xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision 65f8aed5f7eaa1e2ef2ddeffe666264e0669c876)
1 
2 /*
3   Defines matrix-matrix product routines for pairs of MPIAIJ matrices
4           C = A * B
5 */
6 #include <../src/mat/impls/aij/seq/aij.h> /*I "petscmat.h" I*/
7 #include <../src/mat/utils/freespace.h>
8 #include <../src/mat/impls/aij/mpi/mpiaij.h>
9 #include <petscbt.h>
10 #include <../src/mat/impls/dense/mpi/mpidense.h>
11 #include <petsc/private/vecimpl.h>
12 
13 #if defined(PETSC_HAVE_HYPRE)
14 PETSC_INTERN PetscErrorCode MatMatMultSymbolic_AIJ_AIJ_wHYPRE(Mat,Mat,PetscReal,Mat*);
15 #endif
16 
17 PETSC_INTERN PetscErrorCode MatMatMult_MPIAIJ_MPIAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill, Mat *C)
18 {
19   PetscErrorCode ierr;
20 #if defined(PETSC_HAVE_HYPRE)
21   const char     *algTypes[4] = {"scalable","nonscalable","seqmpi","hypre"};
22   PetscInt       nalg = 4;
23 #else
24   const char     *algTypes[3] = {"scalable","nonscalable","seqmpi"};
25   PetscInt       nalg = 3;
26 #endif
27   PetscInt       alg = 1; /* set nonscalable algorithm as default */
28   MPI_Comm       comm;
29   PetscBool      flg;
30 
31   PetscFunctionBegin;
32   if (scall == MAT_INITIAL_MATRIX) {
33     ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
34     if (A->cmap->rstart != B->rmap->rstart || A->cmap->rend != B->rmap->rend) SETERRQ4(comm,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, (%D, %D) != (%D,%D)",A->cmap->rstart,A->cmap->rend,B->rmap->rstart,B->rmap->rend);
35 
36     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
37     PetscOptionsObject->alreadyprinted = PETSC_FALSE; /* a hack to ensure the option shows in '-help' */
38     ierr = PetscOptionsEList("-matmatmult_via","Algorithmic approach","MatMatMult",algTypes,nalg,algTypes[1],&alg,&flg);CHKERRQ(ierr);
39     ierr = PetscOptionsEnd();CHKERRQ(ierr);
40 
41     if (!flg && B->cmap->N > 100000) { /* may switch to scalable algorithm as default */
42       MatInfo     Ainfo,Binfo;
43       PetscInt    nz_local;
44       PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
45 
46       ierr = MatGetInfo(A,MAT_LOCAL,&Ainfo);CHKERRQ(ierr);
47       ierr = MatGetInfo(B,MAT_LOCAL,&Binfo);CHKERRQ(ierr);
48       nz_local = (PetscInt)(Ainfo.nz_allocated + Binfo.nz_allocated);
49 
50       if (B->cmap->N > fill*nz_local) alg_scalable_loc = PETSC_TRUE;
51       ierr = MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm);CHKERRQ(ierr);
52 
53       if (alg_scalable) {
54         alg  = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
55         ierr = PetscInfo2(B,"Use scalable algorithm, BN %D, fill*nz_allocated %g\n",B->cmap->N,fill*nz_local);CHKERRQ(ierr);
56       }
57     }
58 
59     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
60     switch (alg) {
61     case 1:
62       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(A,B,fill,C);CHKERRQ(ierr);
63       break;
64     case 2:
65       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ_seqMPI(A,B,fill,C);CHKERRQ(ierr);
66       break;
67 #if defined(PETSC_HAVE_HYPRE)
68     case 3:
69       ierr = MatMatMultSymbolic_AIJ_AIJ_wHYPRE(A,B,fill,C);CHKERRQ(ierr);
70       break;
71 #endif
72     default:
73       ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ(A,B,fill,C);CHKERRQ(ierr);
74       break;
75     }
76     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
77   }
78   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
79   ierr = (*(*C)->ops->matmultnumeric)(A,B,*C);CHKERRQ(ierr);
80   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
81   PetscFunctionReturn(0);
82 }
83 
84 PetscErrorCode MatDestroy_MPIAIJ_MatMatMult(Mat A)
85 {
86   PetscErrorCode ierr;
87   Mat_MPIAIJ     *a    = (Mat_MPIAIJ*)A->data;
88   Mat_PtAPMPI    *ptap = a->ptap;
89 
90   PetscFunctionBegin;
91   ierr = PetscFree2(ptap->startsj_s,ptap->startsj_r);CHKERRQ(ierr);
92   ierr = PetscFree(ptap->bufa);CHKERRQ(ierr);
93   ierr = MatDestroy(&ptap->P_loc);CHKERRQ(ierr);
94   ierr = MatDestroy(&ptap->P_oth);CHKERRQ(ierr);
95   ierr = MatDestroy(&ptap->Pt);CHKERRQ(ierr);
96   ierr = PetscFree(ptap->api);CHKERRQ(ierr);
97   ierr = PetscFree(ptap->apj);CHKERRQ(ierr);
98   ierr = PetscFree(ptap->apa);CHKERRQ(ierr);
99   ierr = ptap->destroy(A);CHKERRQ(ierr);
100   ierr = PetscFree(ptap);CHKERRQ(ierr);
101   PetscFunctionReturn(0);
102 }
103 
104 PetscErrorCode MatDuplicate_MPIAIJ_MatMatMult(Mat A, MatDuplicateOption op, Mat *M)
105 {
106   PetscErrorCode ierr;
107   Mat_MPIAIJ     *a    = (Mat_MPIAIJ*)A->data;
108   Mat_PtAPMPI    *ptap = a->ptap;
109 
110   PetscFunctionBegin;
111   ierr = (*ptap->duplicate)(A,op,M);CHKERRQ(ierr);
112 
113   (*M)->ops->destroy   = ptap->destroy;   /* = MatDestroy_MPIAIJ, *M doesn't duplicate A's special structure! */
114   (*M)->ops->duplicate = ptap->duplicate; /* = MatDuplicate_MPIAIJ */
115   PetscFunctionReturn(0);
116 }
117 
118 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,Mat C)
119 {
120   PetscErrorCode ierr;
121   Mat_MPIAIJ     *a  =(Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
122   Mat_SeqAIJ     *ad =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
123   Mat_SeqAIJ     *cd =(Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
124   PetscScalar    *cda=cd->a,*coa=co->a;
125   Mat_SeqAIJ     *p_loc,*p_oth;
126   PetscScalar    *apa,*ca;
127   PetscInt       cm   =C->rmap->n;
128   Mat_PtAPMPI    *ptap=c->ptap;
129   PetscInt       *api,*apj,*apJ,i,k;
130   PetscInt       cstart=C->cmap->rstart;
131   PetscInt       cdnz,conz,k0,k1;
132   MPI_Comm       comm;
133   PetscMPIInt    size;
134 
135   PetscFunctionBegin;
136   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
137   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
138 
139   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
140   /*-----------------------------------------------------*/
141   /* update numerical values of P_oth and P_loc */
142   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
143   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
144 
145   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
146   /*----------------------------------------------------------*/
147   /* get data from symbolic products */
148   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
149   p_oth = NULL;
150   if (size >1) {
151     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
152   }
153 
154   /* get apa for storing dense row A[i,:]*P */
155   apa = ptap->apa;
156 
157   api = ptap->api;
158   apj = ptap->apj;
159   for (i=0; i<cm; i++) {
160     /* compute apa = A[i,:]*P */
161     AProw_nonscalable(i,ad,ao,p_loc,p_oth,apa);
162 
163     /* set values in C */
164     apJ  = apj + api[i];
165     cdnz = cd->i[i+1] - cd->i[i];
166     conz = co->i[i+1] - co->i[i];
167 
168     /* 1st off-diagoanl part of C */
169     ca = coa + co->i[i];
170     k  = 0;
171     for (k0=0; k0<conz; k0++) {
172       if (apJ[k] >= cstart) break;
173       ca[k0]      = apa[apJ[k]];
174       apa[apJ[k++]] = 0.0;
175     }
176 
177     /* diagonal part of C */
178     ca = cda + cd->i[i];
179     for (k1=0; k1<cdnz; k1++) {
180       ca[k1]      = apa[apJ[k]];
181       apa[apJ[k++]] = 0.0;
182     }
183 
184     /* 2nd off-diagoanl part of C */
185     ca = coa + co->i[i];
186     for (; k0<conz; k0++) {
187       ca[k0]      = apa[apJ[k]];
188       apa[apJ[k++]] = 0.0;
189     }
190   }
191   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
192   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
193   PetscFunctionReturn(0);
194 }
195 
196 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,PetscReal fill,Mat *C)
197 {
198   PetscErrorCode     ierr;
199   MPI_Comm           comm;
200   PetscMPIInt        size;
201   Mat                Cmpi;
202   Mat_PtAPMPI        *ptap;
203   PetscFreeSpaceList free_space=NULL,current_space=NULL;
204   Mat_MPIAIJ         *a        =(Mat_MPIAIJ*)A->data,*c;
205   Mat_SeqAIJ         *ad       =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
206   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
207   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
208   PetscInt           *lnk,i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi;
209   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
210   PetscBT            lnkbt;
211   PetscScalar        *apa;
212   PetscReal          afill;
213 
214   PetscFunctionBegin;
215   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
216   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
217 
218   /* create struct Mat_PtAPMPI and attached it to C later */
219   ierr = PetscNew(&ptap);CHKERRQ(ierr);
220 
221   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
222   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
223 
224   /* get P_loc by taking all local rows of P */
225   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
226 
227   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
228   pi_loc = p_loc->i; pj_loc = p_loc->j;
229   if (size > 1) {
230     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
231     pi_oth = p_oth->i; pj_oth = p_oth->j;
232   } else {
233     p_oth = NULL;
234     pi_oth = NULL; pj_oth = NULL;
235   }
236 
237   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
238   /*-------------------------------------------------------------------*/
239   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
240   ptap->api = api;
241   api[0]    = 0;
242 
243   /* create and initialize a linked list */
244   ierr = PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt);CHKERRQ(ierr);
245 
246   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
247   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space);CHKERRQ(ierr);
248   current_space = free_space;
249 
250   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
251   for (i=0; i<am; i++) {
252     /* diagonal portion of A */
253     nzi = adi[i+1] - adi[i];
254     for (j=0; j<nzi; j++) {
255       row  = *adj++;
256       pnz  = pi_loc[row+1] - pi_loc[row];
257       Jptr = pj_loc + pi_loc[row];
258       /* add non-zero cols of P into the sorted linked list lnk */
259       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
260     }
261     /* off-diagonal portion of A */
262     nzi = aoi[i+1] - aoi[i];
263     for (j=0; j<nzi; j++) {
264       row  = *aoj++;
265       pnz  = pi_oth[row+1] - pi_oth[row];
266       Jptr = pj_oth + pi_oth[row];
267       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
268     }
269 
270     apnz     = lnk[0];
271     api[i+1] = api[i] + apnz;
272 
273     /* if free space is not available, double the total space in the list */
274     if (current_space->local_remaining<apnz) {
275       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
276       nspacedouble++;
277     }
278 
279     /* Copy data into free space, then initialize lnk */
280     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
281     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
282 
283     current_space->array           += apnz;
284     current_space->local_used      += apnz;
285     current_space->local_remaining -= apnz;
286   }
287 
288   /* Allocate space for apj, initialize apj, and */
289   /* destroy list of free space and other temporary array(s) */
290   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
291   apj  = ptap->apj;
292   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
293   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
294 
295   /* malloc apa to store dense row A[i,:]*P */
296   ierr = PetscCalloc1(pN,&apa);CHKERRQ(ierr);
297 
298   ptap->apa = apa;
299 
300   /* create and assemble symbolic parallel matrix Cmpi */
301   /*----------------------------------------------------*/
302   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
303   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
304   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
305 
306   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
307   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
308 
309   ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(Cmpi, apj, api, dnz, onz);CHKERRQ(ierr);
310   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
311   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
312   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
313 
314   ptap->destroy        = Cmpi->ops->destroy;
315   ptap->duplicate      = Cmpi->ops->duplicate;
316   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
317   Cmpi->ops->destroy   = MatDestroy_MPIAIJ_MatMatMult;
318   Cmpi->ops->duplicate = MatDuplicate_MPIAIJ_MatMatMult;
319 
320   /* attach the supporting struct to Cmpi for reuse */
321   c       = (Mat_MPIAIJ*)Cmpi->data;
322   c->ptap = ptap;
323 
324   *C = Cmpi;
325 
326   /* set MatInfo */
327   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
328   if (afill < 1.0) afill = 1.0;
329   Cmpi->info.mallocs           = nspacedouble;
330   Cmpi->info.fill_ratio_given  = fill;
331   Cmpi->info.fill_ratio_needed = afill;
332 
333 #if defined(PETSC_USE_INFO)
334   if (api[am]) {
335     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
336     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
337   } else {
338     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
339   }
340 #endif
341   PetscFunctionReturn(0);
342 }
343 
344 PETSC_INTERN PetscErrorCode MatMatMult_MPIAIJ_MPIDense(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
345 {
346   PetscErrorCode ierr;
347 
348   PetscFunctionBegin;
349   if (scall == MAT_INITIAL_MATRIX) {
350     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
351     ierr = MatMatMultSymbolic_MPIAIJ_MPIDense(A,B,fill,C);CHKERRQ(ierr);
352     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
353   }
354   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
355   ierr = MatMatMultNumeric_MPIAIJ_MPIDense(A,B,*C);CHKERRQ(ierr);
356   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
357   PetscFunctionReturn(0);
358 }
359 
360 typedef struct {
361   Mat         workB;
362   PetscScalar *rvalues,*svalues;
363   MPI_Request *rwaits,*swaits;
364 } MPIAIJ_MPIDense;
365 
366 PetscErrorCode MatMPIAIJ_MPIDenseDestroy(void *ctx)
367 {
368   MPIAIJ_MPIDense *contents = (MPIAIJ_MPIDense*) ctx;
369   PetscErrorCode  ierr;
370 
371   PetscFunctionBegin;
372   ierr = MatDestroy(&contents->workB);CHKERRQ(ierr);
373   ierr = PetscFree4(contents->rvalues,contents->svalues,contents->rwaits,contents->swaits);CHKERRQ(ierr);
374   ierr = PetscFree(contents);CHKERRQ(ierr);
375   PetscFunctionReturn(0);
376 }
377 
378 /*
379     This is a "dummy function" that handles the case where matrix C was created as a dense matrix
380   directly by the user and passed to MatMatMult() with the MAT_REUSE_MATRIX option
381 
382   It is the same as MatMatMultSymbolic_MPIAIJ_MPIDense() except does not create C
383 */
384 PetscErrorCode MatMatMultNumeric_MPIDense(Mat A,Mat B,Mat C)
385 {
386   PetscErrorCode         ierr;
387   PetscBool              flg;
388   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*) A->data;
389   PetscInt               nz   = aij->B->cmap->n;
390   PetscContainer         container;
391   MPIAIJ_MPIDense        *contents;
392   VecScatter             ctx   = aij->Mvctx;
393   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
394   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
395 
396   PetscFunctionBegin;
397   ierr = PetscObjectTypeCompare((PetscObject)B,MATMPIDENSE,&flg);CHKERRQ(ierr);
398   if (!flg) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Second matrix must be mpidense");
399 
400   /* Handle case where where user provided the final C matrix rather than calling MatMatMult() with MAT_INITIAL_MATRIX*/
401   ierr = PetscObjectTypeCompare((PetscObject)A,MATMPIAIJ,&flg);CHKERRQ(ierr);
402   if (!flg) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"First matrix must be MPIAIJ");
403 
404   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
405 
406   ierr = PetscNew(&contents);CHKERRQ(ierr);
407   /* Create work matrix used to store off processor rows of B needed for local product */
408   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,B->cmap->N,NULL,&contents->workB);CHKERRQ(ierr);
409   /* Create work arrays needed */
410   ierr = PetscMalloc4(B->cmap->N*from->starts[from->n],&contents->rvalues,
411                       B->cmap->N*to->starts[to->n],&contents->svalues,
412                       from->n,&contents->rwaits,
413                       to->n,&contents->swaits);CHKERRQ(ierr);
414 
415   ierr = PetscContainerCreate(PetscObjectComm((PetscObject)A),&container);CHKERRQ(ierr);
416   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
417   ierr = PetscContainerSetUserDestroy(container,MatMPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
418   ierr = PetscObjectCompose((PetscObject)C,"workB",(PetscObject)container);CHKERRQ(ierr);
419   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
420 
421   ierr = (*C->ops->matmultnumeric)(A,B,C);CHKERRQ(ierr);
422   PetscFunctionReturn(0);
423 }
424 
425 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat A,Mat B,PetscReal fill,Mat *C)
426 {
427   PetscErrorCode         ierr;
428   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*) A->data;
429   PetscInt               nz   = aij->B->cmap->n;
430   PetscContainer         container;
431   MPIAIJ_MPIDense        *contents;
432   VecScatter             ctx   = aij->Mvctx;
433   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
434   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
435   PetscInt               m     = A->rmap->n,n=B->cmap->n;
436 
437   PetscFunctionBegin;
438   ierr = MatCreate(PetscObjectComm((PetscObject)B),C);CHKERRQ(ierr);
439   ierr = MatSetSizes(*C,m,n,A->rmap->N,B->cmap->N);CHKERRQ(ierr);
440   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
441   ierr = MatSetType(*C,MATMPIDENSE);CHKERRQ(ierr);
442   ierr = MatMPIDenseSetPreallocation(*C,NULL);CHKERRQ(ierr);
443   ierr = MatAssemblyBegin(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
444   ierr = MatAssemblyEnd(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
445 
446   (*C)->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
447 
448   ierr = PetscNew(&contents);CHKERRQ(ierr);
449   /* Create work matrix used to store off processor rows of B needed for local product */
450   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,B->cmap->N,NULL,&contents->workB);CHKERRQ(ierr);
451   /* Create work arrays needed */
452   ierr = PetscMalloc4(B->cmap->N*from->starts[from->n],&contents->rvalues,
453                       B->cmap->N*to->starts[to->n],&contents->svalues,
454                       from->n,&contents->rwaits,
455                       to->n,&contents->swaits);CHKERRQ(ierr);
456 
457   ierr = PetscContainerCreate(PetscObjectComm((PetscObject)A),&container);CHKERRQ(ierr);
458   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
459   ierr = PetscContainerSetUserDestroy(container,MatMPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
460   ierr = PetscObjectCompose((PetscObject)(*C),"workB",(PetscObject)container);CHKERRQ(ierr);
461   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
462   PetscFunctionReturn(0);
463 }
464 
465 /*
466     Performs an efficient scatter on the rows of B needed by this process; this is
467     a modification of the VecScatterBegin_() routines.
468 */
469 PetscErrorCode MatMPIDenseScatter(Mat A,Mat B,Mat C,Mat *outworkB)
470 {
471   Mat_MPIAIJ             *aij = (Mat_MPIAIJ*)A->data;
472   PetscErrorCode         ierr;
473   PetscScalar            *b,*w,*svalues,*rvalues;
474   VecScatter             ctx   = aij->Mvctx;
475   VecScatter_MPI_General *from = (VecScatter_MPI_General*) ctx->fromdata;
476   VecScatter_MPI_General *to   = (VecScatter_MPI_General*) ctx->todata;
477   PetscInt               i,j,k;
478   PetscInt               *sindices,*sstarts,*rindices,*rstarts;
479   PetscMPIInt            *sprocs,*rprocs,nrecvs;
480   MPI_Request            *swaits,*rwaits;
481   MPI_Comm               comm;
482   PetscMPIInt            tag  = ((PetscObject)ctx)->tag,ncols = B->cmap->N, nrows = aij->B->cmap->n,imdex,nrowsB = B->rmap->n;
483   MPI_Status             status;
484   MPIAIJ_MPIDense        *contents;
485   PetscContainer         container;
486   Mat                    workB;
487 
488   PetscFunctionBegin;
489   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
490   ierr = PetscObjectQuery((PetscObject)C,"workB",(PetscObject*)&container);CHKERRQ(ierr);
491   if (!container) SETERRQ(comm,PETSC_ERR_PLIB,"Container does not exist");
492   ierr = PetscContainerGetPointer(container,(void**)&contents);CHKERRQ(ierr);
493 
494   workB = *outworkB = contents->workB;
495   if (nrows != workB->rmap->n) SETERRQ2(comm,PETSC_ERR_PLIB,"Number of rows of workB %D not equal to columns of aij->B %D",nrows,workB->cmap->n);
496   sindices = to->indices;
497   sstarts  = to->starts;
498   sprocs   = to->procs;
499   swaits   = contents->swaits;
500   svalues  = contents->svalues;
501 
502   rindices = from->indices;
503   rstarts  = from->starts;
504   rprocs   = from->procs;
505   rwaits   = contents->rwaits;
506   rvalues  = contents->rvalues;
507 
508   ierr = MatDenseGetArray(B,&b);CHKERRQ(ierr);
509   ierr = MatDenseGetArray(workB,&w);CHKERRQ(ierr);
510 
511   for (i=0; i<from->n; i++) {
512     ierr = MPI_Irecv(rvalues+ncols*rstarts[i],ncols*(rstarts[i+1]-rstarts[i]),MPIU_SCALAR,rprocs[i],tag,comm,rwaits+i);CHKERRQ(ierr);
513   }
514 
515   for (i=0; i<to->n; i++) {
516     /* pack a message at a time */
517     for (j=0; j<sstarts[i+1]-sstarts[i]; j++) {
518       for (k=0; k<ncols; k++) {
519         svalues[ncols*(sstarts[i] + j) + k] = b[sindices[sstarts[i]+j] + nrowsB*k];
520       }
521     }
522     ierr = MPI_Isend(svalues+ncols*sstarts[i],ncols*(sstarts[i+1]-sstarts[i]),MPIU_SCALAR,sprocs[i],tag,comm,swaits+i);CHKERRQ(ierr);
523   }
524 
525   nrecvs = from->n;
526   while (nrecvs) {
527     ierr = MPI_Waitany(from->n,rwaits,&imdex,&status);CHKERRQ(ierr);
528     nrecvs--;
529     /* unpack a message at a time */
530     for (j=0; j<rstarts[imdex+1]-rstarts[imdex]; j++) {
531       for (k=0; k<ncols; k++) {
532         w[rindices[rstarts[imdex]+j] + nrows*k] = rvalues[ncols*(rstarts[imdex] + j) + k];
533       }
534     }
535   }
536   if (to->n) {ierr = MPI_Waitall(to->n,swaits,to->sstatus);CHKERRQ(ierr);}
537 
538   ierr = MatDenseRestoreArray(B,&b);CHKERRQ(ierr);
539   ierr = MatDenseRestoreArray(workB,&w);CHKERRQ(ierr);
540   ierr = MatAssemblyBegin(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
541   ierr = MatAssemblyEnd(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
542   PetscFunctionReturn(0);
543 }
544 extern PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat,Mat,Mat);
545 
546 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat A,Mat B,Mat C)
547 {
548   PetscErrorCode ierr;
549   Mat_MPIAIJ     *aij    = (Mat_MPIAIJ*)A->data;
550   Mat_MPIDense   *bdense = (Mat_MPIDense*)B->data;
551   Mat_MPIDense   *cdense = (Mat_MPIDense*)C->data;
552   Mat            workB;
553 
554   PetscFunctionBegin;
555   /* diagonal block of A times all local rows of B*/
556   ierr = MatMatMultNumeric_SeqAIJ_SeqDense(aij->A,bdense->A,cdense->A);CHKERRQ(ierr);
557 
558   /* get off processor parts of B needed to complete the product */
559   ierr = MatMPIDenseScatter(A,B,C,&workB);CHKERRQ(ierr);
560 
561   /* off-diagonal block of A times nonlocal rows of B */
562   ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A);CHKERRQ(ierr);
563   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
564   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
565   PetscFunctionReturn(0);
566 }
567 
568 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
569 {
570   PetscErrorCode ierr;
571   Mat_MPIAIJ     *a   = (Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
572   Mat_SeqAIJ     *ad  = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
573   Mat_SeqAIJ     *cd  = (Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
574   PetscInt       *adi = ad->i,*adj,*aoi=ao->i,*aoj;
575   PetscScalar    *ada,*aoa,*cda=cd->a,*coa=co->a;
576   Mat_SeqAIJ     *p_loc,*p_oth;
577   PetscInt       *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
578   PetscScalar    *pa_loc,*pa_oth,*pa,valtmp,*ca;
579   PetscInt       cm          = C->rmap->n,anz,pnz;
580   Mat_PtAPMPI    *ptap       = c->ptap;
581   PetscScalar    *apa_sparse = ptap->apa;
582   PetscInt       *api,*apj,*apJ,i,j,k,row;
583   PetscInt       cstart = C->cmap->rstart;
584   PetscInt       cdnz,conz,k0,k1,nextp;
585   MPI_Comm       comm;
586   PetscMPIInt    size;
587 
588   PetscFunctionBegin;
589   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
590   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
591 
592   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
593   /*-----------------------------------------------------*/
594   /* update numerical values of P_oth and P_loc */
595   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
596   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
597 
598   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
599   /*----------------------------------------------------------*/
600   /* get data from symbolic products */
601   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
602   pi_loc = p_loc->i; pj_loc = p_loc->j; pa_loc = p_loc->a;
603   if (size >1) {
604     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
605     pi_oth = p_oth->i; pj_oth = p_oth->j; pa_oth = p_oth->a;
606   } else {
607     p_oth = NULL; pi_oth = NULL; pj_oth = NULL; pa_oth = NULL;
608   }
609 
610   api = ptap->api;
611   apj = ptap->apj;
612   for (i=0; i<cm; i++) {
613     apJ = apj + api[i];
614 
615     /* diagonal portion of A */
616     anz = adi[i+1] - adi[i];
617     adj = ad->j + adi[i];
618     ada = ad->a + adi[i];
619     for (j=0; j<anz; j++) {
620       row = adj[j];
621       pnz = pi_loc[row+1] - pi_loc[row];
622       pj  = pj_loc + pi_loc[row];
623       pa  = pa_loc + pi_loc[row];
624       /* perform sparse axpy */
625       valtmp = ada[j];
626       nextp  = 0;
627       for (k=0; nextp<pnz; k++) {
628         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
629           apa_sparse[k] += valtmp*pa[nextp++];
630         }
631       }
632       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
633     }
634 
635     /* off-diagonal portion of A */
636     anz = aoi[i+1] - aoi[i];
637     aoj = ao->j + aoi[i];
638     aoa = ao->a + aoi[i];
639     for (j=0; j<anz; j++) {
640       row = aoj[j];
641       pnz = pi_oth[row+1] - pi_oth[row];
642       pj  = pj_oth + pi_oth[row];
643       pa  = pa_oth + pi_oth[row];
644       /* perform sparse axpy */
645       valtmp = aoa[j];
646       nextp  = 0;
647       for (k=0; nextp<pnz; k++) {
648         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
649           apa_sparse[k] += valtmp*pa[nextp++];
650         }
651       }
652       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
653     }
654 
655     /* set values in C */
656     cdnz = cd->i[i+1] - cd->i[i];
657     conz = co->i[i+1] - co->i[i];
658 
659     /* 1st off-diagoanl part of C */
660     ca = coa + co->i[i];
661     k  = 0;
662     for (k0=0; k0<conz; k0++) {
663       if (apJ[k] >= cstart) break;
664       ca[k0]        = apa_sparse[k];
665       apa_sparse[k] = 0.0;
666       k++;
667     }
668 
669     /* diagonal part of C */
670     ca = cda + cd->i[i];
671     for (k1=0; k1<cdnz; k1++) {
672       ca[k1]        = apa_sparse[k];
673       apa_sparse[k] = 0.0;
674       k++;
675     }
676 
677     /* 2nd off-diagoanl part of C */
678     ca = coa + co->i[i];
679     for (; k0<conz; k0++) {
680       ca[k0]        = apa_sparse[k];
681       apa_sparse[k] = 0.0;
682       k++;
683     }
684   }
685   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
686   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
687   PetscFunctionReturn(0);
688 }
689 
690 /* same as MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(), except using LLCondensed to avoid O(BN) memory requirement */
691 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat *C)
692 {
693   PetscErrorCode     ierr;
694   MPI_Comm           comm;
695   PetscMPIInt        size;
696   Mat                Cmpi;
697   Mat_PtAPMPI        *ptap;
698   PetscFreeSpaceList free_space = NULL,current_space=NULL;
699   Mat_MPIAIJ         *a         = (Mat_MPIAIJ*)A->data,*c;
700   Mat_SeqAIJ         *ad        = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
701   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
702   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
703   PetscInt           i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi,*lnk,apnz_max=0;
704   PetscInt           am=A->rmap->n,pn=P->cmap->n,pm=P->rmap->n,lsize=pn+20;
705   PetscReal          afill;
706   PetscScalar        *apa;
707 
708   PetscFunctionBegin;
709   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
710   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
711 
712   /* create struct Mat_PtAPMPI and attached it to C later */
713   ierr = PetscNew(&ptap);CHKERRQ(ierr);
714 
715   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
716   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
717 
718   /* get P_loc by taking all local rows of P */
719   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
720 
721   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
722   pi_loc = p_loc->i; pj_loc = p_loc->j;
723   if (size > 1) {
724     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
725     pi_oth = p_oth->i; pj_oth = p_oth->j;
726   } else {
727     p_oth  = NULL;
728     pi_oth = NULL; pj_oth = NULL;
729   }
730 
731   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
732   /*-------------------------------------------------------------------*/
733   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
734   ptap->api = api;
735   api[0]    = 0;
736 
737   ierr = PetscLLCondensedCreate_Scalable(lsize,&lnk);CHKERRQ(ierr);
738 
739   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
740   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space);CHKERRQ(ierr);
741   current_space = free_space;
742   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
743   for (i=0; i<am; i++) {
744     /* diagonal portion of A */
745     nzi = adi[i+1] - adi[i];
746     for (j=0; j<nzi; j++) {
747       row  = *adj++;
748       pnz  = pi_loc[row+1] - pi_loc[row];
749       Jptr = pj_loc + pi_loc[row];
750       /* Expand list if it is not long enough */
751       if (pnz+apnz_max > lsize) {
752         lsize = pnz+apnz_max;
753         ierr = PetscLLCondensedExpand_Scalable(lsize, &lnk);CHKERRQ(ierr);
754       }
755       /* add non-zero cols of P into the sorted linked list lnk */
756       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
757       apnz     = *lnk; /* The first element in the list is the number of items in the list */
758       api[i+1] = api[i] + apnz;
759       if (apnz > apnz_max) apnz_max = apnz;
760     }
761     /* off-diagonal portion of A */
762     nzi = aoi[i+1] - aoi[i];
763     for (j=0; j<nzi; j++) {
764       row  = *aoj++;
765       pnz  = pi_oth[row+1] - pi_oth[row];
766       Jptr = pj_oth + pi_oth[row];
767       /* Expand list if it is not long enough */
768       if (pnz+apnz_max > lsize) {
769         lsize = pnz + apnz_max;
770         ierr = PetscLLCondensedExpand_Scalable(lsize, &lnk);CHKERRQ(ierr);
771       }
772       /* add non-zero cols of P into the sorted linked list lnk */
773       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
774       apnz     = *lnk;  /* The first element in the list is the number of items in the list */
775       api[i+1] = api[i] + apnz;
776       if (apnz > apnz_max) apnz_max = apnz;
777     }
778     apnz     = *lnk;
779     api[i+1] = api[i] + apnz;
780     if (apnz > apnz_max) apnz_max = apnz;
781 
782     /* if free space is not available, double the total space in the list */
783     if (current_space->local_remaining<apnz) {
784       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
785       nspacedouble++;
786     }
787 
788     /* Copy data into free space, then initialize lnk */
789     ierr = PetscLLCondensedClean_Scalable(apnz,current_space->array,lnk);CHKERRQ(ierr);
790     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
791 
792     current_space->array           += apnz;
793     current_space->local_used      += apnz;
794     current_space->local_remaining -= apnz;
795   }
796 
797   /* Allocate space for apj, initialize apj, and */
798   /* destroy list of free space and other temporary array(s) */
799   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
800   apj  = ptap->apj;
801   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
802   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
803 
804   /* create and assemble symbolic parallel matrix Cmpi */
805   /*----------------------------------------------------*/
806   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
807   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
808   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
809   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
810   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
811 
812   /* malloc apa for assembly Cmpi */
813   ierr = PetscCalloc1(apnz_max,&apa);CHKERRQ(ierr);
814   ptap->apa = apa;
815 
816   ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(Cmpi, apj, api, dnz, onz);CHKERRQ(ierr);
817   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
818   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
819   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
820 
821   ptap->destroy             = Cmpi->ops->destroy;
822   ptap->duplicate           = Cmpi->ops->duplicate;
823   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ;
824   Cmpi->ops->destroy        = MatDestroy_MPIAIJ_MatMatMult;
825   Cmpi->ops->duplicate      = MatDuplicate_MPIAIJ_MatMatMult;
826 
827   /* attach the supporting struct to Cmpi for reuse */
828   c       = (Mat_MPIAIJ*)Cmpi->data;
829   c->ptap = ptap;
830   *C = Cmpi;
831 
832   /* set MatInfo */
833   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
834   if (afill < 1.0) afill = 1.0;
835   Cmpi->info.mallocs           = nspacedouble;
836   Cmpi->info.fill_ratio_given  = fill;
837   Cmpi->info.fill_ratio_needed = afill;
838 
839 #if defined(PETSC_USE_INFO)
840   if (api[am]) {
841     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
842     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
843   } else {
844     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
845   }
846 #endif
847   PetscFunctionReturn(0);
848 }
849 
850 /* This function is needed for the seqMPI matrix-matrix multiplication.  */
851 /* Three input arrays are merged to one output array. The size of the    */
852 /* output array is also output. Duplicate entries only show up once.     */
853 static void Merge3SortedArrays(PetscInt  size1, PetscInt *in1,
854                                PetscInt  size2, PetscInt *in2,
855                                PetscInt  size3, PetscInt *in3,
856                                PetscInt *size4, PetscInt *out)
857 {
858   int i = 0, j = 0, k = 0, l = 0;
859 
860   /* Traverse all three arrays */
861   while (i<size1 && j<size2 && k<size3) {
862     if (in1[i] < in2[j] && in1[i] < in3[k]) {
863       out[l++] = in1[i++];
864     }
865     else if(in2[j] < in1[i] && in2[j] < in3[k]) {
866       out[l++] = in2[j++];
867     }
868     else if(in3[k] < in1[i] && in3[k] < in2[j]) {
869       out[l++] = in3[k++];
870     }
871     else if(in1[i] == in2[j] && in1[i] < in3[k]) {
872       out[l++] = in1[i];
873       i++, j++;
874     }
875     else if(in1[i] == in3[k] && in1[i] < in2[j]) {
876       out[l++] = in1[i];
877       i++, k++;
878     }
879     else if(in3[k] == in2[j] && in2[j] < in1[i])  {
880       out[l++] = in2[j];
881       k++, j++;
882     }
883     else if(in1[i] == in2[j] && in1[i] == in3[k]) {
884       out[l++] = in1[i];
885       i++, j++, k++;
886     }
887   }
888 
889   /* Traverse two remaining arrays */
890   while (i<size1 && j<size2) {
891     if (in1[i] < in2[j]) {
892       out[l++] = in1[i++];
893     }
894     else if(in1[i] > in2[j]) {
895       out[l++] = in2[j++];
896     }
897     else {
898       out[l++] = in1[i];
899       i++, j++;
900     }
901   }
902 
903   while (i<size1 && k<size3) {
904     if (in1[i] < in3[k]) {
905       out[l++] = in1[i++];
906     }
907     else if(in1[i] > in3[k]) {
908       out[l++] = in3[k++];
909     }
910     else {
911       out[l++] = in1[i];
912       i++, k++;
913     }
914   }
915 
916   while (k<size3 && j<size2)  {
917     if (in3[k] < in2[j]) {
918       out[l++] = in3[k++];
919     }
920     else if(in3[k] > in2[j]) {
921       out[l++] = in2[j++];
922     }
923     else {
924       out[l++] = in3[k];
925       k++, j++;
926     }
927   }
928 
929   /* Traverse one remaining array */
930   while (i<size1) out[l++] = in1[i++];
931   while (j<size2) out[l++] = in2[j++];
932   while (k<size3) out[l++] = in3[k++];
933 
934   *size4 = l;
935 }
936 
937 /* This matrix-matrix multiplication algorithm divides the multiplication into three multiplications and  */
938 /* adds up the products. Two of these three multiplications are performed with existing (sequential)      */
939 /* matrix-matrix multiplications.  */
940 #undef __FUNCT__
941 #define __FUNCT__ "MatMatMultSymbolic_MPIAIJ_MPIAIJ_seqMPI"
942 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_seqMPI(Mat A, Mat P, PetscReal fill, Mat *C)
943 {
944   PetscErrorCode     ierr;
945   MPI_Comm           comm;
946   PetscMPIInt        size;
947   Mat                Cmpi;
948   Mat_PtAPMPI        *ptap;
949   PetscFreeSpaceList free_space_diag=NULL, current_space=NULL;
950   Mat_MPIAIJ         *a        =(Mat_MPIAIJ*)A->data;
951   Mat_SeqAIJ         *ad       =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc;
952   Mat_MPIAIJ         *p        =(Mat_MPIAIJ*)P->data;
953   Mat_MPIAIJ         *c;
954   Mat_SeqAIJ         *adpd_seq, *p_off, *aopoth_seq;
955   PetscInt           adponz, adpdnz;
956   PetscInt           *pi_loc,*dnz,*onz;
957   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,rstart=A->rmap->rstart;
958   PetscInt           *lnk,i, i1=0,pnz,row,*adpoi,*adpoj, *api, *adpoJ, *aopJ, *apJ,*Jptr, aopnz, nspacedouble=0,j,nzi,
959                      *apj,apnz, *adpdi, *adpdj, *adpdJ, *poff_i, *poff_j, *j_temp, *aopothi, *aopothj;
960   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n, p_colstart, p_colend;
961   PetscBT            lnkbt;
962   PetscScalar        *apa;
963   PetscReal          afill;
964   PetscMPIInt        rank;
965   Mat                adpd, aopoth;
966 
967   PetscFunctionBegin;
968   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
969   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
970   ierr = MPI_Comm_rank(comm, &rank);CHKERRQ(ierr);
971   ierr = MatGetOwnershipRangeColumn(P, &p_colstart, &p_colend); CHKERRQ(ierr);
972 
973   /* create struct Mat_PtAPMPI and attached it to C later */
974   ierr = PetscNew(&ptap);CHKERRQ(ierr);
975 
976   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
977   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
978 
979   /* get P_loc by taking all local rows of P */
980   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
981 
982 
983   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
984   pi_loc = p_loc->i;
985 
986   /* Allocate memory for the i arrays of the matrices A*P, A_diag*P_off and A_offd * P */
987   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
988   ierr      = PetscMalloc1(am+2,&adpoi);CHKERRQ(ierr);
989 
990   adpoi[0]    = 0;
991   ptap->api = api;
992   api[0] = 0;
993 
994   /* create and initialize a linked list, will be used for both A_diag * P_loc_off and A_offd * P_oth */
995   ierr = PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt);CHKERRQ(ierr);
996   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
997 
998   /* Symbolic calc of A_loc_diag * P_loc_diag */
999   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(a->A, p->A, fill, &adpd);CHKERRQ(ierr);
1000   adpd_seq = (Mat_SeqAIJ*)((adpd)->data);
1001   adpdi = adpd_seq->i; adpdj = adpd_seq->j;
1002   p_off = (Mat_SeqAIJ*)((p->B)->data);
1003   poff_i = p_off->i; poff_j = p_off->j;
1004 
1005   /* j_temp stores indices of a result row before they are added to the linked list */
1006   ierr = PetscMalloc1(pN+2,&j_temp);CHKERRQ(ierr);
1007 
1008 
1009   /* Symbolic calc of the A_diag * p_loc_off */
1010   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
1011   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space_diag);CHKERRQ(ierr);
1012   current_space = free_space_diag;
1013 
1014   for (i=0; i<am; i++) {
1015     /* A_diag * P_loc_off */
1016     nzi = adi[i+1] - adi[i];
1017     for (j=0; j<nzi; j++) {
1018       row  = *adj++;
1019       pnz  = poff_i[row+1] - poff_i[row];
1020       Jptr = poff_j + poff_i[row];
1021       for(i1 = 0; i1 < pnz; i1++) {
1022         j_temp[i1] = p->garray[Jptr[i1]];
1023       }
1024       /* add non-zero cols of P into the sorted linked list lnk */
1025       ierr = PetscLLCondensedAddSorted(pnz,j_temp,lnk,lnkbt);CHKERRQ(ierr);
1026     }
1027 
1028     adponz     = lnk[0];
1029     adpoi[i+1] = adpoi[i] + adponz;
1030 
1031     /* if free space is not available, double the total space in the list */
1032     if (current_space->local_remaining<adponz) {
1033       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(adponz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1034       nspacedouble++;
1035     }
1036 
1037     /* Copy data into free space, then initialize lnk */
1038     ierr = PetscLLCondensedClean(pN,adponz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1039 
1040     current_space->array           += adponz;
1041     current_space->local_used      += adponz;
1042     current_space->local_remaining -= adponz;
1043   }
1044 
1045   /* Symbolic calc of A_off * P_oth */
1046   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(a->B, ptap->P_oth, fill, &aopoth);CHKERRQ(ierr);
1047   aopoth_seq = (Mat_SeqAIJ*)((aopoth)->data);
1048   aopothi = aopoth_seq->i; aopothj = aopoth_seq->j;
1049 
1050   /* Allocate space for apj, adpj, aopj, ... */
1051   /* destroy lists of free space and other temporary array(s) */
1052 
1053   ierr = PetscMalloc1(aopothi[am] + adpoi[am] + adpdi[am]+2, &ptap->apj);CHKERRQ(ierr);
1054   ierr = PetscMalloc1(adpoi[am]+2, &adpoj);CHKERRQ(ierr);
1055 
1056   /* Copy from linked list to j-array */
1057   ierr = PetscFreeSpaceContiguous(&free_space_diag,adpoj);CHKERRQ(ierr);
1058   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1059 
1060   adpoJ = adpoj;
1061   adpdJ = adpdj;
1062   aopJ = aopothj;
1063   apj  = ptap->apj;
1064   apJ = apj; /* still empty */
1065 
1066   /* Merge j-arrays of A_off * P, A_diag * P_loc_off, and */
1067   /* A_diag * P_loc_diag to get A*P */
1068   for (i = 0; i < am; i++) {
1069     aopnz  =  aopothi[i+1] -  aopothi[i];
1070     adponz = adpoi[i+1] - adpoi[i];
1071     adpdnz = adpdi[i+1] - adpdi[i];
1072 
1073     /* Correct indices from A_diag*P_diag */
1074     for(i1 = 0; i1 < adpdnz; i1++) {
1075       adpdJ[i1] += p_colstart;
1076     }
1077     /* Merge j-arrays of A_diag * P_loc_off and A_diag * P_loc_diag and A_off * P_oth */
1078     Merge3SortedArrays(adponz, adpoJ, adpdnz, adpdJ, aopnz, aopJ, &apnz, apJ);
1079     ierr = MatPreallocateSet(i+rstart, apnz, apJ, dnz, onz); CHKERRQ(ierr);
1080 
1081     aopJ += aopnz;
1082     adpoJ += adponz;
1083     adpdJ += adpdnz;
1084     apJ += apnz;
1085     api[i+1] = api[i] + apnz;
1086   }
1087 
1088   /* malloc apa to store dense row A[i,:]*P */
1089   ierr = PetscCalloc1(pN+2,&apa);CHKERRQ(ierr);
1090 
1091   ptap->apa = apa;
1092   /* create and assemble symbolic parallel matrix Cmpi */
1093   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1094   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1095   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
1096 
1097   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1098   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1099 
1100 
1101   ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(Cmpi, apj, api, dnz, onz);CHKERRQ(ierr);
1102   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1103   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1104   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1105 
1106 
1107   ptap->destroy        = Cmpi->ops->destroy;
1108   ptap->duplicate      = Cmpi->ops->duplicate;
1109   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1110   Cmpi->ops->destroy   = MatDestroy_MPIAIJ_MatMatMult;
1111   Cmpi->ops->duplicate = MatDuplicate_MPIAIJ_MatMatMult;
1112 
1113   /* attach the supporting struct to Cmpi for reuse */
1114   c       = (Mat_MPIAIJ*)Cmpi->data;
1115   c->ptap = ptap;
1116   *C = Cmpi;
1117 
1118   /* set MatInfo */
1119   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
1120   if (afill < 1.0) afill = 1.0;
1121   Cmpi->info.mallocs           = nspacedouble;
1122   Cmpi->info.fill_ratio_given  = fill;
1123   Cmpi->info.fill_ratio_needed = afill;
1124 
1125 #if defined(PETSC_USE_INFO)
1126   if (api[am]) {
1127     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1128     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
1129   } else {
1130     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1131   }
1132 #endif
1133 
1134   ierr = MatDestroy(&aopoth);CHKERRQ(ierr);
1135   ierr = MatDestroy(&adpd);CHKERRQ(ierr);
1136   ierr = PetscFree(j_temp);CHKERRQ(ierr);
1137   ierr = PetscFree(adpoj);CHKERRQ(ierr);
1138   ierr = PetscFree(adpoi);CHKERRQ(ierr);
1139   PetscFunctionReturn(0);
1140 }
1141 
1142 
1143 /*-------------------------------------------------------------------------*/
1144 PetscErrorCode MatTransposeMatMult_MPIAIJ_MPIAIJ(Mat P,Mat A,MatReuse scall,PetscReal fill,Mat *C)
1145 {
1146   PetscErrorCode ierr;
1147   const char     *algTypes[3] = {"scalable","nonscalable","matmatmult"};
1148   PetscInt       aN=A->cmap->N,alg=1; /* set default algorithm */
1149   PetscBool      flg;
1150 
1151   PetscFunctionBegin;
1152   if (scall == MAT_INITIAL_MATRIX) {
1153     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
1154     PetscOptionsObject->alreadyprinted = PETSC_FALSE; /* a hack to ensure the option shows in '-help' */
1155     ierr = PetscOptionsEList("-mattransposematmult_via","Algorithmic approach","MatTransposeMatMult",algTypes,3,algTypes[1],&alg,&flg);CHKERRQ(ierr);
1156     ierr = PetscOptionsEnd();CHKERRQ(ierr);
1157 
1158     ierr = PetscLogEventBegin(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
1159     switch (alg) {
1160     case 1:
1161       if (!flg && aN > 100000) { /* may switch to scalable algorithm as default */
1162         MatInfo     Ainfo,Pinfo;
1163         PetscInt    nz_local;
1164         PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
1165         MPI_Comm    comm;
1166 
1167         ierr = MatGetInfo(A,MAT_LOCAL,&Ainfo);CHKERRQ(ierr);
1168         ierr = MatGetInfo(P,MAT_LOCAL,&Pinfo);CHKERRQ(ierr);
1169         nz_local = (PetscInt)(Ainfo.nz_allocated + Pinfo.nz_allocated); /* estimated local nonzero entries */
1170 
1171         if (aN > fill*nz_local) alg_scalable_loc = PETSC_TRUE;
1172         ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1173         ierr = MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm);CHKERRQ(ierr);
1174 
1175         if (alg_scalable) {
1176           alg  = 0; /* scalable algorithm would slower than nonscalable algorithm */
1177           ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(P,A,fill,C);CHKERRQ(ierr);
1178           break;
1179         }
1180       }
1181       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(P,A,fill,C);CHKERRQ(ierr);
1182       break;
1183     case 2:
1184     {
1185       Mat         Pt;
1186       Mat_PtAPMPI *ptap;
1187       Mat_MPIAIJ  *c;
1188       ierr = MatTranspose(P,MAT_INITIAL_MATRIX,&Pt);CHKERRQ(ierr);
1189       ierr = MatMatMult(Pt,A,MAT_INITIAL_MATRIX,fill,C);CHKERRQ(ierr);
1190       c        = (Mat_MPIAIJ*)(*C)->data;
1191       ptap     = c->ptap;
1192       ptap->Pt = Pt;
1193       (*C)->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult;
1194       PetscFunctionReturn(0);
1195     }
1196       break;
1197     default: /* scalable algorithm */
1198       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(P,A,fill,C);CHKERRQ(ierr);
1199       break;
1200     }
1201     ierr = PetscLogEventEnd(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
1202   }
1203   ierr = PetscLogEventBegin(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
1204   ierr = (*(*C)->ops->mattransposemultnumeric)(P,A,*C);CHKERRQ(ierr);
1205   ierr = PetscLogEventEnd(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
1206   PetscFunctionReturn(0);
1207 }
1208 
1209 /* This routine only works when scall=MAT_REUSE_MATRIX! */
1210 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult(Mat P,Mat A,Mat C)
1211 {
1212   PetscErrorCode ierr;
1213   Mat_MPIAIJ     *c=(Mat_MPIAIJ*)C->data;
1214   Mat_PtAPMPI    *ptap= c->ptap;
1215   Mat            Pt=ptap->Pt;
1216 
1217   PetscFunctionBegin;
1218   ierr = MatTranspose(P,MAT_REUSE_MATRIX,&Pt);CHKERRQ(ierr);
1219   ierr = MatMatMultNumeric(Pt,A,C);CHKERRQ(ierr);
1220   PetscFunctionReturn(0);
1221 }
1222 
1223 PetscErrorCode MatDuplicate_MPIAIJ_MatPtAP(Mat,MatDuplicateOption,Mat*);
1224 
1225 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1226 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,PetscReal fill,Mat *C)
1227 {
1228   PetscErrorCode      ierr;
1229   Mat_PtAPMPI         *ptap;
1230   Mat_MPIAIJ          *p=(Mat_MPIAIJ*)P->data,*c;
1231   MPI_Comm            comm;
1232   PetscMPIInt         size,rank;
1233   Mat                 Cmpi;
1234   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1235   PetscInt            pn=P->cmap->n,aN=A->cmap->N,an=A->cmap->n;
1236   PetscInt            *lnk,i,k,nsend;
1237   PetscBT             lnkbt;
1238   PetscMPIInt         tagi,tagj,*len_si,*len_s,*len_ri,icompleted=0,nrecv;
1239   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1240   PetscInt            len,proc,*dnz,*onz,*owners,nzi;
1241   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1242   MPI_Request         *swaits,*rwaits;
1243   MPI_Status          *sstatus,rstatus;
1244   PetscLayout         rowmap;
1245   PetscInt            *owners_co,*coi,*coj;    /* i and j array of (p->B)^T*A*P - used in the communication */
1246   PetscMPIInt         *len_r,*id_r;    /* array of length of comm->size, store send/recv matrix values */
1247   PetscInt            *Jptr,*prmap=p->garray,con,j,Crmax;
1248   Mat_SeqAIJ          *a_loc,*c_loc,*c_oth;
1249   PetscTable          ta;
1250 
1251   PetscFunctionBegin;
1252   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1253   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1254   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1255 
1256   /* create symbolic parallel matrix Cmpi */
1257   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1258   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
1259 
1260   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1261 
1262   /* create struct Mat_PtAPMPI and attached it to C later */
1263   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1264   ptap->reuse = MAT_INITIAL_MATRIX;
1265 
1266   /* (0) compute Rd = Pd^T, Ro = Po^T  */
1267   /* --------------------------------- */
1268   ierr = MatTranspose_SeqAIJ(p->A,MAT_INITIAL_MATRIX,&ptap->Rd);CHKERRQ(ierr);
1269   ierr = MatTranspose_SeqAIJ(p->B,MAT_INITIAL_MATRIX,&ptap->Ro);CHKERRQ(ierr);
1270 
1271   /* (1) compute symbolic A_loc */
1272   /* ---------------------------*/
1273   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&ptap->A_loc);CHKERRQ(ierr);
1274 
1275   /* (2-1) compute symbolic C_oth = Ro*A_loc  */
1276   /* ------------------------------------ */
1277   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Ro,ptap->A_loc,fill,&ptap->C_oth);CHKERRQ(ierr);
1278 
1279   /* (3) send coj of C_oth to other processors  */
1280   /* ------------------------------------------ */
1281   /* determine row ownership */
1282   ierr = PetscLayoutCreate(comm,&rowmap);CHKERRQ(ierr);
1283   rowmap->n  = pn;
1284   rowmap->bs = 1;
1285   ierr   = PetscLayoutSetUp(rowmap);CHKERRQ(ierr);
1286   owners = rowmap->range;
1287 
1288   /* determine the number of messages to send, their lengths */
1289   ierr = PetscMalloc4(size,&len_s,size,&len_si,size,&sstatus,size+2,&owners_co);CHKERRQ(ierr);
1290   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1291   ierr = PetscMemzero(len_si,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1292 
1293   c_oth = (Mat_SeqAIJ*)ptap->C_oth->data;
1294   coi   = c_oth->i; coj = c_oth->j;
1295   con   = ptap->C_oth->rmap->n;
1296   proc  = 0;
1297   for (i=0; i<con; i++) {
1298     while (prmap[i] >= owners[proc+1]) proc++;
1299     len_si[proc]++;               /* num of rows in Co(=Pt*A) to be sent to [proc] */
1300     len_s[proc] += coi[i+1] - coi[i]; /* num of nonzeros in Co to be sent to [proc] */
1301   }
1302 
1303   len          = 0; /* max length of buf_si[], see (4) */
1304   owners_co[0] = 0;
1305   nsend        = 0;
1306   for (proc=0; proc<size; proc++) {
1307     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1308     if (len_s[proc]) {
1309       nsend++;
1310       len_si[proc] = 2*(len_si[proc] + 1); /* length of buf_si to be sent to [proc] */
1311       len         += len_si[proc];
1312     }
1313   }
1314 
1315   /* determine the number and length of messages to receive for coi and coj  */
1316   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&nrecv);CHKERRQ(ierr);
1317   ierr = PetscGatherMessageLengths2(comm,nsend,nrecv,len_s,len_si,&id_r,&len_r,&len_ri);CHKERRQ(ierr);
1318 
1319   /* post the Irecv and Isend of coj */
1320   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1321   ierr = PetscPostIrecvInt(comm,tagj,nrecv,id_r,len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1322   ierr = PetscMalloc1(nsend+1,&swaits);CHKERRQ(ierr);
1323   for (proc=0, k=0; proc<size; proc++) {
1324     if (!len_s[proc]) continue;
1325     i    = owners_co[proc];
1326     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1327     k++;
1328   }
1329 
1330   /* (2-2) compute symbolic C_loc = Rd*A_loc */
1331   /* ---------------------------------------- */
1332   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Rd,ptap->A_loc,fill,&ptap->C_loc);CHKERRQ(ierr);
1333   c_loc = (Mat_SeqAIJ*)ptap->C_loc->data;
1334 
1335   /* receives coj are complete */
1336   for (i=0; i<nrecv; i++) {
1337     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1338   }
1339   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1340   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
1341 
1342   /* add received column indices into ta to update Crmax */
1343   a_loc = (Mat_SeqAIJ*)(ptap->A_loc)->data;
1344 
1345   /* create and initialize a linked list */
1346   ierr = PetscTableCreate(an,aN,&ta);CHKERRQ(ierr); /* for compute Crmax */
1347   MatRowMergeMax_SeqAIJ(a_loc,ptap->A_loc->rmap->N,ta);
1348 
1349   for (k=0; k<nrecv; k++) {/* k-th received message */
1350     Jptr = buf_rj[k];
1351     for (j=0; j<len_r[k]; j++) {
1352       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
1353     }
1354   }
1355   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
1356   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
1357 
1358   /* (4) send and recv coi */
1359   /*-----------------------*/
1360   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1361   ierr   = PetscPostIrecvInt(comm,tagi,nrecv,id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1362   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1363   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1364   for (proc=0,k=0; proc<size; proc++) {
1365     if (!len_s[proc]) continue;
1366     /* form outgoing message for i-structure:
1367          buf_si[0]:                 nrows to be sent
1368                [1:nrows]:           row index (global)
1369                [nrows+1:2*nrows+1]: i-structure index
1370     */
1371     /*-------------------------------------------*/
1372     nrows       = len_si[proc]/2 - 1; /* num of rows in Co to be sent to [proc] */
1373     buf_si_i    = buf_si + nrows+1;
1374     buf_si[0]   = nrows;
1375     buf_si_i[0] = 0;
1376     nrows       = 0;
1377     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1378       nzi = coi[i+1] - coi[i];
1379       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1380       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1381       nrows++;
1382     }
1383     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1384     k++;
1385     buf_si += len_si[proc];
1386   }
1387   for (i=0; i<nrecv; i++) {
1388     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1389   }
1390   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1391   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
1392 
1393   ierr = PetscFree4(len_s,len_si,sstatus,owners_co);CHKERRQ(ierr);
1394   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1395   ierr = PetscFree(swaits);CHKERRQ(ierr);
1396   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1397 
1398   /* (5) compute the local portion of Cmpi      */
1399   /* ------------------------------------------ */
1400   /* set initial free space to be Crmax, sufficient for holding nozeros in each row of Cmpi */
1401   ierr          = PetscFreeSpaceGet(Crmax,&free_space);CHKERRQ(ierr);
1402   current_space = free_space;
1403 
1404   ierr = PetscMalloc3(nrecv,&buf_ri_k,nrecv,&nextrow,nrecv,&nextci);CHKERRQ(ierr);
1405   for (k=0; k<nrecv; k++) {
1406     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1407     nrows       = *buf_ri_k[k];
1408     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1409     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1410   }
1411 
1412   ierr = MatPreallocateInitialize(comm,pn,an,dnz,onz);CHKERRQ(ierr);
1413   ierr = PetscLLCondensedCreate(Crmax,aN,&lnk,&lnkbt);CHKERRQ(ierr);
1414   for (i=0; i<pn; i++) {
1415     /* add C_loc into Cmpi */
1416     nzi  = c_loc->i[i+1] - c_loc->i[i];
1417     Jptr = c_loc->j + c_loc->i[i];
1418     ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1419 
1420     /* add received col data into lnk */
1421     for (k=0; k<nrecv; k++) { /* k-th received message */
1422       if (i == *nextrow[k]) { /* i-th row */
1423         nzi  = *(nextci[k]+1) - *nextci[k];
1424         Jptr = buf_rj[k] + *nextci[k];
1425         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1426         nextrow[k]++; nextci[k]++;
1427       }
1428     }
1429     nzi = lnk[0];
1430 
1431     /* copy data into free space, then initialize lnk */
1432     ierr = PetscLLCondensedClean(aN,nzi,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1433     ierr = MatPreallocateSet(i+owners[rank],nzi,current_space->array,dnz,onz);CHKERRQ(ierr);
1434   }
1435   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1436   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1437   ierr = PetscFreeSpaceDestroy(free_space);CHKERRQ(ierr);
1438 
1439   /* local sizes and preallocation */
1440   ierr = MatSetSizes(Cmpi,pn,an,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1441   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(P->cmap->bs));CHKERRQ(ierr);
1442   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1443   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1444 
1445   /* members in merge */
1446   ierr = PetscFree(id_r);CHKERRQ(ierr);
1447   ierr = PetscFree(len_r);CHKERRQ(ierr);
1448   ierr = PetscFree(buf_ri[0]);CHKERRQ(ierr);
1449   ierr = PetscFree(buf_ri);CHKERRQ(ierr);
1450   ierr = PetscFree(buf_rj[0]);CHKERRQ(ierr);
1451   ierr = PetscFree(buf_rj);CHKERRQ(ierr);
1452   ierr = PetscLayoutDestroy(&rowmap);CHKERRQ(ierr);
1453 
1454   /* attach the supporting struct to Cmpi for reuse */
1455   c = (Mat_MPIAIJ*)Cmpi->data;
1456   c->ptap         = ptap;
1457   ptap->duplicate = Cmpi->ops->duplicate;
1458   ptap->destroy   = Cmpi->ops->destroy;
1459 
1460   /* Cmpi is not ready for use - assembly will be done by MatPtAPNumeric() */
1461   Cmpi->assembled        = PETSC_FALSE;
1462   Cmpi->ops->destroy     = MatDestroy_MPIAIJ_PtAP;
1463   Cmpi->ops->duplicate   = MatDuplicate_MPIAIJ_MatPtAP;
1464   *C                     = Cmpi;
1465   PetscFunctionReturn(0);
1466 }
1467 
1468 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,Mat C)
1469 {
1470   PetscErrorCode    ierr;
1471   Mat_MPIAIJ        *p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1472   Mat_SeqAIJ        *c_seq;
1473   Mat_PtAPMPI       *ptap = c->ptap;
1474   Mat               A_loc,C_loc,C_oth;
1475   PetscInt          i,rstart,rend,cm,ncols,row;
1476   const PetscInt    *cols;
1477   const PetscScalar *vals;
1478 
1479   PetscFunctionBegin;
1480   ierr = MatZeroEntries(C);CHKERRQ(ierr);
1481 
1482   if (ptap->reuse == MAT_REUSE_MATRIX) {
1483     /* These matrices are obtained in MatTransposeMatMultSymbolic() */
1484     /* 1) get R = Pd^T, Ro = Po^T */
1485     /*----------------------------*/
1486     ierr = MatTranspose_SeqAIJ(p->A,MAT_REUSE_MATRIX,&ptap->Rd);CHKERRQ(ierr);
1487     ierr = MatTranspose_SeqAIJ(p->B,MAT_REUSE_MATRIX,&ptap->Ro);CHKERRQ(ierr);
1488 
1489     /* 2) compute numeric A_loc */
1490     /*--------------------------*/
1491     ierr = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&ptap->A_loc);CHKERRQ(ierr);
1492   }
1493 
1494   /* 3) C_loc = Rd*A_loc, C_oth = Ro*A_loc */
1495   A_loc = ptap->A_loc;
1496   ierr = ((ptap->C_loc)->ops->matmultnumeric)(ptap->Rd,A_loc,ptap->C_loc);CHKERRQ(ierr);
1497   ierr = ((ptap->C_oth)->ops->matmultnumeric)(ptap->Ro,A_loc,ptap->C_oth);CHKERRQ(ierr);
1498   C_loc = ptap->C_loc;
1499   C_oth = ptap->C_oth;
1500 
1501   /* add C_loc and Co to to C */
1502   ierr = MatGetOwnershipRange(C,&rstart,&rend);CHKERRQ(ierr);
1503 
1504   /* C_loc -> C */
1505   cm    = C_loc->rmap->N;
1506   c_seq = (Mat_SeqAIJ*)C_loc->data;
1507   cols = c_seq->j;
1508   vals = c_seq->a;
1509   for (i=0; i<cm; i++) {
1510     ncols = c_seq->i[i+1] - c_seq->i[i];
1511     row = rstart + i;
1512     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1513     cols += ncols; vals += ncols;
1514   }
1515 
1516   /* Co -> C, off-processor part */
1517   cm    = C_oth->rmap->N;
1518   c_seq = (Mat_SeqAIJ*)C_oth->data;
1519   cols  = c_seq->j;
1520   vals  = c_seq->a;
1521   for (i=0; i<cm; i++) {
1522     ncols = c_seq->i[i+1] - c_seq->i[i];
1523     row = p->garray[i];
1524     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1525     cols += ncols; vals += ncols;
1526   }
1527   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1528   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1529 
1530   ptap->reuse = MAT_REUSE_MATRIX;
1531   PetscFunctionReturn(0);
1532 }
1533 
1534 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ(Mat P,Mat A,Mat C)
1535 {
1536   PetscErrorCode      ierr;
1537   Mat_Merge_SeqsToMPI *merge;
1538   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1539   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
1540   Mat_PtAPMPI         *ptap;
1541   PetscInt            *adj;
1542   PetscInt            i,j,k,anz,pnz,row,*cj,nexta;
1543   MatScalar           *ada,*ca,valtmp;
1544   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
1545   MPI_Comm            comm;
1546   PetscMPIInt         size,rank,taga,*len_s;
1547   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
1548   PetscInt            **buf_ri,**buf_rj;
1549   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
1550   MPI_Request         *s_waits,*r_waits;
1551   MPI_Status          *status;
1552   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
1553   PetscInt            *ai,*aj,*coi,*coj,*poJ,*pdJ;
1554   Mat                 A_loc;
1555   Mat_SeqAIJ          *a_loc;
1556 
1557   PetscFunctionBegin;
1558   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1559   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1560   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1561 
1562   ptap  = c->ptap;
1563   merge = ptap->merge;
1564 
1565   /* 2) compute numeric C_seq = P_loc^T*A_loc */
1566   /*------------------------------------------*/
1567   /* get data from symbolic products */
1568   coi    = merge->coi; coj = merge->coj;
1569   ierr   = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
1570   bi     = merge->bi; bj = merge->bj;
1571   owners = merge->rowmap->range;
1572   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
1573 
1574   /* get A_loc by taking all local rows of A */
1575   A_loc = ptap->A_loc;
1576   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
1577   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1578   ai    = a_loc->i;
1579   aj    = a_loc->j;
1580 
1581   for (i=0; i<am; i++) {
1582     anz = ai[i+1] - ai[i];
1583     adj = aj + ai[i];
1584     ada = a_loc->a + ai[i];
1585 
1586     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1587     /*-------------------------------------------------------------*/
1588     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1589     pnz = po->i[i+1] - po->i[i];
1590     poJ = po->j + po->i[i];
1591     pA  = po->a + po->i[i];
1592     for (j=0; j<pnz; j++) {
1593       row = poJ[j];
1594       cj  = coj + coi[row];
1595       ca  = coa + coi[row];
1596       /* perform sparse axpy */
1597       nexta  = 0;
1598       valtmp = pA[j];
1599       for (k=0; nexta<anz; k++) {
1600         if (cj[k] == adj[nexta]) {
1601           ca[k] += valtmp*ada[nexta];
1602           nexta++;
1603         }
1604       }
1605       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1606     }
1607 
1608     /* put the value into Cd (diagonal part) */
1609     pnz = pd->i[i+1] - pd->i[i];
1610     pdJ = pd->j + pd->i[i];
1611     pA  = pd->a + pd->i[i];
1612     for (j=0; j<pnz; j++) {
1613       row = pdJ[j];
1614       cj  = bj + bi[row];
1615       ca  = ba + bi[row];
1616       /* perform sparse axpy */
1617       nexta  = 0;
1618       valtmp = pA[j];
1619       for (k=0; nexta<anz; k++) {
1620         if (cj[k] == adj[nexta]) {
1621           ca[k] += valtmp*ada[nexta];
1622           nexta++;
1623         }
1624       }
1625       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1626     }
1627   }
1628 
1629   /* 3) send and recv matrix values coa */
1630   /*------------------------------------*/
1631   buf_ri = merge->buf_ri;
1632   buf_rj = merge->buf_rj;
1633   len_s  = merge->len_s;
1634   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1635   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1636 
1637   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1638   for (proc=0,k=0; proc<size; proc++) {
1639     if (!len_s[proc]) continue;
1640     i    = merge->owners_co[proc];
1641     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1642     k++;
1643   }
1644   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1645   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1646 
1647   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1648   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1649   ierr = PetscFree(coa);CHKERRQ(ierr);
1650 
1651   /* 4) insert local Cseq and received values into Cmpi */
1652   /*----------------------------------------------------*/
1653   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1654   for (k=0; k<merge->nrecv; k++) {
1655     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1656     nrows       = *(buf_ri_k[k]);
1657     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1658     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1659   }
1660 
1661   for (i=0; i<cm; i++) {
1662     row  = owners[rank] + i; /* global row index of C_seq */
1663     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1664     ba_i = ba + bi[i];
1665     bnz  = bi[i+1] - bi[i];
1666     /* add received vals into ba */
1667     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1668       /* i-th row */
1669       if (i == *nextrow[k]) {
1670         cnz    = *(nextci[k]+1) - *nextci[k];
1671         cj     = buf_rj[k] + *(nextci[k]);
1672         ca     = abuf_r[k] + *(nextci[k]);
1673         nextcj = 0;
1674         for (j=0; nextcj<cnz; j++) {
1675           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1676             ba_i[j] += ca[nextcj++];
1677           }
1678         }
1679         nextrow[k]++; nextci[k]++;
1680         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1681       }
1682     }
1683     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1684   }
1685   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1686   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1687 
1688   ierr = PetscFree(ba);CHKERRQ(ierr);
1689   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1690   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1691   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1692   PetscFunctionReturn(0);
1693 }
1694 
1695 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P,Mat A,PetscReal fill,Mat *C)
1696 {
1697   PetscErrorCode      ierr;
1698   Mat                 Cmpi,A_loc,POt,PDt;
1699   Mat_PtAPMPI         *ptap;
1700   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1701   Mat_MPIAIJ          *p=(Mat_MPIAIJ*)P->data,*a=(Mat_MPIAIJ*)A->data,*c;
1702   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1703   PetscInt            nnz;
1704   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1705   PetscInt            am  =A->rmap->n,pn=P->cmap->n;
1706   MPI_Comm            comm;
1707   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1708   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1709   PetscInt            len,proc,*dnz,*onz,*owners;
1710   PetscInt            nzi,*bi,*bj;
1711   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1712   MPI_Request         *swaits,*rwaits;
1713   MPI_Status          *sstatus,rstatus;
1714   Mat_Merge_SeqsToMPI *merge;
1715   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1716   PetscReal           afill  =1.0,afill_tmp;
1717   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Armax;
1718   PetscScalar         *vals;
1719   Mat_SeqAIJ          *a_loc,*pdt,*pot;
1720   PetscTable          ta;
1721 
1722   PetscFunctionBegin;
1723   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1724   /* check if matrix local sizes are compatible */
1725   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend) SETERRQ4(comm,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, A (%D, %D) != P (%D,%D)",A->rmap->rstart,A->rmap->rend,P->rmap->rstart,P->rmap->rend);
1726 
1727   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1728   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1729 
1730   /* create struct Mat_PtAPMPI and attached it to C later */
1731   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1732 
1733   /* get A_loc by taking all local rows of A */
1734   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1735 
1736   ptap->A_loc = A_loc;
1737   a_loc       = (Mat_SeqAIJ*)(A_loc)->data;
1738   ai          = a_loc->i;
1739   aj          = a_loc->j;
1740 
1741   /* determine symbolic Co=(p->B)^T*A - send to others */
1742   /*----------------------------------------------------*/
1743   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1744   pdt  = (Mat_SeqAIJ*)PDt->data;
1745   pdti = pdt->i; pdtj = pdt->j;
1746 
1747   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1748   pot  = (Mat_SeqAIJ*)POt->data;
1749   poti = pot->i; potj = pot->j;
1750 
1751   /* then, compute symbolic Co = (p->B)^T*A */
1752   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1753                          >= (num of nonzero rows of C_seq) - pn */
1754   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
1755   coi[0] = 0;
1756 
1757   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1758   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(poti[pon],ai[am]));
1759   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1760   current_space = free_space;
1761 
1762   /* create and initialize a linked list */
1763   ierr = PetscTableCreate(A->cmap->n + a->B->cmap->N,aN,&ta);CHKERRQ(ierr);
1764   MatRowMergeMax_SeqAIJ(a_loc,am,ta);
1765   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
1766 
1767   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
1768 
1769   for (i=0; i<pon; i++) {
1770     pnz = poti[i+1] - poti[i];
1771     ptJ = potj + poti[i];
1772     for (j=0; j<pnz; j++) {
1773       row  = ptJ[j]; /* row of A_loc == col of Pot */
1774       anz  = ai[row+1] - ai[row];
1775       Jptr = aj + ai[row];
1776       /* add non-zero cols of AP into the sorted linked list lnk */
1777       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1778     }
1779     nnz = lnk[0];
1780 
1781     /* If free space is not available, double the total space in the list */
1782     if (current_space->local_remaining<nnz) {
1783       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1784       nspacedouble++;
1785     }
1786 
1787     /* Copy data into free space, and zero out denserows */
1788     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1789 
1790     current_space->array           += nnz;
1791     current_space->local_used      += nnz;
1792     current_space->local_remaining -= nnz;
1793 
1794     coi[i+1] = coi[i] + nnz;
1795   }
1796 
1797   ierr = PetscMalloc1(coi[pon]+1,&coj);CHKERRQ(ierr);
1798   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1799   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr); /* must destroy to get a new one for C */
1800 
1801   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1802   if (afill_tmp > afill) afill = afill_tmp;
1803 
1804   /* send j-array (coj) of Co to other processors */
1805   /*----------------------------------------------*/
1806   /* determine row ownership */
1807   ierr = PetscNew(&merge);CHKERRQ(ierr);
1808   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1809 
1810   merge->rowmap->n  = pn;
1811   merge->rowmap->bs = 1;
1812 
1813   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1814   owners = merge->rowmap->range;
1815 
1816   /* determine the number of messages to send, their lengths */
1817   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
1818   ierr = PetscMalloc1(size,&merge->len_s);CHKERRQ(ierr);
1819 
1820   len_s        = merge->len_s;
1821   merge->nsend = 0;
1822 
1823   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
1824   ierr = PetscMemzero(len_s,size*sizeof(PetscMPIInt));CHKERRQ(ierr);
1825 
1826   proc = 0;
1827   for (i=0; i<pon; i++) {
1828     while (prmap[i] >= owners[proc+1]) proc++;
1829     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1830     len_s[proc] += coi[i+1] - coi[i];
1831   }
1832 
1833   len          = 0; /* max length of buf_si[] */
1834   owners_co[0] = 0;
1835   for (proc=0; proc<size; proc++) {
1836     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1837     if (len_si[proc]) {
1838       merge->nsend++;
1839       len_si[proc] = 2*(len_si[proc] + 1);
1840       len         += len_si[proc];
1841     }
1842   }
1843 
1844   /* determine the number and length of messages to receive for coi and coj  */
1845   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1846   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1847 
1848   /* post the Irecv and Isend of coj */
1849   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1850   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1851   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
1852   for (proc=0, k=0; proc<size; proc++) {
1853     if (!len_s[proc]) continue;
1854     i    = owners_co[proc];
1855     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1856     k++;
1857   }
1858 
1859   /* receives and sends of coj are complete */
1860   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
1861   for (i=0; i<merge->nrecv; i++) {
1862     PetscMPIInt icompleted;
1863     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1864   }
1865   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1866   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1867 
1868   /* add received column indices into table to update Armax */
1869   /* Armax can be as large as aN if a P[row,:] is dense, see src/ksp/ksp/examples/tutorials/ex56.c! */
1870   for (k=0; k<merge->nrecv; k++) {/* k-th received message */
1871     Jptr = buf_rj[k];
1872     for (j=0; j<merge->len_r[k]; j++) {
1873       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
1874     }
1875   }
1876   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
1877   /* printf("Armax %d, an %d + Bn %d = %d, aN %d\n",Armax,A->cmap->n,a->B->cmap->N,A->cmap->n+a->B->cmap->N,aN); */
1878 
1879   /* send and recv coi */
1880   /*-------------------*/
1881   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1882   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1883   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1884   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1885   for (proc=0,k=0; proc<size; proc++) {
1886     if (!len_s[proc]) continue;
1887     /* form outgoing message for i-structure:
1888          buf_si[0]:                 nrows to be sent
1889                [1:nrows]:           row index (global)
1890                [nrows+1:2*nrows+1]: i-structure index
1891     */
1892     /*-------------------------------------------*/
1893     nrows       = len_si[proc]/2 - 1;
1894     buf_si_i    = buf_si + nrows+1;
1895     buf_si[0]   = nrows;
1896     buf_si_i[0] = 0;
1897     nrows       = 0;
1898     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1899       nzi               = coi[i+1] - coi[i];
1900       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1901       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1902       nrows++;
1903     }
1904     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1905     k++;
1906     buf_si += len_si[proc];
1907   }
1908   i = merge->nrecv;
1909   while (i--) {
1910     PetscMPIInt icompleted;
1911     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1912   }
1913   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1914   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1915   ierr = PetscFree(len_si);CHKERRQ(ierr);
1916   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1917   ierr = PetscFree(swaits);CHKERRQ(ierr);
1918   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1919   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1920 
1921   /* compute the local portion of C (mpi mat) */
1922   /*------------------------------------------*/
1923   /* allocate bi array and free space for accumulating nonzero column info */
1924   ierr  = PetscMalloc1(pn+1,&bi);CHKERRQ(ierr);
1925   bi[0] = 0;
1926 
1927   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
1928   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(pdti[pn],PetscIntSumTruncate(poti[pon],ai[am])));
1929   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1930   current_space = free_space;
1931 
1932   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1933   for (k=0; k<merge->nrecv; k++) {
1934     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1935     nrows       = *buf_ri_k[k];
1936     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1937     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th recieved i-structure  */
1938   }
1939 
1940   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
1941   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1942   rmax = 0;
1943   for (i=0; i<pn; i++) {
1944     /* add pdt[i,:]*AP into lnk */
1945     pnz = pdti[i+1] - pdti[i];
1946     ptJ = pdtj + pdti[i];
1947     for (j=0; j<pnz; j++) {
1948       row  = ptJ[j];  /* row of AP == col of Pt */
1949       anz  = ai[row+1] - ai[row];
1950       Jptr = aj + ai[row];
1951       /* add non-zero cols of AP into the sorted linked list lnk */
1952       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1953     }
1954 
1955     /* add received col data into lnk */
1956     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1957       if (i == *nextrow[k]) { /* i-th row */
1958         nzi  = *(nextci[k]+1) - *nextci[k];
1959         Jptr = buf_rj[k] + *nextci[k];
1960         ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
1961         nextrow[k]++; nextci[k]++;
1962       }
1963     }
1964     nnz = lnk[0];
1965 
1966     /* if free space is not available, make more free space */
1967     if (current_space->local_remaining<nnz) {
1968       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1969       nspacedouble++;
1970     }
1971     /* copy data into free space, then initialize lnk */
1972     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1973     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1974 
1975     current_space->array           += nnz;
1976     current_space->local_used      += nnz;
1977     current_space->local_remaining -= nnz;
1978 
1979     bi[i+1] = bi[i] + nnz;
1980     if (nnz > rmax) rmax = nnz;
1981   }
1982   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1983 
1984   ierr      = PetscMalloc1(bi[pn]+1,&bj);CHKERRQ(ierr);
1985   ierr      = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1986   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
1987   if (afill_tmp > afill) afill = afill_tmp;
1988   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
1989   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
1990 
1991   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1992   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1993 
1994   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
1995   /*----------------------------------------------------------------------------------*/
1996   ierr = PetscCalloc1(rmax+1,&vals);CHKERRQ(ierr);
1997 
1998   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1999   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
2000   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
2001   ierr = MatSetType(Cmpi,MATMPIAIJ);CHKERRQ(ierr);
2002   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
2003   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
2004   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
2005   for (i=0; i<pn; i++) {
2006     row  = i + rstart;
2007     nnz  = bi[i+1] - bi[i];
2008     Jptr = bj + bi[i];
2009     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,vals,INSERT_VALUES);CHKERRQ(ierr);
2010   }
2011   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2012   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2013   ierr = PetscFree(vals);CHKERRQ(ierr);
2014 
2015   merge->bi        = bi;
2016   merge->bj        = bj;
2017   merge->coi       = coi;
2018   merge->coj       = coj;
2019   merge->buf_ri    = buf_ri;
2020   merge->buf_rj    = buf_rj;
2021   merge->owners_co = owners_co;
2022 
2023   /* attach the supporting struct to Cmpi for reuse */
2024   c = (Mat_MPIAIJ*)Cmpi->data;
2025 
2026   c->ptap     = ptap;
2027   ptap->api   = NULL;
2028   ptap->apj   = NULL;
2029   ptap->merge = merge;
2030   ptap->apa   = NULL;
2031   ptap->destroy   = Cmpi->ops->destroy;
2032   ptap->duplicate = Cmpi->ops->duplicate;
2033 
2034   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ;
2035   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
2036   Cmpi->ops->duplicate               = MatDuplicate_MPIAIJ_MatPtAP;
2037 
2038   *C = Cmpi;
2039 #if defined(PETSC_USE_INFO)
2040   if (bi[pn] != 0) {
2041     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
2042     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
2043   } else {
2044     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
2045   }
2046 #endif
2047   PetscFunctionReturn(0);
2048 }
2049