xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision 1873fc9fedfc3bd8fc2f82ccd29ab8b44a80f1db)
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 #include <petsc/private/vecscatterimpl.h>
13 
14 #if defined(PETSC_HAVE_HYPRE)
15 PETSC_INTERN PetscErrorCode MatMatMultSymbolic_AIJ_AIJ_wHYPRE(Mat,Mat,PetscReal,Mat*);
16 #endif
17 
18 PETSC_INTERN PetscErrorCode MatMatMult_MPIAIJ_MPIAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill, Mat *C)
19 {
20   PetscErrorCode ierr;
21 #if defined(PETSC_HAVE_HYPRE)
22   const char     *algTypes[4] = {"scalable","nonscalable","seqmpi","hypre"};
23   PetscInt       nalg = 4;
24 #else
25   const char     *algTypes[3] = {"scalable","nonscalable","seqmpi"};
26   PetscInt       nalg = 3;
27 #endif
28   PetscInt       alg = 1; /* set nonscalable algorithm as default */
29   MPI_Comm       comm;
30   PetscBool      flg;
31 
32   PetscFunctionBegin;
33   if (scall == MAT_INITIAL_MATRIX) {
34     ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
35     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);
36 
37     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)A),((PetscObject)A)->prefix,"MatMatMult","Mat");CHKERRQ(ierr);
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     if (alg == 0 || alg == 1) {
79       Mat_MPIAIJ *c  = (Mat_MPIAIJ*)(*C)->data;
80       Mat_APMPI  *ap = c->ap;
81       ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)(*C)),((PetscObject)(*C))->prefix,"MatFreeIntermediateDataStructures","Mat");CHKERRQ(ierr);
82       ap->freestruct = PETSC_FALSE;
83       ierr = PetscOptionsBool("-mat_freeintermediatedatastructures","Free intermediate data structures", "MatFreeIntermediateDataStructures",ap->freestruct,&ap->freestruct, NULL);CHKERRQ(ierr);
84       ierr = PetscOptionsEnd();CHKERRQ(ierr);
85     }
86   }
87 
88   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
89   ierr = (*(*C)->ops->matmultnumeric)(A,B,*C);CHKERRQ(ierr);
90   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
91   PetscFunctionReturn(0);
92 }
93 
94 PetscErrorCode MatDestroy_MPIAIJ_MatMatMult(Mat A)
95 {
96   PetscErrorCode ierr;
97   Mat_MPIAIJ     *a    = (Mat_MPIAIJ*)A->data;
98   Mat_APMPI      *ptap = a->ap;
99 
100   PetscFunctionBegin;
101   ierr = PetscFree2(ptap->startsj_s,ptap->startsj_r);CHKERRQ(ierr);
102   ierr = PetscFree(ptap->bufa);CHKERRQ(ierr);
103   ierr = MatDestroy(&ptap->P_loc);CHKERRQ(ierr);
104   ierr = MatDestroy(&ptap->P_oth);CHKERRQ(ierr);
105   ierr = MatDestroy(&ptap->Pt);CHKERRQ(ierr);
106   ierr = PetscFree(ptap->api);CHKERRQ(ierr);
107   ierr = PetscFree(ptap->apj);CHKERRQ(ierr);
108   ierr = PetscFree(ptap->apa);CHKERRQ(ierr);
109   ierr = ptap->destroy(A);CHKERRQ(ierr);
110   ierr = PetscFree(ptap);CHKERRQ(ierr);
111   PetscFunctionReturn(0);
112 }
113 
114 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,Mat C)
115 {
116   PetscErrorCode ierr;
117   Mat_MPIAIJ     *a  =(Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
118   Mat_SeqAIJ     *ad =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
119   Mat_SeqAIJ     *cd =(Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
120   PetscScalar    *cda=cd->a,*coa=co->a;
121   Mat_SeqAIJ     *p_loc,*p_oth;
122   PetscScalar    *apa,*ca;
123   PetscInt       cm   =C->rmap->n;
124   Mat_APMPI      *ptap=c->ap;
125   PetscInt       *api,*apj,*apJ,i,k;
126   PetscInt       cstart=C->cmap->rstart;
127   PetscInt       cdnz,conz,k0,k1;
128   MPI_Comm       comm;
129   PetscMPIInt    size;
130 
131   PetscFunctionBegin;
132   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
133   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
134 
135   if (!ptap->P_oth && size>1) SETERRQ(comm,PETSC_ERR_ARG_WRONGSTATE,"AP cannot be reused. Do not call MatFreeIntermediateDataStructures() or use '-mat_freeintermediatedatastructures'");
136 
137   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
138   /*-----------------------------------------------------*/
139   /* update numerical values of P_oth and P_loc */
140   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
141   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
142 
143   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
144   /*----------------------------------------------------------*/
145   /* get data from symbolic products */
146   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
147   p_oth = NULL;
148   if (size >1) {
149     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
150   }
151 
152   /* get apa for storing dense row A[i,:]*P */
153   apa = ptap->apa;
154 
155   api = ptap->api;
156   apj = ptap->apj;
157   for (i=0; i<cm; i++) {
158     /* compute apa = A[i,:]*P */
159     AProw_nonscalable(i,ad,ao,p_loc,p_oth,apa);
160 
161     /* set values in C */
162     apJ  = apj + api[i];
163     cdnz = cd->i[i+1] - cd->i[i];
164     conz = co->i[i+1] - co->i[i];
165 
166     /* 1st off-diagoanl part of C */
167     ca = coa + co->i[i];
168     k  = 0;
169     for (k0=0; k0<conz; k0++) {
170       if (apJ[k] >= cstart) break;
171       ca[k0]      = apa[apJ[k]];
172       apa[apJ[k++]] = 0.0;
173     }
174 
175     /* diagonal part of C */
176     ca = cda + cd->i[i];
177     for (k1=0; k1<cdnz; k1++) {
178       ca[k1]      = apa[apJ[k]];
179       apa[apJ[k++]] = 0.0;
180     }
181 
182     /* 2nd off-diagoanl part of C */
183     ca = coa + co->i[i];
184     for (; k0<conz; k0++) {
185       ca[k0]      = apa[apJ[k]];
186       apa[apJ[k++]] = 0.0;
187     }
188   }
189   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
190   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
191 
192   if (ptap->freestruct) {
193     ierr = MatFreeIntermediateDataStructures(C);CHKERRQ(ierr);
194   }
195   PetscFunctionReturn(0);
196 }
197 
198 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,PetscReal fill,Mat *C)
199 {
200   PetscErrorCode     ierr;
201   MPI_Comm           comm;
202   PetscMPIInt        size;
203   Mat                Cmpi;
204   Mat_APMPI          *ptap;
205   PetscFreeSpaceList free_space=NULL,current_space=NULL;
206   Mat_MPIAIJ         *a        =(Mat_MPIAIJ*)A->data,*c;
207   Mat_SeqAIJ         *ad       =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
208   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
209   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
210   PetscInt           *lnk,i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi;
211   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
212   PetscBT            lnkbt;
213   PetscReal          afill;
214   MatType            mtype;
215 
216   PetscFunctionBegin;
217   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
218   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
219 
220   /* create struct Mat_APMPI and attached it to C later */
221   ierr = PetscNew(&ptap);CHKERRQ(ierr);
222 
223   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
224   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
225 
226   /* get P_loc by taking all local rows of P */
227   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
228 
229   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
230   pi_loc = p_loc->i; pj_loc = p_loc->j;
231   if (size > 1) {
232     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
233     pi_oth = p_oth->i; pj_oth = p_oth->j;
234   } else {
235     p_oth = NULL;
236     pi_oth = NULL; pj_oth = NULL;
237   }
238 
239   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
240   /*-------------------------------------------------------------------*/
241   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
242   ptap->api = api;
243   api[0]    = 0;
244 
245   /* create and initialize a linked list */
246   ierr = PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt);CHKERRQ(ierr);
247 
248   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
249   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space);CHKERRQ(ierr);
250   current_space = free_space;
251 
252   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
253   for (i=0; i<am; i++) {
254     /* diagonal portion of A */
255     nzi = adi[i+1] - adi[i];
256     for (j=0; j<nzi; j++) {
257       row  = *adj++;
258       pnz  = pi_loc[row+1] - pi_loc[row];
259       Jptr = pj_loc + pi_loc[row];
260       /* add non-zero cols of P into the sorted linked list lnk */
261       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
262     }
263     /* off-diagonal portion of A */
264     nzi = aoi[i+1] - aoi[i];
265     for (j=0; j<nzi; j++) {
266       row  = *aoj++;
267       pnz  = pi_oth[row+1] - pi_oth[row];
268       Jptr = pj_oth + pi_oth[row];
269       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
270     }
271 
272     apnz     = lnk[0];
273     api[i+1] = api[i] + apnz;
274 
275     /* if free space is not available, double the total space in the list */
276     if (current_space->local_remaining<apnz) {
277       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
278       nspacedouble++;
279     }
280 
281     /* Copy data into free space, then initialize lnk */
282     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
283     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
284 
285     current_space->array           += apnz;
286     current_space->local_used      += apnz;
287     current_space->local_remaining -= apnz;
288   }
289 
290   /* Allocate space for apj, initialize apj, and */
291   /* destroy list of free space and other temporary array(s) */
292   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
293   apj  = ptap->apj;
294   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
295   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
296 
297   /* malloc apa to store dense row A[i,:]*P */
298   ierr = PetscCalloc1(pN,&ptap->apa);CHKERRQ(ierr);
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 = MatGetType(A,&mtype);CHKERRQ(ierr);
307   ierr = MatSetType(Cmpi,mtype);CHKERRQ(ierr);
308   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
309 
310   ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(Cmpi, apj, api);CHKERRQ(ierr);
311   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
312   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
313   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
314 
315   ptap->destroy        = Cmpi->ops->destroy;
316   ptap->duplicate      = Cmpi->ops->duplicate;
317   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
318   Cmpi->ops->destroy   = MatDestroy_MPIAIJ_MatMatMult;
319   Cmpi->ops->freeintermediatedatastructures = MatFreeIntermediateDataStructures_MPIAIJ_AP;
320 
321   /* attach the supporting struct to Cmpi for reuse */
322   c       = (Mat_MPIAIJ*)Cmpi->data;
323   c->ap = ptap;
324 
325   *C = Cmpi;
326 
327   /* set MatInfo */
328   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
329   if (afill < 1.0) afill = 1.0;
330   Cmpi->info.mallocs           = nspacedouble;
331   Cmpi->info.fill_ratio_given  = fill;
332   Cmpi->info.fill_ratio_needed = afill;
333 
334 #if defined(PETSC_USE_INFO)
335   if (api[am]) {
336     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
337     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
338   } else {
339     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
340   }
341 #endif
342   PetscFunctionReturn(0);
343 }
344 
345 PETSC_INTERN PetscErrorCode MatMatMult_MPIAIJ_MPIDense(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
346 {
347   PetscErrorCode ierr;
348 
349   PetscFunctionBegin;
350   if (scall == MAT_INITIAL_MATRIX) {
351     *C = NULL;
352     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
353     ierr = MatMatMultSymbolic_MPIAIJ_MPIDense(A,B,fill,C);CHKERRQ(ierr);
354     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
355   }
356   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
357   ierr = MatMatMultNumeric_MPIAIJ_MPIDense(A,B,*C);CHKERRQ(ierr);
358   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
359   PetscFunctionReturn(0);
360 }
361 
362 typedef struct {
363   Mat          workB,Bb,Cb,workB1,Bb1,Cb1;
364   MPI_Request  *rwaits,*swaits;
365   PetscInt     numBb;  /* num of Bb matrices */
366   PetscInt     nsends,nrecvs;
367   MPI_Datatype *stype,*rtype;
368 } MPIAIJ_MPIDense;
369 
370 PetscErrorCode MatMPIAIJ_MPIDenseDestroy(void *ctx)
371 {
372   MPIAIJ_MPIDense *contents = (MPIAIJ_MPIDense*)ctx;
373   PetscErrorCode  ierr;
374   PetscInt        i;
375 
376   PetscFunctionBegin;
377   ierr = MatDestroy(&contents->workB);CHKERRQ(ierr);
378 
379   if (contents->numBb) {
380     ierr = MatDestroy(&contents->Bb);CHKERRQ(ierr);
381     ierr = MatDestroy(&contents->Cb);CHKERRQ(ierr);
382 
383     ierr = MatDestroy(&contents->workB1);CHKERRQ(ierr);
384     ierr = MatDestroy(&contents->Bb1);CHKERRQ(ierr);
385     ierr = MatDestroy(&contents->Cb1);CHKERRQ(ierr);
386   }
387   for (i=0; i<contents->nsends; i++) {
388     ierr = MPI_Type_free(&contents->stype[i]);CHKERRQ(ierr);
389   }
390   for (i=0; i<contents->nrecvs; i++) {
391     ierr = MPI_Type_free(&contents->rtype[i]);CHKERRQ(ierr);
392   }
393   ierr = PetscFree4(contents->stype,contents->rtype,contents->rwaits,contents->swaits);CHKERRQ(ierr);
394   ierr = PetscFree(contents);CHKERRQ(ierr);
395   PetscFunctionReturn(0);
396 }
397 
398 /*
399     This is a "dummy function" that handles the case where matrix C was created as a dense matrix
400   directly by the user and passed to MatMatMult() with the MAT_REUSE_MATRIX option
401 
402   It is the same as MatMatMultSymbolic_MPIAIJ_MPIDense() except does not create C
403 */
404 PETSC_INTERN PetscErrorCode MatMatMultNumeric_MPIDense(Mat A,Mat B,Mat C)
405 {
406   PetscBool      flg;
407   PetscErrorCode ierr;
408 
409   PetscFunctionBegin;
410   ierr = PetscObjectTypeCompare((PetscObject)A,MATNEST,&flg);CHKERRQ(ierr);
411   if (flg) {
412     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
413     ierr = MatMatMultSymbolic_Nest_Dense(A,B,PETSC_DEFAULT,&C);CHKERRQ(ierr);
414     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
415     C->ops->matmultnumeric = MatMatMultNumeric_Nest_Dense;
416   } else {
417     ierr = MatMatMultSymbolic_MPIAIJ_MPIDense(A,B,PETSC_DEFAULT,&C);CHKERRQ(ierr);
418   }
419   ierr = (*C->ops->matmultnumeric)(A,B,C);CHKERRQ(ierr);
420   PetscFunctionReturn(0);
421 }
422 
423 /*
424   Create Bb, Cb, Bb1 and Cb1 matrices to be used by MatMatMult_MPIAIJ_MPIDense().
425   These matrices are used as wrappers for sub-columns of B and C, thus their own matrix operations are not used.
426   Modified from MatCreateDense().
427 */
428 PETSC_STATIC_INLINE PetscErrorCode MatCreateSubMPIDense_private(MPI_Comm comm,PetscInt m,PetscInt n,PetscInt M,PetscInt N,PetscInt rbs,PetscInt cbs,PetscScalar *data,Mat *A)
429 {
430   PetscErrorCode ierr;
431 
432   PetscFunctionBegin;
433   ierr = MatCreate(comm,A);CHKERRQ(ierr);
434   ierr = MatSetSizes(*A,m,n,M,N);CHKERRQ(ierr);
435   ierr = MatSetBlockSizes(*A,rbs,cbs);CHKERRQ(ierr);
436   ierr = MatSetType(*A,MATMPIDENSE);CHKERRQ(ierr);
437   ierr = MatMPIDenseSetPreallocation(*A,data);CHKERRQ(ierr);
438   (*A)->assembled = PETSC_TRUE;
439   PetscFunctionReturn(0);
440 }
441 
442 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat A,Mat B,PetscReal fill,Mat *C)
443 {
444   PetscErrorCode  ierr;
445   Mat_MPIAIJ      *aij=(Mat_MPIAIJ*)A->data;
446   Mat_MPIDense    *b=(Mat_MPIDense*)B->data;
447   Mat_SeqDense    *bseq=(Mat_SeqDense*)(b->A)->data;
448   PetscInt        nz=aij->B->cmap->n,nsends,nrecvs,i,nrows_to,j,lda=bseq->lda;
449   PetscContainer  container;
450   MPIAIJ_MPIDense *contents;
451   VecScatter      ctx=aij->Mvctx;
452   PetscInt        Am=A->rmap->n,Bm=B->rmap->n,Bn=B->cmap->n,BN=B->cmap->N,Bbn,Bbn1,bs,nrows_from;
453   MPI_Comm        comm;
454   MPI_Datatype    type1,*stype,*rtype;
455   const PetscInt  *sindices,*sstarts,*rstarts;
456   PetscMPIInt     *disp;
457 
458   PetscFunctionBegin;
459   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
460   if (!(*C)) {
461     ierr = MatCreate(comm,C);CHKERRQ(ierr);
462     ierr = MatSetSizes(*C,Am,Bn,A->rmap->N,BN);CHKERRQ(ierr);
463     ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
464     ierr = MatSetType(*C,MATMPIDENSE);CHKERRQ(ierr);
465     ierr = MatMPIDenseSetPreallocation(*C,NULL);CHKERRQ(ierr);
466     ierr = MatAssemblyBegin(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
467     ierr = MatAssemblyEnd(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
468   } else {
469     /* Check matrix size */
470     if ((*C)->rmap->n != A->rmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Matrix local row dimensions are incompatible, %D != %D",(*C)->rmap->n,A->rmap->n);
471     if ((*C)->cmap->n != B->cmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Matrix local column dimensions are incompatible, %D != %D",(*C)->cmap->n,B->cmap->n);
472   }
473 
474   (*C)->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
475 
476   ierr = PetscNew(&contents);CHKERRQ(ierr);
477   contents->numBb = 0;
478 
479   ierr = VecScatterGetRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,NULL,NULL);CHKERRQ(ierr);
480   ierr = VecScatterGetRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,NULL,NULL);CHKERRQ(ierr);
481 
482   /* Create column block of B and C for memory scalability when BN is too large */
483   /* Estimate Bbn, column size of Bb */
484   if (nz) {
485     Bbn1 = 2*Am*BN/nz;
486   } else Bbn1 = BN;
487 
488   bs = PetscAbs(B->cmap->bs);
489   Bbn1 = Bbn1/bs *bs; /* Bbn1 is a multiple of bs */
490   if (Bbn1 > BN) Bbn1 = BN;
491   ierr = MPI_Allreduce(&Bbn1,&Bbn,1,MPIU_INT,MPI_MAX,comm);CHKERRQ(ierr);
492 
493   /* Enable runtime option for Bbn */
494   ierr = PetscOptionsBegin(comm,((PetscObject)*C)->prefix,"MatMatMult","Mat");CHKERRQ(ierr);
495   ierr = PetscOptionsInt("-matmatmult_Bbn","Number of columns in Bb","MatMatMult",Bbn,&Bbn,NULL);CHKERRQ(ierr);
496   if (Bbn > BN) SETERRQ2(comm,PETSC_ERR_ARG_SIZ,"Bbn=%D cannot be larger than %D, column size of B",Bbn,BN);
497   ierr = PetscOptionsEnd();CHKERRQ(ierr);
498 
499   if (Bbn < BN) {
500     contents->numBb = BN/Bbn;
501     Bbn1 = BN - contents->numBb*Bbn;
502   }
503 
504   if (contents->numBb) {
505     PetscScalar data[1]; /* fake array for Bb and Cb */
506     ierr = PetscInfo3(*C,"use Bb, BN=%D, Bbn=%D; numBb=%D\n",BN,Bbn,contents->numBb);CHKERRQ(ierr);
507     ierr = MatCreateSubMPIDense_private(comm,B->rmap->n,PETSC_DECIDE,A->rmap->N,Bbn,B->rmap->bs,B->cmap->bs,data,&contents->Bb);CHKERRQ(ierr);
508     ierr = MatCreateSubMPIDense_private(comm,Am,PETSC_DECIDE,A->rmap->N,Bbn,(*C)->rmap->bs,(*C)->cmap->bs,data,&contents->Cb);CHKERRQ(ierr);
509 
510     if (Bbn1) { /* Create Bb1 and Cb1 for the remaining columns */
511       ierr = PetscInfo2(*C,"use Bb1, BN=%D, Bbn1=%D\n",BN,Bbn1);CHKERRQ(ierr);
512       ierr = MatCreateSubMPIDense_private(comm,B->rmap->n,PETSC_DECIDE,A->rmap->N,Bbn1,B->rmap->bs,B->cmap->bs,data,&contents->Bb1);CHKERRQ(ierr);
513       ierr = MatCreateSubMPIDense_private(comm,Am,PETSC_DECIDE,A->rmap->N,Bbn1,(*C)->rmap->bs,(*C)->cmap->bs,data,&contents->Cb1);CHKERRQ(ierr);
514 
515       /* Create work matrix used to store off processor rows of B needed for local product */
516       ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,Bbn1,NULL,&contents->workB1);CHKERRQ(ierr);
517     }
518   }
519 
520   /* Create work matrix used to store off processor rows of B needed for local product */
521   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,Bbn,NULL,&contents->workB);CHKERRQ(ierr);
522 
523   /* Use MPI derived data type to reduce memory required by the send/recv buffers */
524   ierr = PetscMalloc4(nsends,&stype,nrecvs,&rtype,nrecvs,&contents->rwaits,nsends,&contents->swaits);CHKERRQ(ierr);
525   contents->stype  = stype;
526   contents->nsends = nsends;
527 
528   contents->rtype  = rtype;
529   contents->nrecvs = nrecvs;
530 
531   ierr = PetscMalloc1(Bm+1,&disp);CHKERRQ(ierr);
532   for (i=0; i<nsends; i++) {
533     nrows_to = sstarts[i+1]-sstarts[i];
534     for (j=0; j<nrows_to; j++){
535       disp[j] = sindices[sstarts[i]+j]; /* rowB to be sent */
536     }
537     ierr = MPI_Type_create_indexed_block(nrows_to,1,(const PetscMPIInt *)disp,MPIU_SCALAR,&type1);CHKERRQ(ierr);
538 
539     ierr = MPI_Type_create_resized(type1,0,lda*sizeof(PetscScalar),&stype[i]);CHKERRQ(ierr);
540     ierr = MPI_Type_commit(&stype[i]);CHKERRQ(ierr);
541     ierr = MPI_Type_free(&type1);CHKERRQ(ierr);
542   }
543 
544   for (i=0; i<nrecvs; i++) {
545     /* received values from a process form a (nrows_from x Bbn) row block in workB (column-wise) */
546     nrows_from = rstarts[i+1]-rstarts[i];
547     disp[0] = 0;
548     ierr = MPI_Type_create_indexed_block(1, nrows_from, (const PetscMPIInt *)disp, MPIU_SCALAR, &type1);CHKERRQ(ierr);
549     ierr = MPI_Type_create_resized(type1, 0, nz*sizeof(PetscScalar), &rtype[i]);CHKERRQ(ierr);
550     ierr = MPI_Type_commit(&rtype[i]);CHKERRQ(ierr);
551     ierr = MPI_Type_free(&type1);CHKERRQ(ierr);
552   }
553 
554   ierr = PetscFree(disp);CHKERRQ(ierr);
555   ierr = VecScatterRestoreRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,NULL,NULL);CHKERRQ(ierr);
556   ierr = VecScatterRestoreRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,NULL,NULL);CHKERRQ(ierr);
557 
558   ierr = PetscContainerCreate(comm,&container);CHKERRQ(ierr);
559   ierr = PetscContainerSetPointer(container,contents);CHKERRQ(ierr);
560   ierr = PetscContainerSetUserDestroy(container,MatMPIAIJ_MPIDenseDestroy);CHKERRQ(ierr);
561   ierr = PetscObjectCompose((PetscObject)(*C),"workB",(PetscObject)container);CHKERRQ(ierr);
562   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
563   PetscFunctionReturn(0);
564 }
565 
566 extern PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat,Mat,Mat);
567 /*
568     Performs an efficient scatter on the rows of B needed by this process; this is
569     a modification of the VecScatterBegin_() routines.
570 
571     Input: Bbidx = 0: B = Bb
572                  = 1: B = Bb1, see MatMatMultSymbolic_MPIAIJ_MPIDense()
573 */
574 PetscErrorCode MatMPIDenseScatter(Mat A,Mat B,PetscInt Bbidx,Mat C,Mat *outworkB)
575 {
576   Mat_MPIAIJ        *aij = (Mat_MPIAIJ*)A->data;
577   PetscErrorCode    ierr;
578   const PetscScalar *b;
579   PetscScalar       *rvalues;
580   VecScatter        ctx = aij->Mvctx;
581   const PetscInt    *sindices,*sstarts,*rstarts;
582   const PetscMPIInt *sprocs,*rprocs;
583   PetscInt          i,nsends,nrecvs,nrecvs2;
584   MPI_Request       *swaits,*rwaits;
585   MPI_Comm          comm;
586   PetscMPIInt       tag=((PetscObject)ctx)->tag,ncols=B->cmap->N,nrows=aij->B->cmap->n,imdex,nsends_mpi,nrecvs_mpi;
587   MPI_Status        status;
588   MPIAIJ_MPIDense   *contents;
589   PetscContainer    container;
590   Mat               workB;
591   MPI_Datatype      *stype,*rtype;
592 
593   PetscFunctionBegin;
594   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
595   ierr = PetscObjectQuery((PetscObject)C,"workB",(PetscObject*)&container);CHKERRQ(ierr);
596   if (!container) SETERRQ(comm,PETSC_ERR_PLIB,"Container does not exist");
597   ierr = PetscContainerGetPointer(container,(void**)&contents);CHKERRQ(ierr);
598 
599   ierr = VecScatterGetRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,&sprocs,NULL/*bs*/);CHKERRQ(ierr);
600   ierr = VecScatterGetRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,&rprocs,NULL/*bs*/);CHKERRQ(ierr);
601   ierr = PetscMPIIntCast(nsends,&nsends_mpi);CHKERRQ(ierr);
602   ierr = PetscMPIIntCast(nrecvs,&nrecvs_mpi);CHKERRQ(ierr);
603   if (Bbidx == 0) {
604     workB = *outworkB = contents->workB;
605   } else {
606     workB = *outworkB = contents->workB1;
607   }
608   if (nrows != workB->rmap->n) SETERRQ2(comm,PETSC_ERR_PLIB,"Number of rows of workB %D not equal to columns of aij->B %D",workB->cmap->n,nrows);
609   swaits  = contents->swaits;
610   rwaits  = contents->rwaits;
611 
612   ierr = MatDenseGetArrayRead(B,&b);CHKERRQ(ierr);
613   ierr = MatDenseGetArray(workB,&rvalues);CHKERRQ(ierr);
614 
615   /* Post recv, use MPI derived data type to save memory */
616   rtype = contents->rtype;
617   for (i=0; i<nrecvs; i++) {
618     ierr = MPI_Irecv(rvalues+(rstarts[i]-rstarts[0]),ncols,rtype[i],rprocs[i],tag,comm,rwaits+i);CHKERRQ(ierr);
619   }
620 
621   stype = contents->stype;
622   for (i=0; i<nsends; i++) {
623     ierr = MPI_Isend(b,ncols,stype[i],sprocs[i],tag,comm,swaits+i);CHKERRQ(ierr);
624   }
625 
626   nrecvs2 = nrecvs;
627   while (nrecvs2) {
628     ierr = MPI_Waitany(nrecvs_mpi,rwaits,&imdex,&status);CHKERRQ(ierr);
629     nrecvs2--;
630   }
631   if (nsends) {ierr = MPI_Waitall(nsends_mpi,swaits,MPI_STATUSES_IGNORE);CHKERRQ(ierr);}
632 
633   ierr = VecScatterRestoreRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,&sprocs,NULL);CHKERRQ(ierr);
634   ierr = VecScatterRestoreRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,&rprocs,NULL);CHKERRQ(ierr);
635   ierr = MatDenseRestoreArrayRead(B,&b);CHKERRQ(ierr);
636   ierr = MatDenseRestoreArray(workB,&rvalues);CHKERRQ(ierr);
637   ierr = MatAssemblyBegin(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
638   ierr = MatAssemblyEnd(workB,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
639   PetscFunctionReturn(0);
640 }
641 
642 /*
643   Compute Cb = A*Bb
644 */
645 PETSC_STATIC_INLINE PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense_private(Mat A,Mat Bb,PetscInt Bbidx,PetscInt start,Mat C,const PetscScalar *barray,PetscScalar *carray,Mat Cb)
646 {
647   PetscErrorCode  ierr;
648   PetscInt        start1;
649   Mat             workB;
650   Mat_MPIAIJ      *aij = (Mat_MPIAIJ*)A->data;
651   Mat_MPIDense    *cbdense = (Mat_MPIDense*)Cb->data;
652 
653   PetscFunctionBegin;
654   /* Place barray to Bb */
655   start1 = start*Bb->rmap->n;
656   ierr = MatDensePlaceArray(Bb,barray+start1);CHKERRQ(ierr);
657 
658   /* get off processor parts of Bb needed to complete Cb=A*Bb */
659   ierr = MatMPIDenseScatter(A,Bb,Bbidx,C,&workB);CHKERRQ(ierr);
660   ierr = MatDenseResetArray(Bb);CHKERRQ(ierr);
661 
662   /* off-diagonal block of A times nonlocal rows of Bb */
663   /* Place carray to Cb */
664   start1 = start*Cb->rmap->n;
665   ierr = MatDensePlaceArray(Cb,carray+start1);CHKERRQ(ierr);
666   ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cbdense->A);CHKERRQ(ierr);
667   ierr = MatDenseResetArray(Cb);CHKERRQ(ierr);
668   PetscFunctionReturn(0);
669 }
670 
671 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat A,Mat B,Mat C)
672 {
673   PetscErrorCode  ierr;
674   Mat_MPIAIJ      *aij    = (Mat_MPIAIJ*)A->data;
675   Mat_MPIDense    *bdense = (Mat_MPIDense*)B->data;
676   Mat_MPIDense    *cdense = (Mat_MPIDense*)C->data;
677   Mat             workB;
678   MPIAIJ_MPIDense *contents;
679   PetscContainer  container;
680   MPI_Comm        comm;
681   PetscInt        numBb;
682 
683   PetscFunctionBegin;
684   /* diagonal block of A times all local rows of B*/
685   ierr = MatMatMultNumeric_SeqAIJ_SeqDense(aij->A,bdense->A,cdense->A);CHKERRQ(ierr);
686 
687   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
688   ierr = PetscObjectQuery((PetscObject)C,"workB",(PetscObject*)&container);CHKERRQ(ierr);
689   if (!container) SETERRQ(comm,PETSC_ERR_PLIB,"Container does not exist");
690   ierr = PetscContainerGetPointer(container,(void**)&contents);CHKERRQ(ierr);
691   numBb = contents->numBb;
692 
693   if (!numBb) {
694     /* get off processor parts of B needed to complete C=A*B */
695     ierr = MatMPIDenseScatter(A,B,0,C,&workB);CHKERRQ(ierr);
696 
697     /* off-diagonal block of A times nonlocal rows of B */
698     ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A);CHKERRQ(ierr);
699 
700   } else {
701     const PetscScalar *barray;
702     PetscScalar       *carray;
703     Mat               Bb=contents->Bb,Cb=contents->Cb;
704     PetscInt          BbN=Bb->cmap->N,start,i;
705 
706     ierr = MatDenseGetArrayRead(B,&barray);CHKERRQ(ierr);
707     ierr = MatDenseGetArray(C,&carray);CHKERRQ(ierr);
708     for (i=0; i<numBb; i++) {
709       start = i*BbN;
710       ierr = MatMatMultNumeric_MPIAIJ_MPIDense_private(A,Bb,0,start,C,barray,carray,Cb);CHKERRQ(ierr);
711     }
712 
713     if (contents->Bb1) {
714       Bb = contents->Bb1; Cb = contents->Cb1;
715       start = i*BbN;
716       ierr = MatMatMultNumeric_MPIAIJ_MPIDense_private(A,Bb,1,start,C,barray,carray,Cb);CHKERRQ(ierr);
717     }
718     ierr = MatDenseRestoreArrayRead(B,&barray);CHKERRQ(ierr);
719     ierr = MatDenseRestoreArray(C,&carray);CHKERRQ(ierr);
720   }
721   PetscFunctionReturn(0);
722 }
723 
724 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
725 {
726   PetscErrorCode ierr;
727   Mat_MPIAIJ     *a   = (Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
728   Mat_SeqAIJ     *ad  = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
729   Mat_SeqAIJ     *cd  = (Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
730   PetscInt       *adi = ad->i,*adj,*aoi=ao->i,*aoj;
731   PetscScalar    *ada,*aoa,*cda=cd->a,*coa=co->a;
732   Mat_SeqAIJ     *p_loc,*p_oth;
733   PetscInt       *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
734   PetscScalar    *pa_loc,*pa_oth,*pa,valtmp,*ca;
735   PetscInt       cm    = C->rmap->n,anz,pnz;
736   Mat_APMPI      *ptap = c->ap;
737   PetscScalar    *apa_sparse;
738   PetscInt       *api,*apj,*apJ,i,j,k,row;
739   PetscInt       cstart = C->cmap->rstart;
740   PetscInt       cdnz,conz,k0,k1,nextp;
741   MPI_Comm       comm;
742   PetscMPIInt    size;
743 
744   PetscFunctionBegin;
745   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
746   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
747 
748   if (!ptap->P_oth && size>1) {
749     SETERRQ(comm,PETSC_ERR_ARG_WRONGSTATE,"AP cannot be reused. Do not call MatFreeIntermediateDataStructures() or use '-mat_freeintermediatedatastructures'");
750   }
751   apa_sparse = ptap->apa;
752 
753   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
754   /*-----------------------------------------------------*/
755   /* update numerical values of P_oth and P_loc */
756   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
757   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
758 
759   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
760   /*----------------------------------------------------------*/
761   /* get data from symbolic products */
762   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
763   pi_loc = p_loc->i; pj_loc = p_loc->j; pa_loc = p_loc->a;
764   if (size >1) {
765     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
766     pi_oth = p_oth->i; pj_oth = p_oth->j; pa_oth = p_oth->a;
767   } else {
768     p_oth = NULL; pi_oth = NULL; pj_oth = NULL; pa_oth = NULL;
769   }
770 
771   api = ptap->api;
772   apj = ptap->apj;
773   for (i=0; i<cm; i++) {
774     apJ = apj + api[i];
775 
776     /* diagonal portion of A */
777     anz = adi[i+1] - adi[i];
778     adj = ad->j + adi[i];
779     ada = ad->a + adi[i];
780     for (j=0; j<anz; j++) {
781       row = adj[j];
782       pnz = pi_loc[row+1] - pi_loc[row];
783       pj  = pj_loc + pi_loc[row];
784       pa  = pa_loc + pi_loc[row];
785       /* perform sparse axpy */
786       valtmp = ada[j];
787       nextp  = 0;
788       for (k=0; nextp<pnz; k++) {
789         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
790           apa_sparse[k] += valtmp*pa[nextp++];
791         }
792       }
793       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
794     }
795 
796     /* off-diagonal portion of A */
797     anz = aoi[i+1] - aoi[i];
798     aoj = ao->j + aoi[i];
799     aoa = ao->a + aoi[i];
800     for (j=0; j<anz; j++) {
801       row = aoj[j];
802       pnz = pi_oth[row+1] - pi_oth[row];
803       pj  = pj_oth + pi_oth[row];
804       pa  = pa_oth + pi_oth[row];
805       /* perform sparse axpy */
806       valtmp = aoa[j];
807       nextp  = 0;
808       for (k=0; nextp<pnz; k++) {
809         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
810           apa_sparse[k] += valtmp*pa[nextp++];
811         }
812       }
813       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
814     }
815 
816     /* set values in C */
817     cdnz = cd->i[i+1] - cd->i[i];
818     conz = co->i[i+1] - co->i[i];
819 
820     /* 1st off-diagoanl part of C */
821     ca = coa + co->i[i];
822     k  = 0;
823     for (k0=0; k0<conz; k0++) {
824       if (apJ[k] >= cstart) break;
825       ca[k0]        = apa_sparse[k];
826       apa_sparse[k] = 0.0;
827       k++;
828     }
829 
830     /* diagonal part of C */
831     ca = cda + cd->i[i];
832     for (k1=0; k1<cdnz; k1++) {
833       ca[k1]        = apa_sparse[k];
834       apa_sparse[k] = 0.0;
835       k++;
836     }
837 
838     /* 2nd off-diagoanl part of C */
839     ca = coa + co->i[i];
840     for (; k0<conz; k0++) {
841       ca[k0]        = apa_sparse[k];
842       apa_sparse[k] = 0.0;
843       k++;
844     }
845   }
846   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
847   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
848 
849   if (ptap->freestruct) {
850     ierr = MatFreeIntermediateDataStructures(C);CHKERRQ(ierr);
851   }
852   PetscFunctionReturn(0);
853 }
854 
855 /* same as MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(), except using LLCondensed to avoid O(BN) memory requirement */
856 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat *C)
857 {
858   PetscErrorCode     ierr;
859   MPI_Comm           comm;
860   PetscMPIInt        size;
861   Mat                Cmpi;
862   Mat_APMPI          *ptap;
863   PetscFreeSpaceList free_space = NULL,current_space=NULL;
864   Mat_MPIAIJ         *a         = (Mat_MPIAIJ*)A->data,*c;
865   Mat_SeqAIJ         *ad        = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
866   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
867   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
868   PetscInt           i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi,*lnk,apnz_max=0;
869   PetscInt           am=A->rmap->n,pn=P->cmap->n,pm=P->rmap->n,lsize=pn+20;
870   PetscReal          afill;
871   MatType            mtype;
872 
873   PetscFunctionBegin;
874   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
875   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
876 
877   /* create struct Mat_APMPI and attached it to C later */
878   ierr = PetscNew(&ptap);CHKERRQ(ierr);
879 
880   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
881   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
882 
883   /* get P_loc by taking all local rows of P */
884   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
885 
886   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
887   pi_loc = p_loc->i; pj_loc = p_loc->j;
888   if (size > 1) {
889     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
890     pi_oth = p_oth->i; pj_oth = p_oth->j;
891   } else {
892     p_oth  = NULL;
893     pi_oth = NULL; pj_oth = NULL;
894   }
895 
896   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
897   /*-------------------------------------------------------------------*/
898   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
899   ptap->api = api;
900   api[0]    = 0;
901 
902   ierr = PetscLLCondensedCreate_Scalable(lsize,&lnk);CHKERRQ(ierr);
903 
904   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
905   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space);CHKERRQ(ierr);
906   current_space = free_space;
907   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
908   for (i=0; i<am; i++) {
909     /* diagonal portion of A */
910     nzi = adi[i+1] - adi[i];
911     for (j=0; j<nzi; j++) {
912       row  = *adj++;
913       pnz  = pi_loc[row+1] - pi_loc[row];
914       Jptr = pj_loc + pi_loc[row];
915       /* Expand list if it is not long enough */
916       if (pnz+apnz_max > lsize) {
917         lsize = pnz+apnz_max;
918         ierr = PetscLLCondensedExpand_Scalable(lsize, &lnk);CHKERRQ(ierr);
919       }
920       /* add non-zero cols of P into the sorted linked list lnk */
921       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
922       apnz     = *lnk; /* The first element in the list is the number of items in the list */
923       api[i+1] = api[i] + apnz;
924       if (apnz > apnz_max) apnz_max = apnz;
925     }
926     /* off-diagonal portion of A */
927     nzi = aoi[i+1] - aoi[i];
928     for (j=0; j<nzi; j++) {
929       row  = *aoj++;
930       pnz  = pi_oth[row+1] - pi_oth[row];
931       Jptr = pj_oth + pi_oth[row];
932       /* Expand list if it is not long enough */
933       if (pnz+apnz_max > lsize) {
934         lsize = pnz + apnz_max;
935         ierr = PetscLLCondensedExpand_Scalable(lsize, &lnk);CHKERRQ(ierr);
936       }
937       /* add non-zero cols of P into the sorted linked list lnk */
938       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
939       apnz     = *lnk;  /* The first element in the list is the number of items in the list */
940       api[i+1] = api[i] + apnz;
941       if (apnz > apnz_max) apnz_max = apnz;
942     }
943     apnz     = *lnk;
944     api[i+1] = api[i] + apnz;
945     if (apnz > apnz_max) apnz_max = apnz;
946 
947     /* if free space is not available, double the total space in the list */
948     if (current_space->local_remaining<apnz) {
949       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
950       nspacedouble++;
951     }
952 
953     /* Copy data into free space, then initialize lnk */
954     ierr = PetscLLCondensedClean_Scalable(apnz,current_space->array,lnk);CHKERRQ(ierr);
955     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
956 
957     current_space->array           += apnz;
958     current_space->local_used      += apnz;
959     current_space->local_remaining -= apnz;
960   }
961 
962   /* Allocate space for apj, initialize apj, and */
963   /* destroy list of free space and other temporary array(s) */
964   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
965   apj  = ptap->apj;
966   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
967   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
968 
969   /* create and assemble symbolic parallel matrix Cmpi */
970   /*----------------------------------------------------*/
971   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
972   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
973   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
974   ierr = MatGetType(A,&mtype);CHKERRQ(ierr);
975   ierr = MatSetType(Cmpi,mtype);CHKERRQ(ierr);
976   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
977 
978   /* malloc apa for assembly Cmpi */
979   ierr = PetscCalloc1(apnz_max,&ptap->apa);CHKERRQ(ierr);
980 
981   ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(Cmpi, apj, api);CHKERRQ(ierr);
982   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
983   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
984   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
985 
986   ptap->destroy             = Cmpi->ops->destroy;
987   ptap->duplicate           = Cmpi->ops->duplicate;
988   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ;
989   Cmpi->ops->destroy        = MatDestroy_MPIAIJ_MatMatMult;
990   Cmpi->ops->freeintermediatedatastructures = MatFreeIntermediateDataStructures_MPIAIJ_AP;
991 
992   /* attach the supporting struct to Cmpi for reuse */
993   c       = (Mat_MPIAIJ*)Cmpi->data;
994   c->ap = ptap;
995   *C = Cmpi;
996 
997   /* set MatInfo */
998   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
999   if (afill < 1.0) afill = 1.0;
1000   Cmpi->info.mallocs           = nspacedouble;
1001   Cmpi->info.fill_ratio_given  = fill;
1002   Cmpi->info.fill_ratio_needed = afill;
1003 
1004 #if defined(PETSC_USE_INFO)
1005   if (api[am]) {
1006     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1007     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
1008   } else {
1009     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1010   }
1011 #endif
1012   PetscFunctionReturn(0);
1013 }
1014 
1015 /* This function is needed for the seqMPI matrix-matrix multiplication.  */
1016 /* Three input arrays are merged to one output array. The size of the    */
1017 /* output array is also output. Duplicate entries only show up once.     */
1018 static void Merge3SortedArrays(PetscInt  size1, PetscInt *in1,
1019                                PetscInt  size2, PetscInt *in2,
1020                                PetscInt  size3, PetscInt *in3,
1021                                PetscInt *size4, PetscInt *out)
1022 {
1023   int i = 0, j = 0, k = 0, l = 0;
1024 
1025   /* Traverse all three arrays */
1026   while (i<size1 && j<size2 && k<size3) {
1027     if (in1[i] < in2[j] && in1[i] < in3[k]) {
1028       out[l++] = in1[i++];
1029     }
1030     else if(in2[j] < in1[i] && in2[j] < in3[k]) {
1031       out[l++] = in2[j++];
1032     }
1033     else if(in3[k] < in1[i] && in3[k] < in2[j]) {
1034       out[l++] = in3[k++];
1035     }
1036     else if(in1[i] == in2[j] && in1[i] < in3[k]) {
1037       out[l++] = in1[i];
1038       i++, j++;
1039     }
1040     else if(in1[i] == in3[k] && in1[i] < in2[j]) {
1041       out[l++] = in1[i];
1042       i++, k++;
1043     }
1044     else if(in3[k] == in2[j] && in2[j] < in1[i])  {
1045       out[l++] = in2[j];
1046       k++, j++;
1047     }
1048     else if(in1[i] == in2[j] && in1[i] == in3[k]) {
1049       out[l++] = in1[i];
1050       i++, j++, k++;
1051     }
1052   }
1053 
1054   /* Traverse two remaining arrays */
1055   while (i<size1 && j<size2) {
1056     if (in1[i] < in2[j]) {
1057       out[l++] = in1[i++];
1058     }
1059     else if(in1[i] > in2[j]) {
1060       out[l++] = in2[j++];
1061     }
1062     else {
1063       out[l++] = in1[i];
1064       i++, j++;
1065     }
1066   }
1067 
1068   while (i<size1 && k<size3) {
1069     if (in1[i] < in3[k]) {
1070       out[l++] = in1[i++];
1071     }
1072     else if(in1[i] > in3[k]) {
1073       out[l++] = in3[k++];
1074     }
1075     else {
1076       out[l++] = in1[i];
1077       i++, k++;
1078     }
1079   }
1080 
1081   while (k<size3 && j<size2)  {
1082     if (in3[k] < in2[j]) {
1083       out[l++] = in3[k++];
1084     }
1085     else if(in3[k] > in2[j]) {
1086       out[l++] = in2[j++];
1087     }
1088     else {
1089       out[l++] = in3[k];
1090       k++, j++;
1091     }
1092   }
1093 
1094   /* Traverse one remaining array */
1095   while (i<size1) out[l++] = in1[i++];
1096   while (j<size2) out[l++] = in2[j++];
1097   while (k<size3) out[l++] = in3[k++];
1098 
1099   *size4 = l;
1100 }
1101 
1102 /* This matrix-matrix multiplication algorithm divides the multiplication into three multiplications and  */
1103 /* adds up the products. Two of these three multiplications are performed with existing (sequential)      */
1104 /* matrix-matrix multiplications.  */
1105 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_seqMPI(Mat A, Mat P, PetscReal fill, Mat *C)
1106 {
1107   PetscErrorCode     ierr;
1108   MPI_Comm           comm;
1109   PetscMPIInt        size;
1110   Mat                Cmpi;
1111   Mat_APMPI          *ptap;
1112   PetscFreeSpaceList free_space_diag=NULL, current_space=NULL;
1113   Mat_MPIAIJ         *a        =(Mat_MPIAIJ*)A->data;
1114   Mat_SeqAIJ         *ad       =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc;
1115   Mat_MPIAIJ         *p        =(Mat_MPIAIJ*)P->data;
1116   Mat_MPIAIJ         *c;
1117   Mat_SeqAIJ         *adpd_seq, *p_off, *aopoth_seq;
1118   PetscInt           adponz, adpdnz;
1119   PetscInt           *pi_loc,*dnz,*onz;
1120   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,rstart=A->rmap->rstart;
1121   PetscInt           *lnk,i, i1=0,pnz,row,*adpoi,*adpoj, *api, *adpoJ, *aopJ, *apJ,*Jptr, aopnz, nspacedouble=0,j,nzi,
1122                      *apj,apnz, *adpdi, *adpdj, *adpdJ, *poff_i, *poff_j, *j_temp, *aopothi, *aopothj;
1123   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n, p_colstart, p_colend;
1124   PetscBT            lnkbt;
1125   PetscReal          afill;
1126   PetscMPIInt        rank;
1127   Mat                adpd, aopoth;
1128   MatType            mtype;
1129   const char         *prefix;
1130 
1131   PetscFunctionBegin;
1132   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1133   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1134   ierr = MPI_Comm_rank(comm, &rank);CHKERRQ(ierr);
1135   ierr = MatGetOwnershipRangeColumn(P, &p_colstart, &p_colend); CHKERRQ(ierr);
1136 
1137   /* create struct Mat_APMPI and attached it to C later */
1138   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1139 
1140   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
1141   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
1142 
1143   /* get P_loc by taking all local rows of P */
1144   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
1145 
1146 
1147   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
1148   pi_loc = p_loc->i;
1149 
1150   /* Allocate memory for the i arrays of the matrices A*P, A_diag*P_off and A_offd * P */
1151   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
1152   ierr      = PetscMalloc1(am+2,&adpoi);CHKERRQ(ierr);
1153 
1154   adpoi[0]    = 0;
1155   ptap->api = api;
1156   api[0] = 0;
1157 
1158   /* create and initialize a linked list, will be used for both A_diag * P_loc_off and A_offd * P_oth */
1159   ierr = PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt);CHKERRQ(ierr);
1160   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
1161 
1162   /* Symbolic calc of A_loc_diag * P_loc_diag */
1163   ierr = MatGetOptionsPrefix(A,&prefix);CHKERRQ(ierr);
1164   ierr = MatSetOptionsPrefix(a->A,prefix);CHKERRQ(ierr);
1165   ierr = MatAppendOptionsPrefix(a->A,"inner_diag_");CHKERRQ(ierr);
1166   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(a->A, p->A, fill, &adpd);CHKERRQ(ierr);
1167   adpd_seq = (Mat_SeqAIJ*)((adpd)->data);
1168   adpdi = adpd_seq->i; adpdj = adpd_seq->j;
1169   p_off = (Mat_SeqAIJ*)((p->B)->data);
1170   poff_i = p_off->i; poff_j = p_off->j;
1171 
1172   /* j_temp stores indices of a result row before they are added to the linked list */
1173   ierr = PetscMalloc1(pN+2,&j_temp);CHKERRQ(ierr);
1174 
1175 
1176   /* Symbolic calc of the A_diag * p_loc_off */
1177   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
1178   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space_diag);CHKERRQ(ierr);
1179   current_space = free_space_diag;
1180 
1181   for (i=0; i<am; i++) {
1182     /* A_diag * P_loc_off */
1183     nzi = adi[i+1] - adi[i];
1184     for (j=0; j<nzi; j++) {
1185       row  = *adj++;
1186       pnz  = poff_i[row+1] - poff_i[row];
1187       Jptr = poff_j + poff_i[row];
1188       for(i1 = 0; i1 < pnz; i1++) {
1189         j_temp[i1] = p->garray[Jptr[i1]];
1190       }
1191       /* add non-zero cols of P into the sorted linked list lnk */
1192       ierr = PetscLLCondensedAddSorted(pnz,j_temp,lnk,lnkbt);CHKERRQ(ierr);
1193     }
1194 
1195     adponz     = lnk[0];
1196     adpoi[i+1] = adpoi[i] + adponz;
1197 
1198     /* if free space is not available, double the total space in the list */
1199     if (current_space->local_remaining<adponz) {
1200       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(adponz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1201       nspacedouble++;
1202     }
1203 
1204     /* Copy data into free space, then initialize lnk */
1205     ierr = PetscLLCondensedClean(pN,adponz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1206 
1207     current_space->array           += adponz;
1208     current_space->local_used      += adponz;
1209     current_space->local_remaining -= adponz;
1210   }
1211 
1212   /* Symbolic calc of A_off * P_oth */
1213   ierr = MatSetOptionsPrefix(a->B,prefix);CHKERRQ(ierr);
1214   ierr = MatAppendOptionsPrefix(a->B,"inner_offdiag_");CHKERRQ(ierr);
1215   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(a->B, ptap->P_oth, fill, &aopoth);CHKERRQ(ierr);
1216   aopoth_seq = (Mat_SeqAIJ*)((aopoth)->data);
1217   aopothi = aopoth_seq->i; aopothj = aopoth_seq->j;
1218 
1219   /* Allocate space for apj, adpj, aopj, ... */
1220   /* destroy lists of free space and other temporary array(s) */
1221 
1222   ierr = PetscMalloc1(aopothi[am] + adpoi[am] + adpdi[am]+2, &ptap->apj);CHKERRQ(ierr);
1223   ierr = PetscMalloc1(adpoi[am]+2, &adpoj);CHKERRQ(ierr);
1224 
1225   /* Copy from linked list to j-array */
1226   ierr = PetscFreeSpaceContiguous(&free_space_diag,adpoj);CHKERRQ(ierr);
1227   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1228 
1229   adpoJ = adpoj;
1230   adpdJ = adpdj;
1231   aopJ = aopothj;
1232   apj  = ptap->apj;
1233   apJ = apj; /* still empty */
1234 
1235   /* Merge j-arrays of A_off * P, A_diag * P_loc_off, and */
1236   /* A_diag * P_loc_diag to get A*P */
1237   for (i = 0; i < am; i++) {
1238     aopnz  =  aopothi[i+1] -  aopothi[i];
1239     adponz = adpoi[i+1] - adpoi[i];
1240     adpdnz = adpdi[i+1] - adpdi[i];
1241 
1242     /* Correct indices from A_diag*P_diag */
1243     for(i1 = 0; i1 < adpdnz; i1++) {
1244       adpdJ[i1] += p_colstart;
1245     }
1246     /* Merge j-arrays of A_diag * P_loc_off and A_diag * P_loc_diag and A_off * P_oth */
1247     Merge3SortedArrays(adponz, adpoJ, adpdnz, adpdJ, aopnz, aopJ, &apnz, apJ);
1248     ierr = MatPreallocateSet(i+rstart, apnz, apJ, dnz, onz); CHKERRQ(ierr);
1249 
1250     aopJ += aopnz;
1251     adpoJ += adponz;
1252     adpdJ += adpdnz;
1253     apJ += apnz;
1254     api[i+1] = api[i] + apnz;
1255   }
1256 
1257   /* malloc apa to store dense row A[i,:]*P */
1258   ierr = PetscCalloc1(pN+2,&ptap->apa);CHKERRQ(ierr);
1259 
1260   /* create and assemble symbolic parallel matrix Cmpi */
1261   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1262   ierr = MatSetSizes(Cmpi,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1263   ierr = MatSetBlockSizesFromMats(Cmpi,A,P);CHKERRQ(ierr);
1264   ierr = MatGetType(A,&mtype);CHKERRQ(ierr);
1265   ierr = MatSetType(Cmpi,mtype);CHKERRQ(ierr);
1266   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1267 
1268 
1269   ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(Cmpi, apj, api);CHKERRQ(ierr);
1270   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1271   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1272   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1273 
1274 
1275   ptap->destroy        = Cmpi->ops->destroy;
1276   ptap->duplicate      = Cmpi->ops->duplicate;
1277   Cmpi->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1278   Cmpi->ops->destroy   = MatDestroy_MPIAIJ_MatMatMult;
1279 
1280   /* attach the supporting struct to Cmpi for reuse */
1281   c       = (Mat_MPIAIJ*)Cmpi->data;
1282   c->ap = ptap;
1283   *C = Cmpi;
1284 
1285   /* set MatInfo */
1286   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
1287   if (afill < 1.0) afill = 1.0;
1288   Cmpi->info.mallocs           = nspacedouble;
1289   Cmpi->info.fill_ratio_given  = fill;
1290   Cmpi->info.fill_ratio_needed = afill;
1291 
1292 #if defined(PETSC_USE_INFO)
1293   if (api[am]) {
1294     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1295     ierr = PetscInfo1(Cmpi,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
1296   } else {
1297     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
1298   }
1299 #endif
1300 
1301   ierr = MatDestroy(&aopoth);CHKERRQ(ierr);
1302   ierr = MatDestroy(&adpd);CHKERRQ(ierr);
1303   ierr = PetscFree(j_temp);CHKERRQ(ierr);
1304   ierr = PetscFree(adpoj);CHKERRQ(ierr);
1305   ierr = PetscFree(adpoi);CHKERRQ(ierr);
1306   PetscFunctionReturn(0);
1307 }
1308 
1309 
1310 /*-------------------------------------------------------------------------*/
1311 PetscErrorCode MatTransposeMatMult_MPIAIJ_MPIAIJ(Mat P,Mat A,MatReuse scall,PetscReal fill,Mat *C)
1312 {
1313   PetscErrorCode ierr;
1314   const char     *algTypes[3] = {"scalable","nonscalable","matmatmult"};
1315   PetscInt       aN=A->cmap->N,alg=1; /* set default algorithm */
1316   PetscBool      flg;
1317 
1318   PetscFunctionBegin;
1319   if (scall == MAT_INITIAL_MATRIX) {
1320     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)A),((PetscObject)A)->prefix,"MatTransposeMatMult","Mat");CHKERRQ(ierr);
1321     ierr = PetscOptionsEList("-mattransposematmult_via","Algorithmic approach","MatTransposeMatMult",algTypes,3,algTypes[1],&alg,&flg);CHKERRQ(ierr);
1322     ierr = PetscOptionsEnd();CHKERRQ(ierr);
1323 
1324     ierr = PetscLogEventBegin(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
1325     switch (alg) {
1326     case 1:
1327       if (!flg && aN > 100000) { /* may switch to scalable algorithm as default */
1328         MatInfo     Ainfo,Pinfo;
1329         PetscInt    nz_local;
1330         PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
1331         MPI_Comm    comm;
1332 
1333         ierr = MatGetInfo(A,MAT_LOCAL,&Ainfo);CHKERRQ(ierr);
1334         ierr = MatGetInfo(P,MAT_LOCAL,&Pinfo);CHKERRQ(ierr);
1335         nz_local = (PetscInt)(Ainfo.nz_allocated + Pinfo.nz_allocated); /* estimated local nonzero entries */
1336 
1337         if (aN > fill*nz_local) alg_scalable_loc = PETSC_TRUE;
1338         ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1339         ierr = MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm);CHKERRQ(ierr);
1340 
1341         if (alg_scalable) {
1342           alg  = 0; /* scalable algorithm would slower than nonscalable algorithm */
1343           ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(P,A,fill,C);CHKERRQ(ierr);
1344           break;
1345         }
1346       }
1347       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(P,A,fill,C);CHKERRQ(ierr);
1348       break;
1349     case 2:
1350     {
1351       Mat         Pt;
1352       Mat_APMPI   *ptap;
1353       Mat_MPIAIJ  *c;
1354       ierr = MatTranspose(P,MAT_INITIAL_MATRIX,&Pt);CHKERRQ(ierr);
1355       ierr = MatMatMult(Pt,A,MAT_INITIAL_MATRIX,fill,C);CHKERRQ(ierr);
1356       c        = (Mat_MPIAIJ*)(*C)->data;
1357       ptap     = c->ap;
1358       if (ptap) {
1359        ptap->Pt = Pt;
1360        (*C)->ops->freeintermediatedatastructures = MatFreeIntermediateDataStructures_MPIAIJ_AP;
1361       }
1362       (*C)->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult;
1363       PetscFunctionReturn(0);
1364     }
1365       break;
1366     default: /* scalable algorithm */
1367       ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(P,A,fill,C);CHKERRQ(ierr);
1368       break;
1369     }
1370     ierr = PetscLogEventEnd(MAT_TransposeMatMultSymbolic,P,A,0,0);CHKERRQ(ierr);
1371 
1372     {
1373       Mat_MPIAIJ *c  = (Mat_MPIAIJ*)(*C)->data;
1374       Mat_APMPI  *ap = c->ap;
1375       ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)(*C)),((PetscObject)(*C))->prefix,"MatFreeIntermediateDataStructures","Mat");CHKERRQ(ierr);
1376       ap->freestruct = PETSC_FALSE;
1377       ierr = PetscOptionsBool("-mat_freeintermediatedatastructures","Free intermediate data structures", "MatFreeIntermediateDataStructures",ap->freestruct,&ap->freestruct, NULL);CHKERRQ(ierr);
1378       ierr = PetscOptionsEnd();CHKERRQ(ierr);
1379     }
1380   }
1381 
1382   ierr = PetscLogEventBegin(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
1383   ierr = (*(*C)->ops->mattransposemultnumeric)(P,A,*C);CHKERRQ(ierr);
1384   ierr = PetscLogEventEnd(MAT_TransposeMatMultNumeric,P,A,0,0);CHKERRQ(ierr);
1385   PetscFunctionReturn(0);
1386 }
1387 
1388 /* This routine only works when scall=MAT_REUSE_MATRIX! */
1389 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult(Mat P,Mat A,Mat C)
1390 {
1391   PetscErrorCode ierr;
1392   Mat_MPIAIJ     *c=(Mat_MPIAIJ*)C->data;
1393   Mat_APMPI      *ptap= c->ap;
1394   Mat            Pt;
1395 
1396   PetscFunctionBegin;
1397   if (!ptap->Pt) {
1398     MPI_Comm comm;
1399     ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1400     SETERRQ(comm,PETSC_ERR_ARG_WRONGSTATE,"PtA cannot be reused. Do not call MatFreeIntermediateDataStructures() or use '-mat_freeintermediatedatastructures'");
1401   }
1402 
1403   Pt=ptap->Pt;
1404   ierr = MatTranspose(P,MAT_REUSE_MATRIX,&Pt);CHKERRQ(ierr);
1405   ierr = MatMatMultNumeric(Pt,A,C);CHKERRQ(ierr);
1406 
1407   /* supporting struct ptap consumes almost same amount of memory as C=PtAP, release it if C will not be updated by A and P */
1408   if (ptap->freestruct) {
1409     ierr = MatFreeIntermediateDataStructures(C);CHKERRQ(ierr);
1410   }
1411   PetscFunctionReturn(0);
1412 }
1413 
1414 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1415 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,PetscReal fill,Mat *C)
1416 {
1417   PetscErrorCode      ierr;
1418   Mat_APMPI           *ptap;
1419   Mat_MPIAIJ          *p=(Mat_MPIAIJ*)P->data,*c;
1420   MPI_Comm            comm;
1421   PetscMPIInt         size,rank;
1422   Mat                 Cmpi;
1423   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1424   PetscInt            pn=P->cmap->n,aN=A->cmap->N,an=A->cmap->n;
1425   PetscInt            *lnk,i,k,nsend;
1426   PetscBT             lnkbt;
1427   PetscMPIInt         tagi,tagj,*len_si,*len_s,*len_ri,icompleted=0,nrecv;
1428   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1429   PetscInt            len,proc,*dnz,*onz,*owners,nzi;
1430   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1431   MPI_Request         *swaits,*rwaits;
1432   MPI_Status          *sstatus,rstatus;
1433   PetscLayout         rowmap;
1434   PetscInt            *owners_co,*coi,*coj;    /* i and j array of (p->B)^T*A*P - used in the communication */
1435   PetscMPIInt         *len_r,*id_r;    /* array of length of comm->size, store send/recv matrix values */
1436   PetscInt            *Jptr,*prmap=p->garray,con,j,Crmax;
1437   Mat_SeqAIJ          *a_loc,*c_loc,*c_oth;
1438   PetscTable          ta;
1439   MatType             mtype;
1440   const char          *prefix;
1441 
1442   PetscFunctionBegin;
1443   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1444   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1445   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1446 
1447   /* create symbolic parallel matrix Cmpi */
1448   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
1449   ierr = MatGetType(A,&mtype);CHKERRQ(ierr);
1450   ierr = MatSetType(Cmpi,mtype);CHKERRQ(ierr);
1451 
1452   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1453 
1454   /* create struct Mat_APMPI and attached it to C later */
1455   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1456   ptap->reuse = MAT_INITIAL_MATRIX;
1457 
1458   /* (0) compute Rd = Pd^T, Ro = Po^T  */
1459   /* --------------------------------- */
1460   ierr = MatTranspose_SeqAIJ(p->A,MAT_INITIAL_MATRIX,&ptap->Rd);CHKERRQ(ierr);
1461   ierr = MatTranspose_SeqAIJ(p->B,MAT_INITIAL_MATRIX,&ptap->Ro);CHKERRQ(ierr);
1462 
1463   /* (1) compute symbolic A_loc */
1464   /* ---------------------------*/
1465   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&ptap->A_loc);CHKERRQ(ierr);
1466 
1467   /* (2-1) compute symbolic C_oth = Ro*A_loc  */
1468   /* ------------------------------------ */
1469   ierr = MatGetOptionsPrefix(A,&prefix);CHKERRQ(ierr);
1470   ierr = MatSetOptionsPrefix(ptap->Ro,prefix);CHKERRQ(ierr);
1471   ierr = MatAppendOptionsPrefix(ptap->Ro,"inner_offdiag_");CHKERRQ(ierr);
1472   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Ro,ptap->A_loc,fill,&ptap->C_oth);CHKERRQ(ierr);
1473 
1474   /* (3) send coj of C_oth to other processors  */
1475   /* ------------------------------------------ */
1476   /* determine row ownership */
1477   ierr = PetscLayoutCreate(comm,&rowmap);CHKERRQ(ierr);
1478   rowmap->n  = pn;
1479   rowmap->bs = 1;
1480   ierr   = PetscLayoutSetUp(rowmap);CHKERRQ(ierr);
1481   owners = rowmap->range;
1482 
1483   /* determine the number of messages to send, their lengths */
1484   ierr = PetscMalloc4(size,&len_s,size,&len_si,size,&sstatus,size+2,&owners_co);CHKERRQ(ierr);
1485   ierr = PetscArrayzero(len_s,size);CHKERRQ(ierr);
1486   ierr = PetscArrayzero(len_si,size);CHKERRQ(ierr);
1487 
1488   c_oth = (Mat_SeqAIJ*)ptap->C_oth->data;
1489   coi   = c_oth->i; coj = c_oth->j;
1490   con   = ptap->C_oth->rmap->n;
1491   proc  = 0;
1492   for (i=0; i<con; i++) {
1493     while (prmap[i] >= owners[proc+1]) proc++;
1494     len_si[proc]++;               /* num of rows in Co(=Pt*A) to be sent to [proc] */
1495     len_s[proc] += coi[i+1] - coi[i]; /* num of nonzeros in Co to be sent to [proc] */
1496   }
1497 
1498   len          = 0; /* max length of buf_si[], see (4) */
1499   owners_co[0] = 0;
1500   nsend        = 0;
1501   for (proc=0; proc<size; proc++) {
1502     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1503     if (len_s[proc]) {
1504       nsend++;
1505       len_si[proc] = 2*(len_si[proc] + 1); /* length of buf_si to be sent to [proc] */
1506       len         += len_si[proc];
1507     }
1508   }
1509 
1510   /* determine the number and length of messages to receive for coi and coj  */
1511   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&nrecv);CHKERRQ(ierr);
1512   ierr = PetscGatherMessageLengths2(comm,nsend,nrecv,len_s,len_si,&id_r,&len_r,&len_ri);CHKERRQ(ierr);
1513 
1514   /* post the Irecv and Isend of coj */
1515   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1516   ierr = PetscPostIrecvInt(comm,tagj,nrecv,id_r,len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1517   ierr = PetscMalloc1(nsend+1,&swaits);CHKERRQ(ierr);
1518   for (proc=0, k=0; proc<size; proc++) {
1519     if (!len_s[proc]) continue;
1520     i    = owners_co[proc];
1521     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1522     k++;
1523   }
1524 
1525   /* (2-2) compute symbolic C_loc = Rd*A_loc */
1526   /* ---------------------------------------- */
1527   ierr = MatSetOptionsPrefix(ptap->Rd,prefix);CHKERRQ(ierr);
1528   ierr = MatAppendOptionsPrefix(ptap->Rd,"inner_diag_");CHKERRQ(ierr);
1529   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Rd,ptap->A_loc,fill,&ptap->C_loc);CHKERRQ(ierr);
1530   c_loc = (Mat_SeqAIJ*)ptap->C_loc->data;
1531 
1532   /* receives coj are complete */
1533   for (i=0; i<nrecv; i++) {
1534     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1535   }
1536   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1537   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
1538 
1539   /* add received column indices into ta to update Crmax */
1540   a_loc = (Mat_SeqAIJ*)(ptap->A_loc)->data;
1541 
1542   /* create and initialize a linked list */
1543   ierr = PetscTableCreate(an,aN,&ta);CHKERRQ(ierr); /* for compute Crmax */
1544   MatRowMergeMax_SeqAIJ(a_loc,ptap->A_loc->rmap->N,ta);
1545 
1546   for (k=0; k<nrecv; k++) {/* k-th received message */
1547     Jptr = buf_rj[k];
1548     for (j=0; j<len_r[k]; j++) {
1549       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
1550     }
1551   }
1552   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
1553   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
1554 
1555   /* (4) send and recv coi */
1556   /*-----------------------*/
1557   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1558   ierr   = PetscPostIrecvInt(comm,tagi,nrecv,id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1559   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1560   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1561   for (proc=0,k=0; proc<size; proc++) {
1562     if (!len_s[proc]) continue;
1563     /* form outgoing message for i-structure:
1564          buf_si[0]:                 nrows to be sent
1565                [1:nrows]:           row index (global)
1566                [nrows+1:2*nrows+1]: i-structure index
1567     */
1568     /*-------------------------------------------*/
1569     nrows       = len_si[proc]/2 - 1; /* num of rows in Co to be sent to [proc] */
1570     buf_si_i    = buf_si + nrows+1;
1571     buf_si[0]   = nrows;
1572     buf_si_i[0] = 0;
1573     nrows       = 0;
1574     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1575       nzi = coi[i+1] - coi[i];
1576       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1577       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1578       nrows++;
1579     }
1580     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1581     k++;
1582     buf_si += len_si[proc];
1583   }
1584   for (i=0; i<nrecv; i++) {
1585     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1586   }
1587   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1588   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
1589 
1590   ierr = PetscFree4(len_s,len_si,sstatus,owners_co);CHKERRQ(ierr);
1591   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1592   ierr = PetscFree(swaits);CHKERRQ(ierr);
1593   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1594 
1595   /* (5) compute the local portion of Cmpi      */
1596   /* ------------------------------------------ */
1597   /* set initial free space to be Crmax, sufficient for holding nozeros in each row of Cmpi */
1598   ierr          = PetscFreeSpaceGet(Crmax,&free_space);CHKERRQ(ierr);
1599   current_space = free_space;
1600 
1601   ierr = PetscMalloc3(nrecv,&buf_ri_k,nrecv,&nextrow,nrecv,&nextci);CHKERRQ(ierr);
1602   for (k=0; k<nrecv; k++) {
1603     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1604     nrows       = *buf_ri_k[k];
1605     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1606     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1607   }
1608 
1609   ierr = MatPreallocateInitialize(comm,pn,an,dnz,onz);CHKERRQ(ierr);
1610   ierr = PetscLLCondensedCreate(Crmax,aN,&lnk,&lnkbt);CHKERRQ(ierr);
1611   for (i=0; i<pn; i++) {
1612     /* add C_loc into Cmpi */
1613     nzi  = c_loc->i[i+1] - c_loc->i[i];
1614     Jptr = c_loc->j + c_loc->i[i];
1615     ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1616 
1617     /* add received col data into lnk */
1618     for (k=0; k<nrecv; k++) { /* k-th received message */
1619       if (i == *nextrow[k]) { /* i-th row */
1620         nzi  = *(nextci[k]+1) - *nextci[k];
1621         Jptr = buf_rj[k] + *nextci[k];
1622         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1623         nextrow[k]++; nextci[k]++;
1624       }
1625     }
1626     nzi = lnk[0];
1627 
1628     /* copy data into free space, then initialize lnk */
1629     ierr = PetscLLCondensedClean(aN,nzi,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1630     ierr = MatPreallocateSet(i+owners[rank],nzi,current_space->array,dnz,onz);CHKERRQ(ierr);
1631   }
1632   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1633   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1634   ierr = PetscFreeSpaceDestroy(free_space);CHKERRQ(ierr);
1635 
1636   /* local sizes and preallocation */
1637   ierr = MatSetSizes(Cmpi,pn,an,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1638   if (P->cmap->bs > 0) {ierr = PetscLayoutSetBlockSize(Cmpi->rmap,P->cmap->bs);CHKERRQ(ierr);}
1639   if (A->cmap->bs > 0) {ierr = PetscLayoutSetBlockSize(Cmpi->cmap,A->cmap->bs);CHKERRQ(ierr);}
1640   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
1641   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1642 
1643   /* members in merge */
1644   ierr = PetscFree(id_r);CHKERRQ(ierr);
1645   ierr = PetscFree(len_r);CHKERRQ(ierr);
1646   ierr = PetscFree(buf_ri[0]);CHKERRQ(ierr);
1647   ierr = PetscFree(buf_ri);CHKERRQ(ierr);
1648   ierr = PetscFree(buf_rj[0]);CHKERRQ(ierr);
1649   ierr = PetscFree(buf_rj);CHKERRQ(ierr);
1650   ierr = PetscLayoutDestroy(&rowmap);CHKERRQ(ierr);
1651 
1652   /* attach the supporting struct to Cmpi for reuse */
1653   c = (Mat_MPIAIJ*)Cmpi->data;
1654   c->ap         = ptap;
1655   ptap->destroy = Cmpi->ops->destroy;
1656 
1657   /* Cmpi is not ready for use - assembly will be done by MatPtAPNumeric() */
1658   Cmpi->assembled        = PETSC_FALSE;
1659   Cmpi->ops->destroy     = MatDestroy_MPIAIJ_PtAP;
1660   Cmpi->ops->freeintermediatedatastructures = MatFreeIntermediateDataStructures_MPIAIJ_AP;
1661 
1662   *C                     = Cmpi;
1663   PetscFunctionReturn(0);
1664 }
1665 
1666 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,Mat C)
1667 {
1668   PetscErrorCode    ierr;
1669   Mat_MPIAIJ        *p=(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1670   Mat_SeqAIJ        *c_seq;
1671   Mat_APMPI         *ptap = c->ap;
1672   Mat               A_loc,C_loc,C_oth;
1673   PetscInt          i,rstart,rend,cm,ncols,row;
1674   const PetscInt    *cols;
1675   const PetscScalar *vals;
1676 
1677   PetscFunctionBegin;
1678   if (!ptap->A_loc) {
1679     MPI_Comm comm;
1680     ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1681     SETERRQ(comm,PETSC_ERR_ARG_WRONGSTATE,"PtA cannot be reused. Do not call MatFreeIntermediateDataStructures() or use '-mat_freeintermediatedatastructures'");
1682   }
1683 
1684   ierr = MatZeroEntries(C);CHKERRQ(ierr);
1685 
1686   if (ptap->reuse == MAT_REUSE_MATRIX) {
1687     /* These matrices are obtained in MatTransposeMatMultSymbolic() */
1688     /* 1) get R = Pd^T, Ro = Po^T */
1689     /*----------------------------*/
1690     ierr = MatTranspose_SeqAIJ(p->A,MAT_REUSE_MATRIX,&ptap->Rd);CHKERRQ(ierr);
1691     ierr = MatTranspose_SeqAIJ(p->B,MAT_REUSE_MATRIX,&ptap->Ro);CHKERRQ(ierr);
1692 
1693     /* 2) compute numeric A_loc */
1694     /*--------------------------*/
1695     ierr = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&ptap->A_loc);CHKERRQ(ierr);
1696   }
1697 
1698   /* 3) C_loc = Rd*A_loc, C_oth = Ro*A_loc */
1699   A_loc = ptap->A_loc;
1700   ierr = ((ptap->C_loc)->ops->matmultnumeric)(ptap->Rd,A_loc,ptap->C_loc);CHKERRQ(ierr);
1701   ierr = ((ptap->C_oth)->ops->matmultnumeric)(ptap->Ro,A_loc,ptap->C_oth);CHKERRQ(ierr);
1702   C_loc = ptap->C_loc;
1703   C_oth = ptap->C_oth;
1704 
1705   /* add C_loc and Co to to C */
1706   ierr = MatGetOwnershipRange(C,&rstart,&rend);CHKERRQ(ierr);
1707 
1708   /* C_loc -> C */
1709   cm    = C_loc->rmap->N;
1710   c_seq = (Mat_SeqAIJ*)C_loc->data;
1711   cols = c_seq->j;
1712   vals = c_seq->a;
1713   for (i=0; i<cm; i++) {
1714     ncols = c_seq->i[i+1] - c_seq->i[i];
1715     row = rstart + i;
1716     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1717     cols += ncols; vals += ncols;
1718   }
1719 
1720   /* Co -> C, off-processor part */
1721   cm    = C_oth->rmap->N;
1722   c_seq = (Mat_SeqAIJ*)C_oth->data;
1723   cols  = c_seq->j;
1724   vals  = c_seq->a;
1725   for (i=0; i<cm; i++) {
1726     ncols = c_seq->i[i+1] - c_seq->i[i];
1727     row = p->garray[i];
1728     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1729     cols += ncols; vals += ncols;
1730   }
1731   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1732   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1733 
1734   ptap->reuse = MAT_REUSE_MATRIX;
1735 
1736   /* supporting struct ptap consumes almost same amount of memory as C=PtAP, release it if C will not be updated by A and P */
1737   if (ptap->freestruct) {
1738     ierr = MatFreeIntermediateDataStructures(C);CHKERRQ(ierr);
1739   }
1740   PetscFunctionReturn(0);
1741 }
1742 
1743 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ(Mat P,Mat A,Mat C)
1744 {
1745   PetscErrorCode      ierr;
1746   Mat_Merge_SeqsToMPI *merge;
1747   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data,*c=(Mat_MPIAIJ*)C->data;
1748   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
1749   Mat_APMPI           *ptap;
1750   PetscInt            *adj;
1751   PetscInt            i,j,k,anz,pnz,row,*cj,nexta;
1752   MatScalar           *ada,*ca,valtmp;
1753   PetscInt            am  =A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
1754   MPI_Comm            comm;
1755   PetscMPIInt         size,rank,taga,*len_s;
1756   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
1757   PetscInt            **buf_ri,**buf_rj;
1758   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
1759   MPI_Request         *s_waits,*r_waits;
1760   MPI_Status          *status;
1761   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
1762   PetscInt            *ai,*aj,*coi,*coj,*poJ,*pdJ;
1763   Mat                 A_loc;
1764   Mat_SeqAIJ          *a_loc;
1765 
1766   PetscFunctionBegin;
1767   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1768   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1769   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1770 
1771   ptap  = c->ap;
1772   if (!ptap->A_loc) SETERRQ(comm,PETSC_ERR_ARG_WRONGSTATE,"PtA cannot be reused. Do not call MatFreeIntermediateDataStructures() or use '-mat_freeintermediatedatastructures'");
1773   merge = ptap->merge;
1774 
1775   /* 2) compute numeric C_seq = P_loc^T*A_loc */
1776   /*------------------------------------------*/
1777   /* get data from symbolic products */
1778   coi    = merge->coi; coj = merge->coj;
1779   ierr   = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
1780   bi     = merge->bi; bj = merge->bj;
1781   owners = merge->rowmap->range;
1782   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
1783 
1784   /* get A_loc by taking all local rows of A */
1785   A_loc = ptap->A_loc;
1786   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
1787   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1788   ai    = a_loc->i;
1789   aj    = a_loc->j;
1790 
1791   for (i=0; i<am; i++) {
1792     anz = ai[i+1] - ai[i];
1793     adj = aj + ai[i];
1794     ada = a_loc->a + ai[i];
1795 
1796     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1797     /*-------------------------------------------------------------*/
1798     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1799     pnz = po->i[i+1] - po->i[i];
1800     poJ = po->j + po->i[i];
1801     pA  = po->a + po->i[i];
1802     for (j=0; j<pnz; j++) {
1803       row = poJ[j];
1804       cj  = coj + coi[row];
1805       ca  = coa + coi[row];
1806       /* perform sparse axpy */
1807       nexta  = 0;
1808       valtmp = pA[j];
1809       for (k=0; nexta<anz; k++) {
1810         if (cj[k] == adj[nexta]) {
1811           ca[k] += valtmp*ada[nexta];
1812           nexta++;
1813         }
1814       }
1815       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1816     }
1817 
1818     /* put the value into Cd (diagonal part) */
1819     pnz = pd->i[i+1] - pd->i[i];
1820     pdJ = pd->j + pd->i[i];
1821     pA  = pd->a + pd->i[i];
1822     for (j=0; j<pnz; j++) {
1823       row = pdJ[j];
1824       cj  = bj + bi[row];
1825       ca  = ba + bi[row];
1826       /* perform sparse axpy */
1827       nexta  = 0;
1828       valtmp = pA[j];
1829       for (k=0; nexta<anz; k++) {
1830         if (cj[k] == adj[nexta]) {
1831           ca[k] += valtmp*ada[nexta];
1832           nexta++;
1833         }
1834       }
1835       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1836     }
1837   }
1838 
1839   /* 3) send and recv matrix values coa */
1840   /*------------------------------------*/
1841   buf_ri = merge->buf_ri;
1842   buf_rj = merge->buf_rj;
1843   len_s  = merge->len_s;
1844   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1845   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1846 
1847   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1848   for (proc=0,k=0; proc<size; proc++) {
1849     if (!len_s[proc]) continue;
1850     i    = merge->owners_co[proc];
1851     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1852     k++;
1853   }
1854   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1855   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1856 
1857   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1858   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1859   ierr = PetscFree(coa);CHKERRQ(ierr);
1860 
1861   /* 4) insert local Cseq and received values into Cmpi */
1862   /*----------------------------------------------------*/
1863   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1864   for (k=0; k<merge->nrecv; k++) {
1865     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1866     nrows       = *(buf_ri_k[k]);
1867     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1868     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1869   }
1870 
1871   for (i=0; i<cm; i++) {
1872     row  = owners[rank] + i; /* global row index of C_seq */
1873     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1874     ba_i = ba + bi[i];
1875     bnz  = bi[i+1] - bi[i];
1876     /* add received vals into ba */
1877     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1878       /* i-th row */
1879       if (i == *nextrow[k]) {
1880         cnz    = *(nextci[k]+1) - *nextci[k];
1881         cj     = buf_rj[k] + *(nextci[k]);
1882         ca     = abuf_r[k] + *(nextci[k]);
1883         nextcj = 0;
1884         for (j=0; nextcj<cnz; j++) {
1885           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1886             ba_i[j] += ca[nextcj++];
1887           }
1888         }
1889         nextrow[k]++; nextci[k]++;
1890         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1891       }
1892     }
1893     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1894   }
1895   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1896   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1897 
1898   ierr = PetscFree(ba);CHKERRQ(ierr);
1899   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1900   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1901   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1902 
1903   if (ptap->freestruct) {
1904     ierr = MatFreeIntermediateDataStructures(C);CHKERRQ(ierr);
1905   }
1906   PetscFunctionReturn(0);
1907 }
1908 
1909 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P,Mat A,PetscReal fill,Mat *C)
1910 {
1911   PetscErrorCode      ierr;
1912   Mat                 Cmpi,A_loc,POt,PDt;
1913   Mat_APMPI           *ptap;
1914   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1915   Mat_MPIAIJ          *p=(Mat_MPIAIJ*)P->data,*a=(Mat_MPIAIJ*)A->data,*c;
1916   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1917   PetscInt            nnz;
1918   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1919   PetscInt            am  =A->rmap->n,pn=P->cmap->n;
1920   MPI_Comm            comm;
1921   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1922   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1923   PetscInt            len,proc,*dnz,*onz,*owners;
1924   PetscInt            nzi,*bi,*bj;
1925   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1926   MPI_Request         *swaits,*rwaits;
1927   MPI_Status          *sstatus,rstatus;
1928   Mat_Merge_SeqsToMPI *merge;
1929   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1930   PetscReal           afill  =1.0,afill_tmp;
1931   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Armax;
1932   Mat_SeqAIJ          *a_loc,*pdt,*pot;
1933   PetscTable          ta;
1934   MatType             mtype;
1935 
1936   PetscFunctionBegin;
1937   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1938   /* check if matrix local sizes are compatible */
1939   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);
1940 
1941   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1942   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1943 
1944   /* create struct Mat_APMPI and attached it to C later */
1945   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1946 
1947   /* get A_loc by taking all local rows of A */
1948   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1949 
1950   ptap->A_loc = A_loc;
1951   a_loc       = (Mat_SeqAIJ*)(A_loc)->data;
1952   ai          = a_loc->i;
1953   aj          = a_loc->j;
1954 
1955   /* determine symbolic Co=(p->B)^T*A - send to others */
1956   /*----------------------------------------------------*/
1957   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1958   pdt  = (Mat_SeqAIJ*)PDt->data;
1959   pdti = pdt->i; pdtj = pdt->j;
1960 
1961   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1962   pot  = (Mat_SeqAIJ*)POt->data;
1963   poti = pot->i; potj = pot->j;
1964 
1965   /* then, compute symbolic Co = (p->B)^T*A */
1966   pon    = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1967                          >= (num of nonzero rows of C_seq) - pn */
1968   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
1969   coi[0] = 0;
1970 
1971   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1972   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(poti[pon],ai[am]));
1973   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1974   current_space = free_space;
1975 
1976   /* create and initialize a linked list */
1977   ierr = PetscTableCreate(A->cmap->n + a->B->cmap->N,aN,&ta);CHKERRQ(ierr);
1978   MatRowMergeMax_SeqAIJ(a_loc,am,ta);
1979   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
1980 
1981   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
1982 
1983   for (i=0; i<pon; i++) {
1984     pnz = poti[i+1] - poti[i];
1985     ptJ = potj + poti[i];
1986     for (j=0; j<pnz; j++) {
1987       row  = ptJ[j]; /* row of A_loc == col of Pot */
1988       anz  = ai[row+1] - ai[row];
1989       Jptr = aj + ai[row];
1990       /* add non-zero cols of AP into the sorted linked list lnk */
1991       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1992     }
1993     nnz = lnk[0];
1994 
1995     /* If free space is not available, double the total space in the list */
1996     if (current_space->local_remaining<nnz) {
1997       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1998       nspacedouble++;
1999     }
2000 
2001     /* Copy data into free space, and zero out denserows */
2002     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
2003 
2004     current_space->array           += nnz;
2005     current_space->local_used      += nnz;
2006     current_space->local_remaining -= nnz;
2007 
2008     coi[i+1] = coi[i] + nnz;
2009   }
2010 
2011   ierr = PetscMalloc1(coi[pon]+1,&coj);CHKERRQ(ierr);
2012   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
2013   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr); /* must destroy to get a new one for C */
2014 
2015   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
2016   if (afill_tmp > afill) afill = afill_tmp;
2017 
2018   /* send j-array (coj) of Co to other processors */
2019   /*----------------------------------------------*/
2020   /* determine row ownership */
2021   ierr = PetscNew(&merge);CHKERRQ(ierr);
2022   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
2023 
2024   merge->rowmap->n  = pn;
2025   merge->rowmap->bs = 1;
2026 
2027   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
2028   owners = merge->rowmap->range;
2029 
2030   /* determine the number of messages to send, their lengths */
2031   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
2032   ierr = PetscCalloc1(size,&merge->len_s);CHKERRQ(ierr);
2033 
2034   len_s        = merge->len_s;
2035   merge->nsend = 0;
2036 
2037   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
2038 
2039   proc = 0;
2040   for (i=0; i<pon; i++) {
2041     while (prmap[i] >= owners[proc+1]) proc++;
2042     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
2043     len_s[proc] += coi[i+1] - coi[i];
2044   }
2045 
2046   len          = 0; /* max length of buf_si[] */
2047   owners_co[0] = 0;
2048   for (proc=0; proc<size; proc++) {
2049     owners_co[proc+1] = owners_co[proc] + len_si[proc];
2050     if (len_si[proc]) {
2051       merge->nsend++;
2052       len_si[proc] = 2*(len_si[proc] + 1);
2053       len         += len_si[proc];
2054     }
2055   }
2056 
2057   /* determine the number and length of messages to receive for coi and coj  */
2058   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
2059   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
2060 
2061   /* post the Irecv and Isend of coj */
2062   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
2063   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
2064   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
2065   for (proc=0, k=0; proc<size; proc++) {
2066     if (!len_s[proc]) continue;
2067     i    = owners_co[proc];
2068     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
2069     k++;
2070   }
2071 
2072   /* receives and sends of coj are complete */
2073   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
2074   for (i=0; i<merge->nrecv; i++) {
2075     PetscMPIInt icompleted;
2076     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
2077   }
2078   ierr = PetscFree(rwaits);CHKERRQ(ierr);
2079   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
2080 
2081   /* add received column indices into table to update Armax */
2082   /* Armax can be as large as aN if a P[row,:] is dense, see src/ksp/ksp/examples/tutorials/ex56.c! */
2083   for (k=0; k<merge->nrecv; k++) {/* k-th received message */
2084     Jptr = buf_rj[k];
2085     for (j=0; j<merge->len_r[k]; j++) {
2086       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
2087     }
2088   }
2089   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
2090   /* 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); */
2091 
2092   /* send and recv coi */
2093   /*-------------------*/
2094   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
2095   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
2096   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
2097   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
2098   for (proc=0,k=0; proc<size; proc++) {
2099     if (!len_s[proc]) continue;
2100     /* form outgoing message for i-structure:
2101          buf_si[0]:                 nrows to be sent
2102                [1:nrows]:           row index (global)
2103                [nrows+1:2*nrows+1]: i-structure index
2104     */
2105     /*-------------------------------------------*/
2106     nrows       = len_si[proc]/2 - 1;
2107     buf_si_i    = buf_si + nrows+1;
2108     buf_si[0]   = nrows;
2109     buf_si_i[0] = 0;
2110     nrows       = 0;
2111     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
2112       nzi               = coi[i+1] - coi[i];
2113       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
2114       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
2115       nrows++;
2116     }
2117     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
2118     k++;
2119     buf_si += len_si[proc];
2120   }
2121   i = merge->nrecv;
2122   while (i--) {
2123     PetscMPIInt icompleted;
2124     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
2125   }
2126   ierr = PetscFree(rwaits);CHKERRQ(ierr);
2127   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
2128   ierr = PetscFree(len_si);CHKERRQ(ierr);
2129   ierr = PetscFree(len_ri);CHKERRQ(ierr);
2130   ierr = PetscFree(swaits);CHKERRQ(ierr);
2131   ierr = PetscFree(sstatus);CHKERRQ(ierr);
2132   ierr = PetscFree(buf_s);CHKERRQ(ierr);
2133 
2134   /* compute the local portion of C (mpi mat) */
2135   /*------------------------------------------*/
2136   /* allocate bi array and free space for accumulating nonzero column info */
2137   ierr  = PetscMalloc1(pn+1,&bi);CHKERRQ(ierr);
2138   bi[0] = 0;
2139 
2140   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
2141   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(pdti[pn],PetscIntSumTruncate(poti[pon],ai[am])));
2142   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
2143   current_space = free_space;
2144 
2145   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
2146   for (k=0; k<merge->nrecv; k++) {
2147     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
2148     nrows       = *buf_ri_k[k];
2149     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
2150     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th recieved i-structure  */
2151   }
2152 
2153   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
2154   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
2155   rmax = 0;
2156   for (i=0; i<pn; i++) {
2157     /* add pdt[i,:]*AP into lnk */
2158     pnz = pdti[i+1] - pdti[i];
2159     ptJ = pdtj + pdti[i];
2160     for (j=0; j<pnz; j++) {
2161       row  = ptJ[j];  /* row of AP == col of Pt */
2162       anz  = ai[row+1] - ai[row];
2163       Jptr = aj + ai[row];
2164       /* add non-zero cols of AP into the sorted linked list lnk */
2165       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
2166     }
2167 
2168     /* add received col data into lnk */
2169     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
2170       if (i == *nextrow[k]) { /* i-th row */
2171         nzi  = *(nextci[k]+1) - *nextci[k];
2172         Jptr = buf_rj[k] + *nextci[k];
2173         ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
2174         nextrow[k]++; nextci[k]++;
2175       }
2176     }
2177     nnz = lnk[0];
2178 
2179     /* if free space is not available, make more free space */
2180     if (current_space->local_remaining<nnz) {
2181       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
2182       nspacedouble++;
2183     }
2184     /* copy data into free space, then initialize lnk */
2185     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
2186     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
2187 
2188     current_space->array           += nnz;
2189     current_space->local_used      += nnz;
2190     current_space->local_remaining -= nnz;
2191 
2192     bi[i+1] = bi[i] + nnz;
2193     if (nnz > rmax) rmax = nnz;
2194   }
2195   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
2196 
2197   ierr      = PetscMalloc1(bi[pn]+1,&bj);CHKERRQ(ierr);
2198   ierr      = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
2199   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
2200   if (afill_tmp > afill) afill = afill_tmp;
2201   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
2202   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
2203 
2204   ierr = MatDestroy(&POt);CHKERRQ(ierr);
2205   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
2206 
2207   /* create symbolic parallel matrix Cmpi - why cannot be assembled in Numeric part   */
2208   /*----------------------------------------------------------------------------------*/
2209   ierr = MatCreate(comm,&Cmpi);CHKERRQ(ierr);
2210   ierr = MatSetSizes(Cmpi,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
2211   ierr = MatSetBlockSizes(Cmpi,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
2212   ierr = MatGetType(A,&mtype);CHKERRQ(ierr);
2213   ierr = MatSetType(Cmpi,mtype);CHKERRQ(ierr);
2214   ierr = MatMPIAIJSetPreallocation(Cmpi,0,dnz,0,onz);CHKERRQ(ierr);
2215   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
2216   ierr = MatSetBlockSize(Cmpi,1);CHKERRQ(ierr);
2217   for (i=0; i<pn; i++) {
2218     row  = i + rstart;
2219     nnz  = bi[i+1] - bi[i];
2220     Jptr = bj + bi[i];
2221     ierr = MatSetValues(Cmpi,1,&row,nnz,Jptr,NULL,INSERT_VALUES);CHKERRQ(ierr);
2222   }
2223   ierr = MatAssemblyBegin(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2224   ierr = MatAssemblyEnd(Cmpi,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2225   merge->bi        = bi;
2226   merge->bj        = bj;
2227   merge->coi       = coi;
2228   merge->coj       = coj;
2229   merge->buf_ri    = buf_ri;
2230   merge->buf_rj    = buf_rj;
2231   merge->owners_co = owners_co;
2232 
2233   /* attach the supporting struct to Cmpi for reuse */
2234   c = (Mat_MPIAIJ*)Cmpi->data;
2235 
2236   c->ap       = ptap;
2237   ptap->api   = NULL;
2238   ptap->apj   = NULL;
2239   ptap->merge = merge;
2240   ptap->apa   = NULL;
2241   ptap->destroy   = Cmpi->ops->destroy;
2242   ptap->duplicate = Cmpi->ops->duplicate;
2243 
2244   Cmpi->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ;
2245   Cmpi->ops->destroy                 = MatDestroy_MPIAIJ_PtAP;
2246   Cmpi->ops->freeintermediatedatastructures = MatFreeIntermediateDataStructures_MPIAIJ_AP;
2247 
2248   *C = Cmpi;
2249 #if defined(PETSC_USE_INFO)
2250   if (bi[pn] != 0) {
2251     ierr = PetscInfo3(Cmpi,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
2252     ierr = PetscInfo1(Cmpi,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
2253   } else {
2254     ierr = PetscInfo(Cmpi,"Empty matrix product\n");CHKERRQ(ierr);
2255   }
2256 #endif
2257   PetscFunctionReturn(0);
2258 }
2259