xref: /petsc/src/mat/impls/aij/mpi/mpimatmatmult.c (revision bef158480efac06de457f7a665168877ab3c2fd7)
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 MatProductSymbolic_AB_MPIAIJ_MPIAIJ(Mat C)
19 {
20   PetscErrorCode      ierr;
21   Mat_Product         *product = C->product;
22   Mat                 A=product->A,B=product->B;
23   MatProductAlgorithm alg=product->alg;
24   PetscReal           fill=product->fill;
25   PetscBool           flg;
26 
27   PetscFunctionBegin;
28   /* scalable */
29   ierr = PetscStrcmp(alg,"scalable",&flg);CHKERRQ(ierr);
30   if (flg) {
31     ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ(A,B,fill,C);CHKERRQ(ierr);
32     PetscFunctionReturn(0);
33   }
34 
35   /* nonscalable */
36   ierr = PetscStrcmp(alg,"nonscalable",&flg);CHKERRQ(ierr);
37   if (flg) {
38     ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(A,B,fill,C);CHKERRQ(ierr);
39     PetscFunctionReturn(0);
40   }
41 
42   /* seqmpi */
43   ierr = PetscStrcmp(alg,"seqmpi",&flg);CHKERRQ(ierr);
44   if (flg) {
45     ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ_seqMPI(A,B,fill,C);CHKERRQ(ierr);
46     PetscFunctionReturn(0);
47   }
48 
49 #if defined(PETSC_HAVE_HYPRE)
50   ierr = PetscStrcmp(alg,"hypre",&flg);CHKERRQ(ierr);
51   if (flg) {
52     ierr = MatMatMultSymbolic_AIJ_AIJ_wHYPRE(A,B,fill,C);CHKERRQ(ierr);
53     PetscFunctionReturn(0);
54   }
55 #endif
56   SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_SUP,"Mat Product Algorithm is not supported");
57 }
58 
59 PetscErrorCode MatDestroy_MPIAIJ_MatMatMult(void *data)
60 {
61   PetscErrorCode ierr;
62   Mat_APMPI      *ptap = (Mat_APMPI*)data;
63 
64   PetscFunctionBegin;
65   ierr = PetscFree2(ptap->startsj_s,ptap->startsj_r);CHKERRQ(ierr);
66   ierr = PetscFree(ptap->bufa);CHKERRQ(ierr);
67   ierr = MatDestroy(&ptap->P_loc);CHKERRQ(ierr);
68   ierr = MatDestroy(&ptap->P_oth);CHKERRQ(ierr);
69   ierr = MatDestroy(&ptap->Pt);CHKERRQ(ierr);
70   ierr = PetscFree(ptap->api);CHKERRQ(ierr);
71   ierr = PetscFree(ptap->apj);CHKERRQ(ierr);
72   ierr = PetscFree(ptap->apa);CHKERRQ(ierr);
73   ierr = PetscFree(ptap);CHKERRQ(ierr);
74   PetscFunctionReturn(0);
75 }
76 
77 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,Mat C)
78 {
79   PetscErrorCode ierr;
80   Mat_MPIAIJ     *a  =(Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
81   Mat_SeqAIJ     *ad =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
82   Mat_SeqAIJ     *cd =(Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
83   PetscScalar    *cda=cd->a,*coa=co->a;
84   Mat_SeqAIJ     *p_loc,*p_oth;
85   PetscScalar    *apa,*ca;
86   PetscInt       cm =C->rmap->n;
87   Mat_APMPI      *ptap;
88   PetscInt       *api,*apj,*apJ,i,k;
89   PetscInt       cstart=C->cmap->rstart;
90   PetscInt       cdnz,conz,k0,k1;
91   MPI_Comm       comm;
92   PetscMPIInt    size;
93 
94   PetscFunctionBegin;
95   MatCheckProduct(C,3);
96   ptap = (Mat_APMPI*)C->product->data;
97   if (!ptap) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be computed. Missing data");
98   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
99   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
100 
101   if (!ptap->P_oth && size>1) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"AP cannot be reused. Do not call MatProductClear()");
102 
103   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
104   /*-----------------------------------------------------*/
105   /* update numerical values of P_oth and P_loc */
106   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
107   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
108 
109   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
110   /*----------------------------------------------------------*/
111   /* get data from symbolic products */
112   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
113   p_oth = NULL;
114   if (size >1) {
115     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
116   }
117 
118   /* get apa for storing dense row A[i,:]*P */
119   apa = ptap->apa;
120 
121   api = ptap->api;
122   apj = ptap->apj;
123   for (i=0; i<cm; i++) {
124     /* compute apa = A[i,:]*P */
125     AProw_nonscalable(i,ad,ao,p_loc,p_oth,apa);
126 
127     /* set values in C */
128     apJ  = apj + api[i];
129     cdnz = cd->i[i+1] - cd->i[i];
130     conz = co->i[i+1] - co->i[i];
131 
132     /* 1st off-diagonal part of C */
133     ca = coa + co->i[i];
134     k  = 0;
135     for (k0=0; k0<conz; k0++) {
136       if (apJ[k] >= cstart) break;
137       ca[k0]      = apa[apJ[k]];
138       apa[apJ[k++]] = 0.0;
139     }
140 
141     /* diagonal part of C */
142     ca = cda + cd->i[i];
143     for (k1=0; k1<cdnz; k1++) {
144       ca[k1]      = apa[apJ[k]];
145       apa[apJ[k++]] = 0.0;
146     }
147 
148     /* 2nd off-diagonal part of C */
149     ca = coa + co->i[i];
150     for (; k0<conz; k0++) {
151       ca[k0]      = apa[apJ[k]];
152       apa[apJ[k++]] = 0.0;
153     }
154   }
155   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
156   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
157   PetscFunctionReturn(0);
158 }
159 
160 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat A,Mat P,PetscReal fill,Mat C)
161 {
162   PetscErrorCode     ierr;
163   MPI_Comm           comm;
164   PetscMPIInt        size;
165   Mat_APMPI          *ptap;
166   PetscFreeSpaceList free_space=NULL,current_space=NULL;
167   Mat_MPIAIJ         *a=(Mat_MPIAIJ*)A->data;
168   Mat_SeqAIJ         *ad=(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
169   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
170   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
171   PetscInt           *lnk,i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi;
172   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n;
173   PetscBT            lnkbt;
174   PetscReal          afill;
175   MatType            mtype;
176 
177   PetscFunctionBegin;
178   MatCheckProduct(C,4);
179   if (C->product->data) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Extra product struct not empty");
180   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
181   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
182 
183   /* create struct Mat_APMPI and attached it to C later */
184   ierr = PetscNew(&ptap);CHKERRQ(ierr);
185 
186   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
187   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
188 
189   /* get P_loc by taking all local rows of P */
190   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
191 
192   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
193   pi_loc = p_loc->i; pj_loc = p_loc->j;
194   if (size > 1) {
195     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
196     pi_oth = p_oth->i; pj_oth = p_oth->j;
197   } else {
198     p_oth = NULL;
199     pi_oth = NULL; pj_oth = NULL;
200   }
201 
202   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
203   /*-------------------------------------------------------------------*/
204   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
205   ptap->api = api;
206   api[0]    = 0;
207 
208   /* create and initialize a linked list */
209   ierr = PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt);CHKERRQ(ierr);
210 
211   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
212   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space);CHKERRQ(ierr);
213   current_space = free_space;
214 
215   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
216   for (i=0; i<am; i++) {
217     /* diagonal portion of A */
218     nzi = adi[i+1] - adi[i];
219     for (j=0; j<nzi; j++) {
220       row  = *adj++;
221       pnz  = pi_loc[row+1] - pi_loc[row];
222       Jptr = pj_loc + pi_loc[row];
223       /* add non-zero cols of P into the sorted linked list lnk */
224       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
225     }
226     /* off-diagonal portion of A */
227     nzi = aoi[i+1] - aoi[i];
228     for (j=0; j<nzi; j++) {
229       row  = *aoj++;
230       pnz  = pi_oth[row+1] - pi_oth[row];
231       Jptr = pj_oth + pi_oth[row];
232       ierr = PetscLLCondensedAddSorted(pnz,Jptr,lnk,lnkbt);CHKERRQ(ierr);
233     }
234 
235     apnz     = lnk[0];
236     api[i+1] = api[i] + apnz;
237 
238     /* if free space is not available, double the total space in the list */
239     if (current_space->local_remaining<apnz) {
240       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
241       nspacedouble++;
242     }
243 
244     /* Copy data into free space, then initialize lnk */
245     ierr = PetscLLCondensedClean(pN,apnz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
246     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
247 
248     current_space->array           += apnz;
249     current_space->local_used      += apnz;
250     current_space->local_remaining -= apnz;
251   }
252 
253   /* Allocate space for apj, initialize apj, and */
254   /* destroy list of free space and other temporary array(s) */
255   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
256   apj  = ptap->apj;
257   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
258   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
259 
260   /* malloc apa to store dense row A[i,:]*P */
261   ierr = PetscCalloc1(pN,&ptap->apa);CHKERRQ(ierr);
262 
263   /* set and assemble symbolic parallel matrix C */
264   /*---------------------------------------------*/
265   ierr = MatSetSizes(C,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
266   ierr = MatSetBlockSizesFromMats(C,A,P);CHKERRQ(ierr);
267 
268   ierr = MatGetType(A,&mtype);CHKERRQ(ierr);
269   ierr = MatSetType(C,mtype);CHKERRQ(ierr);
270   ierr = MatMPIAIJSetPreallocation(C,0,dnz,0,onz);CHKERRQ(ierr);
271 
272   ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(C, apj, api);CHKERRQ(ierr);
273   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
274   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
275   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
276 
277   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
278   C->ops->productnumeric = MatProductNumeric_AB;
279 
280   /* attach the supporting struct to C for reuse */
281   C->product->data    = ptap;
282   C->product->destroy = MatDestroy_MPIAIJ_MatMatMult;
283 
284   /* set MatInfo */
285   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
286   if (afill < 1.0) afill = 1.0;
287   C->info.mallocs           = nspacedouble;
288   C->info.fill_ratio_given  = fill;
289   C->info.fill_ratio_needed = afill;
290 
291 #if defined(PETSC_USE_INFO)
292   if (api[am]) {
293     ierr = PetscInfo3(C,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
294     ierr = PetscInfo1(C,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
295   } else {
296     ierr = PetscInfo(C,"Empty matrix product\n");CHKERRQ(ierr);
297   }
298 #endif
299   PetscFunctionReturn(0);
300 }
301 
302 /* ------------------------------------------------------- */
303 static PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat,Mat,PetscReal,Mat);
304 static PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat,Mat,Mat);
305 
306 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_MPIDense_AB(Mat C)
307 {
308   Mat_Product *product = C->product;
309   Mat         A = product->A,B=product->B;
310 
311   PetscFunctionBegin;
312   if (A->cmap->rstart != B->rmap->rstart || A->cmap->rend != B->rmap->rend)
313     SETERRQ4(PETSC_COMM_SELF,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);
314 
315   C->ops->matmultsymbolic = MatMatMultSymbolic_MPIAIJ_MPIDense;
316   C->ops->productsymbolic = MatProductSymbolic_AB;
317   PetscFunctionReturn(0);
318 }
319 /* -------------------------------------------------------------------- */
320 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_MPIDense_AtB(Mat C)
321 {
322   Mat_Product *product = C->product;
323   Mat         A = product->A,B=product->B;
324 
325   PetscFunctionBegin;
326   if (A->rmap->rstart != B->rmap->rstart || A->rmap->rend != B->rmap->rend)
327     SETERRQ4(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, (%D, %D) != (%D,%D)",A->rmap->rstart,A->rmap->rend,B->rmap->rstart,B->rmap->rend);
328 
329   C->ops->transposematmultsymbolic = MatTransposeMatMultSymbolic_MPIAIJ_MPIDense;
330   C->ops->productsymbolic          = MatProductSymbolic_AtB;
331   PetscFunctionReturn(0);
332 }
333 
334 /* --------------------------------------------------------------------- */
335 PETSC_INTERN PetscErrorCode MatProductSetFromOptions_MPIAIJ_MPIDense(Mat C)
336 {
337   PetscErrorCode ierr;
338   Mat_Product    *product = C->product;
339 
340   PetscFunctionBegin;
341   switch (product->type) {
342   case MATPRODUCT_AB:
343     ierr = MatProductSetFromOptions_MPIAIJ_MPIDense_AB(C);CHKERRQ(ierr);
344     break;
345   case MATPRODUCT_AtB:
346     ierr = MatProductSetFromOptions_MPIAIJ_MPIDense_AtB(C);CHKERRQ(ierr);
347     break;
348   default:
349     break;
350   }
351   PetscFunctionReturn(0);
352 }
353 /* ------------------------------------------------------- */
354 
355 typedef struct {
356   Mat          workB,workB1;
357   MPI_Request  *rwaits,*swaits;
358   PetscInt     nsends,nrecvs;
359   MPI_Datatype *stype,*rtype;
360   PetscInt     blda;
361 } MPIAIJ_MPIDense;
362 
363 PetscErrorCode MatMPIAIJ_MPIDenseDestroy(void *ctx)
364 {
365   MPIAIJ_MPIDense *contents = (MPIAIJ_MPIDense*)ctx;
366   PetscErrorCode  ierr;
367   PetscInt        i;
368 
369   PetscFunctionBegin;
370   ierr = MatDestroy(&contents->workB);CHKERRQ(ierr);
371   ierr = MatDestroy(&contents->workB1);CHKERRQ(ierr);
372   for (i=0; i<contents->nsends; i++) {
373     ierr = MPI_Type_free(&contents->stype[i]);CHKERRQ(ierr);
374   }
375   for (i=0; i<contents->nrecvs; i++) {
376     ierr = MPI_Type_free(&contents->rtype[i]);CHKERRQ(ierr);
377   }
378   ierr = PetscFree4(contents->stype,contents->rtype,contents->rwaits,contents->swaits);CHKERRQ(ierr);
379   ierr = PetscFree(contents);CHKERRQ(ierr);
380   PetscFunctionReturn(0);
381 }
382 
383 static PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIDense(Mat A,Mat B,PetscReal fill,Mat C)
384 {
385   PetscErrorCode  ierr;
386   Mat_MPIAIJ      *aij=(Mat_MPIAIJ*)A->data;
387   PetscInt        nz=aij->B->cmap->n,nsends,nrecvs,i,nrows_to,j,blda,clda;
388   MPIAIJ_MPIDense *contents;
389   VecScatter      ctx=aij->Mvctx;
390   PetscInt        Am=A->rmap->n,Bm=B->rmap->n,BN=B->cmap->N,Bbn,Bbn1,bs,nrows_from,numBb;
391   MPI_Comm        comm;
392   MPI_Datatype    type1,*stype,*rtype;
393   const PetscInt  *sindices,*sstarts,*rstarts;
394   PetscMPIInt     *disp;
395   PetscBool       cisdense;
396 
397   PetscFunctionBegin;
398   MatCheckProduct(C,4);
399   if (C->product->data) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_PLIB,"Product data not empty");
400   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
401   ierr = PetscObjectBaseTypeCompare((PetscObject)C,MATMPIDENSE,&cisdense);CHKERRQ(ierr);
402   if (!cisdense) {
403     ierr = MatSetType(C,((PetscObject)B)->type_name);CHKERRQ(ierr);
404   }
405   ierr = MatSetSizes(C,Am,B->cmap->n,A->rmap->N,BN);CHKERRQ(ierr);
406   ierr = MatSetBlockSizesFromMats(C,A,B);CHKERRQ(ierr);
407   ierr = MatSetUp(C);CHKERRQ(ierr);
408   ierr = MatDenseGetLDA(B,&blda);CHKERRQ(ierr);
409   ierr = MatDenseGetLDA(C,&clda);CHKERRQ(ierr);
410   ierr = PetscNew(&contents);CHKERRQ(ierr);
411 
412   ierr = VecScatterGetRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,NULL,NULL);CHKERRQ(ierr);
413   ierr = VecScatterGetRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,NULL,NULL);CHKERRQ(ierr);
414 
415   /* Create column block of B and C for memory scalability when BN is too large */
416   /* Estimate Bbn, column size of Bb */
417   if (nz) {
418     Bbn1 = 2*Am*BN/nz;
419   } else Bbn1 = BN;
420 
421   bs = PetscAbs(B->cmap->bs);
422   Bbn1 = Bbn1/bs *bs; /* Bbn1 is a multiple of bs */
423   if (Bbn1 > BN) Bbn1 = BN;
424   ierr = MPI_Allreduce(&Bbn1,&Bbn,1,MPIU_INT,MPI_MAX,comm);CHKERRQ(ierr);
425 
426   /* Enable runtime option for Bbn */
427   ierr = PetscOptionsBegin(comm,((PetscObject)C)->prefix,"MatMatMult","Mat");CHKERRQ(ierr);
428   ierr = PetscOptionsInt("-matmatmult_Bbn","Number of columns in Bb","MatMatMult",Bbn,&Bbn,NULL);CHKERRQ(ierr);
429   ierr = PetscOptionsEnd();CHKERRQ(ierr);
430   Bbn  = PetscMin(Bbn,BN);
431 
432   if (Bbn > 0 && Bbn < BN) {
433     numBb = BN/Bbn;
434     Bbn1 = BN - numBb*Bbn;
435   } else numBb = 0;
436 
437   if (numBb) {
438     ierr = PetscInfo3(C,"use Bb, BN=%D, Bbn=%D; numBb=%D\n",BN,Bbn,numBb);CHKERRQ(ierr);
439     if (Bbn1) { /* Create workB1 for the remaining columns */
440       ierr = PetscInfo2(C,"use Bb1, BN=%D, Bbn1=%D\n",BN,Bbn1);CHKERRQ(ierr);
441       /* Create work matrix used to store off processor rows of B needed for local product */
442       ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,Bbn1,NULL,&contents->workB1);CHKERRQ(ierr);
443     } else contents->workB1 = NULL;
444   }
445 
446   /* Create work matrix used to store off processor rows of B needed for local product */
447   ierr = MatCreateSeqDense(PETSC_COMM_SELF,nz,Bbn,NULL,&contents->workB);CHKERRQ(ierr);
448 
449   /* Use MPI derived data type to reduce memory required by the send/recv buffers */
450   ierr = PetscMalloc4(nsends,&stype,nrecvs,&rtype,nrecvs,&contents->rwaits,nsends,&contents->swaits);CHKERRQ(ierr);
451   contents->stype  = stype;
452   contents->nsends = nsends;
453 
454   contents->rtype  = rtype;
455   contents->nrecvs = nrecvs;
456   contents->blda   = blda;
457 
458   ierr = PetscMalloc1(Bm+1,&disp);CHKERRQ(ierr);
459   for (i=0; i<nsends; i++) {
460     nrows_to = sstarts[i+1]-sstarts[i];
461     for (j=0; j<nrows_to; j++){
462       disp[j] = sindices[sstarts[i]+j]; /* rowB to be sent */
463     }
464     ierr = MPI_Type_create_indexed_block(nrows_to,1,(const PetscMPIInt *)disp,MPIU_SCALAR,&type1);CHKERRQ(ierr);
465 
466     ierr = MPI_Type_create_resized(type1,0,blda*sizeof(PetscScalar),&stype[i]);CHKERRQ(ierr);
467     ierr = MPI_Type_commit(&stype[i]);CHKERRQ(ierr);
468     ierr = MPI_Type_free(&type1);CHKERRQ(ierr);
469   }
470 
471   for (i=0; i<nrecvs; i++) {
472     /* received values from a process form a (nrows_from x Bbn) row block in workB (column-wise) */
473     nrows_from = rstarts[i+1]-rstarts[i];
474     disp[0] = 0;
475     ierr = MPI_Type_create_indexed_block(1, nrows_from, (const PetscMPIInt *)disp, MPIU_SCALAR, &type1);CHKERRQ(ierr);
476     ierr = MPI_Type_create_resized(type1, 0, nz*sizeof(PetscScalar), &rtype[i]);CHKERRQ(ierr);
477     ierr = MPI_Type_commit(&rtype[i]);CHKERRQ(ierr);
478     ierr = MPI_Type_free(&type1);CHKERRQ(ierr);
479   }
480 
481   ierr = PetscFree(disp);CHKERRQ(ierr);
482   ierr = VecScatterRestoreRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,NULL,NULL);CHKERRQ(ierr);
483   ierr = VecScatterRestoreRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,NULL,NULL);CHKERRQ(ierr);
484   ierr = MatSetOption(C,MAT_NO_OFF_PROC_ENTRIES,PETSC_TRUE);CHKERRQ(ierr);
485   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
486   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
487 
488   C->product->data = contents;
489   C->product->destroy = MatMPIAIJ_MPIDenseDestroy;
490   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIDense;
491   PetscFunctionReturn(0);
492 }
493 
494 PETSC_INTERN PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat,Mat,Mat,const PetscBool);
495 /*
496     Performs an efficient scatter on the rows of B needed by this process; this is
497     a modification of the VecScatterBegin_() routines.
498 
499     Input: Bbidx = 0: B = Bb
500                  = 1: B = Bb1, see MatMatMultSymbolic_MPIAIJ_MPIDense()
501 */
502 PetscErrorCode MatMPIDenseScatter(Mat A,Mat B,PetscInt Bbidx,Mat C,Mat *outworkB)
503 {
504   Mat_MPIAIJ        *aij = (Mat_MPIAIJ*)A->data;
505   PetscErrorCode    ierr;
506   const PetscScalar *b;
507   PetscScalar       *rvalues;
508   VecScatter        ctx = aij->Mvctx;
509   const PetscInt    *sindices,*sstarts,*rstarts;
510   const PetscMPIInt *sprocs,*rprocs;
511   PetscInt          i,nsends,nrecvs;
512   MPI_Request       *swaits,*rwaits;
513   MPI_Comm          comm;
514   PetscMPIInt       tag=((PetscObject)ctx)->tag,ncols=B->cmap->N,nrows=aij->B->cmap->n,nsends_mpi,nrecvs_mpi;
515   MPIAIJ_MPIDense   *contents;
516   Mat               workB;
517   MPI_Datatype      *stype,*rtype;
518   PetscInt          blda;
519 
520   PetscFunctionBegin;
521   MatCheckProduct(C,4);
522   if (!C->product->data) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_PLIB,"Product data empty");
523   contents = (MPIAIJ_MPIDense*)C->product->data;
524   ierr = VecScatterGetRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,&sprocs,NULL/*bs*/);CHKERRQ(ierr);
525   ierr = VecScatterGetRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,&rprocs,NULL/*bs*/);CHKERRQ(ierr);
526   ierr = PetscMPIIntCast(nsends,&nsends_mpi);CHKERRQ(ierr);
527   ierr = PetscMPIIntCast(nrecvs,&nrecvs_mpi);CHKERRQ(ierr);
528   if (Bbidx == 0) {
529     workB = *outworkB = contents->workB;
530   } else {
531     workB = *outworkB = contents->workB1;
532   }
533   if (nrows != workB->rmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Number of rows of workB %D not equal to columns of aij->B %D",workB->cmap->n,nrows);
534   swaits = contents->swaits;
535   rwaits = contents->rwaits;
536 
537   ierr = MatDenseGetArrayRead(B,&b);CHKERRQ(ierr);
538   ierr = MatDenseGetLDA(B,&blda);CHKERRQ(ierr);
539   if (blda != contents->blda) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Cannot reuse an input matrix with lda %D != %D",blda,contents->blda);
540   ierr = MatDenseGetArray(workB,&rvalues);CHKERRQ(ierr);
541 
542   /* Post recv, use MPI derived data type to save memory */
543   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
544   rtype = contents->rtype;
545   for (i=0; i<nrecvs; i++) {
546     ierr = MPI_Irecv(rvalues+(rstarts[i]-rstarts[0]),ncols,rtype[i],rprocs[i],tag,comm,rwaits+i);CHKERRQ(ierr);
547   }
548 
549   stype = contents->stype;
550   for (i=0; i<nsends; i++) {
551     ierr = MPI_Isend(b,ncols,stype[i],sprocs[i],tag,comm,swaits+i);CHKERRQ(ierr);
552   }
553 
554   if (nrecvs) {ierr = MPI_Waitall(nrecvs_mpi,rwaits,MPI_STATUSES_IGNORE);CHKERRQ(ierr);}
555   if (nsends) {ierr = MPI_Waitall(nsends_mpi,swaits,MPI_STATUSES_IGNORE);CHKERRQ(ierr);}
556 
557   ierr = VecScatterRestoreRemote_Private(ctx,PETSC_TRUE/*send*/,&nsends,&sstarts,&sindices,&sprocs,NULL);CHKERRQ(ierr);
558   ierr = VecScatterRestoreRemoteOrdered_Private(ctx,PETSC_FALSE/*recv*/,&nrecvs,&rstarts,NULL,&rprocs,NULL);CHKERRQ(ierr);
559   ierr = MatDenseRestoreArrayRead(B,&b);CHKERRQ(ierr);
560   ierr = MatDenseRestoreArray(workB,&rvalues);CHKERRQ(ierr);
561   PetscFunctionReturn(0);
562 }
563 
564 static PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIDense(Mat A,Mat B,Mat C)
565 {
566   PetscErrorCode  ierr;
567   Mat_MPIAIJ      *aij    = (Mat_MPIAIJ*)A->data;
568   Mat_MPIDense    *bdense = (Mat_MPIDense*)B->data;
569   Mat_MPIDense    *cdense = (Mat_MPIDense*)C->data;
570   Mat             workB;
571   MPIAIJ_MPIDense *contents;
572 
573   PetscFunctionBegin;
574   MatCheckProduct(C,3);
575   if (!C->product->data) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_PLIB,"Product data empty");
576   contents = (MPIAIJ_MPIDense*)C->product->data;
577   /* diagonal block of A times all local rows of B */
578   /* TODO: this calls a symbolic multiplication every time, which could be avoided */
579   ierr = MatMatMult(aij->A,bdense->A,MAT_REUSE_MATRIX,PETSC_DEFAULT,&cdense->A);CHKERRQ(ierr);
580   if (contents->workB->cmap->n == B->cmap->N) {
581     /* get off processor parts of B needed to complete C=A*B */
582     ierr = MatMPIDenseScatter(A,B,0,C,&workB);CHKERRQ(ierr);
583 
584     /* off-diagonal block of A times nonlocal rows of B */
585     ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A,PETSC_TRUE);CHKERRQ(ierr);
586   } else {
587     Mat      Bb,Cb;
588     PetscInt BN=B->cmap->N,n=contents->workB->cmap->n,i;
589 
590     for (i=0; i<BN; i+=n) {
591       ierr = MatDenseGetSubMatrix(B,i,PetscMin(i+n,BN),&Bb);CHKERRQ(ierr);
592       ierr = MatDenseGetSubMatrix(C,i,PetscMin(i+n,BN),&Cb);CHKERRQ(ierr);
593 
594       /* get off processor parts of B needed to complete C=A*B */
595       ierr = MatMPIDenseScatter(A,Bb,i+n>BN,C,&workB);CHKERRQ(ierr);
596 
597       /* off-diagonal block of A times nonlocal rows of B */
598       cdense = (Mat_MPIDense*)Cb->data;
599       ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(aij->B,workB,cdense->A,PETSC_TRUE);CHKERRQ(ierr);
600 
601       ierr = MatDenseRestoreSubMatrix(B,&Bb);CHKERRQ(ierr);
602       ierr = MatDenseRestoreSubMatrix(C,&Cb);CHKERRQ(ierr);
603     }
604   }
605   PetscFunctionReturn(0);
606 }
607 
608 PetscErrorCode MatMatMultNumeric_MPIAIJ_MPIAIJ(Mat A,Mat P,Mat C)
609 {
610   PetscErrorCode ierr;
611   Mat_MPIAIJ     *a   = (Mat_MPIAIJ*)A->data,*c=(Mat_MPIAIJ*)C->data;
612   Mat_SeqAIJ     *ad  = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data;
613   Mat_SeqAIJ     *cd  = (Mat_SeqAIJ*)(c->A)->data,*co=(Mat_SeqAIJ*)(c->B)->data;
614   PetscInt       *adi = ad->i,*adj,*aoi=ao->i,*aoj;
615   PetscScalar    *ada,*aoa,*cda=cd->a,*coa=co->a;
616   Mat_SeqAIJ     *p_loc,*p_oth;
617   PetscInt       *pi_loc,*pj_loc,*pi_oth,*pj_oth,*pj;
618   PetscScalar    *pa_loc,*pa_oth,*pa,valtmp,*ca;
619   PetscInt       cm    = C->rmap->n,anz,pnz;
620   Mat_APMPI      *ptap;
621   PetscScalar    *apa_sparse;
622   PetscInt       *api,*apj,*apJ,i,j,k,row;
623   PetscInt       cstart = C->cmap->rstart;
624   PetscInt       cdnz,conz,k0,k1,nextp;
625   MPI_Comm       comm;
626   PetscMPIInt    size;
627 
628   PetscFunctionBegin;
629   MatCheckProduct(C,3);
630   ptap = (Mat_APMPI*)C->product->data;
631   if (!ptap) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be computed. Missing data");
632   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
633   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
634   if (!ptap->P_oth && size>1) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"AP cannot be reused. Do not call MatProductClear()");
635 
636   apa_sparse = ptap->apa;
637 
638   /* 1) get P_oth = ptap->P_oth  and P_loc = ptap->P_loc */
639   /*-----------------------------------------------------*/
640   /* update numerical values of P_oth and P_loc */
641   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_REUSE_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
642   ierr = MatMPIAIJGetLocalMat(P,MAT_REUSE_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
643 
644   /* 2) compute numeric C_loc = A_loc*P = Ad*P_loc + Ao*P_oth */
645   /*----------------------------------------------------------*/
646   /* get data from symbolic products */
647   p_loc = (Mat_SeqAIJ*)(ptap->P_loc)->data;
648   pi_loc = p_loc->i; pj_loc = p_loc->j; pa_loc = p_loc->a;
649   if (size >1) {
650     p_oth = (Mat_SeqAIJ*)(ptap->P_oth)->data;
651     pi_oth = p_oth->i; pj_oth = p_oth->j; pa_oth = p_oth->a;
652   } else {
653     p_oth = NULL; pi_oth = NULL; pj_oth = NULL; pa_oth = NULL;
654   }
655 
656   api = ptap->api;
657   apj = ptap->apj;
658   for (i=0; i<cm; i++) {
659     apJ = apj + api[i];
660 
661     /* diagonal portion of A */
662     anz = adi[i+1] - adi[i];
663     adj = ad->j + adi[i];
664     ada = ad->a + adi[i];
665     for (j=0; j<anz; j++) {
666       row = adj[j];
667       pnz = pi_loc[row+1] - pi_loc[row];
668       pj  = pj_loc + pi_loc[row];
669       pa  = pa_loc + pi_loc[row];
670       /* perform sparse axpy */
671       valtmp = ada[j];
672       nextp  = 0;
673       for (k=0; nextp<pnz; k++) {
674         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
675           apa_sparse[k] += valtmp*pa[nextp++];
676         }
677       }
678       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
679     }
680 
681     /* off-diagonal portion of A */
682     anz = aoi[i+1] - aoi[i];
683     aoj = ao->j + aoi[i];
684     aoa = ao->a + aoi[i];
685     for (j=0; j<anz; j++) {
686       row = aoj[j];
687       pnz = pi_oth[row+1] - pi_oth[row];
688       pj  = pj_oth + pi_oth[row];
689       pa  = pa_oth + pi_oth[row];
690       /* perform sparse axpy */
691       valtmp = aoa[j];
692       nextp  = 0;
693       for (k=0; nextp<pnz; k++) {
694         if (apJ[k] == pj[nextp]) { /* column of AP == column of P */
695           apa_sparse[k] += valtmp*pa[nextp++];
696         }
697       }
698       ierr = PetscLogFlops(2.0*pnz);CHKERRQ(ierr);
699     }
700 
701     /* set values in C */
702     cdnz = cd->i[i+1] - cd->i[i];
703     conz = co->i[i+1] - co->i[i];
704 
705     /* 1st off-diagonal part of C */
706     ca = coa + co->i[i];
707     k  = 0;
708     for (k0=0; k0<conz; k0++) {
709       if (apJ[k] >= cstart) break;
710       ca[k0]        = apa_sparse[k];
711       apa_sparse[k] = 0.0;
712       k++;
713     }
714 
715     /* diagonal part of C */
716     ca = cda + cd->i[i];
717     for (k1=0; k1<cdnz; k1++) {
718       ca[k1]        = apa_sparse[k];
719       apa_sparse[k] = 0.0;
720       k++;
721     }
722 
723     /* 2nd off-diagonal part of C */
724     ca = coa + co->i[i];
725     for (; k0<conz; k0++) {
726       ca[k0]        = apa_sparse[k];
727       apa_sparse[k] = 0.0;
728       k++;
729     }
730   }
731   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
732   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
733   PetscFunctionReturn(0);
734 }
735 
736 /* same as MatMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(), except using LLCondensed to avoid O(BN) memory requirement */
737 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ(Mat A,Mat P,PetscReal fill,Mat C)
738 {
739   PetscErrorCode     ierr;
740   MPI_Comm           comm;
741   PetscMPIInt        size;
742   Mat_APMPI          *ptap;
743   PetscFreeSpaceList free_space = NULL,current_space=NULL;
744   Mat_MPIAIJ         *a  = (Mat_MPIAIJ*)A->data;
745   Mat_SeqAIJ         *ad = (Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc,*p_oth;
746   PetscInt           *pi_loc,*pj_loc,*pi_oth,*pj_oth,*dnz,*onz;
747   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,*aoj=ao->j,rstart=A->rmap->rstart;
748   PetscInt           i,pnz,row,*api,*apj,*Jptr,apnz,nspacedouble=0,j,nzi,*lnk,apnz_max=0;
749   PetscInt           am=A->rmap->n,pn=P->cmap->n,pm=P->rmap->n,lsize=pn+20;
750   PetscReal          afill;
751   MatType            mtype;
752 
753   PetscFunctionBegin;
754   MatCheckProduct(C,4);
755   if (C->product->data) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Extra product struct not empty");
756   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
757   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
758 
759   /* create struct Mat_APMPI and attached it to C later */
760   ierr = PetscNew(&ptap);CHKERRQ(ierr);
761 
762   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
763   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
764 
765   /* get P_loc by taking all local rows of P */
766   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
767 
768   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
769   pi_loc = p_loc->i; pj_loc = p_loc->j;
770   if (size > 1) {
771     p_oth  = (Mat_SeqAIJ*)(ptap->P_oth)->data;
772     pi_oth = p_oth->i; pj_oth = p_oth->j;
773   } else {
774     p_oth  = NULL;
775     pi_oth = NULL; pj_oth = NULL;
776   }
777 
778   /* first, compute symbolic AP = A_loc*P = A_diag*P_loc + A_off*P_oth */
779   /*-------------------------------------------------------------------*/
780   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
781   ptap->api = api;
782   api[0]    = 0;
783 
784   ierr = PetscLLCondensedCreate_Scalable(lsize,&lnk);CHKERRQ(ierr);
785 
786   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
787   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space);CHKERRQ(ierr);
788   current_space = free_space;
789   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
790   for (i=0; i<am; i++) {
791     /* diagonal portion of A */
792     nzi = adi[i+1] - adi[i];
793     for (j=0; j<nzi; j++) {
794       row  = *adj++;
795       pnz  = pi_loc[row+1] - pi_loc[row];
796       Jptr = pj_loc + pi_loc[row];
797       /* Expand list if it is not long enough */
798       if (pnz+apnz_max > lsize) {
799         lsize = pnz+apnz_max;
800         ierr = PetscLLCondensedExpand_Scalable(lsize, &lnk);CHKERRQ(ierr);
801       }
802       /* add non-zero cols of P into the sorted linked list lnk */
803       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
804       apnz     = *lnk; /* The first element in the list is the number of items in the list */
805       api[i+1] = api[i] + apnz;
806       if (apnz > apnz_max) apnz_max = apnz;
807     }
808     /* off-diagonal portion of A */
809     nzi = aoi[i+1] - aoi[i];
810     for (j=0; j<nzi; j++) {
811       row  = *aoj++;
812       pnz  = pi_oth[row+1] - pi_oth[row];
813       Jptr = pj_oth + pi_oth[row];
814       /* Expand list if it is not long enough */
815       if (pnz+apnz_max > lsize) {
816         lsize = pnz + apnz_max;
817         ierr = PetscLLCondensedExpand_Scalable(lsize, &lnk);CHKERRQ(ierr);
818       }
819       /* add non-zero cols of P into the sorted linked list lnk */
820       ierr = PetscLLCondensedAddSorted_Scalable(pnz,Jptr,lnk);CHKERRQ(ierr);
821       apnz     = *lnk;  /* The first element in the list is the number of items in the list */
822       api[i+1] = api[i] + apnz;
823       if (apnz > apnz_max) apnz_max = apnz;
824     }
825     apnz     = *lnk;
826     api[i+1] = api[i] + apnz;
827     if (apnz > apnz_max) apnz_max = apnz;
828 
829     /* if free space is not available, double the total space in the list */
830     if (current_space->local_remaining<apnz) {
831       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(apnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
832       nspacedouble++;
833     }
834 
835     /* Copy data into free space, then initialize lnk */
836     ierr = PetscLLCondensedClean_Scalable(apnz,current_space->array,lnk);CHKERRQ(ierr);
837     ierr = MatPreallocateSet(i+rstart,apnz,current_space->array,dnz,onz);CHKERRQ(ierr);
838 
839     current_space->array           += apnz;
840     current_space->local_used      += apnz;
841     current_space->local_remaining -= apnz;
842   }
843 
844   /* Allocate space for apj, initialize apj, and */
845   /* destroy list of free space and other temporary array(s) */
846   ierr = PetscMalloc1(api[am]+1,&ptap->apj);CHKERRQ(ierr);
847   apj  = ptap->apj;
848   ierr = PetscFreeSpaceContiguous(&free_space,ptap->apj);CHKERRQ(ierr);
849   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
850 
851   /* create and assemble symbolic parallel matrix C */
852   /*----------------------------------------------------*/
853   ierr = MatSetSizes(C,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
854   ierr = MatSetBlockSizesFromMats(C,A,P);CHKERRQ(ierr);
855   ierr = MatGetType(A,&mtype);CHKERRQ(ierr);
856   ierr = MatSetType(C,mtype);CHKERRQ(ierr);
857   ierr = MatMPIAIJSetPreallocation(C,0,dnz,0,onz);CHKERRQ(ierr);
858 
859   /* malloc apa for assembly C */
860   ierr = PetscCalloc1(apnz_max,&ptap->apa);CHKERRQ(ierr);
861 
862   ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(C, apj, api);CHKERRQ(ierr);
863   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
864   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
865   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
866 
867   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ;
868   C->ops->productnumeric = MatProductNumeric_AB;
869 
870   /* attach the supporting struct to C for reuse */
871   C->product->data    = ptap;
872   C->product->destroy = MatDestroy_MPIAIJ_MatMatMult;
873 
874   /* set MatInfo */
875   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
876   if (afill < 1.0) afill = 1.0;
877   C->info.mallocs           = nspacedouble;
878   C->info.fill_ratio_given  = fill;
879   C->info.fill_ratio_needed = afill;
880 
881 #if defined(PETSC_USE_INFO)
882   if (api[am]) {
883     ierr = PetscInfo3(C,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
884     ierr = PetscInfo1(C,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
885   } else {
886     ierr = PetscInfo(C,"Empty matrix product\n");CHKERRQ(ierr);
887   }
888 #endif
889   PetscFunctionReturn(0);
890 }
891 
892 /* This function is needed for the seqMPI matrix-matrix multiplication.  */
893 /* Three input arrays are merged to one output array. The size of the    */
894 /* output array is also output. Duplicate entries only show up once.     */
895 static void Merge3SortedArrays(PetscInt  size1, PetscInt *in1,
896                                PetscInt  size2, PetscInt *in2,
897                                PetscInt  size3, PetscInt *in3,
898                                PetscInt *size4, PetscInt *out)
899 {
900   int i = 0, j = 0, k = 0, l = 0;
901 
902   /* Traverse all three arrays */
903   while (i<size1 && j<size2 && k<size3) {
904     if (in1[i] < in2[j] && in1[i] < in3[k]) {
905       out[l++] = in1[i++];
906     }
907     else if (in2[j] < in1[i] && in2[j] < in3[k]) {
908       out[l++] = in2[j++];
909     }
910     else if (in3[k] < in1[i] && in3[k] < in2[j]) {
911       out[l++] = in3[k++];
912     }
913     else if (in1[i] == in2[j] && in1[i] < in3[k]) {
914       out[l++] = in1[i];
915       i++, j++;
916     }
917     else if (in1[i] == in3[k] && in1[i] < in2[j]) {
918       out[l++] = in1[i];
919       i++, k++;
920     }
921     else if (in3[k] == in2[j] && in2[j] < in1[i])  {
922       out[l++] = in2[j];
923       k++, j++;
924     }
925     else if (in1[i] == in2[j] && in1[i] == in3[k]) {
926       out[l++] = in1[i];
927       i++, j++, k++;
928     }
929   }
930 
931   /* Traverse two remaining arrays */
932   while (i<size1 && j<size2) {
933     if (in1[i] < in2[j]) {
934       out[l++] = in1[i++];
935     }
936     else if (in1[i] > in2[j]) {
937       out[l++] = in2[j++];
938     }
939     else {
940       out[l++] = in1[i];
941       i++, j++;
942     }
943   }
944 
945   while (i<size1 && k<size3) {
946     if (in1[i] < in3[k]) {
947       out[l++] = in1[i++];
948     }
949     else if (in1[i] > in3[k]) {
950       out[l++] = in3[k++];
951     }
952     else {
953       out[l++] = in1[i];
954       i++, k++;
955     }
956   }
957 
958   while (k<size3 && j<size2)  {
959     if (in3[k] < in2[j]) {
960       out[l++] = in3[k++];
961     }
962     else if (in3[k] > in2[j]) {
963       out[l++] = in2[j++];
964     }
965     else {
966       out[l++] = in3[k];
967       k++, j++;
968     }
969   }
970 
971   /* Traverse one remaining array */
972   while (i<size1) out[l++] = in1[i++];
973   while (j<size2) out[l++] = in2[j++];
974   while (k<size3) out[l++] = in3[k++];
975 
976   *size4 = l;
977 }
978 
979 /* This matrix-matrix multiplication algorithm divides the multiplication into three multiplications and  */
980 /* adds up the products. Two of these three multiplications are performed with existing (sequential)      */
981 /* matrix-matrix multiplications.  */
982 PetscErrorCode MatMatMultSymbolic_MPIAIJ_MPIAIJ_seqMPI(Mat A, Mat P, PetscReal fill, Mat C)
983 {
984   PetscErrorCode     ierr;
985   MPI_Comm           comm;
986   PetscMPIInt        size;
987   Mat_APMPI          *ptap;
988   PetscFreeSpaceList free_space_diag=NULL, current_space=NULL;
989   Mat_MPIAIJ         *a  =(Mat_MPIAIJ*)A->data;
990   Mat_SeqAIJ         *ad =(Mat_SeqAIJ*)(a->A)->data,*ao=(Mat_SeqAIJ*)(a->B)->data,*p_loc;
991   Mat_MPIAIJ         *p  =(Mat_MPIAIJ*)P->data;
992   Mat_SeqAIJ         *adpd_seq, *p_off, *aopoth_seq;
993   PetscInt           adponz, adpdnz;
994   PetscInt           *pi_loc,*dnz,*onz;
995   PetscInt           *adi=ad->i,*adj=ad->j,*aoi=ao->i,rstart=A->rmap->rstart;
996   PetscInt           *lnk,i, i1=0,pnz,row,*adpoi,*adpoj, *api, *adpoJ, *aopJ, *apJ,*Jptr, aopnz, nspacedouble=0,j,nzi,
997                      *apj,apnz, *adpdi, *adpdj, *adpdJ, *poff_i, *poff_j, *j_temp, *aopothi, *aopothj;
998   PetscInt           am=A->rmap->n,pN=P->cmap->N,pn=P->cmap->n,pm=P->rmap->n, p_colstart, p_colend;
999   PetscBT            lnkbt;
1000   PetscReal          afill;
1001   PetscMPIInt        rank;
1002   Mat                adpd, aopoth;
1003   MatType            mtype;
1004   const char         *prefix;
1005 
1006   PetscFunctionBegin;
1007   MatCheckProduct(C,4);
1008   if (C->product->data) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Extra product struct not empty");
1009   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1010   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1011   ierr = MPI_Comm_rank(comm, &rank);CHKERRQ(ierr);
1012   ierr = MatGetOwnershipRangeColumn(P, &p_colstart, &p_colend);CHKERRQ(ierr);
1013 
1014   /* create struct Mat_APMPI and attached it to C later */
1015   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1016 
1017   /* get P_oth by taking rows of P (= non-zero cols of local A) from other processors */
1018   ierr = MatGetBrowsOfAoCols_MPIAIJ(A,P,MAT_INITIAL_MATRIX,&ptap->startsj_s,&ptap->startsj_r,&ptap->bufa,&ptap->P_oth);CHKERRQ(ierr);
1019 
1020   /* get P_loc by taking all local rows of P */
1021   ierr = MatMPIAIJGetLocalMat(P,MAT_INITIAL_MATRIX,&ptap->P_loc);CHKERRQ(ierr);
1022 
1023 
1024   p_loc  = (Mat_SeqAIJ*)(ptap->P_loc)->data;
1025   pi_loc = p_loc->i;
1026 
1027   /* Allocate memory for the i arrays of the matrices A*P, A_diag*P_off and A_offd * P */
1028   ierr      = PetscMalloc1(am+2,&api);CHKERRQ(ierr);
1029   ierr      = PetscMalloc1(am+2,&adpoi);CHKERRQ(ierr);
1030 
1031   adpoi[0]    = 0;
1032   ptap->api = api;
1033   api[0] = 0;
1034 
1035   /* create and initialize a linked list, will be used for both A_diag * P_loc_off and A_offd * P_oth */
1036   ierr = PetscLLCondensedCreate(pN,pN,&lnk,&lnkbt);CHKERRQ(ierr);
1037   ierr = MatPreallocateInitialize(comm,am,pn,dnz,onz);CHKERRQ(ierr);
1038 
1039   /* Symbolic calc of A_loc_diag * P_loc_diag */
1040   ierr = MatGetOptionsPrefix(A,&prefix);CHKERRQ(ierr);
1041   ierr = MatProductCreate(a->A,p->A,NULL,&adpd);CHKERRQ(ierr);
1042   ierr = MatGetOptionsPrefix(A,&prefix);CHKERRQ(ierr);
1043   ierr = MatSetOptionsPrefix(adpd,prefix);CHKERRQ(ierr);
1044   ierr = MatAppendOptionsPrefix(adpd,"inner_diag_");CHKERRQ(ierr);
1045 
1046   ierr = MatProductSetType(adpd,MATPRODUCT_AB);CHKERRQ(ierr);
1047   ierr = MatProductSetAlgorithm(adpd,"sorted");CHKERRQ(ierr);
1048   ierr = MatProductSetFill(adpd,fill);CHKERRQ(ierr);
1049   ierr = MatProductSetFromOptions(adpd);CHKERRQ(ierr);
1050   ierr = MatProductSymbolic(adpd);CHKERRQ(ierr);
1051 
1052   adpd_seq = (Mat_SeqAIJ*)((adpd)->data);
1053   adpdi = adpd_seq->i; adpdj = adpd_seq->j;
1054   p_off = (Mat_SeqAIJ*)((p->B)->data);
1055   poff_i = p_off->i; poff_j = p_off->j;
1056 
1057   /* j_temp stores indices of a result row before they are added to the linked list */
1058   ierr = PetscMalloc1(pN+2,&j_temp);CHKERRQ(ierr);
1059 
1060 
1061   /* Symbolic calc of the A_diag * p_loc_off */
1062   /* Initial FreeSpace size is fill*(nnz(A)+nnz(P)) */
1063   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(adi[am],PetscIntSumTruncate(aoi[am],pi_loc[pm]))),&free_space_diag);CHKERRQ(ierr);
1064   current_space = free_space_diag;
1065 
1066   for (i=0; i<am; i++) {
1067     /* A_diag * P_loc_off */
1068     nzi = adi[i+1] - adi[i];
1069     for (j=0; j<nzi; j++) {
1070       row  = *adj++;
1071       pnz  = poff_i[row+1] - poff_i[row];
1072       Jptr = poff_j + poff_i[row];
1073       for (i1 = 0; i1 < pnz; i1++) {
1074         j_temp[i1] = p->garray[Jptr[i1]];
1075       }
1076       /* add non-zero cols of P into the sorted linked list lnk */
1077       ierr = PetscLLCondensedAddSorted(pnz,j_temp,lnk,lnkbt);CHKERRQ(ierr);
1078     }
1079 
1080     adponz     = lnk[0];
1081     adpoi[i+1] = adpoi[i] + adponz;
1082 
1083     /* if free space is not available, double the total space in the list */
1084     if (current_space->local_remaining<adponz) {
1085       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(adponz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1086       nspacedouble++;
1087     }
1088 
1089     /* Copy data into free space, then initialize lnk */
1090     ierr = PetscLLCondensedClean(pN,adponz,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1091 
1092     current_space->array           += adponz;
1093     current_space->local_used      += adponz;
1094     current_space->local_remaining -= adponz;
1095   }
1096 
1097   /* Symbolic calc of A_off * P_oth */
1098   ierr = MatSetOptionsPrefix(a->B,prefix);CHKERRQ(ierr);
1099   ierr = MatAppendOptionsPrefix(a->B,"inner_offdiag_");CHKERRQ(ierr);
1100   ierr = MatCreate(PETSC_COMM_SELF,&aopoth);CHKERRQ(ierr);
1101   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(a->B, ptap->P_oth, fill, aopoth);CHKERRQ(ierr);
1102   aopoth_seq = (Mat_SeqAIJ*)((aopoth)->data);
1103   aopothi = aopoth_seq->i; aopothj = aopoth_seq->j;
1104 
1105   /* Allocate space for apj, adpj, aopj, ... */
1106   /* destroy lists of free space and other temporary array(s) */
1107 
1108   ierr = PetscMalloc1(aopothi[am] + adpoi[am] + adpdi[am]+2, &ptap->apj);CHKERRQ(ierr);
1109   ierr = PetscMalloc1(adpoi[am]+2, &adpoj);CHKERRQ(ierr);
1110 
1111   /* Copy from linked list to j-array */
1112   ierr = PetscFreeSpaceContiguous(&free_space_diag,adpoj);CHKERRQ(ierr);
1113   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1114 
1115   adpoJ = adpoj;
1116   adpdJ = adpdj;
1117   aopJ = aopothj;
1118   apj  = ptap->apj;
1119   apJ = apj; /* still empty */
1120 
1121   /* Merge j-arrays of A_off * P, A_diag * P_loc_off, and */
1122   /* A_diag * P_loc_diag to get A*P */
1123   for (i = 0; i < am; i++) {
1124     aopnz  =  aopothi[i+1] -  aopothi[i];
1125     adponz = adpoi[i+1] - adpoi[i];
1126     adpdnz = adpdi[i+1] - adpdi[i];
1127 
1128     /* Correct indices from A_diag*P_diag */
1129     for (i1 = 0; i1 < adpdnz; i1++) {
1130       adpdJ[i1] += p_colstart;
1131     }
1132     /* Merge j-arrays of A_diag * P_loc_off and A_diag * P_loc_diag and A_off * P_oth */
1133     Merge3SortedArrays(adponz, adpoJ, adpdnz, adpdJ, aopnz, aopJ, &apnz, apJ);
1134     ierr = MatPreallocateSet(i+rstart, apnz, apJ, dnz, onz);CHKERRQ(ierr);
1135 
1136     aopJ += aopnz;
1137     adpoJ += adponz;
1138     adpdJ += adpdnz;
1139     apJ += apnz;
1140     api[i+1] = api[i] + apnz;
1141   }
1142 
1143   /* malloc apa to store dense row A[i,:]*P */
1144   ierr = PetscCalloc1(pN+2,&ptap->apa);CHKERRQ(ierr);
1145 
1146   /* create and assemble symbolic parallel matrix C */
1147   ierr = MatSetSizes(C,am,pn,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1148   ierr = MatSetBlockSizesFromMats(C,A,P);CHKERRQ(ierr);
1149   ierr = MatGetType(A,&mtype);CHKERRQ(ierr);
1150   ierr = MatSetType(C,mtype);CHKERRQ(ierr);
1151   ierr = MatMPIAIJSetPreallocation(C,0,dnz,0,onz);CHKERRQ(ierr);
1152 
1153 
1154   ierr = MatSetValues_MPIAIJ_CopyFromCSRFormat_Symbolic(C, apj, api);CHKERRQ(ierr);
1155   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1156   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1157   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1158 
1159 
1160   C->ops->matmultnumeric = MatMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1161   C->ops->productnumeric = MatProductNumeric_AB;
1162 
1163   /* attach the supporting struct to C for reuse */
1164   C->product->data    = ptap;
1165   C->product->destroy = MatDestroy_MPIAIJ_MatMatMult;
1166 
1167   /* set MatInfo */
1168   afill = (PetscReal)api[am]/(adi[am]+aoi[am]+pi_loc[pm]+1) + 1.e-5;
1169   if (afill < 1.0) afill = 1.0;
1170   C->info.mallocs           = nspacedouble;
1171   C->info.fill_ratio_given  = fill;
1172   C->info.fill_ratio_needed = afill;
1173 
1174 #if defined(PETSC_USE_INFO)
1175   if (api[am]) {
1176     ierr = PetscInfo3(C,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
1177     ierr = PetscInfo1(C,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
1178   } else {
1179     ierr = PetscInfo(C,"Empty matrix product\n");CHKERRQ(ierr);
1180   }
1181 #endif
1182 
1183   ierr = MatDestroy(&aopoth);CHKERRQ(ierr);
1184   ierr = MatDestroy(&adpd);CHKERRQ(ierr);
1185   ierr = PetscFree(j_temp);CHKERRQ(ierr);
1186   ierr = PetscFree(adpoj);CHKERRQ(ierr);
1187   ierr = PetscFree(adpoi);CHKERRQ(ierr);
1188   PetscFunctionReturn(0);
1189 }
1190 
1191 /*-------------------------------------------------------------------------*/
1192 /* This routine only works when scall=MAT_REUSE_MATRIX! */
1193 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult(Mat P,Mat A,Mat C)
1194 {
1195   PetscErrorCode ierr;
1196   Mat_APMPI      *ptap;
1197   Mat            Pt;
1198 
1199   PetscFunctionBegin;
1200   MatCheckProduct(C,3);
1201   ptap = (Mat_APMPI*)C->product->data;
1202   if (!ptap) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be computed. Missing data");
1203   if (!ptap->Pt) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtA cannot be reused. Do not call MatProductClear()");
1204 
1205   Pt   = ptap->Pt;
1206   ierr = MatTranspose(P,MAT_REUSE_MATRIX,&Pt);CHKERRQ(ierr);
1207   ierr = MatMatMultNumeric_MPIAIJ_MPIAIJ(Pt,A,C);CHKERRQ(ierr);
1208   PetscFunctionReturn(0);
1209 }
1210 
1211 /* This routine is modified from MatPtAPSymbolic_MPIAIJ_MPIAIJ() */
1212 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,PetscReal fill,Mat C)
1213 {
1214   PetscErrorCode      ierr;
1215   Mat_APMPI           *ptap;
1216   Mat_MPIAIJ          *p=(Mat_MPIAIJ*)P->data;
1217   MPI_Comm            comm;
1218   PetscMPIInt         size,rank;
1219   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1220   PetscInt            pn=P->cmap->n,aN=A->cmap->N,an=A->cmap->n;
1221   PetscInt            *lnk,i,k,nsend;
1222   PetscBT             lnkbt;
1223   PetscMPIInt         tagi,tagj,*len_si,*len_s,*len_ri,icompleted=0,nrecv;
1224   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1225   PetscInt            len,proc,*dnz,*onz,*owners,nzi;
1226   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1227   MPI_Request         *swaits,*rwaits;
1228   MPI_Status          *sstatus,rstatus;
1229   PetscLayout         rowmap;
1230   PetscInt            *owners_co,*coi,*coj;    /* i and j array of (p->B)^T*A*P - used in the communication */
1231   PetscMPIInt         *len_r,*id_r;    /* array of length of comm->size, store send/recv matrix values */
1232   PetscInt            *Jptr,*prmap=p->garray,con,j,Crmax;
1233   Mat_SeqAIJ          *a_loc,*c_loc,*c_oth;
1234   PetscTable          ta;
1235   MatType             mtype;
1236   const char          *prefix;
1237 
1238   PetscFunctionBegin;
1239   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1240   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1241   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1242 
1243   /* create symbolic parallel matrix C */
1244   ierr = MatGetType(A,&mtype);CHKERRQ(ierr);
1245   ierr = MatSetType(C,mtype);CHKERRQ(ierr);
1246 
1247   C->ops->transposematmultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable;
1248 
1249   /* create struct Mat_APMPI and attached it to C later */
1250   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1251   ptap->reuse = MAT_INITIAL_MATRIX;
1252 
1253   /* (0) compute Rd = Pd^T, Ro = Po^T  */
1254   /* --------------------------------- */
1255   ierr = MatTranspose_SeqAIJ(p->A,MAT_INITIAL_MATRIX,&ptap->Rd);CHKERRQ(ierr);
1256   ierr = MatTranspose_SeqAIJ(p->B,MAT_INITIAL_MATRIX,&ptap->Ro);CHKERRQ(ierr);
1257 
1258   /* (1) compute symbolic A_loc */
1259   /* ---------------------------*/
1260   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&ptap->A_loc);CHKERRQ(ierr);
1261 
1262   /* (2-1) compute symbolic C_oth = Ro*A_loc  */
1263   /* ------------------------------------ */
1264   ierr = MatGetOptionsPrefix(A,&prefix);CHKERRQ(ierr);
1265   ierr = MatSetOptionsPrefix(ptap->Ro,prefix);CHKERRQ(ierr);
1266   ierr = MatAppendOptionsPrefix(ptap->Ro,"inner_offdiag_");CHKERRQ(ierr);
1267   ierr = MatCreate(PETSC_COMM_SELF,&ptap->C_oth);CHKERRQ(ierr);
1268   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Ro,ptap->A_loc,fill,ptap->C_oth);CHKERRQ(ierr);
1269 
1270   /* (3) send coj of C_oth to other processors  */
1271   /* ------------------------------------------ */
1272   /* determine row ownership */
1273   ierr = PetscLayoutCreate(comm,&rowmap);CHKERRQ(ierr);
1274   rowmap->n  = pn;
1275   rowmap->bs = 1;
1276   ierr   = PetscLayoutSetUp(rowmap);CHKERRQ(ierr);
1277   owners = rowmap->range;
1278 
1279   /* determine the number of messages to send, their lengths */
1280   ierr = PetscMalloc4(size,&len_s,size,&len_si,size,&sstatus,size+2,&owners_co);CHKERRQ(ierr);
1281   ierr = PetscArrayzero(len_s,size);CHKERRQ(ierr);
1282   ierr = PetscArrayzero(len_si,size);CHKERRQ(ierr);
1283 
1284   c_oth = (Mat_SeqAIJ*)ptap->C_oth->data;
1285   coi   = c_oth->i; coj = c_oth->j;
1286   con   = ptap->C_oth->rmap->n;
1287   proc  = 0;
1288   for (i=0; i<con; i++) {
1289     while (prmap[i] >= owners[proc+1]) proc++;
1290     len_si[proc]++;               /* num of rows in Co(=Pt*A) to be sent to [proc] */
1291     len_s[proc] += coi[i+1] - coi[i]; /* num of nonzeros in Co to be sent to [proc] */
1292   }
1293 
1294   len          = 0; /* max length of buf_si[], see (4) */
1295   owners_co[0] = 0;
1296   nsend        = 0;
1297   for (proc=0; proc<size; proc++) {
1298     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1299     if (len_s[proc]) {
1300       nsend++;
1301       len_si[proc] = 2*(len_si[proc] + 1); /* length of buf_si to be sent to [proc] */
1302       len         += len_si[proc];
1303     }
1304   }
1305 
1306   /* determine the number and length of messages to receive for coi and coj  */
1307   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&nrecv);CHKERRQ(ierr);
1308   ierr = PetscGatherMessageLengths2(comm,nsend,nrecv,len_s,len_si,&id_r,&len_r,&len_ri);CHKERRQ(ierr);
1309 
1310   /* post the Irecv and Isend of coj */
1311   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1312   ierr = PetscPostIrecvInt(comm,tagj,nrecv,id_r,len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1313   ierr = PetscMalloc1(nsend+1,&swaits);CHKERRQ(ierr);
1314   for (proc=0, k=0; proc<size; proc++) {
1315     if (!len_s[proc]) continue;
1316     i    = owners_co[proc];
1317     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1318     k++;
1319   }
1320 
1321   /* (2-2) compute symbolic C_loc = Rd*A_loc */
1322   /* ---------------------------------------- */
1323   ierr = MatSetOptionsPrefix(ptap->Rd,prefix);CHKERRQ(ierr);
1324   ierr = MatAppendOptionsPrefix(ptap->Rd,"inner_diag_");CHKERRQ(ierr);
1325   ierr = MatCreate(PETSC_COMM_SELF,&ptap->C_loc);CHKERRQ(ierr);
1326   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(ptap->Rd,ptap->A_loc,fill,ptap->C_loc);CHKERRQ(ierr);
1327   c_loc = (Mat_SeqAIJ*)ptap->C_loc->data;
1328 
1329   /* receives coj are complete */
1330   for (i=0; i<nrecv; i++) {
1331     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1332   }
1333   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1334   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
1335 
1336   /* add received column indices into ta to update Crmax */
1337   a_loc = (Mat_SeqAIJ*)(ptap->A_loc)->data;
1338 
1339   /* create and initialize a linked list */
1340   ierr = PetscTableCreate(an,aN,&ta);CHKERRQ(ierr); /* for compute Crmax */
1341   MatRowMergeMax_SeqAIJ(a_loc,ptap->A_loc->rmap->N,ta);
1342 
1343   for (k=0; k<nrecv; k++) {/* k-th received message */
1344     Jptr = buf_rj[k];
1345     for (j=0; j<len_r[k]; j++) {
1346       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
1347     }
1348   }
1349   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
1350   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
1351 
1352   /* (4) send and recv coi */
1353   /*-----------------------*/
1354   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1355   ierr   = PetscPostIrecvInt(comm,tagi,nrecv,id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1356   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1357   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1358   for (proc=0,k=0; proc<size; proc++) {
1359     if (!len_s[proc]) continue;
1360     /* form outgoing message for i-structure:
1361          buf_si[0]:                 nrows to be sent
1362                [1:nrows]:           row index (global)
1363                [nrows+1:2*nrows+1]: i-structure index
1364     */
1365     /*-------------------------------------------*/
1366     nrows       = len_si[proc]/2 - 1; /* num of rows in Co to be sent to [proc] */
1367     buf_si_i    = buf_si + nrows+1;
1368     buf_si[0]   = nrows;
1369     buf_si_i[0] = 0;
1370     nrows       = 0;
1371     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1372       nzi = coi[i+1] - coi[i];
1373       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1374       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1375       nrows++;
1376     }
1377     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1378     k++;
1379     buf_si += len_si[proc];
1380   }
1381   for (i=0; i<nrecv; i++) {
1382     ierr = MPI_Waitany(nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1383   }
1384   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1385   if (nsend) {ierr = MPI_Waitall(nsend,swaits,sstatus);CHKERRQ(ierr);}
1386 
1387   ierr = PetscFree4(len_s,len_si,sstatus,owners_co);CHKERRQ(ierr);
1388   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1389   ierr = PetscFree(swaits);CHKERRQ(ierr);
1390   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1391 
1392   /* (5) compute the local portion of C      */
1393   /* ------------------------------------------ */
1394   /* set initial free space to be Crmax, sufficient for holding nozeros in each row of C */
1395   ierr          = PetscFreeSpaceGet(Crmax,&free_space);CHKERRQ(ierr);
1396   current_space = free_space;
1397 
1398   ierr = PetscMalloc3(nrecv,&buf_ri_k,nrecv,&nextrow,nrecv,&nextci);CHKERRQ(ierr);
1399   for (k=0; k<nrecv; k++) {
1400     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1401     nrows       = *buf_ri_k[k];
1402     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1403     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1404   }
1405 
1406   ierr = MatPreallocateInitialize(comm,pn,an,dnz,onz);CHKERRQ(ierr);
1407   ierr = PetscLLCondensedCreate(Crmax,aN,&lnk,&lnkbt);CHKERRQ(ierr);
1408   for (i=0; i<pn; i++) {
1409     /* add C_loc into C */
1410     nzi  = c_loc->i[i+1] - c_loc->i[i];
1411     Jptr = c_loc->j + c_loc->i[i];
1412     ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1413 
1414     /* add received col data into lnk */
1415     for (k=0; k<nrecv; k++) { /* k-th received message */
1416       if (i == *nextrow[k]) { /* i-th row */
1417         nzi  = *(nextci[k]+1) - *nextci[k];
1418         Jptr = buf_rj[k] + *nextci[k];
1419         ierr = PetscLLCondensedAddSorted(nzi,Jptr,lnk,lnkbt);CHKERRQ(ierr);
1420         nextrow[k]++; nextci[k]++;
1421       }
1422     }
1423     nzi = lnk[0];
1424 
1425     /* copy data into free space, then initialize lnk */
1426     ierr = PetscLLCondensedClean(aN,nzi,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
1427     ierr = MatPreallocateSet(i+owners[rank],nzi,current_space->array,dnz,onz);CHKERRQ(ierr);
1428   }
1429   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1430   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1431   ierr = PetscFreeSpaceDestroy(free_space);CHKERRQ(ierr);
1432 
1433   /* local sizes and preallocation */
1434   ierr = MatSetSizes(C,pn,an,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1435   if (P->cmap->bs > 0) {ierr = PetscLayoutSetBlockSize(C->rmap,P->cmap->bs);CHKERRQ(ierr);}
1436   if (A->cmap->bs > 0) {ierr = PetscLayoutSetBlockSize(C->cmap,A->cmap->bs);CHKERRQ(ierr);}
1437   ierr = MatMPIAIJSetPreallocation(C,0,dnz,0,onz);CHKERRQ(ierr);
1438   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1439 
1440   /* members in merge */
1441   ierr = PetscFree(id_r);CHKERRQ(ierr);
1442   ierr = PetscFree(len_r);CHKERRQ(ierr);
1443   ierr = PetscFree(buf_ri[0]);CHKERRQ(ierr);
1444   ierr = PetscFree(buf_ri);CHKERRQ(ierr);
1445   ierr = PetscFree(buf_rj[0]);CHKERRQ(ierr);
1446   ierr = PetscFree(buf_rj);CHKERRQ(ierr);
1447   ierr = PetscLayoutDestroy(&rowmap);CHKERRQ(ierr);
1448 
1449   /* attach the supporting struct to C for reuse */
1450   C->product->data    = ptap;
1451   C->product->destroy = MatDestroy_MPIAIJ_PtAP;
1452 
1453   /* C is not ready for use - assembly will be done by MatPtAPNumeric() */
1454   C->assembled = PETSC_FALSE;
1455   PetscFunctionReturn(0);
1456 }
1457 
1458 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_nonscalable(Mat P,Mat A,Mat C)
1459 {
1460   PetscErrorCode    ierr;
1461   Mat_MPIAIJ        *p=(Mat_MPIAIJ*)P->data;
1462   Mat_SeqAIJ        *c_seq;
1463   Mat_APMPI         *ptap;
1464   Mat               A_loc,C_loc,C_oth;
1465   PetscInt          i,rstart,rend,cm,ncols,row;
1466   const PetscInt    *cols;
1467   const PetscScalar *vals;
1468 
1469   PetscFunctionBegin;
1470   MatCheckProduct(C,3);
1471   ptap = (Mat_APMPI*)C->product->data;
1472   if (!ptap) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be computed. Missing data");
1473   if (!ptap->A_loc) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtA cannot be reused. Do not call MatProductClear()");
1474   ierr = MatZeroEntries(C);CHKERRQ(ierr);
1475 
1476   if (ptap->reuse == MAT_REUSE_MATRIX) {
1477     /* These matrices are obtained in MatTransposeMatMultSymbolic() */
1478     /* 1) get R = Pd^T, Ro = Po^T */
1479     /*----------------------------*/
1480     ierr = MatTranspose_SeqAIJ(p->A,MAT_REUSE_MATRIX,&ptap->Rd);CHKERRQ(ierr);
1481     ierr = MatTranspose_SeqAIJ(p->B,MAT_REUSE_MATRIX,&ptap->Ro);CHKERRQ(ierr);
1482 
1483     /* 2) compute numeric A_loc */
1484     /*--------------------------*/
1485     ierr = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&ptap->A_loc);CHKERRQ(ierr);
1486   }
1487 
1488   /* 3) C_loc = Rd*A_loc, C_oth = Ro*A_loc */
1489   A_loc = ptap->A_loc;
1490   ierr = ((ptap->C_loc)->ops->matmultnumeric)(ptap->Rd,A_loc,ptap->C_loc);CHKERRQ(ierr);
1491   ierr = ((ptap->C_oth)->ops->matmultnumeric)(ptap->Ro,A_loc,ptap->C_oth);CHKERRQ(ierr);
1492   C_loc = ptap->C_loc;
1493   C_oth = ptap->C_oth;
1494 
1495   /* add C_loc and Co to to C */
1496   ierr = MatGetOwnershipRange(C,&rstart,&rend);CHKERRQ(ierr);
1497 
1498   /* C_loc -> C */
1499   cm    = C_loc->rmap->N;
1500   c_seq = (Mat_SeqAIJ*)C_loc->data;
1501   cols = c_seq->j;
1502   vals = c_seq->a;
1503   for (i=0; i<cm; i++) {
1504     ncols = c_seq->i[i+1] - c_seq->i[i];
1505     row = rstart + i;
1506     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1507     cols += ncols; vals += ncols;
1508   }
1509 
1510   /* Co -> C, off-processor part */
1511   cm    = C_oth->rmap->N;
1512   c_seq = (Mat_SeqAIJ*)C_oth->data;
1513   cols  = c_seq->j;
1514   vals  = c_seq->a;
1515   for (i=0; i<cm; i++) {
1516     ncols = c_seq->i[i+1] - c_seq->i[i];
1517     row = p->garray[i];
1518     ierr = MatSetValues(C,1,&row,ncols,cols,vals,ADD_VALUES);CHKERRQ(ierr);
1519     cols += ncols; vals += ncols;
1520   }
1521   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1522   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1523 
1524   ptap->reuse = MAT_REUSE_MATRIX;
1525   PetscFunctionReturn(0);
1526 }
1527 
1528 PetscErrorCode MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ(Mat P,Mat A,Mat C)
1529 {
1530   PetscErrorCode      ierr;
1531   Mat_Merge_SeqsToMPI *merge;
1532   Mat_MPIAIJ          *p =(Mat_MPIAIJ*)P->data;
1533   Mat_SeqAIJ          *pd=(Mat_SeqAIJ*)(p->A)->data,*po=(Mat_SeqAIJ*)(p->B)->data;
1534   Mat_APMPI           *ptap;
1535   PetscInt            *adj;
1536   PetscInt            i,j,k,anz,pnz,row,*cj,nexta;
1537   MatScalar           *ada,*ca,valtmp;
1538   PetscInt            am=A->rmap->n,cm=C->rmap->n,pon=(p->B)->cmap->n;
1539   MPI_Comm            comm;
1540   PetscMPIInt         size,rank,taga,*len_s;
1541   PetscInt            *owners,proc,nrows,**buf_ri_k,**nextrow,**nextci;
1542   PetscInt            **buf_ri,**buf_rj;
1543   PetscInt            cnz=0,*bj_i,*bi,*bj,bnz,nextcj;  /* bi,bj,ba: local array of C(mpi mat) */
1544   MPI_Request         *s_waits,*r_waits;
1545   MPI_Status          *status;
1546   MatScalar           **abuf_r,*ba_i,*pA,*coa,*ba;
1547   PetscInt            *ai,*aj,*coi,*coj,*poJ,*pdJ;
1548   Mat                 A_loc;
1549   Mat_SeqAIJ          *a_loc;
1550 
1551   PetscFunctionBegin;
1552   MatCheckProduct(C,3);
1553   ptap = (Mat_APMPI*)C->product->data;
1554   if (!ptap) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtAP cannot be computed. Missing data");
1555   if (!ptap->A_loc) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"PtA cannot be reused. Do not call MatProductClear()");
1556   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
1557   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1558   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1559 
1560   merge = ptap->merge;
1561 
1562   /* 2) compute numeric C_seq = P_loc^T*A_loc */
1563   /*------------------------------------------*/
1564   /* get data from symbolic products */
1565   coi    = merge->coi; coj = merge->coj;
1566   ierr   = PetscCalloc1(coi[pon]+1,&coa);CHKERRQ(ierr);
1567   bi     = merge->bi; bj = merge->bj;
1568   owners = merge->rowmap->range;
1569   ierr   = PetscCalloc1(bi[cm]+1,&ba);CHKERRQ(ierr);
1570 
1571   /* get A_loc by taking all local rows of A */
1572   A_loc = ptap->A_loc;
1573   ierr  = MatMPIAIJGetLocalMat(A,MAT_REUSE_MATRIX,&A_loc);CHKERRQ(ierr);
1574   a_loc = (Mat_SeqAIJ*)(A_loc)->data;
1575   ai    = a_loc->i;
1576   aj    = a_loc->j;
1577 
1578   for (i=0; i<am; i++) {
1579     anz = ai[i+1] - ai[i];
1580     adj = aj + ai[i];
1581     ada = a_loc->a + ai[i];
1582 
1583     /* 2-b) Compute Cseq = P_loc[i,:]^T*A[i,:] using outer product */
1584     /*-------------------------------------------------------------*/
1585     /* put the value into Co=(p->B)^T*A (off-diagonal part, send to others) */
1586     pnz = po->i[i+1] - po->i[i];
1587     poJ = po->j + po->i[i];
1588     pA  = po->a + po->i[i];
1589     for (j=0; j<pnz; j++) {
1590       row = poJ[j];
1591       cj  = coj + coi[row];
1592       ca  = coa + coi[row];
1593       /* perform sparse axpy */
1594       nexta  = 0;
1595       valtmp = pA[j];
1596       for (k=0; nexta<anz; k++) {
1597         if (cj[k] == adj[nexta]) {
1598           ca[k] += valtmp*ada[nexta];
1599           nexta++;
1600         }
1601       }
1602       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1603     }
1604 
1605     /* put the value into Cd (diagonal part) */
1606     pnz = pd->i[i+1] - pd->i[i];
1607     pdJ = pd->j + pd->i[i];
1608     pA  = pd->a + pd->i[i];
1609     for (j=0; j<pnz; j++) {
1610       row = pdJ[j];
1611       cj  = bj + bi[row];
1612       ca  = ba + bi[row];
1613       /* perform sparse axpy */
1614       nexta  = 0;
1615       valtmp = pA[j];
1616       for (k=0; nexta<anz; k++) {
1617         if (cj[k] == adj[nexta]) {
1618           ca[k] += valtmp*ada[nexta];
1619           nexta++;
1620         }
1621       }
1622       ierr = PetscLogFlops(2.0*anz);CHKERRQ(ierr);
1623     }
1624   }
1625 
1626   /* 3) send and recv matrix values coa */
1627   /*------------------------------------*/
1628   buf_ri = merge->buf_ri;
1629   buf_rj = merge->buf_rj;
1630   len_s  = merge->len_s;
1631   ierr   = PetscCommGetNewTag(comm,&taga);CHKERRQ(ierr);
1632   ierr   = PetscPostIrecvScalar(comm,taga,merge->nrecv,merge->id_r,merge->len_r,&abuf_r,&r_waits);CHKERRQ(ierr);
1633 
1634   ierr = PetscMalloc2(merge->nsend+1,&s_waits,size,&status);CHKERRQ(ierr);
1635   for (proc=0,k=0; proc<size; proc++) {
1636     if (!len_s[proc]) continue;
1637     i    = merge->owners_co[proc];
1638     ierr = MPI_Isend(coa+coi[i],len_s[proc],MPIU_MATSCALAR,proc,taga,comm,s_waits+k);CHKERRQ(ierr);
1639     k++;
1640   }
1641   if (merge->nrecv) {ierr = MPI_Waitall(merge->nrecv,r_waits,status);CHKERRQ(ierr);}
1642   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,s_waits,status);CHKERRQ(ierr);}
1643 
1644   ierr = PetscFree2(s_waits,status);CHKERRQ(ierr);
1645   ierr = PetscFree(r_waits);CHKERRQ(ierr);
1646   ierr = PetscFree(coa);CHKERRQ(ierr);
1647 
1648   /* 4) insert local Cseq and received values into Cmpi */
1649   /*----------------------------------------------------*/
1650   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1651   for (k=0; k<merge->nrecv; k++) {
1652     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1653     nrows       = *(buf_ri_k[k]);
1654     nextrow[k]  = buf_ri_k[k]+1;  /* next row number of k-th recved i-structure */
1655     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* poins to the next i-structure of k-th recved i-structure  */
1656   }
1657 
1658   for (i=0; i<cm; i++) {
1659     row  = owners[rank] + i; /* global row index of C_seq */
1660     bj_i = bj + bi[i];  /* col indices of the i-th row of C */
1661     ba_i = ba + bi[i];
1662     bnz  = bi[i+1] - bi[i];
1663     /* add received vals into ba */
1664     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1665       /* i-th row */
1666       if (i == *nextrow[k]) {
1667         cnz    = *(nextci[k]+1) - *nextci[k];
1668         cj     = buf_rj[k] + *(nextci[k]);
1669         ca     = abuf_r[k] + *(nextci[k]);
1670         nextcj = 0;
1671         for (j=0; nextcj<cnz; j++) {
1672           if (bj_i[j] == cj[nextcj]) { /* bcol == ccol */
1673             ba_i[j] += ca[nextcj++];
1674           }
1675         }
1676         nextrow[k]++; nextci[k]++;
1677         ierr = PetscLogFlops(2.0*cnz);CHKERRQ(ierr);
1678       }
1679     }
1680     ierr = MatSetValues(C,1,&row,bnz,bj_i,ba_i,INSERT_VALUES);CHKERRQ(ierr);
1681   }
1682   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1683   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1684 
1685   ierr = PetscFree(ba);CHKERRQ(ierr);
1686   ierr = PetscFree(abuf_r[0]);CHKERRQ(ierr);
1687   ierr = PetscFree(abuf_r);CHKERRQ(ierr);
1688   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1689   PetscFunctionReturn(0);
1690 }
1691 
1692 PetscErrorCode MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(Mat P,Mat A,PetscReal fill,Mat C)
1693 {
1694   PetscErrorCode      ierr;
1695   Mat                 A_loc,POt,PDt;
1696   Mat_APMPI           *ptap;
1697   PetscFreeSpaceList  free_space=NULL,current_space=NULL;
1698   Mat_MPIAIJ          *p=(Mat_MPIAIJ*)P->data,*a=(Mat_MPIAIJ*)A->data;
1699   PetscInt            *pdti,*pdtj,*poti,*potj,*ptJ;
1700   PetscInt            nnz;
1701   PetscInt            *lnk,*owners_co,*coi,*coj,i,k,pnz,row;
1702   PetscInt            am  =A->rmap->n,pn=P->cmap->n;
1703   MPI_Comm            comm;
1704   PetscMPIInt         size,rank,tagi,tagj,*len_si,*len_s,*len_ri;
1705   PetscInt            **buf_rj,**buf_ri,**buf_ri_k;
1706   PetscInt            len,proc,*dnz,*onz,*owners;
1707   PetscInt            nzi,*bi,*bj;
1708   PetscInt            nrows,*buf_s,*buf_si,*buf_si_i,**nextrow,**nextci;
1709   MPI_Request         *swaits,*rwaits;
1710   MPI_Status          *sstatus,rstatus;
1711   Mat_Merge_SeqsToMPI *merge;
1712   PetscInt            *ai,*aj,*Jptr,anz,*prmap=p->garray,pon,nspacedouble=0,j;
1713   PetscReal           afill  =1.0,afill_tmp;
1714   PetscInt            rstart = P->cmap->rstart,rmax,aN=A->cmap->N,Armax;
1715   Mat_SeqAIJ          *a_loc,*pdt,*pot;
1716   PetscTable          ta;
1717   MatType             mtype;
1718 
1719   PetscFunctionBegin;
1720   ierr = PetscObjectGetComm((PetscObject)A,&comm);CHKERRQ(ierr);
1721   /* check if matrix local sizes are compatible */
1722   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);
1723 
1724   ierr = MPI_Comm_size(comm,&size);CHKERRQ(ierr);
1725   ierr = MPI_Comm_rank(comm,&rank);CHKERRQ(ierr);
1726 
1727   /* create struct Mat_APMPI and attached it to C later */
1728   ierr = PetscNew(&ptap);CHKERRQ(ierr);
1729 
1730   /* get A_loc by taking all local rows of A */
1731   ierr = MatMPIAIJGetLocalMat(A,MAT_INITIAL_MATRIX,&A_loc);CHKERRQ(ierr);
1732 
1733   ptap->A_loc = A_loc;
1734   a_loc       = (Mat_SeqAIJ*)(A_loc)->data;
1735   ai          = a_loc->i;
1736   aj          = a_loc->j;
1737 
1738   /* determine symbolic Co=(p->B)^T*A - send to others */
1739   /*----------------------------------------------------*/
1740   ierr = MatTransposeSymbolic_SeqAIJ(p->A,&PDt);CHKERRQ(ierr);
1741   pdt  = (Mat_SeqAIJ*)PDt->data;
1742   pdti = pdt->i; pdtj = pdt->j;
1743 
1744   ierr = MatTransposeSymbolic_SeqAIJ(p->B,&POt);CHKERRQ(ierr);
1745   pot  = (Mat_SeqAIJ*)POt->data;
1746   poti = pot->i; potj = pot->j;
1747 
1748   /* then, compute symbolic Co = (p->B)^T*A */
1749   pon = (p->B)->cmap->n; /* total num of rows to be sent to other processors
1750                          >= (num of nonzero rows of C_seq) - pn */
1751   ierr   = PetscMalloc1(pon+1,&coi);CHKERRQ(ierr);
1752   coi[0] = 0;
1753 
1754   /* set initial free space to be fill*(nnz(p->B) + nnz(A)) */
1755   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(poti[pon],ai[am]));
1756   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1757   current_space = free_space;
1758 
1759   /* create and initialize a linked list */
1760   ierr = PetscTableCreate(A->cmap->n + a->B->cmap->N,aN,&ta);CHKERRQ(ierr);
1761   MatRowMergeMax_SeqAIJ(a_loc,am,ta);
1762   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
1763 
1764   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
1765 
1766   for (i=0; i<pon; i++) {
1767     pnz = poti[i+1] - poti[i];
1768     ptJ = potj + poti[i];
1769     for (j=0; j<pnz; j++) {
1770       row  = ptJ[j]; /* row of A_loc == col of Pot */
1771       anz  = ai[row+1] - ai[row];
1772       Jptr = aj + ai[row];
1773       /* add non-zero cols of AP into the sorted linked list lnk */
1774       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1775     }
1776     nnz = lnk[0];
1777 
1778     /* If free space is not available, double the total space in the list */
1779     if (current_space->local_remaining<nnz) {
1780       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1781       nspacedouble++;
1782     }
1783 
1784     /* Copy data into free space, and zero out denserows */
1785     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1786 
1787     current_space->array           += nnz;
1788     current_space->local_used      += nnz;
1789     current_space->local_remaining -= nnz;
1790 
1791     coi[i+1] = coi[i] + nnz;
1792   }
1793 
1794   ierr = PetscMalloc1(coi[pon]+1,&coj);CHKERRQ(ierr);
1795   ierr = PetscFreeSpaceContiguous(&free_space,coj);CHKERRQ(ierr);
1796   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr); /* must destroy to get a new one for C */
1797 
1798   afill_tmp = (PetscReal)coi[pon]/(poti[pon] + ai[am]+1);
1799   if (afill_tmp > afill) afill = afill_tmp;
1800 
1801   /* send j-array (coj) of Co to other processors */
1802   /*----------------------------------------------*/
1803   /* determine row ownership */
1804   ierr = PetscNew(&merge);CHKERRQ(ierr);
1805   ierr = PetscLayoutCreate(comm,&merge->rowmap);CHKERRQ(ierr);
1806 
1807   merge->rowmap->n  = pn;
1808   merge->rowmap->bs = 1;
1809 
1810   ierr   = PetscLayoutSetUp(merge->rowmap);CHKERRQ(ierr);
1811   owners = merge->rowmap->range;
1812 
1813   /* determine the number of messages to send, their lengths */
1814   ierr = PetscCalloc1(size,&len_si);CHKERRQ(ierr);
1815   ierr = PetscCalloc1(size,&merge->len_s);CHKERRQ(ierr);
1816 
1817   len_s        = merge->len_s;
1818   merge->nsend = 0;
1819 
1820   ierr = PetscMalloc1(size+2,&owners_co);CHKERRQ(ierr);
1821 
1822   proc = 0;
1823   for (i=0; i<pon; i++) {
1824     while (prmap[i] >= owners[proc+1]) proc++;
1825     len_si[proc]++;  /* num of rows in Co to be sent to [proc] */
1826     len_s[proc] += coi[i+1] - coi[i];
1827   }
1828 
1829   len          = 0; /* max length of buf_si[] */
1830   owners_co[0] = 0;
1831   for (proc=0; proc<size; proc++) {
1832     owners_co[proc+1] = owners_co[proc] + len_si[proc];
1833     if (len_si[proc]) {
1834       merge->nsend++;
1835       len_si[proc] = 2*(len_si[proc] + 1);
1836       len         += len_si[proc];
1837     }
1838   }
1839 
1840   /* determine the number and length of messages to receive for coi and coj  */
1841   ierr = PetscGatherNumberOfMessages(comm,NULL,len_s,&merge->nrecv);CHKERRQ(ierr);
1842   ierr = PetscGatherMessageLengths2(comm,merge->nsend,merge->nrecv,len_s,len_si,&merge->id_r,&merge->len_r,&len_ri);CHKERRQ(ierr);
1843 
1844   /* post the Irecv and Isend of coj */
1845   ierr = PetscCommGetNewTag(comm,&tagj);CHKERRQ(ierr);
1846   ierr = PetscPostIrecvInt(comm,tagj,merge->nrecv,merge->id_r,merge->len_r,&buf_rj,&rwaits);CHKERRQ(ierr);
1847   ierr = PetscMalloc1(merge->nsend+1,&swaits);CHKERRQ(ierr);
1848   for (proc=0, k=0; proc<size; proc++) {
1849     if (!len_s[proc]) continue;
1850     i    = owners_co[proc];
1851     ierr = MPI_Isend(coj+coi[i],len_s[proc],MPIU_INT,proc,tagj,comm,swaits+k);CHKERRQ(ierr);
1852     k++;
1853   }
1854 
1855   /* receives and sends of coj are complete */
1856   ierr = PetscMalloc1(size,&sstatus);CHKERRQ(ierr);
1857   for (i=0; i<merge->nrecv; i++) {
1858     PetscMPIInt icompleted;
1859     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1860   }
1861   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1862   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1863 
1864   /* add received column indices into table to update Armax */
1865   /* Armax can be as large as aN if a P[row,:] is dense, see src/ksp/ksp/tutorials/ex56.c! */
1866   for (k=0; k<merge->nrecv; k++) {/* k-th received message */
1867     Jptr = buf_rj[k];
1868     for (j=0; j<merge->len_r[k]; j++) {
1869       ierr = PetscTableAdd(ta,*(Jptr+j)+1,1,INSERT_VALUES);CHKERRQ(ierr);
1870     }
1871   }
1872   ierr = PetscTableGetCount(ta,&Armax);CHKERRQ(ierr);
1873   /* 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); */
1874 
1875   /* send and recv coi */
1876   /*-------------------*/
1877   ierr   = PetscCommGetNewTag(comm,&tagi);CHKERRQ(ierr);
1878   ierr   = PetscPostIrecvInt(comm,tagi,merge->nrecv,merge->id_r,len_ri,&buf_ri,&rwaits);CHKERRQ(ierr);
1879   ierr   = PetscMalloc1(len+1,&buf_s);CHKERRQ(ierr);
1880   buf_si = buf_s;  /* points to the beginning of k-th msg to be sent */
1881   for (proc=0,k=0; proc<size; proc++) {
1882     if (!len_s[proc]) continue;
1883     /* form outgoing message for i-structure:
1884          buf_si[0]:                 nrows to be sent
1885                [1:nrows]:           row index (global)
1886                [nrows+1:2*nrows+1]: i-structure index
1887     */
1888     /*-------------------------------------------*/
1889     nrows       = len_si[proc]/2 - 1;
1890     buf_si_i    = buf_si + nrows+1;
1891     buf_si[0]   = nrows;
1892     buf_si_i[0] = 0;
1893     nrows       = 0;
1894     for (i=owners_co[proc]; i<owners_co[proc+1]; i++) {
1895       nzi               = coi[i+1] - coi[i];
1896       buf_si_i[nrows+1] = buf_si_i[nrows] + nzi;  /* i-structure */
1897       buf_si[nrows+1]   = prmap[i] -owners[proc]; /* local row index */
1898       nrows++;
1899     }
1900     ierr = MPI_Isend(buf_si,len_si[proc],MPIU_INT,proc,tagi,comm,swaits+k);CHKERRQ(ierr);
1901     k++;
1902     buf_si += len_si[proc];
1903   }
1904   i = merge->nrecv;
1905   while (i--) {
1906     PetscMPIInt icompleted;
1907     ierr = MPI_Waitany(merge->nrecv,rwaits,&icompleted,&rstatus);CHKERRQ(ierr);
1908   }
1909   ierr = PetscFree(rwaits);CHKERRQ(ierr);
1910   if (merge->nsend) {ierr = MPI_Waitall(merge->nsend,swaits,sstatus);CHKERRQ(ierr);}
1911   ierr = PetscFree(len_si);CHKERRQ(ierr);
1912   ierr = PetscFree(len_ri);CHKERRQ(ierr);
1913   ierr = PetscFree(swaits);CHKERRQ(ierr);
1914   ierr = PetscFree(sstatus);CHKERRQ(ierr);
1915   ierr = PetscFree(buf_s);CHKERRQ(ierr);
1916 
1917   /* compute the local portion of C (mpi mat) */
1918   /*------------------------------------------*/
1919   /* allocate bi array and free space for accumulating nonzero column info */
1920   ierr  = PetscMalloc1(pn+1,&bi);CHKERRQ(ierr);
1921   bi[0] = 0;
1922 
1923   /* set initial free space to be fill*(nnz(P) + nnz(AP)) */
1924   nnz           = PetscRealIntMultTruncate(fill,PetscIntSumTruncate(pdti[pn],PetscIntSumTruncate(poti[pon],ai[am])));
1925   ierr          = PetscFreeSpaceGet(nnz,&free_space);CHKERRQ(ierr);
1926   current_space = free_space;
1927 
1928   ierr = PetscMalloc3(merge->nrecv,&buf_ri_k,merge->nrecv,&nextrow,merge->nrecv,&nextci);CHKERRQ(ierr);
1929   for (k=0; k<merge->nrecv; k++) {
1930     buf_ri_k[k] = buf_ri[k]; /* beginning of k-th recved i-structure */
1931     nrows       = *buf_ri_k[k];
1932     nextrow[k]  = buf_ri_k[k] + 1;  /* next row number of k-th recved i-structure */
1933     nextci[k]   = buf_ri_k[k] + (nrows + 1); /* points to the next i-structure of k-th received i-structure  */
1934   }
1935 
1936   ierr = PetscLLCondensedCreate_Scalable(Armax,&lnk);CHKERRQ(ierr);
1937   ierr = MatPreallocateInitialize(comm,pn,A->cmap->n,dnz,onz);CHKERRQ(ierr);
1938   rmax = 0;
1939   for (i=0; i<pn; i++) {
1940     /* add pdt[i,:]*AP into lnk */
1941     pnz = pdti[i+1] - pdti[i];
1942     ptJ = pdtj + pdti[i];
1943     for (j=0; j<pnz; j++) {
1944       row  = ptJ[j];  /* row of AP == col of Pt */
1945       anz  = ai[row+1] - ai[row];
1946       Jptr = aj + ai[row];
1947       /* add non-zero cols of AP into the sorted linked list lnk */
1948       ierr = PetscLLCondensedAddSorted_Scalable(anz,Jptr,lnk);CHKERRQ(ierr);
1949     }
1950 
1951     /* add received col data into lnk */
1952     for (k=0; k<merge->nrecv; k++) { /* k-th received message */
1953       if (i == *nextrow[k]) { /* i-th row */
1954         nzi  = *(nextci[k]+1) - *nextci[k];
1955         Jptr = buf_rj[k] + *nextci[k];
1956         ierr = PetscLLCondensedAddSorted_Scalable(nzi,Jptr,lnk);CHKERRQ(ierr);
1957         nextrow[k]++; nextci[k]++;
1958       }
1959     }
1960     nnz = lnk[0];
1961 
1962     /* if free space is not available, make more free space */
1963     if (current_space->local_remaining<nnz) {
1964       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(nnz,current_space->total_array_size),&current_space);CHKERRQ(ierr);
1965       nspacedouble++;
1966     }
1967     /* copy data into free space, then initialize lnk */
1968     ierr = PetscLLCondensedClean_Scalable(nnz,current_space->array,lnk);CHKERRQ(ierr);
1969     ierr = MatPreallocateSet(i+owners[rank],nnz,current_space->array,dnz,onz);CHKERRQ(ierr);
1970 
1971     current_space->array           += nnz;
1972     current_space->local_used      += nnz;
1973     current_space->local_remaining -= nnz;
1974 
1975     bi[i+1] = bi[i] + nnz;
1976     if (nnz > rmax) rmax = nnz;
1977   }
1978   ierr = PetscFree3(buf_ri_k,nextrow,nextci);CHKERRQ(ierr);
1979 
1980   ierr      = PetscMalloc1(bi[pn]+1,&bj);CHKERRQ(ierr);
1981   ierr      = PetscFreeSpaceContiguous(&free_space,bj);CHKERRQ(ierr);
1982   afill_tmp = (PetscReal)bi[pn]/(pdti[pn] + poti[pon] + ai[am]+1);
1983   if (afill_tmp > afill) afill = afill_tmp;
1984   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
1985   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
1986 
1987   ierr = MatDestroy(&POt);CHKERRQ(ierr);
1988   ierr = MatDestroy(&PDt);CHKERRQ(ierr);
1989 
1990   /* create symbolic parallel matrix C - why cannot be assembled in Numeric part   */
1991   /*-------------------------------------------------------------------------------*/
1992   ierr = MatSetSizes(C,pn,A->cmap->n,PETSC_DETERMINE,PETSC_DETERMINE);CHKERRQ(ierr);
1993   ierr = MatSetBlockSizes(C,PetscAbs(P->cmap->bs),PetscAbs(A->cmap->bs));CHKERRQ(ierr);
1994   ierr = MatGetType(A,&mtype);CHKERRQ(ierr);
1995   ierr = MatSetType(C,mtype);CHKERRQ(ierr);
1996   ierr = MatMPIAIJSetPreallocation(C,0,dnz,0,onz);CHKERRQ(ierr);
1997   ierr = MatPreallocateFinalize(dnz,onz);CHKERRQ(ierr);
1998   ierr = MatSetBlockSize(C,1);CHKERRQ(ierr);
1999   for (i=0; i<pn; i++) {
2000     row  = i + rstart;
2001     nnz  = bi[i+1] - bi[i];
2002     Jptr = bj + bi[i];
2003     ierr = MatSetValues(C,1,&row,nnz,Jptr,NULL,INSERT_VALUES);CHKERRQ(ierr);
2004   }
2005   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2006   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
2007   merge->bi        = bi;
2008   merge->bj        = bj;
2009   merge->coi       = coi;
2010   merge->coj       = coj;
2011   merge->buf_ri    = buf_ri;
2012   merge->buf_rj    = buf_rj;
2013   merge->owners_co = owners_co;
2014 
2015   /* attach the supporting struct to C for reuse */
2016   C->product->data    = ptap;
2017   C->product->destroy = MatDestroy_MPIAIJ_PtAP;
2018   ptap->merge         = merge;
2019 
2020   C->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ;
2021 
2022 #if defined(PETSC_USE_INFO)
2023   if (bi[pn] != 0) {
2024     ierr = PetscInfo3(C,"Reallocs %D; Fill ratio: given %g needed %g.\n",nspacedouble,(double)fill,(double)afill);CHKERRQ(ierr);
2025     ierr = PetscInfo1(C,"Use MatTransposeMatMult(A,B,MatReuse,%g,&C) for best performance.\n",(double)afill);CHKERRQ(ierr);
2026   } else {
2027     ierr = PetscInfo(C,"Empty matrix product\n");CHKERRQ(ierr);
2028   }
2029 #endif
2030   PetscFunctionReturn(0);
2031 }
2032 
2033 /* ---------------------------------------------------------------- */
2034 static PetscErrorCode MatProductSymbolic_AtB_MPIAIJ_MPIAIJ(Mat C)
2035 {
2036   PetscErrorCode ierr;
2037   Mat_Product    *product = C->product;
2038   Mat            A=product->A,B=product->B;
2039   PetscReal      fill=product->fill;
2040   PetscBool      flg;
2041 
2042   PetscFunctionBegin;
2043   /* scalable */
2044   ierr = PetscStrcmp(product->alg,"scalable",&flg);CHKERRQ(ierr);
2045   if (flg) {
2046     ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ(A,B,fill,C);CHKERRQ(ierr);
2047     goto next;
2048   }
2049 
2050   /* nonscalable */
2051   ierr = PetscStrcmp(product->alg,"nonscalable",&flg);CHKERRQ(ierr);
2052   if (flg) {
2053     ierr = MatTransposeMatMultSymbolic_MPIAIJ_MPIAIJ_nonscalable(A,B,fill,C);CHKERRQ(ierr);
2054     goto next;
2055   }
2056 
2057   /* matmatmult */
2058   ierr = PetscStrcmp(product->alg,"at*b",&flg);CHKERRQ(ierr);
2059   if (flg) {
2060     Mat       At;
2061     Mat_APMPI *ptap;
2062 
2063     ierr = MatTranspose(A,MAT_INITIAL_MATRIX,&At);CHKERRQ(ierr);
2064     ierr = MatMatMultSymbolic_MPIAIJ_MPIAIJ(At,B,fill,C);CHKERRQ(ierr);
2065     ptap = (Mat_APMPI*)C->product->data;
2066     if (ptap) {
2067       ptap->Pt = At;
2068       C->product->destroy = MatDestroy_MPIAIJ_PtAP;
2069     }
2070     C->ops->transposematmultnumeric = MatTransposeMatMultNumeric_MPIAIJ_MPIAIJ_matmatmult;
2071     goto next;
2072   }
2073 
2074   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"MatProduct type is not supported");
2075 
2076 next:
2077   C->ops->productnumeric = MatProductNumeric_AtB;
2078   PetscFunctionReturn(0);
2079 }
2080 
2081 /* ---------------------------------------------------------------- */
2082 /* Set options for MatMatMultxxx_MPIAIJ_MPIAIJ */
2083 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_AB(Mat C)
2084 {
2085   PetscErrorCode ierr;
2086   Mat_Product    *product = C->product;
2087   Mat            A=product->A,B=product->B;
2088 #if defined(PETSC_HAVE_HYPRE)
2089   const char     *algTypes[4] = {"scalable","nonscalable","seqmpi","hypre"};
2090   PetscInt       nalg = 4;
2091 #else
2092   const char     *algTypes[3] = {"scalable","nonscalable","seqmpi"};
2093   PetscInt       nalg = 3;
2094 #endif
2095   PetscInt       alg = 1; /* set nonscalable algorithm as default */
2096   PetscBool      flg;
2097   MPI_Comm       comm;
2098 
2099   PetscFunctionBegin;
2100   /* Check matrix local sizes */
2101   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
2102   if (A->cmap->rstart != B->rmap->rstart || A->cmap->rend != B->rmap->rend) SETERRQ4(PETSC_COMM_SELF,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);
2103 
2104   /* Set "nonscalable" as default algorithm */
2105   ierr = PetscStrcmp(C->product->alg,"default",&flg);CHKERRQ(ierr);
2106   if (flg) {
2107     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2108 
2109     /* Set "scalable" as default if BN and local nonzeros of A and B are large */
2110     if (B->cmap->N > 100000) { /* may switch to scalable algorithm as default */
2111       MatInfo     Ainfo,Binfo;
2112       PetscInt    nz_local;
2113       PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
2114 
2115       ierr = MatGetInfo(A,MAT_LOCAL,&Ainfo);CHKERRQ(ierr);
2116       ierr = MatGetInfo(B,MAT_LOCAL,&Binfo);CHKERRQ(ierr);
2117       nz_local = (PetscInt)(Ainfo.nz_allocated + Binfo.nz_allocated);
2118 
2119       if (B->cmap->N > product->fill*nz_local) alg_scalable_loc = PETSC_TRUE;
2120       ierr = MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm);CHKERRQ(ierr);
2121 
2122       if (alg_scalable) {
2123         alg  = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
2124         ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2125         ierr = PetscInfo2(B,"Use scalable algorithm, BN %D, fill*nz_allocated %g\n",B->cmap->N,product->fill*nz_local);CHKERRQ(ierr);
2126       }
2127     }
2128   }
2129 
2130   /* Get runtime option */
2131   if (product->api_user) {
2132     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatMatMult","Mat");CHKERRQ(ierr);
2133     ierr = PetscOptionsEList("-matmatmult_via","Algorithmic approach","MatMatMult",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2134     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2135   } else {
2136     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_AB","Mat");CHKERRQ(ierr);
2137     ierr = PetscOptionsEList("-matproduct_ab_via","Algorithmic approach","MatMatMult",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2138     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2139   }
2140   if (flg) {
2141     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2142   }
2143 
2144   C->ops->productsymbolic = MatProductSymbolic_AB_MPIAIJ_MPIAIJ;
2145   PetscFunctionReturn(0);
2146 }
2147 
2148 /* Set options for MatTransposeMatMultXXX_MPIAIJ_MPIAIJ */
2149 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_AtB(Mat C)
2150 {
2151   PetscErrorCode ierr;
2152   Mat_Product    *product = C->product;
2153   Mat            A=product->A,B=product->B;
2154   const char     *algTypes[3] = {"scalable","nonscalable","at*b"};
2155   PetscInt       nalg = 3;
2156   PetscInt       alg = 1; /* set default algorithm  */
2157   PetscBool      flg;
2158   MPI_Comm       comm;
2159 
2160   PetscFunctionBegin;
2161   /* Check matrix local sizes */
2162   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
2163   if (A->rmap->rstart != B->rmap->rstart || A->rmap->rend != B->rmap->rend) SETERRQ4(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, A (%D, %D) != B (%D,%D)",A->rmap->rstart,A->rmap->rend,B->rmap->rstart,B->rmap->rend);
2164 
2165   /* Set default algorithm */
2166   ierr = PetscStrcmp(C->product->alg,"default",&flg);CHKERRQ(ierr);
2167   if (flg) {
2168     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2169   }
2170 
2171   /* Set "scalable" as default if BN and local nonzeros of A and B are large */
2172   if (alg && B->cmap->N > 100000) { /* may switch to scalable algorithm as default */
2173     MatInfo     Ainfo,Binfo;
2174     PetscInt    nz_local;
2175     PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
2176 
2177     ierr = MatGetInfo(A,MAT_LOCAL,&Ainfo);CHKERRQ(ierr);
2178     ierr = MatGetInfo(B,MAT_LOCAL,&Binfo);CHKERRQ(ierr);
2179     nz_local = (PetscInt)(Ainfo.nz_allocated + Binfo.nz_allocated);
2180 
2181     if (B->cmap->N > product->fill*nz_local) alg_scalable_loc = PETSC_TRUE;
2182     ierr = MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm);CHKERRQ(ierr);
2183 
2184     if (alg_scalable) {
2185       alg  = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
2186       ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2187       ierr = PetscInfo2(B,"Use scalable algorithm, BN %D, fill*nz_allocated %g\n",B->cmap->N,product->fill*nz_local);CHKERRQ(ierr);
2188     }
2189   }
2190 
2191   /* Get runtime option */
2192   if (product->api_user) {
2193     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatTransposeMatMult","Mat");CHKERRQ(ierr);
2194     ierr = PetscOptionsEList("-mattransposematmult_via","Algorithmic approach","MatTransposeMatMult",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2195     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2196   } else {
2197     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_AtB","Mat");CHKERRQ(ierr);
2198     ierr = PetscOptionsEList("-matproduct_atb_via","Algorithmic approach","MatTransposeMatMult",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2199     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2200   }
2201   if (flg) {
2202     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2203   }
2204 
2205   C->ops->productsymbolic = MatProductSymbolic_AtB_MPIAIJ_MPIAIJ;
2206   PetscFunctionReturn(0);
2207 }
2208 
2209 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_PtAP(Mat C)
2210 {
2211   PetscErrorCode ierr;
2212   Mat_Product    *product = C->product;
2213   Mat            A=product->A,P=product->B;
2214   MPI_Comm       comm;
2215   PetscBool      flg;
2216   PetscInt       alg=1; /* set default algorithm */
2217 #if !defined(PETSC_HAVE_HYPRE)
2218   const char     *algTypes[4] = {"scalable","nonscalable","allatonce","allatonce_merged"};
2219   PetscInt       nalg=4;
2220 #else
2221   const char     *algTypes[5] = {"scalable","nonscalable","allatonce","allatonce_merged","hypre"};
2222   PetscInt       nalg=5;
2223 #endif
2224   PetscInt       pN=P->cmap->N;
2225 
2226   PetscFunctionBegin;
2227   /* Check matrix local sizes */
2228   ierr = PetscObjectGetComm((PetscObject)C,&comm);CHKERRQ(ierr);
2229   if (A->rmap->rstart != P->rmap->rstart || A->rmap->rend != P->rmap->rend) SETERRQ4(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, Arow (%D, %D) != Prow (%D,%D)",A->rmap->rstart,A->rmap->rend,P->rmap->rstart,P->rmap->rend);
2230   if (A->cmap->rstart != P->rmap->rstart || A->cmap->rend != P->rmap->rend) SETERRQ4(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, Acol (%D, %D) != Prow (%D,%D)",A->cmap->rstart,A->cmap->rend,P->rmap->rstart,P->rmap->rend);
2231 
2232   /* Set "nonscalable" as default algorithm */
2233   ierr = PetscStrcmp(C->product->alg,"default",&flg);CHKERRQ(ierr);
2234   if (flg) {
2235     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2236 
2237     /* Set "scalable" as default if BN and local nonzeros of A and B are large */
2238     if (pN > 100000) {
2239       MatInfo     Ainfo,Pinfo;
2240       PetscInt    nz_local;
2241       PetscBool   alg_scalable_loc=PETSC_FALSE,alg_scalable;
2242 
2243       ierr = MatGetInfo(A,MAT_LOCAL,&Ainfo);CHKERRQ(ierr);
2244       ierr = MatGetInfo(P,MAT_LOCAL,&Pinfo);CHKERRQ(ierr);
2245       nz_local = (PetscInt)(Ainfo.nz_allocated + Pinfo.nz_allocated);
2246 
2247       if (pN > product->fill*nz_local) alg_scalable_loc = PETSC_TRUE;
2248       ierr = MPIU_Allreduce(&alg_scalable_loc,&alg_scalable,1,MPIU_BOOL,MPI_LOR,comm);CHKERRQ(ierr);
2249 
2250       if (alg_scalable) {
2251         alg = 0; /* scalable algorithm would 50% slower than nonscalable algorithm */
2252         ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2253       }
2254     }
2255   }
2256 
2257   /* Get runtime option */
2258   if (product->api_user) {
2259     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatPtAP","Mat");CHKERRQ(ierr);
2260     ierr = PetscOptionsEList("-matptap_via","Algorithmic approach","MatPtAP",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2261     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2262   } else {
2263     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_PtAP","Mat");CHKERRQ(ierr);
2264     ierr = PetscOptionsEList("-matproduct_ptap_via","Algorithmic approach","MatPtAP",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2265     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2266   }
2267   if (flg) {
2268     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2269   }
2270 
2271   C->ops->productsymbolic = MatProductSymbolic_PtAP_MPIAIJ_MPIAIJ;
2272   PetscFunctionReturn(0);
2273 }
2274 
2275 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_RARt(Mat C)
2276 {
2277   Mat_Product *product = C->product;
2278   Mat         A = product->A,R=product->B;
2279 
2280   PetscFunctionBegin;
2281   /* Check matrix local sizes */
2282   if (A->cmap->n != R->cmap->n || A->rmap->n != R->cmap->n) SETERRQ4(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Matrix local dimensions are incompatible, A local (%D, %D), R local (%D,%D)",A->rmap->n,A->rmap->n,R->rmap->n,R->cmap->n);
2283 
2284   C->ops->productsymbolic = MatProductSymbolic_RARt_MPIAIJ_MPIAIJ;
2285   PetscFunctionReturn(0);
2286 }
2287 
2288 /*
2289  Set options for ABC = A*B*C = A*(B*C); ABC's algorithm must be chosen from AB's algorithm
2290 */
2291 static PetscErrorCode MatProductSetFromOptions_MPIAIJ_ABC(Mat C)
2292 {
2293   PetscErrorCode ierr;
2294   Mat_Product    *product = C->product;
2295   PetscBool      flg = PETSC_FALSE;
2296   PetscInt       alg = 1; /* default algorithm */
2297   const char     *algTypes[3] = {"scalable","nonscalable","seqmpi"};
2298   PetscInt       nalg = 3;
2299 
2300   PetscFunctionBegin;
2301   /* Set default algorithm */
2302   ierr = PetscStrcmp(C->product->alg,"default",&flg);CHKERRQ(ierr);
2303   if (flg) {
2304     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2305   }
2306 
2307   /* Get runtime option */
2308   if (product->api_user) {
2309     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatMatMatMult","Mat");CHKERRQ(ierr);
2310     ierr = PetscOptionsEList("-matmatmatmult_via","Algorithmic approach","MatMatMatMult",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2311     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2312   } else {
2313     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_ABC","Mat");CHKERRQ(ierr);
2314     ierr = PetscOptionsEList("-matproduct_abc_via","Algorithmic approach","MatProduct_ABC",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2315     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2316   }
2317   if (flg) {
2318     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2319   }
2320 
2321   C->ops->matmatmultsymbolic = MatMatMatMultSymbolic_MPIAIJ_MPIAIJ_MPIAIJ;
2322   C->ops->productsymbolic    = MatProductSymbolic_ABC;
2323   PetscFunctionReturn(0);
2324 }
2325 
2326 PETSC_INTERN PetscErrorCode MatProductSetFromOptions_MPIAIJ(Mat C)
2327 {
2328   PetscErrorCode ierr;
2329   Mat_Product    *product = C->product;
2330 
2331   PetscFunctionBegin;
2332   switch (product->type) {
2333   case MATPRODUCT_AB:
2334     ierr = MatProductSetFromOptions_MPIAIJ_AB(C);CHKERRQ(ierr);
2335     break;
2336   case MATPRODUCT_AtB:
2337     ierr = MatProductSetFromOptions_MPIAIJ_AtB(C);CHKERRQ(ierr);
2338     break;
2339   case MATPRODUCT_PtAP:
2340     ierr = MatProductSetFromOptions_MPIAIJ_PtAP(C);CHKERRQ(ierr);
2341     break;
2342   case MATPRODUCT_RARt:
2343     ierr = MatProductSetFromOptions_MPIAIJ_RARt(C);CHKERRQ(ierr);
2344     break;
2345   case MATPRODUCT_ABC:
2346     ierr = MatProductSetFromOptions_MPIAIJ_ABC(C);CHKERRQ(ierr);
2347     break;
2348   default:
2349     break;
2350   }
2351   PetscFunctionReturn(0);
2352 }
2353