.0*apnz);CHKERRQ(ierr);
1079   }
1080 
1081   /* 3) C_loc = Rd*AP_loc, C_oth = Ro*AP_loc */
1082   ierr = ((ptap->C_loc)->ops->matmultnumeric)(ptap->Rd,AP_loc,ptap->C_loc);CHKERRQ(ierr);
1083   ierr = ((ptap->C_oth)->ops->matmultnumeric)(ptap->Ro,AP_loc,ptap->C_oth);CHKERRQ(ierr);
1084   C_loc = ptap->C_loc;
1085   C_oth = ptap->C_oth;
1086 
1087   /* add C_loc and Co to to C */
1088   ierr = MatGetOwnershipRange(C,&rstart,&rend);CHKERRQ(ierr);
1089 
1090   /* C_loc -> C */
1091   cm    = C_loc->rmap->N;
1092   c_seq = (Mat_SeqAIJ*)C_loc->data;
1093   cols = c_seq->j;
1094   vals = c_seq->a;
1095 
1096 
1097   /* The (fast) MatSetValues_MPIAIJ_CopyFromCSRFormat function can only be used when C->was_assembled is PETSC_FALSE and */
1098   /* when there are no off-processor parts.  */
1099   /* If was_assembled is true, then the statement aj[rowstart_diag+dnz_row] = mat_j[col] - cstart; in MatSetValues_MPIAIJ_CopyFromCSRFormat */
1100   /* is no longer true. Then the more complex function MatSetValues_MPIAIJ() has to be used, where the column index is looked up from */
1101   /* a table, and other, more complex stuff has to be done. */
1102   if (C->assembled) {
1103     C->was_assembled = PETSC_TRUE;
1104     C->assembled     = PETSC_FALSE;
1105   }
1106   if (C->was_assembled) {
1107     for (i=0; i<cm; i++) {
1108       ncols = c_seq->i[i+1] - c_seq->i[i];
1109       row = rstart + i;
1110       ierr = MatSetValues_MPIAIJ(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1111       cols += ncols; vals += ncols;
1112     }
1113   } else {
1114     ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat(C,c_seq->j,c_seq->i,c_seq->a);CHKERRQ(ierr);
1115   }
1116 
1117   /* Co -> C, off-processor part */
1118   cm = C_oth->rmap->N;
1119   c_seq = (Mat_SeqAIJ*)C_oth->data;
1120   cols = c_seq->j;
1121   vals = c_seq->a;
1122   for (i=0; i<cm; i++) {
1123     ncols = c_seq->i[i+1] - c_seq->i[i];
1124     row = p->garray[i];
1125     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1126     cols += ncols; vals += ncols;
1127   }
1128 
1129   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1130   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1131 
1132   ptap->reuse = MAT_REUSE_MATRIX;
1133 
1134   /* supporting struct ptap consumes almost same amount of memory as C=PtAP, release it if C will not be updated by A and P */
1135   if (ptap->freestruct) {
1136     ierr = MatFreeIntermediateDataStructures(C);CHKERRQ(ierr);
1137   }
1138   PetscFunctionReturn(0);
1139 }
1140